Introduction: Hilbert's Axiomatics As 'Symbolic Form'?

Home , David Hilbert, Foundations of geometry

Introduction: Hilbert’s Axiomatics as ‘Symbolic Form’?

Rossella Lupacchini University of Bologna

According to Ernst Cassirer, symbolic forms “are not different modes in which an independent reality manifests itself to the human spirit but roads which the spirit proceeds towards its objectivization, i.e., its self- revelation” (1923, p. 78). Can Hilbert’s axiomatics be viewed as a symbolic form? The aim of this introductory essay is to frame the question in a ‘map’ and to provide a few signs to orient it. Each of the essays collected in this volume may contribute a different set of instructions to reach an answer. Both Hilbert’s axiomatics and Cassirer’s philosophy of symbolic forms have their roots in Leibniz’s idea of a ‘universal characteristic,’ and grow on Hertz’s ‘principles of mechanics,’ and Dedekind’s ‘foundations of arithmetic’. As Cassirer recalls in the introduction to his Philosophy of Symbolic Forms, it was the discovery of the analysis of inªnity that led Leibniz to focus on “the universal problem inherent in the function of symbolism, and to raise his ‘universal characteristic’ to a truly philosophical plane.” In Leibniz’s view, the logic of ‘things’ cannot be separated from the logic of ‘signs,’ as “the sign is no mere accidental cloak of the idea, but its necessary and essential organ.” Every ‘law’ of nature assumes for our thinking the form of a universal ‘formula’—and a formula can be expressed only by a combina- tion of universal and speciªc signs. [. . .] The content of the spirit is disclosed only in its manifestations; the ideal form is known only by and in the aggregate of the sensible signs which it uses for its expression. (Cassirer 1923, p. 86)

I am grateful to Wilfried Sieg for his detailed remarks on the ªrst draft of this essay.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 2 Hilbert’s Axiomatics as ‘Symbolic Form’?

It was by means of a “grammar of the symbolic function” that Leibniz was able to broaden the traditional and historical concept of idealism. “Ideal- ism,” Cassirer observes, “has always aimed at juxtaposing to the mundus sensibilis another cosmos, the mundus intelligibilis, and at deªning the boundary between these two worlds” (Cassirer 1923, p. 87). But for the ‘universal characteristic’ this opposition is no longer irreconcilable and ex- clusive: “the senses and the spirit are now joined in a new form of reciproc- ity and correlation.” The question at issue, in the language of Kantian philosophy, is “the share which thought, on the one hand, and experience, on the other, have in our knowledge.” Hilbert addressed it at Königsberg, in 1930, in order to highlight the core of his scientiªc carrier. He entrusted his answer to the axiomatic method, “a general method for the theoretical treatment of questions in the natural sciences,” and even more: already in daily life methods and concept-formations are used that demand a great measure of abstraction and that can be understood only by the unconscious application of axiomatic methods. For example, the general process of negation, and in particular the concept “inªnite.” (1930, p. 1159) Precisely in the concept of ‘inªnite,’ Hilbert identiªed “an important parallelism between nature and thought, a fundamental agreement between experience and theory.” The inªnite is not an ‘observable’ of nature; it is nowhere realized, and it is not to be admitted in our thought “without special precautions”. As far as the concept ‘inªnite’ is concerned, we must be clear to ourselves that ‘inªnite’ has no intuitive meaning [anschauliche Bedeutung] and that without more detailed investigation it has absolutely no sense. For everywhere there are only ªnite things. [. . .] There is absolutely nothing continuous that can be divided inªn- itely often. Even light has atomic structure [. . .] inªnity, because it is the negation of a condition that prevails everywhere, is a gigantic abstraction—practicable only through the conscious or unconscious application of the axiomatic method. (1930, p. 1159) It is mathematics that mediates between theory and practice, between thought and observation; it is mathematics that builds the connecting bridges. Hilbert also noticed that even though “the laws of the world can be acquired in no other way than through experience,” conic sections were studied “long before one suspected that our planets or even electrons move in such a course.” This is a very old and eloquent example of “an occur- rence which is virtually an embodiment and realization of mathematical

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 3

thoughts.” Mathematics, however, does not merge with nature, nor with thought. As Hilbert explained: We can understand this agreement between nature and thought, between experience and theory, only if we take into account both the formal element and the mechanism that is connected with it; and we must do this both for nature and for our understanding. (1930, p. 1160; my emphasis)

