Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences

17-1 | 2013 The Epistemological Thought of Otto Hölder

Paola Cantù et Oliver Schlaudt (dir.)

Édition électronique URL : http://journals.openedition.org/philosophiascientiae/811 DOI : 10.4000/philosophiascientiae.811 ISSN : 1775-4283

Éditeur Éditions Kimé

Édition imprimée Date de publication : 1 mars 2013 ISBN : 978-2-84174-620-0 ISSN : 1281-2463

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SOMMAIRE

General Introduction Paola Cantù et Oliver Schlaudt

Intuition and Reasoning in Geometry Inaugural Academic Lecture held on July 22, 1899. With supplements and notes Otto Hölder

Otto Hölder’s 1892 “Review of Robert Graßmann’s 1891 Theory of Number”. Introductory Note Mircea Radu

Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891) Otto Hölder

Between Kantianism and Empiricism: Otto Hölder’s Philosophy of Geometry Francesca Biagioli

Hölder, Mach, and the Law of the Lever: A Case of Well-founded Non-controversy Oliver Schlaudt

Otto Hölder’s Interpretation of David Hilbert’s Axiomatic Method Mircea Radu

Geometry and Measurement in Otto Hölder’s Epistemology Paola Cantù

Varia

Hat Kurt Gödel Thomas von Aquins Kommentar zu Aristoteles’ De anima rezipiert? Eva-Maria Engelen

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General Introduction

Paola Cantù and Oliver Schlaudt

1 The epistemology of Otto Hölder

1 This special issue is devoted to the philosophical ideas developed by Otto Hölder (1859-1937), a mathematician who made important contributions to analytic functions and group theory. Hölder’s substantial work on the foundations of mathematics and the general philosophical conception outlined in this work are, however, still largely unknown. Up to the present, philosophical interest in Hölder’s work has been limited to his axiomatic formulation of a theory of measurable quantities published in 1901 in the article The Axioms of Quantity in the Theory of Measurement. This article attracted the interest, among others, of Louis Couturat, Ernst Nagel, and Patrick Suppes. More recent historical studies ([Ehrlich 2006], [Radu 2003]) have explored other aspects of Hölder’s rich conception of the foundations of mathematics, against the historical background of the 19th century Grundlagenkrise, emphasizing the rich interdependence between Hölder’s conception and ideas proposed by Immanuel Kant, Hermann v. Helmholtz, Moritz Pasch, David Hilbert, Hermann and Robert Graßmann, Giuseppe Veronese, and Rodolfo Bettazzi. These studies opened up a broad field for further historical research. A deeper understanding of Hölder’s ideas requires a more detailed analysis of his logico-philosophical conception of the foundations of arithmetic, geometry, and physics. It is our conviction that this goal would benefit substantially, on the one hand, from a more widespread knowledge of Hölder’s earliest methodological publications and, on the other hand, from a closer examination of the ideas found in Hölder’s last and most elaborate philosophical work.

2 Hölder’s interest for the modern reader is both historical and theoretical: from a historical point of view, Hölder belongs to a tradition of mathematicians who, like Poincaré or Weyl, investigated the epistemology of mathematics by taking the results of physics more into account than the results of logic. The general oblivion into which his epistemology fell might be related to the increased interest in the logical foundations of mathematics and set theory. Hölder’s interest in other branches of mathematics, as discussed in The Mathematical Method, are one of the reasons for the

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interest in his approach: on the one hand it is oriented to case-studies and to an investigation of the practice of mathematics; on the other hand it is centered on the notion of deduction, but does not reduce deduction to the relation of logical consequence. For this reason, reading Hölder might be especially interesting for the investigation of mathematical practice and the deductive inferences that are at play in informal mathematical proofs.

2 Biography

3 Otto Hölder was born on December 22, 1859 in Stuttgart. He took up his studies in his native town and then moved to Berlin, where he studied mathematics with Weierstraß, Kronecker and Kummer. Hölder then moved to Tübingen, where he received his PhD in mathematics in 1882 under the direction of Paul du Bois-Reymond with a thesis in potential theory (the study of harmonic functions). In 1884 he went to Göttingen, where he obtained a PhD in philosophy and wrote his Habilitationsschrift on Fourier series. In 1882 he met Felix Klein in Leipzig, where he later became associate professor in 1889, after having held such a position for seven years in Tübingen and then for a further three years in Königsberg, where he became the successsor of Minkowski in 1896. In Leipzig, where he received the chair of Lie, he was the advisor of students such as Emil Artin and Oskar Becker, and became member of the Sächsische Akademie der Wissenschaften. Hölder died on August 29, 1937.

4 Hölder made important contributions to analysis and group theory, but he also worked on geometry, algebra, measurement theory and epistemology. His two main mathematical results—Hölder’s inequality and the Jordan-Hölder-theorem—are still named after him. An interest in philosophical questions was already present at the beginning of his career, but from 1914 on, he concentrated almost exclusively on the foundations of geometry and arithmetic and the role of deduction in mathematics and physics. The results of this work merged ten years later in a comprehensive work Die Mathematische Methode – Logisch erkenntnistheoretische Untersuchung im Gebiete der Mathematik, Mechanik und Physik ( The Mathematical Method. Logico-philosophical Investigations in the Domain of Mathematics, Mechanics, and Physics), published in 1924. In this book Hölder reveals his doubts concerning the possibility of expressing all mathematics through logical symbolism. From 1924 onwards, Hölder turned again to complex analysis and number theory, but continued publishing on the foundations of mathematics.

3 Content of this volume

5 This issue contains English translations of two early epistemological texts by Hölder— Intuition and Reasoning in Geometry from 1900 and his review of Robert Graßmann’s Formenlehre, published in 1892—as well as four previously unpublished articles discussing Hölder’s ideas in the light of 19th and early 20th century philosophy of mathematics.

6 Intuition and Reasoning in Geometry (1900) is Hölder’s inaugural lecture given in Leipzig at the beginning of the academic year 1899. The published text includes extensive notes added by Hölder that constitute almost two-thirds of the volume (comparable only to

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Alexander von Humboldt’s Aspects of Nature, notorious for its excessive annotation). Here, Hölder studies philosophical problems raised by the methods used in geometry and, to some degree, in mechanics. Hölder claims that no serious investigation of mathematical method—e.g., of the special character of the reasoning used in mathematical deduction from the axioms—is available. While mathematicians, on the one hand, are concerned only with the construction of various systems of axioms, and philosophers, on the other hand, are only looking for the sources of the axioms, to be found either in experience or intuition, no one investigates the deduction from axioms. In his inaugural lecture, Hölder begins a case by case investigation of deductive methods used in mathematics and mechanics. Even if a richer set of examples will be given only in his later work on The Mathematical Method, the inaugural lecture already contains some fundamental examples, emphasizing the use of auxiliary lines in geometric constructions and of arithmetic in the method of exhaustion. He comes to the conclusion that mathematics and the exact sciences have a logic of their own: although purely formal and completely exempt from intuition, deduction is not analytical. Rather, according to Hölder, in deduction we perform a thought experiment which replaces real experiments and no longer deals with the objects themselves, but with their mutual relations in thought.

7 It seems that there are not many traces of Intuition and Reasoning in Geometry in the secondary literature. In a review of The Mathematical Method, Edgar Zilsel praised the originality of the former work: “Already a quarter century ago, the author, with his inaugural lecture Intuition and Reasoning in Geometry, became engaged in the philosophical discussion on the foundations of mathematics—and raised its level considerably” [Zilsel(1926), 646]. Nevertheless, Intuition and Reasoning in Geometry made no waves in the literature and even today is known only to a small number of specialists. The only direct trace in the literature seems to be the dispute with Ernst Mach on the proof of the law of the lever (analyzed by Oliver Schlaudt in this volume). We think however that Hölder’s paper rightly deserves attention today, in particular in view of recent trends in philosophy of mathematics that focus on case studies and mathematical practice, rejecting the identification between the inference structure of mathematics and the deductive calculi developed in classical logic.

8 The same holds for Hölder’s review of Robert Graßmann’s Theory of Number or Arithmetic, which is—as the translator Mircea Radu rightly remarks—less a confrontation with Robert Graßmann, than a first outline of Hölder’s own account of axiomatics and of its limits in the foundation of mathematics. As in the case of the inaugural lecture, this review already contains a preliminary sketch of what Hölder will later develop in the 1924 volume. Hölder compares two interpretations of Robert Graßmann’s Theory of Number, which was meant as part of a project to restore the organic unity of the system of mathematics by examining the specific process of sign- construction involved in producing mathematical knowledge.

9 Biagioli’s paper provides a general introduction to the philosophy of geometry at the end of the 19th century and presents the main topics developed by Hölder in the inaugural lecture. The analysis of Hölder’s answers to the charge of circularity raised by the Kantians to Helmholtz’s empiricism reveals that he considers geometrical concepts to be empirical inasmuch as they are derived from empirical experience, and as conditions of experience, inasmuch as they are rules that allow us to infer further facts. Yet, Hölder not only reacts to the epistemology of Kant and Helmholtz, inheriting

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the idea of some kind of a priori knowledge from the former and the empirical origin of several geometrical concepts from the second. He also discusses the classical distinction between geometry, which has axioms, and arithmetic, which is on the contrary based on definitions, and accepts the new trend in philosophy of mathematics based on the rejection of intuition in the development of rigorous proofs, further investigating the role of deduction in geometrical constructions. Biagioli’s paper investigates how these features of Hölder’s epistemology can be interpreted with respect to Kant’s own conception and to the neo-Kantian perspective of Cassirer. On the one hand, there are some limits to Hölder’s understanding of Kant’s remarks on pure intuition; on the other hand, Hölder’s logical analysis of mathematical reasoning as a concatenation of relations has some resemblance to Cassirer’s conception of mathematics. Biagioli claims that Hölder, like Cassirer, aims to develop a methodological synthesis of Kantianism and empiricism that is grounded on a relational conception of the a priori.

10 Analyzing Hölder’s reply to Mach’s critique of the Archimedean proof of the Law of the Lever, Oliver Schlaudt offers an exhaustive presentation of one of the relevant case studies investigated in Intuition and Reasoning in Geometry. Hölder’s interest in this mechanical example is related to his understanding of deduction in science: like Mach he believed that in the Archimedean proof the conclusion was implicitly contained in the premises. Yet, while Mach argued that the Archimedean proof was circular— meaning that it could not provide any further knowledge beyond the empirical facts assumed in the premises—Hölder claimed that the proof is sound because it makes explicit something that is contained in the premises but is not epistemically accessible as such. Deduction is investigated as a means to order mathematical truths, and the difference between the premises and the conclusion is related to the nature of the relational concepts contained in them. Schlaudt associates the case study analyzed in 1899 with some remarks made by Hölder in the 1924 volume, where he develops a theory of synthesis, claiming that synthetic concepts have “hidden properties” that can be brought out in further deductive steps. The detailed analysis of quantitative and qualitative concepts and of metrical relations contained in the proof of the Law of the Lever allows Schlaudt to make a connection between the critique of Mach and Hölder’s understanding of logic, and in particular of the role of concept formation, which is a much more complicated process than nominal definitions. The paper clearly shows how a bottom-up investigation of a specific mathematical proof leads Hölder to develop an argumentative analysis of mathematical inferences that is nearer to Vailati’s analysis of mechanics than to foundational efforts to reformulate scientific theories as logical calculi.

11 The same idea is at the core of Hölder’s review of Robert Graßmann’s volume, as Mircea Radu shows in his contribution: disclaiming Graßmann’s idea that rigor can be reached only by means of a transformation of scientific theories into a thinking calculus— Leibniz’s utopian project—Hölder’s main aim is to criticize the widespread confidence in symbolism and limit its possible uses in mathematics. This is shown by Radu through an analysis of Hölder’s conception of axiomatics and proof: Hölder believes that logic could not itself be axiomatized, given that this would amount to the infinite regress of giving a deductive account of the science of deduction; the same holds for metamathematics, for a meta-mathematical proof of the consistency of mathematics would run into the same regress, because at some point it would require another system of axioms. Although Hölder does not support his claim with fully developed

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philosophical reasonings, Radu shows convincingly how he derives some arguments from the analysis of various mathematical proofs occurring in geometry and arithmetic. Relying on several examples taken from The Mathematical Method, Radu evaluates Hölder’s defense of the genetic against the axiomatic method in the case of arithmetic. The two methods were opposed by Hilbert in Über den Zahlbegriff [On the concept of number], where he remarked that even if the genetic method is mostly used in arithmetic, whereas the axiomatic method is used in geometry, the axiomatic method should be considered as preferable in both disciplines. According to the genetic method, the general notion of real number is generated by successive extensions of the concept of natural number: in particular one first defines a number—the unit—and then further generates the other numbers together with their rules. According to the axiomatic method, a set of entities is assumed as existent from the beginning, together with their relations, and one must afterwards prove the consistency of the axioms that state such relations. Radu shows that Hölder’s idea that the genetic method should be primary is related to a radically different conception of mathematical proof, based on the synthesis of concepts and on the idea that sequences of concepts of different order are used in mathematical proofs. The genetic method is typical of mathematics and logic, which—unlike empirical sciences—are characterized by an unclear distinction between the content of the theory and its form: e.g., the sequence-concept is at the same time the content and the form of our thinking.

12 The difference between geometry and arithmetic is further explored in the paper by Paola Cantù, where the distinction between given and constructed concepts is analyzed in order to show that the fundamental difference between the two disciplines does not concern the kind of objects these sciences are applied to, but rather the different origin and formulation of their concepts. While geometry contains some concepts that cannot be defined synthetically, the arithmetical concepts are all synthetic—i. e., concepts that have their source in some activity, as in the genetic method described above. More precisely, arithmetical concepts are constructed in deductive processes themselves, because “the abstraction of new general concepts and rules thus forms in a way a constituent of deduction”. This analysis of the opposition “given-constructed” allows Cantù to reconstruct Hölder’s position with respect to Pasch, Helmholtz and Kant and provides several reasons why Hölder, although evidently sharing an empiricist account of geometry, did not attempt to advance a definitive argument against the Kantian understanding of mathematics. The difference between arithmetic and geometry and the role played by axioms and experience in it are then used to understand Hölder’s choice of primitive propositions in his axiomatic formulation of the theory of magnitudes, and in particular of the Archimedean axiom as a relevant property of magnitudes—even if he had developed an original non-Archimedean model. Comparing Hölder’s and Veronese’s mathematical theories, Cantù shows that the different choices made at the axiomatic level reflect different epistemological frameworks: although they both took inspiration from Pasch, Veronese was interested in the mathematical construction of a non-standard order relation that is nonetheless compatible with our intuition of physical objects, while Hölder’s was exclusively interested in the empiricalfoundation of measurement.

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4 A Bibliography of Hölder’s writings1

13 1. Beiträge zur Potentialtheorie, PhD-thesis, Tübingen, 1882.

14 2. Beweis des Satzes, daß eine eindeutige analytische Function in unendlicher Nähe einer wesentlich singulären Stelle jedem Wert beliebig nahe kommt, Math. Annalen, 20, 1882, 138–142.

15 3. Grenzwerte von Reihen an der Convergenzgrenze, Math. Annalen, 20, 1882, 535–549.

16 4. Zum Invariantenbegriff, Math.-naturwiss. Mitteilungen, 1, 1884, 59–65.

17 5. Zur Theorie der trigonometrischen Reihen, Math. Annalen, 24, 1884, 181–216.

18 6. Über eine neue hinreichende Bedingung für die Darstellbarkeit einer Function durch die Fouriersche Reihe, Sitzungsber. preuß. Akad. Berlin, 1885, 419–434.

19 7. Bemerkung zu der Mitteilung des Herrn Weierstraß: Zur Theorie der aus n Haupteinheiten gebildeten komplexen Größen, Nachr. Ges. Wiss. Göttingen, 1886, 241–244.

20 8. Über eine transcendente Function, Nachr. Ges. Wiss. Göttingen, 1886, 514–522.

21 9. Über die Eigenschaft der Gammafunktion, keiner algebraischen Differentialgleichung zu genügen, Math. Annalen, 28, 1886, 1–13.

22 10. Über eine Function, welche keiner algebraischen Functionalgleichung genügt, Nachr. Ges. Wiss. Göttingen, 1887, 662–676.

23 11. Zurückführung einer beliebigen algebraischen Gleichung auf eine Kette von Gleichungen, Math. Annalen, 34, 1889, 26–56.

24 12. Bemerkungen zur Quaternionentheorie, Nachr. Ges. Wiss. Göttingen, 1889, 34–38.

25 13. Über einen Mittelwerthssatz, Nachr. Ges. Wiss. Göttingen, 1889, 38–47.

26 14. Über den Söderbergschen Beweis des Galoisschen Fundamentalsatzes, Math. Annalen, 34, 1889, 454–462.

27 15. Über den Casus irreducibilis bei der Gleichung dritten Grades, Math. Annalen, 38, 1891, 307–312.

28 16. Die einfachen Gruppen im ersten und zweiten Hundert der Ordnungszahlen, Math. Annalen, 40, 1892, 55–88.

29 17. R. Graßmann, Die Zahlenlehre oder Arithmetik, Göttingische gelehrte Anzeigen 15, 1892, 585–595 [Eng. trans. in this volume, 57–70].

30 18. Die Gruppen der Ordnungen p3, pq2, pqr, p4, Math. Annalen, 43, 1893, 301–412.

31 19. Bildung zusammengesetzter Gruppen, Math. Annalen, 46, 1895, 321–422.

32 20. Die Gruppen mit quadratfreier Ordnungszahl, Nachr. Ges. Wiss. Göttingen, 1895, 211– 229.

33 21. Über die Prinzipien von Hamilton und Maupertuis, Nachr. Ges. Wiss. Göttingen, 1896, 122–157.

34 22. Weierstraß, Mathematische Werke, zweiter Band, Göttinger gelehrte Anzeigen, 1896, 769–773.

35 23. Herleitung der elliptischen Funktionen, Schriften phys.-ökon. Ges. Königsberg, 38, 1897, 53–57.

36 24. Galoissche Theorie mit Anwendungen, Enzykl. d. math. Wiss., 1, 1899, 480–520.

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37 25. Anschauung und Denken in der Geometrie. Akademische Antrittsvorlesung gehalten am 22. Juli 1899. Mit Zusätzen, Anmerkungen und einem Register, B. G. Teubner, Leipzig, 1900 [Eng. trans. in this volume, 15–52].

38 26. Die Axiome der Quantität und die Lehre vom Maß, Ber. sächs. Akad. Leipzig 53, 1901, 1–64 [Eng. trans.: The axioms of quantity and the theory of measurement. Translated from the 1901 German original and with notes by Joel Michell and Catherine Ernst, with an introduction by Michell, Part 1, J. Math. Psych., 40(3), 1996, 235–252; Part 2, J. Math. Psych., 41(4), 1997, 345–356].

39 27. Die Zahlenskala auf der projektiven Geraden und die independente Geometrie dieser Geraden, Math. Annalen, 65, 1908, 161–260.

40 28. Adolf Mayer, Nekrolog, gesprochen in der öffentlichen Gesamtsitzung beider Klassen der K. Sächs. Ges. d. Wiss. am 14. Nov. 1908, Ber. sächs. Ges. Wiss. Leipzig, 60, 1908, 353–373.

41 29. Streckenrechnung und projektive Geometrie, Ber. sächs. Ges. Wiss. Leipzig, 63, 1911, 65–183.

42 30. Bedingungen des analytischen Charakters für reelle Funktionen reellen Arguments, Ber. sächs. Ges. Wiss. Leipzig, 63, 1911, 388–401.

43 31. Die Cauchysche Randwertaufgabe für den Kreis in der Potentialtheorie, Ber. sächs. Ges. Wiss. Leipzig, 63, 1911, 477–500.

44 32. Über einige Determinanten, Ber. sächs. Ges. Wiss. Leipzig, 65, 1913, 110–120.

45 33. Neues Verfahren zur Herleitung der Differentialgleichung für das relative Extremum eines Integrals, Annali di Mat. (3rd ser.), 20, 1913, 171–184.

46 34. Über einige Determinanten. Zweite Mitteilung, Ber. sächs. Ges. Wiss. Leipzig, 66, 1914, 98–102.

47 35. Die Arithmetik in strenger Begründung, B. G. Teubner, Leipzig, 1914.

48 36. Abschätzungen in der Theorie der Differentialgleichungen, in: C. Carathéodory, G. Hessenberg, E. Landau, L. Lichtenstein, dir., Mathematische Abhandlungen. Hermann Amandus Schwarz zu seinem fünfzigjährigen Doktorjubiläum am 6. Aug. 1914 gewidmet von Freunden und Schülern, Springer, Berlin, 1914, 116–132.

49 37. Karl Rohn, Nekrolog, Ber. sächs. Ges. Wiss. Leipzig, 72, 1921, 109–127.

50 38. Carl Neumann zum 90. Geburtstag, Math. Annalen, 86, 1922, 161–162.

51 39. Die mathematische Methode. Logisch-erkenntnistheoretische Untersuchungen im Gebiete der Mathematik, Mechanik und Physik, Springer, Berlin, 1924.

52 40. Das Volumen in einer Riemannschen Mannigfaltigkeit und seine Invarianteneigenschaft, Math. Zeitschr., 20, 1924, 7–20.

53 41. Berichtigung zu der Abhandlung: Das Volumen in einer Riemannschen Mannigfaltigkeit, Bd 20, S. 7–20, Math. Zeitschr., 21, 1924, 160.

54 42. Über gewisse Hilfssätze der Potentialtheorie und das alternierende Verfahren von Schwarz, Ber. sächs. Ges. Wiss. Leipzig, 77, 1925, 61–73.

55 43. C. Neumann, Nachruf, gesprochen am 14. November 1925 in der öffentlichen Sitzung beider Klassen, Ber. sächs. Ges. Wiss. Leipzig, 77, 1925, 154–172.

56 44. Der angebliche circulus vitiosus und die sogenannte Grundlagenkrise in der Analysis, Ber. sächs. Ges, Wiss. Leipzig, 78, 1926, 243–250.

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57 45. Bemerkungen zu meinem Aufsatz: Über gewisse Hilfssätze der Potentialtheorie, Ber. sächs. Ges. Wiss. Leipzig, 78, 1926, 240–242.

58 46. Carl Neumann, Math. Annalen, 96, 1927, 1–25.

59 47. Über einen Grenzübergang in Abels Recherches sur les Functions Elliptiques, Journ. f. reine u. angew. Math., 157, 1927, 171–188.

60 48. Über einige trigonometrische Reihen, S.-B. bayer. Akad. Wiss. München, 1928, 83–96.

61 49. Bemerkungen über die Herleitung einiger elementarer Formeln, Ber. sächs. Ges. Wiss. Leipzig, 80, 1928, 117–121.

62 50. Über eine von Abel untersuchte Transzendente und eine merkwürdige Funktionalbeziehung, Ber. sächs. Ges. Wiss. Leipzig, 80, 1928, 312–325.

63 51. Der zweite Mittelwertsatz der Integralrechnung für komplexe Größen, Math. Annalen, 100, 1928, 438–444.

64 52. Der indirekte Beweis in der Mathematik, Ber. sächs. Ges. Wiss. Leipzig, 81, 1929, 201– 216.

65 53. Ein Versuch im Gebiet der höheren Mächtigkeiten, Ber. sächs. Ges. Wiss. Leipzig, 82, 1930, 83–96.

66 54. Nachtrag zu meinem Aufsatz über den indirekten Beweis, Ber. sächs. Ges. Wiss. Leipzig, 82, 1930, 97–104.

67 55. Einige Sätze über die größten Ganzen, Ber. sächs. Ges. Wiss. Leipzig, 82, 1930, 159–170.

68 56. Über gewisse Teilsummen von ΣΦ(n), Ber. sächs. Ges. Wiss. Leipzig, 83, 1931, 175–178.

69 57. Axiome, empirische Gesetze und mathematische Konstruktionen, Scientia, 49, 1931, 317–326.

70 58. Zur Theorie der zahlentheoretischen Funktion μ(n), Ber. sächs. Ges. Wiss., 83, 1932, 321–328.

71 59. Über eine Art von Reziprozität bei summatorischen Funktionen, Ber. sächs. Ges. Wiss., 83, 1932, 329–332.

72 60. Über einen asymptotischen Ausdruck, Acta math., 59, 1932, 79–89.

73 61. Über gewisse der Möbiusschen Funktion μ(n) verwandte zahlentheoretische Funktionen, die Dirichletsche Multiplikation und eine Verallgemeinerung der Umkehrformeln, Ber. sächs. Ges. Wiss., 85, 1933, 3–28.

74 62. Zusätzliche Gleichungen zur Hermiteschen Formel, Math. Annalen, 108, 1933, 605– 614.

75 63. Verallgemeinerung einer Dirichletschen Summenumformung, Math. Zeitschr., 38, 1934, 476–482.

76 64. Zur Theorie der Gaußschen Summen, Ber. sächs. Ges. Wiss. Leipzig, 87, 1935, 27–36.

77 65. Bemerkungen zu einer Dirichletschen Frage, Ber. sächs. Ges. Wiss., 87, 1935, 81–84.

78 66. Verallgemeinerung einer Formel von Hacks, Math. Zeitschr., 40, 1936, 463–468.

79 67. Zur Theorie der Kreisteilungsgleichung Km(x) = 0, Prace mat.-fiz., 43, 1936, 13–23. 80 68. Über eine Verallgemeinerung der binomischen Formel, Ber. sächs. Ges. Wiss., 88, 1936, 61–66.

81 69. Elementare Herleitung einer dem binomischen Satz verwandten Formel, Ber. sächs. Ges. Wiss., 88, 1936, 133–134.

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82 70. Über eine Darstellung der Eulerschen Konstanten, Ber. sächs. Ges. Wiss., 89, 1937, 167– 170.

Acknowledgements

83 Some of the papers of this volume were read at a workshop on Otto Hölder’s epistemology at the Archives Henri Poincaré at Nancy (France) in 2010. We would like to thank the Archives and Région Lorraine for financial support. We are also indebted to all the participants of the workshop on Hölder and to the referees who kindly accepted to review the papers submitted for publication. Finally, a special thanks goes to Andrew Haigh and to Kelly F. Baldessari for their help with the English revision.

BIBLIOGRAPHY

EHRLICH, PHILIP 2006 The emergence of non-Archimedean Grössensysteme. The rise of non-Archimedean mathematics and the roots of a misconception, Archive for History of Exact Sciences, 60(1), 1–121.

RADU, MIRCEA 2003 A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann, Historia Mathematica, 30, 341–377.

WAERDEN, BARTEL L. VAN DER 1939 Nachruf auf Otto Hölder, Mathematische Annalen, 116, 157–165.

ZILSEL, EDGAR 1926 Otto Hölder, Die mathematische Methode, Die Naturwissenschaften, 14(27), 646.

NOTES

1. Cf. the bibliography provided by [Waerden(1939)], which we completed according to our best knowledge.

AUTHORS

PAOLA CANTÙ Aix-Marseille Université – CNRS, CEPERC, UMR 7304, Aix-en-Provence (France)

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OLIVER SCHLAUDT Universität Heidelberg (Germany) Laboratoire d’Histoire des Sciences et de Philosophie, Archives H. Poincaré, UMR 7117, Université de Lorraine – CNRS, Nancy (France)

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Intuition and Reasoning in Geometry Inaugural Academic Lecture held on July 22, 1899. With supplements and notes

Otto Hölder Translation : Paola Cantù and Oliver Schlaudt

EDITOR'S NOTE

The translation is a work of joint authorship, but Paola Cantù mainly contributed to the translation of the first half (up to page 23, endnote (42) included), while Oliver Schlaudt mainly contributed to the second half of Hölder’s inaugural lecture.

1 The way in which geometrical knowledge has been obtained has always attracted the attention of philosophers. The fact that there is a science that concerns things outside our thinking and that proceeds inferentially appeared striking, and gave rise to specific theories of experience and space. Nonetheless, the geometrical method has not yet been sufficiently investigated. Philosophers who investigate the theory of knowledge discuss the question of whether geometry is an empirical science, but do not investigate in detail the technical concepts used by geometers. Yet, this would be appropriate. Mathematicians involved in the study of the foundations of geometry, and in particular of the so-called non-Euclidean geometry, which has often been rejected from a philosophical perspective, investigate what follows from certain assumptions. Even if geometers are aware of the fact that they did not make those assumptions arbitrarily, and that they could not make them completely arbitrarily, it is quite natural for them not to worry about the sources from which those assumptions flew out of. Geometers are only concerned with inferring consequences, and do not pay attention to the different mental activities involved in the exercise of this. A combination of different skills acting in unison would most likely solve the problem posed for the theory of knowledge by the existence of geometry.

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2 Surveying the concepts used by geometers, one might notice a fundamental difference. Some concepts are defined by means of a construction, others are basically taken as given. The concept of a square is never introduced without a construction, no matter how well-known the figure is.1 You start from an arbitrary side of the square, add at a right angle a second side that is equal in length to the first, then a third, and finally you close the figure. In such a construction one presupposes the notions of straight line, right angle, and equal lengths. The concept of a straight line is not defined by this construction.2 The idea of a straight line appears thus as primary. Like the straight line, the point is also a given concept in geometry. The definition given by Euclid, i.e., that which has no parts, is not appropriate to give the idea of a point to someone who does not know what it is; indeed, this definition is not used by Euclid afterwards.3 Beside these concepts, which are available to the geometer as a given material, some assertions of a more general kind (which hold or should hold for these concepts) are also established from the beginning. These fundamental facts, being presupposed without proof, are called axioms or postulates.4 For example, one uses all the time the axiom that a straight line can be traced between any two points and be thus fully determined.5

3 Where do the concepts that are taken for granted in geometry come from? And where do the axioms come from? These were the first questions raised as one began to investigate the theory of geometrical knowledge. Some answered: “From the intuition of space”, others answered: “From experience”. As is well-known, Kant saw the source of our geometrical knowledge in intuition. He tried to explain that this knowledge is— as he said—necessary,6 by assuming a “pure” intuition, independent from experience. To explain that the laws of this pure intuition—that is proper to us—can be applied to the world of phenomena, Kant claimed that this intuition of ours is the condition set by our own nature for having a sensory experience of spatially extended objects. For this reason, some inner laws impose themselves on the form of our perception. So, according to Kant, intuition makes external experience possible and thus can not be explained as deriving from experience.7 Unlike Kant, other authors considered the intuitive representations of spatial objects that we come across in our imagination just as mnemonical representations of visual and tactile sense perceptions. We can arrange these representations in succession, and maybe also partially modify them, yet, they derive from sense perception. Intuition is considered as the source of geometrical knowledge but it appears here, conversely, as a result of experience.8

4 Geometry is immediately related to experience by those who consider axioms as the straightforward results of observations or measurements made on physical bodies. Thus the primitive concepts are also explained in a more physical way; for example, the straight line is for them the line of sight, or the position taken by a thread in a state of tension. This view was defended by outstanding natural scientists: it has recently been made prominent by Helmholtz, but it had already been fostered by Newton.9 This same view has been attacked fervidly by Kant’s followers, who considered it a vicious circle to ground on experience facts that are contained in intuition, because they saw intuition as a condition of external experience. If one wants to hold the above- mentioned view without running into fallacies, one should at any rate limit oneself to the observations that are not themselves grounded in geometrical considerations.10 One should use facts of immediate perception.

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5 An example of an immediate perception of this kind could be that a succession of smaller objects might overlap from a given point of view, whereas they do not overlap from another viewer’s standpoint, and that these objects that overlap with respect to a given standpoint do not overlap partially from any other viewpoint.11 From such perceptions one can draw12 the concept of the straight line and also the fact that there is one and only one straight line through two points. I have used here the facts related to the line of sight. We could also consider the taut thread. We must assume that the visual images which characterize the taut thread have a certain immediate13 similarity to one another, and differ from those of a thread which is not taut. The simultaneous occurrence of those first visual images with some tactile sensations, and with muscular sensations provoked in us by the effort of making the thread taut, the experiences that we have when we pluck the taut thread, or when we look along it, build up a factual basis from which the concept of the straight line, and also the mechanical concept of tension can be drawn. Since the repeated attempt to stretch a thread always reaches the same position, experience teaches us that between two points there can be one and only one straight line.

6 Another concept that should be explained in a similar way is the concept of equality of line segments and angles. Whether two line segments are equal in size can be empirically tested by repeated application of a mathematical compass or a ruler. The compass, or the ruler, should not be modified in the meanwhile, but should be rigid. For this reason, according to many empiricists, the concept of equality of lengths is based on the physical concept of a rigid body, or on something similar.14 Similarly, the comparison of angles presupposes that one moves a rigid body. One would fall into a vicious circle, if one wanted to define at the same time a rigid body as a body whose points have invariable distance from each other—as happens when one presupposes geometry. To understand how one could get to the pure concept of a rigid body independently from geometry, and in full accordance with experience, we should imagine for ourselves the perceptions that we experience when on the one hand we moved back and forth a so-called rigid object and when on the other hand we worked a piece of wax or clay. In both cases some variable visual representations, which are accompanied by tactile sensations and muscular feelings letting us perceive the effort of our action, follow one another. At the same time, it occurs to us that we govern the modifications by means of our will, and so we immediately become aware of the impulses of our will. One will remark that in the first case the same visual representations can always be reproduced easily, whereas in the second case we manage to do so only through very hard efforts. The two cases of rigid versus flexible bodies can be distinguished by the ease with which the visual representationscan be reproduced.

7 One could similarly arrive, in pure accordance with experience, to the concept of an invariable, rigid body. If one admits that a comparison of two distances by means of a compass is an observation that needs no developed intuition nor any developed geometrical concepts, then one will also concede that there is what Helmholtz called physical geometry,15 namely a geometry that is in pure accordance with experience, because one can obtain theorems just from the comparison of distances. One could, for example, imagine two physical models, each one with five prominent points that could be distinguished in both models by the same five colors. Now, let’s compare the distance between two points on one model and the distance between two points having

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the same color in the other model. A similar comparison might be carried out ten times: given that one finds a correspondence between the distances nine times, then it follows that one finds the correspondence the tenth time too. This experience repeats itself whenever one repeats the test with other models. So, a theorem would follow inductively from measurement.16

8 Certainly, a strict follower of Kant would still suspect that any of those results obtained through measurement were already influenced by ready and exact intuition taken as a condition of any experience. Maybe, if he wants to hold on to his point of view, he could not be refuted. On the other hand, it cannot be denied that his standpoint is artificial, and it seems legitimate to drop that standpoint, if one believes one can do without it. It seems to me that the empiricist view already has an advantage in the fact that it allows a detailed explanation of intuition, while the hypothesis assumed by the follower of Kant cuts off all the rest. Indeed, it is always an advantage toset up new problems.

9 On the nature of intuition and its relation to geometry one might want to take into account the history of geometry, or even the development of intuition in a single individual. If the history of geometry could nowadays establish in a more precise way how the fundamental concepts of this science developed in mankind, this would surely shed light on these questions. Probably, in ancient times geometry had the character of an experimental science in the strict sense of the word, because at that time it had mainly practical ends. The geometry of Egyptians was probably limited to some empirical rules applied by master builders and land measurers, while only the Greeks gradually learned to prove all truths of geometry from a small number of axioms.17 But this ancient development of geometry is so cloaked in darkness that we cannot draw completely certain conclusions.

10 Given also that the development of the intuition of space in a single person can hardly be observed, there remains just one way to further explore the nature of intuition: an analysis of the simplest18 facts of perception, which might in some cases be supported by the knowledge of the construction of our sense organs. 19 Maybe such a psychological and physiological analysis will lead to results that will be later acknowledged once and for all, or maybe the problem in question will always receive different answers depending on the individual philosophical standpoint. Anyway, it is possible to consider the whole construction of geometry independently from that question, investigating what assumptions are effectively used by geometry, no matter where they come from, and observing how other knowledge is derived from these assumptions by a succession of smaller and more certain steps, namely, how geometers work deductively. To this purpose, one should go through all proofs of geometry, decompose them into their smaller steps, and pay attention to all the assumptions that are thereby made, explicitly or tacitly. It is sufficient to do this for elementary geometry, because higher geometry would yield no essentially new results with respect to the mentioned purpose. A series of important investigations has been made by mathematicians on the foundations of geometry, with the purpose of establishing all the axioms that are used, reducing them to the smallest number possible, and showing that the remaining axioms are independent from each other. There has also been a successful attempt to modify one of the axioms, and to build a non-Euclidean geometry such that the usual geometry is in some sense a special case of it or, more precisely, a borderline case.20 Less complete is the investigation of geometrical deduction itself.

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11 I will study the latter by means of a simple example, recalling the proof of the theorem of the sum of the angles in a triangle. This proof rests on two facts that hold for parallel lines, and that can be formulated as follows:21 1. When two straight lines lying on a plane do not intersect, no matter how far they are extended—i.e. when they are parallel lines—then they form equal angles with a straight line that intersects them in some way, and in particular the angles that lie between the parallel lines on opposite sides of the intersecting straight line are equal. 2. For any point, there is a line that is parallel to a given straight line.

12 The first fact, though slightly differently formulated, corresponds to the axiom by Euclid that we simply call the axiom of parallel lines and that is dropped in non- Euclidean geometries.22 The second fact appears in Euclid as a theorem that can be proved on the basis of certain axioms; it corresponds to Proposition 31 of Book I. For the present purpose, we can consider this fact as an axiom.

13 Now, to prove the theorem of the sum of the angles, I will—for the sake of simplicity— consider a triangle which is upright and whose surface is directly in front of me. Let’s consider that one side is horizontal and let’s take it as the base line. From the apex let’s trace a line parallel to the base: according to the assumed representation, this line is also an horizontal straight line. There appear now three angles at the apex: the median is an angle of the triangle, while the other two have both been generated just now. From the first of the aforementioned propositions about parallel lines it results that these two angles are equal to the angles situated on the triangle’s base line, and in particular the angle situated to the left of the apex is equal to the angle situated on the left-hand side of the triangle’s base, while the angle situated to the right of the apex is equal to the angle situated on the right-hand side of the triangle’s base. Now the three angles of the triangle are represented by means of three angles situated at the apex; these angles together fill up the flat space on the underside of the line we traced parallel to the base line, and thus their sum equals two right angles.

14 The parallel line traced through the apex of the triangle is an essential step of this proof. It has already been remarked many times that in a geometrical proof the figure is almost always expanded through auxiliary lines.23 The possibility to trace the auxiliary line rests here on the second proposition about parallel lines, saying that through any point there is a parallel to any straight line. This is an existential proposition. It has already been suggested with insistence that existential propositions play a role in geometry.24 Concerning the conclusion inferred here, it seems to me that it does not fit the usual forms of scholastic logic.25 I would rather describe it in the following way. We imagine a series of geometrical elements that should be built in a determinate sequence according to certain axioms; relations between these elements are thereby also given. Now, we have several axioms at our disposal saying that if certain relations occur between several elements, then further relations must subsist between these elements alone, or between these elements and other elements. By means of these rules we can derive further properties of our figure. Besides, since existential propositions allow the introduction of an unlimited number of new elements in the figure, we can find many other relations from the known relations in the figure. Even if the execution of this procedure follows certain rules, its results cannot be foreseen in full detail, unless one carries out the procedure in a given way. We are thus obliged to seek and fumble: we make some kind of experiment on whose account we finally predict how the measurements that we carried out on a precisely

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drawn figure, or on a model, should turn out to be. We have thus replaced a real experiment by a thought experiment: this is what deduction consists in.26

15 Now, what role does intuition play in deduction itself? Are the elements being combined in the deduction, and the rules according to which they are combined, all what is taken from intuition, or respectively from experience? Or does intuition co- operate also in the single steps of deduction? The latter point of view has been mainly defended by philosophers and seems to have been also Kant’s point of view, as he saw in intuition the essential principle for obtaining geometrical knowledge.27 This is also the reason why most philosophers reject non-Euclidean geometries. They consider Euclid’s axiom of parallel lines as necessarily given within intuition,28 and consider any geometry that operates intuitively with some other assumption as an absurdity.

16 To clarify the role of intuition in a proof, I’ll now go back to the previous example. One must concede that we have here taken from intuition that those three angles on the triangle’s apex fill up the angular space of two right angles, and that the angles near the parallel lines are located in such a way that they are equal, and do not complement each other so as to form two right angles. Something else is often taken to be extracted immediately from intuition. For example, one will readily concede, on the basis of intuition, that the bi-secants of two of the angles of the triangle intersect at a point inside the triangle, whereas one would not assume by a mere appeal to intuition the fact that the bi-secant of the third angle goes through the same point.

17 It has already been suggested that we judge the intuitive picture somehow immediately, and that these immediate judgments play a role in a geometrical proof, although only rough judgements are endorsed in that way, whereas we deal with finer judgements by the artificial method of deduction.29 A rough judgement on the mental image of a triangle tells us that the bi-secant lines of two angles of the triangle intersect inside the triangle, but it cannot be proved with certainty on the basis of our intuitive picture that the third bi-secant goes through the same intersection point, because our intuitive picture always has a certain indetermination.

18 As a matter of fact, we usually behave as described above. The procedure remains valid also in practice, but from a theoretical point of view it has a double disadvantage. Firstly, we might be unsure as to the distinction between rough judgments and fine judgments. Secondly, the intuitive picture that we have before us, is just one single picture, but it is taken to be typical for all configurations that the figure of the theorem to be proved could produce. Thus, the geometrical proof in this form actually contains aninference by analogy.

19 On the other hand, the latter remark has also been considered wrong, because we imagine the figure on which we execute a proof, as in motion, and we disregard all the configurations that it might assume. It would thus be a complete induction rather than an inference by analogy, and we would be certain of the general validity of our observation. Those who consider intuition as a very special principle of knowledge attach great weight to the power of our inner intuition to set its own pictures in motion.30 But if one actually tries to represent a similar process of movement case by case, one discovers that it works only with very simple figures. If one considers two straight lines that intersect, one in a fixed position, and the other rotating around the intersection point, one would easily see that in all cases four angles are generated, namely two pairs of equal angles. In complicated cases, a similar thing would be nearly

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impossible: one never actually conjures up a complex figure in its entirety as in motion, but only singular parts of it—and only in certain kinds of proofs.

20 One would have to assume the cooperation of an inference by analogy in a geometrical proof, were it not possible to give a different form to this proof. But this is indeed the case.31 The use that we make of intuition in the proof can be reduced to certain rules: this amounts to the formulation of some axioms that are commonly used, and that were already tacitly used by Euclid. For example, one needs the following axiom: if A, B, C, D are points of a straight line, and B lies between A and D, and C lies between B and D, then B lies between A and C, and C lies between A and D. Axioms of this kind have recently been called “axioms of order” by Hilbert.32 Euclid makes a particular use of intuition in the proof of the first proposition on congruence. This proposition says that if two sides and the included angle of two triangles are congruent, then the other parts of the triangles are congruent too. This is proved by hinting at the fact that if a triangle is moved forward without alteration of its form and size, it overlaps the other. This proof does not correspond to the previous description of deduction. On the contrary, it consists in an immediate hint to intuition, or, if we prefer, to the experiences made through the movement of rigid bodies.33 The theorem of congruence under discussion expresses thus a primarily intuitive content, and must then be considered as axiomatic. 34 By means of such a formulation of all intuitive assumptions, one can divest geometrical deduction itself of intuition. If one denotes the geometrical elements and the operations on these elements through symbols and through successions of symbols —just as in algebra one symbolically represents the numerical magnitudes and the operations to be carried out on them—so, in the simplest cases, it is possible to entirely reduce the geometrical deduction into a calculus that is carried out on symbols, just as in algebra the inferences concerning numerical magnitudes are carried out by means of the so-called letter-calculus.35 Such a calculus has already been accomplished for single domains of geometry; Peano has assembled these symbolical procedures.36 This is a realization of an idea that Leibniz had first enunciated in 1679 in a letter to the famous physicist Huyghens. In this passage Leibniz expresses himself as follows:37 “I have found some principles of a new language of symbols that is completely different from algebra, and that allows us to advantageously represent in thought, in a precise and adequate manner, and without figures, anything that depends on intuition.” In a following passage of the same letter,38 Leibniz suggests a much broader application of his language of symbols, showing his characteristic overestimation of conceptual operations.39 Huyghens proves to be, from the very beginning, very skeptical about the new method.40 But it cannot be denied that such a purely formal inference method, which is dressed-up as a calculus, can be adequately applied onlyto very limited domains.

21 If, in the aforementioned manner, we divest the geometrical inferences of intuition, we can say with full certainty that something follows from given assumptions, and we can do it even if these assumptions contradict intuition. So, it follows with certainty from Lobachevsky’s assumptions that in no triangle can the sum of the angles be greater than two right angles, and that if it is smaller than two right angles in one triangle, then it must really be smaller than two right angles in any triangle.41

22 The mentioned form of deduction, to some extent its pure representation, is not the usual way of geometrical discovery. The latter is mainly guided by intuitive pictures, and sometimes by observations that are made in a series of cases, i.e., guided by

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analogy and induction, just as often in all mathematical domains some results obtained inductively show the direction along which the deductive discovery will be made.42

23 We distinguished in geometry between primary concepts (taken from intuition or experience) on the one hand, and geometrically constructed concepts on the other hand. But in fact it also happens that concepts which were clearly immediately abstracted from intuition, can later be deduced. The different treatments of the theory of proportions provide an instructive example for that. The concept of the proportion of lengths is closely related to the concept of similarity of bodies and plane figures. It may be taken as an immediate intuitive fact that we can reproduce a body with its proper form but on a different scale. In this case the edges of the reproduction have the same ratio to each other as the edges of the original body; also all angles remained unchanged. If one admits from the outset the possibility of constructing for any body a similar body at any scale,43 all propositions of the theory of proportions can easily be found. We might even have formed the concept of proportion according to the fact that there are similar bodies.44 On the contrary, Euclid bases the existence of geometrically similar bodies on the theory of proportions.45 He however reduced even the concept of the proportion to more simple concepts. I shall now enlarge on the concept of the measure of length in order to appreciate these ideas developed by Euclid46 which formerly were not understood for a long time and were held to be dispensable.

24 In geometry one sometimes starts with the concept of the measure of length without properly having established this concept. In doing so it is taken for granted that all the lengths occurring in a figure have a determinate numerical ratio to one of them chosen by us. The length chosen is then called the unit of length. The number indicating how often the unit is contained in a line segment is called the measure-number of this line segment. The line segment to be measured can be either commensurable or incommensurable to the unit; in the latter case the measure-number will be irrational. The measure-number of a square’s diagonal for example is irrational if the side of the square is chosen as the unit.

25 A closer look however reveals a difficulty in the concept of measure. For sake of simplicity let us consider two commensurable line segments. In order to conceive their length ratio as a numerical ratio we have to imagine an additional line segment both given line segments are multiples of. It lies in commensurability that such a third line segment exists; there is however more than one such line segment. What would we do if we could yield different numerical ratios as measuring results when choosing a different third line segment? In order to establish the concept of measure, we have to prove that this is not possible. In fact we could admit this concept without foundation; this is equivalent to the assumption of new and basically complicated axioms.

26 The proof mentioned can nevertheless be done by means of the two facts that equals added to equals yield equals and that the greater added to the greater yields the greater.47 These facts may be regarded as axioms. 48 Adding the Archimedean axiom, according to which we can exceed any line segment by means of multiplication of any given line,49 and taking into account also the propositions on parallels and congruence allows us to construct in the style of Euclid the whole theory of proportions and similar figures.50 The definition of proportion used by Euclid 51 and by means of which he reduces, as mentioned above, this concept to simpler ones can be stated as follows: Two line segments a and b have the same ratio as a′ and b′ if any multiple μa of a is, in comparison, smaller than, equal to, or greater than any multiple νb of b, depending on

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whether the according multiple μa′ of a′ is smaller than, equal to, or greater than the according multiple νb′ of b′.52 This definition applies to two incommensurable line segments as well as to commensurable ones.

27 Galilei put forward a different treatment of proportions, which aimed at simplifying the Euclidean account. In doing so, however, he had to postulate the concept of proportion along with several axioms referring to it.53 The way he carries out the proof does not satisfy the claim of presupposing as little as possible. For this reason, from a scientific point of view, Euclid’s account has to be considered as the more complete one.

28 Recently Hilbert presented a completely new derivation of the theory of proportions.54 In this account even the Archimedean axiom is avoided.

29 In this derivation—as in the Euclidean one—the concept “equal” appears as a primary concept that several axioms in addition are held to apply to. The concept of the ratio of lengths of two line segments as well as the concept of the proportion of four line segments are constructed;55 the propositions about proportions appear to be provable.56

30 The concept of content of plane figures and bodies has not been constructed deductively by Euclid, but it is possible to do so.a In the case of bodies, one may regard the content as an empirical concept extracted from the usage of measures of capacity in measuring fluids, and in this way one may also get to the content of plane figures. Euclid takes it for granted that figures have a content, and that to them also the axioms apply that equals added to equals yield equals and the greater added to the greater yields the greater. If however one does not want to presuppose the concept of content, but to establish it geometrically, one first and foremost has to show that figures, in particular figures of different form, can be compared as to their magnitude. As a first step congruent figures will be declared to be equal; the figure that completely comprises the other figure will be called the greater one. In order to compare any two figures, they have however to be cut through, and it must be proved that the result of the comparison does not depend on the actualway this is done.57

31 The construction of the proportions of line segments as well as of the concept of content presupposes that certain geometrical operations are repeated an indefinite number of times. Indeed, in the consideration of proportion, we had to take the nth multiple of a line segment, n being an indeterminate number. Into such considerations general concepts of arithmetic enter and such questions cannot always be solved in a mechanical calculus.58 Together with arithmetical, or in general combinatorical, concepts, more complicated modes of inference occur within the ambit of geometry.

32 The method of exhaustion which permits us to evaluate figures of curvilinear boundary provides an example for this kind of inference.59 Let me take the surface of a circle as an example. Its content equals the content of a triangle having the circle’s circumference as its base and the radius as its height. The comparison of the contents, required in the proof, cannot be done by means of a number of decompositions. Rather certain comparative figures have to be found which serve as an intermediary and only approximately represent the circular surface. An inscribed regular octagon for instance can be compared first to the circular surface and then to the triangle, by which it is found that the triangle and the circular surface in any case do not differ by more than the half of the latter. The difficulties involved in this comparison here might be passed over. Hereafter a regular hexadecagon might be inscribed in the circle, and thus it can

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be shown that the triangle and the circular surface differ by less than a quarter of the circular surface. This goes on in the same manner. By means of the regular inscribed polygon of 2n+2 edges, it is shown that the triangle differs from the circular surface by less than a 2nth of the latter. Everyone will understand instinctively that from this the exact equality of the circular and the triangular surfaces can be concluded.

33 For the sake of logical rigor it should be noted that we are actually dealing with an infinity of results. The difference between the figures in question is less than ½, than ¼, than ⅛ of the circle. We are sure of all of these results, notwithstanding their unlimited number, for they arise in a uniform manner; the series of considerations they result from proceeds according to a law and thus can be covered by a single view. All of these infinite results are only approximative. The result concluded from them consists in a perfectly exact proposition. In order to get to this proposition, things have to be put otherwise. The fact we make use of here is that the difference of the contents of two surfaces—the surface of the triangle and the surface of the circle—can be represented as a determinate surface. The consideration of this surface leads into a contradiction. The proof is thus indirect.60

34 I shall now elaborate on this proof. The surface mentioned is duplicated in thought, the surface thus obtained duplicated again and so forth. By a sufficient number of repetitions of duplications any given piece of a surface will necessarily be exceeded. This follows from the Archimedean axiom as applied to surfaces. The surface in question shall now be duplicated n times with n chosen such that the surface of the circle will be exceeded. Consequently, the 2nth multiple of the surface is greater than the circle, and, accordingly, the surface—i.e., the difference between the surface of the circle and the surface of the triangle—is greater than a 2nth of the surface of the circle. This however contradicts the result obtained before by means of the inscribed polygon of 2n+2 edges. From this contradiction follows the incorrectness of the only uncertain assumption introduced, the assumption namely that the surface of the circle and the surface of the triangle differ in their content. I think that this example at once shows that geometry cannot do without indirect proofs.61

35 Wherever it can be applied in the exact sciences, deduction appears in a way quite similar to how I characterized its role in geometry. This is particularly clear in mechanics. I hold most of the proofs appearing in mechanics to be as good proofs as any proof of geometry, even though they require suppositions which are generally held to originate from experience.62 Empiricists indeed also hold the geometrical axioms to originate from experience, without rejecting the deductive method by virtue of which science augments its knowledge. Such an empirical fact which plays in mechanics the role of a postulate (axiom) is, e. g., that the point where a force acts on a rigid body can be shifted in the line of the force without altering its effect. Galilei’s principle of inertia as well as Newton’s “leges”63 are such postulates. These postulates and the concepts— force (attractive and repulsive force, pressure, tension), rigid connection, mass, time, place—between which the postulates establish relations, result by way of a certain generalization and idealization64 from empirical observations. They may count as evident insofar as most people let themselves be determined by everyday life alone to recognize the concepts and connections of concepts as valid.65

36 Neither the postulates (axioms) nor the primitive concepts of mechanics are of logical necessity. All attempts to define satisfactorily the primitive concepts failed. They can be explained only by reference to the simple facts they were extracted from.66

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37 We make use of the primitive concepts of mechanics in quite a similar way to the primitive concepts of geometry.67 In deductions only a formal use is made of these concepts. The real application of the concepts according to their content would be the application to the objects that the concepts refer to. In spatial geometry, in order to prove the mutual relations of points, straight lines and planes a purely formal use is made of the fact that a straight line which has two points in common with a plane completely lies in this plane. This fact is applied in the real world by the stone-cutter who applies his rod to the block he is working on in order to achieve a flat surface. In the same sense we make a formal use of the concept of force in the theory of the equilibrium, whereas we really apply the concept in any material device involving the production of pressure or tension. In deduction we thus do not operate with the objects themselves; but rather we operate theoretically with the mutual relations of the objects.

