Fig. III.2A. Gauss - Weber collage according to an idea of Friedrich Zöllner. Reproduced from [Zöllner 1878], page v, by courtesy of NSUB Göttingen. III.2 ‘O θo`ςαριθμητιζ´ ι — The Rise of Pure Mathematics as Arithmetic with Gauss JOSE´ FERREIROS1 The beautiful picture facing this page, which displays Carl Gauss and his colleague physicist Wilhelm Weber, is noteworthy not just for the quality and accuracy of the portraits, but also because of the mottos it includes. It was composed upon the inspiration of Friedrich Zöllner (1834–1882), an astrophysicist, professor at Berlin, and close follower of Weber.2 Among the mottos, one originates with Gauss himself: o θo`ςαριθμητιζ´ ι, God does arithmetic, or more literally “God arithmetizes” – presumably meaning that in his thoughts God is always dealing with numbers and number-relations. This motto is an adaptation of a sentence attributed to Plato: o θoςαιγωμτρι´ , “God geometrizes eternally.”3 Although this sentence is not found in Plato’s dialogues, nevertheless they offer declarations in the same spirit (see for instance the dialogue The Republic). Interestingly, the very same sentence, in Greek, is quoted by Kepler in his first work, Mysterium cosmographicum (1596), a book that Gauss must have been familiar with. Plato’s words can be found in the crucial chapter where Kepler describes the main guidelines of his peculiarly platonic 1. Universidad de Sevilla. I am grateful to participants in the Gauss Tagung at Oberwolfach for their reactions, and more particularly to Catherine Goldstein, Norbert Schappacher and Jeremy Gray for their criticism and suggestions based on a previous version of this paper. My work was partially supported by the Ministerio de Ciencia y Tecnología, Spain, grant BFF2000–0097–C02–02. 2. See [Zöllner 1878]. The portrait of Gauss was based on the well-known oil painting by Jensen and on the medal prepared for the 1877 centenary by the Göttingen Academie der Wissenschaften. Weber was portrayed on the basis of a photograph taken in 1877. The drawing is by A. Weger. See op. cit., v. 3. See [Plutarch], book VIII, quest. 2, where Plutarch indicates that the phrase cannot be found in extant writings of Plato. 207 208 III. The German reception of the D.A. reconstruction of the architecture of the universe, based on a clever combination of the regular solids.4 Several witnesses attest to the fact that the motto stems directly from Gauss, among them his friend the Göttingen professor Sartorius von Waltershausen, and the physician who treated him, Dr. Wilhelm Baum. Thus, Dr. Baum wrote to Alexander von Humboldt immediately after Gauss’s death: The last days of his life were often very painful owing to the aggravated complaint of dropsy, which the hypertrophy of his heart produced — but still he always main- tained his freedom and greatness of spirit, the strongest conviction of his personal permanence, the firmest hope in the still deeper intelligent insight into the number- relationships, which God places in matter and which he would perhaps be able to recognize in the intensive magnitudes, for he used to say ‘o θo`ςαριθμητ´ιζ ι.’5 In fact, Gauss employed those words when talking about questions that lay beyond the reach of human knowledge, at least in its present stage. Waltershausen explains it as follows: By science he understood only that rigorous logical edifice, closed in itself, whose foundations rest on some truths generally acknowledged by the human mind, truths which, once admitted, open for us an immense field of most intricate researches, linked to one another by an iron chain of thoughts. Therefore, as we have already mentioned, he placed arithmetic at the top, and, in connection with questions that we cannot ascertain scientifically, he loved to employ the words: ‘O θo`ςαριθμητ´ιζ ι, with which he acknowledged the logic that goes through the whole cosmos, also for those domains in which our mind is not allowed to penetrate.6 The Gaussian motto implies that there is a theological explanation for the im- pressive applicability and effectiveness of mathematics: God’s creation bears the mark of His thought, which we are able to grasp because we are of His lineage. But this point is perhaps less interesting than the way in which his adaptation of Plato’s words reflects changing perceptions of mathematical knowledge. The above motto and related statements document the end of the thousand year long domination of geometry in Western images of mathematics, and the corresponding rise of arith- metic as the paradigmatic mathematical discipline. As Hilbert wrote at the close 4. See [Kepler 1858], p. 124. 5. Quote from [Dunnington 1927], p. ***. I should mention that M. Kline has wrongly attributed the motto “God ever arithmetizes” to Jacobi; see [Kline 1972], vol. 3, 1026. While the attribution is wrong factually, it seems however quite right in spirit (see below). 