1. The ‘Mechanism’ of Physical Knowledge The axiomatic design, brought about by mathematical constructions, also reveals an underlying connecting structure between various disciplines.1 This helps clarify why “the most important bearers of mathematical thought have always...cultivated the relations to the neighbouring sciences, especially to the great empires of physics and epistemology” (Hilbert 1918, p. 1107). This helps clarify in what way the most rigorous treatment of a problem, speciªc to one science, can extract beneªts from results and concepts that originate from neighbouring sciences. In Hilbert’s vision, rigor goes hand in hand with open-mindedness. While insisting on rigour as a requirement for a perfect solution of a problem, I should like [. . .] to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment [. . .] Such a one-sided interpretation of the requirement of rigour would soon lead to the ignoring of all concepts arising from geometry, mechanics, and physics, to a stoppage of the ºow of new material from the outside world, and ªnally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an important nerve, vital to mathematical science, would be cut by the extir- pation of geometry and mathematical physics! (Hilbert 1900b, p. 1100) But unlike Hilbert, who saw in mathematics the key to understanding nature and considered the link between mathematics, geometry, and physics as vital, important bearers of physical thought at the end of the 19th century concerned themselves instead with keeping physics separate from mathematics. In the 1880s, the ‘atomist’ Boltzmann was accused of substituting mathematical deduction for physical thought, of reducing the physical science to an empty game of abstract symbols and monstrous algorithms. 1. Along this lines, Marletto and Rasetti’s essay in this volume is an eloquent development of Hilbert’s view.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 4 Hilbert’s Axiomatics as ‘Symbolic Form’?

The accuser was Peter G. Tait, co-author with Lord Kelvin of the Treatise on Natural Philosophy (1867) one of the most inºuential works of the 19th century. Located in the tradition of Newtonian empiricism, the Trea- tise supports energetics in the ‘atomistics vs. energetics’ dispute, which fu- eled the epistemological debate. The ‘crisis’ of atomism was claimed at the 1895 Lübeck meeting of the German Association of Scientists and Physicians,2 and corroborated by Ernst Mach’s phenomenalism. According to Mach, the variety of natural processes and phenomena is already an argument against the primacy of mechanistic atomism. He admits that our efforts to mirror the world in thought would be futile if we were not able to grasp something stable in the ºow of experience. Yet, when we separate an object from its changeable environment, what we really do is isolate a group of sensations to which we ascribe a greater stability than to the others. On this basic group of sensations, we ªx our thoughts. Since the object does not lose its identity, while single elements are extracted from the group, we are led to consider that, after removing all the elements, something still remains. Here comes the illusion of a ‘substance’ distinct from its attributes, a Ding-an-sich, and the useless attempt to reduce all phenomena to the motion of atoms. The physical model, proposed as an alter- native to classical mechanistic philosophy, is that which results from the collection of our sensory representations of physical data, mathematically organized according to a “principle of economy.” All of the conceptual instruments involved in the cognitive process are regarded as devices for the economy of thought. But the advantage of logical-scientiªc thought, notices Husserl con- temporaneously, lies in the capacity of going beyond the limits of what is given in intuition. Logic ought to come before any economy of thought; hence founding the former on the latter would be nonsense (1900, Prole- gomena to Pure Logic IX). Husserl underlines a fundamental difference between natural and logical theories: what makes a theory “logical” is an ideal nexus of necessity which prevails within it, whereas any properly theoretical component remains foreign to a “natural” theory. Thus, in the frame of phenomenalism, a physical theory is viewed as a natural theory. It was for mathematical reasons that Boltzmann intervened in the dispute between the physics of atoms and the physics of energy, or the physics of the continuum. He highlights that if our perceptions are adequately represented by the picture of the continuum, then atomism goes beyond the facts; if, instead, we think of the world in atomistic terms, then the position is completely reversed: it is the idea of the continuum that goes