38 It may be that we cannot present deduction in mechanics in as a pure way as in geometry. It may be that in mechanics—having a rather material content—we unconsciously use certain empirical analogies in addition to the axioms. In any case it is correct that in mechanical deduction, too, we completely abandon—at least temporarily—the object investigated and perform independently from this object a thought experiment, relying on the fact that its result must finally be in accordance with the object. That we can successfully apply such a procedure rests on the exact— and in a certain sense verified, though not properly provable68—validity of the laws which we found by virtue of experience and which we considered as generally true. This is the reason why this kind of deduction can be applied in the exact sciences; nonetheless it would only be in vain to try to apply to the forms of organic life or even to the historical sciences a procedure which can be applied to mathematics, mechanics, and certain parts of astronomy and physics.69

39 The procedure of deduction consists in inferences of a peculiar form as has been shown in the preceding examples. The mathematical sciences thus indeed have a particular method. Reasoning in itself of course is always the same and always consists in the very same simple mental activities. The particularity of the mathematical sciences rests in the object which permits us to perform a long series of thought operations in characteristic connections so that in this way particular forms of conclusions emerge. In this sense it can be said that mathematics and the exact sciences have a logic of their own.

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BELTRAMI, EUGENIO 1868 Saggio di interpretazione della geometria non-euclidea, Giornale di Matematiche, 6, 284–322.

CANTOR, GEORG 1880 Vorlesungen über Geschichte der Mathematik, vol. 1, Leipzig: Teubner.

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NOTES

1. Euclid constructs the square in Prop. 46 of Book I (cf. [Euclid & Heiberg 1883]). 2. Euclid’s definition of a straight line does not contain any construction and is to be conceived as a mere nominal definition (Book I, Definition 4); indeed, no use is made of this definition afterwards. 3. In his Lectures on Modern Geometry, Pasch explains the notion of a point as follows: “The bodies whose division is incompatible with the limits of observation are always called points, whereas in geometry the word ‘body’ is reserved for a different use.” Pasch alludes here to the abstraction by means of which we are able to think of the concept of a point [Pasch 1882]. Anyway, in the construction of geometry this definition has been used as rarely as any other definition of a point that has ever been given. 4. As a matter of fact, we hardly distinguish between axioms and postulates; when we do, we simply want to express a nuance in our conception: whereas by the word ‘axiom’ we want to emphasize the fact that it is a very certain assumption, by the word ‘postulate’ we emphasize the fact that the generality of the rule that has been put forward is actually something that is only assumed by us. In Euclid there are two groups of principles: the aitémata (postulata) and the koina ennoia (communes animi conceptiones), the latter being called axiómata by Proclus. According to the latter, Euclid considered the former principles as postulates, and the latter as self-evident truths. This seems to be quite plausible also with respect to the content of the principles, if one considers the repartition of the principles to be found in Heiberg’s edition of the axioms, which is based on a rigorous analysis of the sources [Euclid & Heilberg 1883]. In other editions the repartition is different (cf. e.g., the 1781 German translation by Lorenz based on the 1703 Oxford edition [Lorenz 1781].) 5. The first of the mentioned facts is included by Euclid in the first postulate, the second is missing in Heiberg’s edition, but it appears as the 12th axiom in other editions. This axiom, which might thus constitute a later addition, says that two straight lines cannot encompass any space. If one could join two points in a plane by means of two different straight line segments, these would encompass a surface; thus, it follows from axiom 12 that there can be only one straight line that joins two points. Facts of the kind here considered are called by Hilbert axioms of connection [Hilbert 1889, 5]. 6. One could contest that the axioms of geometry are necessary in the Kantian sense; they might be necessary only in the same sense in which one calls natural laws necessary. But if one wants to infer some kind of inner necessity of the axioms—from the mutual agreement of different consequences that can be derived from them—then one could reply that the non-Euclidean geometry is in itself as consistent as the Euclidean geometry (compare note 20). In this regard, mathematical investigations on non-Euclidean geometry have supported an empirical conception in the theory of knowledge. For example Benno Erdmann claims that the new geometrical knowledge has a negative impact on the rationalist theory [Erdmann 1877, 116]. According to the empirical theory, the axioms cannot be called strictly necessary; something that is strictly necessary might of course follow from them, but what is required is the universal validity of the axioms (compare also [Mill 1869, 254]). 7. In the Kritik der reinen Vernunft [Critique of Pure Reason], 1st ed., 1781, Doctrine of Elements, Part 1 [The Transcendental Aesthetic], Section 1 [On Space], § 2 [Metaphysical Exposition of this Concept, A 24] Kant says that space is a “necessary and a priori” idea that grounds all external intuitions and should be regarded “as the condition for the possibility of phenomena, and not as a determination depending on them” [Kant 1781-1787, 175]. In an add-on from the 2nd edition of 1787 (Doctrine of Elements, Part 1, Section 1, § 3 [Transcendental Exposition of the Concept of Space, B 41]) it is stated even more clearly that the idea of space is “a pure, non empirical intuition”, and that this intuition “has its place merely in the subject, as its formal quality of

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being affected by objects and thereby receive immediate representation, i.e., intuition, of them” [Kant 1781-1787, 176]. Then, on this necessity of the idea of space is grounded the “apodeictic certainty of all geometrical principles” [Kant 1781-1787, 1st ed., nr. 3–4]. It is fully clear here, that Kant takes the single axioms to actually arise from pure intuition, because he says in the mentioned passage: “Thus also all geometric principles, e.g., that in a triangle two sides taken together are always greater than the third, are never derived from the general concepts of line and triangle, but rather derived from intuition, and indeed derived a priori and with apodeictic certainty.” (This is a fact that I would not even rate among the principles.) 8. It is not my aim to go into all the conceptions that the inventors of different philosophical systems held of space and spatial intuition. Compare here [Baumann 1869], and [Wundt 1883, 85– 96]. 9. Compare [Newton 1687, Preface], and [Helmholtz 1884, vol. 2, 217]. Of course, the advocates of this empirical viewpoint will not deny that any elaboration of experience comes out from assumptions, at least from the assumption of a certain conformity to laws of the investigated object, which we could not otherwise grasp conceptually (compare in particular [Helmholtz 1884, vol. 2, 266ff.]). Indeed, any single fact of experience, if expressed by means of concepts—and how could one want to express it otherwise—is the result of a mental elaboration of experience. But unlike Kant’s conception, empiricism emphasizes the fact that according to the empirical viewpoint, no law referring to external objects comes about independently from external experience. Kant says on the contrary that geometrical knowledge comes from intuition and that intuition is independent of experience. 10. If one uses for example a micrometer screw in measurement, then one infers the displacement of the longitudinal axis of the screw from the rotations and their fractional parts. In measurements of this kind one already uses a set of relations from the science of geometry. So, the observations on which one would like to ground geometry should anyway have a much simpler nature than a measurement of that kind. 11. When the observer steps in between the objects, he will certainly be able to see only one part at a time; but certainly, if from his standpoint two of them are overlapping, then all the objects he can see will be overlapping. 12. For a Kantian, the fact that, from a determinate viewer’s standpoint, certain small bodies overlap, is the reason why the viewer—following the intuition that he has independently from experience and by means of which the essence of the straight line is fully given to him—designs the places of those bodies as rectilinear. For the empiricist, the straight line is nothing else but a concept that is abstracted from the occurrence of the overlapping of small bodies, or maybe from other similar occurrences. When, following a well-known linguistic use, I designate the process of concept construction as an “abstracting” or “extracting” process, I do not want to be associated with a theory that claims that the concept arises from a succession of ideas when we “leave out” the features that differentiate ideas and keep the features they have in common. When we build a concept from a succession of ideas, the ground will always be that we perceive those ideas as immediately similar, and that, in accordance with the linguistic use of other people, we learned the use of a word that expresses what is typical in those ideas. We say that we have mastered the concept, when we believe that we will be able, in future cases, to apply it with confidence and in agreement with other people. Basically, we can never prove but only claim that we are able to do this (compare [Volkelt 1886, 181ff.]). In particular, we require an absolutely safe use of the concepts in the scientific treatment. For example, in geometry we act not only as if we could pinpoint exact points and exact lines, but also as if we could always distinguish, given a straight line and a point, whether the point lies on the straight line or not, and in doing so the concept of this difference is also considered as unconditionally applicable.

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If considered from the empiricist point of view, the geometrical concepts appear at first as hardly distinguishable from other empirical concepts obtained by abstraction. The difference between the former and the latter concepts, which was strongly accentuated by Wundt [Wundt 1883, vol. 2, 95] 1883qe, emerges only afterwards in the scientific use of geometrical concepts, which influenced also their practical use (cf. note 64). As we can extract or abstract a concept from a succession of ideas, we can also abstract a rule from a succession of combinations of ideas, when we find that these combinations are not only similar one to the other but also to a complex process of consciousness, which thereby represents their connection and thus exhibits their rule. For example, we might have noticed, in a series of cases, that a thread stretched between two small rings takes a fully determinate position. We compare these observations to a conscious process, when we distinguish three elements and consider one of them as determined by the other two. Besides, if we postulate that our thoughts are universally valid for all cases of the observed kind, we get to the rule: “Through two points there always passes one and only one straight line.” Since the common concept appears unitary in thought, and accordingly occurs mostly in association with a single word, a rule of that kind is a “concept of higher order”, that displays a logical structure [Volkelt 1886, 384]. 13. ‘Immediate’ should simply mean here that we immediately judge to be similar the successions of sensory perceptions, on which are based the visual images that first appear as unitary to our consciousness. This does not contradict a conception of physiology, according to which those comparisons of visual images are possible only on the basis of eye-movements. 14. [Helmholtz 1884, vol. 2, 260]. One can also use the thread to compare distances. An objection has been raised to this however, that a comparison of distances is possible only by means of an inelastic thread, and that there is indeed no feature that can distinguish an inelastic from an elastic thread apart from the invariability of the length of the former: so one would end up moving in a circle. Against this objection, I would like to remark, that in the case of a piece of inelastic thread kept stretched between two hands, we cannot generate any visible variation by merely intensifying our stretching effort, whereas in the case of an elastic thread we can do that. So, we have here a feature of the inelastic thread that does not presuppose the given concept of length. One could probably try to define the concept of equality of line segments also in many other ways (cf. Helmholtz’s Bemerkungen über ‘physische Gleichwerthigkeit’ von Raumgrössen, [Helmholtz 1884, vol. 2]). The above explanation just aims at showing that one can indeed consider the concepts ‘equal’, ‘bigger’, ‘smaller’ as extracted from certain empirical activities, without having to concede a claim that has already been advanced [Zindler 1889, 11], that any comparative activity of that kind already contains the concepts of length and equality. Basically, the following appear thus as facts of experience: that two line segments being equal to a third are equal to each other; and that given three line segments such that the second is smaller than the first, and the third is smaller than the second, then the third has also to be smaller than the first. The fact that two magnitudes being equal to one and the same third magnitude are equal to each other is described by Sigwart [Sigwart 1889, vol. 1, 414] as an analytical fact, i.e., it is described as a proposition that follows from the concept of equality. In some sense, this will be conceded by everybody, insofar as the mentioned fact is apparently inextricably associated in our consciousness with the concept of ‘equal’. However, one should not forget that a different intuitive content or, if one prefers, empirical content can be attributed to this proposition, depending on the use we make of it. The equality of line segments is something different from the equality of angles, or even from the equality of surface portions of different form. Besides, the mentioned proposition does not appear to me to be a consequence of the so-called principle of identity, because objects that are declared to be equal do always differ in some other respect, and are thus not logically identical. We could most correctly say, with Helmholtz [Helmholtz 1887, vn vol. 3, 356ff., 375–376] that by the term ‘equal’ we designate a relation

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between two things such that, whenever the relation subsists between a and b, one can infer that the same relation subsists between b and a, and whenever the relation subsists at the same time between a and b and between b and c, one can infer that it subsists between a and c. However, whether a relation is of that kind, might result, in the mentioned case, only from the natureof the concerned objects. In certain circumstances, in particular, one could as well say that a fact is abstracted from experience, or that it arises immediately from a concept, inasmuch as the concept is taken, together with the fact associated to it, from experience or intuition. However, there can never be any disagreement whether a fact can or cannot, in the proper sense of the word, be deduced from others, i.e., whether an already obtained deduction of that kind is binding or not. 15. [Helmholtz 1896, vol. 2, 260ff.], where the author gives a convincing example. 16. This would be an induction that does not rest on any deduction. On the relation between induction and deduction one finds apparently opposite views. Some say that deduction is always grounded in facts that are discovered inductively [Wundt 1883, vol. 2, 27], others say that induction relies on the results of demonstrative logic, i.e., on deduction [Lotze 1874, 340], [Sigwart 1893, vol. 2, 384–385]. The two theses can be combined, up to a certain degree. The first one, which anyway seems to me inappropriate for arithmetic, might be correct for all sciences that concern objects of experience in the strict sense of the word. It is correct, insofar as certain facts of a general kind (laws) are inductively obtained from experience, and exactly these facts provide the rules along which deduction proceeds. On the other hand, there are complicated inductions that become possible, only after one has obtained certain concepts that arise from a deductive elaboration of a domain of knowledge. When, for example, we induce from many observations that in the case of a falling body the spaces traversed from the beginning of the fall are proportional to the squares of the falling times, then we must possess not only a general concept of the process that we call ‘to fall’, but also the concept of a square number, which has its source in the deductively elaborated domain of arithmetic. On the contrary, it seems to me that the intellectual activity that from the beginning goes hand in hand with experience, when we build concepts of experience, is a preliminary activity that should not be assigned neither to deduction nor to induction. 17. [Cantor 1880, vol. 1, 46ff.], [Hankel 1874, 88]. Compare also the interesting remarks by Kant in the Preface to the 2nd edition of the Critique of Pure Reason. 18. That is, of the facts that are most simple for our consciousness. 19. Localization is a problem whose solution, requiring psychology and physiology, should greatly clarify the nature of our intuition of space. The importance of the localization problem will certainly be denied by those who consider the pure intuition of space as a condition of any perception. But they must certainly admit that the empirical part of my perception, i.e., also my sensory perceptions, must contain something that allows me, in a single case, to locate an object in this way, and not otherwise, namely to see the object in this determinate position and not in some other position with respect to my body (cf. the theory of ‘local signs’ by Lotze, [Lotze 1884, 547ff.]). Similarly, there must be an empirical reason why I see an object, in a given instant, exactly in the form in which it appears to me, and not in one of the other forms that would be possible according to the laws of ‘pure intuition’. This concession could already be considered as conflicting with Kant’s conception, according to which the form of the perceptions should have its source in the intuition of space, which is independent from experience. One could at any rate be tempted to understand Kant in such a way that pure intuition, i.e., the form of perception, has nothing to do with the empirical content of perception. But if, as Kant says, the geometrical ‘knowledge’ (laws)—which, according to him, must have its origin a priori—“perhaps serve[s] only to establish a connection among our sensory representations” [Kant 1781–1787, 1st edition, Introduction, 128], then one should undoubtedly understand Kant as saying that in a singular case we obtain from the senses certain empirical data, and that from these data, under the action

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of the laws of intuition, we delineate the form of the object that stands before us, and we determine its position. Anyway, since the observation of the object can be arbitrarily extended, it cannot be ignored that then we should be able to avoid an arbitrary increase of those empirical data, or we would obtain many more data than those that are necessary to determine the intuitive picture of the object. Anyway, all these empirical data, with the help of the laws of intuition, produce a corresponding picture of the object: this can only be explained, according to me, by the assumption that the laws of intuition have at the same time the function of exhibiting a valid order for the empirical part of perception. But then we would basically be back to the empirical point of view. 20. The task of building a simple and complete system of independent axioms for Euclidean geometry has recently been solved by Hilbert Hilbertys. The geometries that differ from the usual Euclidean geometry had their point of departure in the efforts to prove Euclid’s axiom of parallel lines. The latter is the 11th axiom in Heiberg’s edition, and the 5th axiom in other editions, cf. [Lobachevsky & Engel 1899, 373–383]. For the purpose of that proof, it was at first tentatively assumed that the axiom of parallel lines was not satisfied, originally with the intention to derive a contradiction from this assumption. But the contradiction did not ensue, and non-Euclidean geometries were developed from such considerations. Lobachevsky was the first (1829) to publish a fully accomplished geometry without the mentioned axiom (this geometry, which was also discovered by Gauß, was discovered independently by other authors; for further details, cf. [Lobachevsky & Engel 1899,). Lobachevsky used the so-called synthetic method, i.e., the usual geometrical method. One could also use the ‘analytic’ method. This rests on the possibility—which can be proved on the basis of Euclidean geometry—to establish the position of a point in space through three measurements (this is the property of space that we express by saying that space has three dimensions). One can thus relate the totality of points in space to the totality of number triplets, which is expressed in words as follows: the space can be conceived as a threefold infinite numerical manifold. If one lets space correspond in such a way to a numerical manifold, then the relations of position, and so on, that subsist in space between points, straight lines, and surfaces are reflected into relations between numerical structures, and the same laws hold in both cases. This conception opens up a new way to answer the question of possible geometries—a way that was first developed by Riemann ([Riemann 1867]; [Riemann 1876, 254]), and afterwards further pursued especially by Helmholtz ([Helmholtz 1883, vol. 2, 610, 618] ; [Helmholtz 1884, vol. 2, 3]). These inquiries presuppose that the relations of space might be exactly represented through certain dominant relations in a numerical manifold (this requirement has been emphasized especially by Sophus Lie, who also corrected the mentioned inquiries on several issues; cf. [Lie & Engel 1893, 393]), and are therefore only concerned with the structures of the numerical manifold. As a result, in other similar threefold numerical multiplicities structures have been defined whose reciprocal relations show laws that are both analogous to, and different from, the laws into which usual geometry is reflected: one such new case—discovered in that way—corresponds precisely to Lobachevsky’s geometry. Anyway, these inquiries ensured that Lobachevsky’s geometry is in itself consistent, while the original, pure synthetic treatment of this geometry did not at first exclude the possibility that further additional considerations might later lead to contradictions. It is sufficient to consider, for the moment, that by the words ‘point’, ‘straight line’, and so on, we intend to designate just those numerical structures in the new numerical multiplicities. Thus, the axioms postulated for the new system are satisfied, and therefore they cannot, at any rate, contradict one another. The relations of Lobachevsky’s geometry can also be represented with the help of certain complicated conceptual constructions that are tied to Euclidean geometry ([Beltrami 1868, vol. 6, 284], [Klein 1871, vol. 4, 573ff.], [Klein 1873, vol. 6, 112ff.], [Cayley 1872, vol. 5, 630ff.]). So, if one assumes that Euclidean geometry is in itself consistent, the consistency of the new geometry is

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proved. However, the mentioned assumption—generally and tacitly assumed before the discovery of non-Euclidean geometries—primarily rested merely on the belief that the Euclidean geometry is the true expression of certain objective relations. Of course, one could also prove the consistency of Euclidean geometry by pointing to the numerical manifold that represents it, as was done by Hilbert [Hilbert 1899, 19–21]. The consistency of the non-Euclidean geometry, in which the Euclidean postulate does not hold but the other axioms are satisfied, ensues that no axiom can be inferentially derived from the others. This result is maybe the most important advantage obtained through the inquiries into non-Euclidean geometries. 21. Some people try to elude the difficulties that lie in the parallel theory by introducing from the very beginning the concept of ‘direction’ (cf. [Zindler 1889, 7]), and by defining the angle as ‘difference of direction’. In this case, certain axioms are set by the way the concepts of ‘direction’ and ‘difference of direction’ are used. Firstly, in this conception one tacitly postulates the fact that is highlighted in note 25, namely that two straight lines lying on a surface and forming equal angles with an intersecting straight line, form equal angles with any other intersecting line. Besides, in the mentioned conception one also tends to tacitly presuppose the fact that straight lines that lie on a surface and do no not intersect have the same direction, i.e., that they form equal angles with a third straight line; but then one already presupposes the axiom of parallel lines. However, with the the help of assumptions that are anyway used, Euclid showed that the first fact follows from the second. Thus, if one assumes the here-delineated conception, which is legitimate, provided one explicitly assumes the axiom (cf. the remarks by Clebsch-Lindemann, [Clebsch 1891, vol. 2, part 1, 543, note]), then one has not only deprived himself of the non- Euclidean geometry, but also built the Euclidean geometry on a larger-than-necessary number of axioms. Mill has suggested to modify the Euclidean definition of parallel straight lines ([Mill 1869, vol. 2, 156]). He wants to define two straight lines a and b as parallel, if and only if they everywhere have the same distance one from the other. Given that the distance between the parallel lines is measured by means of a line segment that is perpendicular to both, Mill thus basically requires two properties: any straight line that is perpendicular to a should also be perpendicular to b, and vice versa; and the parts of all these perpendiculars that lie between a and b should be equal to one another. Yet (in the usual plane geometry) it is also the case that two straight line are parallel, when a third straight line is perpendicular to both. But this fact cannot be derived from Mill’s definition, so it would have to be assumed axiomatically. Anyway, from this fact and from Mill’s definition result two facts that can be expressed even without the concept of parallel lines: 1) given two straight lines a and b, if a third straight line is perpendicular to them, then any straight line that is perpendicular to a is perpendicular to b; 2) in the supposed case, all the straight lines that lie between a and b, and that are perpendicular to them, are equal. It seems now appropriate to let these two facts come first, and add the definition of parallel lines afterwards. Of the two facts 1) and 2), the second follows from the first with the help of the axioms that are used in Euclidean geometry, with the exception of the axiom of parallel lines. I do not doubt that if Mill had known this state of affairs, he would have carried out a reduction of the second fact to the first, because he acknowledges [Mill 1869, vol. 2, 156, note, 146, note] the procedure by means of which one selects some properties of a concept so as to derive its other properties. However fact number 1) remains axiomatic. No matter how one approaches the issue, the usual theory of parallel lines (apart from some inessential changes) must be derived just as Euclid did it. It is not sufficient to assume a definition of parallel lines; one also needs an axiom. If one does not assume an axiom of parallel lines, then one obtains, beside the usual geometry, the non-Euclidean geometry too. 22. Let’s imagine one traces all possible rays through a point—i.e., straight lines that begin in the point and that go to infinity on one side—, and let g be a straight line that does not go through

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the point, and that goes to infinity on both sides. All the rays that intersect g thus fill an angle whose sides do not intersect the straight line g. Now, whereas the Euclidean geometry consists in the assumption that this angle be equal to two right angles, Lobachevsky assumes it to be smaller than two right angles (cf. also note 41). It is also evident that the Euclidean geometry is a borderline case of Lobachevsky’s geometry. Now, introducing without proof the fact that those rays actually fill up an angle is equivalent to assuming an axiom. Lobachevsky gives a proof of this fact that cannot be considered as fully rigorous, at least not in the given version [Lobachevsky 1840, Th. 16, 7ff.]. Nonetheless, there is no doubt that the fact can be proved rigorously from more simple axioms, which must include the axioms of order (note 32) and the continuity axioms (note 49). 23. [Sigwart 1893, .vol. 2, 275, 280] 24. [Zindler 1889, 33]. The demonstrable existential proposition introduced in the text rests, amongst others, on an axiom that establishes existence. Geometrical existence is of course something different from the physical existence of an object, yet I think it can be expressed more clearly by the word ‘existence’ rather than by the word ‘possibility’. 25. The one-sidedness of the usual ‘syllogistic’ has been pointed out especially by Lotze [Lotze 1874, 133]; it is not suitable neither for the various kinds of inferences that are drawn in a science such as mathematics, nor for the various kinds of assertions that occur here. A geometrical assertion, properly, always refers to reciprocal properties of several geometrical elements (points, straight lines, and so on). I think that mathematical inferences, should they be once represented in a purely formal way, can be better conceived from the point of view adopted by English authors in their “Logic of relatives” (cf. [Jevons 1877, 122]), rather than from the point of view of the Aristotelian [syllogistic] figures. According to me, the so-called principle of identity does not clarify mathematical inferences either. For example, in many cases one could conceive the inferences that are drawn from the propositions on parallel lines in the following way. From the propositions on the parallel lines mentioned in the text together with the theorem it follows that if two straight lines in a plane form—in the previously indicated way—equal angles with a given straight line, then they form equal angles with any straight line that intersects them. Here, we can also substitute in a given assertion a straight line for another straight line. The mentioned theorem expresses thus a principle of substitution that constitutes the rule according to which certain inferences proceed. However, the inferences that can be drawn from this principle of substitution are not analogous to the inferences that can be drawn from the logical principle of identity, because here we do not substitute identical for identical, but rather different things for different things. 26. [Kroman 1883, 26, 139]. 27. [Kant 1781-1787, 1st edition 1781, 2nd part, 1st section, 2nd book, 2nd chapter, § 3.1, 288]: “On this successive synthesis of the productive imagination, in the generation of shapes, is grounded the mathematics of extension (geometry) with its axioms...” Cf. also [Kant 1781-1787, part 1, § 1.4]. By the way, from the mentioned passages it cannot be derived with full clarity, whether Kant admits that only the principles have their source in intuition, or rather assumes that intuition is at stake in any step of a proof. 28. Sigwart says that the metric relations of space “are grounded on a necessity of our spatial intuition that cannot be further analyzed”, whereas he understands by ‘metric relations of space’ the metric relations that Euclid has assumed as laws of space [Sigwart 1893, vol. 2, 82, note]. 29. [Kroman 188, 392ff.]. 30. [Kroman 1883, 74–79]. Sigwart too finds ([Sigwart 1893, vol. 2, 226]) that the movement of points and straight lines in space is operant in all syntheses (constructions), and when he speaks about the multiplicity of constructions that extends itself into the inconceivable, he says: “but everywhere there is the same requirement: to run through the whole extension of the assumed possibilities by means of a conceptual formula (what is meant here is a prescription of the

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construction), obtaining at the same time the borderline cases, and tracing the limits of the extension for the purpose of a classification” [Sigwart 1893, 227]. 31. Cf. the examples given in notes 47 and 48. 32. Which facts concerning order one wants to assume as axioms, so as to prove the others from them, is up to a certain degree arbitrary. Cf. [Pasch 1884, 5–7], where similar axioms are established for the first time, and [Hilbert 1899, 6]. To the facts concerning order belongs another fact that has often been considered as an axiom (cf. [Klein 1873, vol. 6, 113]): the infinite length of straight lines. This infinite length distinguishes in quite an essential way the straight line from the circle, which returns upon itself and has a finite extension. In exact terms, the fact that holds for the straight line should be expressed as follows. Given two points A0 and A1 on a straight line, we can imagine an unlimited succession of points A2, A3, A4,… constructed in such a way that A1 lies between A0 and A2, A2 lies between A1 and

A3, A3 lies between A2 and A4 and so on, generally Aν lies between Aν-1 and Aν+1. We further imagine that the line segments A0A1, A1A2, A2A3, A3A4,… are all equal. Then, one can generally assert for any value 2, 3, 4,... of the number ν that neither Aν nor any point situated between Aν-1 and Aν coincides with A0, nor with A1, nor with a point situated between A0 and A1. The first part of this assertion, expressing the possibility of that construction, follows from the axioms of equality (see note 48); the second part can be proved from the axioms of order, as formulated by Pasch and Hilbert. 33. Lobachevsky too presupposes the facts of congruence in his geometry. Apart from that geometry, the facts of congruence also hold in another non-Euclidean geometry, which can be obtained when, besides abandoning the axiom of parallel lines, one also modifies the axioms of order (note 32, cf. also note 41). According to the philosophical perspective, the determinate facts of congruence are sometimes inserted in certain—quite vaguely formulated—properties of space. For example, Lotze says that the usual conception can only imagine space as homogeneous in its whole extension [Lotze 1884, 261]; to this regard, he uses the words: “On the contrary, space itself, as an impartial arena that features all these events, cannot have local differences in its own nature, because they would prevent that everything that exists or happens around one of its points, could be repeated without variation around any other point.” It is remarkable that Mill does not acknowledge in his analysis of Prop. 5 of Book I of Euclid’s Elements ([Mill 1869, vol. 1, 242ff.]), that he himself used the first theorem of congruence (Prop. 4 of the Book I by Euclid) in his presentation, without substituting it with a deduction. However, this is concealed by the vague way in which he expresses himself in this passage. He already used the principle that equal line segments overlap in the widened sense that a pair of unequal line segments must overlap with another pair of line segments that are respectively equal to the former. Thereby, he obviously imagines a pair of line segments in motion, so rigidly tied to each other that the included angle, which was initially identical to the angle between the other line segments, remains equal during the motion. He is able to draw his conclusion only because he assumes that the pair in motion overlaps the other pair; here one has to be guided by some representations of the relative position, and these representations used by Mill are fully equivalent to the application of thetheorem of congruence. 34. Strictly speaking, one needs, as Hilbert [Hilbert 1899; 12–14] has shown, to assume as an axiom just a part of the assertion of the congruence’s theorem; the remaining part can be proved afterwards. The second theorem of congruence, which says that two triangles coincide in all their parts if they coincide in one side and two angles, had already been proved—in the proper sense of the word—by Euclid: it is Prop. 26 of Book I. Among the axioms of congruence Hilbert counts also some facts that ground the addition of line segments and angles, and that are tacitly used in the customary and most common way to carry out geometrical proofs (cf. note 48). 35. Sometimes one experiences—and the first algebra lesson, being often too formalistic, is partly responsible for this—that people readily operate with the rules of the letter-calculus but

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encounter difficulties when they have to substitute numbers in the formulas. In such a case, there is a lack of knowledge of the meaning of the algebraic symbols. But, more generally, what is missing is an understanding of why the algebraic transformations are legitimate. So, the general formulas on which the transformations are grounded, such as a · b = b ·a or a(b + c) = ab + ac, have a content that is fundamentally non self-evident. These are theorems, the first of which says that on the whole I obtain as many objects—let’s say, for example, spheres—when I form a heaps of b spheres or vice versa b heaps of a spheres. Algebra consists in the fact that we have such theorems, and use them to make inferences; the letters thereby only provide an adequate denomination, and the form of the calculus given to inferences has only the function of clarifying the overall view. Each formula and each transformation of formulas can be expressed in words, and there is no fundamental difference between tasks that, as they say, can be solved by ‘raisonnement’, and tasks whose solution necessarily requires calculus. One can thus designate the procedure of the letter-calculus as an inference process that is ruled by formulas, such as ab = ba, a(b + c) =ab + ac used to derive inferences. This is again an inference process that corresponds to the “Logic of relatives” (note 25). If one considers those rules as axioms, then the analogy with geometry is complete. Anyway, an essential difference with respect to geometry consists in the fact that we can prove, inasmuch we go back from the algebraic formalism to the number concept, that ab = ba, and precisely, we can prove it not simply by a confirmation of the given formula by examples, as may have been the case when it was originally discovered by induction, but by a deductive procedure (note 58). 36. [Peano 1888]. 37. [Leibniz 1899, 570]. 38. [Leibniz 1899, 575]. 39. On this cf. [Baumann 1869, vol. 2, 56–63]. 40. Cf. the above-mentioned Briefwechsel [Correspondence], [Leibniz 1899, 577]. 41. [Lobachevsky 1840, 13–17]. I think that among the axioms of Lobachevsky one should also count the axioms of order (note 32), tacitly used by him in a similar form to the one adopted by Hilbert [Hilbert 1899, 6–7]. One can then rigorously prove that the sum of the angles of the triangle is not bigger than two right angles, and that the angle mentioned in note 22 must be equal or smaller than two right angles. 42. Even in arithmetic, which is, in my opinion, purely deductive (see note 58), the results are usually first found by induction. An example is provided by the arithmetical theorem, according to which a prime number that is divided by 4 gives a remainder of 1, can always be represented as the sum of two squares. This theorem was found by Fermat, undoubtedly by induction, and afterwards first proved by Euler. 43. It roughly amounts to the same as saying that from the beginning geometry is only concerned with relative ratios of magnitudes [Zindler 1889, 5]; for this claim could only be justified by the preceding acquaintance with the phenomenon of similarity which Euclid rightly deduces from the more simple facts of equality, congruence, etc. In non-Euclidean geometry, there are no similar non-congruent figures; there is a length proper to any of these geometries which takes a particular position in relation to all other lengths, as the right angle does in Euclid in relation to all other angles. 44. Cf. Helmholtz’s remark on the intuitive knowledge of the typical behavior of bodies, obtained in the observation of spatial relations [Helmholtz 1884, vol. 2, 30]. 45. Book VI, Def. 1, propositions 4–7. 46. In Book V. 47. Euclid asserts all these axioms as applying to any kind of magnitudes (Book I). From the point of view of geometry, the content of these axioms differs according to the kind of magnitude taken into consideration; in the present paper I will restrict myself to line segments.

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The proof mentioned additionally requires the properties of the whole numbers (which by the way can be deduced) as well as some commonly known axioms. This is how the proof is set down: take a straight line g and two points A0 and A1 on g. Now there is exactly one point A2 on g so that

A1 lies between A0 and A2 and the line segment A1A2 is equal to a given line segment; this fact has to be considered as an axiom ([Hilbert 1899, 10]). We take A1A2 as being equal to A0A1 and then assume a point A3 on g such that A2 lies between A1 and A3 and such that A2A3 is equal to A1A2 and thus—according to the former as well as to the known axiom (note 14)—to A0A1. Moreover we choose A4 such that A3 lies between A2 and A4 and at the same time the line segment A3A4 is equal to the line segments A2A3, A1A2, and A0A1. For our next step we choose A5 in the same manner and so on. As a consequence of the axioms of order, e.g., not only does A2 lie between A1 and A3—as it does according to the construction—, but also between A0 and A4 and so on. Given that the line segments between any two consecutive points are equal, we can conclude the equality of the line segments A0Am and AmA2, m being a whole number, by virtue of the axiom which states that equals added to equals yield equals (this axiom being asserted here only for the addition of line segments and the latter being conceived purely geometrically as concatenation [Hilbert 1899, 11]). Now it is possible—by virtue of the definition of the concatenation of line segments and by virtue of the axioms of order—to conceive the line segment A0Anm, n and m being whole numbers, both as the sum of the line segments A0Am, AmA2m, A2mA3m and so on, and as the sum of the line segments A0A1, A1A2, A2A3, … Anm-1Anm, from what follows that for the line segment A0A1, i.e., for any line segment, the proposition holds that its nmth multiple is equal to the nth multiple of its mth multiple. These considerations had to be stated before. If one takes now two given line segments AB and A′B′—being respectively the mth and m′th multiple of the line segment CD—it follows from what was said before that the m′th multiple of AB is equal to mth multiple of A′B′; for the former is the m′mth, the latter the mm′th multiple of CD, and we know that for numbers (see note 58) it holds that m′m = mm′. Now if, at the same time, AB and A′B′ equal respectively the nth and the n′th multiple of one and the same line segment EF, the n′th multiple of AB also equals the nth multiple of A′B′. In the next step one has to show that using CD or using EF in measuring AB and A′B′ yields the same numerical ratio, i.e., it has to be shown that , viz. that mn′ =nm′. If one takes now the nm′th multiple of AB, this equals the nth multiple of the m′th multiple of AB, i.e., it equals the nth multiple of the mth multiple of A′B′, thus it is equal to the nmth multiple, i.e., to the mnth multiple of A′B′. This in turn is the mth multiple of the nth multiple of A′B′, i.e., it is equal to the mth multiple of the n′th multiple of AB, thus to the mn′th multiple of AB. We have now shown that the nm′th multiple of AB equals the mn′th multiple of the same line segment. From this follows that mn′ =nm′. The rigorous proof for that has to be set out indirectly. If it were true e.g., that mn′ > nm′, one would take the (nm′ + 1)th and the (nm′ + 2)th multiple of AB and so forth. The axiom that the greater added to the greater yields the greater—which I hold to contain that adding something to the greater or to the equal yields something greater than adding nothing to the smaller or the the equal—permits us to conclude that the (nm′ + 1)th multiple, the (nm′ + 2)th multiple and so forth and finally the mn′th multiple of AB would be greater than the nm′th multiple of this line segment, which contradicts the result found above. In this somewhat circumstantial proof it becomes clear that nowhere has any use of the geometric intuition been made in an immediate way, but only in a mediate way in implementing a couple of axioms which can be stated in a perfectly precise manner. 48. In the case of line segments, the axioms of equality can be put in such a way ([Hilbert 1899, 10–11]) that from them together with the axioms of order follows the fact concerning the addition of the greater to the greater. This of course presupposes the reduction of the concept “greater” to the concepts “between” and “equal” by giving the following definition: The line

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segment AB is greater than CD if there is on the first line segment a point E between A and B such that AE equals CD. The difficulty of such a proof for the untrained is that one has to put out of one’s mind in an artificial way certain ideas which are taken for granted due to a long process of familiarization. That it is nevertheless possible to unobjectionably set down the proof might be shown by the following considerations. The axioms concerning the equality of line segments posited by Hilbert can be stated in the following way: (1) The line segment AB always equals the line segment BA (to this axiom corresponds the fact that if two rods, one having the ends A and B, the other A and B′, can be superposed in such a way that A′ coincides with A and B′ with B, it is also possible to superpose them inversely such that A′ coincides with B and B′ with A). (2) If the line segment AB equals the line segment A′B′, and if the latter equals the line segment A″B″, it is also true that AB = A″B″ (see note 14). (3) If A, B, and C are points on a straight line, and if A′, B′, and C′ also are points on a straight line, and if furthermore B lies between A and C and B′ between A′ and C′, it follows from AB = A′B′ and BC = B′C together that always AC = A′C′ (axiom of the addition of equals to equals). (4) Given a point A on a straight line g and a second point B on the same line, and furthermore somewhere a line segment CD; then there will be one and only one point E with the twofold property that the line segment AE equals CD and that E is placed on the same side of A as B (i.e., such that A does not lie between B and E); moreover there is also one and only one point E′ with the twofold property that AE′ = CD and that B and E′ are placed on different sides with respect to A (i.e., such that A lies between B and E′). Having given the definition of “greater”, I firstly note that the proposition AB > (greater) CD, corresponding to the way we put it, excludes the proposition AB = CD. Indeed according to the definition there is a point E between A and B (and, thus, different from A and B) such that AE = CD. If it were true in addition that AB = CD, it would follow from axiom (2) that also AE = AB; but since E and B lie on the same side with regard to A, E and B must coincide according to axiom (4), in contradiction to what we said before. Next we suppose that A′B′ = AB and AB > CD and show that from this it follows that A′B′ = > CD. In order to do so we take the point E as before; in addition we take a point E′ on the straight line g′ given by A′ and B′ (axiom (4)) such that A′E′ = CD and such that E′ lies on the same side of A as B′. According to the axioms of order E′ either lies between A′ and B′ or coincides with B′ or it takes such a position that B′ lies between A′ and E′. It has to be proven that the first case is true. Suppose that there is a second point B″ on g′ such that E′B″ = EB and that E′ lies between A′ and B″. Now from AE = CD = A′E′ (cf. axiom (2)) and from EB = E′B″ follows that according to axiom (3) AB also equals A′B″. The points B′ and B″ thus have the same properties; it holds that AB = A′B′, that AB = AB″, and by virtue of the assumptions as well as the axioms of order both B′ and B″ lie on the same side with regard to A′. Therefore B′ and B″ must coincide according to (4). Thus E′, being located between A′ and B″, also lies between A′ and B′; according to the definition A′B′ therefore is greater than CD being equal to the line segment A′E′. If AB > CD, it follows from axiom (1) and from what was proved above that BA > CD (this result states—put in an intuitive way—that, if the line segment CD can be cut off from the line segment AB at its end A, this can also be done at its end B; and from the preceding consideration it also follows that both ways of cutting off yield an equal line segment). Now one might assume that AB is greater than CD and that CD is greater than FG; it is to be proved that also AB > FG. As a result of the assumptions just made there is a point E between A and B with AE = CD. Given that also CD > FG, and that according to what has already been proved AE > FG,

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there is a point H between A and E with AH = FG. From the axioms of order now follows that H also lies between A and B which, together with the last equation, means that AB > FG. From this it results that, according to the way we put the concepts in this note, the propositions AB > CD and CD > AB exclude each other; for from both taken together would follow that AB > AB, whereas the concepts “greater” and “equal” are incompatible (vide supra). On the basis of these preliminaries the proof we are interested in can be set down. The proof can immediately be reduced to the following arrangement. Given a point B between A and C, a point B′ both between A and B and between A and C′; if furthermore BC > B′C′, it is to be proved that C′ lies between A and C. From the assumptions together with the axioms of order it follows that C′ and C are located on the same side in regard to B′; hence C′ either lies between B′ and C, or it coincides with C, or it is located in such a way that C lies between B′ and C′. In the second case, it follows from the arrangement of the points together with the definition of the concept “greater” that B′C′ > BC (or, actually, first of all that C′B′ > CB), which contradicts the original assumption that BC > B′C′. In the third case, the same reasoning yields that both B′C′ > B′C and B′C > BC, for which reason according to what has been proved before B′C′ > BC would hold, which again contradicts the original assumption. The first case hence applies, i.e., C′ lies between B′ and C and thus, according to the axioms of order, between A and C, which was to be demonstrated. For the sake of simplicity and clarity I restricted myself to line segments, the axioms concerning the order of points being at our disposal in the proofs. Euclid always asserts the relevant facts in a general manner for all magnitudes. The concept of a magnitude or a quantum could indeed be conceived as given; in this case it would be possible to show that, insofar as we postulate some of the facts commonly attached to this concept, we can prove the remaining facts. This is what Mill aims to do and what to some extent he indeed carries out [Mill 1869, 146, 147, note]. He is however not aware of all the assumptions he actually makes use of; he thus assumes that adding b to (a − b) + c yields a + c, i.e., he presupposes that (a − b) + c = (a + c) − b. It only appears as if Mill only refers to numbers, and for numbers this assumption can indeed be proved (see note 58). But if it is to be shown deductively that any objects can be conceived numerically, i. e., that they are in numerical relations to each other, certain facts have to be presupposed as axioms in order to deduce the other required facts from them. Considerations of this kind are also of importance as a preliminary for the measurement of a physical state. They make it clear which experiments suffice to prove that the physical state— e.g., the charge of an electrical body—can be conceived quantitatively, cf. [Maxwell 1873,35]. It is also helpful to note that we cannot speak of quantity properly in all cases where we are aware of graduation. We can for example construct a hardness scale by indicating whether two given bodies are equally hard and, in case they are not, which of them is the harder one. Given three bodies we consequently are also able to indicate which one lies between the two others, but we cannot say that one body is twice as hard as a different one, or that two bodies taken together are as hard as a third one. We thus cannot quantitatively conceive hardness. 49. If one adds the “axiom of continuity”, the Archimedean axiom can be proved on the basis of the axioms of order and the axioms of equality. The proof can take the following form. If all points located between A and B are distributed in any way on two categories such that any point belongs to a determinate category, that there are points in both categories, that furthermore, given a point X from the first category and an point Y from the second, X always lies between A and Y, there is in any case between A and B a point C such that between A and C there are only points of the first category and between C and B there are only points of the second category. The point C itself might belong to one category or the other, as the case might be (cf. the different versions of this axiom in [Hilbert 1895, 92]). The axiom of continuity provides a number of additional conclusions. If we divide the points between A and B into two categories, e. g. by assigning a point X to the first category if the double of AX is smaller than AB, and by assigning a point Y to the second category if the double of AY is

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greater than or equal to AB, it is easy to show that both categories bear the properties which are necessary for the application of the axiom of continuity. This axiom then leads to the existence of a point C. The assumption that the double of AC is smaller than AB can be reduced ad absurdum by virtue of the propositions mentioned; the same holds for the assumption that the double of AC is greater than AB. From this it follows finally, that the double of AC is exactly equal to AB. It is thus proved that for a given line segment there is a second one which represents exactly half of the former. In the same way the existence of the third, the quarter and so on can be proved, and thus a great many facts have been reduced to a single axiom. 50. Beside the axioms of order of course the “axioms of concatenation” have to be presupposed. 51. Book V, definition 5. 52. Both μ and ν are whole numbers and I reemphasize that the concept of multiplication of a line segment presupposes the concept of equality, but not the concept of measure. E. g. AB equals the threefold of CD if there are two points M and N between A and B such that M lies between A and N, N between M and B, and AM = MN = NM = CD. 53. [Galilei, Fifth Day, 22ff.]. The different facts admitted as immediately clear by the people in the “Discorsi” imply the assumption of as many axioms. 54. [Hilbert 1899, 26ff.]. 55. The measure of length established in such a way leads to assigning the positive and negative numerical magnitudes to the points of a straight line, fixing arbitrarily the points the numbers 0 and 1 are to correspond to. The so called “projective” geometry leads to a different manner of distributing the positive and negative numerical magnitudes to the points of a straight line, fixing arbitrarily the points three determinate numbers are to correspond to. As one repeatedly concatenates equal line segments in the treatment concerning length, so in this projective treatment an operation has to be repeated several times, which is merely based on the connection of points and the partition of straight lines, without ever equalizing two distances or comparing them as to their equality. It is thus possible to establish a sort of measure in geometry without making use of the concept of length or even of the the concept of equality of line segments (cf. [Staudt 1857, 166ff.]; [Klein 1873, 112ff.]). 56. It was already remarked that Euclid proves too much when sometimes deriving from evident facts other facts which are also said to be evident [Zindler 1889, 36ff.]. Because of the fact, however, that we can avoid intuition itself in the proper proof, the attempt at deducing the whole geometry from a minimum of intuitive facts—or empirical facts—assumed as axiomatic, gains the highest importance. All deductions of this kind are of interest for the interrelation of knowledge—even in the case that we are able to deduce both of two intuitive facts one from the other, using the axioms which have been introduced anyway (in this case it is in fact an arbitrary decision which of the intuitive facts should be added to the axioms). In any case there will be quarrels about what self-evident means, while there is normally no dissent in mathematics whether a given proof is stringent or not. 57. [Schur 1892, section 5, 5], [Killing 1898], [Hilbert, 40ff]. According to the approach indicated in the text, we do not start with a definition of the content that is common to equal figures, but with a definition of a procedure the result of which is taken to be decisive in determining whether two figures (or bodies) can be said to be “equal” or different. Since the definition chosen by us can be proved to be such that two figures which are equal to a third one also have to be said to be equal to each other, it holds that bringing together into one totality all figures equal to a given one will provide a totality of figures equal to each other. This collective concept replaces effectively the concept of content. This method is often applied in mathematics; in our case we can easily pass over to the definition of the measure of area, i.e., to the proper concept of content. The approach set out here can well be compared with Euclid’s theory of proportions. As we did not define content, but merely declared under which conditions two figures are to be said to be equal, so Euclid did not define the ratio of two line segments as a number, but merely indicated

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under which conditions two line segments are to be said to be in the same ratio as two different line segments. 58. Arithmetical concepts can be used in deduction—also in geometrical deduction—in a way that geometrical concepts are never used within deduction. The geometrical concepts of point, line segment, angle, and so forth are handled in proofs in the purely “formal” manner already explained in the text. We imagine a plurality of geometrical elements bearing certain relative properties and can derive from this new relative properties of the elements, operating not really with the geometrical concepts, but only with the axioms relating them. If one has to prove a proposition concerning a square or a hexagon, one will achieve this by repeating four times or six times a certain thought operation. But if one has to prove a proposition concerning the general n-sided polygon or the general polygon with an even number of sides, one has to conceive a thought operation as repeated an indeterminate number of times. In doing so general concepts have to be associated with the repetition of this thought operation as well as with their order. These general concepts of a secondary kind are number-concepts or combinatorical concepts, and in the proof these concepts are used not only “formally”, but “contentually”. The difference between the formal and the contentual use of concepts can be made clear by comparing algebra—in the ordinary sense of the word—to arithmetics. In showing e.g., that (a − b)(a² + ab + b²) = a3− b3, one operates merely with algebraic symbols according to external rules comprising the rule that ab = ba. But if one wants to prove that the content expressed by the formula ab = ba is true, one has to take into account the nature of the concepts of number and multiplication. We have to show that we are dealing with the same number of objects, e.g., spheres, whether there are a collections of b spheres or b collections of a spheres. As is well known, the proof goes like this: one begins with conceiving a collections of b spheres. Next one takes away one sphere from each collection, which results in a collections of b − 1 spheres and, by bringing together the removed spheres, a new collection of a spheres. Now once again one takes away one sphere from each of the a collections, what results in a collections of b − 2 spheres and, together with the collection previously formed, two collections of a spheres. By proceeding in such a manner finally all of the a collections will be exhausted at the same time, and the spheres originally given are now parted in b collections of a spheres. We thus provided a proof of ab = ba in terms of rearranging spheres. It is difficult to say on which ground we conceive the general possibility of this rearrangement; one cause may be seen in the fact that we actually already carried out operationsof this kind. As was noticed for the first time by Schröder ([Schröder 1873]) and as then was stressed by Helmholtz ([Helmholtz 1895, 358]), the concept of the cardinal number which the preceding consideration is based on presupposes a certain fact which I nevertheless hold to be provable, cf. [Stolz 1885, 9, 10]. Helmholtz, following the example of H. Graßmann ([Graßmann 1860]), presented a different proof of the propositions concerning numbers (cf. also [Kronecker 1887, 263ff.]). He makes use amongst others of the formula ab = (a − 1)b + b which he considers to be an “axiom of arithmetics”. This axiom (of Graßmann) however is—as Helmholtz himself put it in an analogous case (mentioned in the above quoted “Philosophische Aufsätze” ([Helmholtz 1887, 24] and ([Helmholtz 1895, 363])—merely a description of the procedure of multiplication. The formula can indeed be conceived as the definition of multiplication, presupposing that the concept of addition is known. For given the case that someone does not know what a times b is, we could explain to him: 1 times b is b, 2 times b is 1 times b increased by b, 3 times b is 2 times b increased by b, and so forth; that is to say that in general a times b equals (a − 1) times b increased by b, or, as a formula, ab = (a − 1)b + b. In the same manner any procedure that can be repeated indefinitely yields a general concept as well as a rule for the application of this concept, the rightness of which we understand together with the reality of the concept. For example, we are able to multiply 1 by 2, the result by 3, the

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result obtained by 4, and so forth. The result obtained in the multiplication by a will be designated by a!, and to this concept applies the rule that a! equals (a − 1)! a. Such rules may appear as axioms, since they cannot be formally proved, but are immediately abstracted from the procedure in question in order to form the basis of formal deductions. But it is in fact clear that in arithmetic we would get an infinite number of axioms, because any procedure that can be indefinitely continued yields such an axiom. For this reason I would not like to call the formulas thus obtained axioms; but there is still another reason for this: we frequently introduce into arithmetic a preceding or a combinatorical procedure in order to find, with the help of the new concepts and rules yielded in such a way, certain hidden properties of already known concepts or, insofar as these properties have already been inductively found, to prove them. We can prove for instance the theorem of arithmetics mentioned in note 42 by means of a procedure of “reduction” of the so called “quadratic forms”. In arithmetic the abstraction of new general concepts and rules thus forms in a way a constituent of deduction, and the procedure providing the basis of abstraction is a process we do not count among experience in a narrower sense of the word, that is to say in opposition to reasoning. Having said this I want to assert that arithmetic—at least in so far as it concerns whole and rational numbers (as regards rational numbers cf. [Stolz 1885, 25ff.] and a review by the author in Göttinger gelehrte Anzeigen, [Hölder 1892, 592ff.]—has no axioms in the proper sense, and that the arithmetical proof process is not a purely formal process, but a mixed and extremely complex process. b Cf. Mircea Radu’s translation of this text in this volume, pp. 57–70) 59. The method of exhaustion, usually attributed to Archimedes, has already been used by Euclid in the proof that two circular surfaces stand in the same ratio as the squares of their radii (Book XII, no. 2). 60. The content of the circular surface can also be determined as follows. The circle is said to be a regular polygon of an infinite number of infinitesimally small sides which can be divided into an infinite number of infinitely narrow triangles; the proposition in question follows from composition of these triangles, having the radius as their height. Inferences of this type are called “method of infinity”. The ideas involved in this method of infinity basically have to be considered as auxiliary ideas used for the sake of brevity. Such considerations get a precise sense when interpreted—as done in the text—in terms of the method of exhaustion (i.e., the method of inclusion in successively narrower limits). In this way strength is given to these considerations, and it goes without saying that in mathematics the method of infinity does not contain a new (transcendental) principle. Moreover, in the method of infinity the indirect procedure only seems to be avoided. 61. There are a couple of mathematical proofs which have never successfully been set down in a direct manner, and it might be possible to show that they can actually only be set down as an indirect (apagogic) proof. The proof given in the text for instance has been reached only by virtue of a theoretical twist rendering the proof indirect. The concerns raised by some authors thus seem to me unjustified. The indirect proof rests on the principle that the contrary of an assumption from which a contradiction can be derived must be true. The principle is quite a useful version of the principle of contradiction, whereas the so-called identity principle which is said to comprise the principle of contradiction is definitely unfruitful. 62. I hold the Archimedean proof of the Law of the Lever to be among these proofs in mechanics. In this regard I disagree with the remarks of Mach in Mach1883ul. Mach advances the view that in Archimedes it is already presupposed that the effect that a weight exerts on the lever depends on the product of the magnitudes of the weight and the arm of the lever. This is not the case. Archimedes ([Archimedes 1881, 142]) indeed presupposes that there is no equilibrium in the case of equal weights and unequal arms of the lever, and that an existing equilibrium will be disturbed when one of the weights is increased; it is implied therein that the equilibrium depends on the