6. See [Waltershausen 1856], p. 97: Unter Wissenschaft verstand er allein jenes streng in sich abgeschlossene logische Gebäude, dessen Fundamente auf gewissen vom menschlichen Geist allgemein anerkannten Wahrheiten beruhe, die ein Mal zugegeben ein unabse- hbares Feld der verwickeltsten durch eine eiserne Gedankenkette mit einander zusam- menhängenden Forschungen gestatte. Er stellte daher wie schon bemerkt die Arithmetik an die Spitze und pflegte in Bezug auf Fragen die für uns wissenschaftlich nicht zu ergründen sind die Worte zu gebrauchen: ‘O θo`ςαριθμητ´ιζ ι, womit er die durchs ganze Weltall gehende Logik auch für solche Gebiete anerkannte, in welche einzudringen unserm Geiste nicht verstattet ist. 2. FERREIROS´ : Pure mathematics as arithmetic 209 of the century,7 mathematics in the XIXth century had developed “under the sign of number,” a topic that was discussed by Klein, Poincaré and others under the name of “arithmetization”.8 The purpose of this chapter is to discuss the complex of ideas and viewpoints related to the Gaussian conception of arithmetic and its role in mathematics: the new demarcation between pure and applied mathematics; aspects of the modern transformation of mathematics such as arithmetization and the conceptual approach; the role of conceptual developments, new cultural values, and epistemological views in the process. It will be argued that already in the case of Gauss, philosophical motives played a role in the rise of arithmetic as a paradigm, and that the intellectual atmosphere of Neohumanism promoted such approaches. It is obvious from this short summary that the picture I am about to draw is a complex, multifaceted one. While that may seem confusing to some readers, in my opinion it is essential to provide such a multifaceted analysis in order to account for the emergence of modern mathematics. 1. Pure mathematics: Arithmetic vs. geometry In order to understand what is behind Gauss’s motto, it is worthwhile to pause over the following sentences that the great mathematician wrote to Olbers in April 1817: I come more and more to the conviction, that the necessity of our [Euclidean] ge- ometry cannot be proven, at least not by human understanding nor for human un- derstanding. Perhaps in another life we come to different insights into the essence of space, that are now impossible for us to reach. Until then, we should not put geometry on the same rank with arithmetic, which stands purely a priori, but say with mechanics.9 Three aspects of this quote deserve to be emphasized. First, and as will become clearer soon, Gauss uses a rather philosophical language to express his views. Sec- ond, although the passage is usually quoted in connection with the foundations of geometry, it also presents us with a very characteristic view on the foundations of arithmetic. This branch of pure mathematics “stands purely a priori,” it consists of a priori knowledge. The language is Kantian, although the viewpoint might also be understood in the sense of Leibniz (see §5). A letter to Friedrich Wilhelm Bessel written in April 1830 is even more striking: According to my most intimate conviction, the theory of space has a completely different position with regards to our knowledge a priori, than the pure theory of 7. See [Hilbert 1897], p. 66. 8. See also chapter V.2 below. 9. See [Gauss 1900], p. 177: Ich komme immer mehr zu der Überzeugung, dass die Noth- wendigkeit unserer Geometrie nicht bewiesen werden kann, wenigstens nicht vom men- schlichen Verstande noch für den menschlichen Verstand. Vielleicht kommen wir in einem andern Leben zu andern Einsichten in das Wesen des Raumes, die uns jetzt unerreichbar sind. Bis dahin müsste man die Geometrie nicht mit der Arithmetik, die rein a priori steht, sondern etwa mit der Mechanik in gleichen Rang setzen. 210 III. The German reception of the D.A. magnitudes. Our knowledge of the former lacks completely that absolute conviction of its necessity (and therefore of its absolute truth) which is characteristic of the latter. We must humbly acknowledge that, whereas number is just a product of our minds, space also has a reality outside our minds, whose laws we cannot prescribe a priori.10 Here again, a clear distinction between geometric and arithmetic knowledge is drawn by characterizing the latter as “knowledge a priori,” consisting of necessary, certain, absolute truths. Concepts and terminology seem to be taken from Kant, who em- phasized that true science consists of necessary, certain, absolute truths, not just of empirical knowledge (see §5). The third aspect is that Gauss draws a neat distinction between what might be called, using 18th century language, pure and mixed mathematics. Mixed mathe- matics is that part of the discipline which deals with knowledge having (at least partly) empirical origins; specifically Gauss indicates that geometry and mechanics belong here. Pure mathematics is that part of mathematical knowledge which stands completely a priori.