2. In the debate, L. Boltzmann’s atomic theory of matter contrasted with G. Helm and G. Ostwald’s position on Energetics.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 5

beyond the facts. In his 1900 inaugural lecture at the University of Leipzig, he put the issue into these words: The old Kantian antinomy opposing inªnite divisibility of matter to atomistic constitution still keeps science in suspense, except that today we do not regard the two views as infected by internal logical inconsistencies arising from the laws of thought, but we take each as a mental picture we have constructed and we ask which can be developed more clearly and more easily while most correctly and deªnitely reproducing the laws of phenomena. (1900, p. 145) In Boltzmann’s opinion, the inseparable link between atomism and the concept of the continuum is of a mathematical nature. Without atomistic conceptions, the very concept of a limit would be senseless. The deªnite integrals that represent the solution of a differential equation demand, in general, division into a ªnite number of parts to be calculated. “Those who imagine they have got rid of atomism by means of differential equations fail to see the wood for the trees” (Boltzmann 1897, p. 43). To recon- cile the mechanicistic atomism with the physics of the continuum, Boltz- mann sought an explanation of the irreversibility of thermodynamics in the calculus of probability and statistical mechanics. And yet, by asserting the existence of a mathematical link between atomism and the physics of the continuum, he became ‘guilty’ of founding physical knowledge on algorithms rather than on experiments. From the point of view of the phe- nomenalistic empiricism, Boltzmann’s subtle mathematizations led physics astray. From a completely different perspective, for Hilbert (1900b): Boltzmann’s work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua. In his Leipzig lecture, Boltzmann referred to “the beautiful philosophic foundation of the Hertzian mechanics”. It was in Hertz’s Principles of Me- chanics, indeed, that a physical theory acquired the character of a “logical theory” in the sense of Husserl. We form for ourselves images or symbols of external objects; and the form which we give them is such that the logically necessary consequents of the images in thought are always the images of necessary consequents in nature of the things pictured. In order that this requirement may be satisªed, there must be a certain conformity between nature and our thought. [. . .] The images which we here speak of are our conceptions of things. With the things them-

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 6 Hilbert’s Axiomatics as ‘Symbolic Form’?

selves they are in conformity in one important respect, namely, in satisfying the above-mentioned requirement. For our purpose it is not necessary that they should be in conformity with the things in any other respect whatever. (Hertz 1894, pp. 1–2) The images which we form of things are not determined without ambigu- ity. Images of the same objects may differ in various respects; nevertheless they must satisfy three fundamental requirements: logical permissibility [Zulässigkeit], correctness [Richtigkeit], i.e., empirical adequacy, and appro- priateness [Zweckmässigkeit]. The questions of whether an image is permissible or not, and whether it is correct or not, are both ‘decidable’. The ap- propriateness of an image, however, may be a matter of opinion. Given two images of the same object, the more appropriate is the one which pic- tures more of the essential relations of the object: namely, the more ‘distinct’. Given two images of equal distinctness, the more appropriate is the one which contains the smaller number of superºuous relations:3 namely, the simpler of the two. Simplicity does not apply to a single sentence of a theory, or to a class of sentences, but to the whole picture. It means that the different steps in the construction of the theory must account for the largest number of facts with the smallest number of assumptions. “Only by gradually testing many images can we ªnally succeed in obtaining the most appropriate.” Putting aside both the concepts of force and energy, previously in dispute as essential constituents of mechanics, Hertz establishes his Principles by assuming only three independent basic notions: space, time, and mass. The idea of a physical theory emerging from sensory experience is shaken. It is mathematics that presents reality with a ‘physical form’. The elements of this picture, Hertz’s notion of Bild, are not (necessarily) drawn from observation. Instead, they map possible sequences of observable events. Any selection of propositions from which the whole theory of mechanics can be drawn by purely deductive reasoning, without any further appeal to experience, corresponds to what we have in mind when we talk of the principles of mechanics. Simplicity is both a heuristic principle and a criterion for selection. Moreover, that physical processes should conform to simple mathematical equations guarantees their intelligibility.4 The 3. Hertz (1894, p. 2) explains that “Empty relations cannot altogether be avoided: they enter into the images because they are simply images,—images produced by our mind and necessarily affected by the characteristics of its mode of portrayal.” 4. Cf. Weyl (1932, pp. 57–58): “The astonishing thing is not that there exist natural laws, but that the further the analysis proceeds, the ªner the details, the ªner the elements to which the phenomena are reduced, the simpler—and not the more complicated, as one would originally expect—the fundamental relations become and the more exactly do they describe the actual occurrences.”