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arm of lever on the one hand and on the weight itself on the other hand, but it is not implied that an existing equilibrium is preserved in the case that for instance one of the weights is triplicated and at the same time the fulcrum is shifted in such a way that the arm of the lever is reduced to a third. In addition to these precisely stated assumptions, Archimedes makes use of another assumption. He assumes that the apparent use he makes of the center of gravity amounts to nothing else—and this also seems to be Mach’s opinion—than that in an arrangement of any number of weights two equal weights suspended at different points can always be replaced by a single one of double weight which is suspended at the mid-point between the points of suspension of the two weights. Archimedes additionally presupposes that the inverse substitution also is feasible. The assumption can be thought of as being motivated on the one hand by the consideration that equal weights at equal arms of the lever are in equilibrium and exert exactly the same force on the point of suspension as these two weights would do if conjoined below the point of suspension; and on the other hand by the assumption that the observed equivalence also holds under different circumstances. The latter assumption can again be reduced to the following one, that a system of forces acting on a rigid body is in equilibrium, if a part of the forces considered separately is in equilibrium and if the same also holds of the other part of the forces. This is a postulate (an axiomatic assumption)—motivated, of course, by experience. This assumption can be said to contain a wholly general idea of the fact that effects of forces can be composed, and hence one may be led to the opinion that the assumption is equivalent to the physical principle of “superposition” (on superposition, cf. [Volkmann 1894, 21] and [Volkmann 1896, 69ff.]). An essential difference however consists in the fact that the application of the principle of superposition presupposes a particular metrical law according to which the effects of any two forces can be composed. The assumption made here is easier. Whatever one might think about Archimedes’ suppositions, it cannot be denied that it is not possible to immediately, i.e., by means of the mere power of judgment, perceive in them the Law of the Lever, though this metrical law necessarily follows from these suppositions in a mathematical reasoning. Such proofs (deductions) are always of value for the deeper knowledge of the scientific interrelation they provide; this is true even if the result to be deduced has previously been found inductively—a claim we cannot assert in the case of the Law of the Lever since the history of its discovery is unknown to us. In any case, in order to immediately establish in an inductive way the Law of the Lever, numerous and more precise measurements would have to precede, whereas already the experiences of every day life might lead us to postulate the Archimedean assumptions, which recommends these assumptions as the simpler foundations. These foundations nevertheless are inductive; but this fact does not prevent us from using them deductively, indeed just as we also use empirical laws of nature deductively. The real method of mathematical physics consists of a conjunction of the inductive and the deductive procedure (cf. [Wundt 1883, 66]). 63. Cf. Philosophiae naturalis principia mathematica [Newton 1687]. 64. It is of utmost interest that the mechanical concepts, though they are widely considered as extracted from experience, show in their scientific use the same ideal perfection as geometrical concepts. As regards the latter, the circumstance in question was exploited for the benefit of a philosophical theory, in particular for the benefit of Kant’s conception of space. It is usually said, then, that the empirical objects never perfectly represent the geometrical concepts, for in experience there are merely little bodies instead of points, whereas the concepts and propositions of geometry are held to be exactly valid for the perfect laws of the intuition proper to us and thus basically apply to intuition rather than to experience. In this regard I have to object that I cannot credit intuition neither with perfection nor with exact correspondence to geometry insofar as this intuition is not merely postulated, but is actually to be observed in us. In fact we are only acquainted with the faculty of our imagination,

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to represent images of spatial objects with a vividness close to that of actually seeing these objects. On the contrary, I cannot pretend to be able to imagine two completely equal line segments. I am however completely able to picture two approximately equal images of line segments and, by doing so, to decide that I will consider them as equal, i. e., I am able to conceive them as equal. Nor I could pretend to imagine with sensual vividness a straight line as infinitely extended. I rather imagine a limited straight line augmented by a segment, and the thought that the same segment can be added to the straight line again and again makes me conceiving of it as infinite. Similarly in mechanics the thought of a “material point” results from the idea of a body whose mass is taken into consideration whereas its extension is so little or can be made so little by compression that it is no longer taken into consideration. As the mechanical concepts are thus abstractions on the basis of experience, so also the geometrical concepts are likely to have resulted from abstraction—be it from experience or from intuition—, and geometrical and mechanical concepts thus hardly differ from ordinary empirical concepts. But there is more. After having found, by virtue of experience, laws which interrelate the geometrical concepts, these laws fuse in our consciousness with the concepts, and from this results the will to retain these laws given any future experiences; this however will also affect the particular application of the concepts. In mechanics, this is indeed the case. It has already often been remarked and has often been objected against an empirical origin of geometry that we usually correct experience by geometry, not the other way round. It is true that we often do so. If we put a ruler on the table and if the ruler does not fit the table in all directions, we will conclude from this that the table is not perfectly plane, but not that the propositions applying to straight lines and planes are incorrect. I nevertheless think that this point does not refute the empirical origin of geometry. In mechanics we act precisely in this way; if, e. g., a comet shows an inexplicable deceleration of its motion, we assume for the sake of the law of inertia that the space is filled with a resisting medium. We can understand this point as follows: the primitive concepts of geometry and mechanics as well as the simplest concepts of this science are abstracted according to experience, in the case of geometry probably according to quite raw experiences. Only after the laws, obtained by experience, proved to be true on the whole and to be useful in application, one attempted to postulate them as exactly valid for points and straight lines, for masses that are perfect points, for perfectly constant forces and so forth. With these postulates we thus ascend to unlimitedprecision (cf. [Volkmann 1894, 17]). The interrelated concepts of the geometric elements—point, straight line and so forth—together with the relations which can hold between these elements (e.g., a point can lie on a straight line) and the laws (axioms) postulated by us and permitting us to conclude the existence of certain relations from the existence of other relations, amount to a completely elaborated conceptual image of the real and apparent order of things which is called “space” (this concept of space is a concept of higher order which shows a logical structure and permits us to draw conclusions from, cf. [Volkelt 1886, 384]). But since we are able, as we have seen, to develop from different assumptions different coherent geometric systems, the concept of space can also be formulated in different ways. I think that the earlier philosophers who concerned themselves with the essence of space in geometry all had in mind the Euclidean concept of space, and their theories differ only in how they thought about the relation between this concept and reality. It is comprehensible that some wanted to credit such a consequential concept with a higher truth in regards to reality, whereas Kant considered it as subjective, assuming this concept to be identical with intuition and this “pure” intuition to be the form that things have to appear in because of our mental organization. If one takes into consideration that we have a similarly consequential concept of the motions of a body and of the play of forces governing this motion, which probably rests on experience, one

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will be apt to believe that also the concept of space has been formed with the help of experience. It will no longer appear contradictory that, though we use this concept in some cases in order to interpret experience, we nevertheless consider it possible to check this concept—whose adequacy is hypothetical—for correspondence with experience in order to reshape the concept, if necessary, as we do with physical concepts (in the domain of physical knowledge we admittedly form concepts by the way of trial and error, reshaping them in the case of non-confirmation, cf. [Sigwart 1893, 242]). The lack of exactness which, in regard to geometry, inheres in empirical objects, does not necessarily obstruct verification. Euclid’s geometry allows us to draw conclusions for extended bodies, i.e., also for points that are not exactly points, straight lines that are not exactly straight lines, and so forth, for which reason we can verify whether experiences obtained with such imprecise objects agree, within accuracy limits, with conclusions from the axioms. The same holds for the mechanical postulates which we check only in a mediate way. Until now there was accordance between Euclidean geometry and experience; this geometry has been verified in innumerable cases, since practically any empirical application of this geometry is such a verification. At least as regards application, there is no reason to deviate from Euclid’s geometry. Riemann presumes that possibly in the future new facts will be found which will determine such a deviation, and already Lobachevsky held that astronomical measurements might givereasons for this. Such assertions have provoked rejection in philosophy (cf. [Lotze 1884, 248]; [Sigwart 1893, 82]). There is no doubt that a contradiction in astronomical measurements and calculations might be interpreted in quite different ways. We could assume that the light ray which falls in the observer’s telescope is not completely straight; for even if we originally abstracted the concept of the straight line from the line of vision, it is still possible to subsequently relate the concept of the exactly straight line to the Euclidean axioms and then to distinguish between the straight line and the line of vision. But, since the astronomical measurement in question must be made at different moments and since the celestial bodies meanwhile have moved, the error might also be corrected by assuming different laws for the motion of the celestial bodies. In order to take into consideration in our hypothetical case all possibilities, the whole system of physico-geometrical concepts, insofar as it is relevant for astronomy, must be reorganized (the work of Lipschitz, who developed a mechanical theory for non-Euclidean geometry, can be regarded from this point of view, cf. [Lipschitz 1872, 116]). It is surely extremely improbable that geometry would finally be the object of this reorganization—but it is nonetheless not impossible. 65. The sensation which makes us declare something as evident seems in my opinion to rest on this. 66. The fact that it is impossible to define properly, i.e., constructively, the primitive concepts of mechanics has also resulted in the complete abandoning of any definition of these concepts, e.g., in [Kirchhoff 1877]. At the same time Kirchhoff declared the problem of mechanics to consist in a complete and most simple description of motions in nature. The descriptions he has in mind are effectuated in terms of equations; he considered the magnitudes of the forces and masses to be nothing else than numerical coefficients figuring in these equations [Kirchhoff 1877, 5, 12]. This conception does not do justice to the physical content of the concepts and within this conception it seems inexplicable why those numerical coefficients, which are called masses, are independent from time. Kirchhoff’s claim that mechanics merely describes has been repeated too often. In a certain sense explaining of course amounts to describing, but nonetheless it fundamentally differs from a purely external description, and we should thus retain the distinguishing term. Paul Du Bois-Reymond says : “The derivation of a manifold phenomenic domain from the simplest elements of appearance is no description. I think it is better called the synthesis, the construction or the build-up of the phenomenic domain from simplest mechanisms” [du Bois- Reymond 1890, 14].

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67. As has already been remarked, the similarity between geometrical and mechanical concepts as well as between geometrical and mechanical deduction provides evidence for the empirical origin of geometry, unless certain concepts of mechanics also are considered as “a prioric”, i.e., as independent from experience. This seems to be the case in Kant where he speaks of “pure” science. For example, the persistence of substance he counts among “the pure and completely a priori laws of nature” ([Kant 1781–1787, Pt. II, Div. I, Bk. II, Ch. II, 3., A, 301]). What should be understood, according to Kant, by persistence of substance becomes evident from the example given by him in the passage in question: “A philosopher was asked: How much does the smoke weigh? He replied: If you take away from the weight of the wood that was burnt the weight of the ashes that are left over, you will have the weight of the smoke. He thus assumed as incontrovertible that even in fire the matter (substance) never disappears but rather only suffers an alteration in its form” [Kant 1781–1787, 302]. Kant hence wants the precise law of the conservation of matter to be included in the principle of persistence. He however had a different opinion about different mechanical concepts, e.g., when he considers motion to be an empirical concept (Doctrine of Elements, Pt. I, Div. I, Bk. I, Ch. III, 3., [Kant 1781–1787]). In addition to this similarity, an immediate interrelation between geometrical and mechanical concepts can be found. Helmholtz indeed assumes (annotation 14) that the geometrical concept of congruence presupposes a physical concept, e.g., the concept of the rigid body; and Benno Erdmann considers this in his Axiome der Geometrie ([Erdmann 1877, 91, 147, 148]) as a proof of the empirical view. 68. On “experience as a verification of correctness of knowledge”, cf. [Volkelt 1886, 256]. 69. In both sciences it becomes evident that a collection and examination of empirical facts has to come first in order to inductively find the laws which make possible deduction in the particular parts of the science. This becomes evident because still today empirical facts are— intentionally—collected by means of measurement and experiment.

ENDNOTES a. As regards the concept of content [Inhalt], Joel Michell remarks in a note to his and Catherine Ernst’s translation of Hölder’s The Axioms of Quantity: “Hölder’s term Inhalt literally means contents and, so, is more abstract than either of the English terms area or volume (Flächeninhalt and Rauminhalt in German, respectively). Since contents is not a specifically quantitative concept in English, translating Inhalt as contents would cause Hölder’s term to lose an essential ingredient. Spatial magnitude might be a better choice, but there is evidence [...] that he may have meant area”, [Hölder 1996, 251].

AUTHORS

OTTO HÖLDER (1859-1937)

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Otto Hölder’s 1892 “Review of Robert Graßmann’s 1891 Theory of Number”. Introductory Note

Mircea Radu

1 In 1891, the year in which Robert Graßmann’s Theory of Number was published, Otto Hölder (1859-1937) was 32 whereas Robert Graßmann (1815-1901) was 76 years old. Graßmann’s book is the closing chapter of a long and remarkable mathematical and philosophical tradition. This tradition goes back to the mathematical and philosophical ideas that had been gradually developed by the elder Justus Graßmann (1779-1852) beginning with 1817.2 This line of thinking was picked up and substantially developed by Hermann Graßmann (1809-1877), Robert Graßmann’s elder brother. The most important achievements of this tradition are found in the first edition of Hermann Graßmann’s pioneering work The Calculus of Extension (Die Ausdehnungslehre) published in 1844 and also in the Treatise on Arithmetic (Lehrbuch der Arihtmetik), which was published in 1861. The latter was planned and written together with a series of other books in close collaboration by the two brothers.3

2 An essential component of this tradition is the quest for restoring the organic unity of the system of mathematics and related to that, the idea that such unity can only be uncovered by examining the specific process of sign-construction involved in producing mathematical knowledge independently of any other disciplines. Such a treatment of mathematics must be reached without presupposing logic, language and, of course, independently of sense experience. Reorganizing the body of mathematical knowledge in accordance with this methodological ideal was the main objective of Robert Graßmann’s mathematical writings. The outcome of this endeavour are found in The Theory of Forms or Mathematics (Die Formenlehre oder Mathematik) published in 1872 and in the Theory of number (Die Zahlenlehre oder Arithmetik)published in 1891.

3 We do not know how Hölder came to be interested in Robert Graßmann’s book. It is, indeed, a bit surprising that a book mainly concerned with fairly elementary mathematical ideas attracted his attention. Hölder was a careful student of Helmholtz’s work and Helmholtz praised Graßmann’s 1861 treatment of arithmetic in it. This may

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have been Hölder’s first contact with the name “Graßmann”. Another link between Hölder and Graßmann may have been reached through Paul du Bois-Reymond, Hölder’s Doktorvater. In his Theory of Number, in the effort to clarify the nature of his own approach, Graßmann provides a brief and harsh criticism of du Boys-Reymond’s ideas (see below). On the other hand, from an early stage in his career on, Hölder was regarded as a scholar with a strong interest in matters of method. A project such as Robert Graßmann’s with its self-declared objective of revolutionizing the presentation of mathematics may therefore have attractedHölder’s attention.4

4 Be this as it may, Hölder used the Review as an opportunity of presenting his own ideas concerning the foundations of mathematics. Hölder’s Review is the expression of the clash between the genetic and the axiomatic method in dealing with the foundations of mathematics, a long time before David Hilbert’s discussion of the distinction and the debate triggered by Hilbert’s position were published. Hölder’s discussion of the distinction provided in the Review contains a series of remarkable considerations about things like the relationship between existence and consistency in mathematics and about the general requirements that should be met by an axiomatic approach to the foundations of pure mathematics. It also contains some important remarks about the relationship between logic, proof, and arithmetic.

5 Hölder basically argues that Graßmann’s conception can be given both a formal- axiomatic as well as a genetic interpretation. Hölder raises a series of objections to the former and endorses the latter. In a sense, Hölder’s Review is less a confrontation with Robert Graßmann. It is rather an outline of Hölder’s own conception of axiomatics, of its proper place and limits in dealing with the foundations of mathematics. Hölder’s Review is the first statement of an original critical philosophy of mathematics whose mature expression enfolded in a series of subsequent writings.

6 One difficulty in truly appreciating Hölder’s ideas is caused by the style of the Review. In the Review one often finds only short fragments, hints, and allusions, where more elaborate explanations would have been needed. The other difficulty is linked to the fact that, in a sense, the Review requires a reader, familiar with the broad tenets of Robert Graßmann’s work. In some cases I have included Hölder’s original terminology in German along with the English translation. In all these cases I preserved the orthography found in the German original document, which in many cases differs from modern German orthography.

7 In order to render the significance of some of Hölder’s statements clearer, I have included a series of footnotes in which I provide some context to Hölder’s statements. The notes at the end of the document belong to Hölder.

BIBLIOGRAPHY

GRASSMANN, HERMANN 1862 Die Ausdehnungslehre, Berlin: Enslin.

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2000 Extension Theory (1862), The American Mathematical Society, translated by Lloyd Kannenberg.

GRASSMANN, JUSTUS 2011 On the concept and extent of the pure theory of number (1827), in Hermann Graßmann—From Past to Future: Graßmann’s Work in Context. Graßmann Bicentennial Conference, September 2009, edited by PETSCHE, H.-J., LEWIS, C. A., LIESEN, J., & RUSS, S., Basel: Birkhäuser, 455–488, translated by Lloyd Kannenberg.

GRASSMANN, ROBERT 1872 Die Formenlehre oder Mathematik, Hildesheim: Georg Olms Verlagsbuchhandlung. 1891 Die Zahlenlehre oder Arithmetik streng wissenschaftlich in strenger Formelentwicklung, Stettin: Verlag R. Graßmann. 2009 Unpublished description of the life of Hermann Graßmann by his brother Robert (1877), in Hermann Graßmann–Roots and Traces, edited by PETSCHE, H.-J., KANNENBERG, L., KESSLER, G., & LISKOWACKA, J., Basel: Birkhäuser, 203–221.

HÖLDER, OTTO 1892 Review of Robert Graßmann, Die Zahlenlehre oder Arithmetik, streng wissenschaftlich in strenger Formelentwicklung, Göttingische gelehrte Anzeigen, 154(15), 585–595, Engl. tr. by M. Radu, this volume 57–70. 1924 Die mathematische Methode. Logisch erkenntnistheoretische Untersuchung im Gebiete der Mathematik, Mechanik und Physik, Berlin: Springer.

RADU, MIRCEA 2000 Justus Graßmann’s Contributions to the Foundations of Mathematics—Mathematical and Philosophical Aspects, Historia Mathematica, 27, 4–35. 2003 A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann, Historia Mathematica, 30, 341–377. 2011 Axiomatics and self-reference—Reflections about Hermann Graßmann’s contribution to axiomatics, in Hermann Graßmann—From Past to Future: Graßmann’s Work in Context. Graßmann Bicentennial Conference, September 2009, edited by PETSCHE, H.-J., LEWIS, C. A., LIESEN, J., & RUSS, S., Basel: Birkhäuser, 101–116.

NOTES

2. Justus Graßmann was the father of Hermann and of Robert Graßmann. Beginning with 1817, Justus Graßmann published a series of books in which he treated several mathematical subjects and, at the same time, articulated a uniquely interesting conception about the foundations of mathematics in which philosophical as well as pedagogical ideas played a very important part. One of the most important of Justus Graßmann's works was published in 1827. Since 2009 an English translation of this work due to Lloyd Kannenberg is available [J. Graßmann 2011]. A detailed analysis of Justus Graßmann’s ideas is provied in [Radu 2000]. 3. Compare, for instance, [R. Graßmann 2009, 214]. 4. Compare [Radu 2003, 342ff.]. 1. I thank Dr. David Marshall for native speaker advice.

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ABSTRACTS

Hölder’s review of Robert Graßmann’s Theory of Number provides the first statement of Hölder’s most significant tenets concerning the distinction between the genetic and axiomatic presentation of mathematics, the mature expression of which is found in Hölder’s book The Mathematical Method of 1924. By translating Hölder’s review into English, I hope to make this unique document known to a wider public. In my introductory note I provide some context to Hölder’s paper and a few other remarks concerning the translation work.1

Le compte rendu de la Théorie du nombre de Robert Graßmann constitue la première affirmation des idées de Hölder sur la distinction entre présentation génétique et présentation axiomatique des mathématiques, qui trouvera sa formulation mature dans le livre La Méthode mathématique (1924). J’espère que cette traduction en anglais du compte rendu de Hölder permettra de faire connaître ce document unique à un public plus large. Dans la note introductive, je précise le contexte de l’écrit de Hölder et j’ajoute quelques commentaires concernant la traduction.

AUTHOR

MIRCEA RADU Universität Bielefeld (Germany)

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Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891)

Otto Hölder Traduction : Mircea Radu

RÉFÉRENCE

Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas. Stettin, Robert Graßmann Pub. Co. 1891, XII and 242 pages. Price: 5 Marks.

1 The author of this book pursues the objective of treating the whole of pure mathematics [die ganze reine Mathematik] in four sections [Abtheilungen].a One half of the first of these sections is dedicated to arithmetic and is already available. The other half of the first section “A heuristic treatise on number [Zahlenlehre in freier Gedankenentwicklung]” which treats the same discipline is supposed to follow.b The author may have opted for such an unusual separation [of the treatment of arithmetic – M. R.] in the assumption that this would underpin his strong commitment to rigor [Strenge], something of great importance to him.c

2 The Theory of Numberd begins with an extensive introductory chapter [Einleitung in die Zahlenlehre]. It contains a theory of operations with abstract magnitudes in the most general sense of the term [Theorie der Größenoperationen im weitesten Sinne des Worts]. Even at this early stage of the presentation, great care is taken to accommodate multiplication as an operation, which, in addition to not satisfying the commutative law, does not even satisfy the associative law. The introduction also provides, right

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from the beginning, a discussion of the various forms of proof [Beweisformen], particularly of the so-called mathematical induction [vollständige Induction].e

3 The introduction is followed by the treatment of arithmetic in four chapters. The first chapter contains the four fundamental operations with whole—positive and negative— numbers and with fractions. Results concerning prime-factor decomposition of integers are also included. Beside that, this chapter also contains a detailed exposition of practical calculations with decimal fractions and with named numbers [benannte Zahlen]. The second chapter deals with powers, roots and logarithms; one also finds the binomial and the geometrical series, as well as the arithmetical series here. The third chapter contains a combined treatment of trigonometry and of the complex numbers, which also includes solving algebraic equations up to the fourth degree. Approximation methods for determining higher order roots [of polynomials – M. R.] arealso presented.

4 Selection of the material covered is determined by the author's aim of writing a book for teachers [Lehrer] that at the same time meets the highest standards of scientific rigor. One can accept that the elementary teaching of arithmetic should be pursued in a more rigorous way. It is often dealt with mechanically, with insufficient attention dedicated to the justification of the methods presented. One would, however, not deny that it is possible to go too far in the pursuit of rigor.

5 I would not recommend Graßmann's treatment of arithmetic for teaching purposes. The treatment of the operations [Rechnungsoperationen] in the most general terms possible is too abstract for the student. The concepts dealt with [in schools – M. R.] should rather be introduced starting with the positive whole numbers [die positiven ganzen Zahlen]. I do not attach great weight to the objection already raised by Mr. Graßmann in his book that, in this case, one would have to repeat the same steps over and over again. Transposing a proof [Beweis] from a special case to a more general one is a useful exercise. In the case of frequent repetition, one can invoke the analogy and proof may be omitted. A presentation which begins with concrete examples also has the advantage that certain objects and operations satisfying certain computation laws are known beforehand, so that one has a firm ground under one's feet, right from the start.

6 I now come to my fundamental criticism of the author. He holds his approach to be a milestone on the path towards a rigorous treatment of arithmetic and scorns alternative treatments. It is thus only reasonable to judge his work by the same high standards. I do not, however, believe that his work can pass such a test.

7 It is perhaps not so easy to do justice to the author, because his conception differs so strongly from the commonly held views. His conception is, so it seems, underpinned by a peculiar philosophical principle.

8 By magnitude [Größe] he understands (No. 2) “everything that is or can become an object of thinking [Gegenstand des Denkens], in that it only possesses one rather than multiple values [Werthe]”. He calls equal (No. 10) “two magnitudes which can be replaced the one for the other in the connections [Knüpfungen] of the theory of magnitudes [Größenlehre] without, however, changing the value of the connection”.f

9 I do not wish to claim that the term “magnitude” was defined by replacing it with the synonymous term “value”. Requiring of a magnitude to have a unique value is, I think, simply the expression of the determination of the principle used when comparing magnitudes. But should there be no objects of thinking that can be compared based on

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criteria not involving value? And how are we supposed to apply the above criterion of equality? Conceptual formulations of this kind remind us of Scholastic philosophy, which is not yet fully extinct in Germany.g

10 In my view, one should begin by describing the objects to be dealt with. One should afterwards explicitly name the criteria to be used for comparing objects, for deciding if two objects are declared equal or unequal.h Then, after having clarified the operations involved, one can move on to proving, say, that equals added to equals yields equals. In this manner, the proposition according to which equals can be substituted for equals [daß man Gleiches für Gleiches setzen kann] would become a non-tautological proposition [Lehrsatz], and only such propositions allow fruitful applications. My remarks concerning the treatment of fractions [Bruchlehre] provides an illustration of this.

11 Still, I do not wish to attach too much importance to the definitions mentioned above. They are only an external robe [äußeres Gewand]. In fact, there is, despite these definitions, little to object to with regard to the consistency [Folgerichtigkeit] of the inferences given in the general introductory part. Solely, under this kind of derivation [Herleitung], it is only possible to assign a hypothetical validity [eine hypothetische Giltigkeit] to the results thus derived.i If, for instance, an addition operation is defined on a domain of magnitudes, and this operation admits an inverse operation defined without restrictions (that is, if it leads to a subtraction that can be always carried out, i.e., the outcome of which is uniquely defined), Mr. Graßmann calls this addition a separable connection [trennbare Knüpfung].j However, the possibility of a separable connection should have been proved beforehand. In the case of a treatment such as that of Mr. Graßmann, which takes only a single operation as its starting point, this result may almost be seen as obvious, and the requirement of proving it overlooked. The matter would look rather different if two interconnected operations were considered. This happens when a multiplication or a separable multiplication is added to the separable addition. In this case, multiple relations between the two operations are required, and it is not immediately obvious, whether an iteration of these operations would not lead to contradictions. And even if the system as a whole were to be proved consistent [widerspruchslos] one would have to require a justification of its applicability.k

12 Such a justification should be given in the first subsection right after the general introduction. Here, however, something seems to be missing in the introduction of negative numbers and of fractions (No. 165): “division is the name of the separation corresponding to the multiplication of the numbers” and immediately afterwards the term “dividing-magnitude” [Theilgröße] is introduced without any critical examination.l Concerns about the existence of such a magnitude are simply overlooked. It goes without saying that, according to the spirit of Mr. Graßmann's own approach, a foundation of the theory of fractions must be laid independently of any considerations of external intuition [äußere Anschauung]1. Thus, our difficulty cannot be put to rest simply, say, by calling upon the fact that it is always possible to divide continuous extended magnitudes.

13 It would, however, be possible to hold another position, which, even though nowhere explicitly stated in the book discussed here, may express the actual conception of its author, namely, that the peculiar basic formulas [besondere Grundformeln]m posited at the beginning of the treatment are so chosen as to define both the numbers and the operations at the same time.n This is to some extent correct. To clarify this, I must

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emphasize two distinct concepts standing behind the expression whole number [ganze Zahl].o

14 The concept of cardinal number [Cardinalzahl oder Anzahlbegriff] emerges based on comparing aggregates of discrete objects. This is done by associating one individual of one of the aggregates to an individual of the other aggregate. While doing this, one examines whether, while performing this procedure of comparing the two aggregates, both aggregates are simultaneously exhausted or not. As Mr. v. Helmholtz pointed out2, Mr. Schröder3 was the first to recognize that here an additional assumption is tacitly made, namely, that the outcome of this comparison is independent of the manner in which it is carried out.p Because this fact can be easily proved 4, a treatment of arithmetic taking the concept of cardinal number as its starting point, looks unproblematic to me. I also do not accept the idea that the cardinal number concept requires external experience, for I am able to count things given just in thinking— names stored in my memory, or things like that. In doing this, one only needs such psychological and logical actions as are required by any presentation of arithmetic and by any mathematical deduction.

15 Another possibility is to take the ordinal number concept as a starting point. It is possible to carry out the addition, subtraction, and multiplication of the whole numbers without previously introducing the cardinal number concept5. In this case one regards the number sequence 1, 2, 3, 4, 5… as made of arbitrary signs, which gain their specific meaning only from their fixed order within the sequence. To add 1 to a number thus means nothing other than moving on to the next member of the sequence, which is then denoted by a + 1 in addition to its original notation6. It now becomes possible to deduce all the laws of addition based on the formula:

a + (b + 1) = (a + b) + 1 (1)

16 This formula can be used as a definition of addition, because it explains what it means to add , assuming that one already knows what it means to add . Since adding 1 is fully defined, everything is uniquely determined. In this sense, it is possible to say that taken together, the formulas

1 · a = a

(b + 1) (2)

define multiplication (in b · a the number b should be seen as the multiplying factor), and all the laws of multiplication can be deduced [deducieren] based on these formulas.

17 The formulas I have labeled by 1) and 2) were introduced as a foundation for arithmetic by the Graßmann brothers a long time ago7, and founding arithmetic on such simple principles is a great accomplishment of the Graßmann brothers. It is possible, as pointed out by Mr. v. Helmholtz, to move on directly to the negative numbers, by

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pursuing the number sequence backwards. One then gets the bi-directional infinite sequence …5′, 4′, 3′,2′,1′, 0, 1, 2, 3, 4, 5,…

18 Every sign introduced to the left or to the right of the sequence is taken to be distinct from all the others. By holding the condition governing the addition of 1 to be generally true, and by requiring equation 1) to hold successively for b = 0′, 1′, 2′, 3′ one obtains, one after the other, the definitions [Erklärungen] ruling the addition of 0, of 1′, of 2′ and so forth. It then becomes possible to take a as a positive or as a negative number in formulas 2). By positing b = 0′, 1′, 2′, 3′,… one gets conditions, which are sufficient for defining multiplication for negative multiplication-factors. Developing subtraction then raises not the slightest difficulty. One must define as a fully determined number satisfying the equation [Gleichung]

(a - b) + b = a (3)

and, with this, subtraction becomes possible in all cases.

19 Frequently negative numbers are introduced by directly allowing symbols of the form a - b, which are supposed to satisfy equation 3). In this respect, the treatment of Arithmetic discussed here follows this practice. Equivalence relations hold between symbols of the form a - b. These equivalences are fully determined by equations 1) and 3), as soon as we also add that the equation x + b = a always has a unique solution. Equation 3), for instance, implies that (3 - 5) + 5 = 3.

20 By adding 1 we next get ((3 - 5) + 5) + 1 = 4.

21 Then, by applying equation 1) on the left side of the former equation, we obtain (3 - 5) + 6 = 4.

22 This proves that 3 - 5 is the solution of the equation x + 6 = 4, and therefore, one must posit 3 - 5 = 4 - 6

23 It is, however, not entirely obvious that this way of introducing the symbols a - b is justified. It seems conceivable that, taken together, equations 1), 2), and 3) may also lead to equivalences of a quite different kind. It would then no longer be permissible simply to posit that (Graßmann No. 114) “Numbers generated by successively adjoining of 1, are all taken as distinctfrom each other”.

24 As soon as this difficulty is eliminated in the way proposed aboveq or in any other way, it becomes possible to regard equations 1), 2), and 3) as definitory relations [definierende Relationen]8 for the positive numbers, for the negative numbers, and for the operations of addition, subtraction, and multiplication to be carried out on these numbers.

25 The path adopted in the book under scrutiny here is in essence the path just described, which is rooted in the ordinal number concept.r With respect to multiplication, we must, however, emphasize that once the formula (b + 1) · a = b · a + a has been postulated, it is inappropriate to adopt the equation

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a · (b + 1) = a · b + a as an additional postulate. I hope I have managed to show that the former equation suffices for establishing the definition of multiplication, and, for that reason, the latter equation can no longer be introduced by arbitrary stipulation. Such a practice cannot exclude the emergence of contradictions. If the former equation is regarded as the definition of multiplication, it then becomes necessary to regard the latter as a proposition [Lehrsatz]. I have reached the conviction that it can be proved. The difficulties facing introducing fractions [gebrochene Zahlen] without relying on geometric premises can be easily overcome based on an idea generally used by Mr. Stolz [einen Gedanken, der in allgemeiner Weise von Herrn Stolz durchgeführt worden ist]. The theory of integers and its first three operations can thereby be taken as accomplished. The propositions ruling over the relations “greater than” and “smaller than” are

subsequently introduced without difficulty. Expressions of the form are considered next, a and b being positive or negative integers, b being different from 0. Such a symbol has no meaning attached to it yet. It is simply a mere form inside which we only can distinguish two numerical values: a and b. We are obviously able to call the numbers and numerator and denominator. Two such symbols seem, at a first glance, different, if their numerators and denominators are not respectively identical. We

then, however, explicitly stipulate that two such symbols and should be seen as equivalent only if ab′ - a′b = 0 holds. It is now possible to prove that, if two such symbols are equivalent to a third, then they are equivalent to each other. If all symbols equivalent to each other are united to form a single category, it follows that all symbols of one category are equivalent to each other. We have thus created a new concept, for which we use the term “value”. We attribute the same value to all mutually equivalent symbols. Addition and multiplication are next defined directly through the formulas

and the only thing left to prove (and this is an easy exercise) is that the value of the sum depends only on the value, not on the form, of themagnitudes added.

26 Using this approach, propositions such as “if two magnitudes are equal to a third, then they are equal to each other” and “equals can always be substituted for equals” are not tautological [tautologisch]. These propositions were made redundant by the ordinal construction of the whole numbers [ganze Zahlen], because, in that case, distinct objects of the same value were not available.

The earlier whole number is now equivalent with the symbol . The associative and commutative laws for addition and multiplication of fractions can now be easily proved. This also holds for the law connecting the two operations—namely, the distributive law. There is no difficulty in moving on to proving the laws governing subtraction and division. The relations “greater than” and “smaller than” can be easily

defined as well. By definition it is stipulated that , if ab′ > a′b, whereby the numbers a, a′, b, b′ are positive integers. Propositions such as the following then become provable: of two different numbers, one is greater than the other; if a number is greater than a second one, and the latter is greater than a third, then the first is also

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greater than the third; each number can be repeatedly multiplied by another until it becomes greater than some given other number;9 adding greater numbers to equal or greater numbers leads to greater numbers, etc.s

27 What was previously said regarding the introduction of the negative integers and of the fractions in the new book of Mr. Graßmann applies, of course, even more so, with respect to the introduction of the roots and the logarithms in the second chapter of the book. Graßmann however introduces even the irrationals without any justification. He simply provides a definition (No. 82): “Irrational numbers are magnitudes that are not rational numbers [nicht Endzahlen sind].t The laws ruling the comparison of rational numbers also hold for the irrational numbers.” This definition is followed by a proposition (No. 383): “All propositions found in arithmetic, which hold for arbitrary integers and for fractions, also hold for the irrational numbers.” If the existence theorems are taken for granted10, then Mr. Graßmann's treatment is basically consistent.u

I do not wish to discuss Graßmann's introduction of in the third subsection. Nor do I wish to raise the question of whether the straight forward introduction of the concept of “oriented angle” [Winkel der Richteinheit] in No. 435 is consistent with the point of view initially adopted by its author. It seems that, in this case, considerations of angle-measurement were taken into account. I wish, however, to use another example, to show that the author, who is so severe in criticizing others for their “fallacies” [Trugschlüsse], is not himself fully free of them. In No. 451 the formula (cos α + i sin α)(cos β + i sin β) = cos(α + β) + i sin (α + β) is proved as follows: Calculating the left side of the equation leads to cos α cos β - sin α sin β + i(sin α cos β + cos α sin β)

28 This magnitude should also be a fundamental unit, or, as it could be expressed, its module should be 1. It therefore can also be brought to the form

cos γ + i sin γ (4)

in which γ depends on α and β. The author, therefore, posits

γ = α ∘ β (5)

29 and calls this a connection [Verknüpfung] of α and β. After showing that becomes α ∘ β for α = 0 and α for β = 0, the deduction is pursued in this way: “The connection α ∘ β is, therefore, the connection having zero [die Null] as its non-changing magnitude, that is, the connection is the addition.” No. 71 is invoked as a basis for this inference. There, however, one only finds the following: Definition. The non-changing magnitude [nicht ändernde Größe] of addition [Fügung] is called zero [Null]. The sign of zero is 0.v Zero is thus that magnitude which can be connected to any other, without thereby changing the value of the latter magnitude; or The connection having zero as its non-changing magnitude is the Fügung or addition.w

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30 To this I emphasize that according to Mr. Graßmann's terminology a connection [Knüpfung] is the most general term used to describe a combination of magnitudes. In No. 5 we read: A connection of magnitudes is any combination or union of magnitudes that is in the reach of the human mind, in as far as its outcome has just one not multiple values.

31 Basically, this means that, every single valued function of two variables F(α,β), with F(0,α) = F(α,0) = α, must coincide with α + β. I leave the criticism of such a claim to the reader. However, although the author did not indicate this, it is possible that F(α,β) = F(β,α), F(α, F(β,γ)) = F(F(α,β),γ) can be deduced. Even so, one would have yet to prove that F(α,β) = α + β.

32 Tübingen Otto Hölder Göttingische gelehrte Anzeigen, Nr. 15, 1892, 585–595.

BIBLIOGRAPHIE

DYCK, WALTHER 1882 Gruppentheoretische Studien, Mathematische Annalen, 20, 1–44.

GRASSMANN, HERMANN 1860 Lehrbuch der Arithmetik für höhere Lehranstalten, Berlin: Th. Chr. Fr. Enslin, 2nd edition, 1861.

GRASSMANN, ROBERT 1872 Die Formenlehre oder Mathematik, Hildesheim: Georg Olms Verlagsbuchhandlung. 1891 Die Zahlenlehre oder Arithmetik streng wissenschaftlich in strenger Formelentwicklung, Stettin: Verlag R. Graßmann.

HELMHOLTZ, HERMANN VON 1887 Zählen und Messen, erkenntnistheoretisch betrachtet Philosophische Aufsätze. Eduard Zeller gewidmet, Fues's Verlag.

RADU, MIRCEA 2003 A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann, Historia Mathematica, 30, 341–377. 2011 Axiomatics and self-reference—Reflections about Hermann Graßmann's contribution to axiomatics, in Hermann Graßmann—From Past to Future: Graßmann's Work in Context. Graßmann Bicentennial Conference, September 2009, edited by PETSCHE, H.-J., LEWIS, C. A., LIESEN, J., & RUSS, S., Basel: Birkhäuser, 101–116.

SCHRÖDER, ERNST 1873 Lehrbuch der Arithmetik und Algebra für Lehrer und Studirende, vol. 1, Leipzig: Teubner.

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STOLZ, OTTO 1885 Allgemeines und Arithmetik der reellen Zahlen, Vorlesungen über allgemeine Arithmetik, vol. 1, Leipzig: Teubner.

NOTES a. Robert Graßmann opens the preface of his book with the following words: “The theory of forms or mathematics will be presented (...) free of any logical fallacy [Trugschluss]. It will cover all branches [Zweige] of pure mathematics: the theory of numbers or arithmetic [Zahlenlehre], the theory of functions [Funktionenlehre], the calculus of extension [Ausdehnungslehre], and the Erweiterungslehre”, [Graßmann 1891, Preface]. I found no adequate translation for Graßmann’s “Erweiterungslehre”. Robert Graßmann explains that the “Erweiterungslehre” is a development of ideas put forward by Hermann in his “Ausdehnungslehre” under No. 410-527. This description only fits the second edition of Hermann Graßmann’s Calculus of Extension of 1862. There Hermann Graßmann treats fundamental subjects belonging to analysis. Kannenberg translates Hermann Graßmann’s title of the corresponding chapter by “Theory of Functions”. These dificulties need not concern us here, since Hölder’s Review only covers arithmetic. b. Graßmann emphasizes the need to provide two complementary presentations [die Zahlenlehre in zwei Formen darstellen] of arithmetic. He calls the first presentation a “strict development by formulas” [strenge Formelentwicklung]. The second is called “freie Gedankenentwicklung”, a rather uncommon expression, not so easy to translate appropriately. One should perhaps translate this by “a speculative conceptual development” or perhaps a “heuristic treatment” of arithmetic. Graßmann’s distinction, is the consequence of his general concept of method. His main methodological concern was to accomplish an abstract and logically stringent development of arithmetic by formulas, the only possible strictly scientific [wissenschaftliche] presentation of arithmetic (in his opinion). Graßmann claims that the “Formulas” (they are special aggregations of signs) used by him are direct expressions of the thinking operations [Denkoperationen], icons of the mental operations involved in arithmetic. In Graßmann’s view, such a strictly formal treatment of arithmetic is elementary and scientific at the same time, and is therefore also appropriate and, indeed, necessary for teaching. Graßmann’s understanding of his own account of the mathematical inscriptions is reminiscent of Leibniz’s famous ideal of developing a Calculus Ratiocinator, an ideal that Hölder considered impossible to achieve. The second “speculative” or “heuristic” treatment of arithmetic mentioned by Graßmann seems to have a genuine pedagogical function. It is supposed to deal with an aspect of arithmetic that the strict scientific treatment cannot deal with, namely, additional explanations and descriptions of alternative paths of proving the same result [Graßmann 1891, Preface, v ff.]. Hölder, who also emphasizes conceptual clarity, regards Graßmann’s identification of symbolic expression and conceptual content as a blind alley and Graßmann’s obsession with it a methodological evil. Splitting the treatment of arithmetic in a scientific and a separate didactic part only becomes necessary because of Graßmann’s belief that in mathematics abstract sign aggregates are the only direct and therefore the only appropriate expression of thinking. As far as I know, Graßmann never wrote the second book. c. The word “Strenge” occurs twice in the title of Graßmann’s book: “The Theory of Number or Arithmetic in strict scientific presentation by strict use of formulas [Die Zahlenlehre oder Arithmetik streng wissenschaftlich in strenger Formelnentwicklung].” d. Due to the meaning that the expressions “number theory” and “theory of numbers” have today, it would be misleading to use any of them as a translation of Graßmann’s “Zahlenlehre”. In Graßmann’s work, this term refers to the general construction of the number systems beginning with the system of the natural through to that of the complex numbers. Graßmann does not use

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set-theory in his work, so that it would also be misleading to use the word “set” in describing his work. The same difficulties hold for the term “Arithmetik”. To avoid misunterstandings I have translated Graßmann’s term “Zahlenlehre” by “Theory of Number”. e. Graßmann calls the theory presented in his introductory chapter “Theory of Magnitudes [Größenlehre]”. This theory of magnitudes includes, among other things, a general discussion of algebraic operations and structures. It extends over fifty (!) pages. To the modern reader, the “Größenlehre” looks like a general theory of algebraic structures. In it Graßmann harshly criticizes Paul du Bois-Reymond, “who fails to recognize the purely formal nature of mathematics” [Graßmann 1891, 2]. More importantly Graßmann writes “Mr. Paul du Bois- Reymond (...) regards the pure theory of number [Zahlenlehre], which generates its numbers and number-magnitudes by itself, as nothing more than a symbolic game, comparable to chess, not a science” [Graßmann 1891, 3] . This reminds of Hermann Weyl’s ciritque of formal axiomatics. In his Review, Hölder never explicitly mentions Graßmann’s critique of du Bois-Reymond. Much of Hölder’s criticism of Graßmann and indeed many of Hölder’s subsequent writings, however, attach great importance to the distinction between the symbolic statement of mathematical results and methods (their mere form, as it were), on the one hand, and their true conceptual origin, on the other [Radu 2003, 365 ff]. In a sense, Hölder’s Review can be seen as a reinforcement of the position attributed to du Bois-Reymond by Graßmann. f. These formulations are an interesting expression of the difficulties still being encountered in 1892 with describing concepts and results that, from our point of view, belong to the standard description of algebraic structures. They also represent a historic relic insofar as they repeat a terminology originally developed by Hermann Graßmann and published in the 1844 edition of his famous Calculus of Extension. In this book Hermann Graßmann makes a great effort to create a language suitable for a presentation of vector-algebra. Hermann Graßmann’s terminology was developed further in the writings of his brother Robert. g. Graßmann is defining magnitudes, value, and operation. His definitions are not easy to use. Could Graßmann have introduced all these notions as primitive ones, as Hölder seems to be suggesting here? The term “magnitude” might perhaps have been taken as a primitive notion. What about “value” and “operation”? Graßmann’s definition of his operation concept expresses a perfectly legitimate requirement also found in modern definitions of the concept. Today we state this simply by saying that an algebraic operation is a function. Graßmann does not think of operations in these terms, and states his definition in alternative ones. I find Graßmann’s definition of operation adequate. h. As will be seen, Hölder returns several times to this issue. A better understanding of Hölder’s intentions can be reached by taking his subsequent writings into account. It concerns the relationship between equality as a general logical concept and equivalence. Hölder explains that a general concept of equality, defined by means of general axioms is useless in mathematics because, in his view, such a concept would be completely vacuous. Hölder argues that equality makes sense only as equality between well defined equivalence-classes in respect to some explicitly defined equivalence relation. Thus the general axioms of equality only make sense if they are proved as theorems based on equivalence relations defined in advance. i. Here Hölder aknowledges that Graßmann’s “Einleitung” provides a general axiomatic treatment of the theory of operations. j. A “separable connection” is Graßmann’s phrase for an invertible operation. k. I have discussed these and other important claims in greater detail in [Radu 2003]. l. Graßmann defines “Teilgröße” as any magnitude that is preceded by the division sign “:”. The term stands for the inverse element of an element a in respect to an algebraic operation, something now commonly denoted by a-1.

Graßmann writes instead, [Graßmann 1891, 71].

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m. Hölder calls certain equations—such as a + (b + 1) =(a + b) + 1—that are used by Graßmann “basic formulas”, “formulas”, and also “equations”. This will become clearer in the sequel. n. In Hölder’s view, such an interpretation of the basic formulas used by Graßmann would make it possible to regard the formulas as constructive procedures and not as descriptive axioms. Hölder is right, when emphasizing that Graßmann does not discuss this distinction. It is indeed not easy to establish whether Graßmann is clearly aware of this distinction and of its significance. Reading Robert Graßmann’s writings, one gets the impression that the genetic and the axiomatic standpoint are conflated. o. Hölder has positive natural numbers in mind. p. It is independent of the order in which the elements of the two sets are associated to each other. q. By defining natural number taking the cardinal-number concept as a starting point. r. Here Hölder seems to be oscillating between his sharp criticism of Graßmann’s approach as something fundamentally based on, as it were, abusive use of the permanence principle and an alternative interpretation according to which “in essence” Graßmann’s approach is simply a genetic approach similar to that advocated by Hölder himself. This uncertainty is rooted in the way in which Graßmann’s book reviewed by Hölder is written. The two tendencies seem to be competing with each other. This also holds for all the other books written by the Graßmann brothers, and it is responsible for the ongoing debate concerning the place of axiomatics in the writings of the Graßmann brothers. Compare [Radu 2003]; [Radu 2011]. s. The way in which Hölder expresses these well-known laws is a bit surprising. His formulations are rather cumbersome. Indeed they sound very much like the formulations of these laws found in Euclid’s books. It would have been a great deal easier to express these laws symbolically. Contrary to Graßmann, Hölder, however, is not happy with symbolically expressed laws. In his subsequent methodological writings, he constantly emphasizes the “conceptual” nature of these laws and the fact that symbolic abbreviations are nothing more than a more economic expression of laws originally stated in conceptual terms. (Compare [Radu 2003, 345–359]). t. Graßmann often uses terms no longer common today. Rational numbers are called “Rationalzahlen” and also “Endzahlen”. Irrational numbers are called “Irrationalzahlen” and also “Unzahlen”. Graßmann defines irrational numbers as numbers that are not rational. Rational numbers are defined in the usual way, [Graßmann 1891, 182]. u. Hölder’s comments on Graßmann’s treatment are surprisingly mild. Graßmann’s “definition” 382 is anything but harmless. Graßmann does not just define irrational number, but also explains that certain comparison laws hold for the newly introduced numbers as well. These laws are those already quoted by Hölder in the present Review in his discussion of the comparison of rational numbers. Such a claim, however, should be a theorem rather than an axiom. It certainly cannot be a definition. Also, Graßmann’s proposition 383 reproduced by Hölder is adopted without any proof by Graßmann. v. An alternative translation of Graßmann’s term “nicht ändernde Größe” would be non-value- changing magnitude. w. Obviously, Graßmann’s term “non-value-changing magnitude [nicht ändernde Größe]” stands for the modern term “neutral element” with respect to an internal algebraic operation.

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NOTES DE FIN

1. One should compare pages 2 and 3 of the introduction, where reference to measuring lengths is explicitly rejected. 2. v. Helmholtz, Zählen und Messen, Philosophische Aufsätze, Eduard Zeller zu seinem fünfzigsten Doctorjubiläum gewidmet, Leipzig, 1887, [Helmholtz 1887, 19]. 3. Lehrbuch der Arithmetik und Algebra, Leipzig, 1873 [Schröder 1873]. 4. Compare O. Stolz, Vorlesungen über allgemeine Arithmetik, Leipzig 1885, [Stolz 1885, 9–10]. 5. Compare the already mentioned work of Mr. v. Helmholtz and the differing presentation given by Kronecker in the same collection of papers. 6. In essence, this view had already been advocated by Leibniz. He used to define a + 2 through (a + 1) and could then provide his famous proof of the formula 2 + 2 = 4. 7. Hermann Graßmann, Arithmetik, Stettin 1860, Berlin 1861 [Graßmann 1860], Robert Graßmann, Die Formenlehre oder Mathematik, Stettin 1872 [Graßmann 1872]. 8. A similar view has been developed in the group-theory of Mr. Dyck: Mathematische Annalen, Bd. 20 [Dyck 1882]. In this case, one would, however, have to add the relation expressed by the associative law, which is generally valid, to the definitory relations explicitly stated in group- theory (and which only contain special symbols) in order to preserve the analogy to the text discussed here. I do not wish to call these generating relations axioms. It seems to me that here the situation is different from that in geometry. The primitive concepts [Grundbegriffe] and first principles [Grundsätze] of geometry do not have their origin in sequential processes [fortschreitenden Parcessen (sic!)], of the kind used in geometric and arithmetic deductions alike. One may regard the first principles of geometry as rooted in intuitive evidence, or, perhaps more appropriately, as principles abstracted from external experience. In both cases, the first principles and the primitive concepts of geometry come from a foreign realm [a realm foreign to pure reason – M. R.]. If one wishes to regard the above mentioned formulas as axioms, one would then have to introduce similar axioms for innumerable concepts of arithmetic and analysis, and the number of arithmetical axioms would become infinite. 9. This proposition may look fully obvious here. I emphasize it because it plays a great role when we treat limit-processes. In order to justify using arithmetical propositions in geometry, it is necessary to introduce in geometry results analogous to some of the fundamental arithmetical theorems as geometric axioms. This holds for the proposition mentioned here. The importance of this proposition was strongly emphasized by Mr. Stolz. 10. I think that the foundations of the theory of irrational numbers has been completed thanks to the works of Weierstraß, Dedekind, Lipschitz, and Cantor. It is possible to define such numbers on a purely arithmetical basis, introduce the concepts equal and greater than, define the operations and prove all propositions. The introduction of an irrational magnitude requires, of course, a specific law. I therefore wish to express my doubt with respect to the legitimacy of bringing all rational and all irrational numbers under a single general concept. I am not so sure that it is possible to construct the continuum, on which certain chapters of the theory of functions rely, in a purely arithmetical manner.

AUTEURS

OTTO HÖLDER (1859-1937)

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Between Kantianism and Empiricism: Otto Hölder’s Philosophy of Geometry

Francesca Biagioli

Introduction

1 In 1899 Otto Hölder succeeded Sophus Lie in the chair of mathematics at the University of Leipzig. In the extended version of his inaugural lecture, Anschauung und Denken in der Geometrie, which was printed in 1900, Hölder first presents the proof that the so- called Archimedean axiom can be derived from Dedekind’s continuity. This proof is the starting point of the axiomatic theory of quantity he develops in 1901 and of the theory of proportions he uses in 1908 to construct a numerical scale on a projective straight line. These developments are to be considered in connection with Hölder’s philosophy of geometry. In fact, in Anschauung und Denken, Hölder takes part in a debate about the foundations of geometry which had begun in the second half of the 19th century, after non-Euclidean geometry was rediscovered.