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 7

point at issue is overlooked if, in line with Mach, the mathematical form of natural laws has only value for the economy of thought. Along the Kantian tradition, by constrast, Hertz’s ‘axiomatics’ and, as we are going to see, Hilbert’s, make clear that the form of scientiªc knowledge is moulded by mathematics. The naïve copy theory of knowledge is discredited. The fundamental concepts of each science, the instruments with which it propounds its questions and formulates its solutions, are regarded no longer as passive images of something given but as symbols created by the in- tellect itself. Mathematicians and physicists were ªrst to gain a clear aware- ness of this symbolic character of their basic implements. (Cassirer 1923, p. 75) The new ideal of knowledge as a ‘symbolic construction of forms,’ according to Cassirer, was brilliantly formulated by Hertz. This ideal, which has its roots in Leibniz’s ‘characteristica,’ would be sharpened through Hilbert’s axiomatics, and would be pursued by Cassirer himself for the whole of cul- tural activity.5

2. Hilbert’s Axiomatics In his 1917 Zurich lecture on Axiomatic Thought, Hilbert states that “anything at all that can be the object of scientiªc thought becomes dependent on the axiomatic method, and thereby indirectly on mathematics, as soon as it is ripe for the formation of a theory.” What is a “theory”? When we assemble the facts of a deªnite, more-or-less comprehensive ªeld of knowledge, we soon notice that these facts are capable of being ordered. This ordering always comes about with the help of a certain framework of concepts [Fachwerk von Begriffen]inthe following way: a concept of this framework corresponds to each individual object of the ªeld of knowledge, and a logical relation between concepts corresponds to every fact within the ªeld of knowledge. The framework of concepts is nothing other than the theory of the ªeld of knowledge. (Hilbert 1918, p. 1107) 5. “Every authentic function of the human spirit has this decisive characteristic in common with cognition: it does not merely copy but rather embodies an original, formative power. [. . .] This is as true of art as it is of cognition; it is as true of myth as of religion. All live in particular image-worlds, which do not merely reºect the empirically given, but which rather produce it in accordance with an independent principle. Each of these functions creates its own symbolic forms [. . .] They are not different modes in which an independent reality manifests itself to the human spirit but roads by which the spirit proceeds towards its objectivation, i.e., its self-revelation.” (Cassirer 1923, p. 78)

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 8 Hilbert’s Axiomatics as ‘Symbolic Form’?

The basic idea of axiomatics rests on the fact that a few propositions sufªce for the construction of the entire ediªce of the theory. But the selection of the axioms is not unique, there is a variety of possible choices. Moreover, Hilbert clariªes that once the problem of grounding the individual ªeld of knowledge has found a solution, this solution is always temporary: “the need arose to ground the fundamental axiomatic propositions themselves.” Therefore: The procedure of the axiomatic method [. . .] amounts to a deepening of the foundations of the individual domains of knowledge— a deepening that is necessary for every ediªce that one wishes to expand and to build higher while preserving its stability. The distinction between the axiomatic presentation of a theory (as a Fachwerk von Begriffen) and the axiomatic method as a sophisticated procedure of logical inquiry is lucidly explained by Ulrich Majer (in this volume, pp. 56– 79). The ªrst eloquent example of the interplay between these two aspects of axiomatics is provided by Hilbert’s work on geometry. Since Euclid’s time, Hilbert recalls in the introduction to Die Grund- lagen der Geometrie, the choice of the axioms of geometry and the investigation of their mutual connections has been an issue discussed in many ex- cellent works of the mathematical literature. Why another work on the subject? Hilbert’s vision of geometry as a ‘physical-mathematical’ science helps answer this question. In his 1898–99 lectures on mechanics, he de- scribes geometry as follows: Also geometry [like mechanics] emerges from the observation of nature, from experience. To this extent, it is an experimental science . . . But its experimental foundations are so irrefutably and so generally acknowledged, they have been conªrmed to such a degree, that no further proof of them is deemed necessary. Moreover, all that is needed is to derive these foundations from a minimal set of independent axioms and thus to construct the whole building of geometry by purely logical means. In this way [i.e., by means of the axiomatic treatment] geometry is turned into a pure mathematical science. (Quoted in Corry 1999, p. 152) In this way, mechanics had been already addressed by Hertz. Through the axiomatic analysis, Hertz’s Principles of Mechanics shows that neither the concept of force nor the concept of energy can draw their ‘truth’ from the observation of nature. The meaning of a physical concept is ªxed by an axiom, and an axiom is an image in our mind.6 Hilbert shares Hertz’s view 6. In his 1893 lecture notes on geometry, Hilbert wrote: “The axioms are, as Hertz