2 Already during the 1820s, János Bolyai and Nikolaj Lobačevskij, independently of each other, developed a new geometry that is based upon the denial of Euclid’s fifth postulate. Such development had been anticipated by mathematicians such as Gerolamo Saccheri, Johann Heinrich Lambert, and Adrien-Marie Legendre, who sought to prove Euclid’s fifth postulate by denying it and finding a contradiction. Since these and many other attempts to prove Euclid’s fifth postulate failed, the theory of parallel lines had lost credibility at the time Bolyai and Lobačevskij wrote. For this reason, their works remained largely unknown at that time and models of non-Euclidean geometry were first developed many years later, namely Eugenio Beltrami’s model in 1868, Felix Klein’s model in 1871, and Henri Poincaré’s model in 1882.

3 One of the first to recognize the importance of non-Euclidean geometry was Carl Friedrich Gauss. However, his appreciation of the works of Bolyai and Lobačevskij is to be found only in his private correspondence, which was published posthumously in the

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second half of the 19th century. At the same time, Gauss called into question Kant’s a priori geometry and developed the conviction that the geometry of space should be determined a posteriori. His opinion was that the necessity of Euclid’s geometry cannot be proved. For this reason, in a letter to Olbers dated 28 April 1817, Gauss claimed that “for now geometry must stand, not with arithmetic which is pure a priori, but with mechanics” (Gauß 1900, 177] Eng. trans. in [Gray 2006, 63]). In a letter to Bessel dated 9 April 1830, Gauss wrote: According to my most sincere conviction the theory of space has an entirely different place in knowledge from that occupied by pure mathematics. There is lacking throughout our knowledge of it the complete persuasion of necessity (also of absolute truth) which is common to the latter; we must add in humility that if number is exclusively the product of our mind, space has a reality outside our mind and we cannot completely prescribe its laws. [Gauß 1900, 201]; Eng. trans. in [Kline 1980, 87]

4 It is tempting to relate Gauss’ opinion to his later claim that, since 1792, he had been developing the conviction that a non-Euclidean geometry would be consistent (see Gauss’ letter to Schumacher dated 28 November 1846 in [Gauß 1900, 238]). However, Gauss’ knowledge about non-Euclidean geometry before his reading of the works of Bolyai and Lobačevskij is hard to reconstruct, and the question as to whether his views about space and geometry presuppose the assumption of non-Euclidean geometry is controversial. The problem is that Gauss could hardly have possessed the concept of a non-Euclidean, three-dimensional space. Nevertheless, his empiricist insights were seminal in 19th-century philosophy of geometry and, at the time Hölder wrote, it was quite natural to associate Gauss’ claims with a view which follows from Bernhard Riemann’s classification of hypotheses underlying geometry [Riemann 1854] and which can be summarised as follows: since different geometries are logically possible, none of them can be deemed necessary or given a priori in our conception of space.

5 A related question is whether geometrical propositions can be empirically tested. Lobačevskij sought to detect whether the sum of the angles in a triangle is equal to or less than 180 degrees by means of measurements, and Sartorius von Waltershausen reports that Gauss also made such an attempt during his geodetic work [Sartorius von Waltershausen 1856, 81].1

6 A more refined kind of empiricism was developed by the German physiologist and physicist Hermann von Helmholtz in a series of epistemological writings. The most important of them were reedited by Paul Hertz and Moritz Schlick in 1921 [Helmholtz 1921]. They are: Über die Tatsachen, die der Geometrie zugrunde liegen [Helmholtz 1868], Über den Ursprung und die Bedeutung der geometrischen Axiome [Helmholtz 1870], Die Tatsachen in der Wahrnehmung [Helmholtz 1878], Zählen und Messen, erkenntnistheoretisch betrachtet [Helmholtz 1887]. Helmholtz maintains that geometric knowledge rests upon general facts to be induced by experience, especially our experiences with solid bodies. Such experiences are required in order to determine the most basic kind of relationship between spatial magnitudes, namely, their congruence. If two such magnitudes are to be proved to be congruent, they must be brought to coincidence. The condition for observing this particular outcome is the well-known experience that solid bodies can be displaced without changes in shape and size. In mathematical terms, if spatial magnitudes are to be measured, their points must remain fixedly-linked during displacement. Helmholtz deems those bodies whose elements satisfy this requirement rigid, and maintains that the geometric properties we attribute to physical space

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depend on facts such as the free mobility of rigid bodies. Since physical bodies are only approximately rigid, and free mobility is presupposed as a precise, mathematical condition of measurement, it is clear that geometrical propositions cannot be directly tested. Moreover, in 1869, after his correspondence with Eugenio Beltrami, Helmholtz realizes that a non-Euclidean metric might satisfy the same conditions as Euclid’s metric. He concludes that the geometry of our space is only approximately Euclidean or not notably different from Euclidean geometry. But he does not exclude that under different circumstances Bolyai-Lobačevskij’s geometry might also be adopted.

7 In his 1870 paper Helmholtz discusses the consequences of his analysis of geometric knowledge for the Kantian philosophy of geometry as follows. He maintains that the principles of geometry cannot be a priori synthetic judgements in Kant’s sense. Unlike a priori judgements, they are not necessarily valid, because their validity with regard to the empirical manifold is only hypothetical. Therefore Helmholtz’s principles are supposed to be derived from experience. As an alternative, a strict Kantian might consider the concept of fixed geometric structure a transcendental one, in which case, however, the axioms of geometry would follow analytically from the same concept, because only structures satisfying those axioms could be acknowledged as fixed ones [Helmholtz 1921, 24, Eng. trans. 25].

8 Poincaré’s well-known objection is that Helmholtz’s definition of congruence between spatial magnitudes already presupposes geometry. Helmholtz believes that his definition follows from the fact that two such magnitudes can be brought into congruent coincidence. Apparently, the observation of this particular outcome presupposes some kind of displacements. Poincaré’s question is: how are such displacements to be defined? If they are supposed to leave the magnitudes to be compared unvaried in shape and size, the concept of rigid figure is presupposed from the outset and geometrical empiricism is unable to escape circularity [Poincaré 1902, 60]. So Poincaré rejects Helmholtz’s attempt to show how geometric notions can be derived from some facts: the foundations of geometry are to be better understood as general rules. On the other hand, he agrees with Helmholtz that geometrical axioms cannot be a priori synthetic judgements. Poincaré’s view is that rules such as the free mobility of rigid bodies follow from the definition of such bodies and must be stipulated. Therefore he maintains that one geometry cannot be more true than another; it can only be more convenient [Poincaré 1902, 91].

9 Poincaré develops his argument in a series of papers which he collects in 1902 in his La Science et l’Hypothèse [Poincaré 1902]. Hölder does not seem to be acquainted with Poincaré’s earlier papers at the time of Anschauung und Denken. Nevertheless, he takes into account a similar objection to Helmholtz, and yet develops a different conception of geometry. Poincaré’s argument for the conventionality of geometry presupposes his rejection of both Kantianism and empiricism. I do not think that he rules out all solutions other than conventionalism. Hölder proceeds in a different way. He does not emphasize the opposition between Kantianism and empiricism, because he seeks to profit from Kantian objections in order to develop a consistent empiricism. His argument for empiricism depends on his analysis of mathematical method rather than on Helmholtz’s physiological conception of spatial intuition. Hölder’s analysis is strictly connected with his approach to the theory of measurement. Moreover, he develops his methodological considerations in his later work Die mathematische Methode: Logisch erkenntnistheoretische Untersuchungen im Gebiete der Mathematik [Holder 1924]. There he

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makes explicit that he is also influenced by the Kantian view that mathematics is synthetic.

10 In the following, I will present Hölder’s emendation of Helmholtz’s empiricism. Moreover, I will analyze Hölder’s methodological considerations and discuss his relationship with Kant. In the last section of my paper, I will suggest a comparison with the Marburg School of neo-Kantianism, especially Ernst Cassirer. Both Hölder and Cassirer develop a synthetic conception of geometry which is compatible with the use of deductive method, and which enables them to take into account non-Euclidean geometry.

1 Helmholtz’s empiricism revised

11 Geometrical empiricism (broadly construed) was popular among German mathematicians in the second half of the 19th century. In particular, Hölder was influenced by Helmholtz and by Moritz Pasch. I will focus on Helmholtz, because his epistemological writings were widely discussed not only among mathematicians but also among philosophers and his assessment of Kant’s philosophy of geometry was seminal for the debate Hölder refers to.

12 Hölder summarizes the different views on spatial intuition and geometry as follows. Kant’s view is that a pure intuition endowed with subjective laws independently of experience makes experience first possible. By contrast, 19th-century thinkers, such as Julius Baumann and Wilhelm Wundt, maintain that spatial intuition can be induced by experience. Both standpoints look at intuition as a source of geometric knowledge. According to Hölder, Helmholtz avoids any commitment to the supposedly intuitive character of geometrical axioms and explains basic geometric concepts “in a more physical way” [Holder 1900, 3, Eng. trans. 16]. At the same time, this explanation requires that our perceptions show some regularity, so that experiences form a general system in Kant’s sense. Hölder writes: Of course, the advocates of this empirical viewpoint will not deny that any elaboration of experience comes out from assumptions, at least from the assumption of a certain conformity to laws of the investigated object, which we could not otherwise grasp conceptually (compare in particular [Helmholtz 1921 , 148ff.]). Indeed, any single fact of experience, if expressed by means of concepts— and how could one want to express it otherwise—is the result of a mental elaboration of experience. But unlike Kant’s conception, empiricism emphasizes the fact that according to the empirical viewpoint no law referring to external objects comes about independently from external experience. Kant says on the contrary that geometrical knowledge comes from intuition and that intuition is independent from experience.[Holder 1900, 4, note 9, Eng. trans. 28]

13 Hölder emphasizes that the recognition of some facts presupposes both observation and reasoning. He therefore rejects the view that empirical laws can be generalized by induction. He writes: On the contrary, it seems to me that the intellectual activity that from the beginning goes hand in hand with experience, when we build concepts of experience, is a preliminary activity that should not be assigned neither to deduction nor to induction. [Holder 1900, 6, note 16, Eng. trans. 31]

14 Hölder’s account enables him to defend Helmholtz’s geometrical empiricism from the charge of circularity. Hölder presents such objection as a Kantian one and summarizes

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it as follows. Since geometrical axioms are both exact and general rules, the role of single experiences in their development is only occasional. Their source, according to Kant, is a kind of intuition in which there is nothing that belongs to sensation. At the same time, Kant’s pure intuition is a condition of experience. From this point of view, any attempt to infer geometrical propositions from experience would entail a vicious circle. For instance, Hölder mentions Helmholtz’s definition of congruence. This definition had already been called into question by Poincaré. Hölder does not seem to be acquainted with Poincaré’s remark. Nevertheless, a similar objection had been formulated also by Konrad Zindler in a paper entitled Beiträge zur Theorie der mathematischen Erkenntnis, from which Hölder quotes. Zindler maintains that Helmholtz’s attempt to define the congruence of spatial magnitudes physically (i.e., by comparison of rigid bodies) already presupposes the geometric concept of congruence [Zindler 1889, 11].

15 Hölder’s reply is that a consistent description of the empirical origin of geometric concepts is possible. Such description should be based on observations which do not presuppose geometry. For instance, approximately rigid bodies may be distinguished from non-rigid ones, because the former can be easily brought back to their initial position after displacement [Holder 1900, 5ff., Eng. trans. 17], [Holder 1924, 371].2 Helmholtz’s free mobility can be derived from the fact that parts of two such bodies can be brought to coincidence and that such experience can be repeated at any time. Hölder’s point is that, since facts are basically singular, Helmholtz’s facts underlying geometry are to be better understood as rules for inferring facts in any further cases [Holder 1900, 4, note 12, Eng. trans. 29].3

16 The core of Hölder’s geometrical empiricism is the claim that geometric concepts do not differ from empirical concepts such as those of mechanics and optics. All of these concepts are supposed to be related to the empirical manifold in a twofold way: they can be said to be derived from experience, and yet, once they have been defined, they make our understanding of experiences possible. Therefore general laws such as geometrical axioms have to be postulated. More precisely, our requirement is that such laws are valid with regard to the empirical manifold. On the one hand, geometrical axioms are exact statements and enable deductive inferences; on the other, their validity with regard to experience is but hypothetical. I think that Hölder recognizes a relativized a priori in the following consideration about the concept of space: It will no longer appear contradictory that, though we use this concept in some cases in order to interpret experience, we nevertheless consider it possible to check this concept—whose adequateness is hypothetical—for correspondence with experience in order to reshape the concept, if necessary, as we do with physical concepts. [Holder 1900, 21, note 64; Eng. trans. 46]

17 In the following, Hölder points out that Kant also speaks of a priori with regard to principles of mechanics such as the conservation of matter and distinguishes between a pure part of natural science and an empirical one [Holder 1900, 21, note 67, Eng. trans. 47]. It seems that space and geometry may be deemed a priori in the same sense.4

18 In Die mathematische Methode Hölder analyzes Archimedes’ proof of the Law of the Lever to show that the underlying suppositions play the role of axioms. Hölder admits that it may not be possible to develop the whole of mechanics by deductive means [Holder 1924, 50]. His argument is that geometry and mechanics are not different in

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principle, and this argument depends on his view that geometry and physics form a single system of concepts.

19 Hölder’s view is made explicit in his discussion of Lobačevskij’s experiment. Lobačevskij’s conjecture is that the sum of the angles in a triangle might be proved to be less than 180 degrees, provided that the chosen triangle is sufficiently large. Therefore he needs astronomical measurements. At the time Hölder wrote, such procedure had already been criticized by Hermann Lotze and Poincaré. Hölder summarizes earlier criticisms as follows. In order to measure an angle directly, one should be placed at its vertex. Since the angles of an astronomical triangle cannot be measured simultaneously, it is clear that the result of measurement largely rests upon the adopted theories, including both optics and mechanics. Thus, if the sum of the angles is less than 180 degrees, we are not compelled to adopt Bolyai-Lobačevskij’s geometry instead of Euclid’s geometry. We may either suppose that light does not travel rigorously in a straight line or change the laws of motion, see [Poincaré 1902, 83ff.]. Hölder’s remark is that our entire system of concepts is involved: changes are not arbitrary, but for instance, in order to change the optical part of our system, we should find out that light travels according to the laws of the chosen geometry. Hölder emphasizes that not every geometry can be used to describe the same observations [Holder 1924, 400]. So the view that geometry can be empirically tested is to be rejected. Nevertheless, the criticism of Lobačevskij’s experiment does not provide a counterargument against Hölder’s claim that geometry can be put to the test indirectly, namely, along with the remaining parts of our system of concepts.

20 To sum up, Hölder maintains that geometry does not differ from mechanics: they both need axioms, which are exact and general propositions, and yet are only hypothetically valid with regard to the empirical manifold. On the other hand, he draws a fundamental distinction between geometry and arithmetic, because arithmetic does not need axioms in the above mentioned sense. As we shall see, Hölder’s motivation for his classification of the sciences is to be found in his analysis of mathematical method. But first of all, his considerations in this matter entail a rejection of intuition both as a source of truth and as an indispensable tool for proof.

2 Deductive method in geometry and Hölder’s approach to the theory of proportions

21 The claim that arithmetic has no axioms goes back to Kant. Hölder quotes from the following passage of the Kritik der reinen Vernunft: On this successive synthesis of the productive imagination, in the generation of shapes, is grounded the mathematics of extension (geometry) with its axioms, which express the conditions of sensible intuition a priori, under which alone the schema of a pure concept of outer appearance can come about; e.g., between two points only one straight line is possible; two straight lines do not enclose a space, etc. These are the axioms that properly concern only magnitudes (quanta) as such. But concerning magnitude (quantitas), i.e., the answer to the question “How big something is?”, although various of these propositions are synthetic and immediately certain (indemonstrabilia), there are nevertheless no axioms in the proper sense. [Kant 1787, 204ff.; Eng. trans. 288]

22 Since geometry and arithmetic deal with different objects, namely, with spatial magnitudes and the magnitude of a quantity respectively, they also differ with regard

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to their methods. Kant’s claim is that geometric objects are to be constructed, whereas the answer to the question: “How big something is?” is to be found by means of calculations. Geometrical axioms are finite in number and express those conditions of outer intuition a priori which are required for spatial magnitudes to be constructed. Kant maintains that arithmetic has no axioms in the proper sense, because it rests upon analytic judgements, not synthetic ones. According to him, arithmetic also entails a priori synthetic judgements. However, he calls these propositions numerical formulas and distinguishes them from axioms, because they are infinite in number and there is only one way to accomplish the corresponding calculation. By contrast, one and the same geometric construction can be realized in infinitely many ways in principle—for example by varying the length of the lines and the size of the angles in a given figure whilst leaving some given proportions unvaried.

23 Note that in this passage Kant makes it clear that the pure intuition of space as analyzed in the Transcendental Aesthetic does not entail geometrical axioms. These are now said to be grounded in the synthesis of the productive imagination, which is distinguished from the empirical, receptive one, because of its spontaneity. Therefore Kant sometimes (e.g., at B 155) identifies the productive imagination with the understanding. More precisely, the corresponding synthesis is required for the concepts of the understanding to be applied to the manifold of pure intuition. So Kant is not committed to the claim that our spatial intuition alone entails Euclid’s axioms, even though he might have thought that these are the only possible ones.

24 Hölder, as most of his contemporaries, overlooks this aspect. Nevertheless, he is right to point out that Kant’s characterisation of geometry apparently presupposes Euclid’s method and the use of intuition as an indispensable tool for proof. By contrast, Hölder argues that intuition, although useful in praxis, can be replaced by a formal use of concepts. All proofs in geometry presuppose objects such as points, lines, etc. and their relationships to one another, and consist in an inference from the relations between the given objects to the relations between new objects. Intuitively, the proof may require that auxiliary objects be constructed and the inference is made by analogy: singular figures are supposed to exemplify all similar cases and such supposition depends on a case by case basis. By contrast, the consequences of a deductive inference follow necessarily from general premises to be postulated, namely, the axioms. Hölder’s claim is that intuitive suppositions and concrete constructions in geometry are then replaced by geometrical axioms. He mentions, for example, the proof that the sum of the angles in a triangle is equal to 180 degrees. Given the sides of a triangle ABC and its angles α,β and γ, the auxiliary object to be constructed is the parallel line to the base of the triangle from the opposite vertex. At the same time, two new angles γ′ and α′ are constructed, which obviously form two right angles, if summed with the angle β . So α should be identified with α′, and γ with γ′. Hölder points out that the construction of the parallel line presupposes the existence of such a line through any given point, namely, Euclid’s fifth postulate. The conclusion then follows from the assumption that alternate angles formed by a transversal of two parallel lines are equal, which is Euclid’s proposition I.31.

25 The formulation of geometrical axioms enables rigorous proofs and makes the lack of precision of intuition inessential. Moreover Hölder maintains that geometric proofs can be developed deductively also in the case that the underlying assumptions contradict our intuitions. For example, Lobačevskij’s assumptions entail that the sum of the angles

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in a triangle cannot be more than 180 degrees, and that such sum is less than 180 degrees in all triangles, if it is so in any given triangle.

26 Hölder gives further examples in Die mathematische Methode [Holder 1924, 18ff.]. They show that mathematical reasoning consists of a “concatenation of relations” (Verkettung der Relationen) [Holder 1924, 24]. In fact, already in Anschauung und Denken, [Holder 1900] Hölder points out that the axiomatic method does not deal with objects, but rather with their relationships to one another. Some objects must be presupposed as given in intuition or experience, and yet they can be studied independently of their origin. By contrast, Hölder maintains that arithmetic objects exist independently of experience, because they depend on the indefinite repetition of an operation of our thought. Therefore arithmetic does not need axioms, but rather recurrent definitions. For example, the formula ab = (a-1)b + b provides a definition of multiplication, provided that the concept of sum is already known. This definition is recurrent, because the concept at stake is the generalized rule that governs a series of prepositions like: one times b is b, two times b is one times b plus b, three times b is two

times b plus b, etc. [Holder 1924, 19, note 58]. Similarly, Grassmann’s formula a + (b + 1) =(a + b) + 1 provides a definition of sum and means that the sum of a number a and the successor b + 1 of a number b is the successor of the number a + b in the series of integernumbers [Holder 1901, 1ff.].

27 A similar classification of the sciences is to be found in Hermann Grassmann’s introduction to his Ausdehnungslehre [Graßmann 1844]. Grassmann distinguishes between real and formal sciences: the former ones reproduce something real and their truth depends on the correspondence between thought and its objects; the objects of the latter ones are produced by the acts of our thought and truth depends on the coherence of such acts with one another. These kinds of sciences differ at the very beginning: formal sciences begin with definitions whilst real sciences presuppose some principles. In this regard, Grassmann mentions the distinction between arithmetic and geometry [Graßmann, XIX, and note].

28 Hölder agrees with Grassmann that the objects of geometry are given in intuition or experience, whereas the objects of arithmetic must be constructed. On the other hand, Hölder maintains that given objects can also be constructed. His example is the construction of the concept of measurable magnitude (Maß). He therefore presupposes the axioms of order, those of equality, and an equivalent formulation for Dedekind’s continuity: If all quantities Q are divided into two classes such that each quantity belongs to one and only one class, and each member of the first class is less than each member of the second class, then there is a member ξ of Q such that each ξ′ < ξ belongs to the first class, and each ξ″ < ξ belongs to the second class. Dedekind calls cuts all partitions of rational numbers in two such classes. Each rational number corresponds to one and only one cut and irrational numbers can be defined by requiring that each cut corresponds to one and only onenumber [Dedekind 1872, 13].

29 Hölder shows how to use the Dedekind’s continuity to deduce the divisibility of a given segment into equal parts and to prove a theorem which is equivalent to the Archimedean axiom: Given two segments a and b, and a < b, there is an integer number n such that na > b (see [Holder 1900, 17, note 49, Eng. trans. 40]).

30 This proof is supposed to provide us with a construction of the concept of magnitude, because Hölder considers divisibility and being Archimedean as conditions of measurement. This way of proceeding enables him to improve Euclid’s theory of

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proportions by making the requirement that the laws of addition hold for segments explicit. Once the Archimedean axiom has been deduced from the Dedekind’s continuity and the theory of numbers is presupposed, Euclid’s definition of proportions can be extended to non-commensurable segments and reformulated as follows: The ratio of a to b equals the ratio of a′ to b′, if any multiple μa of a is equal to, less than or more than any multiple νb of b, when μa′ is equal to, less than or more than νb′. The concept of measurable magnitude is not presupposed from the outset, because μ and ν are not considered as the magnitudes of a given quantity (Maßzahlen), but rather as positive integers whatsoever [Holder 1900, 17, Eng. trans. 24, and note 52].

31 Hölder’s theory of proportions is developed in detail in the first part of his 1901 paper Die Axiome der Quantität und die Lehre vom Mass. He formulates a minimal set of axioms of quantity, including the axiom of continuity, by means of which he proves divisibility and the Archimedean property. He then presents the theory of irrational numbers and proceeds as follows. Suppose that a and b are quantities of the same kind and that μ∕ν is

called a lower fraction in relation to their ratio a : b, if νa > μb; and an upper fraction, if νa ≤ μb. By the Archimedean property, there exist both a positive integer ν, such that νa > 1b, and a positive integer μ, such that 1a < μb . Thus, in relation to the ratio a:b , there exist both lower fractions and upper ones, and in every case the lower fractions are less than the upper ones. It follows from the axiom of continuity that: For every ratio of quantities a:b, i.e., for each pair of members of Q which are given in a determined order, there is one and only one cut, i.e., a determined number in the general sense of the word [Holder 1901, §10].

32 This theorem justifies Newton’s requirement that the ratio of quantities of the same kind be expressed by positive real numbers. Since order and operations with cuts can be developed arithmetically, the laws of addition apply to all objects that satisfy the axiom of continuity and the remaining axioms of quantity.

33 In the second part of the paper, Hölder presents a model of a non-Archimedean continuum. This is the most original part of the paper. So it is surprising that, in his introductory remarks, Hölder emphasizes above all the importance of the said theorem for the analysis of measurement. Nevertheless his remarks are interesting for our present topic, because they reflect his philosophical assumptions. Since Hölder develops a formal theory of quantity, nothing prevents us from considering his results independently of such assumptions. Nevertheless he vindicates a non-formalistic approach and maintains that the general concepts of arithmetic, such as the concept of number and that of sum, are developed by “purely logical” means and cannot be reduced to a symbolic calculus [Holder 1901, 2, and note].

34 This claim is made clearer in Die mathematische Methode. Hölder maintains that those thoughts which may lead to the introduction of new symbols cannot be represented by another symbolic computation. Otherwise there would be an infinite regress. Hölder’s view is that symbols represent concepts, and these can be constructed. He deems concepts whose development does not require assumptions other than the operations of our thought, such as the concept of coordination, series, number, and group, purely synthetic, and distinguishes them from the concepts of geometry and mechanics, which are deemed hypothetico-synthetic or simply synthetic and require special assumptions [Holder 1924, 5ff., 295].

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35 Hölder’s approach to the theory of proportions shows the attempt to replace given magnitudes with arithmetical reasoning. Similarly, he maintains that projective metric must be complemented by the theory of irrational numbers [Holder 1901, 19, note].

36 Already in 1847 Christian von Staudt had excluded metric concepts from the foundations of projective geometry by defining the harmonic relationship among four points by means of incidence relationships alone. This way of proceeding enables him to develop a projective metric independently of the axioms of congruence. In order to give projective coordinates, Staudt presupposes the fundamental theorem of projective geometry and the properties of involutions, and develops his calculus of jets, see [Nabonnand 2008, 202ff.]. As an alternative, a projective scale on a straight line can be constructed by determining the fourth harmonic to three given points. The resulting points can be coordinated with the series of rational numbers. As regards irrational points, their introduction requires that each point corresponds to one and only one number, which can be either rational or irrational. One such construction is sketched by Klein in the second part of his second paper Über die sogenannte Nicht-Euklidische Geometrie [Klein 1873, §7], and developed by him in detail in his lectures on non- Euclidean geometry [Klein 1893, 315ff.]. Hölder himself deals with that subject-matter in an essay entitled Die Zahlenskala auf der projektiven Geraden und die independente Geometrie dieser Geraden [Holder 1908]. In the introduction, he writes: The fundamental feature of inquiries such as those presented here lies in the development of proofs, which is to some extent purely logical: the entire, intuitive content of the theory must be put in the postulates. [Holder 1908, 168]

37 A few years later, in 1911, Hölder also presents a calculus of segments which proceeds as the calculus Hilbert developed in the fifth chapter of his Grundlagen der Geometrie [Hilbert 1899]. In order to define operations with segments and give coordinates, Hilbert uses not so much the fundamental theorem of projective geometry, as, more simply, the theorem of Desargues. He presupposes all axioms of linear and plane geometry (connection, order, and the axiom of parallel lines), except the axioms of congruence and the Archimedean axiom [Hilbert 1899, 55ff.]. According to Hölder, this approach also enables the development of projective geometry in the manner of Staudt, that is, without metric foundations [Holder 1911, 67, and note].

38 Hölder’s approach sheds some light on his connection with Helmholtz. In the paper Hölder refers to, Zählen und Messen, erkenntnistheoretisch betrachtet [Helmholtz 1887], Helmholtz formulates additive principles for physical magnitudes. He does not presuppose Euclidean space, but rather the laws of arithmetic, on the one hand, and conditions such as the free mobility of rigid bodies, on the other. Despite the fact that Helmholtz considers the laws of arithmetic as axioms, Hölder agrees with him that arithmetic is to be connected with the theory of quantity ([Holder 1901, 1, note]; on Hölder’s connection with Helmholtz, see also [Michell 1993, 196]). Nevertheless he does not agree with Helmholtz’s claim that numbers are but arbitrarily chosen symbols, on the one side, and that arithmetic is a method constructed upon psychological facts, on the other [Helmholtz 1921, 72]. Helmholtz believes that the series of natural numbers depends on our capacity to retain in our memory the sequence in which the acts of consciousness occurred in time. The time sequence is supposed to provide us with a natural basis for numerical symbolism, because it requires us to adopt a system of symbols allowing neither interruptions nor repetitions, as in the decimal system. Hölder’s philosophy of arithmetic differs from Helmholtz’s one, because it entails not

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so much a psychological approach to the theory of numbers, as a logical one. Since Hölder also rejects formalism, his conception of logic shows a philosophical aspect which has much in common with Kant’stranscendental logic, see [Radu 2003, 343].

3 Hölder’s connection with Kant: From Anschauung und Denken to Mathematische Methode

39 Since Hölder in Anschauung und Denken rejects both the assumption of geometric intuition and the apriority of Euclidean geometry, it might seem that he rejects Kant’s standpoint altogether. Moreover, even if Hölder acknowledges that synthetic reasoning plays a role in the development of some concepts, he maintains that Kant’s much discussed classification of judgements has become pointless [Holder 1924, 8, 292, note]. Nevertheless, after introducing his distinction between hypothetico-synthetic and pure synthetic concepts in Die mathematische Methode, Hölder vindicates a Kantian conception of mathematics. Of course Hölder’s view that geometrical axioms are only hypothetically valid for the empirical manifold contradicts Kant’s claim that the axioms of geometry, including Euclid’s fifth postulate, are necessary conditions of knowledge. On the other hand, Hölder rejects Helmholtz’s more radical view that the laws of our thought depend on our habits and claims that mathematics is an a priori science in Plato’s and Kant’s sense. So Hölder admits a priori knowledge. What he calls into question is the border between a priori knowledge and experience as drawn by Kant. As Hölder puts it, the borderline must be drawn somewhere else [Holder 1924, 7].

40 A priori knowledge in Hölder’s sense is restricted to progression and continuity [Holder 1924, 391]. Let us begin with Hölder’s reflections on continuity. He maintains that this concept cannot be generated arithmetically. Dedekind’s continuity entails that the totality of cuts corresponds to all the points of a straight line or linear continuity. By definition each cut requires a particular rule. Hölder’s objection is that the totality of such rules might be contradictory [Holder 1924, 193ff., 325, and note]. In order to prevent this problem, Hölder considers Dedekind’s continuity as a definition in the theory of irrational numbers and as an axiom in the theory of quantity. Therefore he reintroduces Kant’s pure intuition. Since one-dimensional continuum enables the construction of two-, three- or more dimensional continuum, it can be regarded as a form. And since it can neither be derived from purely synthetic concepts nor proved to be consistent, it must be givena priori [Holder 1924, 351].

41 On the other hand, Hölder maintains that progression does not require further justification, because it is a product of thought. As he puts it, progression is the form of that process of thought which enables us to explain one concept by means of one other or to use the consequence of an inference as a premise for a further inference in those cases in which each inference presupposes the preceding one. For example, the principle of mathematical induction as formulated by Giuseppe Peano [Peano 1889] can be described as that inference from n to n + 1 which generalizes a chain of inferences from 1 to 2, 2 to 3, etc., see also [Poincaré 1902, 20ff.]. According to Hölder, recurrent definitions occurring in arithmetic are grounded in progression, and the same holds true for those geometric constructions whose development requires us to indefinitely repeat some operations [Holder 1924, 338].

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42 The basic concepts of arithmetic (including the theory of irrational numbers) are supposed to exist owing to their purely logical development. This suggests that the consistency of geometry can be derived from that of arithmetic. By contrast, the consistency of a one-dimensional continuum cannot be taken for granted. Therefore the theory of quantity presupposes both progression as a form of thought and continuity as a form of intuition.

43 Hölder’s view is problematic, for several reasons. Firstly, he does not take into account different conceptions of continuity such as Veronese’s. As a consequence, the borderline he draws between a priori knowledge and experience might seem arbitrary. Secondly, it is questionable whether this way to reconsider Kant’s philosophy of mathematics is appropriate. I think that Hölder prepares a subtler strategy when he points out that a priori knowledge in Kant’s sense might turn out to be hypothetical. This suggests that the fundamental feature of a priori knowledge lies not so much in its intrinsic necessity, as in its constitutive function. Such function is predominant in Hölder’s considerations about progression, but at least continuity is said to be both necessary and intuitive because it cannot be constructed. It is likely that Hölder might have had some hesitation about reintroducing Kant’s pure intuition. In fact, he agrees with interpreters such as Cassirer, who points out that Kant uses the expression “form of intuition” in a very general sense: arithmetic is grounded not so much in our conception of time, which is only a kind of progression, as in the form of progression [Holder 1924, 339, and note]; see also [Cassirer 1907, 34, note]. In this case it is clear that Kant’s form of intuition might be described as a form of thought to be analyzed in mathematical logic. Kant might have sought for intuitive foundations of mathematics, rather than logical ones, because the syllogistic logic of his time did not suffice for such analysis. For that reason he used transcendental logic. We already mentioned that Hölder overlooks the role of Kant’s productive imagination in the development of geometric knowledge and interprets Kant’s form of outer intuition as it entailed Euclid’s geometry. Hölder’s distinction between time and progression enables him to avoid a similar misunderstanding at least with regard to the form of inner intuition.

44 For the above mentioned reasons, I think that Hölder’s most helpful insights into the Kantian conception of a priori knowledge are to be found not so much in his general distinction between a priori and empirical knowledge, but rather in his analysis of mathematical method. In this regard, it is worth noting that there might be a connection between Hölder and the Marburg School of neo-Kantianism.

4 Hölder and the Marburg School of neo-Kantianism: Natorp and Cassirer

45 There is evidence that Hölder appreciates the philosophical logic of Paul Natorp, who studies the principles of the exact sciences in order to prove the possibility of knowledge. Not only is Natorp’s major work on that subject-matter, Die logischen Grundlagen der exakten Wissenschaften [Natorp 1910], often quoted by Hölder in Die mathematische Methode, but it is mentioned by him in the preface as an example of methodological inquiry next to Bertrand Russell and Louis Couturat.5

46 Hölder might agree with Natorp that Kant’s forms of intuition depend not so much on the assumption of pure intuitions, as on some processes of thought. For instance,

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Natorp analyzes the development of the series of natural numbers [Natorp 1910, 98ff.]. This section of Natorp’s work is mentioned by Hölder in support of his own analysis of progression. Moreover he openly agrees with Natorp’s rejection of formalism [Holder 1924, 2ff.]. Natorp acknowledges that logic is the science of deduction, and yet his claim is that logic does not need to be developed deductively [Natorp 1910, 5]. Both Hölder and Natorp focus on methodological issues. On the other hand, they disagree on the solution to many such issues, especially in the philosophy of geometry. Natorp’s attempt is to show that if spatial order is to be determined in one and the same way, the geometry of space must be Euclidean. He does not deny the formal-logical possibility of non-Euclidean geometry. Nevertheless he deems Euclid’s axioms necessary conditions of possible experience [Natorp 1910, 312]. By contrast, it follows from Hölder’s analysis of deductive method in geometry that geometers only infer those consequences which follow necessarily from some assumptions. Since the development of non-Euclidean geometry shows that such assumption cannot be necessary themselves, Hölder maintains that the constitutive function of the geometry underlying the interpretation of measurements is relative to scientific theories.

47 Hölder’s views may have more in common with the philosophy of geometry developed by Cassirer in Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik [Cassirer 1910]. In the second chapter, which is devoted to space and geometry, Cassirer maintains that the development of geometric concepts shows the characteristics of the deductive method. Individuals in axiomatic systems are replaced by their relationships to one another and cease to be objects of intuition. Moreover Cassirer deems initial assumptions hypothetical, because they might be true only with regard to their consequences [Cassirer 1907, 29], [Cassirer 1910, 123ff.].

48 Note that Cassirer’s example for the realization of the deductive method in geometry is the same as Hölder’s example, namely, the development of projective geometry in the manner of Staudt-Klein. Cassirer writes: [...] without any application of metrical concepts, a fundamental relation of position is established by a procedure which uses merely the drawing of straight lines. The logical ideal of a purely projective construction of geometry is thus reduced to a simpler requirement; it would be fulfilled by showing the possibility of deducing all the points of space in determinate order as members of a systematic totality, by means merely of this fundamental relation and its repeated application. [Cassirer 1910, 113, Eng. trans, 86]

49 Such possibility is realized by the construction of a projective scale on a straight line. Cassirer does not seem to be acquainted with Hölder’s construction of 1908. Nonetheless he mentions the construction developed by Klein in his lectures on non- Euclidean geometry, [Klein 1893, 15]. Moreover Cassirer refers to projective metric as developed by Arthur Cayley [Cayley 1859] and henceforth applied to non-Euclidean geometry by Klein [Klein 1871]. Cassirer maintains that projective metric entails a deduction of the concept of space in its most general form, independently of the theory of parallel lines. In order to distinguish between Euclid’s geometry and non-Euclidean geometries (parabolic, elliptic, and hyperbolic geometry, in Klein’s terminology), special assumptions are required. So the choice among equivalent geometries cannot be determined by our conception of space. Since Euclid’s geometry is included in a more general classification of hypotheses, Cassirer does not exclude the possibility that a different geometry might be used inphysics [Cassirer 1910, 147].

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50 There is no evidence that Hölder and Cassirer read each other’s works at that time. And still in Die mathematische Methode Hölder mentions only Cassirer’s earlier paper Kant und die moderne Mathematik [Cassirer 1907]. Nonetheless, since at least 1936-1937, Cassirer has been appreciating Hölder’s description of mathematical reasoning and he has been adopting Hölder’s expression “concatenation of relations” [Cassirer 1936-1937, 47], [Cassirer 1940, 82]. Cassirer’s point is that mathematics provides knowledge, because its general concepts are developed progressively. By contrast, syllogistic inferences presuppose a hierarchy of concepts, some of which are given. Hölder’s approach is mentioned by Cassirer as an example of purely logical approach, because it shows how the top-down direction of syllogistic reasoning can be reversed, so that given concepts can be constructed.

Concluding remarks

51 The comparison with Cassirer may shed some light on Hölder’s claim that his approach to the theory of quantity is purely logical, and yet not formalistic: it is logical because of his constructive character, and it is not formalistic because of his bottom-up direction. As regards geometry, Cassirer’s views have much in common with Hölder’s views because Cassirer, unlike other neo-Kantians, acknowledges the hypothetical character of geometrical axioms. There is no evidence that Cassirer and Hölder influenced each other in the development of their views. Nonetheless, their agreement shows that Hölder’s attempt to defend a conception of geometry other than conventionalism, and yet compatible with the use of axiomatic method, was not isolated. Both Hölder and Cassirer show that there might be an unexpected affinity between Kantianism and empiricism, provided that methodological issues are discussed independently of ontological assumptions. From both points of view, empirical measurements require more general laws, and Kantians and empiricists might also agree that the search for such laws must be freed from unjustified presuppositions such as habits or intellectual authorities. Of course such a comparison between Kantianism and empiricism is only possible after Kant’s claim that the axioms of (Euclid’s) geometry are necessary conditions of experience has been rejected. Nonetheless, Kant’s a priori knowledge can be relativized, so that geometry can have a constitutive function relative to scientific theories. I think that a relativized conception of the a priori follows from Hölder’s holistic view that geometry can provide rules for the interpretation of measurements only as a part of a complex system of concepts also including optics and mechanics. Hölder’s related empiricist view is that such system can be empirically tested as a whole.

52 Despite the various aspects of Hölder’s epistemology, I think that he cannot be charged with eclecticism. His classification of mathematical objects rests upon his analysis of mathematical method. It seems to me that this way of proceeding is basically Kantian. At the same time, especially as regards the problems concerning space, Hölder admits solutions other than those provided by Kant, because he takes into account scientific developments such as non-Euclidean geometry, projective geometry, and axiomatic method. In these regards, he finds points of agreement with various thinkers, such as Hermann Grassmann, Helmholtz, and Pasch, and reformulates their arguments from an original point of view.

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CASSIRER, ERNST 1907 Kant und die moderne Mathematik, Kant Studien, 12, 1–40. 1910 Substanzbegriff und Funktionsbegriff: Untersuchungen über die Grundfragen der Erkenntniskritik, Berlin: B. Cassirer, cited according to the English translation by Swabey, W. C. and Swabey, M. C.: Substance and Function and Einstein’s Theory of Relativity, Chicago-London: The Open Court Publishing Company, 1923. 1936-1937 Ziele und Wege der Wirklichkeitserkenntnis, in Nachgelassene Manuskripte und Texte, edited by KÖHNKE, K. C. & KROIS, J. M., Hamburg: Meiner, vol. 2, 1999. 1940 Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, Von Hegels Tod bis zur Gegenwart (1832-1932), vol. 4, Stuttgart: Kohlhammer, 1957.

CAYLEY, ARTHUR 1859 A sixth memoir upon quantics, in The collected mathematical papers of Arthur Cayley, Cambridge: At the University Press, 561–592.

DEDEKIND, RICHARD 1872 Stetigkeit und irrationale Zahlen, Braunschweig: Vieweg.

FRIEDMAN, MICHAEL 1999 Reconsidering Logical Positivism, Cambridge: Cambridge University Press.

GAUSS, CARL FRIEDRICH 1900 Werke, vol. 8, Göttingen: Königliche Gesellschalft der Wissenschaften zu Göttingen.

GRASSMANN, HERMANN 1844 Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch die Statistik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, Leipzig: Wigand.

GRAY, JEREMY 2006 Gauss and non-Euclidean geometry, in Non-Euclidean Geometries: Jànos Bolyai Memorial Volume, edited by PRÉKOPA, A. & MOLNÀR, E., New York: Springer, 61–80.

HELMHOLTZ, HERMANN VON 1868 Über die Tatsachen, die der Geometrie zugrunde liegen, in Schriften zur Erkenntnistheorie, Berlin: Springer, 38–55, 1921. 1870 Über den Ursprung und die Bedeutung der geometrischen Axiome, in Schriften zur Erkenntnistheorie, Berlin: Springer, 1–24, 1921. 1878 Die Tatsachen in der Wahrnehmung, in Schriften zur Erkenntnistheorie, Berlin: Springer, 109– 152, 1921. 1887 Zählen und Messen, erkenntnistheoretisch betrachtet, in Schriften zur Erkenntnistheorie, Berlin: Springer, 70–97, 1921. 1921 Schriften zur Erkenntnistheorie, Berlin: Springer, edited by Hertz, P. & SCHLICK, M., cited according to the English translation by M. F. Lowe: Epistemological Writings, Dordrecht-Boston: Reidel, 1977.

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HILBERT, DAVID 1899 Die Grundlagen der Geometrie, Leipzig: Teubner.

HÖLDER, OTTO 1900 Anschauung und Denken in der Geometrie. Akademische Antrittsvorlesung gehalten am 22. Juni 1899 mit Zusätzen, Anmerkungen und einem Register, Leipzig: Teubner, Eng. trans. Intuition and Reasoning in Geometry by P. Cantù and O. Schlaudt, this volume 15–52. 1901 Die Axiome der Quantität und die Lehre vom Mass, Berichten der mathematisch-physischen Classe der Königl. Sächs. Gesellschaft der Wissenschaften zu Leipzig, 53, 1–63. 1908 Die Zahlenskala auf der projektiven Geraden und die independente Geometrie dieser Geraden, Mathematische Annalen, 65, 161–260. 1911 Streckenrechnung und projektive Geometrie, Berichte über die Verhandlungen der Königl. Sächs. Gesellschaft der Wissenschaften, 63, 65–183. 1924 Die mathematische Methode. Logisch erkenntnistheoretische Untersuchungen im Gebiete der Mathematik, Mechanik und Physik, Berlin: Springer.

KANT, IMMANUEL 1787 Kritik der reinen Vernunft, Riga: Hartknoch, cited according to the English translation by Guyer, P. and Wood, A. W.: Critique of Pure Reason, Cambridge: Cambridge University Press, 1998.

KLEIN, FELIX 1871 Über die sogenannte Nicht-Euklidische Geometrie, Mathematische Annalen, 4, 573–625. 1873 Über die sogenannte Nicht-Euklidische Geometrie, Mathematische Annalen, 6, 112–145. 1893 Nicht-Euklidische Geometrie, vol. 1. Vorlesungen gehalten während des Wintersemesters 1889-1890, ausgearbeitet von F. Schilling.

KLINE, MORRIS 1980 Mathematics: The Loss of Certainty, Oxford: Oxford University Press.

MICHELL, JOEL 1993 The origins of the representational theory of measurement: Helmholtz, Hölder, and Russell, Studies in History and Philosophy of Science, 24, 185–206.

NABONNAND, PHILIPPE 2008 La théorie des Würfe de von Staudt – Une irruption de l’algèbre dans la géométrie pure, Archive for History of Exact Sciences, 62, 201–242.

NATORP, PAUL 1910 Die logischen Grundlagen der exakten Wissenschaften, Leipzig-Berlin: Teubner.

PASCH, MORITZ 1882 Vorlesungen über neuere Geometrie, Leipzig: Teubner.

PEANO, GIUSEPPE 1889 Arithmetices principia nova methodo exposita, Torino: Bocca.

POINCARÉ, HENRI 1902 La Science et l’Hypothèse, Paris: Flammarion, Eng. trans., Science and Hypothesis, London: Walter Scott Pubishing, 1905.

RADU, MIRCEA 2003 A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann, Historia Mathematica, 30, 341–377.

REICHENBACH, HANS 1920 Relativitätstheorie und Erkenntnis apriori, Berlin: Springer.

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RIEMANN, BERNHARD 1854 Über die Hypothesen, welche der Geometrie zu Grunde liegen, in Gesammelte mathematische Werke und wissenschaftlicher Nachlass, Leipzig: Teubner, 254–269, edited by WEBER, H., with the assistance of R. Dedekind

SARTORIUS VON WALTERSHAUSEN, WOLFGANG 1856 Gauß zum Gedächtnis, Stuttgart: Hirzel.

SCHOLZ, ERHARD 2004 C.F. Gauß’ Präzisionsmessungen terrestrischer Dreiecke und seine Überlegungen zur empirischen Fundierung der Geometrie in den 1820er Jahren, in Form, Zahl, Ordnung: Studien zur Wissenschafts- und 24 Technikgeschichte. Ivo Schneider zum 65. Geburtstag, edited by FOLKERTS, M., HASHAGEN, U., & SEISING, R., Stuttgart: Franz Steiner Verlag, 355–380.

ZINDLER, KONRAD 1889 Beiträge zur Theorie der mathematischen Erkenntnis, Sitzungsberichte der phil.-hist. Classe der K. Akademie der Wissenschaften zu Wien, 118, 1–98.

NOTES

1. It might be questioned whether Gauss deliberately undertook an empirical test of Euclid’s geometry. Ernst Breitenberger convincingly argues that the measurement Sartorius refers to does not suffice to draw such a conclusion. More probably, Gauss might have mentioned such measurement in his inner circle, because it incidentally confirms his conviction that Euclid’s geometry is true within the limits of the best observational error of his time. The question of the empirical test is to be related to that of Gauss’ views on space and geometry, because such a test would presuppose the assumption of non-Euclidean geometry as a possible alternative to Euclidean geometry. In fact, to support his reconstruction, Breitenberger maintains that Gauss could not have possessed the notion of a curved, three-dimensional space as developed only in 1854 by Bernhard Riemann [Breitenberger 1984, 285]. More recently, this point has been developed by Jeremy Gray, who points out that Gauss calls into question the necessity of Euclid’s geometry, because he focuses on the problems concerning the definition of the plane and that of parallel lines. But he did not start with non-Euclidean three-dimensional space as Bolyai and Lobačevskij did [Gray 2006, 75]. On the other hand, the interpretation of Gauss’ measurement as an empirical test of Euclid’s geometry has been recently defended by Erhard Scholz. Scholz’s claim is that, even though Gauss could not have known non-Euclidean geometry at that time, his study of the geometric properties of surfaces enabled him to make heuristic assumptions about physical space. Regarding Breitenberger’s objection, Scholz writes: “Es bedeutet dabei keine starke Zuspitzung, eine solche Genauigkeitsschranke als informelle Fassung der Abschätzung einer mit den Messergebnissen verträglichen oberen Schranke für den Betrag der Raumkrümmung zu interpretieren” [Scholz 2004, 364ff.]. 2. In 1924, Hölder recalls that Poincaré developed a similar explanation in La Science et l’Hypothèse [Poincaré 1902, 79]: “Parmi les objects qui nous entourent, il y en a qui éprouvent fréquemment des déplacements susceptibles d’être [...] corrigés par un mouvement corrélatif de notre propre corps, ce sont le corps solides.” 3. Even though Pasch’s Vorlesungen über neuere Geometrie [Pasch 1882] are not mentioned in this regard by Hölder, he shows elsewhere that he is well acquainted with Pasch’s writing and appreciates his attempt to provide an empirical explanation of primitive terms [Holder 1900, 2, note 3, Eng. trans. 27]. Hölder might also have been influenced by Pasch’s following consideration regarding geometrical principles: “Die Grundsätze sollen das von der Mathematik zu

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verarbeitende empirische Material vollständig umfassen, so dass man nach ihrer Aufstellung auf die Sinneswahrnehmungen nicht mehr zurückzugehen braucht” [Pasch 1882, 17]. 4. Similarly, Hans Reichenbach distinguishes between axioms of coordination and axioms of connection. Axioms of connection are empirical and presuppose sufficiently well-defined concepts. Axioms of coordination are those non-empirical principles which make such definitions possible. The latter ones are deemed a priori relative to a theory, insofar as they can be distinguished from the former ones [Reichenbach 1920, 46ff.]. See also [Friedman 1999, 61]: “Thus, for example, Gauss’s proposed ‘experiment’ to determine the geometry of physical space presupposes the notion of ‘straight line,’ which notion is simply not well defined independently of the geometrical and optical principles supposedly being tested. The inadequacy of such an attempt thus makes it clear that at least some geometrical principles must be laid down antecedently as axioms of coordination before any empirical determination of space even makes sense. [...] Physical geometry is [...] revisable and can evolve and change with the progress of science. No geometry is necessary and true for all time. Nevertheless, at a given time and in the context of given scientific theory, the axioms of coordination—at that time and relative to the theory—are still quite distinct from the axioms of connection.” 5. It is astonishing that Hölder does not mention more recent studies. However, he declares that his writing was completed during the First World War. Since at that time he found difficulties to publish, Die mathematische Methode was published only in 1924 by Springer [Holder 1924, IV].

ABSTRACTS

Hölder’s philosophy of geometry might appear to be the most problematic part of his epistemology. He maintains that geometry depends on experience also after Poincaré’s fundamental criticism of Helmholtz. Nevertheless, I think that Hölder’s view is worth discussing, for two reasons. Firstly, the related methodological considerations were crucial for the development of his epistemology. Secondly, Poincaré uses the opposition between Kantianism and empiricism to argue for his geometrical conventionalism. Nevertheless, Hölder shows that an alternative strategy is not excluded: he profits from Kantian objections in order to develop a consistent empiricism. At the same time, especially in Die mathematische Methode [Holder 1924], he vindicates the Kantian view that mathematics is synthetic. In this paper, I will consider Hölder’s defence of the deductive method in geometry in Anschauung und Denken in der Geometrie [Holder 1900] in connection with his approach to the theory of quantity [Holder 1901]. Moreover, I will discuss his connection with Kant. My suggestion is that Hölder’s methodological considerations enable him to foreshadow a relativized conception of the a priori.

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il

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adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori.

AUTHOR

FRANCESCA BIAGIOLI Università degli Studi di Torino (Italy)

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Hölder, Mach, and the Law of the Lever: A Case of Well-founded Non- controversy

Oliver Schlaudt

1 Introduction

1 The Law of the Lever was among the first laws of nature to be formulated in quantitative terms. It dates back at least to Archimedes’ On the Equilibrium of Planes and possibly even to Aristotle’s Mechanics. Shortly after its first formulation, scholars like Archimedes and Euclid were already seeking to prove it by means of deduction from general axioms and postulates. The Law of the Lever thus composes the very core of rational mechanics. However, in 1883 Ernst Mach accused Archimedes’ proof of circularity, thereby provoking a discussion about the role of proof in mechanics, and indeed about the nature of the foundations of mechanics as a whole. This discussion is associated first and foremost with the name of Giovanni Vailati, though Otto Hölder was, in fact, the first to reply to Mach’s criticism.1 Hölder was interested in the proof of the Law of the Lever as a case study on deduction in mathematical physics. He agreed with Mach on experience being the only source of knowledge. But unlike Mach, Hölder did not view deduction as a mere test of agreement between the premises and the conclusion: For Hölder, deduction constructively bridges a qualitative difference between the premises and the conclusion. He attempted to explicate this concept of deduction in terms of “synthesis”.