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 9

of mechanics and is aware of the difªculties of its axiomatization, as shown through the following passage in a letter to Frege: After a concept has been ªxed completely and unequivocally, it is on my view completely illicit and illogical to add an axiom—a mistake made very frequently, especially by physicists. By setting up one new axiom after another in the course of their investigations, without confronting them with the assumptions they made earlier, and without showing that they do not contradict a fact that follows from the axioms they set up earlier, physicists often allow sheer nonsense to appear in their investigations. One of the main sources of mistakes and misunderstandings in modern physical investigations is precisely the procedure of setting up an axiom, appealing to its truth (?), and inferring from this that it is compati- ble with the deªned concepts. One of the main purposes of my Festschrift was to avoid this mistake. (Letter to Frege, 29 December 1899; in Frege 1980, p. 40) Hilbert identiªes the task of his Festschrift as “the logical analysis of our intuition of space.” The task is accomplished by constructing geometry as an axiom-system, complete and as simple as possible. Concerning the truth of the axioms, in the same letter to Frege, Hilbert explains: If the arbitrary chosen axioms do not contradict each other with all their consequences, then they are true and the things deªned by the axioms exist. That for me is the criterion of truth and existence. If axioms and theories are images or symbols, there must be a certain conformity between nature and our thought. If a theory is “acceptable,” however, it is not because its axioms are true on the basis of ‘visual’ evidence or sensory experience, but because its axioms are, as Hertz would say, “logically permissible,” i.e., consistent. Long before non-Euclidean geometries were born, the ‘visual evidence’ of the parallel axiom was questioned by Renaissance artists, and revised by projective geometry. Assumptions based on visual evidence, implicit in Euclid’s theorems, were also well known. It was in the 19th century, however, that a clarifying logical analysis of Euclidean geometry was pursued ªrst by Pasch (1882), in terms of projective geometry, and then by the Italian ‘formalists’ Fano, Veronese, Peano, and Pieri (1895). According to Gray (1999, p. 65), Pieri’s work in particular stands out from Pasch’s in

would say, images or symbols in our mind, such that consequents of the images are again images of the consequences, i.e., what we can logically deduce from the images is itself valid in nature.” (Quoted in Corry 1999, p. 151)

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 10 Hilbert’s Axiomatics as ‘Symbolic Form’?

that it “is the complete abandonment of any intention to formalize what is given in experience” to constrain his geometry to any ‘physical interpretation’ of the premises. Pieri’s goal is a purely deductive and abstract treatment of geometry. Does Hilbert’s work on geometry pursue the same goal? Here is what he said in Paris: The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those ªgures; and in order that these geometrical ªgures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content. (Hilbert 1900b, pp. 1100–1101) Hilbert’s goal seems not so much to translate geometry into logic as to safeguard the use of “geometrical signs”—together with the other mathematical ªgures—as a means of strict mathematical proof: The arithmetical symbols are written diagrams and the geometrical ªgures are graphic formulae; and no mathematician could spare these graphic formulae, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs. (1900b, p. 1100) For Hilbert, mathematics has the meaning of a world-view (Bernays 1922, p. 190), and to proceed axiomatically means to think with consciousness.7 The point is neither to encompass factual knowledge within mathematics, nor to abstract mathematics from experience; it is rather to be mindful about the boundaries of mathematics, and “to secure a sphere of inºuence for mathematical thought that is as comprehensive as possible.” At the boundary between thought and experience, geometry offers an ideal perspective on those methodological questions. Although the epistemic character of propositions about ‘terms’ deªned by expressions like “a point is that which has no part” might well be disputable, still in his 1920–21 lectures on Anschauliche Geometrie, Hilbert maintains that “intuitive understanding” plays a major role in geometry: In mathematics, as in any scientiªc research, we ªnd two tendencies present. On the one hand, the tendency toward abstraction seeks to crystallize the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and 7. Cf. Hilbert (1922, p. 1120). The passage continues: “although in earlier times without the axiomatic method it happened that one naïvely believed in certain interconnections as though they were dogmas, the axiomatic method eliminates this naïveté, but leaves us the advantage of belief.”

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 11

orderly manner. On the other hand, the tendency toward intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations. As to geometry, in particular, the abstract tendency has here led to the magniªcent systematic theories of Algebraic Geometry, of Riemannian Geometry, and of Topology [. . .] Notwithstanding this, it is still true today as it ever was that intuitive understanding plays a major role in geometry. (Hilbert and Cohn-Vossen 1932, p. iii)

But the question of the epistemic character of axioms does not under- mine the endurance of the axiomatic method, for mathematical and epistemological problems must be kept distinct. Hilbert’s axiom-system for geometry is set “in such a way that the signiªcance [Bedeutung] of the different groups of axioms and the scope [Tragweite] of the consequences, to be derived from individual axioms, come to light clearly”; logic and spatial intuition are separate from the beginning. Each group of axioms captures a peculiar relation between homogeneous elementary facts of our intuition: incidence, order, congruence, parallelism, and continuity. Each axiom captures a single ‘intuitive’ notion. Nevertheless, the axiom-system itself does not represent any visual intuition, or anything factual; indeed, the geometrical axioms of order and of congruence can be applied so much to Euclid’s points, lines, and planes as to beer mugs or little fruit ºies.8 Like Hertz’s mechanics, Hilbert’s geometry presents a possible theory that must be mathematically investigated according to its internal properties. As explains Bernays (1922, pp. 192–193):

The spatial relationships are, as it were, projected into the sphere of the mathematical-abstract in which the structure of their connections appears as an object of pure mathematical thought. This structure is subjected to a mode of investigation that concentrates only on the logical relations and is indifferent to the question of the factual truth, that is, the question whether the geometrical connec- 8. As in Hilbert’s biography: “He [Hilbert] began by explaining to his audience that Euclid’s deªnitions of point, straight line and plane were really mathematically insig- niªcant. They would come into focus only by their connection with whatever axioms were chosen. In other words, whether they were called points, straight lines, planes or were called tables, chairs, beer mugs, they would be those objects for which the relationships expressed by the axioms were true. In a way this was rather like saying that the meaning of an un- known word becomes increasingly clear as it appears in various contexts” (Reid 1996, p. 60). See also Hilbert’s Königsberg lecture (1930, p. 1159).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 12 Hilbert’s Axiomatics as ‘Symbolic Form’?

tions determined by the axioms are found in reality (or even in our spatial intuition). Once the theory turns into an object of pure mathematical thought, the independence of its axioms, its consistency, and its adequacy as a model of the relevant ªeld of knowledge is to be established through an axiomatic investigation. If the theory of a ªeld of knowledge—that is, the framework of concepts that represents it—is to serve its purpose of orienting and ordering, then it must satisfy two requirements above all: ªrst it should give us an overview of the independence and dependence of the propositions of the theory; second, it should give us a guarantee of the consistency of all the propositions of the theory. In particular, the axioms of each theory are to be examined from these two points of view. (Hilbert 1918, p. 1109) As to the ªrst requirement, the parallel axiom in geometry is a classic example of the independence of an axiom. It is the characteristic axiom of Euclidean geometry, and Euclid’s method of investigation has been the paradigm for axiomatic research since around 300 BCE. Questioning visual evidence, “the real challenge to Euclid emerged from disquiet over the parallel axiom” (Stillwell 2005, p. 174). And yet neither the parallel axiom, nor its possible negations are consequences of Euclid’s other axioms. Thus, Euclidean and non-Euclidean geometries are both ‘consistent,’ for all of their respective axioms are logically independent of each other. Through the axiomatic analysis, Hilbert is led to focus on projective geometry, which is more basic than Euclidean because it does not involve the concept of length. In particular, by investigating the role of Pascal, alias Pappus, and Desargues’ theorems in Euclid’s theory of proportions, Hil- bert ªnds a deeper link between geometry and algebra. These theorems capture, respectively, the commutative and associative laws of multiplica- tion; hence, they allow Hilbert to build a model of a complete ordered ªeld, i.e., his theory of real numbers.9 But a thorough analysis of “our spatial intuition” must also come to terms with the notion of ‘continuity’. Geometry cannot base any axiom of continuity on sensory experience. Neither can mathematics guarantee any theory of the continuum, if all of its ‘constructions’ must originate from natural numbers. Yet, to override the ban of the law of continuum from pure mathematics, decreed by Kronecker, Hilbert wanted the system of geometry to be ‘closed,’ i.e., unable to be enlarged. To grasp the essence of 9. Hilbert (1900a). For more on this see Stillwell’s essay in this volume.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 13