2 In this paper, an outline of the discussion between Mach and Hölder will be given, with particular stress on Hölder’s contributions and his epistemological thought. The paper consists of two parts: In the first part (Section 2), I shall present a rigorous analysis of Mach and Hölder’s assumptions, aims, and strategies, given that they were at cross- purposes in this discussion, and they themselves never clearly elaborated on their conflict of opinion. It will be shown that Hölder tried to refocus the discussion on the role of proof in the mathematical sciences. In the second part (Section 3), some key

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aspects of Hölder’s account of deduction in the mathematical sciences will be discussed. I will argue that Hölder provides a pertinent defence of Archimedes’ proof against Mach’s critique.

2 The Hölder-Mach-controversy

2.1 The Law of the Lever

3 In modern terms, the Law of the Lever is a special case of conservation of angular momentum: “If the total external torque is zero, then the total vector angular momentum of the system is a constant” [Feynman, Leighton & Sands, 1963, 20–25]. When applied to the “lever“, that is, to a rigid rod pivoted on a fulcrum, this is to state that the lever is in equilibrium if the weights put on it are reciprocally proportional to their distances to the fulcrum. This is a descriptive law which can, however, be translated into a normative rule by saying that reciprocal displacements of weights on both sides of the fulcrum do not disturb equilibria. In this paper, the normative version will often be more appropriate, since equilibrium-conserving manipulations of configurations of masses will be in the main focus.

4 The Law of the Lever can be sited for the first time in Aristotle’s Mechanics, be it as an original element of the text or as a later addition. This text illustrates, as Beisenherz and Renn et al. pointed out, the recent appearance of the lever with unequal arms in everyday life in ancient Greece. This instrument allowed for determining weight by displacing a given counterweight on the opposite arm of the lever ([Beisenherz 1980, 455–456]; [Renn, Damerow & McLaughlin 2003, 47]). According to Renn et al., the lever with unequal arms led to the generalization of the concept of equilibrium. This generalized idea accounted for the fact that weights evidently can be compensated not only by weights, but also by distances. Concept formations like “force of the weight” (Euclid) or “momentum ponderis” (Virtruvius) reflected this idea [Renn, Damerow & McLaughlin 2003, 55]). I would like to add Stevin’s concept of “evenstaltwichtigheid” as opposed to “evenwichtigheid” (that is, in accordance with Dijksterhuis’s translation, “being of apparent equal weight” and “being of equal weight” respectively), introduced in his De beghinselen der weeghconst from 1586 [Stevin 1955, 38, 105]. Masses which compose a system in equilibrium are of equal apparent weight; equality of weight can be assigned to them only in the case that the system is a symmetric one. We can put this in modern terms by saying that the force F of a body is a function of the weight m and its position relative to the fulcrum d, F = F(m, d). It is clear that F is a linear function of the weight m, since a (homogeneous) body m can be conceived as consisting of parts

mi whose forces together add to the total force F(m) = ∑iF(mi). The way F depends on d is the critical question, because no similar reasoning holds for d: A mass m at distance d cannot be conceived as compounded of n parts of mass m at the distance 1∕n · d. The Law of the Lever states that F is a linear function of d as well. However, it should be noted that all kinds of functions, such as a power law of the form F = m ·dn, are compatible with the special case of the symmetric lever, i. e., the fact that equal weights can be displaced symmetrically without disturbing the equilibrium.

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2.2 Archimedes’ proof of the Law of the Lever

5 Archimedes presented an early proof of the Law of the Lever, that is, situated it within a theory of deductive style. As Vailati Vailati1904, Dijksterhuis (1938, quotations from the second English edition Dijksterhuis1987), and Renn2003 concordantly argue, the Archimedean proof was essentially based on the notion of the center of gravity, that is, according to Dijksterhuis, on the premise that the influence exerted on the equilibrium by a body suspended on a lever is judged exclusively by the gravity of the body and the place of its center of gravity, and that the shape is immaterial [Dijksterhuis 1987, 291] and consequently that a lever equilibrium is not disturbed, if one of the suspended magnitudes is made to change its shape while preserving its weight and center of gravity. [Dijksterhuis 1987, 292]

6 As Dijksterhuis made clear, the above was intended to be a reconstruction of the actual cognitive content of the Archimedean theory [Dijksterhuis 1987, 295]. Hölder and Mach, however, were not interested in a historical reconstruction of Archimedean physics; they were attacking a systematic problem of rational mechanics. In direct terms, their question was whether the Archimedean proof was valid or not.2 Accordingly, in the following I will disregard the question of in how far Hölder and Mach did justice to Archimedes. Rather I shall demonstrate the problem both attacked. In order to do so, I will first present Hölder’s reconstruction of Archimedes’ proof.

Figure 1: Hölder’s reconstruction of the Archimedean proof. Figures 32-36 from [Hölder 1924, 40]

7 Consider a body of weight 10 fixed at position (fig. 1-32). This body is successively cut into pieces of weight 1 which are symmetrically distributed pairwise around position 0 in linearly increasing distances till a uniform distribution of ten unit weights is reached (fig. 1-34). In a next step, the four outer weights on the left hand and the remaining six weights on the right hand are symmetrically recombined at their respective geometric centers (fig. 1-35, 36). This finally produces two bodies of weights 4 and 6 at positions 3 and 2 respectively (fig. 1-36). The idea of the proof is as follows: Since the initial distribution (fig. 1-32) was in equilibrium, and since only symmetrical, i. e., equilibrium conserving manipulations, have been carried out, the final distribution (fig. 1-36) must still be in equilibrium. The final distribution however is no longer symmetrical: The

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distances are inversely proportional to the weights, a result which can easily be generalized for any pair of commensurable weights. It should be emphasized that the Archimedean proof does not deal with actual manipulations, but rather intends to rationalize the Law of the Lever in terms of manipulations “performed in thought”, i. e., manipulations performed by means of a graphical representation. Of course, this representation itself tacitly rests on the notion of the center of gravity, for it seems that in reality, the original weight of 10 at the beginning of the proof was not intended to be a physical body, but rather represents the center of gravity which remains unchanged in the following steps.

8 Before going into the details of the Mach-Hölder controversy, two remarks should be made in order to facilitate the discussion. These remarks concern features of the proof which will not be of importance to our discussion and thus can be dealt with in a short comment. First, it should be noted that the proof only applies to commensurable ratios of lengths and weights. Archimedes, however, generalized it to apply to both, commensurable and incommensurable ratios. The generalized proof essentially consists of a reduction of the problem of incommensurable ratios to commensurable ratios by subtracting an appropriate amount of one of the weights, thus completely resting on the simpler case. Since the point of the whole discussion regards a step of the basic proof for the case of commensurable quantities, I will not refer to the extension to the case of incommensurable quantities in the following. Second, there are three further suppositions tacitly made in the proof which should be mentioned: (1) that the weights are made of a homogeneous material, (2) that the lever is perfectly rigid and of mass zero, and (3) finally that the lever is put into a homogeneous gravity field. All three suppositions may be classified as idealizations not uncommon to mechanics. As regards the third point, concerning the gravity field, Hölder himself parted with the supposition of homogeneity by abstracting from the origin of the acting forces. In fact, he mostly did not speak about weight, but simply about acting forces in general.3 These suppositions are also not of interest to the purpose of the present paper and will not be mentioned again.

2.3 Mach’s critique of the proof

9 Archimedes, as Hölder and Mach understood him, claimed to establish the Law of the Lever by means of a special case, namely by showing that an asymmetric configuration can be reached based on a symmetric one simply by distributing and recombining weights symmetrically. In his Die Mechanik in ihrer Entwicklung, historisch-kritisch dargestellt from 1883, Ernst Mach claimed that such a proof cannot be sound. In fact, the theorem to be proven, i. e., the Law of the Lever, must have been tacitly presupposed— otherwise it would have been impossible to derive it from the given premises, dealing only with the special case of symmetric distributions. Mach analyzed the proof and accurately localized the point where tacit presupposition enters into the proof. According to Mach’s analysis, the first half of the proof (leading from fig. 1-32 to 1-34) is sound. Indeed, the weights are displaced symmetrically with regard to the fulcrum. In the second step (fig. 1-35, 36), on the other hand, the weights are recombined symmetrically with regard to a point located beside the fulcrum. Such an operation is feasible only if an additional condition is fulfilled (fig. 2).

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Figure 2: Mach’s analysis of the Archimedean proof.

10 The two weights together at point d (the distance from the fulcrum) must exert the same force on the lever as both weights when separately located at the points d − h and d + h (that is, the effects of displacing them from position d to d − h and d + h, respectively must compensate for, or cancel out, each other). In mathematical terms, the following functional equation must hold:

2m · f(d) = m · f(d − h) + m ·f(d + h) (1)

11 It is then easy to show that this implies that is a linear function of , i. e., that , and hence that the presupposition is equivalent to the Law of the Lever.4 Thus, Mach concludes, the proof is circular.

2.4 Hölder’s defense of Archimedes’ proof

12 In one of his extensive annotations to Anschauung und Denken in der Geometrie, Hölder’s inaugural lecture at Leipzig hold on June 22, 1899, Hölder responded to Mach’s critique. In that lecture, he explicitly denied that the Archimedean proof presupposes the law to be proven: Mach advances the view that in the Archimedean Proof it is already presupposed that the effect that a weight exerts on the lever depends on the product of the magnitudes of the weight and the arm of the lever. This is not the case. [Hölder 1900, 63, Eng. trans. 44]

13 In order to defend the Archimedean proof, Hölder came up with a detailed analysis of its premises, one which is, I believe, much more subtle than what one finds in Mach and thus deserving a closer look. Hölder, in accordance with Mach, identified the symmetric replacement beside the fulcrum as the problematic point. According to Hölder, this substitution can be “accounted for” based on two arguments: first of all, through experiences with symmetric scales; second by the “assumption”, as Hölder puts it, “that the observed equivalence [i. e., on the symmetric scale] also holds true under different circumstances”. “Under different circumstances” here clearly means: beside the fulcrum. Hölder continues in his analysis and finds that the assumption that

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symmetric displacements are also feasible beside the fulcrum can be reduced to the following: a system of forces acting on a rigid body is in equilibrium, if a part of the forces considered separately is in equilibrium and if the same also holds of the other part of the forces. This is a postulate (an axiomatic assumption)—motivated, of course, by experience. [Hölder 1900, 64, Eng. trans. 44]

14 Hölder thus reduces the second tacit assumption of the Archimedean proof to the superposability of equilibria, as he explicitly makes clear elsewhere [Hölder 1931, 322]. To summarize, Hölder seeks to rationally reconstruct the Archimedean proof based on the following two assumptions which serve as its premises: (P1) Symmetrical configurations are in equilibrium and displacements symmetric to the fulcrum do not disturb equilibria. That is, in fig. 3: Displacement (I*) ↔ (II*) is equilibrium conserving.

Figure 3: Premise 1 in Hölder’s reconstruction.

(P2) Equilibria can be superposed. That is, in fig. 4: If configuration (I**) is in equilibrium, configuration (II**) also is in equilibrium, and vice versa.

Figure 4: Premise 2 in Hölder’s reconstruction.

15 It is clear that these two premises lead to a generalization of the case of the symmetric lever. According to the premises, symmetric displacements of equal weights are feasible at any point on the configuration, and not only when symmetric with regard to the fulcrum. The second premise is equivalent to the assumption that any subsystem can be reduced to its center of gravity. In Hölder’s formulation, this premise is particularly striking, because, when read backwards, it yields a kind of geometrical or even graphical algorithm for the analysis of matter distributions. Applying these rules

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(in the order P2-P1-P2) yields a justification for the critical step in the proof of the Law of the Lever by showing the configurations of fig. 5 to be equivalent pairwise.

Figure 5: Hölder’s analysis of the step (I) → (II) in fig. 2.

16 It seems that Hölder was elaborating here on what Archimedes actually did in his proof, i. e., picking out subsystems and manipulating them symmetrically with regard to their respective center of gravity. In this way, Hölder accounted for the equivalence of (I) and (II) in Mach’s proof (fig. 2) through the equivalence of (I**) and (II**) in (P2), i. e., the assumption that any subsystem can be reduced to its center of gravity. However, as Mach made clear, this premise is precisely equivalent to the Law of the Lever, as becomes obvious when spelled out in mathematical terms as done above in the functional equation (6). The proposition P3 (comprising P1 and P2) (P3) Displacements of equal weights are feasible (i. e., equilibrium conserving) when symmetric to any point on the configuration. is strictly equivalent, as is established in the proof of the Law of the Lever, to the proposition (P4) Displacement of weights is feasible (i. e., equilibrium conserving) when, with regard to the fulcrum, reciprocal to the weights.

17 The latter proposition is, of course, the Law of the Lever. Hölder’s analysis of the premises is certainly more subtle than Mach’s, but in how far then does it challenge Mach’s criticism? Here is the conclusion Hölder himself drew from his analysis: Whatever one might think about Archimedes’ suppositions [P3], it cannot be denied that it is not possible to immediately, i.e., by means of the mere power of judgment, perceive in them the Law of the Lever [P4], though this metrical law necessarily follows from these suppositions in a mathematical reasoning. [Hölder 1900, 65, Eng. trans. 44]

18 Hölder hence willingly admits that the Law of the Lever follows from the premises. This indeed amounts to nothing other than stating that the proof is sound. Hölder’s point is clearly a different one. He does not deny that the conclusion is contained in the premises—and how could he have done this? On the contrary, Hölder emphasizes the way the conclusion is contained in the premises: The law to be proven is contained in the premises in such a way that it is impossible to conceive it through “the mere power of judgment”. It is difficult to say what this power of judgment consists in exactly, but if one accepts talk of “containment” of the conclusion in the (conjunction of the) premises—as did Mach, Hölder, and Vailati—one arrives at a peculiar diagnosis for the Mach-Hölder controversy: Clearly, both, Hölder and Mach, admitted that the conclusion, i. e., the Law of the Lever, is implicitly contained in the premises, but each drew contrary consequences from this. Mach states that the conclusion is (implicitly)

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contained in the premises—hence the proof is circular. Hölder, on the other hand, says that the conclusion is (implicitly) contained in the premises—hence the proof is sound.

2.5 A failed dialogue

19 The paradoxical diagnosis characterized above is most certainly due to diverging conceptions of logic—Mach still thought of syllogistic logic, whereas Hölder already approached mathematical logic—and thus to the misguiding metaphor of “containment”, a defect which we will later see how to avoid in the reconstruction of Hölder’s argument (Section 3.2). But there is a different point I would like to draw attention to: besides diverging conceptions of logic, at least a partial explanation for the peculiar incommensurability of Hölder and Mach’s arguments can be found in the fact that both had completely different questions in mind when tackling the problem of deduction in mathematical physics, illustrated by the Archimedean proof of the Law of the Lever. When discussing this proof, both made clear what their intentions respectively consisted of. For Mach, the leading question of his critical survey of mechanics was whether there are non-empirical sources of knowledge. The answer he gave was an emphatic “No!”, as he repeatedly highlights: But the aim of my whole book is to convince the reader that we cannot make up properties of nature with the help of self-evident suppositions, but that these suppositions must be taken from experience. [Mach 1915, 2, 7th German edition] If we were unable philosophically and a priori to excogitate the simple fact of the dependence of equilibrium on weight and distance, but were obliged to go for that result to experience, in how much less a degree shall we be able, by speculative methods, to discover the form of this dependence, the proportionality! [Mach 1919, 14, 4th German ed.]

20 Mach’s point clearly is that experience is the only source of knowledge and that linking propositions by means of deduction cannot change that fact. Hölder, however, completely agreed with Mach on the empirical nature of scientific knowledge, a point which he repeatedly emphasized: he more or less took it for granted that all scientific knowledge is empirical;5 Hölder instead raised the question of what, then, is the purpose of deduction and deductively organizing empirical knowledge. In his comprehensive study Die mathematische Methode from 1924, Hölder programmatically declared: To clarify deductive reasoning by means of analyzing examples is the main concern of my treatise. [Hölder 1924, 3]

21 And already in 1900 he had stressed that this aim can be achieved without touching epistemological questions about the source of our knowledge, i. e., the kind of questions Mach was interested in [Hölder 1900, 8, Eng. trans. 19]. This discrepancy in Hölder’s and Mach’s aims might be at least a partial explanation of why they could end up with such contrary conclusions, notwithstanding a quite considerable agreement on the premises, namely that the Law of the Lever is implicitly contained in Archimede’s assumptions, and that these assumptions as well as the law deduced from them, are of empirical character. Hölder tried to refocus the discussion toward the more important and more promising issue of the function of deductively ordering empirical knowledge. With regard to Mach, however, Hölder failed: Mach, though he appreciated Hölder’s critique6 and to some extent adopted Hölder’s terminology,7 did not readjust the focus

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of his critique, and his response to Hölder in the fourth edition of Die Mechanik did not provide any substantial argument: I cannot agree with O. Hölder who upholds the correctness of the Archimedean deductions against my criticism in his essay Anschauung und Denken in der Geometrie, although I am greatly pleased with the extent of our agreement as to the nature of the exact sciences and their foundations. It would seem as if Archimedes (De aequiponderantibus) regarded it as a general experience that two equal weights may under all circumstances be replaced by one equal to their combined weight at the center (Theorem 5, Corollary 2). In such an event, his long deduction (Theorem 6) would be [un]necessary [my correction of the translation, O. S.], for the reason sought follows immediately (see pp. 14, 513). Archimedes’ mode of expression is not in favor of this view. Nevertheless, a theorem of this kind cannot be regarded as a priori evident; and the view advanced on pp. 14, 513 appears to me to still be uncontroverted. [Mach 1919, 514–515]

22 Here Mach, when referring to Archimedes’ “mode of expression“, provided more philological than systematical evidence and did not at all challenge Hölder’s point. In particular, Mach did not attempt to analyze the origin of his dissent from Hölder. The controversy thus ended before having revealed its real subject matter.

2.6 The real issue: Mach and Hölder on logic and proof in mathematical physics

23 What, then, should have been the substance of the Mach-Hölder-controversy? It is clear from the preceding analysis that the underlying argument between Mach and Hölder is not about non-empirical sources of knowledge, but about the interest in structuring empirical knowledge in a deductive manner. Mach and Hölder had different ideas about the role of logic and deduction, indeed.

24 Mach was quite clear about what he conceives as the role of logic. He developed his account of deduction in a short article, “Deduktion und Induktion in psychologischer Beleuchtung”, published as a chapter of Erkenntnis und Irrtum in 1905. It is there that he essentially repeated Mill’s critique of syllogistic reasoning, i. e., that the general premise cannot be held without already asserting the conclusion. He stressed that syllogisms hence do not provide new knowledge, though one might have that impression: only experience is able to provide knowledge. He comes to the following conclusion: This alone shows that the opening up of new sources of knowledge cannot be the task of the rules of logic, which rather serve only to examine whether findings drawn from other sources agree or disagree, and, if the latter, to point to the need to secure full agreement. [Mach 1976, 225]

25 Here Mach offers an instrumentalist account of logic, the latter serving as a mere corrective: In particular, there are no proofs in physics for Mach. Logic does not uncover truth; it only indicates disagreement and thus error when applied to knowledge drawn from experience.

26 Hölder, as we have seen, agrees with the point that experience is the only source of knowledge. He nevertheless remains at odds with Mach as regards the role of proof and deduction. His arguments are not self-evidently drawn from the discussion with Mach since they failed to push the analysis of their dissent that far. It at least seems to be clear, though, that Hölder, when claiming that it is not possible, through the mere power of judgment, to immediately perceive the Law of the Lever in the premises of the

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Archimedean proof [Hölder 1900, 65, Eng. trans. 44], directly answers Mach’s argument that a syllogism’s conclusion “could have directly been seen” in the premises [Mach 1976, 227]. There is indeed a major difference between the proof of the Law of the Lever and the ordinary syllogisms Mach refers to. Certainly it is impossible to maintain that all men are mortal without already asserting that (the man) Socrates is mortal. It is also true that the conclusion of this syllogism can directly be seen in the general premise, the latter being understood simply as a conjunction of all special cases. But does the same hold true for the proof of the Law of the Lever? Hölder, being one generation younger than Mach, indeed referred to Jevons’ “Logic of relatives” which he thought more adequate than Aristotelian syllogisms to capture mathematical inferences in general [Hölder 1900, 41, n. 25, Eng. trans. 34]. As regards the Law of the Lever, too, Hölder clearly believes that its proof cannot be modeled on a syllogism. Deriving the Law of the Lever from its premises as Hölder reconstructed them (P1 and P2) takes work. Vailati, who also used the metaphor of containment, made this clear in the following impressive illustration: Finally, the assertion that in Archimedes’ premises there is already implicitly contained the conclusion that he deduces from them, seems to me true only in the sense in which it is true of any mathematical proof, insofar as in any mathematical proof the truth of the proposition being proved appears as a simple consequence of certain operations of selection, linking, and coordination performed on fundamental propositions that are the basis of the treatment. And this, to use a celebrated Aristotelian paragon, is as far from diminishing the value of their discovery and of their proof as it would be far from diminishing the value of a sculptor to claim that the statue he created was already within the marble from which he crafted it, and that he has done nothing but to remove the superfluous parts from the block of marble. [Vailati 1904], quotation from the English translation in [Palmieri 2009, 321]

27 We see, then, that Vailati agrees with what I have worked out to be Hölder’s main point, 8 as did Dijksterhuis, it should be mentioned.9 But in order to emphasize the strength of Hölder’s perspective, I should restate the issue in more abstract terms.

28 We saw that Mach thought logic to merely provide a check for consistency. For example, if two empirical thoughts are in logical contradiction, at least one of them must be false. However, “agreement” and “disagreement” of empirical knowledge are symmetric relations, whereas deduction establishes a hierarchy which is essentially asymmetric. Hölder took this hierarchy seriously and could therefore use it in defending the Archimedean proof. This hierarchy indicates a qualitative difference between the premises and the conclusions—a difference which is bridged in deduction. This is precisely what Hölder was after. Put into Vailatian terms, Hölder’s main concern was to explicate what it means to get the statue out of the marble block, that is, to explicate (1.) in what the difference between the premises and the conclusion consists and (2.) how this difference is bridged in deduction. I will study Hölder’s answers to these questions in the next section.

3 Hölder’s account of deduction in physics

29 The hierarchy between the premises and the conclusion must result from a difference between them. In general, it is not easy to say what this difference actually consists of, though a first guess would be generality. But given Poincaré’s discussion on the principle of recurrence [Poincaré 1902], the answer that the premises are simply more

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general must be ruled out as naive. This was held true by Poincaré at least for arithmetical proofs involving induction. But in the case of the proof of the Law of the Lever, it may also hold that the premises (P3 in my reconstruction given above, p. 101) are not necessarily more general than the conclusion (P4).10 Hölder’s answer is that the difference lies in degrees of epistemic simplicity. This will be explained in Section 3.1. In deduction, this difference is then bridged by synthesis, which Hölder parallels to, or even identifies with, deduction, as will be explained in Section 3.2. It will then be shown in Section 3.3, that this account holds true, according to Hölder, not for the whole of physics, but only for those parts which can be reconstructed on the basis of axioms expressing simple experiences. Aside from these parts, there are different domains which do not rest on axioms, but on hypotheses.

3.1 Axioms in mechanics

30 As I have already emphasized, Hölder, like Mach, believed both premises and conclusions, to be empirical. Hölder’s analysis of deduction thus had nothing to do with the quest for non-empirical sources of knowledge, the main target of Mach’s empiricist criticism. However, as already seen in Anschauung und Denken in der Geometrie from 1900, Hölder suggested that the premises and the conclusion might be the result of different kinds of experience, which will then turn out to be his main point about the difference between axioms and theorems: already the experience of every-day life might lead us to postulate the Archimedean assumptions [P1 and P2], which recommands these assumptions as the simpler foundations. [Hölder 1900, 65, Eng. trans. 44]

31 The difference between the premises and the conclusion is hence reduced to the difference between two different kinds of experience that they originate from; the former are the result of everyday experiences, whereas the latter require a somehow more demanding kind of experience. Hölder took up this issue in several of his writings [Hölder 1914, 1924, 1931] from which it is possible to achieve a quite precise picture of what he had in mind when distinguishing between different kinds of experience.

32 In Die Arithmetik in strenger Begründung of 1914 Hölder gives an important hint when using the tacit premises of the Law of the Lever worked out in Anschauung und Denken in order to provide a metrization of weight—an issue also present in Anschauung und Denken when discussing the problem of the rigid body, a prominent question in the contemporary discussion on the foundations of geometry and measurement.11 It appears that Hölder did not work out the details of this idea;12 but from his remarks it already becomes clear, that the difference between quality and quantity is somehow supposed to be in line with the difference between the premises in question and the conclusion, the Law of the Lever, for metrization consists in establishing quantity on a fundament which is not quantitative. In Die mathematische Methode from 1924 Hölder finally explicitly applies the terms of quality and quantity in order to make clear his point and to definitively refute Mach’s critique: It should also be mentioned that the assumptions were not of quantitative, but of purely qualitative nature [...]. Hence the criticisms Mach directed against the famous Archimedean proof of the Law of the Lever should be considered as invalidated. [Hölder 1924, 318]

33 It is thus clear that Hölder considered deduction of the Law of the Lever to be the same as establishing a metrical law on a purely qualitative fundament. This fundament

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consists in the qualitative axioms first that symmetrical configurations are in equilibrium and second, that equilibria can be superposed, while the conclusion states in quantitative terms that, in equilibrium, length is inversely proportional to weight. In order to discuss this more closely I quote the premises as Hölder explained them in 1931, when touching on the problem for the last time [Hölder 1931 322]: 1. Two forces of equal direction and equal magnitude, when acting on a rigid body, can be equilibrated by a force of opposite direction and twofold magnitude acting on the body at the midpoint between these two forces. 2. Equilibria, when superposed, result in equilibria. 3. Two forces acting on a body at the same point are equivalent to their sum when they are coming from the same direction, and cancel when they are coming from opposite directions.

34 It is obvious that these axioms do not dispose of quantity tout court. They presuppose quantitative notions of force (or, in the more special case of Archimedean statics, of weight) and of distance. Taken at face value Hölder’s claim is hence wrong. But apart from the implausibility of such an obvious mistake there are other reasons for considering Hölder’s claim more carefully. For there is indeed a difference between the premises and the conclusion which is linked to establishing a metrical law: The premises take length and weight (force) into consideration separately while the conclusion contains a statement about their functional dependence. Thus the issue is not establishing single quantities, as indicated in the 1914 paper [Hölder 1914], but establishing functional dependencies between different quantities. The confusion between establishing quantity and establishing quantitative relations becomes less problematic when one takes into account that Hölder, as we will see, regarded them as successive steps in the establishment of mechanics. These steps fit into Hölder’s larger idea of the make-up of mathematical sciences which he circumscribed by the notion of synthesis, as will be explained next.

3.2 Deduction and synthesis

35 In his comprehensive study Die mathematische Methode from 1924 Hölder developed an overall view of mathematical sciences with which his considerations on the Law of the Lever fit well. The Law of the Lever undeniably plays the role of a case study for Hölder’s approach. The correlation with geometry, for example, is striking in the following quotation: Geometry also develops metrical dependencies based on certain simple facts concerning equality and some other relations. [Hölder 1924, 45]

36 In this book Hölder develops a general theory of concept formation by means of “synthesis”. He adopted this term from his former teacher Paul du Bois-Reymond who used it synonymously with “Construction” (construction) and “Aufbau” (built-up) [Hölder 1900, 71–72, Eng. trans. 47], [du Bois-Reymond, 1890, 14]. In [Hölder 1924, 5–6], synthesis consists of the construction of new concepts with the goal to identify and to substantiate hidden properties of already known concepts. Synthesis is clearly paralleled to, or even identified with, deduction which, as we have seen, also brings out implicit knowledge: The construction [Aufbau] of concepts here even constitutes a part of the proof. [Hölder 1924, 5]

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37 In geometry and mechanics, as Hölder explains, the concepts synthesized are constructed on the basis of intuitive or empirical concepts (as “point”, “straight line”, “plane”, “angle”, “force”, “mass”). These concepts serve as a basis for a conceptual “super-structure” achieved by way of synthesis. Hölder stresses that this super- structure can again serve as a basis for iterated acts of synthesis ([Hölder 1924, 318]; cf. also the hints given in [Waerden 1938, 100]).

38 Hölder then expands this view to all mathematical disciplines, including arithmetic, geometry, and mechanics, by distinguishing between pure synthetic and hypothetico- synthetic concepts, [Hölder 1924, 295]. The former are products of an intellectual activity such as setting and canceling, unifying and separating, sequencing and coordinating [Hölder 1924, 296]. In particular, they yield numbers and operations on numbers [Hölder 1924, 161ff.]. With hypothetico-synthetic concepts, on the other hand, something empirical adheres to them insofar as the elements they are constructed from have an empirical meaning and insofar as the axioms involved are “occasioned“, as Hölder states, by experience [Hölder 1924, 295]. The counterparts to intellectual activities are geometrical and physical operations such as the concatenation of quantities [Hölder 1924, 297]. Hypothetico-synthetic concepts in particular are the concept of measure in general [Hölder 1924, 51], the measure of the arm of the lever [318]Hölder1924, and the concept of the center of gravity [Hölder 1924, 294]. The simple elements which are used in the construction or synthesis of the concept of measure in particular are the relation of order, equality, and addition [Hölder 1924, 67]. As regards the elements of mechanics, Hölder makes clear that both logical deduction and a formalist treatment fail (on Hölder’s critique of formalism in general cf. [Radu 2003]), thus it holds true that they have to be exemplified in the empirical context of their original abstraction [Hölder 1924, 403]. He stressed that this context is situated in daily life and paid special attention to manual production procedures like the so-called three-plate procedure in the production of steel planes.13

39 The distinction between intellectual and physical operations as the basis of arithmetic on the one hand and geometry as well as mechanics on the other hand makes sense. Indeed, in counting, one brings together the objects counted simply by successively considering them. For example when counting the planets of the solar system, the actual planets may stay where they are; it is sufficient to consider them as planets in order to count them and to determine their number. Counting thus involves purely intellectual activity and no causal manipulation of the objects counted. In measurement, on the other hand, where arithmetic is somehow put into practice, things are much more complicated, for measurement is something more than counting: It is counting so as to determine the magnitude of a property. Here the objects really (physically) have to be brought together in a specific way so as to concatenate, say, two unit quantities to a quantity of double magnitude. In order to do that, one basically needs an empirical operation of comparison and an empirical operation of addition.

40 To conclude, we recognize two synthetical steps in the formulation of the Law of the Lever, leading from qualitative to quantitative notions and from quantitative notions to metrical laws correlating two quantities:

basic layer qualitative concepts (Lebenswelt) order, equality, addition

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first layer quantitative concepts weight, distance

second metrical laws and concepts comprising center of gravity, measure of the lever layer several quantities f(m,d) = m · d

41 Weight and distance are two quantitative concepts synthesized from the respective physical operations of comparison and addition. Adding the simple, pre-metrical fact that equilibria can be superposed, yields, again by way of synthesis, the Law of the Lever, codifying knowledge that before was somehow hidden in the simple elements. In Archimedes, this knowledge enters into the concept of the center of gravity, and it is interesting to see that scientific concepts are not mere composites of arbitrary conceptual elements, but actually codify empirical knowledge. Hölder did not work with the concept of the center of gravity. In Hölder, the knowledge codified in this concept rather provided a kind of (representational or graphical) rule in the analysis of equilibria which is then used in the proof of the Law of the Lever. This rule, however, brings about the same result, since it allows for the reduction of arbitrary subsystems to their respective center of gravity. In establishing the Law of the Lever, two quantities—mass and length—are thus tied together by forming a new concept—center of gravity—which is constituted from both and which comprises their relation, i. e., the Law of the Lever.

42 It is not inappropriate to refer to this kind of concept formation as “synthesis”. Moreover, I believe that Hölder’s account, by which deduction is identified with synthesis, indeed challenges Mach’s critique of the proof of the Law of the Lever. We saw that both, Mach and Hölder, used the perhaps misleading metaphor of “containment” of the conclusion within the premises. Here is a definition of circularity from more recent literature which avoids this metaphor: What is standardly taken to make an argument circular is, that every reason for doubting the conclusion is an equally strong reason for doubting one or more of the premises or inferences which lead up to the conclusion. [Detlefsen 1986, 60]

43 Applying this to Hölder’s account indeed shows that reasons for doubting a proposition involving synthesized concepts of level n (testing the Law of the Lever in measurement) are not equally strong reasons for doubting the premises which are built from synthesized concepts of level n − 1 and which thus rest on a simpler, less sophisticated kind of experience (the qualitative experience expressed in the premises P1 and P2). This seems to me to be the argument behind Hölder’s short commentary from Die mathematische Methode already quoted above (cf. p. 107), namely that the assumptions of the proof are of purely qualitative nature and that hence Mach’s critique is invalid [Hölder 1924, 318]. Therefore, even if Mach is right in that the proof of the Law of the Lever does not change our epistemological situation (in the sense that it does not dispose of experience, but merely shows different experiences to be compatible), Hölder succeeds in showing that our discursive situation definitively changes (for the proof provides stronger arguments by linking given experiences to experiences of a simpler kind).14

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3.3 Deduction and explanation

44 The above account of deduction in terms of synthesis, however, was not meant to hold true for the whole of physics. Indeed, there is evidence in Die mathematische Methode that Hölder admitted a special status to the Law of the Lever and mechanics in general within mathematical physics. This becomes evident when distinguishing more carefully between axioms and hypotheses or assumptions. In fact, it was common during Hölder’s time to distinguish between hypotheses and axioms as regards their epistemological function and status. Nadine de Courtenay substantiated this distinction in a recent study in particular in Meinong, whose work was well known to, and casually cited by, Hölder [de Courtenay 2010]. De Courtenay reconstructs the distinction between hypotheses and axioms in terms of representational modes:15 “Assumptions and hypotheses both appear as indirect modes of presentation in contrast to the direct mode akin to axioms.” The direct mode of representation arises, according to de Courtenay, from abstraction from the phenomena investigated while the indirect mode of representation depends on the means of description and may even lead to propositions contrary to the facts, in which case the corresponding representations cannot refer to any existing thing [de Courtenay 2010, 45]. De Courtenay stresses that the different modes of representation carry with them different conceptions of explanation: In contrast to axioms, hypotheses lead to more than a mere classification of facts, but “[imply] a causal relation: A stipulated external object or system, exemplifying the properties denoted by the alien system of concepts, is responsible for the causal production of the empirical phenomena observed” [de Courtenay 2010, 44– 45]. The point regarding the modes of presentation fits quite well with Hölder’s account, as will become clear. We have already seen that Hölder did not think of axioms as self-evident judgments which are neither empirically verifiable nor, in the case of mathematics, demonstrable. From his analysis of the Archimedean proof, it is clear that Hölder thought of axioms as expressions of simple empirical facts (whereby “simple” is to be understood relatively to the next layer of synthesized concepts). This does not only apply to Archimedean statics, but also hold true for an astonishingly large part of mechanics. Hölder explicitly mentions Newton’s third law (equality of action and reaction) as well as d’Alembert’s principle,16 which are considered to be abstracted from simple experiences and then generalized by way of analogy [Hölder 1924, 430– 431]. Hölder, however, recognizes a limitation in this method. In particular thermodynamics, electrodynamics, the theory of atoms, and relativity theory are said to rest on hypotheses which essentially differ from axioms [Hölder 1924, §151, §155– 165]. According to Hölder, hypotheses are assumptions which postulate a basically non-empirical relation, a relation often even in contradiction with experience to a certain degree. [Hölder 1924, 434] Hypotheses postulate the existence of objects whose properties are commonly not found together in experience. So the atom, which has physical mass without being divisible, and the ether, which bears physical properties without mass. [Hölder 1924, 381, note 2]

45 Hölder’s distinction between axioms expressing simple empirical facts and hypotheses postulating facts essentially beyond experience and intuition somehow fits with the distinction Vailati made between two ends of deduction, namely deduction as a means of proof and deduction as a means of research [Vailati 1898]. In the first case, deduction consists in reducing the disputable to the indisputable, that is, the axioms; in the

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second case, the indisputable phenomenon takes the place of the conclusion, which is linked by means of deduction to the hypotheses; hypotheses are essentially disputable, but serve to explain the phenomenon. Vailati thus probably conceives proof and explanation as an exclusive alternative, as did Hölder. Thus it is clear that the lessons drawn from the Law of the Lever were not intended to be generalized onto physics in general. Mechanics seems to hold a special status where deduction does not yield an explanation derived from non-empirical microscopic entities (postulated by hypotheses), but rather a synthetic construction of the theory derived from basic experiences (expressed in the axioms).

4 Conclusion

46 Hölder’s account of deduction in mathematical physics is of interest for two reasons:

47 First he provided a more subtle analysis of the proof of the Law of the Lever than Mach. Mach had, indeed, called attention to a weak point of the proof; however, primarily due to the anti-apriorist motivation of his critique and his Aristotelian conception of logic, he ended up with the result that deduction in physics plays a rather marginal role since it does not create non-empirical knowledge, thus does not help disposing of experience. This result is plainly inadequate, for it does not explain why empirical knowledge is arranged in deductively structured theories at all.

48 The second point of interest is the theory of concept formation implied in Hölder’s account of deduction. His theory of synthesis is admittedly difficult to understand, and perhaps not even consistently worked out in his magnum opus Die mathematische Methode from 1924; nonetheless, it offers a quite fascinating account. According to Hölder, concepts are synthesized either from empirical elements (including equality, order, etc.) or from formerly synthesized concepts. As such they have, as Hölder says, “hidden properties”, brought out in further synthetical steps. I believe that this feature should be assessed in comparison with theories which explain concept formation in terms of nominal definitions. Hölder’s analysis of deduction in mechanics provides a substantial argument for the view that concept formation is a much more complicated process than the amalgamation of an enumerable set of properties as one single notion for the sake of brevity. Hölder aimed at a reconstruction of scientific concepts on the basis of everyday linguistic and manual competences (as Hölder stresses in [Hölder 1900, 65, Eng. trans. 44] and [Hölder 1924, 403]), while paying special attention to manual production procedures like the so-called three-plate procedure in the production of steel planes [Hölder 1924, 370]. This account bears a certain resemblance to Hugo Dingler’s approach (cf. [Dingler 1988]), notwithstanding the apparent incompatibility on a rhetoric level, namely empiricist in the case of Hölder and rationalist in the case of Dingler. Both argue for non-holistic semantics and hold quantitative concepts to be definable in (or synthesizable from) purely qualitative concepts. In particular, this involves the problem of defining the rigid body in qualitative terms, which Hölder explicitly considered possible without resorting to a solution in terms of conventions [Hölder 1924, 370–371]. Nevertheless there is also a strong affinity to Poincaré’s conventionalist approach. For, unlike Dingler, Hölder stressed the role of experience in concept formation and agreed with Poincaré on the rejection of naive empiricism: concept formations, according to Poincaré and Hölder, do not match or copy experience, but are “occasioned by experience“. It thus becomes

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clear that Hölder pursued an original path between rationalism and empiricism fully deserving of attention.

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POINCARÉ, HENRI 1902 La Science et l’Hypothèse, Paris: Flammarion, Eng. trans., Science and Hypothesis, London: Walter Scott Pubishing, 1905.

RADU, MIRCEA 2003 A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann, Historia Mathematica, 30, 341–377.

RENN, JÜRGEN, DAMEROW, PETER & MCLAUGHLIN, PETER 2003 Aristotle, Archimedes, Euclid, and the origin of mechanics: The perspective of historical epistemology, in Symposium Archímedes, edited by MONTESINOS SIRERA, J. L., Berlin: Max-Planck- Institut für Wissenschaftsgeschichte, Preprint, vol. 239, 43–59.

STEVIN, SIMON 1955 Mechanics, The Principles Works of Simon Stevin. Edited by Ernst Crone, E. J. Dijksterhuis, R. J. Forbes, M. G. J. Minnaert, A. Pannekoek, vol. I, Amsterdam: C. V. Swets & Zeitlinger, edited by E. J. Dijksterhuis.

VAILATI, GIOVANNI 1897a Del concetto di Centro di Gravità nella Statica di Archimede, Atti della R. Accademia delle Scienze di Torino, XXXII(13), 742–758, [reprinted in Vailati 1911, 79–90]. 1897b Di una dimostrazione del Principio della Leva, attribuita ad Euclide, Bollettino di Storia e Bibliografia Matematica, nov.-dic., 21–22, [reprinted in Vailati 1911, 115-117]. 1898 La méthode déductive comme instrument de recherche, Revue de Métaphysique et de Morale, IV, 667–703. 1904 La dimostrazione del principio della Leva data da Archimede nel libro primo sull’equilibrio delle figure piane, in Atti del Congresso Internazionale di Scienze storiche, Roma 1903, vol. XII, 243–249, English translation in [Palmieri 2009].

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1911 Scritti (1863-1909), Leipzig and Firenze: Johann Ambrosius Barth (Leipzig) and Successori B. Seeber (Firenze).

VAN DYCK, MAARTEN 2009 On the epistemological foundations of the law of the lever, Studies in the History and Philosophy of Science, 40, 315–318.

VERONESE, GIUSEPPE 1891 Fondamenti di Geometria a più dimensioni e a più specie die unità rettilinee esposti in forma elementare. Lezioni per la Scuola di magistero in Matematica, Padova: Tipografia del Seminario.

WAERDEN, BARTEL L. VAN DER 1938 Otto Hölder. Nachruf gesprochen in der öffentlichen Sitzung vom 25. Juni 1938, Berichte der math.-phys. Klasse der sächs. Akademie der Wissenschaften, XC, 97–101.

ZINDLER, KONRAD 1889 Beiträge zur Theorie der mathematischen Erkenntnis, Sitzungsberichte der phil.-hist. Classe der K. Akademie der Wissenschaften zu Wien, 118, 1–98.

NOTES

1. Vailati already in 1897 published two papers on the proof of the Law of the Lever and its critics [Vailati 1897a], [Vailati 1897b]. These texts however do not contain any reference to Mach. 2. As regards this question, it was disputed by [Beisenherz 1980] whether Dijksterhuis’ reconstruction is a plausible candidate for a systematic defense of Archimedes. Beisenherz’ main point is that Archimedean statics should be read as an attempt to establish a metrization of weight and that the center of gravity cannot be used within this attempt, since this notion depends on the metrization of weight [Beisenherz 1980, 457–459]. 3. Palmieri recently took up this issue in a paper from 2008, showing that the actual form of the Law of the Lever is contingent on the specific form of the earth’s gravity field. I believe Hölder and Mach were perfectly aware of this. In his response to Palmieri, Van Dyck additionally provides evidence that the problem of the uniqueness of the center of gravity of bodies on the curved surface of the earth was already discussed in the sixteenth century by Simon Stevin and Guidobaldo del Monte [Van Dyck 2009, 318]. Palmieri’s overall aim in his two papers, [Palmieri 2008], [Palmieri 2009] is to defend Mach’s rejection of non-empirical sources of knowledge. In light of the Mach-Hölder debate, however, it becomes clear that this point actually was not at issue. 4. I should mention that the first to use these mathematical terms was actually Vailati in his article from 1904. However, from Mach’s texts it is clear that the argument presented here was the one intended by him and that he was aware of the fact that the compensation of the effects of a symmetric displacement beside the fulcrum is equivalent to the Law of the Lever—otherwise his arguments would make no sense. I stress this point because of a paper published by Palmieri in 2008 containing circumstantial proof of the equivalence in question, where the author claims to be the first one to make this mathematical argument. Actually, Hölder explicitly indicated how to prove the equivalence in Die mathematische Methode [Hölder 1924, 44–45, note 1]. 5. In Anschauung und Denken Hölder takes both intuition and experience into consideration as sources of knowledge, especially for geometric axioms [Hölder 1900, 57, Eng. trans. 41], and finally advocates for the latter [Hölder 1900, 72, Eng. trans. 47]. 6. In the preface to the fourth edition, Mach underlined the following: “New books and criticisms touching on my discussions have received attention in special additions, which in some instances have assumed considerable proportions. On these strictures, O. Hölder’s note on my criticism of

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the Archimedean deduction (Denken und Anschauung in der Geometrie, p. 63, note 62) has been of special value, inasmuch as it afforded me the opportunity of establishing my view on still firmer foundations (see pages 512–517). I do not at all dispute that rigorous demonstrations are as possible in mechanics as in mathematics. But with respect to the Archimedean and certain other deductions, I am still of the opinion that my point is the correct one” [Mach 1901, xi]; [Mach 1919, xv–xvi]. 7. This can be seen in the use of the phrases “the actions of two equal weights to be the same under all circumstances as that of the combined weights” [Mach 1919, 513] (my italics) and “general experience” [Mach 1919, 514]. 8. In fact Vailati already expressed this point in abstracto in his 1898 paper on deduction [Vailati 1898, 690]. 9. Dijksterhuis also shows that the Law of the Lever is the solution to equation (6) and remarks: “Indeed, if it is not necessary to solve the functional equation in this case, when is there any sense in doing so? And if in general the drawing of a conclusion Q from a group of premises P may be called superfluous when P applies only if Q is true, one may just as well reject as superfluous all mathematical proof, for if Q is not true, it is clear that P will not hold!” [Dijksterhuis 1987, 294]. 10. P3 holds true for any given point, but is restricted to equal weights; P4, in turn, holds for any given pair of weights but is restricted to the fulcrum as the point of reference. 11. [Hölder 1900, 5, 30, Eng. trans. 17, 30]; [Hölder 1924, 370–371], where he refers to [Zindler 1889]; [Dingler 1925, 310–330], and, according to [Heath 1926, vol. III, 226–227], [Veronese 1891]; etc. 12. Unlike [Beisenherz 1980], who is quite in agreement with Hölder’s approach. 13. [Hölder 1924, 370]. This idea is known from [Helmholtz 1887], [Helmholtz 1977] and, in particular, from Hugo Dingler. Hölder however does not refer to Dingler, but to [Hjelmslev 1916]. This procedure allows for producing planes without the need for a prototype. For an English description, cf. Wolter’s note to the English translation [Dingler 1988, 385]. 14. It seems that Van Dyck, in his defense of Vailati against a misreading by Palmieri [Palmieri 2008], has a quite similar point in mind when explaining that, according to Vailati, the Law of the Lever is “a more informative way to describe what is allowed by [the premises of its proof]” and that “we can see Archimedes as using mathematics to reorganize the empirical information already at his disposal” [Van Dyck 2009, 316–317]. 15. According to de Courtenay, there is a further distinction between hypotheses and assumptions which lacks a counterpart in Hölder. Hypotheses and assumptions, though they are on a par as regards the mode of representation, differ as to the mental attitude: “Assumptions and hypotheses encapsulate different attitudes of the subject towards his object: Assumptions do not involve belief, whereas hypotheses do involve belief” [de Courtenay 2010, 43]. 16. Cf. also Jourdain’s analysis [Jourdain 1912, 298–301] of Hölder’s work on d’Alembert’s principle and analytical mechanics [Hölder 1896].

ABSTRACTS

Otto Hölder’s reply to Mach’s renowned critique of the Archimedean proof of the Law of the Lever is analyzed and contextutalized as part of Hölder’s epistemological theory. The proof of the Law of the Lever provided for Hölder a case study on deduction in mechanics. He developed in it

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an original understanding of proof as a constructive, synthetical process leading to higher order concepts. This enabled him to avoid the view held by Mach that proof can only lead to analytic truths, and to defend Archimedes against Mach’s critique.

Nous analysons la réponse qu’a donnée Otto Hölder à la célèbre critique de la preuve archimédienne de la loi du levier par Ernst Mach et nous la situons dans sa pensée epistémologique. Pour Hölder, la preuve de la loi du levier fournit une étude de cas sur la déduction en mécanique. Considérant la preuve comme un processus constructif et synthétique montant d’un niveau conceptuel à un niveau supérieur, Hölder parvient à défendre la preuve d’Archimède contre la critique de Mach, pour qui au contraire une preuve n’amène qu’à une vérité analytique.

AUTHOR

OLIVER SCHLAUDT Universität Heidelberg (Germany) Laboratoire d’Histoire des Sciences et de Philosophie, Archives H. Poincaré, UMR 7117, Université de Lorraine – CNRS, Nancy (France)

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Otto Hölder’s Interpretation of David Hilbert’s Axiomatic Method*

Mircea Radu

1 Otto Hölder’s forgotten legacy

1 Hölder’s reputation as a mathematician is well established.1 Things look different with respect to Hölder’s ideas about the philosophy of mathematics.2

2 The most important objective of Hölder’s philosophical thinking is the clarification of the value and of the limitations of sign-use in mathematics [Hölder 1924, 5]. Hölder is concerned about the emergence of a Zeitgeist characterized by a unilateral confidence in symbolic practices, which he sometimes calls “formalism”.

3 The first clear expression of this concern can be found in Hölder’s critique of Robert Graßmann’s conception of rigor [Hölder 1892]. Graßmann claims that rigor can only be reached by putting in place a symbolic practice in which the fundamental unity of concept and symbolic expression, between analysis and synthesis, is conveyed. We recognize here Leibniz’s ideal of developing a thinking calculus, a Calculus Ratiocinator.3 In his 1892 review of Robert Graßmann’s book, Hölder tries to establish that Graßmann’s approach falls short of accomplishing the methodological ideal of putting up a calculus expressing the full harmony between conceptual content and symbolic expression. Hölder’s arguments also suggest, more generally, that the shortcomings of Graßmann’s approach prove that Leibniz’s methodological ideal pursued by Graßmann is utopian.4

4 Hölder’s concern about the best way to understand the relationship between conceptual content and symbolic expression in mathematics was emphasized by van der Waerden. In his Nachruf auf Otto Hölder, which was published in 1939, not long after Hölder’s death, van der Waerden praises Hölder as someone having significantly contributed to an important transformation of mathematics “from the formal to the critical, from mere computation to concept” [Waerden 1939, 55, my translation].5 And van der Waerden goes on to claim that Kurt Gödel’s methods and results were the

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confirmation of Hölder’s conception of method. Unfortunately, van der Waerden’s remarks are brief, so that the precise nature of the link between Hölder’s account of method on the one hand and Gödel’s methods and his philosophy of mathematics, on the other, is left open.

5 The discrepancy between van der Waerden’s suggestion of a link between Hölder and Gödel, on the one hand, and the lack of recognition of Hölder’s philosophy of mathematics by posterity, on the other, is surprising. If van der Waerden is right, modern history and philosophy of mathematics has failed to take notice of an immensely fruitful contribution, provided by a first rate mathematician. How can this be explained?

6 Gödel’s results concerning the limitations of a formal-axiomatic treatment of arithmetic were published in 1931, seven years after the publication of Hölder’s The Mathematical Method [Die mathematische Methode]. As far as I can tell, Gödel nowhere mentions Hölder’s work. In the present context, the interesting question is rather, how did Hölder receive Gödel’s results? This major event could not have escaped Hölder’s attention. So far, however, I have been unable to find any documents recording Hölder’s reaction to Gödel’s results.6

7 Some of the remarks made by Hölder in 1924 concerning the work of Gottlob Frege and of Giuseppe Peano on the foundations of arithmetic provide an interesting hint. Hölder emphasizes that the foundations of arithmetic cannot be secured through a symbolic calculus. Hölder thinks that a careful analysis of the symbolic proofs found in writings such as those of Frege and Peano would reveal a vicious circle: it would turn out that these proofs are based on the same sequential procedures, which they are supposed to justify [Hölder 1924, 349].7

8 Hölder’s rejection of the methods of Frege and Peano and, more generally, his strict rejection of symbolic logic (see below) may be one of the causes having led to the poor reception of his ideas.

9 The structure and style of Hölder’s approach to these issues in his writings may be the other cause. Hölder does not manage to develop a sufficiently clear and rich philosophical conception capable to support his examination of method. He writes not as a philosopher, but as a mathematician interested in a clarification of the mathematical method. This may be seen as a major limitation of Hölder’s approach. This limitation, however, is also an advantage, for Hölder does not try to defend any particular philosophical school, nor does he try to create a philosophical system of his own. He rather observes the mathematical practice and the way it is reflected by philosophers, trying to expose the issues on which the philosophical description fails to do justice to the practice of the working mathematician.

10 The main objective of this paper is to give a brief presentation of some of Hölder’s reflections concerning the limits of formalization and the role of arithmetical intuition in mathematical deduction. I rely mainly but not exclusively on Hölder’s The Mathematical Method of 1924.

2 The limits of formalization

11 The first important remarks on the subject are made in the Introduction to the previously mentioned book. Hölder describes mathematics as a deductive science.

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Mathematical deduction, we are told, cannot be adequately described in terms of syllogistic logic. An adequate description of mathematical deduction requires the logic of relations.8 And then Hölder warns against looking for a deductive presentation of logic. Since logic is the science of deduction, it would be inappropriate to seek a deductive theory of deduction [Hölder 1924, 2].