continuity, Hilbert proceeds along Dedekind’s road. In his essay Continuity and Irrational Numbers (1872), Dedekind sought the ‘true’ origin of continuity in arithmetic, without any appeal to geometrical evidence or to representations suggested by geometry. He writes: When we compared the domain R of rational numbers with the straight line, we found in the former a gappiness, incompleteness, discontinuity; but to the straight line we ascribe absence of gaps, completeness, continuity. In what then does this continuity consist? For a long time he sought a precise answer in vain, but ªnally he found one. Considering that “every point p of the straight line produces a separa- tion of the same into two portions [Stücke] such that every point of one portion lies to the left of every point of the other,” he saw the essence of continuity in the converse, i.e., in the following principle: If all points of the straight line fall into two classes such that every point of the ªrst class [Klasse] lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. (Dedekind 1872, p. 771; my emphasis) No one has the power to adduce any proof for the correctness of this statement. “The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we think continuity into the line” (Dedekind 1872, 771–772). Following Dedekind, the correct conceptual order lies in imposing continuity on our treatment of space, and not in deriving it directly from our ‘intuition’ of space. In addition to the Archimedean axiom (of ‘arithmetic’ nature), stated in the ªrst edition of his Foundations of Geometry (1899), two years later, Hilbert introduces a second axiom of continuity or completeness, according to which the geometric universe of points, straight lines, and planes is complete, i.e., incapable of being extended while continuing to satisfy all the other axioms.10 Though it is not needed to derive any of Euclid’s theorems, the axiom implies that the points of the straight line correspond to real numbers. As Stillwell remarks (in this volume, p. 49), by adding an axiom of continuity to the axioms of geometry, Hilbert evidently wants to show that the real numbers can be put on a geometric foundation. More- 10. Hilbert also utilizes this idea in his axiomatization of the arithmetic of real numbers, where he replaces Dedekind’s axiom of continuity with the Archimedean axiom and an axiom of completeness that expresses the fact according to which “the numbers form a system of things which is incapable of being extended while continuing to satisfy all the axioms” (1900a, p. 1094).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 14 Hilbert’s Axiomatics as ‘Symbolic Form’?

over, Hilbert’s foundations of geometry do not dissolve into physics, nor into ‘pure’ mathematics. As to the second requirement, Hilbert proves the consistency of the axioms of geometry “by showing that any contradiction in the consequences of the geometrical axioms must necessarily appear in the arithmetic of the system of real numbers as well” (1918, p. 1112). It is the search for proofs of consistency that encourages Hilbert to consider procedures ‘reduction- ist’ in nature or spirit. For the ªelds of physical knowledge too, Hilbert observes, “it is clearly sufªcient to reduce the problem of internal consistency to the consistency of the arithmetical axioms.” The question of consistency seems not yet crucial, at least for ‘empirical sciences’. That the example of geometry should pave the way for the axiomatic treatment of physics was one of Hilbert’s expectations for the 20th century.11 In his 1898–99 lectures on mechanics (already mentioned above), he presents the task as follows:

In mechanics it is also the case [as in geometry] that all physicists recognize its most basic facts. But the arrangement of the basic concepts is still subject to a change in perception...andtherefore mechanics cannot yet be described today as a pure mathematical discipline, at least [not] to the same extent that geometry is. We must strive that it becomes one. We must stretch the limits of pure mathematics ever further, on behalf not only of our mathematical interest, but rather of the interest of science in general. (Quoted in Corry 1999, pp. 152–153)

The difference between an axiomatic treatment of geometry and of physics seemed a matter of maturity. In contrast to geometry, physics was “not yet” mature enough for being captured in a rigorous axiom-system. But the struggle of turning physics into a pure mathematical discipline would reveal a real challenge. Hilbert’s interest in physics plays a decisive role in the development of his axiomatic.12 He gave not solely several lecture courses on various domains of physics, but also notable speciªc results such as the solution of the ‘Boltzmann equation’ in the kinetic theory of gases, and the derivation of the ªeld equations from a variational principle in general relativity 11. As he stated in his 1900 Paris lecture on Mathematical Problems (see Majer’s essay and Rédei’s in this volume). 12. As stressed by Corry (1999, p. 159), “Hilbert’s axiomatic approach clearly did not evolve either as an empty game with arbitrary systems of postulates or as a conceptual break with the classical entities and problems of mathematics and empirical science. Rather it sought to improve the mathematician’s understanding of the latter.”