12 The same topic is taken up a few lines further with respect to the logic of relations. Hölder explains that in the hands of mathematicians, the logic of relations was turned into a symbolic calculus known under the name of “Logistic” [Hölder 1924, 4–5]. Hölder points out that in certain cases formalization may prove helpful but only as a heuristic tool. He continues by rejecting the view (which he attributes to Louis Couturat), that the formalization of logic is an essential requirement for providing a truly scientific presentation of logic. He explains his position as follows: (...) when in the course of a mathematical investigation new signs (...) are introduced, as this often happens, it is not possible to justify the reasoning behind the introduction of those new signs, by means of a new symbolic calculus. It is obvious that if this could be done, and if the previous position would be adequate,9 one would then be compelled to justify the latter symbolic calculus by means of a new such calculus, and we would have to pursue this process further indefinitely. We would thus reach (...) an “infinite regress”.10 [Hölder 1924, 5]

13 Hölder relies here on a classical kind of argument often raised to emphasize the limits of justification. One equally classical solution to this type of problem is provided by the axiomatic method.

14 If we desire to provide an axiomatic treatment of a theory, then all we need to do is to identify a finite adequate list of undefined primitive notions, and a finite adequate list of first principles expressing properties of the primitive notions (the axioms of the theory). Based on the chosen primitive notions and axioms we can then state and deduce all other results of the theory in case. One of the major hopes linked to the new axiomatic method, as it was understood, for instance by David Hilbert and his school, was precisely the conviction that a symbolic, axiomatic treatment of pure mathematics and of logic would make it possible to prove the consistency of mathematics.

15 Hölder had carefully reflected on these matters independently of Hilbert, and indeed before Hilbert’s work on the foundations of geometry was published. As said, the first traces of Hölder’s thinking are visible in Hölder’s review of Robert Graßmann’s Treatise on Number [Hölder 1892]. Here we find the first expression of Hölder’s conviction that the axiomatic method cannot be used for proving the consistency of arithmetic.

16 Before returning to Hölder’s arguments concerning the nature of mathematical deduction, it is helpful to emphasize an important aspect of Hilbert’s conception of proof, criticized by Hölder. Niebergall and Schirn describe Hilbert’s conception of proof in the following terms: While nowadays metamathematics is conceived as being formulated and axiomatized in a formal language, Hilbert’s finitist metamathematics qua contentual theory of formalized proof (...) is neither formalized nor axiomatized. It is supposed to allow only intuitive and contentual reasoning, and, as Hilbert (...) stresses, this kind of reasoning is, by its very nature, free from axiomatic assumptions. Due to this sharp distinction between formalized mathematics and contentual metamathematics, the meanings attached to the word “to prove” in the two “disciplines” are fundamentally different. In formalized mathematics, it means to infer according to the formal rules of the calculus; in contentual metamathematics, it means to show by means of contentual intuitively evident inference. It is precisely the

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intuition-based character of metamathematical reasoning that is supposed to guarantee its security and reliability. (...) In a nutshell: Hilbert’s finitist metamathematics is a non-formalized, non-axiomatized, and surveyable “theory” with a non-mathematical vocabulary whose intended domain consists of concrete and surveyable figures, in particular, of formulas (e.g., axioms) and proofs. [Niebergall & Schirn 1998, 274ff.]

17 Hölder believes that the difficulties involved in establishing the consistency of formalized mathematics, cannot be conclusively dealt with, by means of the metamathematical shift proposed by Hilbert. Hölder thinks that the same difficulties would sooner or later surface again in examining the proofs elaborated at the level of contentual metamathematics. We would then, if the ideal of a formal-axiomatic treatment is retained, sooner or later have to mathematize metamathematics. In this case, however, we would run precisely into the infinite regress described by Hölder above.

18 In the next sections of this paper I will present some of Hölder’s key arguments advanced in support of this position. These arguments belong to an alternative conception of mathematical proof outlined by Hölder in his The Mathematical Method of 1924.

3 A genetic interpretation of Hilbert’s axiomatic method

19 Hölder is not just skeptical about the possibility of proving the consistency of arithmetic along the lines proposed by Hilbert. He finds no merit whatsoever in seeking an axiomatic treatment of arithmetic. The main reason for this is Hölder’s understanding of the value and limits of the axiomatic approach in pure mathematics. The fundamental ingredient of this attitude is Hölder’s account of the content/form distinction in mathematics. Let me explain this.

20 An axiomatic approach is needed and indeed natural in those sciences in which there is a clear separation between the content dealt with and the conceptual apparatus used in describing that content, that is, the theoretical form used to shape the content. This is the case in empirical sciences such as geometry or mechanics [Hölder 1924, 465–466]. Hölder’s approach to this matter is typical for the positivist approach to natural sciences. In natural sciences the axioms are hypothetical statements about the properties of sensible objects [Hölder 1924, 374–493].11

21 The situation changes considerably if we shift our attention from empirical science to pure mathematics and particularly to arithmetic and logic. In arithmetic and logic it is impossible to draw a clear distinction between the content of the theory and its form.

22 In order to establish this, Hölder explores a large number of examples. One such example is taken from geometry. We are told that in geometry we often find geometric concepts and geometric proofs, which essentially rely on our logical intuition of the sequence of the natural numbers. In such cases, geometric concepts are used “not just formally but according to their content” [Hölder 1899, 58–59]. Here is one of Hölder’s examples used to illustratethis situation.

23 Take, for instance, Hölder’s discussion of the construction of the concepts square and n- gon. According to Hölder, the definition of the concept square is a specification of a

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finite sequence construction-steps.12 Finite constructions of this kind do not involve arithmetical intuition. The situation changes in the case of general concepts such as the n-gon concept. A definition of this concept must provide a description of a sequential construction procedure. In the case of the n-gon concept, however, the construction involved requires an arbitrary, unspecified number of steps. It is therefore impossible to list all the steps required one by one or to actually carry out the construction.

24 We can easily prove that in any equilateral triangle, square, regular pentagon, etc., the sum of the distances of any interior point to its sides is independent of the position of the point, without using arithmetical intuition. To state and prove a theorem such as: “In any regular n-gon the sum of the distances of an interior point to its sides does not depend on the position of the point” we need to go beyond the finite. The idea is that our ability to frame certain general concepts and general proofs must be rooted in a special faculty of our mind, the arithmetical intuition, our intuition of the sequence of the natural numbers [Hölder 1924, 338ff.].

25 The simplest mathematical expression of arithmetical intuition is the arithmetic of the natural numbers. The core of this is our ability to frame infinite sequential procedures and use them in developing new concepts and proofs. The sequence-concept [Reihenfolgebegriff] is the content and, at the same time, the form of our thinking [Hölder 1924, 292–320].

26 In Über den Zahlbegriff, which was published in 1900, Hilbert contrasted the genetic to the axiomatic approach to arithmetic and remarked that priority should be given to the axiomatic method. Hölder is not against the axiomatic method, but simply against giving it priority over the genetic method (which Hölder prefers to call the synthetic method) in respect to consistency questions.

27 In a section with the heading “The so-called axiomatic presentation of arithmetic illuminated from the perspective of the superposition of concepts [Die sogenannte axiomatische Arithmetik, beleuchtet vom Standpunkt der Überbauung der Begriffe]”, Hölder gives a clearer shape to his fundamental objection to the “formalism” which he attributes, for instance, to Couturat and Hilbert.13 Hölder achieves this by investigating Hilbert’s proof, given in his Grundlagen der Geometrie [Hilbert 1899, 26ff., 72ff.] of the commutative law for the multiplication of certain objects.

28 Hilbert presents a system of axioms for “complex systems of numbers [complexe Zahlensysteme]” [Hilbert 1899, 26], a term already used by Hermann Hankel. Hilbert defines his Zahlensysteme by introducing three groups of axioms. They are: “Sätze der Verknüpfung”; “Sätze der Anordnung”; Archimede’s axiom. Hilbert uses these to define a totally ordered, Archimedean field.

29 Hölder’s general critique of Hilbert’s position is, as said, directed against Hilbert’s belief that the consistency of arithmetic can be proved independently of genetic considerations. At the same time, Hilbert’s 1899 proof of the commutative law of multiplication discussed by Hölder has nothing to do with the consistency of arithmetic. Hölder is aware of it. He obviously believes, however, that his analysis of Hilbert’s proof of the commutative law reveals that Hilbert’s axiomatic treatment of the matter essentially involves contentual arithmetic, or, in other words, that Hilbert’s axiomatic treatment of the complex system of magnitudes, tacitly relies on the genetic approach which it is supposed to avoid. Hölder assumes that any axiomatic treatment

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of arithmetic will require proofs similar to those discussed by him, so that his arguments would count for those proofs as well.14

30 Hölder begins his examination of Hilbert’s proof by reproducing the main principles involved in his discussion of the proof:

a + (b + c) = (a + b) + c (7)

a + b = b + a (8)

a(bc) = (ab)c (9)

a(b + c) = ab + ac (10)

(a + b)c = ac + bc (11)

ab = ba. (12)

31 He describes Hilbert’s approach in the following terms: We imagine certain objects of an arbitrary kind, which are not numbers, but which allow two operations [Kompositionsweisen] to be carried out on them. We call the operations “addition” and “multiplication” (...). It goes without saying that these operations must be distinguished from the addition and multiplication of numbers. The two operations are taken to satisfy formulas (1) to (5) above. A few other obvious principles are (...) also stipulated. Of particular importance is the Archimedean axiom which is also adopted as a principle. The latter states that, provided that a is smaller than b, by adding a to itself sufficiently often, an object a + a + a + … + a can be manufactured [hergestellt], which is greater than b, that is, there is a multiple of a exceeding b. [Hölder 1924, 320ff.]

32 Hölder then reproduces the most important steps of Hilbert’s proof. Here I only discuss those steps of Hilbert’s proof, which according to Hölder involve arithmetical intuition. To establish (6), Hilbert first proves the equality ndmd = mnd. Hilbert’s proof goes like this.

33 Hilbert postulates the existence of a special unit e for which ae = ea = a holds for any magnitude a. By repeatedly using axiom (4) we get:

a(e + e + … + e) = a + a + … + a; (10)

and by using axiom (5) we get:

(e + e + … + e)a = a + a + … + a. (11)

34 This leads immediately to:

a(e + e + … + e) = (e + e + … + e)a. (12)

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35 The proof of ndmd = mnd is completed as follows:15 Md · nd = {(e + e + … + e)d}{[e + e + … + e]d} = {(e + e + … + e)d[e + e + … + e]}d = {(e + e + … + e){d[e + e + … + e]}}d = {(e + e + … + e){[e + e + … + e]d}}d = {{(e + e + … + e)[e + e + … + e]}d}d = {(e + e + … + e)[e + e + … + e]}{d²} = mnd²

36 This step of Hilbert’s proof is analyzed by Hölder in the following terms: This may leave the impression that in this proof proper arithmetical reasoning has been switched off [ausgeschaltet]. A closer examination of the genesis [Entstehung] of (10) and (11) leads, however, to the contrary view. For instance, in order to obtain (10) by using rule (4), one must first acknowledge that the repeated application of rule (4) to the expression:

a(e + e + … + e) (13)

should be understood, for instance, as meaning a(…(((e + e) + e) + e) + … + e). The latter operation must produce the object a as its result, as many times as the object e was present in the bracket of (13). One therefore finds oneself counting the steps of the proof procedure [Man denkt sich also wieder die Schritte des Beweisverfahrens selbst gezählt]. [Hölder 1924, 323ff.]

37 Examining the previous proof, one may be perhaps inclined to think that it requires an explicit use of mathematical induction or, at least, that explicit induction might be used to rewrite some of the previous steps. But this would lead to adding the induction- axiom to the axioms actually used by Hilbert. Hilbert, however, does not explicitly require induction here and neither does Hölder.16 Hölder simply emphasizes that this way of using the signs involves counting the steps of the proof while proving. Expressions such as (13) may look finite because the number of iterations in (13) is supposed to be finite. At the same time, however, (13) involves, so to speak, second- order-counting. There is, so to speak, first-order-counting involved in each of the individual steps e;(e + e);(e + e) + e, etc. However, formulas such as (13) involve more than that. In using them we quit the initial counting process, and view it as accomplished. We proceed by counting the counting steps themselves, thus reaching what I have called second-order-counting or perhaps, even better, meta-counting. In this way certain steps of the proof are turned, while proving, into objects of the proof, which proceeds by further counting former steps of the proof. Therefore each proof of this kind is, not simply a proof about operations on magnitudes but rather also a proof about our way of conceptualizing our operations carried out while proving. This is what Hölder means when he speaks of contentual use of arithmetic and by concept- superposition [Überbauung der Begriffe].17

38 Let me now turn to Hölder’s comments on Hilbert’s use of the Archimedean axiom. Hilbert’s proof also relies on the following identities:

md < a ≤ (m + 1)d (8)

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and

nd < b ≤ (n + 1)d. (9)

39 Hölder writes: An exact analysis of the thinking pattern that led to the inequalities (8) and (9) also leads to a notable result. Using the Archimedean axiom only states that not all multiples of d can be smaller than a. From this one deduces that in the sequence of multiples 1d;2d;3d;4d;… there must be a first that is not smaller than a. But since d itself, that is, the first element of the sequence, was taken to be smaller than a, there must be a multiple of that first element of the sequence for which md is still smaller than a, whereas is (m + 1)d greater or equal to a ; this proves (8). [Hölder 1924, 324]

40 Here arithmetical intuition is involved twice. First of all, it is involved in the axiom itself. This axiom is nothing more than the expression of the idea that it is possible to add d an indefinite number of times, until we find some natural number k satisfying a ≤ kd. The axiom is used, so to speak, to cover up arithmetical intuition. The second time it is involved in establishing the existence of the number m. To find m we must conceive another sequential procedure. We pursue the sequence 1d, 2d, 3d,…,kd backwards an indefinite number of times until we find the least natural number (which we denote by m + 1) satisfying a ≤ (m + 1)d. Once found, it is clear that md < a also holds. In order to establish the existence of m + 1 we must appeal to the fact that the set of the natural numbers is well-ordered. The Archimedean axiom and the well-ordering principle depend on our ability to imagine certain sequential procedures as being repeated for an unspecified number of times, and therefore, on contentual arithmetic. The proof described here also involves concept-superposition [Überbauung] of concepts: in a first step, we use a sequential procedure to generate k; this is followed by a second sequential procedure which is attached on top of the first, and which leads to m.

41 Hölder concludes his discussion of Hilbert’s proof with the following interesting general statement. If the consistency of arithmetic can only be proved axiomatically, as Hilbert desires, we would then have to prove the consistency of the system of axioms chosen. If such a proof were to be given independently of genetic considerations, we would have to rely on yet another system of axioms in order to achieve our goal. This however would require a new system of axioms. We would thus get an infinite regress [recursus ad infinitum] [Hölder 1924, 325], so that a complete consistency proof could never be accomplished. It could not be accomplished because, in the course of such a proof, we would be forced to define new infinite procedures and based on them introduce new synthetic concepts [synthetische Begriffe] in the sense explained above. The Archimedean axiom is, as far as Hölder is concerned, not an axiom but, since it depends on contentual arithmetic, a definition.18

Conclusion

42 In the previous pages I presented a small sample of Hölder’s reflections concerning mathematical deduction. The most important aspect found is Hölder’s view that certain proofs involve what I have called second-order-counting or meta-counting, something

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that Hölder calls concept-superposition. According to Hölder, therefore, a deductive examination of certain theorems of arithmetic involves self-referential moments. A sharp separation between deduction and counting, of the kind involved in Hilbert’s distinction between mathematics and metamathematics appears therefore, impossible.

43 Niebergall and Schirn point out that there is a complementarity between Hilbert’s treatment of arithmetic, which requires coding numbers by signs on the one hand, and the post-Gödelian tradition which relies on coding signs by numbers, on the other [Niebergall & Schirn 1998, 174]. Hölder recognized that counting cannot be switched off, that is, that formalization would require, in a second step, mathematization. In this sense, Hölder’s investigations of proof do anticipate Gödel’s methods.

44 Hölder’s methodological reflections have a great deal more to offer. I think that Hölder’s methodological remarks belong to the intuitionist tradition. His views are compatible with most of the seven tenets of intuitionist thinking proposed by Tieszen [Tieszen 2005, 228ff.]. A detailed reconstruction of Hölder’s particular brand of intuitionism is still missing.

BIBLIOGRAPHY

BROUWER, LUITZEN EGBERTUS JAN 1913 Intuitionism and formalism, Bulletin of the American Mathematical Society, 20(2), 81–97.

GRASSMANN, ROBERT 1891 Die Zahlenlehre oder Arithmetik streng wissenschaftlich in strenger Formelentwicklung, Stettin: R. Graßmann.

HILBERT, DAVID 1899 Grundlagen der Mathematik, Leipzig: Teubner.

HÖLDER, OTTO 1892 Graßmann, Robert – Die Zahlenlehre oder Arithmetik, streng wissenschaftlich in strenger Formelentwicklung, in Göttingische gelehrte Anzeigen, Stettin: R. Graßmann, vol. 15, 585–595, see translation into English in this volume pp. 57–70. 1899 Anschauung und Denken in der Geometrie, Stuttgart: Teubner, see translation into English in this volume pp. 15–52. 1924 Die mathematische Methode, Berlin: Springer Verlag. 1926 Der angebliche circulus vitiosus und die sogenannte Grundlagenkrise in der Analysis, in Berichte über den Verhandlungen d. sächsischen Akademie der Wissenschaften zu Leipzig, vol. 78, 243– 250.

NIEBERGALL, KARL-GEORG & SCHIRN, MATTHIAS 1998 Hilbert’s finitism and the notion of infinity, in The Philosophy of Mathematics Today, edited by SCHIRN, M., Oxford: Clarendon Press, 271–306.

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RADU, MIRCEA 2003 A debate about the axiomatization of arithmetic: Otto Hölder against Robert Graßmann, Historia Mathematica, 30, 341–377.

TAPP, CHRISTIAN & LÜCK, UWE 2003 Transfinite Schlussweisen in Hilbertschen Konsistenzbeweisen, in Proceedings der GAP.5, Bielefeld 22.–26.09.2003, 49–61. URL www.gap5.de/proceedings/pdf/049-061_tapp_lueck.pdf .

TIESZEN, RICHARD L. 2005 Phenomenology, Logic, and the Philosophy of Mathematics, Cambridge: Cambridge University Press.

WAERDEN, BARTEL L. VAN DER 1939 Nachruf auf Otto Hölder, Mathematische Annalen, 116, 157–165.

NOTES

*. I thank the anonymous reviewers for their remarks that led to important improvements of an earlier version of this paper. 1. Compare [Waerden 1939]. 2. These ideas can be found in a series of works published between 1892 and 1935. 3. One important consequence of this is that, according to Graßmann, a rigorous treatment of mathematics must be given independently of any theory of logical inference. For Graßmann, a rigorous treatment of logic is only possible if logic is developed as a symbolic calculus, that is, as a chapter of mathematics. Hölder agrees with the former claim, but dismisses the latter (see below). 4. Here, I am, of course, only proposing an interpretation of Hölder’s critique of Graßmann. In the 1892 review of Graßmann’s work, Hölder does not explicitly mention this aspect of Leibniz’s conception. He mentions it, however, in 1924, in a discussion of the place and value of symbolic methods in developing the logic of relations. Hölder calls Leibniz’s idea of a Characteristica Universalis inadequate [Hölder 1924, 274]. 5. Hölder died in August 1937. Van der Waerden’s Nachruf was received by the Annalen in May 1938. It was published in 1939. 6. If I am not mistaken, a detailed comparison between the philosophical ideas of Hölder and those of Gödel is still missing. 7. Arguments of a similar kind are discussed below. 8. Remarks such as these are typical for Hölder’s time. There is nothing original or surprising about them. This also holds for much of Hölder’s detailed discussion of the limits of syllogistic logic and of the virtues of the logic of relations given in the 12th chapter of his book. The 12th chapter has the title “Preliminary general logical remarks” [Hölder 1924, 247–277]. The chapter is, of course, important for a reconstruction of the precise terms in which Hölder views this subject. Such a reconstruction, however, goes beyond the purposes of the present paper. 9. Hölder returns to Couturat’s position later in the book. He criticizes Couturat for regarding the symbolic treatment of logic as a necessary condition for logical completeness. Hölder writes: “One must take into account the fact that each (...) sign requires a rule describing its use; this rule, however, cannot be explained by means of a new symbolic expression, but rather based on the meaning [Bedeutung] of the sign alone” [Hölder 1924, 177]. In a footnote to this comment Hölder also mentions Brouwer’s paper “Intuitionism and Formalism” [Brouwer 1913]. Hölder

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emphasizes that from his own point of view, when it comes to the foundations of mathematics, formalism is an inappropriate conception. 10. All the translations of Hölder’s texts are mine. 11. Hölder quotes, for instance, the views of Moriz Schlick [Hölder 1924, 374-397] and of Ernst Mach [Hölder 1924, 42–45; 464ff.]. I cannot discuss the details of Hölder’s conception of the foundations of natural science here. 12. First construct a line segment. Choose one of its endpoints. Draw a perpendicular line to the segment through the point chosen, etc. 13. In this context it is perhaps interesting to note that Hölder also criticizes Hermann Hankel’s use of the permanence principle in similar terms [Hölder 1924, 210–211]. 14. For similar discussions of other proofs compare [Hölder 1924, 313ff.; 503–510; 511–515]. 15. The variables m and n stand for natural numbers. The letters a and d stand for arbitrary magnitudes. 16. Hölder does discuss the place of the induction-axiom in mathematics and argues that inductive proofs require the contentual [inhaltlich] use of arithmetical intuition. He does this independently of his reconstruction of Hilbert’s proof discussed here [Hölder 1924, 313ff.]. For a very good discussion of the distinction between formal and contentual induction in Hilbert’s work and of the difficulties linked with Hilbert’s concept of finite concistency-proofs, compare [Tapp & Lück 2003]. Tapp and Lück consider Poincaré’s critique of Hilbert and some of Skolem’s ideas on the debate, but they do not mention Hölder. 17. Hölder explores a wide range of examples of this kind, independently of Hilbert’s conception, in his detailed discussion of arithmetic [Hölder 1924, 161–198]. In particular, he claims that the commutative law for the multiplication of the natural numbers already involves what I have called second order counting and thus involves concept-superposition. 18. Hölder also points out that E. Study and L. E. J. Brouwer share similar views [Hölder 1924, 325, footnotes 1 and 3].

ABSTRACTS

In this paper I provide a brief reconstruction of Otto Hölder’s conception of proof. My reconstruction focuses on Hölder’s critical assessment of David Hilbert’s account of axiomatics in general, and of Hilbert’s conception of metamathematics in particular. I argue that Hölder’s analysis of Hilbert’s general methodological ideas and, more importantly, Hölder’s analysis of the logical structure of the proofs provided by Hilbert in his Grundlagen der Geometrie of 1899 are helpful in reaching a clearer understanding of van der Waerden’s claim linking Hölder’s conception of proof to the tradition established by Kurt Gödel.

L’article présente une reconstruction brève de la conception de preuve développée par Otto Hölder. La reconstruction se concentre sur la critique de la conception axiomatique de David Hilbert en général et sur sa conception des métamathématiques en particulier. On affirme que l’analyse faite par Hölder des idées méthodologiques générales de Hilbert et surtout de la structure logique des preuves fournie par Hilbert dans les Grundlagen der Geometrie (1899) est très utile pour comprendre plus clairement la thèse de van der Waerden affirmant le lien entre la conception de la preuve développée par Hölder et la tradition établie par Kurt Gödel.

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AUTHOR

MIRCEA RADU Universität Bielefeld (Germany)

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Geometry and Measurement in Otto Hölder’s Epistemology*

Paola Cantù

1 Introduction

Otto Hölder is known for his results on the theory of groups (see [Gray 1994], [Schlimm 2008]) and for the paper on the axiomatic theory of measurement published in 1901 under the title “The Axioms of Quantity and the Theory of Measurement” (see [Diez 1997], [Darrigol 2003], [Michell 1993]). Less investigated are the epistemological considerations on the relations between geometry, arithmetic and theory of measurement that Hölder developed before and after this paper: most of them are contained either in the paper Intuition and Reasoning [Hölder 1900] or in the volume The Mathematical Method [Hölder 1924]. 1 Further remarks on indirect proofs (see [Hölder 1929], [Hölder 1930]), fallacies of circularity in mathematics (see [Hölder 1926]), the axiomatization of mathematics (see [Hölder 1931], [Radu 2003]), and the paradoxes are contained in several minor articles published between 1926 and 1931.

1.1 Continuity, deduction and axiomatics

Hölder’s epistemology is centered on the investigation of the notion of deduction: its relation to geometry was first investigated in Intuition and Reasoning [Hölder 1900] and further developed in The Mathematical Method [Hölder 1924], which is explicitly presented as a critical reaction to Russell, Couturat and Natorp. The paper investigates some historical and philosophical questions about the distinction between geometry and arithmetic, deduction and experience, synthetic and given. 1) Does Hölder endorse the Grassmannian distinction between formal and empirical sciences and agree with Gauß and Helmholtz about the empirical nature of geometry? 2) How does Hölder combine a rather empirical conception of geometry with the Kantian terminology (‘synthetic’ opposed to ‘given’) used to describe it? 3) Are the primitive concepts of the

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theory of measurable magnitudes given or constructed? What is the status of the Archimedean axiom? Hölder’s approach was devoted to the investigation of the possible relations between deduction and experience—a classical epistemological problem that he investigated in an original perspective, i.e., on the basis of a case by case analysis of the use of deduction in several mathematical disciplines: geometry, mechanics, measurement theory, the theory of continuum, analytical geometry, the theory of manifolds, the theory of integration and differentiation, the theory of real and imaginary numbers, and the theory of prime numbers. In modern terminology, one could say that Hölder was particularly interested in mathematical practice, believing that the logical investigation of the mathematical method should not precede but follow the analysis of mathematical theories and examples, and should not limit itself to number theory, but include all relevant mathematical disciplines [Hölder 1924, 1–3]. This case by case investigation of mathematics, developed during the First World War and published with minor changes more than ten years later, is based on the idea that logic might be field-dependent, i.e., that a general logic might only arise from the investigation of the deductive procedures used in specific mathematical disciplines.2 Hölder, unlike most of his contemporaries, claimed that if logic is the science of deduction, then it cannot itself be built deductively [Hölder 1924, 2–3, 277], and that symbolic language, although useful to express mathematics, is not as useful in logic, where new symbols and new concepts are often introduced, requiring at each time new linguistic clarifications and comments [Hölder 1924, 5]. The ordinary language, thanks to its tonality and suggestive power, was thus praised as the language best suited to investigate deduction [Hölder 1924, 7–8]. According to Hölder, an essential characteristic of mathematics is that it allows the construction of concepts: the aim of his logic of mathematics is thus the investigation of different ways in which concepts might be constructed (as pure synthetic concepts in arithmetic and as hypothetico-synthetic concepts in geometry). The notion of deduction is related to the mathematical activity of construction of concepts, and thus to the Kantian perspective, which is the object of an internal criticism. The main question is still Kantian: “how can pure mathematics be possible?”, but the border between a priori and non a priori knowledge is differently drawn: geometry does not belong to pure mathematics [Hölder 1924, 5–8]. If geometry is a science whose primitive axioms have their origin in experience, and measurement theory, like geometry, “can be based upon a set of facts”, why does the latter include among its axioms the postulate of Dedekind’s continuity? The latter is a consequence of the commonly accepted axiom of Archimedes, but in the 1901 paper, Hölder had developed an original example of a non-Archimedean continuum (containing infinitely great and small quantities): why did he not further develop this intuition? To understand Hölder’s axiomatic choices, a deep understanding of the peculiarity of his epistemological framework is no less useful than a clear understanding of the technical differences between his model and other contemporary approaches to non-Archimedean continuity by Bettazzi and Veronese.

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1.2 Plan of the paper

In section 2, I will briefly sketch Hölder’s conception of ‘given’ concepts in geometry. Hölder’s epistemology was deeply influenced by the debate on the relationship between geometry and experience raised by Gauß, Graßmann and Helmholtz, but also by some remarks by Konrad Zindler, a disciple of Meinong. Hölder’s epistemology can be better understood in the light of his interest for deduction and for case by case analysis of mathematical practice, which led him to oppose logicism as well as Kantianism, but also to discuss some limits of empiricism. In section 3, I will discuss whether Hölder might be considered as a follower of Kant’s philosophy of mathematics and at the same time a supporter of Pasch’s empiricism. In particular, I will focus my attention on the reasons why Hölder used a Kantian terminology (‘synthetic’ as opposed to ‘given’) even if he criticized Kant’s conception. I will claim that in around 1900 Hölder was surely nearer to Pasch’s empirical understanding of mathematics and to Helmholtz’s epistemology rather than to Kant’s philosophy, even if he did not consider the Kantian position as provably wrong, and adopted a terminology that became more Kantian in his later work (see for example the distinction ‘given’–‘constructed’ being substituted by the distinction ‘pure synthetic’–‘hypothetico-synthetic’). In section 4, I will discuss the relations between this idea of ‘given’ and Hölder’s conception of continuity, which is at the base of his theory of measurement. Hölder’s theory of measurable magnitudes is quite different from the theories developed by Veronese, Bettazzi and Hilbert: it is Archimedean, and even Dedekind-continuous. I will discuss the role of the Archimedean axiom and of Dedekind’s postulate with respect to experience. In section 5, I will outline some complex relations between Hölder’s epistemology, as it emerges from the reconstruction made in section 2 and section 3, and the conceptions of other 19th century mathematicians and philosophers. The originality of Hölder’s epistemological approach to geometry and measurement will thus appear more clearly.

2 A demarcation between arithmetic and geometry

Hölder’s philosophy of mathematics is based on the “traditional” idea that there is a fundamental difference between arithmetic and geometry. This idea is traditional inasmuch as it was introduced by the Greeks and maintained unaltered up to late modernity, when the development of algebra began to outline correspondences and similarities between discrete and continuous quantities. The 19th century, from Gauß to Helmholtz, was the time for the introduction of a new distinction, no longer based on a qualitative distinction between the two disciplines—the former being the science of (discrete) plurality and the second the science of (continuous) magnitude—but rather on their relationship to experience. The former is a pure science based on definitions, while the latter is a natural science (it is part of physics) and based on axioms. Once the distinction is related not to the kind of objects the sciences are applied to, but to a difference in the origin of their concepts and in their formulation, the investigation of the relations between intuition, experience and mathematical rigor in geometrical deductions becomes the most relevant epistemological question. Hölder introduced a shift, separating the question of the origin of the concepts, which he

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considered as a question that pertains to philosophy of mathematics, from the question of the demarcation between arithmetic and geometry. The latter is essentially a mathematical question and can be answered by a thorough consideration of several examples of primitive concepts with respect to the activity of construction needed to introduce them.

2.1 What is ‘given’?

The idea that geometrical objects and properties are ‘given’ and not constructed by our thought goes back to Gauß and Graßmann. It is well known that Gauß remarked that space also has a reality outside our mind, and that we cannot describe a priori the laws of that reality, at least not entirely a priori.3 Graßmann distinguished formal and real sciences, according to the fact that they investigate a being that is posited from thought itself or that faces thought as something autonomous.4 Given objects are objects that confront our thought and whose properties are independent from our mental activities. This conception of ‘given’ as some physical reality that exists independently and autonomously from our thought is often associated in the 19th century to the idea that geometry is an empirical science, and more precisely a part of physics (Gauß, Graßmann, Helmholtz) or at most a mixed science, i.e., partly empirical and partly formal (Veronese). In Intuition and Reasoning Hölder held a view that does not entirely agree with Gauß’s perspective: 1) firstly, he used the term ‘given’ in a different sense: this results in the fact that not all geometrical concepts are given; 2) secondly, even when he discussed the philosophical question of the origin of ‘given’, he remarked that it might originate either in experience or in intuition, and did not explicitly refute the Kantian conception.

2.1.1 Three meanings of the term ‘given’

The term ‘given’ occurs very often in Intuition and Reasoning, but its meaning is not always clear. Sometimes it refers to concepts that “are available to the geometer as a given material”,5 sometimes it refers to concepts that are not “defined by means of a construction”,6 sometimes to concepts that are “given within intuition”,7 sometimes to concepts that are not constructed but “taken from intuition or experience”.8 To understand Hölder’s remarks, I will first distinguish between three possible meanings of the term ‘given’: 1) epistemological , 2) axiomatic, 3) genetic. 1. According to the epistemological meaning, something ‘given’ is something that is epistemologically accessible via experience or intuition, i.e., by means of our senses—be they exterior or inner. In Intuition and reasoning, Hölder remarked that what is given in geometry might be either given empirically or given within intuition, but he rather preferred the first option, which guarantees more freedom—claiming that philosophers who consider geometrical concepts as given within intuition tend on the contrary to criticize the choice of non intuitive concepts as primitive concepts [Hölder 1900, 11, Eng. trans. 20] There is no one-to-one correspondence between ‘given’ and primitive concepts, because some primitive concepts might be constructed rather than ‘given’ in intuition or in experience. Moreover, some “concepts which were clearly immediately abstracted from intuition, can later be deduced” [Hölder 1900, 15, Eng. trans. 23].9 This notion of ‘given’ is an epistemological distinction that is relative to the state of our knowledge; it is thus different from the logical notion of primitiveness used in modern axiomatics.10

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2. According to the axiomatic meaning, ‘given’ refers to concepts that occur in the primitive propositions of a mathematical theory, whatever the epistemological access we might have to them, or whatever the kind of construction that might be used to explain them. In this case, all the concepts of an axiomatic theory would be ‘given’, but several concepts that are ‘given’ in a theory might not be ‘given’ in a different axiomatic theory.11 The distinction between given and constructed would thus be reduced to the opposition between primitive and derived, and would of course make sense only inside an axiomatic framework. This distinction is relative (to a given axiomatic theory). Yet, the term ‘given’ does not occur in this meaning in the paper on measurable magnitudes, where Hölder builds an axiomatic theory: on the contrary, ‘given’ is used there only in the technical sense of ‘data’ in a mathematical proof. 3. According to the genetic meaning, a ‘given’ is something whose properties can be determined without any activity, i.e., without executing any operation. According to this meaning, geometrical concepts are partly ‘given’ and partly not. This distinction is best developed in The Mathematical Method, but is already present in Intuition and Reasoning, especially where Hölder makes reference to a paper by Konrad Zindler. My claim will be that Hölder’s use of the term ‘given’ can be understood according to the third sense both in Intuition and Reasoning and in The Mathematical Method, where he discussed mathematical questions such as the difference between arithmetic and geometry, or the nature of primitive propositions in mathematics (see sections 2.1.2 and 2.2). The term is on the contrary used in the epistemological meaning when Hölder makes reference to the philosophical discussion about the genesis of geometrical knowledge (see section 3).

2.1.2 Given versus constructed

The distinction between given and constructed concepts was influenced, as Hölder himself declared, by an article published in 1889 by a student of Meinong: Konrad Zindler, who claimed that geometry takes representations of space as given and does not investigate their origin.12 Zindler believed that the problem of the correspondence between geometrical truths and the external world is a question raised by physics that does not concern geometry.13 Although Hölder mentioned Zindler in several parts of Intuition and Reasoning, he did not explicitly mention Zindler’s distinction between geometrical objects that are taken as given and arithmetical concepts that are not.14 Yet this division is quite similar to Hölder’s own distinction between concepts that are given and concepts that are constructed: in both cases axiomatic concepts are opposed to concepts that occur in the theory of numbers. According to Zindler, in arithmetic there is just one determinate primitive concept from which all others can be considered as developed, whereas in geometry there are several concepts that cannot be inter-defined, such as direction, point, distance, coincidence of figures, area, volume, and space [Zindler 1889, 1–3].15 Hölder, like Zindler, believed geometrical primitive concepts to be axiomatic inasmuch as they are taken as given, but unlike him he distinguished two kinds of axiomatic concepts: concepts that denote objects and concepts that denote relations [Hölder 1901, 3n, Eng. trans., I, 247]. Zindler, on the contrary, had claimed that all primitive concepts of geometry denote comparative relations, quoting Meinong’s article [Meinong 1882] on the theory of relations for the study of compatibility and incompatibility between relations.16

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2.2 The demarcation between geometry and arithmetic

The distinction between given and constructed is explicitly used in The Mathematical Method to demarcate arithmetic from geometry. The difference relies on the fact that arithmetic concerns only constructed, or—as Hölder called them in The Mathematical Method—synthetic concepts, i.e., concepts that have their source in some activity, whereas geometry concerns both constructed (synthetic) and given concepts.17 The examples of given concepts mentioned in 1924 include those mentioned in 1899, i.e., point, line and plane, which are concepts denoting objects, but also concepts that denote relations, such as the relation of betweenness. As an example of a concept that is on the contrary synthetic (and precisely hypothetico-synthetic), Hölder mentions a line segment, because it can be defined as a couple of points [Hölder 1924, 10–11]. Another example of a constructed concept is the square, which can be constructed from the concepts of segment and angle. More generally, all geometrical constructed (hypothetico-synthetic) concepts are obtained by executing a series of operations in a given order: the possibility of executing those operations in the mentioned order is guaranteed by the axioms [Hölder 1924, 10–11]. Synthetic concepts, like given concepts, might denote an object or a relation: for example, the square denotes an object, whereas the property of a plane figure of being obtained by a—perpendicular, parallel or central—projection from another figure is a relation [Hölder 1924, 12]. This insistence on the difference between concepts of objects and concepts of relations is important a) to determine the extent of Meinong’s influence, b) to explain the tension between Hölder’s own conception of geometry and the modern axiomatic conception, and c) to understand how Hölder managed to demarcate arithmetic from geometry: the main difference between the two relies exactly on the fact that the latter contains some concepts that cannot be defined synthetically.18 In arithmetic one considers only things that correspond to our activity together with their relations: mathematical concepts are thus all constructed, but not from given concepts, so they are called ‘pure synthetic concepts’: examples are numbers, but also groups and permutations.19 In arithmetic there are no given objects, so the construction cannot be hypothetico-synthetic, and there cannot be any axioms.20 Even if in arithmetic there are no ‘given’ concepts, there are of course primitive concepts that are used to build other concepts, as happens in the arithmetical activities of composing, removing, assembling and separating that generate concepts of higher order, wholes composed of parts [Hölder 1924, 295–296]. Hölder explicitly compared his pure synthetic concepts with Meinong’s objects of higher order, considered as wholes composed of parts and their relations: e.g., a rectangular triangle whose internal surface is green. But Hölder remarked upon a relevant difference: according to Meinong, the parts and the whole derive their meaning from external experience, while this holds, according to Hölder, only for geometrical concepts [Hölder 1924, 296]. Hölder added that Meinong’s distinction between Zusammenstellung and Zusammensetzung could be interpreted as relying on an opposition between two distinct kinds of operations: on the one hand, to add and to remove, to bring together and to pull apart, to order in a sequence and to assign; on the other hand, to superpose segments, to join bodies, and to apply forces [Hölder 1924, 296–297]. This is interesting,

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because the operations mentioned with respect to Meinong’s Zusammenstellung are the same that occur in the examples of pure synthetic concepts: to order in a sequence is the basic activity for natural numbers taken as positional numbers (Stellenzeichen), whereas the relation of assigning is the activity on which the notion of cardinal number (Anzahl) is based [Hölder 1924, 161ff.].

2.2.1 Hölder and modern axiomatics

Modern axiomatics is based on the idea that different primitives can be used in the axiomatization of the same theory. Furthermore, the properties of the primitives might be altered and different systems obtained. How are these two things possible, if geometry has to begin from ‘given’ concepts, implying that these concepts cannot be altered at will? Hölder’s belief that among the given concepts of geometry there are not only relational concepts, such as betweenness, containment and congruence, but also primitive concepts denoting objects, illustrates a relevant difference between Hölder and Hilbert. With this respect, Hölder is nearer to Veronese than to Hilbert, because he considered certain geometrical data as non alterable, as concepts that cannot be different from what they are taken to be in our representation of space. This might be a reason why he did not consider a non-Archimedean theory of magnitudes, even if he had developed a non-Archimedean model of it (see further section 4). Yet, sometimes Hölder seemed to admit that one might substitute a derived concept for a given one in a system. As a result of a modification of the whole system of geometry it might anyway happen that afterwards a synthetic concept also occurs in place of a concept that was previously treated as given. [Hölder 1924, 12] Does Hölder mean here that given concepts can also be presented as synthetic concepts, provided the axiomatic formulation of the theory is modified? The above passage seems to introduce a tension between the genetic meaning of given used in the previously mentioned passages, and the axiomatic meaning that seems to be used here. Some remarks on Hölder’s conception of the axiomatic method might be helpful to clarify this issue. The preference for the Euclidean and the Kantian use of deduction and the suspiciousness towards the modern construction of “connected chains of inferences” [Hölder 1924, 2] might be related to the fact that an axiomatic system cannot itself describe the content of axiomatic systems having different axioms. This is one reason why deduction cannot be used to study deduction itself. In particular, symbolical logic cannot establish the relation between different axiom systems and experience. Van der Waerden has suggested a Gödelian interpretation of the above-mentioned passage, claiming that Hölder’s mistrust in the possibility of formalizing the whole of mathematics relied on the idea that the concepts and inferences developed at one stage of mathematics immediately underwent a logical analysis (e.g., counting the steps of the deduction), and thereby became themselves objects of investigation [van der Waerden 1938, 161]. This interpretation, although developed in order to show that Hölder had somehow foreseen Gödel’s critical results, might also be used as an explanation of the problem mentioned above: how is it possible that concepts based on some activity assume the role of ‘given’? Concepts of higher order that are evidently derived by means of

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inferences, might themselves become objects of investigation at some higher level (that of logical analysis, for example), and thus play the role of ‘given’: so, for example, something that is obtained by a first-level activity might itself become a ‘given’ in metamathematical investigations, where the kind of activity involved in the construction of synthetic concepts is of a higher kind.

3 The origin of ‘given’: Kantian intuition vs. empirical experience

Paul du Bois-Reymond in the introduction to his The General Theory of Functions [du Bois- Reymond 1882, 138] had referred to the debate on the foundations of mathematics as an opposition between two positions: the idealists, admitting the existence of limits, and the empiricists, trying to limit mathematics to the realm of finite experience. Hölder—attesting thus his familiarity (if not affiliation) with the Kantian tradition— suggested that the fundamental question in philosophy of mathematics is the relation between intuition and deduction. On the one hand Hölder discussed the role of intuition in proofs, as I will briefly outline in section 3.1. On the other hand he discussed the role of intuition in the origin of given geometrical concepts: are they derived from a priori intuition (Kant) or taken from experience (Helmholtz, but also Lotze, Baumann, Volkelt)?21 In the first case, intuition is “considered as the source of geometrical knowledge”, whereas in the second case it “appears, conversely, as a result of experience” from which geometrical knowledge is derived [Hölder 1900, 3, Eng. trans. as a result of experience]. This question, together with its effects on Hölder’s stance on apriorism and empiricism, will be discussed in section 3.2.

3.1 Intuition and deduction

Otto Hölder’s fundamental aim in Intuition and Reasoning, as well as in The Mathematical Method was precisely the analysis of mathematical deduction. He claimed that intuition plays no role in deduction, or rather, as Pasch had rather put it in the Lectures on Modern Geometry (1882), should not play any role in a complete and reliable deductive process.22 Mathematical rigor is Pasch’s main aim: intuition should be eliminated from deduction but its presence in the axioms might be accepted, because it is harmless or even precious in order to understand the empirical origin of mathematics. Yet Hölder raises the question once more and in an explicit way, against those philosophers (i.e., Kant) who believe that intuition might play a role in deduction too. Now, what role does intuition play in deduction itself? Are the elements being combined in the deduction, and the rules according to which they are combined, all what is taken from intuition, or respectively from experience? Or does intuition co- operate also in the single steps of deduction? The latter point of view has been mainly defended by philosophers and seems to have been also Kant’s point of view, as he saw in intuition the essential principle for obtaining geometrical knowledge. [Hölder 1900, 11, Eng. trans. 20] To show that intuition does not necessarily occur in geometrical deduction, Hölder suggests a procedure that differs from the one followed by Pasch:23 the best way to make Euclid’s implicit appeals to intuition explicit, is to transform geometry into a calculus, as Peano did in his Geometrical Calculus [Peano 1888]. 24 Since a purely logical proof in geometry is possible—even if intuition plays a role in the kind of reasoning

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that leads effectively to the discovery of a certain result [Hölder 1900, 15, Eng. trans. 23]—it is useless to assume an a priori intuition of space in order to explain geometrical proofs (the Kantian thesis is not necessary for the understanding of deduction).25 Against the usual objection that geometrical deduction is based on intuition because it makes recourse to auxiliary figures, Hölder argued that this procedure corresponds to the introduction of appropriate existential propositions. To the objection that in certain proofs the geometer reasons by analogy with objects of intuition, Hölder answered that such proofs can be rigorously formulated without any recourse to analogy [Hölder 1900, 10–13, Eng. trans. 20–21].

3.2 A partial criticism to Kant

Even if he suggested several arguments in defense of the empirical origin of geometrical objects, in the 1899 lecture Hölder did not attempt to advance a definitive argument against the Kantian interpretation. There is hardly any doubt that Hölder shared an empirical conception of geometry. His remarks on the origin of geometry from raw experience and the subsequent process of idealization26 do not substantially differ from Pasch’s remarks on the origin of geometrical objects.27 Yet, Hölder did not explicitly adhere to a specific version of empiricism. Although strongly influenced by Helmholtz, as is particularly evident both in the 1901 and in the 1924 essays, Hölder considered two possible ways to explain Pasch’s idea that geometrical objects derive from experience: either they derive from immediate perceptions (Helmholtz) or they derive from memories of perceptions (Baumann, Wundt) [Hölder 1900, 3, Eng. trans. 28]. Furthermore, Hölder’s arguments offered a general defense of empiricism rather than a defense of Helmholtz’s approach. If Hölder evidently shared an empiricist account of geometry, why was he so cautious about the Kantian conception? Why did he not refute it? There are probably some practical reasons: Hölder was presenting the state of the matter concerning the foundational problem of geometry rather than imposing his own point of view with definitive arguments—a reasonable thing to do in an inaugural lecture. This might also explain why he did not defend just one version of empiricism, but discussed the differences between the appeal to immediate perceptions and the explanation based on remembered perceptions. But there might also be methodological and epistemological reasons, as I will claim in the following section.

3.2.1 Empiricism is consistent and preferable

Firstly, Hölder considered himself as a working mathematician that investigated the theory of knowledge of his own discipline rather than as a professional philosopher [Hölder 1924, 1], and thus did not venture in a confutation of a philosophical position.28 He contented himself with arguments showing that empiricism was consistent and preferable. That Hölder held a clear preference for the empiricist viewpoint, is clearly attested by the fact that he put forward some arguments to show that the empirical origin of geometry might be legitimate or even preferred to the idea that the properties of geometrical objects are determined by an a priori intuition. The arguments are not presented as a confutation of Kant, but rather aimed at defending the legitimacy, if not the preferability, of alternative conceptions.

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Secondly, Hölder refuted the objection that the empirical origin of geometrical concepts might be based on a fallacy of circularity: as Helmholtz had shown,29 there are some observations—facts of immediate perception—“that are not themselves grounded in geometrical considerations” [Hölder 1900, 4, Eng. trans. 17]. For example, one might derive the geometrical notion of a straight line from some physical observations concerning the line of sight or a tensioned thread, and the distinction between a plastic and a rigid body from the observation that after manipulation of a given material, it might be more or less easy to recover the original visual representation of the object [Hölder 1900, 5–6, Eng. trans. 16–17]. Hölder also considered the fact that the properties of rigid bodies might be determined by a set of subsequent measurements and the geometrical theorems might be the result of induction, but he acknowledged that one cannot thereby exclude the possibility that the Kantian notion of a priori intuition might be compatible with that explanation. Yet, in favour of the empirical interpretation, he claimed that the Kantian recourse to a general notion of a priori intuition • is more “artificial”, • is not strictly necessary to solve the problem, and in fact it is combined with some empirical understanding of the phenomena, and • does not give an equally detailed explanation.30 Hölder admitted that the Kantian conception could not be refuted, but he was confident that future results in the history of mathematics, psychology and physiology might furnish further arguments in favor of the empirical interpretation. The history of mathematics might prove helpful, because a clarification of the origin of geometrical concepts in the Egyptian culture might prove that they originally had a practical aim. The physiological analysis of the spatial properties of objects might suggest that immediate perceptions need to be decomposed, so that there cannot be a unique a priori intuition but different forms of intuition depending on vision, tact, muscular sensations, and movement [Hölder 1900, 7, Eng. trans. 18]. This section anticipated some remarks that can be found in the fourth chapter of Poincaré’s Science and Hypothesis [Poincaré 1902], but which were already in the essay on Geometry and Space [Poincaré 1895].31 Rather, Hölder’s remarks are inspired by Helmholtz’s writings and Klaus Volkelt’s Experience and Reasoning [Volkelt 1886].32 A further argument advanced by Hölder was a reasoning to the best explanation: empiricism can better explain human’s capacity to localize an object in a determinate point of space. Quoting Lotze [Lotze 1884, 547ff.], Hölder argued that the reference to individual perceptions can better explain the human capacity to locate an object in a singular point of space.33 A general a priori intuition that is assumed as a necessary condition of mathematical judgement either cannot do that or has to be interpreted as merely introducing an order between the empirical data, “but then we would basically be back to the empirical point of view” [Hölder 1900, 17n, Eng. trans. 32]. In 1899, Hölder believed that it cannot be definitively established that geometrical intuition derives from experience, because an a priori intuition might also explain our knowledge of spatial objects. Yet, deduction is not grounded in intuition, so, whatever the philosophical debate on the origin of geometrical concepts, it should have no relevance for the study of mathematical deduction [Hölder 1900, 8, Eng. trans. 19].

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3.2.2 Kant’s thesis can neither be proved nor refuted

Secondly, Hölder considered Kantian geometrical apriorism as a hypothesis that is non provable and non refutable, and as such has poor explanatory power. In The Mathematical Method Hölder advanced the same argument in defense of empiricism that he had already suggested in Intuition and Reasoning (see above, section 3.2.1): the Kantian objection to Helmholtz’s theory does not hold because there is no fallacy of vicious circle. But then he added: I would only like [...] to remark here that this [Kantian] conception seems to be neither strictly provable nor strictly refutable, and not even directly fruitful, because of its abstract nature.[Hölder 1924, 368] The same remark is repeated in § 132, where Hölder discusses criticism towards Kantian apriorism: the only way to defend a priori intuition is to assume a platonic idea of space as a non provable hypothesis.34 Hölder’s answer to this rhetorical question had already been given in Intuition and Reasoning: the Kantian hypothesis does not have any explanatory power, because it gets rid—too easily—of several relevant problems. It seems to me that the empiricist view already has an advantage in the fact that it allows a detailed explanation of intuition, while the hypothesis assumed by the follower of Kant cuts off all the rest. Indeed, it is always an advantage to set up new problems. [Hölder 1900, 6–7, Eng. trans. to set up new problems]

3.2.3 The Kantian epistemological project is still viable and fruitful

Thirdly, even if he did not accept the role attributed by Kant to intuition in geometrical judgements, Hölder still considered Kant’s fundamental question “how is pure mathematics possible” as a fruitful guideline for the epistemological and logical investigation of mathematics [Hölder 1924, 3]. In The Mathematical Method, Hölder was involved in a truly epistemological project and characterized it as an answer to Kant’s question: “How is pure mathematics possible?”, which can be divided, in the case of geometry, into two different questions: 1. How can geometry be deductively built from its premises? and 2. Where do the axioms come from? The two problems identified by Hölder in 1899 and analyzed with respect to the different role of intuition—the origin of geometrical concepts and the investigation of mathematical deduction—are here presented as two aspects of Kant’s foundational program. Furthermore, Hölder’s terminology is even more Kantian, because the expression ‘synthetic’ is here preferred to ‘constructed’.

3.2.4 Kant was right about arithmetic

Finally, Hölder agreed with Kant’s solution as far as arithmetic is concerned. In particular, Hölder preferred Kantian apriorism to the extreme consequences of an empiricist point of view, as for example the application of psychologism to arithmetic, and the abandonment of the distinction between a priori and a posteriori knowledge. The acceptance of an empirical understanding of the origin of geometrical axioms does not imply—according to Hölder—any change in the Kantian and Platonic understanding of arithmetic as a necessary and unconditioned kind of knowledge.35 For this reason,

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Hölder did not share the position of the empiricists who claim that arithmetic also has an empirical origin: I am very far from speaking in favor of an extended empiricism, for example a psychological conception that might transform the laws of reasoning into habits of reasoning. [Hölder 1924, 7] And for the same reason Hölder defended the distinction between a priori and a posteriori knowledge.36

4 The case of measurable magnitudes

The previous remarks about Hölder’s conception of geometry and ‘given’ concepts are necessary to understand Hölder’s approach to the theory of measurement, because the latter is taken to be a science that has axioms, like geometry or mechanics [Hölder 1901, 3, Eng. trans. I, 237]. The axioms express three properties of order—trichotomy, density, and Dedekind’s continuity—and three properties of addition—positivity, solvability and associativity. If the theory of magnitudes contains axioms about ‘given’ concepts, why is Dedekind’s continuity included among them? In the following, I will try to understand why Hölder considered Dedekind’s continuity an indispensable property of measurable magnitudes, and what role was played in the choice of the axioms for continuity both by the philosophical understanding of geometry delineated in section 2 and by some new mathematical results by Hölder himself. This might help us in understanding why he included such a strong axiom as Dedekind’s postulate—which can be derived from the axiom of Archimedes, together with the other axioms for measurable magnitudes—notwithstanding his knowledge of Veronese’s non-Archimedean theory and his own development of an original non- Archimedean model.

4.1 Theory of measurement and geometry

Like geometry and mechanics, the theory of measurement is also based—according to Hölder—on axioms. It is a different matter with geometry and mechanics, where certain axioms based upon sensory experience (or, as some want it, upon intuition) are presupposed. As with geometry and mechanics, the theory of measurable magnitudes can be based upon a set of facts which I will call “axioms of magnitude” or “axioms of quantity.” [Hölder 1901, Eng. trans. I, 237] Hölder did not give any argument to prove the analogy. The latter cannot be based on the fact that the theory of magnitudes shares all axioms with geometry, because Hölder claimed that the axioms of measurement do not occur as such in geometry. Even if the axioms of the theory of measurement hold for the comparison and the addition of straight lines and surfaces, they can be applied to geometry only provided certain further axioms are also assumed: “Special axioms must be formulated again for geometrical applications” [Hölder 1901, Eng. trans. II, 346]. For example, the axioms of the theory of measurable quantities can be applied to equalities between figures, only if the following fact is also assumed, that two lines can be compared, and be either equal or unequal.

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On the contrary, the axioms of the theory of measurable magnitudes satisfied by line segments follow from purely geometrical axioms for points and segments on a line. [Hölder 1901, Eng. trans. I, 237] In particular, the different assumptions that need to be made in the case of lines and in the case of plane figures explain the difference between the equality of lines, that is taken as ‘primary’, and the equality of content of figures, that is taken as ‘constructible’.37 Notwithstanding these differences, the relationship between geometry and theory of measurement is yet so close, that Hölder claimed their relative consistency.38 The strict relationship to geometry is also a reason why Hölder’s axiomatic system was based on the operation of addition.39

4.2 The continuity axiom

According to Hölder, the origin of the axioms of the theory of measurement is similar to the genesis of the axioms of geometry: they are all derived from experience. We are thus justified in believing they will not lead to contradiction. This cannot of course be proved by experience, but the empirical genesis and the empirical fruitfulness in applications are considered by Hölder as reliable reasons for the choice to include an axiom into the system (see section 2.2 above and section 4.3 below). When Hölder published his essay on measurement, there were several other texts discussing the same topic: two of them, written by Giuseppe Veronese and Rodolfo Bettazzi did not include the postulate of Dedekind among the set of axioms that characterize measurable magnitudes. More precisely, neither of them included the axiom of Archimedes, nor, a fortiori, the postulate of Dedekind, which can be derived from the axiom of Archimedes together with the other axioms of the theory of magnitudes.40 Hölder was well acquainted with both essays—he quoted the latter and discussed extensively the former—as well as with the Euclidean theory of proportions, which he declared to be the original nucleus of the theory of measurable magnitudes, containing the axiom of Archimedes as the general form of continuity. So, Hölder must have had some good reasons to include in a theory that declares to have an empirical origin the postulate of Dedekind that cannot be verified empirically, and that was not included in the ancient theory of proportions presented in Euclid’s Elements nor in recent theories of measurement. What were his reasons? The choice of axioms is up to a certain point arbitrary,41 claimed Hölder, yet he had reasons to include Dedekind’s axiom rather than alternative formulations of continuity. Why did he not content himself with the axiom of Archimedes, which is sufficient to compare any two magnitudes? Hölder declared that his approach differed from those of Veronese and Bettazzi, but did not explain what was wrong with their proposals. I suggest that he did not share Veronese’s and Bettazzi’s choice of the axioms because he did not share their epistemological aims, or at least what he took them to be. Hölder interpreted the theory of Veronese as absolutely incompatible with Dedekind’s theory of continuum, and the notion of magnitude developed by Bettazzi as a mathematical formulation of the properties of special kinds of abstract magnitudes rather than a theory that could apply to the usual physical and geometrical magnitudes. I intend only to propose a simple system of axioms from which the properties of the ordinary continuum of magnitudes can be derived; I do not intend to establish special kinds of magnitudes as was done by Bettazzi [...]. [Hölder 1901, Eng. trans. I, 247n]

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Taken as such, Hölder’s argument claims that one should hold to what is usually considered to be a magnitude, and maintain the standard notion of continuity. Hölder’s reasoning seems to be an argument ad populum: let’s conform to what the majority of mathematicians usually do, i.e., to the standard notion of continuity, which was expressed by the Archimedes axiom in Euclid’s time but which is now usually formulated along Dedekind’s lines. If this was the argument, then it would not be so stringent. A brief comparison with Bettazzi, Veronese and Dedekind might help us to clarify Hölder’s point of view.

4.3 Bettazzi’s and Veronese’s approaches

What are the mistakes and the ambiguities that Hölder criticized in recent theories? First of all, Dedekind’s postulate was presented as an arithmetical axiom,42 but according to Hölder the principle used in arithmetic is not an axiom but a definition, whose consistency has to be proved.43 Secondly, a theory of measurement has the principal aim of accounting for physical operations executed on empirical objects: the operation of addition should be in this sense more fundamental than the relation of order. Yet, in the arithmetical analyses of the continuum, the latter is often based on some order relation rather than on addition. But what was wrong with Bettazzi’s and Veronese’s conceptions? Before answering this question, I will briefly summarize the approach developed by Bettazzi and Veronese, who partly shared Hölder’s epistemological aims, but arrived at different conclusions. This will help us to understand that Hölder choice not to follow them was based on epistemological grounds. In Bettazzi’s Theory of Magnitudes [Bettazzi 1890] the properties of order were introduced after the properties of addition, and continuity was taken to be a property of magnitudes and not of numbers. Notwithstanding these similarities, Bettazzi followed an abstract approach based on the construction of structures that did not have an empirical counterpart—a method that Hölder could not accept because it contrasted with his own empirical epistemology. According to Bettazzi’s definition, a class of magnitudes was an Abelian additive semigroup, and could be defined independently from order [Ehrlich 2006], [Cantù 2010]. Different kinds of order relation were taken into account and different abstract classes of measurable magnitudes were introduced: additivity—which suffices, together with a relation of equality or inequality, to determine the notion of a magnitude—was introduced independently from a total order relation, which was considered as necessary to introduce measurement. The order relation was not defined from the outset as Dedekind- continous: on the contrary, different kinds of order allowed measurement, which was defined not only for one-dimensional classes but also for multidimensional classes that can be decomposed into ordered subclasses: in particular, non-Archimedean classes. So, magnitudes were primarily defined as a structure and not by reference to some physical operation that allows us to measure things. In Veronese’s Foundations of Geometry [Veronese 1891], continuity was also taken to be a property of magnitudes rather than of numbers, and geometry was taken to be a mixed science, whose postulates are partly empirical and partly mathematical extensions of empirical facts. This point of view does not seem too far from Hölder’s own epistemological approach: the postulate of continuity is exactly one of these ‘ideal’ or

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purely logical concepts that do not have a counterpart in experience, because it involves the concept of limit. In particular, Veronese believed that Dedekind’s continuity, being defined for numbers and being a part of arithmetic, didn’t need to be the only possible or adequate description of the empirical world. He thus developed a different notion of continuum, discussed in detail by Hölder himself. Veronese’s continuum can be roughly described as a weaker form of continuity that can be geometrically represented by a sheaf of straight lines and that includes Dedekind’s continuity as a special case. Each straight line is Dedekind-continuous, and thus also Archimedean, but the whole sheaf, ordered from bottom to top and from left to right, is not [Cantù 1999, 107ff.]. Veronese’s system is thus a generalization of a Dedekind-continuous subsystem and includes a generalization of the axiom of Archimedes: “for any two magnitudes of the system such that one is smaller than the other, there is a multiple of the smaller that exceeds the bigger”. Only, the notion of multiple is not the usual one, based on natural numbers, but is built on an enlarged set that includes infinitely small and great numbers. Hölder, like most of his contemporaries—with the exception of Hans Hahn [Hahn 1907] and Luitzen Egbertus Jan Brouwer [Brouwer 1907]—overlooked this aspect of Veronese’s theory, but focused only on the fact that Veronese, like Bettazzi, questioned whether the Archimedean axiom might be a necessary feature of magnitudes, and of measurable magnitudes in particular.

4.4 Hölder’s reasons to include Dedekind’s continuity

Hölder admitted that it cannot be verified by our experience that Dedekind’s continuity is an indispensable property of measurable magnitudes, but he considered this assumption as justified on the basis of some empirical experiences and applications. I will have to explain later (§ 99) that the question which one of two concepts ‘precedes’ the other cannot always be definitely answered. But in this case, to defend the Dedekindian axiom of continuity, it would suffice to point out the fact that an axiom gains importance in mathematics by allowing us, together with the other introduced axioms, to dominate a multifaceted factual domain. I will furnish evidence of this in the case of the axiom of continuity; for example, by means of this axiom, significantly extended to time intervals, I will prove that of any two points moving in the same direction on a straight line, the one that happens to have a bigger velocity must overtake the other. [Hölder 1924, 88] Hölder thus advanced pragmatic reasons for the acceptance of Dedekind’s continuity as a property of measurable magnitudes. But why could he not admit a more general and weaker property, allowing Dedekind’s continuity to be, as in the case of Veronese, just a special case of the former? The relevance for the pragmatic applications could have thus been maintained. I suggest that the answer is related both to some significant technical results to be found in Hölder’s 1901 paper and to Hölder’s preference for an empirical conception of geometry discussed in section 3.

4.4.1 Two technical results

As Philip Ehrlich has shown, there are two important results to be found in Hölder’s paper on measurable quantities that concern non-Archimedian continuity: 1) the proof that the axiom of Archimedes can be derived from the postulate of Dedekind’s cuts,

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together with the other axioms of Hölder’s theory of measurable magnitudes, and 2) the construction of a particular model of a non-Archimedian continuum. 1. Hölder showed that the axiom of Archimedes can be derived from the postulate of Dedekind together with the other axioms of measurable magnitudes [Hölder 1901, 10ff., Eng. trans. 248]. Hölder was not the first to prove this result, but he had several predecessors: Pasch, Veronese, Stolz, Bettazzi, Hilbert. Pasch [Pasch 1882, 125–126], in particular, was praised by Hölder as the one who first derived the axiom of Archimedes from a projective formulation of Dedekind’s continuity postulate [Hölder 1924, 89]. Hölder’s proof was already advanced in 1899, and thus before Hilbert’s book on the foundations of geometry, even if it is usually quoted from the 1901 paper [Ehrlich 2006, 59–61]. 2. Hölder himself had developed an original model of a non-Archimedean continuum that differs from those introduced by Veronese, Bettazzi and Hilbert.44 Ehrlich has shown that Hölder’s model of a non-Archimedean continuum differs from the one by Veronese, because it satisfies Veronese’s Principles I-IV but not Veronese’s Principle V, which expresses the generalized version of the Archimedes axiom and allows us to consider Dedekind’s continuity as a special case of the non-Archimedian continuum [Ehrlich 2006, 97].45 Unlike Veronese’s model and like Du Bois-Reymond’s theory, the system developed by Hölder satisfies divisibility [Ehrlich 2006, 94], which is relevant in physical measurements, because it guarantees the divisibility of a magnitude in submultiples. Hölder rightly assumed the counternominal implication of his result on the derivability of the Archimedes axiom from Dedekind’s continuity. He thus held that if the Archimedes axiom A can be derived from the Dedekind’s postulate D and the others axioms of measurable magnitudes Γ, then, if A is negated, D and Γ cannot both hold. 46 So, if one wants Dedekind’s continuity to hold for pragmatic reasons, then one cannot refute the Archimedean axiom. Now, given the original results developed by Hölder, and in particular the fact that he had built a new non-Archimedean model, why did he stick to the Archimedean property in the axiomatic formulation of the theory of measurable magnitudes? I will suggest that this technical result played a fundamental role. A main reason why Hölder assumed from the beginning the validity of the Archimedean axiom, even if he had constructed a consistent non-Archimedaen model, was that he believed that in refuting it, he would have to abandon Dedekind’s continuity altogether. This is true in Hölder’s non-Archimedean model, but not generally in Veronese’s model, for Dedekind’s continuity can be saved as a special case. Yet, why did he overlook the relation between Veronese’s continuity and Dedekind’s continuity? My argument is that his epistemological framework and its difference from Veronese’s perspective might play a role here.

4.4.2 The epistemological framework

The mathematical choice of different axioms in the case of Hölder and Veronese should be related to different epistemological frameworks. Both Veronese and Hölder took inspiration from Pasch, but Veronese was interested in a mathematical hypothesis concerning an order relation that might differ from the standard one, and yet be compatible with our intuition of physical objects (a bundle of parallel sticks, for example). Hölder’s approach was on the contrary the result of an exclusive interest in the empirical foundation of measurement.

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Veronese considered geometry as a mixed science (partly empirical and partly ideal, i.e., going beyond the empirical realm), or—as Hölder would have said—the result of a combination of hypothetico-synthetic concepts and pure synthetic concepts. Hölder, on the contrary, considered all primitive propositions of geometry—and thus of measurement theory—as axioms, i.e., as based on experience. According to him, the hypotheses that Veronese was trying to anchor to experience, did not really apply to experience—and in fact the model that Hölder suggested as a realization of Veronese’s system, was based on functions and not on magnitudes [Hölder 1901, 12n, Eng. trans. 249].47 On the contrary, the Archimedean property, apart from being present in Euclid’s Elements (Book VI, § 9), appeared to him as relevant for empirical applications and as a standard component of our “usual continuum” [Hölder 1901, 11, Eng. trans. 249]. Dedekind’s continuity was praised for its compatibility with the Archimedean axiom but also for its important role in proofs and applications. But there is one more reason: using Dedekind’s continuity, it is possible to establish at the same time a very tight connection between the Euclidean theory of proportions and the the notion of measurement on the one hand, and the arithmetical theory of irrational numbers on the other hand. [Hölder 1901, 19, Eng. trans. 241] Hölder assumed the definition of the arithmetical continuum, and then showed that the modern theory of proportion (or theory of measurement) can be grounded only if one shows not only that to any geometrical measure of a magnitude there corresponds a number, but that also to any geometrical sum of two magnitudes a and b there corresponds the arithmetical sum of the corresponding numbers that measure a and b. He thus introduced, like Bettazzi before him but unlike Veronese, a representation theorem that justifies the fact that we can apply numbers to measure magnitudes. Veronese, on the contrary, did not conceive the idea of a representation theorem, because he was far away from the idea of establishing a tight connection between arithmetic and geometry. On the contrary, he aimed to show that the notion of continuity is different in geometry and in arithmetic, and that, given the fact that the former is fundamental, the latter might not be an adequate description of continuity in general. The radical demarcation between arithmetic and geometry is not only a matter of epistemological preference or philosophical inclination: it affects, and not in a regressive way, Hölder’s mathematical works, because it is at the basis of the representation theorem that he developed in The Axioms of Quantity. Having assumed that real numbers are a symbolical representation of empirical magnitudes, Hölder perceived the necessity to prove that, if the symbolic system should measure (represent) the system of magnitudes, it should have a similar structure. This was not due to the fact that he had overlooked the mathematical investigation of new systems of quantities: on the contrary, he developed an original model of non-Archimedean magnitudes. Rather, having related the theory of quantities to geometry and the latter to certain empirical operations of superposition, Hölder saw no reason for adopting a higher generality in his approach to measurement.

5 Conclusion

I will now summarize some of the complex and interesting relations between Hölder and other mathematicians and geometers of the 19th century, as they emerge from the

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reconstruction of Hölder’s epistemology suggested in this paper. Hölder’s relations to previous or contemporary authors are not linear, but complex and multifaceted: this explains the originality of his position, which could be prima facie assimilated to that of other authors, but reveals, upon deeper investigation, its peculiarity.

5.1 The demarcation between geometry and arithmetic

In section 2, I have investigated the distinction between ‘given’ and ‘constructed’ that Hölder had introduced already in Intuition and reasoning, although he clarified it further in The Mathematical Method: rather than being just a difference concerning the origin of mathematical concepts (as was the case in Gauß, who declared geometry to be an empirical science), it is a difference concerning the possibility of defining the concept by means of a constructive activity. It cannot be reduced to the modern axiomatic distinction between primitive and derived concepts, nor to the epistemological distinction between a priori and a posteriori, but is rather related to the formal genesis of the object itself. While talking of activity, Hölder said something similar to what Hermann Graßmann had described as the genesis of pure mathematical concepts through laws. The same does not hold for geometry, according to Graßmann, because the latter is not a pure mathematical theory but has to assume the three-dimensionality of space as something given. Even if Hölder had a different understanding of what is given in geometry, because he considered points and lines among given concepts, he had a similar understanding of what given is opposed to, i.e., something that can be defined by its own constructive law. Hölder’s use of the distinction given-constructed to demarcate geometry from arithmetic is quite near to Graßmann’s use of the distinction between what is posited by thought and what stands in front of our thought, a distinction that Graßmann used to demarcate formal from real sciences. But there are also relevant differences, because Hölder remarked that not all geometrical concepts are given. The mixed nature of geometry is more similar to the conception of Giuseppe Veronese, who had made an effort to reconcile the belief in the empirical genesis of basic geometrical truths with the ideal constructions required by non-Euclidean, non-Archimedean or hyperspace geometries. Anyway, Hölder did not quote Graßmann, but Zindler, a disciple of Meinong who insisted on the fact that there is only one primitive concept in arithmetic while there are many different concepts in geometry: for this reason geometrical primitives are mainly concepts denoting relations. On this point Hölder differed from Zindler, but also from Veronese, Hilbert and modern axiomatics. Against the tendency introduced by Pasch to identify geometrical primitives as specific relations between elements (e.g., betweenness), Hölder insisted on the necessity of distinguishing two kinds of axiomatic concepts: concepts denoting objects and concepts denoting relations. This remark can be read as a reaction against the tendency to call geometry any abstract theory on relations of betweenness, connection, order that did not take into account certain fundamental properties of geometrical objects. With this respect, Hölder’s position is again similar to that of Veronese, who refused to admit that certain mathematical theories, like Poincaré’s theory of the hyperboloid of one sheet or Hilbert’s non-

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arguesian theory, could be called ‘geometry’, exactly because they did not preserve certain properties of the geometrical object“straight line”. The demarcation between arithmetic and geometry was further developed in The Mathematical Method, where a different terminology was used for constructed concepts, being now called hypothetico-synthetic (in geometry) and pure synthetic (in arithmetic). Furthermore, the distinction is compared with Meinong’s opposition between Zusammenstellung and Zusammensetzung, resulting in a clarification of the different kinds of activities that can be respectively associated to arithmetic (to order in a sequence or to assign) and geometry (to superpose segments). Hölder adhered to Kant’s apriorism as far as arithmetic is concerned, defending the distinction between a priori and a posteriori reasoning processes.

5.2 Axiomatic method and mathematical practice

Pasch’s effort to eliminate intuition from deduction was the starting point of most mathematicians and logicians of the time. Hölder was highly sensitive to this issue and was willing to adopt Peano’s symbolic reconstructions of geometry as a calculus in order to show that intuition does not play any role in deduction. The latter was in effect the center of Hölder’s interest, but in a quite original and different way with respect to the contemporary dominant traditions in axiomatics. Hölder’s interest was directed to the formulation of the primitive propositions of a given theory, and to the analysis of the kind of activities involved in deduction itself rather than to the investigation of independence and consistency results. Although sharing with Couturat and Russell a certain conception of arithmetic as purely formal, Hölder held a radically different conception of logic: it is not a universal theory that precedes all mathematical disciplines, but a disciplinary investigation aiming at isolating the specific deductive patterns that are used in different branches of mathematics. From this perspective, he was much more interested in the analysis of increasing levels of complexities in our mathematical activities, and in the limits of the use of deduction to study deduction itself. For this very reason, he adopted a different approach, based on a case by case investigation of mathematical deductive procedures that cannot itself be developed into a symbolic language, but can be accomplished only by means of the natural language. This idea of a general logic that arises from a never definitively achieved investigation of mathematical deductive activities explains the distance with respect to Hilbert’s effort to prove metamathematical results inside axiomatic systems, but also the radical difference with respect to Peano, who aimed to express all mathematics by means of a unique symbolic language, or even adopting a standardized universal language like the latino sine flexione for scientific communication. Hölder on the contrary, believed that the specificity of mathematics had nothing to do with the symbols used to represent it, but rather with the kind of activity, i.e., deduction, deployed in its practice. For this reason he developed a sort of empirical study of deduction: this is one of the most original aspects of Hölder’s investigation, and probably also a reason why his main book went largely unnoticed in his time.

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5.3 Philosophy of mathematics

The interest in the investigation of mathematical deduction and in the construction of a general logic that can be derived from the examination of specific cases is the core of Hölder’s epistemology of mathematics. Yet, in Intuition and Reasoning, and again in The Mathematical Method he often discussed the position of different philosophers on the questions of apriorism and empiricism. Hölder’s point of view is clear: such questions belong to the philosophy of mathematics but do not have any influence on how mathematical deduction works or is developed. The mentioned questions can be asked with respect to the genesis of mathematical objects, but do not have anything to do with what mathematics actually is. This might be a reason why Hölder’s position was partially unresolved, both with respect to empiricism and Kantian apriorism. Another reason might be the fact that Hölder was not only a mathematician, but had truly philosophical and argumentative skills that urged him to distinguish his own preferences or inclination for a philosophical position from the possibility of corroborating or confuting its claims. The adhesion to the Kantian project was not evident in Intuition and Reasoning, neither in the terminology—no use of the term “synthetic” was made—nor in the conception of geometry, yet Hölder apparently refrained from a too strong and explicit criticism of Kant’s apriorism. On the contrary, in The Mathematical Method there is an explicit revival of the Kantian epistemological project, which is mentioned in the introduction as the starting point of the whole essay. Besides, Hölder’s conception of arithmetic is compatible with the Kantian perspective, even if the criticism towards the Kantian conception of geometry is more explicit than in his inaugural lecture. A similar ambivalent relation was developed by Hölder towards empiricism. On the one hand, he was strongly influenced by Pasch concerning the foundation of geometry, and by Helmholtz concerning the theory of measurement. Notwithstanding the general acceptance of a genetic theory of geometrical objects and measurable magnitudes, Hölder resisted some possible consequences of an empiricist position. For example, he abhorred the application of psychologism to arithmetic, and the extension of empiricism to ‘pure’ mathematics.

5.4 Conclusive remarks

The difficulty of classifying Hölder as well as his ambivalence towards apriorism and empiricism are already a proof of the unconventionality of his position. His ability to analyze advantages and limits of the axiomatic method and of the traditional debate in the philosophy of mathematics is typical of a scholar who was both a practicing mathematician and a practicing philosopher, and believed in the inexhaustibility of both practices. It also reveals a deep knowledge of the contemporary mathematical and philosophical debate, but unfortunately only up until the First World War, because The Mathematical Method appeared in 1924 but in an almost unrevised version with respect to its initial project. Hölder’s epistemological approach does not prove to be generally regressive, if one considers the relation to his own mathematical research: sometimes the effect is innovative, as is the case for the distinction between arithmetic and geometry that allowed him to develop a representation theorem, even if it was sometimes limitative,

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as in the case of the strict association between geometry and measurement, that made the development of a higher generality useless, or pragmatically uninteresting.

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VAN DER WAERDEN, BARTEL L. 1938 Otto Hölder. Nachruf [Obituary], gesprochen in der öffentlichen Sitzung vom 25. Juni 1938, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissesnchaften zu Leipzig, 90, 97–101.

VERONESE, GIUSEPPE 1891 I fondamenti della geometria [Foundations of Geometry], Padova: Tipografia del Seminario.

VOLKELT, JOHANNES 1886 Erfahrung und Denken: kritische Grundlegung der Erkenntnistheorie [Experience and Reasoning], Hamburg: Voss.

ZINDLER, KONRAD 1889 Beiträge zur Theorie der mathematischen Erkenntnis [Contributions to the theory of mathematical knowledge], Sitzungsberichte der phil.-hist. Classe d. K. Akademie der Wissenschaften zu Wien, 118(9), 1–98.

NOTES

*. The author would like to thank two anonymous referees for their valuable constructive criticism of this paper, and Volker Peckhaus, Oliver Schlaudt, Philippe Nabonnand, Gerhard Heinzmann together with all the participants of the Workshop on Hölder held in Nancy in 2010 for their precious comments in relation to a preliminary version. 1. Given that there is an English translation of the 1899 paper published in this issue but no English translation of the 1924 volume, I will often quote more extensively from the latter rather than from the former. Translations from [Hölder 1924] are mine. 2. Among the precursors of this enterprise, Hölder quotes Lambert’s New Organon [Lambert 1764], Pasch’s Lectures on Modern Geometry [Pasch 1882] and Hilbert’s Foundations of Geometry [Hilbert 1899].

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3. See the letters to Bessel dated 9 April 1830 and 27 January 1829. In the latter Gauß remarked that his belief that geometry cannot be grounded entirely a priori had become stronger [Gauß 1900, vol. VIII, 200–201]. 4. “The principal division of the sciences is into the real and the formal. The real represent the existent in thought as existing independently of thought, and their truth consists in the correspondence of the thought with that existent. The formal on the other hand have as their object what has been produced by thought alone, and their truth consists in the correspondence between the thought processes themselves” [Graßmann 1844, xxi, Eng. trans., 23]. 5. “Beside these concepts, which are available to the geometer as a given material, some assertions of a more general kind (which hold or should hold for these concepts) are also established from the beginning; these fundamental facts, being presupposed without proof, are called axioms or postulates” [Hölder 1900, 1, Eng. trans., 16]. Italics in this and in the following two quotations are mine. 6. “Some concepts are defined by means of a construction, others are basically taken as given” [Hölder 1900, 1, Eng. trans., 15]. 7. “[Philosophers] consider Euclid’s axiom of parallel lines as necessarily given within intuition, and consider any geometry that operates intuitively with some other assumption as an absurdity” [Hölder 1900, 6, Eng. trans., 20]. 8. “We distinguished in geometry between primary concepts (taken from intuition or experience) on the one hand, and geometrically constructed concepts on the other hand” [Hölder 1900, 15, Eng. trans., 23]. 9. Hölder refers here to the similitude of figures, which can be either intuited and used to ground the concept of ratio, or derived from the theory of proportions, as Euclid did. Even if “it may be taken as an immediate intuitive fact that we can reproduce a body with its proper form but on a different scale” [Hölder 1900, 15, Eng. trans. 23], Hölder seems to believe that this is the result of a construction from more primary concepts rather than a fact that is ‘given within intuition’. 10. See Hölder’s terminological distinction between ‘primary’ concepts [ursprüngliche Begriffe] and ‘primitive’ concepts [Grundbegriffe]. 11. See for example Pasch’s remarks on geometrical concepts: “These concepts are said to refer to position. They are thus introduced as derived with respect to geometrical fundamental concepts. Yet, if, without adding other geometrical concepts, one derives from them further concepts taken to refer to position, then one might take the derived concepts as stem concepts [Stammbegriffe] that precede the others inside the group of concepts that refer to position” [Pasch 1882, 74, my trans.]. 12. “In mathematics but not in philosophy of mathematics should the axioms be the starting point” [Zindler 1889, 2ff.]. 13. “In mathematics the representation of space should be considered as given. It is unnecessary to take into account how our representation could have originated—geometry is indispensable also in physiological and psychological investigations on the meaning of space and already finds the most extensive application—, how it relates to an objective real space, what sense could this question about an objective real space have and whether this question has any sense at all” [Zindler 1889, 2ff.]. 14. Yet, both in the paper on measurable magnitudes and in The mathematical method Hölder mentioned Zindler’s use of the term axiomatic to denote concepts that occur in the axioms. See [Hölder 1901, 3n, Eng. trans., I, 247], [Hölder 1924, 10] and [Hölder 1924, 10]. 15. “When we look for the primitive geometrical representations we cannot specify a single wholly determined primitive concept that might be considered as the starting point of the development of other concepts, but there are here several fundamental representations that are reciprocally irreducible and undefinable. Such are for example the direction, the point, the

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distance, the concept of coincidence or of superposition of geometrical figures, of surfaces, of volumes, and of course of space itself” [Zindler 1889, 3]. 16. “Objects that fall under one of the previously introduced primitive concepts are accessible comparative relations, as the possibility of their quantitative measurement (measure of lengths, measure of surface, measure of solids, measure of angles) shows” [Zindler 1889, 2ff.]. 17. Hölder distinguished the synthesis that is at stake in the formation of concepts from the synthesis that is at stake in the formation of judgements and that is associated to the distinction between analytic and synthetic propositions. This use of the term ‘synthetic’ is thus different from Kant’s use, as Hölder himself remarked. One would expect Hölder to avoid, for the sake of clarity, to use a Kantian term in a non-Kantian meaning, given that he elsewhere remarked that many philosophical distinctions that were originally introduced with one meaning and are subsequently maintained with a different meaning usually generate more confusion than clarification of concepts [Hölder 1924, 292n, 361]. So, presumably, he must have had some strong reasons for maintaining the Kantian term, as if he wanted his approach to remain Kantian, notwithstanding the differences. 18. “Thereby, between the arithmetico-combinatorial domain and geometry (or respectively mechanics) there is the essential difference that in the former domain one does not assume any such given element as in geometry the point and the line, nor any given and non synthetically definable—i.e., non logically derivable—relation, nor any law (axiom) with respect to those elements and their relations”[Hölder 1924, 295]. 19. “With reference to what I have just said, I would like to distinguish the hypothetico-synthetic concept from the pure synthetic concept, depending on the fact that the concept is or is not set up on assumptions similar to those that are made in geometry and in mechanics. One could express it by saying that there is something empirical that inheres in the hypothetico-synthetic concept, so that the elements that compose it together with the relevant relations have some meaning in experience, and also the previously assumed laws (axioms) originate in experience or intuition” [Hölder 1924, 295]. The examples are quoted in the introduction [Hölder 1924, 6]. 20. “In arithmetic the abstraction of new general concepts and rules thus forms in a way a constituent of deduction, and the procedure providing the basis of abstraction is a process we do not count among experience in a narrower sense of the word, that is to say in opposition to reasoning. Having said this, I want to assert that arithmetic—at least in so far as it concerns whole and rational numbers [...]—has no axioms in the proper sense, and that the arithmetical proof process is not a purely formal process, but a mixed and extremely complex process” [Hölder 1900, 61–62, Eng. trans. 43]. See also [Hölder 1924, 6, 295ff.]. The distinction between arithmetic, that contains definitions but no axioms, and geometry, that contains axioms, was introduced by Hermann Graßmann in a similar way: “Thus proof in the formal sciences does not extend beyond the sphere of thought, but resides purely in the combination of different thought processes. Consequently, the formal sciences cannot begin with axioms, as do the real; rather, definitions comprise their foundation. (Although postulates have been introduced into the formal sciences, for example in arithmetic, this is to be regarded as an error, only to be explained by the corresponding treatment of geometry.)” [Graßmann 1844, 22, Engl. trans. 23]. 21. Here the term ‘given’ is used in the epistemological sense introduced in section 2.1.1. 22. “The fundamental propositions should include all empirical material mathematics has to deal with, so that once those propositions have been set, there is no need to go back to sense perceptions anymore. Actually, in order for geometry to be really deductive, the process of deduction should everywhere be independent from the sense of geometrical concepts, and also independent from figures. Only the relations between the geometrical concepts laid down in the used propositions or definitions can be taken into account. On the contrary, it is certainly admissible and useful for deduction, but in no way necessary, to take into account the meaning of the geometrical concepts that occur in it, so that precisely when this becomes necessary, the

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incompleteness of deduction and, if the gap cannot be filled by some change in the reasoning, the unreliability of the propositions that have been previously used as demonstration means arise” [Pasch 1882, 98, my trans.]. On Pasch’s philosophy of mathematics see in particular [Schlimm 2010]. 23. Pasch did not transform geometry into a calculus, but was rather interested in the duality principle and in the possibility of deriving theorems from primitive propositions by logical means rather than by observation of given figures. This was obtained by identifying as primitives several relevant relations between points and lines, so that it could be possible to interpret the primitive propositions by means of alternative concepts (lines and points respectively) and yet obtain valid theorems [Pasch 1882, 98]. 24. “By means of such a formulation of all intuitive assumptions, one can divest geometrical deduction itself of intuition. If one denotes the geometrical elements and the operations on these elements through symbols and through successions of symbols—just as in algebra one symbolically represents the numerical magnitudes and the operations to be carried out on them —so, in the simplest cases, it is possible to entirely reduce the geometrical deduction into a calculus that is carried out on symbols, just as in algebra the inferences concerning numerical magnitudes are carried out by means of the so-called letter-calculus. Such a calculus has already been accomplished for single domains of geometry; Peano has assembled these symbolical procedures” [Hölder 1900, 14, Eng. trans. 22]. Hölder is well known for his remarks on the limits of symbolism (see e.g., section 2.2.1 above and especially the contribution by Mircea Radu in this volume [Radu 2013]), but this does not mean that he denied the use of symbolical calculi or its merits in certain cases, as it is evident from the above quotation. 25. “The recent knowledge of the possibility to represent a geometrical proof in a purely logical way frees us from the necessity to assume an a priori intuition of space for the sake of the geometrical proof” [Hölder 1924, 28]. 26. “We figured ourselves the geometrical concepts of point, straight line, surface and so on together with their corresponding relations as formed according to experience, certainly a relatively raw experience [...]. Once we arrived at certain regular relations, idealization arose: we imagine that there could be exact points, exact straight lines, and that one might establish exactly whether a given point lies on a given straight line or not, and at the same time we assume that for these exact concepts the regular relations or axioms have unconditioned validity” [Hölder 1924, 397]. 27. “The successful application that geometry encounters continuously both in natural sciences and in practical life is based only on the fact that geometrical concepts originally corresponded exactly to empirical objects. Even if they were gradually overdrawn by a net of artificial concepts in order to promote the theoretical development, and inasmuch as one limits oneself, from the beginning, to the empirical nucleus, geometry maintains the character of a natural science, distinguished from other parts of the science of nature, because it needs to derive immediately from experience only a very small amount of concepts and laws” [Pasch 1882, iii, my trans.]. 28. Like Zindler, Hölder believed that to investigate geometry from a mathematical point of view, it suffices to know that some of its concepts are taken as given: it is not necessary to discuss how they are given to us. 29. See in particular The facts of perception [Helmholtz 1884, 218], but also Die Anwendbarkeit der Axiome auf die physische Welt [The applicability of the axioms to the physical world], supplement to Die neuere Entwickelung von Faraday’s Ideen über Elektricität [The recent development of Faraday’s ideas about electricity] [Helmholtz 1884, 405]. Note that the title itself of Hölder’s lecture is inspired by Helmholtz’s question: “Was ist Wahrheit in unserem Anschauen und Denken? [What is truth in our intuition and reasoning?]” [Helmholtz 1884, 218]. 30. “Certainly, a strict follower of Kant would still suspect that any of those results obtained through measurement were already influenced by ready and exact intuition taken as a condition

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of any experience. Maybe, if he wants to hold on to his point of view, he could not be refuted. On the other hand, it cannot be denied that his standpoint is artificial, and it seems legitimate to drop that standpoint, if one believes one can do without it” [Hölder 1900, 6–7, Eng. trans. 18]. 31. Hölder did not quote Poincaré in 1899, but quoted Science and Hypothesis [Poincaré 1902] in 1924, as he discussed indirect proofs, the analysis of induction, the compatibility of non- Euclidean geometries with experience, and the distinction between the visual, the tactile and the motor space. It can thus be reasonably assumed that he had not read Poincaré’s essay Space and Geometry [Poincaré 1895] before giving the 1899 Inaugural Lecture. 32. In this text, Volkelt developed a theory of knowledge that inherited some features from Descartes, Kant and Locke. He claimed that the starting point of knowledge had to be subjective; yet objective—or rather trans-subjective—knowledge could be reached once experiences were combined with logical deduction. 33. “The importance of the localization problem will certainly be denied by those who consider the pure intuition of space as a condition of any perception. But they must certainly admit that the empirical part of my perception, i.e., also my sensory perceptions, must contain something that allows me, in a single case, to locate an object in this way, and not otherwise, namely to see the object in this determinate position and not in some other position with respect to my body (cf. the theory of ‘local signs’ by Lotze [...])” [Hölder 1900, 16n, Eng. trans. 31]. 34. “Under the given circumstances it seems to me that one can hold the Kantian assumption only if one assumes, so to speak, behind the representative space an a priori intuition that operates in it but is incompletely expressed by it, a sort of platonic idea of space that is the a priori form and the condition of all experience. Such an assumption could hardly be refuted. Actually, it is a hypothesis that is not supported by any evidence. [...] And does it truly explain the applicability of mathematics to experience?” [Hölder 1924, 384–385]. 35. “That arithmetic is an a priori science is certainly assumed by most mathematicians. [...] The point of view that Plato, and among the moderns especially Kant, validated, i.e., that there is unconditioned necessary knowledge that is grounded in what is for us condition and criterion of all knowledge, might well remain in place. But the borderline between what is necessary (a priori) and what belongs to experience is dragged to a different position” [Hölder 1924, 7]. 36. “On the other hand, it is certainly possible to deny an unconditioned distinction between two kinds of knowledge, an a priori and a posteriori kind of knowledge. Yet, even if one does it, one cannot consider all reasoning processes as equivalent” [Hölder 1924, 395]. 37. “While with line segments it must be assumed that they can be compared and necessarily found to be either equal or unequal, one can prove from axioms, in which the word “area” does not occur, and on the basis of certain definitions, that two figures can be compared with respect to area [...]. This is based on the fact that equality of line segments is a primary concept and equality of areas in figures is a derived geometrical concept [...]” [Hölder 1901, 3, Eng. trans. modif. I, 247]. 38. “From the consistency of the axioms of magnitudes one can derive the consistency of the geometrical axioms (a) to (κ) in Part II of this work and vice versa” [Hölder 1901, Eng. trans. I, 248n]. 39. “Geometrical and physical magnitudes, too, such as line segments, areas, times, masses, forces, etc., as given in the first place, can only be added, and addition is only possible with magnitudes of the same kind. Talk of the multiplication of geometrical or physical magnitudes is based upon arbitrary convention. The present paper assumes only such axioms as relate to the comparison and addition of magnitudes. These axioms are sufficient to deduce the Euclidean theory of proportion and the contemporary theory of measurement. The results concerning the multiplication of magnitudes appear here only as corollaries to one of these theories” [Hölder 1901, Eng. trans. I, 250n].

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40. The axiom of Archimedes states that given two quantities a and b such that a < b, there exists a natural number m ∈ ℕ such that ma > b. The postulate of Dedekind asserts that if a system of quantities can be divided in two parts A and B so that each element of A is at the left of each element of B, that is to say, that each element of A is strictly inferior to each element of B, then there is one and only one element that determines this section: it will be the maximum of A or the minimum of B. 41. “Of course, the axioms chosen are arbitrary, up to a point, and it is a matter of convenience whether Dedekind’s axiom of continuity plus axioms I to VI or some other axioms are to be preferred” [Hölder 1901, Eng. trans. I, 248n]. 42. On the contrary, in the theory of quantities Hölder gives a geometric formulation of the postulate. “This last axiom is the axiom of continuity in geometrical form. Dedekind first noted that the nature of the continuity of straight lines can be expressed by this principle [Dedekind 1872, 93]. At the same time, Dedekind put less value upon postulating this axiom for geometry, rather he took it as an opportunity for his definition of the irrational numbers” [Hölder 1901, Eng. trans. II, 354]. 43. “If one considers that the definition of each step requires a specific law, so the concept of the totality of all steps means that we believe we can conceive the totality of all the laws, which nonetheless require a certain stipulation. Yet, such a wholly indeterminate totality could represent an inadmissible concept [Hölder 1924, 193–194]. 44. Whereas Hölder discussed in detail the model by Veronese, which is also based on a geometrical understanding of magnitudes, he did not extensively analyze Bettazzi’s model, which appears to him as too abstract, nor Hilbert’s model, which was based on arithmetical functions. 45. Veronese’s absolute continuity also known as ‘Archimedes generalized’ is formulated as follows: “if a and b are two intervals of the system and a < b, there is a determined symbol of multiplication (number) η such that ηa > b”. Hölder’s property of divisibility requires that “for each member A of the system and each positive integer n, there is an X in the system such that nX = A”. 46. If (Γ ∧ D) ⊦ A, then ¬A ⊦ ¬(Γ ∧ D). 47. That they could not be falsified by experience itself was not a sufficient argument, because neither Veronese’s continuum nor Dedekind’s continuum can be.

ABSTRACTS

The aim of the paper is to analyze Hölder’s understanding of geometry and measurement presented in Intuition and Reasoning [Hölder 1900], “The Axioms of Quantity and the Theory of Measurement” [Hölder 1901], and The Mathematical Method [Hölder 1924]. The paper explores the relations between a) Hölder’s demarcation of geometry from arithmetic based on the notion of given concepts, b) his philosophical stance towards Kantian apriorism and empiricism, and c) the choice of Dedekind’s continuity in the axiomatic formulation of the theory of magnitudes. The paper shows that the choices made at the axiomatic level reflect Hölder’s epistemological framework, and individuates the originality of his approach in the case by case analysis of deductive procedures.

L’article a pour but d’analyser la conception de la géométrie et de la mesure présentée dans Intuition et Raisonnement [Hölder 1900], « Les axiomes de la grandeur et la théorie de la

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mensuration » [Hölder 1901] et La Méthode mathématique [Hölder 1924]. L’article examine les relations entre a) la démarcation introduite par Hölder entre géométrie et arithmétique à partir de la notion de ‘concept donné’, b) sa position philosophique par rapport à l’apriorisme kantien et à l’empirisme et c) le choix du postulat de la continuité de Dedekind parmi les axiomes de la théorie des grandeurs. L’article montre que les choix faits au niveau axiomatique sont reliés au cadre épistémologique et que l’originalité de Hölder consiste surtout dans son analyse au cas par cas des procédures déductives en mathématiques.

AUTHOR

PAOLA CANTÙ Aix-Marseille Université – CNRS, CEPERC, UMR 7304, Aix-en-Provence (France)

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Varia

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Hat Kurt Gödel Thomas von Aquins Kommentar zu Aristoteles’ De anima rezipiert?*

Eva-Maria Engelen

Die Frage, ob einer der größten Mathematiker und Logiker des 20. Jahrhunderts einen der größten Theologen und Philosophen des Mittelalters rezipiert hat, mag zunächst seltsam und vielleicht sogar ein wenig abwegig erscheinen. Sie ist es allerdings nicht.1

1 Was berechtigt zu dieser Frage?

Von 1934-1955, also in einem Zeitraum von 22 Jahren, hat Kurt Gödel 16 Hefte mit Philosophischen Bemerkungen hinterlassen, von denen 15 erhalten sind und eines verloren gegangen ist.2 Diese Hefte liegen gemeinsam mit dem übrigen umfangreichen Nachlass des großen Logikers im The Shelby White and Leon Levy Archives Center des Institute for Advanced Study in Princeton, und sind, da sie in der nur noch von einigen Experten auf der Welt gelesenen Kurzschrift Gabelsberger verfasst sind, noch nicht veröffentlicht. Zu diesen Heften hat Cheryl Dawson, die bei der Herausgabe der Collected Works von Kurt Gödel mitgearbeitet hat, ein Namensglossar erstellt, in dem zu Thomas von Aquin 15 Namensnennungen vermerkt sind.3 Verglichen mit der Namensnennung von Gottlob Frege und Bertrand Russell, je 28 Mal, oder mit Aristoteles, 35 Mal, mag das nicht viel sein, zählt man aber die Häufigkeit, mit der zum Beispiel der Logiker Alfred Tarski (10 Mal), der Philosoph Ludwig Wittgenstein (4 Mal) oder die Mathematiker David Hilbert (5 Mal) und Herrmann Weyl (4 Mal) genannt werden, ändert sich das Bild. Die Frage nach einer möglichen Rezeption von Thomas’ Kommentar zu De anima lässt sich dadurch aber noch nicht rechtfertigen. Diese ergibt sich vielmehr aus zwei Nennungen des Kommentars in den Maximen Philosophie-Heften.4 Das erste Mal wird Thomas bereits in Heft 0 genannt, wo Gödel ein Lektüreprogramm für sich aufstellt, das er in die Punkte I und II aufteilt. Unter I führt er Sekundärliteratur zur Geschichte der

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Philosophie und Überblickswerke auf und unter II Primärliteratur. Auf Seite 20 ist dort unter anderem zu lesen: Progr: Bücher, die durchzusehen sind: I. […] II. Originalarbeiten: Thomas: Summa, Opuscula (Quaestiones?) Aristoteles, Comm d Thomas, Kant Hegel (Wissenschaft der Logik), Sigwart (Logik) Brentano?, B. Erdmann? Husserl (Phil d Arithm, Log Unters)5 Diese Liste wird von Gödel noch über anderthalb Seiten fortgesetzt. Die zweite, noch interessantere Notiz Gödels zum Kommentar befindet sich auf dem Deckblatt von Heft VI der Max Phil-Hefte. Dort notiert Gödel in gut lesbarer Langschrift: Thomas Comm. De anima und gleich darunter, gleichfalls in Langschrift Steinthal Sprachtypen. Heft VI, auf dessen Titelblatt dies vermerkt ist, handelt nicht ausschließlich, aber zu einem guten Teil von Themen, die auch in Aristoteles’ Schrift De anima erörtert werden: um Überlegungen zum Verhältnis von Leben und toter Materie, zum Prinzip des Lebens und zum Verhältnis von Materie, Pflanzen, Tieren und Menschen sowie zum Verhältnis von Seele, Verstand und Geist zu den Sinnen, et cetera. Ausgehend von der alten Fragestellung, wie es kommt, dass der Mensch wahrnimmt, dass er wahrnimmt, die von Aristoteles stammend seit Augustin auch als Frage nach dem inneren Sinn [Leinkauf 1995, 627] und [Dewender 1995, 635] und in der folgenden mittelalterlichen Diskussion als die nach dem sensus communis oder Gemeinsinn geführt wurde [Heller-Roazen 2007, 136],6 ergänzt Gödel diese Überlegungen, die er in systematischer Weise aufgreift und fortführt, um das Verhältnis von Seele, Bewusstsein und Ich zueinander. Thomas von Aquin nennt er dabei kein einziges Mal, sondern spricht jeweils nur von Aristoteles. Daraus ergibt sich die Frage, warum Gödel auf dem Deckblatt zu Heft VI seiner Philosophischen Bemerkungen „Thomas Comm. De anima“ und „Steinthal Sprachtypen“ verzeichnet hat. Eine Möglichkeit wäre, dass er sich damit gewissermaßen eine Erinnerungsnotiz machen wollte, um sich bei passender Gelegenheit hinsichtlich der von ihm in Heft VI behandelten Themen auch die beiden genannten Werke anzusehen und seinen Wissens- und Diskussionshorizont damit zu erweitern. Zu dieser Hypothese würde passen, dass der Name von Chaim Heymann Steinthal, einem Sprachforscher in der Tradition Wilhelm von Humboldts, im Verlaufe des Heftes VI nicht mehr genannt wird. Die Nennung von „Thomas Comm. De anima“ und „Steinthal Sprachtypen“ könnte allerdings auch darauf hinweisen, dass Gödel die Angaben nicht nachträglich, quasi als Memorandum, notiert hat, sondern vor Beginn der Niederschrift von Heft VI, um festzuhalten, dass er, ausgehend von der Lektüre dieser beiden Werke, die dann folgenden Überlegungen entwickelt. Die These, Gödel habe Thomas von Aquins Kommentar zu Aristoteles’ Werk De anima von 1268 sowie Chaim Heymann Steinthals Charakteristik der hauptsächlichen Typen des Sprachbaus von 1860 rezipiert, ist freilich begründungsbedürftig. Beginnen wir mit dem selteneren und spezielleren Werk von Steinthal, um zu sehen, was dafür spricht, dass Gödel es tatsächlich verwendet haben könnte, denn da die Bemerkungen zu

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Sprachtypen und Sprachformen in Heft VI zahlenmäßig um einiges geringer ausfallen, als die zu Bewusstsein, dem Ich und der Seele ist es arbeitsökonomisch gesehen sinnvoll, mit der Spurensuche hier zu beginnen. Wäre eine Rezeption nämlich bereits in Bezug auf Steinthals Werk auszuschließen, läge es nahe, die beiden Hinweise auf dem Deckblatt für Erinnerungsstützen zu halten.

2 Hinweise zur Rezeption von Steinthals Werk und eine mögliche thematische Verbindung zu De anima

Steinthal war ein Sprachwissenschaftler, der eine mentalistische Sprachauffassung vertreten hat und keine mechanistische, wie sie zu seiner Zeit weit verbreitet war. Sein Versuch, Psychologie und Sprachwissenschaft zu verbinden, ergibt sich aus dem mentalistischen Ansatz. Er war davon überzeugt, dass es nicht ausreicht, ein Wort auf seine grammatikalischen Besonderheiten hin zu untersuchen, um seine volle Bedeutung zu verstehen. Vielmehr soll für ein umfassendes Verständnis einer Wortbedeutung stets mitgedachtes Hintergrundwissen herangezogen werden. Daher sollen Untersuchungen zum Brauchtum, zu Gesetzen, Religion und Sprachverwendung mit berücksichtigt werden [Christy 2002, 3f u. 16]. Diese Vorgehensweise allein hätte Steinthal bereits zu einem interessanten Autor für Gödel gemacht, der sich wie Steinthal Gedanken über den Beitrag der Psychologie für Denken und Sprache macht; aber dass Steinthal ein Interesse daran hatte, eine mathematische Strenge in die Erforschung mentaler Strukturen zu bringen, hätte sicher sein Übriges getan. Der Ansatzpunkt für Gödel, sich mit Steinthals Charakteristik der hauptsächlichen Typen des Sprachbaus zu beschäftigen, dürfte jedoch noch eher zu Tage treten, wenn man sich näher ansieht, inwiefern Steinthals Ausführungen zu Sprachtypen eine Gemeinsamkeit mit Thomas von Aquins Kommentar zu De anima aufweisen könnten. Eine Passage bei Steinthal, die deshalb auch ausführlicher zitiert werden soll, ist hierfür von besonderem Interesse: Den Reigen beginnen die hinduistischen, die unentwickelten, formlosesten aller Sprachen. Sie entsprechen den Zoophyten der Zoologie. Wie diese den Uebergang aus dem Pflanzenreich in das Thierreich darstellen, so bilden diese Sprachen die Gränzen der menschlichen Rede und nähern sich der Stummheit der Geberdensprache. [...] Diese Sprachen haben keinen Bau, wie die genannten Thiere kein Skelett. [...] Ihr Satzbau ist ein Abbild des niedrigsten mechanischen Vorgangs, des Falls. [Steinthal 1860, 328–329] Steinthal stellt Sprachsystematiken auf und ordnet sie ebenso wie andere Sprachforscher seiner Zeit in vollständigere und unvollständige Sprachen ein. Dafür interessierte sich Gödel wie sein Eingehen auf die Unvollständigkeit altaischer Sprachen und Bantusprachen zeigt.7 Das von Steinthal angeführte Zitat verweist zudem auf Formen und Kriterien der Systematisierung, die damit in Zusammenhang gebracht werden können, was in De anima in ein Verhältnis zueinander gesetzt wird. Denn die Abgrenzungen von Materie und Leben, Pflanzen und Tieren, Tieren und Menschen ist auch Gegenstand der Überlegungen in De anima, und dies, sowie die Gemeinsamkeiten des Lebendigen in eine Systematik zu bringen, ist unter anderem eines der zentralen Anliegen Gödels in Heft VI seiner Philosophischen Bemerkungen. Dazu passen Steinthals Überlegungen, die Systematik der Sprachen mit derjenigen Systematik zu vergleichen, die von seinen Zeitgenossen bei Lebewesen aufgestellt wurden.

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Solche Systematiken aufzustellen, erlaubt nicht nur die Welt zu ordnen, sondern darüber hinaus auch das Verhältnis der Wissenschaften zueinander, die mit Leben, toter Materie, dem Denken und dem Nachdenken darüber beschäftigt sind. Es werden auf diese Weise zwischen Fragestellungen und Herangehensweisen in den Disziplinen Analogien gezogen. Dieses Vorgehen ist für Gödel typisch. Er untersucht philosophische Probleme und Fragestellungen, wie ‚Begriff‘, ‚Bedeutung‘, ‚Bewusstsein‘, ‚Verantwortung‘, ‚Urteilen, ‚Erkenntnis‘ und ‚Freiheit, indem er schaut, ob und was einzelne Disziplinen dazu zu sagen haben, um es dann, oberflächlich betrachtet, nebeneinander zu stellen. Schaut man genauer hin, sieht man, dass Gödel die Begriffe und Fragestellungen auf diese Weise aus mehreren Perspektiven betrachtet, um ihnen einen vollständigeren und tieferen Sinn zu entnehmen.

3 Thomas Kommentar zu De anima und die Bibliotheken in Princeton um 1940

Steinthal und Thomas von Aquin sind also nicht so ganz zufällig in einem Atemzug genannt. Nachdem man daher zeigen kann, dass Gödel Steinthal rezipiert hat,8 lohnt es sich umso eher zu untersuchen, ob sich dasselbe für Thomas von Aquins Kommentar zu De anima sagen lässt. Mittels der eingesehenen Liste der Bücher aus Gödels Privatbibliothek lässt sich diese Annahme zunächst nicht verifizieren, denn dort ist der Kommentar zu De anima von Thomas ebenso wenig vermerkt, wie Steinthals Charakteristik der hauptsächlichen Typen des Sprachbaues. Unter „Thomas von Aquin“ findet man dort zwar die Summa philosophica9 in einer Ausgabe von 1877 als eines der Werke, die sich in seinem persönlichen Besitz befunden haben, nicht jedoch den Kommentar und auch kein Werk von Chajim Heymann Steinthal. Es ist allerdings bekannt, dass Gödel ein überaus emsiger Nutzer der Bibliothek des Institute for Advanced Studies in Princeton gewesen ist, weshalb man sich auf die Suche nach Ausgaben vor 194210 in den dortigen Katalogen machen kann. An dieser Stelle sei darauf hingewiesen, dass es bereits auf Grund von Gödels eigenem Hinweis (Thomas Comm. De anima) nahe liegend ist, dass Gödel Thomas auf Latein gelesen hat und nicht auf Deutsch. Denn der Titel der auch schon zu Gödels Zeit erhältlichen deutschen Übersetzung [Thomas Von Aquin 1268c] enthält nicht wie die lateinische Fassung11 den Hinweis auf einen „Kommentar“, sondern auf „Erklärungen“. Es genügt daher nach lateinischen Ausgaben zu suchen. Lassen sich nun Ausgaben der genannten Bücher in Princetons akademischen Bibliotheken für den Zeitraum vor 1942 nachweisen, sei es im Institute for Advanced Studies (IAS) oder in der Bibliothek der Universität? Die Online-Recherche, die man heute leicht vom heimischen Computer aus durchführen kann, ist nur in Bezug auf Steinthals Werk positiv, nicht jedoch in Bezug auf den Kommentar oder einschlägige Gesamteditionen von Thomas’ Werk. Fernleihen waren nach Auskunft der Bibliothek des IAS zu dieser Zeit auch noch nicht möglich.12 Daher müsste man mit der Suche nach einem möglichen Lektüreexemplar für Gödel in Princeton um 1940 in anderen als den genannten Bibliotheken erfolgreich sein, will man nicht mit der Hypothese arbeiten, Gödel habe die Werke bereits in Wien exzerpiert oder der — in diesem speziellen Fall wohl abwegigen — Annahme folgen, er könne sich das Werk von einem Kollegen geliehen haben.

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Auf dem Weg zwischen Gödels ehemaligem Wohnhaus und dem Institute for Advanced Studies befindet sich die Princeton Theological Seminary Library, zu der Gödel grundsätzlich Zugang gehabt hat13 und in der er sich mit dem De anima-Kommentar von Thomas hätte befassen können. Im Katalog dieser Institution wird man dann auch fündig. Er weist die Opera Omnia aus, die in Parma im Verlagshaus Fiaccadori von 1852-1873 erschienen sind. Der Kommentar von Thomas von Aquin zu Aristoteles’ De anima erschien dort 1866 und ist in der Princeton Theological Seminary Library vorhanden.14 Gödel hatte in Princeton also grundsätzlich die Möglichkeit, sich in Thomas von Aquins Interpretation von De anima zu vertiefen. Wir können uns daher der spannenden Frage zuwenden, welches philosophische Interesse er gehabt haben könnte, dies zu tun.

4 Was könnte Gödel veranlasst haben, sich für Thomas von Aquins Interpretation von De anima zu interessieren?

Sollte man trotz der bisherigen Ergebnisse noch skeptisch sein, ob Gödel je einen Kommentar von Thomas zu De anima in der Hand gehalten hat, mögen einige Überlegungen dazu, warum Gödel ein inhaltliches Interesse gehabt haben könnte, sich mit Thomas von Aquins Interpretation von De anima auseinanderzusetzen, helfen, weitere Überzeugungsarbeit zu leisten. Gödel hatte ein besonderes Interesse am Körper-Geist und ebenso am Maschine-Geist- Problem, weil er diese Problematik im Zusammenhang mit seinem berühmten Unvollständigkeitssatz gesehen hat. Um dies in aller Kürze etwas näher zu erläutern, sei sein Aufsatz von 1951 zitiert: Either mathematics is incomplete in this sense, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified. [Gödel 1951, 310]15 Und anschließend: [...] The philosophical implications prima facie will be disjunctive too; however, under either alternative they are very decidedly opposed to materialistic philosophy. Namely, if the first alternative holds, this seems to imply that the working of the human mind cannot be reduced to the working of the brain, which to all appearances is a finite machine [...]. On the other hand, the second alternative, where there exist absolutely undecidable mathematical propositions, seems to disprove the view that mathematics is only our own creation; for the creator necessarily knows all properties of his creatures [...]. [Gödel 1951, 311] Gödel sieht seine Unvollständigkeits-Theoreme in philosophischer Hinsicht demnach als einen Hinweis darauf, dass die Reduktion oder Eliminierung des Geistes und abstrakter Entitäten aus dem mathematischen Denken nicht möglich ist.16 Sie sind in seiner philosophischen Weltauffassung gegen einen reinen Empirismus und damit gegen einen reinen Szientismus gerichtet, der nichts zulässt als beobachtbare Tatsachen. Die Frage danach, was der Geist ist oder sein könnte, interessiert ihn daher im Zusammenhang dessen, was seine berühmten Theoreme gezeigt haben und auch im Zusammenhang dessen, was Gödel damit zeigen wollte. Dies ergibt ein mögliches philosophisch motiviertes Interesse dafür, dass Gödel sich mit De anima näher

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beschäftigt hat, weil dort dargelegt wird, dass der Intellekt (auf Griechisch ‚νοῦς‘) kein körperliches Organ hat. Zudem hebt Thomas von Aquin, obgleich für ihn der Mensch ein Sinnenwesen ist, genau wie Aristoteles, die besondere Rolle des Intellekts hervor. Dieser erlaubt es, Abstraktionen von Einzelfällen vorzunehmen, abstrakte Begriffe zu bilden und er geht daher mit sprachlichen und symbolischen Fähigkeiten einher. Da der Intellekt unkörperlich ist, lässt sich das, wozu er befähigt, auch nicht auf materielle oder physikalische Grundlagen reduzieren und ist nicht auf das Erkennen von Körperlichem beschränkt. So ist es der Intellekt, der es dem Menschen ermöglicht, das Wesen der Dinge zu erkennen. Darauf, dass Thomas’ Kommentar einen guten Zugang bietet, um den sehr schwierigen Text von Aristoteles zu verstehen, hat nicht nur bereits Pico della Mirandola hingewiesen, sondern auch heute denkende und arbeitende Philosophen wie etwa Martha Nussbaum haben es getan.17 Und dass Gödel genauso gedacht haben könnte, zeigt der bereits zitierte Eintrag in Heft 0 der Max Phil, wo Gödel unter Primärliteratur notiert hat: Progr: Bücher, die durchzusehen sind: I. [… ] II. Originalarbeiten: Thomas: Summa, Opuscula (Quaestiones?) Aristoteles, Comm d Thomas, Kant. Dieser Eintrag lässt sich nämlich auch so interpretieren, dass Gödel an Stelle des Originalwerkes De anima von Aristoteles den Kommentar von Thomas dazu zu studieren gedachte, den er mithin als einen Primärtext eingeordnet hat. Selbstverständlich darf man nicht vergessen, dass Thomas von Aquin neben dem Kommentar zu Aristoteles’ De anima noch weitere Kommentare zu Werken von Aristoteles geschrieben hat, weshalb zu überlegen wäre, ob das „Comm“.. In Heft 0 der Max Phil-Hefte nicht zu „Comm“ zu ergänzen wäre. Da von Gödel im Folgenden aber nur der Kommentar zu De anima erwähnt wird, darf vermutet werden, dass Gödel schon an dieser Stelle lediglich an den De anima Kommentar gedacht hat. Gödel sieht zudem die Lektüre von Thomas’ Werken als eine mögliche Hilfestellung, um die Philosophie seiner Zeit zu verstehen, wie sich aus Max Phil 0, Seite 1 ergibt: Fr: Was muss man kennen, um historisch ein Verständnis für die moderne Philosophie zu gewinnen?: Plato (Neupatonism), Arist , Mittelalter Theol (Thomas und Nominalisten) (Locke, Hume, Mill, Kant, Hegel, Mach. Dieser Hinweis wird verständlicher, wenn man hinzunimmt, dass Gödel Aristoteles und Thomas von Aquin auf Seite 17 desselben Heftes unter anderen als Lehrer von Franz von Brentano nennt.18

5 Äußerungen Gödels zum „Intellekt“

Dass Gödel nun aber tatsächlich nicht ausschließlich Aristoteles’ De anima verwendet hat, sondern auch Schriften von Thomas von Aquin, vermag letztlich nur eine genaue Textanalyse zu klären. Eine der Passagen aus Heft VI der Philosophischen Bemerkungen erweist sich dabei bereits als signifikant (andere Hinweise werden folgen). In ihr wird direkt auf die Rolle des Intellekts (intellectus) eingegangen, womit auch schon die lateinische Übersetzung des ‚νοῦς‘ bei Aristoteles durch Thomas bei

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Gödel ihre Verwendung findet.19 Die angesprochene Stelle steht in Heft VI auf der Manuskript-Seite 404: Bem (Phil): Beweis des Arist, dass der Intellekt nicht körperlich ist und überhaupt kein körperliches Organ hat (und dass er letzten Endes überhaupt keine Natur hat, sondern eine bloße Möglichkeit ist, die alle haben) 20 ist letzten Endes der antinomische Charakter des „Alle“.a (Er sagt aber, wenn er eine bestimmte Natur hätte, würde diese ihn hindern, etwas von dieser Natur Verschiedenes wahrzunehmen.b) Könnte man aber nicht auch Typ des Intellektes annehmen? Der Mangel des Organs wird auch daraus bewiesen, dass der Intellekt und nur dieser der Ort der Formen [species] ist, denn ein Organ wäre dann auch ein Ort dieser

.21 Dieser Passus ist ausführlich zu erläutern, es sei jedoch vorweg darauf hingewiesen, dass die entscheidenden Hinweise dafür, dass Gödel tatsächlich einen Text von Thomas gelesen und rezipiert hat, in der Verwendung des Begriffs ‚species‘ in der erläuternden Klammer hinter ‚Formen‘ zu finden ist, sowie in der Formulierung „sed potentia tantum“. Wie interpretiert Gödel entsprechende Stellen bei Aristoteles und Thomas von Aquin, wonach der Intellekt kein körperliches Organ hat und auch nichts Körperliches ist, sondern eine bloße Möglichkeit, alles zu denken oder zu erkennen? Er greift die entsprechenden Passagen bei Aristoteles auf [Aristoteles 334–323 v Chr., Buch III, Kapitel 4, 429a 18–29]22 und fügt ihnen mit dem Hinweis auf den antinomischen Charakter des ‚Alle‘ eine Interpretation dafür hinzu, was es für einen Logiker und Mathematiker des 20. Jahrhunderts bedeutet, dass der Intellekt alles umfasst und alles erfasst („πάντα νοεῖ“ [Aristoteles 334–323 v Chr., Buch III, Kapitel 4, 429a 18]) wie es bei Aristoteles heißt. Er fügt der Aristotelischen Argumentation damit ein eigenes Argument für die Unkörperlichkeit des Intellekts hinzu, das im Zusammenhang mit der Reichweite der Ergebnisse seiner eigenen Unvollständigkeitsbeweise in der Mathematik zu verstehen ist. Die Grundstruktur des Arguments bei Gödel, das auf der Argumentation in De anima aufruht, ist die folgende: Organe des Körpers (und die entsprechenden Seelenteile) beziehen sich auf etwas in der Welt, das ihnen gewissermaßen entspricht. So reagiert die Nase auf Düfte, die Augen auf Seheindrücke, et cetera. Gäbe es nun so etwas wie ein Organ des Intellekts müsste auch dieses zu etwas Spezifischem in der Welt passen. Es gibt aber keine Menge in der Welt, die alles enthält, welche der Intellekt zum Gegenstand des Denkens machen könnte. Da es so etwas in der Welt nicht gibt, kann der Intellekt, der zumindest potentiell ‚alle‘ wieder zum Objekt machen kann, nichts Körperliches sein. Die Möglichkeit, ‚alle‘ wieder zum Objekt zu machen und damit fortzufahren, zeigt für Gödel die Uferlosigkeit der Mathematik, die man in seinem Aufsatz von 1951 als „inexhaustibility of mathematics“23 wieder findet. Diese, in Heft VI in Bemerkung Philosophie auf Seite 404 angeführte Uferlosigkeit bringt Gödel mit einer Aussage zum Sein Gottes in Verbindung (Est Deus sed potentia tantum), die eine von Thomas sehr häufig verwendete Formulierung (sed potentia tantum) enthält und sich auch in Thomas von Aquins Kommentar zu De anima findet. 24 Klassisch übersetzt lautet die Stelle bei Gödel: „Gott existiert, aber nur als (geistige) Kraft“. Da der Kontext der Bemerkung in Heft VI keine weiteren Aufschlüsse für die Interpretation dieses Zusatzes enthält, kann man Gödels Gottesbeweis heranziehen, um ihn genauer zu deuten. Gödels ontologischer Gottesbeweis endet damit, dass zu den Vollkommenheiten Gottes auch die Existenz gehört. Für die Interpretation der

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fraglichen Textstelle lässt sich insofern daraus direkt nichts ableiten. Der Beweis enthält jedoch Zwischenschritte, die in natürlichsprachlicher Übersetzung lauten: Theorem: Wenn etwas göttlich ist, dann gilt notwendigerweise, dass es etwas Göttliches gibt (G(x) → □∃y G(y)). Also: Wenn es etwas Göttliches gibt, dann gilt notwendigerweise, dass es etwas Göttliches gibt (∃x G(x) → □∃y G(y)); also: Wenn es möglich ist, dass es etwas Göttliches gibt, dann ist es auch möglich, dass notwenigerweise gilt, dass es etwas Göttliches gibt (⋄∃x G(x) → ⋄□∃ G(y)). Wenn es möglich ist, dass es etwas Göttliches gibt, dann gilt notwendigerweise, dass es etwas Göttliches gibt (⋄∃x G(x) → □∃ G(y)) (Bromand & Kreis 2011, 486, 484f., 397]). Das „Est Deus sed potentia tantum“ könnte also auch eine Überlegung Gödels dazu enthalten, inwiefern es der Möglichkeit nach etwas Göttliches gibt und das wäre dann die Uferlosigkeit der Mathematik beziehungsweise des mathematischen Denkens, auch wenn ein Verweis auf die Existenz Gottes als geistige Kraft die zunächst nahe liegende Interpretation ist.25 An dieser Stelle lässt sich hinsichtlich der Ausgangsfrage „Hat Gödel Thomas von Aquins Kommentar zu De anima rezipiert?“ daher lediglich festhalten, dass Gödel für seine Bemerkung mit „sed potentia tantum“ eine Wendung benutzt, die zwar nicht nur bei Thomas von Aquin vorkommt, von diesem aber besonders gerne verwendet wird. Mit „Er sagt aber, wenn er eine bestimmte Natur hätte, würde diese ihn hindern, etwas von dieser Natur Verschiedenes wahrzunehmen“ folgt dann eine Abwandlung des schon vorgestellten Arguments von Aristoteles, nach welchem ein Organ und der ihm zugeordnete Seelenteil nur das ihm Spezifische wahrnehmen kann und nichts anderes. Der dialektische Einwand, der dann folgt („Könnte man aber nicht auch Typ des Intellektes annehmen ?“), verweist darauf, dass verschiedene Typen des Intellekts verschiedene, spezifische Dinge wahrnehmen würden, dass man damit also das Argument entkräften könnte, nach welchem Organe26 der Wahrnehmung von Spezifischem dienen, während das beim Intellekt nicht der Fall sei. An Stelle einer genauen Erwiderung auf diesen Einwand wird aber ein weiteres Argument für die Unkörperlichkeit des Intellekts vorgebracht. Der letzte Abschnitt der Bemerkung („Der Mangel des Organs wird auch daraus bewiesen, dass der Intellekt und nur dieser der „Ort“ der Formen [species] ist, denn ein Organ wäre dann auch ein Ort dieser “) legt mittels eines weiteren, neuen Arguments dar, inwiefern nur der Intellekt der Ort der Ideen (Formen, species) ist. Denn, selbst wenn dieser ein Organ wäre, wäre dieses Organ doch auch wiederum ein Ort der Ideen und damit auf nichts Konkretes in der Welt gerichtet, weshalb ihm nichts Körperliches in der Welt entspräche und es kein Organ für die Wahrnehmung von Körperlichem sein könne. Das Organ wäre vielmehr wiederum der Ort der Form und damit der Idee (oder Vorstellung) von dem Organ selbst. In dieser Argumentation findet man demnach wie im Unvollständigkeitssatz die Gedankenfigur der Selbstbezüglichkeit wieder. Gödel setzt das Denken in Selbstbezüglichkeiten als allgemeine Methode ein, um gegen ein materialistisches, reduktionistisches Weltbild zu argumentieren. Einige der für diese Diskussion relevanten Stellen aus Thomas’ Kommentar zu Aristoteles’ De anima sind die folgenden:27 622 Haec autem est differentia, qua differt cognitio intellectiva a sensitiva, quod sentire est aliquid corporeum. Non enim operatio sensus est sine organo corporali. Intelligere autem non est aliquid corporeum; quia operatio intellectus non est per organum corporeum, ut infra ostendetur.28

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In diesem Abschnitt wird zunächst schlicht konstatiert, dass Denken und Wahrnehmen insofern verschieden sind, als es für Sinneswahrnehmungen Sinnesorgane bedarf, während das hinsichtlich des Denkens nicht der Fall ist. Im Vergleich mit der Wortwahl bei Gödel („dass der Intellekt nicht körperlich ist und überhaupt kein körperliches Organ hat“) fällt die Übereinstimmung auf. Gödel benutzt auch das Wort Intellekt für νοῦς (und nicht etwa Verstand oder Vernunft) und entsprechend dem ‚organo corporali‘ bei Thomas die Wendung ‚körperliches Organ‘, während bei Aristoteles schlicht vom Körperlichen (σωματικόν)29 oder vom Organ (ὄργανόν)30 die Rede ist. Ein weiteres Zitat von Thomas von Aquin ist mit Hinblick auf die bisher angestellten Überlegungen zu Est Deus sed potentia tantum interessant: 683 Et ne quis crederet, quod esset hoc verum de quolibet intellectu, quod sit in potentia ad sua intelligibilia, antequam intelligat; interponit quod nunc loquitur de intellectu quo anima opinatur et intelligit. Et hoc dicit, ut praeservet se ab intellectu divino, qui non est in potentia, sed est quammodo intellectus omnium.31 Wenn bei Gödel die Möglichkeit ‚alle‘ wieder zum Objekt zu machen bedeutet, darauf hindeuten sollte, dass die Uferlosigkeit des mathematischen Denkens der Möglichkeit nach etwas Göttliches ist, besteht eine gewisse Diskrepanz zu der Aussage von Thomas von Aquin, denn dieser weist ausdrücklich darauf hin, dass der göttliche Intellekt nicht bloß möglich ist, sondern gewissermaßen ein Denken von allem ist. Für uns zeigt das etwas genauer, inwiefern Gödel Thomas von Aquin hier rezipiert hat. Es zeichnet sich nämlich ab, dass er tatsächlich mit Thomas’ Kommentar gearbeitet hat, ihn aber nicht in allen Aspekten auch übernommen hat, genauso wenig wie er das bei Aristoteles getan hat. Der nächste Abschnitt von Thomas von Aquin wurde ausgesucht, um auf Grund der Wortwahl, die sich bei Gödel findet, noch besser zeigen zu können, dass sie auf den Wortgebrauch bei Thomas zurückgeht. 686 [...] et dicit, quod ex quo pars intellectiva non habet organum sicut pars sensitiva, iam potest verificari dictum illorum, qui dixerunt, quod anima est locus specierum: quod per similitudinem dicitur, eo quod est specierum receptiva. Quod quidem non verum esset, si quaelibet pars animae haberet organum; quia iam species non reciperentur in anima sola, sed in coniuncto. Non enim visus solum est susceptivus specierum, sed oculus: et ideo non dicendum est, quod tota anima sit locus specierum, sed solum pars intellectiva, quae organum non habet. Nec ita est locus specierum, quod habeat actu species, sed potentia tantum.32 Zunächst wird bei Thomas noch einmal wiederholt, dass der Intellekt kein Organ hat,33 um dann hinzuzufügen, dass der vernünftige Teil der Seele der Ort der Ideen beziehungsweise Formen ist. Bei Gödel lautet die analoge Stelle: Der Mangel des Organs wird auch daraus bewiesen, dass der Intellekt und nur dieser der „Ort“ der Formen [species] ist, denn ein Organ wäre dann auch ein Ort dieser . Gödel spezifiziert ‚Formen‘ mittels des Ausdrucks ‚species‘ in der eckigen Klammer dahinter und verwendet damit einen für Thomas von Aquin typischen Begriff. So hat sich auch Anthony Kenny in seiner Einführung in Thomas von Aquins Philosophie geäußert: Das [...] verwendete Wort species spielt in Thomas’ Theorie des Denkens eine sehr wichtige Rolle. Zunächst taucht es als Synonym für das platonische Wort ‚Idee‘ auf, wird jedoch auch für die tatsächlich denkbaren Gegenstände verwendet [...]. Im Deutschen eignet sich vielleicht am besten das Wort ‚Vorstellung‘ dazu, die

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vielfältigen Facetten des Sinns des lateinischen Wortes species wiederzugeben. [Kenny 1999, 115]34 Im direkten Vergleich mit den Zitatstellen bei Thomas von Aquin und Kurt Gödel sind schließlich auch noch die entscheidenden Äußerungen von Aristoteles in De anima heranzuziehen, um eventuelle inhaltliche oder sprachliche Abweichungen oder wörtliche Übereinstimmungen feststellen zu können: [...] ähnlich wie das Wahrnehmungsfähige sich zum Wahrnehmbaren verhält, so muß sich die Vernunft (der Intellekt) zum Intelligiblen verhalten. Folglich muß sie, da sie alles erfasst, unvermischt sein [...], um zu erkennen. [...] Daher besitzt sie auch keine andere Natur als diese, daß sie vermögend (zum Erkennen) ist. Die sogenannte Vernunft der Seele also [...] ist nichts (identischerweise) von dem Seienden in Wirklichkeit, bevor sie erkennt. Daher hat es auch einen guten Grund, daß sie nicht mit dem Körper vermischt ist; denn dann nähme sie eine bestimmte Beschaffenheit an, [...] und hätte ein körperliches Organ, wie das Wahrnehmungsvermögen. Nun kommt ihr aber nichts (derartiges) zu. Und treffend äußern sich diejenigen, die sagen, die Seele sei der Ort der Formen, nur daß dies nicht die ganze, sondern die vernünftige Seele ist, und daß sie die Formen nicht in Wirklichkeit, sondern in Möglichkeit ist. [Aristoteles 334–323 v. Chr., 167; Hervorhebungen von E.-M. Engelen] […] καὶ ὁμοίως ἔχειν, ὥσπερ τὸ αἰσθητικόν πρὸς τὰ αἰσθητά, οὕτω τὸν νοῦν π ρὸς τὰ νοητά. ἀνάγκη ἄρα, ἐπεὶ πάντα νοεῖ, άμιγῆ εἶναι, […] […] ἵνα γνωρίξῃ […]; ὥστε μηδ΄ αὐτοῦ εἶναι φύσιν μηδεμίαν ἀλλ΄ ἢ ταύτην, ὅτι δυνατόν. ὁ ἄρα καλούμενος τῆς ψυχῆς νοῦς [… ] οὐθέν ἐστιν ἐνεργείᾳ τῶν ὄντων πρὶν νοεῖν;διὸ οὐδὲ μεμῖχθαι εὔλογον αὐτὸν τῷ σώματι; ποιός τις γὰρ ἂν γίγνοιτο, […] κἂν ὄ ργανόν τι εἴη, ὥσπερ τῷ αἰσθητικῷ; νῦν δ΄ οὐθέν ἔστιν. καὶ εὗ δὴ οἱ λέγοντες τὴν ψυχὴν εἶναι τόπον εἰδῶν, πλὴν ὅτι οὔτε λὅη άλλ΄ ἡ νοητική, οὔτε ἐντελεχείᾳ ἀλλὰ δυνάμει τὰ εἴδη. [Aristoteles 334–323 v. Chr., Buch III, Kapitel 4, 429a 16– 29, Hervorhebungen von E.-M. Engelen] Wir sehen hier, dass Gödel alle für seine eigene Bemerkung relevanten Stichpunkte auch direkt bei Aristoteles hätte nachlesen können, etwa, dass es die Vernunft beziehungsweise der Intellekt ist, der alles erkennt, dass dieser nichts Körperliches und insbesondere auch kein körperliches Organ ist, sondern der Ort der Formen beziehungsweise Ideen. Außerdem wird die Seele als der Ort bezeichnet, der die Formen oder Ideen der Möglichkeit nach enthält. Inhaltlich hätte sich Gödel also direkt auf Aristoteles beziehen können. Wie die Analyse des Wortgebrauchs aber zeigt, hat er den Kommentar von Thomas von Aquin zu Aristoteles’ Schrift De anima verwendet. Am deutlichsten sind die Übereinstimmungen mit dem Kommentar im Falle der Wendung „dass der Intellekt und nur dieser der „Ort“ der Formen [species] ist“, wie es bei Gödel heißt, die bei Thomas von Aquin „tota anima sit locus specierum“ lautet und bei Aristoteles „ τὴν ψυχὴν εἶναι τόπον εἰδῶν“. Sowohl Thomas von Aquin, als auch Aristoteles bezeichnen den vernünftigen Teil der Seele (ἡ νοητική bei Aristoteles, pars intellectiva bei Thomas) als den Ort der Vorstellungen oder Ideen, während Gödel nur vom Intellekt spricht. Gödel verwendet dann, ebenso wie Thomas von Aquin, species für εἰδῶν, und nicht ‚Ideen‘ wie es nach dem griechischen Original nahe liegend wäre. Das zeigt, dass Gödel den Kommentar als Lektüreexemplar verwendet hat, es weist aber auch darauf hin, dass Gödels Thema hier allein das Denken ist.

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6 Woher wusste Gödel von Thomas von Aquins Kommentar zu De anima?

Es gehört nicht zum Allgemeinwissen, dass Thomas von Aquin einen Kommentar zu Aristoteles’ De anima geschrieben hat. Wo oder durch wen könnte Kurt Gödel also davon gehört haben? Heinrich Gomperz, den Gödel ausdrücklich als seinen Lehrer in Wien nennt,35 und der zum erweiterten Wiener Kreis zu zählen ist, listet im zweiten Band seines Werkes Weltanschauungslehre unter anderem auf: Thomas von Aquino, „Summa Theologiae“, Turin, 1888, sowie Ch. H. Steinthal, „Einleitung in die Psychologie und Sprachwissenschaft (Abriß der Sprachwissenschaft I.)“, 2. Auflage, Berlin, 1881, [Gomperz 1908, 297]. Daraus ergibt sich zwar weder ein Hinweis auf Steinthals Sprachtypen, noch auf Thomas’ De anima-Kommentar, aber es zeigt, dass Gödel von beiden Autoren durch die Vorlesung von Gomperz, die er 1925-1926 in Wien gehört hat, erfahren haben könnte. Das wird auch durch eine Bemerkung in Heft IV der philosophischen Bemerkungen (Max Phil IV) bestätigt.36 Dort heißt es auf Seite 164: Bem Phil: Wesentliche Philosophen, die zu studieren sind: Plato, Aristot, Thomas, Descartes, Leibniz, Locke, Kant, Hegel, Plotin. (Lit siehe Vorlesung Gomperz).37 Zuvor ist auf Seite 161 Heft IV der Max Phil-Hefte zwar auch zu lesen, dass Gödel den Eindruck gewonnen hat, genug von Thomas von Aquins Werken zur Kenntnis genommen zu haben: Bem (Max): Bei manchen Büchern (Thomas, Fries) habe ich das Gefühl, dass ich ihm bereits „entwachsen“ bin. Daher eigenes Denken fruchtbarer.38 Aber auf Seite 164 führt Gödel ihn dann eben doch wieder als einen der Autoren an, die es zu studieren gilt und verweist auf die Lektüreliste, die Heinrich Gomperz mutmaßlich in seinen Vorlesungen gegeben hat. Dass Gödel Thomas nicht nur als einen reinen Kommentator von Aristoteles gesehen haben mag, deutet auch eine Stelle aus Heft V der Max Phil an, wo er ihn zwar als Epigonen bezeichnet, gleichzeitig aber auch zu erkennen gibt, dass Thomas von Aquin die Schriften des Aristoteles nach Meinung von Gödel aus einem bestimmten Grund verwendet hat: Sonne: Mond Bem Theol: Wirkliche Theol: Thomas = wirkliches Axiom: Euklid (Thomas & Euklid beide Epigonen). Es fehlt das Wesentliche in den Grundlagen [d.h. es wird nicht expl formuliert, sondern anschauungsmäßig vorausgesetzt], aber die Entwicklung in der Breite sehr weit. Aber in der Geom hat jeder die fehlende Anschauung, in der Theol nicht . Zu diesem Zweck Hinweis des Thomas auf Aristoteles. In unserer Zeit gibt es zwei ganz verschiedene Lichter, ein altes unreflektiertes [d.h. wirklichkeitsfernes] die Thomistik und ein neues wirklichkeitsnahes, die Wissenschaft.39 Die Bemerkung, dass die Thomistik unreflektiert sei, ist im Zusammenhang mit der Lichtmetapher zu verstehen. Während die Wissenschaften als eine Weise, die Welt zu erkennen, gesehen wird, deren Systematiken sich gewissermaßen mittels der Anschauung an der Wirklichkeit brechen, und dadurch nicht nur mit ihr in Berührung kommen, sondern sie auch auslotet, fehlt der thomistischen systematischen Theologie dieser Wirklichkeitsbezug. Sie ersetzt diese Leerstelle durch aristotelische Annahmen. Ob man diese Auffassung nun teilt oder nicht, ist in unserem Zusammenhang nicht maßgeblich. Wichtig ist hingegen festzuhalten, dass Gödel, obgleich er Thomas von

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Aquin als einen Epigonen bezeichnet, die thomistische Theologie dennoch als eine systematische Weise anerkennt, die Stellung des Menschen in der Welt zu verstehen. Sie ist allerdings auf eine Ergänzung durch Aristoteles angewiesen. Das bedeutet, dass Gödel die philosophischen Werke des Thomas von Aquin nicht lediglich als eine reine Wiederholung der aristotelischen Philosophie betrachtet, sondern ihnen einen eigenen, wenn auch ergänzungsbedürftigen Stellenwert einräumt.

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NOTES

*. Die Arbeit an diesem Aufsatz wurde von der französischen Forschungsgemeinschaft (Agence Nationale de la Recherche) im Rahmen des Projektes ANR-09-BLA-0313 unter der Leitung von Gabriella Crocco finanziert.

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1. Dass eine solche Fragestellung nicht ganz abwegig ist, mag eine gerade erschienene Untersuchung zu einer möglichen argumentativen Parallele zwischen Gödel und Thomas von Aquins zeigen: „[...] just as in Gödel-Rosser, we are forced at the limits of number theory into the metamathematical proposition that those limits are not closed, so in Aquinas, we are forced at the limits of reason into the metaphysical proposition that reason is open to that which altogether transcends it, that is, to God“. Eine Parallele, die Turner schließlich doch nicht für tragfähig erachtet, weil sich der mathematische und der theologische Begriff der Unvollständigkeit nicht in Deckung bringen lassen [Turner 2011, 279 und 294f.]. Siehe zu möglichen Verbindungen zwischen Gödel und Thomas von Aquin auch [Van Atten 2009, 26–27]. 2. Verloren gegangen ist Heft XIII. 3. Cheryl Dawson hat die Liste dem an der Université de Provence unter der Leitung von Gabriella Crocco angesiedelten Projekt „Kurt Gödel: de la logique à la cosmologie“ freundlicher Weise zur Verfügung gestellt. Gabriella Crocco möchte ich an dieser Stelle für die intensive Diskussion der Erstfassung des hier vorgelegten Aufsatzes danken. 4. Im Folgenden mit Max Phil-Hefte abgekürzt, oder auch als Philosophische Bemerkungen bezeichnet. 5. Transkription von Eva-Maria Engelen. Für die Transkription wurden Kopien verwendet, die mit Geldern des Centre d’ÉPistémologie et d’ERgologie Comparatives (Ceperc), einer Institution des CNRS, angesiedelt an der Université de Provence 2006 erworben worden sind. Die Erlaubnis, die in diesem Aufsatz verwendeten Bemerkungen aus den Kurt Gödel Papers abzudrucken, die sich im Bestand der Princeton University befinden, wurde vom The Shelby White and Leon Levy Archives Center des Institute for Advanced Study, Princeton, NJ, USA, erteilt. Die fettgedruckten Satzzeichen setzt Gödel selbst, die nicht fett gedruckten sind von E.-M. Engelen gesetzt. Die kursiv geschrieben Worte sind von Gödel in lateinischer Langschrift geschrieben. In den eckigen Klammern befinden sich Ergänzungen der Transkriptorin. Unterstreichungen sind solche von Gödel, fett gedruckte Wörter sind von Gödel im Manuskript doppelt unterstrichen oder markiert. All dies gilt auch für die noch im Folgenden abgedruckten Bemerkungen Gödels. 6. Heller-Roazen weist, ebenso wie Leinkauf und Dewender darauf hin, dass der Ausdruck ‚innerer Sinn‘ auf Augustin zurückgeht. Im deutschen Sprachraum wird in Zusammenhang mit ‚innerer Sinn‘ an Immanuel Kants Verwendungsweise gedacht. Vgl. Eisler, Artikel ‚Innerer Sinn‘, Online-Ausgabe [Eisler 1930]. Gödel selbst verwendet den Ausdruck ‚innerer Sinn‘, während Thomas von Aquin vom ‚sensus communis‘ spricht. 7. Max Phil VI, S. 339–440: „Bem (Phil): Keine Sprache enthält ursprünglich Falsch (sondern erst durch Missbrauch und Missverständnis), sondern sie sind nur mehr oder weniger unvollständig; z. B., die [440] altaischen Sprachen kein Behauptungszeichen und kein prädik Verhältnis. Was fehlt den Bantusprachen noch? Und fehlt nicht unserer Sprache etwas im Vergleich zur richtigen [Ur-]Sprache?“ Transkription von Eva-Maria Engelen unter Zuhilfenahme einer ersten Transkription von Cheryl Dawson. Auf das Altaische geht auch Steinthal ein [Steinthal 1860, 328f.]. Ein Hinweis auf Bantu fehlt dort. Er hat sich jedoch an anderen Stellen damit auseinandergesetzt. 8. Ein weiterer Hinweis ergibt sich aus einer Bemerkung Gödels von Seite 384 von Max Phil VI, wo es heißt: Bem (Gr): Der Unterschied der Sprachen, in denen Worte wie „ich“ vorkommen, ist, dass nicht die Formen der Sprachen bezeichnen, sondern die einzelnen Zeichen selbst. Die verschiedenen Weisen Sätze auszudrücken, in denen man auf sich selbst referiert, ohne über ein eigenes Wort wie „ich“ zu verfügen, spricht Steinthal auf vielen Seiten an [Steinthal 1860, 194, 199, 205, 219, 229 und weitere] Transkription von Eva-Maria Engelen unter Zuhilfenahme einer ersten Transkription von Cheryl Dawson. 9. Unter dem Titel Summa Philosophica verbirgt sich die Summa contra gentiles.

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10. Heft VI der Philosophischen Bemerkungen von Gödel stammt von 1942, daher können später erworbene oder erschienene Ausgaben nicht berücksichtigt werden. Mit Hinweisen zu möglichen Ausgaben hat dankenswerter Weise Rolf Schönberger weiter geholfen. 11. Zu Gödels Zeit käme dafür etwa in Frage: Thomas von Aquin, In Aristotelis librum De anima commentarium, [Thomas Von Aquin 1268b]. Diese Ausgabe ist in der Universitätsbibliothek in Princeton aber nur in einer Ausgabe von 1948 vorhanden; die Ausgaben aus den Verlagshäusern Hédde Paris 1912 und Vivès Paris 1872-1880 sowie andere aus dem 19. und frühen 20. Jahrhundert sind nicht nachweisbar. Die Referenzausgabe Editio Leonina beginnt zwar 1882, der Kommentar zu De anima stammt aber erst aus dem Jahre 1984. 12. Karen Downing vom Interlibrary Loan Service des IAS war so freundlich, mir das mitzuteilen. 13. Den Hinweis auf diese Möglichkeit verdanke ich Marcia Tucker von der Bibliothek des IAS: „To further complicate figuring out where Gödel may have consulted Aquinas’ work, he would have had access to the Princeton Theological Seminary Library which is somewhat on the way to the Institute from where Gödel lived. The Institute’s (IAS) library has several older editions, but all were purchased after 1950“. Zu ergänzen ist hier, dass es auch die Ausgaben der University of Princeton erst nach 1948 erlaubt hätten, Thomas’ Kommentar zu De anima zu studieren. 14. Nach Auskunft der für die Princeton Theological Seminary Library zuständigen Bibliothekarin Kate Skrebutenas ist Band XX in einem der im Katalog aufgeführten 13 Folio- Bände enthalten. 15. Hervorhebung von Gödel. Weitere Zitate, in denen er sich in diesem Sinne äußert, lassen sich beispielsweise zwei Briefen entnehmen: „My theorems only show that the mechanization of mathematics, i.e., the elimination of the mind and of abstract entities, is impossible, if one wants to have a satisfactory foundation and system of mathematics. I have not proved that there are mathematical questions undecidable for the human mind, but only that there is no machine that can decide all number theoretical questions [...]“, [Gödel 1962, 176]. Und in einem weiteren Brief schreibt er: „What has been proved is only that the kind of reasoning necessary in mathematics cannot be completely mechanized. Rather constantly renewed appeals to mathematical intuition are necessary. The decision of my „undecidable“ proposition [...] results from such an appeal. [...] Whether every arithmetical yes or no question can be decided with the help of some chain of mathematical intuitions is not known. At any rate it has not been proved that there are arithmetical questions undecidable by the human mind. Rather what has been proved is only this: Either there are such questions or the human mind is more than a machine. In my opinion the second alternative is much more likely“ [Gödel 1967, 162]. 16. Auf diesen Aspekt geht Hilary Putnam nicht ein [Putnam 2011]. Putnam konzentriert sich vor allem darauf zu zeigen, dass die Behauptungen falsch seien, nach welchen Gödels Beweise zeigen sollen, dass der menschliche Geist nicht identisch mit unserem Gehirn oder einem anderen physikalischen System seien (John Lucas) oder dass wir eine andere Physik bräuchten, um zu erklären, inwiefern menschliche Gehirne zu mathematischem Denken fähig sind (Roger Penrose). 17. „Also produced in the thirteenth century is one of the greatest commentaries on the work, by Thomas Aquinas“ [Nussbaum & Rorty 1992, 3]. Siehe zu den Abweichungen von Aristoteles’ Text durch Thomas von Aquin in dessen Kommentar hingegen [Seidl 1995, XXXV–XLII]. 18. Eine weitere Nennung findet sich in Max Phil I, auf Seite 46: „8a. Bibel, Thomas im kleinen Opuscula, Summa contra gentiles“. Transkriptionen jeweils von Eva-Maria Engelen. 19. ‚νοῦς‘ kann auch mit ‚Verstand‘ oder, je nach Kontext, mit ‚Vernunft‘ ins Deutsche übersetzt werden. Alois Mager, der den Kommentar von Thomas von Aquin zu Aristoteles’‘ De anima ins Deutsche übersetzt hat, verwendet für seine Übersetzung an keiner Stelle ‚Intellekt‘. 20. Im Manuskript steht an dieser Stelle ‚werden’, das hier zu ‚haben’ verbessert wurde. Gemeint wären bei dieser Lesart ‚alle Menschen’, nicht jedoch bei den folgenden Vorkommnissen von ‚alle’.

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21. Transkription von Eva-Maria Engelen unter Zuhilfenahme einer ersten Transkription von Cheryl Dawson. 22. Aristoteles bezieht sich an dieser Stelle auf Anaxagoras. 23. „The inexhaustibility of mathematics, however, always is present in some form, no matter which standpoint is taken“ [Gödel 1951, 305]. „[...] not only that the procedure just described can be carried out with any operation, but that moreover there should exist a set closed with respect to it, that is, one which has the property, that, if this procedure (with any application) is applied to elements of this set, it again yields elements of this set“ [Gödel 1951, 307]. 24. Die Wendung „sed potentia tantum“ findet man bei Thomas etwa in [Thomas von Aquin 1268b, Liber III, lectio VII, 677, 686 und Liber III, lectio IX, 722, S. 227, 229, 237]. In der Online-Ausgabe des Corpus Thomisticum, lib. 3 l. 7 n. 7 (Anaxagoras vero dixit intellectum esse simplicem et immixtum, et nulli rerum corporalium habere aliquid commune. Ex eo ergo quod dictum est, quod intellectus non est actu intelligens, sed potentia tantum, concludit quod necesse est intellectum, propter hoc quod intelligit omnia in potentia, non esse mixtum ex rebus corporalibus, sicut Empedocles posuit, sed esse immixtum sicut dixit Anaxagoras.); lib. 3 l. 7 n. 16 (Non enim visus solum est susceptivus specierum, sed oculus: et ideo non dicendum est, quod tota anima sit locus specierum, sed solum pars intellectiva, quae organum non habet. Nec ita est locus specierum, quod habeat actu species, sed potentia tantum.) und lib. 3 l. 9 n. 3 (Intellectus igitur dicitur pati, inquantum est quodammodo in potentia ad intelligibilia, et nihil est actu eorum antequam intelligat. Oportet autem hoc sic esse, sicut contingit in tabula, in qua nihil est actu scriptum, sed plura possunt in ea scribi. Et hoc etiam accidit intellectui possibili, quia nihil intelligibilium est in eo actu, sed potentia tantum). Hervorhebung E.-M. Engelen. 25. Hinsichtlich Gödels Verwendung des Satzes „Est Deus sed potentia tantum“ ist sicherlich noch weitere Forschungsarbeit zu leisten, die den Rahmen der hier verfolgten Fragestellung jedoch sprengen würde. 26. Wenn von ‚Organ‘ die Rede ist, ist das dazugehörige Wahrnehmungsvermögen im Falle von Wahrnehmen stets mitgemeint. 27. Benutzt wurde neben der Ausgabe von P. F. Angeli M. Pirotta von 1925 und die Online- Ausgabe (Corpus Thomisticum). 28. Übersetzung: „Das aber ist der Unterschied zwischen Denken und Sinneserkenntnis, daß Sinneswahrnehmung etwas Körperliches ist. Denken aber ist etwas Unkörperliches. Denn die Denktätigkeit ist an kein körperliches Organ gebunden, wie später gezeigt werden wird [684]“ [Thomas Von Aquin 1268c 327]. Die Übersetzungen von Mager sind für heutige Leser an einigen Stellen problematisch, erlauben aber einen ersten Zugang. Hier hat Mager etwa vergessen, einen Teilsatz zu übersetzen, der daher ergänzt wurde. Die Hervorhebungen hier und im Folgenden von E.-M. Engelen. 29. Thomas von Aquin kommentiert damit den folgenden Passus bei Aristoteles: „πάντες γὰρ οὗτοι τὸ νοεῖν σωματικον ὥσπερ τὸ αἰσθάνεσθαι ὑπολαμβάνουσιν, καὶ αἰσθάνεσθαί τε καὶ φρονεῖν τῷ ὁμοίῳ τὸ ὅμοιον “ [Aristoteles 334-323 v. Chr., Buch III, Kap. 3, 427a 26–29]. Hervorhebung E.-M. Engelen. Alle diese Denker nehmen nämlich das vernünftige Erfassen als Körperliches an, wie das Wahrnehmen, und meinen, daß man das Gleiche durch das Gleiche wahrnehme und einsehe [Aristoteles 334–323 v. Chr., 155]. 30. In dem hier besprochenen Kontext ist auch der folgende Passus einschlägig: „ ὁ ἄρα καλούμενος τῆς ψυχής νοῦς (λέγω δέ νοῦν ᾧ διανοεῖται και ὑπολαμβάνει ἡ ψυχή) οὐθέν ἐστιν ἐνεργείᾳ τῶν ὄντων πρὶν νοεῖν. διὸ οὐδὲ μεμῖχθαι εὔλογον αὐτὸν τῷ σώματι. ποιός τις γὰρ ἂν γίγνοιτο, ἢ ψυχρὸς ἢ θερμός, ἢ κἂν ὄργανόν τι εἴη, ὥσπερ τῷ αἰσθητικῷ. νῦν δ΄ οὐθὲν ἔστιν. καὶ εὗ δὴ οἱ λέγοντες τήν ψυχὴν εἶναι τόπον εἰδῶν, πλὴν ὅτι οὔτε ὅλη ἀλλ΄ ἡ νοητική, οὔτε ἐντελεχείᾳ ἀλλὰ δυνάμει τὰ εἵδη “ [Aristoteles 334–323 v. Chr. Buch III, Kap. 4, 429a 22–29]. Hervorhebung E.-M. Engelen. „Die sogenannte Vernunft der Seele also — ich nenne Vernunft das,

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womit die Seele nachdenkt und Annahmen macht — ist nichts (identischerweise) von dem Seienden in Wirklichkeit, bevor sie erkennt. Daher hat es auch einen guten Grund, dass sie nicht mit dem Körper vermischt ist; denn dann nähme sie eine bestimmte Beschaffenheit an, würde kalt oder warm, und hätte ein (körperliches) Organ wie das Wahrnehmungsvermögen“ [Aristoteles 334–323 v. Chr., 167]. ‚Körperliches‘ ist, wie die Klammersetzung anzeigt, von Theiler zur Erläuterung ergänzt. 31. Übersetzung: „Damit niemand meine, daß das von jedem Denken gelte, dass es, bevor es denkt, der Möglichkeit nach das Gedankliche ist, macht er die Zwischenbemerkung, daß nur von dem Denkvermögen die Rede sei, mit dem unsere Seele überlegt und denkt. Er macht diese Bemerkung, um einen Vorbehalt wegen des göttlichen Denkens zu machen, der [sic!] nie Möglichkeit, sondern gleichsam das Alldenken ist“ [Thomas Von Aquin 1268c, 351]. 32. Übersetzung: „Auf Grund der Tatsache, daß die Denkseele kein körperliches Organ hat, wie die Empfindungsseele, kann die Behauptung derer gerechtfertigt werden, die der Auffassung waren, die Seele wäre der Raum der gedanklichen Wesenheiten. Das ist nur vergleichsweise gesagt, weil die Seele für die gedanklichen Wesenheiten aufnahmefähig ist. Das träfe nicht zu, wenn jeder Seelenbereich ein Organ hätte. Denn die gedanklichen Wesenheiten würden ja nicht ins Seelische allein, sondern in Verbindung von Seelischem und Körperlichem aufgenommen: Denn nicht der Gesichtssinn allein ist aufnahmefähig für die sinnfälligen Formen, sondern das Auge. Deshalb darf man nicht sagen, daß die ganze Seele Raum der gedanklichen Wesenheiten sei, sondern nur die Denkseele, die kein Organ besitzt. Sie ist auch nicht in dem Sinn ihr Raum, daß sie der Wirklichkeit nach diese Wesenheiten in sich schließt, sondern nur der Möglichkeit nach“ [Thomas Von Aquin 1268c, 352]. Mager übersetzt ‚species‘ mit gedanklichen Wesenheiten, besser wäre es ‚species‘ mit ‚Ideen‘ oder ‚Vorstellungen‘ zu übersetzen und ‚locus‘ nicht mit ‚Raum‘, sondern mit ‚Ort‘. 33. Das ist auch noch an anderen Stellen der Fall, etwas in den beiden im Kommentar zu De anima folgenden: 687 „Sed de intellectu hoc accidere non potest, cum organo careat; unde nec per se nec per accidens passibilis est“. (Übersetzung: „Beim Denkvermögen kann das nicht vorkommen, da es kein Organ besitzt. Darum ist es sowohl an sich als auch nebenbei unfähig, etwas zu erleiden“. [Thomas Von Aquin 1268c, 353]); sowie in 699 „Mirum est autem quomodo tam leviter erraverunt, ex hoc quod dicit quod intellectus est separatus, cum ex litera sua huius rei habeatur intellectus, dicit enim separatus intellectus, quia non habet organum, sicut sensus“. (Übersetzung: „Man muß sich aber wundern, daß sie so leicht dem Irrtum verfielen, deshalb, weil er sagt, das Denkvermögen sei leibgetrennt, da doch aus dem Text hervorgeht, daß er mit leibgetrenntem Denkvermögen meint, daß es kein Organ, wie das Sinnesvermögen, hat“ [Thomas Von Aquin 1268c, 355–356].). 34. Im englischen Original heißt es bei Kenny: „The word species which is transliterated in the above passage is a word which plays a very important role in Aquina’s theory of thought. It appears first as a synonym for the Platonic word `idea’, but it is used also for the actually thinkable objects which figure in the Aristotelian theory. The English word `idea’ is indeed the best to capture the multiple facets of the sense of the Latin word species [Kenny 1980, 69] 80. Vgl. auch: Form: „that, by its presence, makes what exists in potentiality exist in actuality“. Species: „Aquinas most general term for any sort of cognitive representation within either sense or intellect“ [Pasnau & Shields 2004, 244]. 35. In seinem Kurzlebenslauf für den Promotionsantrag verweist Gödel darauf, dass er bei Schlick, Gomperz und Carnap Vorlesungen gehört hat. Zuletzt zitiert in: [Siegmund 2011, 76]. Siehe auch [Wang 1988, 8]. 36. Auf Grund des Namenindex von Cheryl Dawson konnten diejenigen Stellen, in denen Thomas von Aquin genannt wird, selbst in denjenigen Max Phil-Heften ausfindig gemacht werden, die noch nicht transkribiert sind. Der Namensindex konnte bereits provisorisch erstellt werden, weil Namen fast immer in lateinischer Langschrift geschrieben sind.

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37. Transkription Eva-Maria Engelen. Eine erste kursorische Durchsicht von Gödels gleichfalls in Gabelsberger geschriebenen Vorlesungsmitschrift, die mit den übrigen Manuskript-Konvoluten im Archiv in Princeton liegt, hat leider keinen Hinweis auf eine Lektüreliste oder auf sonstige Angaben zu Werken von Thomas von Aquin ergeben. Auch in dem Werk Griechische Denker [Gomperz 1896–1902] von Theodor Gomperz, dem Vater von Heinrich, lässt sich ein solcher Literaturhinweis nicht finden. 38. Transkription von Eva-Maria Engelen. 39. Transkription von Eva-Maria Engelen unter Zuhilfenahme einer ersten Transkription von Cheryl Dawson.

NOTES DE FIN a. Oder besser, die Möglichkeit „alle“ wieder zum Objekt zu machen und darüber hinauszugehen (die Uferlosigkeit) → Est Deus sed potentia tantum. b. Oder zumindest würde man alles in diesem Lichte sehen.

RÉSUMÉS

La recherche d’une réponse à la question qui constitue le titre a conduit à des éclaircissements concernant la réception critique d’œuvres philosophiques majeures par Kurt Gödel. Cela illustre la manière dont il utilise des argumentations philosophiques d’auteurs classiques et les change en des aspects nouveaux pour sa propre argumentation philosophique. Dans le cas qui nous concerne, Gödel emploie un argument classique d’Aristote pour l’immatérialité de l’âme afin d’ajouter certains éléments à son propre raisonnement concernant l’inexhaustibilité des mathématiques, le problème corps-esprit, et ses considérations à propos de Dieu. Et réciproquement, il fournit de nouveaux éclaircissements concernant un argument dont la réception a une longue histoire.

The search for an answer to the question that constitutes the title has led to some insightful results concerning Kurt Gödel’s critical reception of major philosophical works. It shows how he uses philosophical argumentations of classical authors and turns them into new aspects for his own philosophical argumentation. In the case at hand a classical argument by Aristotle for the immaterialness of the soul is used by Gödel in order to add considerations to his own reasoning for the inexhaustibility of mathematics, the mind-body-problem and his considerations about God; and vice versa to give new insights into an argument with a long history of reception.

AUTEUR

EVA-MARIA ENGELEN Universität Konstanz (Deutschland) Aix Marseille Université, CNRS – Délégation Provence et Corse (Frankreich)

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