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 15

theory. Though most physicists did not fully appreciate or grasp the meaning of his axiomatic approach, Max Born (1922) did:

The physicist set out to explore how things are in nature; experiment and theory are thus for him only a means to attain an aim. Conscious of the inªnite complexities of the phenomena with which he is confronted in every experiment, he resists the idea of considering a theory as something deªnite. He therefore abhors the word ‘axiom,’ which in its usual usage evokes the idea of deªnite truth. The physicist is thus acting in accordance with his healthy instinct, that dogmatism is the worst enemy of natural science. The mathematician, on the contrary, has no business with factual phenomena, but rather with logic interrelations. In Hilbert’s language the axiomatic treatment of a discipline implies in no sense a deªn- itive formulation of speciªc axioms as eternal truth, but rather the following methodological demand: specify the assumptions at the beginning of your deliberation, stop for a moment and investigate whether or not these assumptions are partly superºuous or contradict each other. (Translated in Corry 1999, p. 181)

The provisional character of the foundations of physical theories urges an axiomatic analysis to support and fortify them. As Corry (1999, p. 172) reports, Hilbert was very attracted to the mechanistic reduction of Hertz or Boltzmann until around 1913. Then he became involved with electro- magnetism. In his 1913 lectures on electron theory, he developed an axiomatic analysis of electrodynamics. It shows that a complete foundation of the theory requires the concept of a rigid body, i.e., an electron, to be added to the Maxwell equations and to the concept of energy. In principle, the Maxwell equations, the concept of energy, and the concept of rigidity allow all of the laws of physics, except gravitation, to be derived. Lec- turing on electromagnetic oscillations, the following year, Hilbert once more calls attention to geometry as a model (for physics) of a theoretical science grounded in experience:

From antiquity the discipline of geometry is a part of mathematics. The experimental grounds necessary to build it are so suggestive and generally acknowledged, that from the outset it has immediately appeared as a theoretical science. I believe that the highest glory that such a science can attain is to be assimilated by mathematics, and that theoretical physics is presently on the verge of at- taining this glory. This is valid, in the ªrst place for relativistic mechanics, or four-dimensional electrodynamics, which I have been

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 16 Hilbert’s Axiomatics as ‘Symbolic Form’?

convinced for a long time belongs to mathematics. (Quoted in Corry 1999, p. 175) According to Hilbert, Einstein’s general theory of relativity enables physics to entwine its roots with those of geometry, and then to bring out the “empirical nature” of geometry,13 or to accentuate it. His 1916–17 lecture notes on the foundations of physics report: In the past, physics adopted the conclusions of geometry without hesitation. This was justiªed to the extent that not only the rough, but also the ªnest physical facts conªrmed those conclusions. This was also the case when Gauss measured the sum of the angles in a triangle and found that it equals the sum of two right angles. That is no longer the case for recent physics. Modern physics must draw geometry into the realm of its investigations. This is logical and natural: every science grows like a tree, where not only do the branches con- tinually expand, but the roots also penetrate deeper. [. . .] the physicist must become a geometer, for otherwise he runs the risk of ceasing to be a physicist and vice versa. (Sauer and Majer 2009, pp. 167–168) But quantum theory compels physics to face some genuinely new problems. Hilbert’s 1926–27 lecture notes on quantum physics, revised by Nordheim and von Neumann, resulted in the joint paper Über die Grund- lagen der Quantenmechanik (1927). This was Hilbert’s last publication in physics, and likely a unique document of his involvement with quantum mechanics. Despite his interest in the atomic theory of matter, the ‘principle of discontinuity,’ and other important issues related to quantum physics, Hilbert remained hesitant about the new theory. His biography reports that at a lecture of Schrödinger on the ‘new physics,’ in 1928 or 1929, he grumbled: “I don’t see how anybody understands what is hap- pening in physics today” (Reid 1996, p. 183). In 1926–27, the equivalence between the two different forms of quantum theory, that of the matrix mechanics and that of Schrödinger’s wave function, was proved. This result pointed out a ‘deepening in foundations’ which encouraged further investigations. Hilbert, Nordheim, and von Neumann declare the aim of their paper at the very beginning: 13. Cf. the following passage from the Königsberg lecture: “We can say: in recent times the conception of the empirical nature of geometry, as represented by Gauss and Helm- holtz, has become a secure result of science. It must today serve as a ªrm point of support for all philosophical speculations that concern space and time. The Einsteinian theory of gravitation makes it manifest: geometry is nothing more than a branch of physics; geometric truths are in no principled way whatsoever different from physical truths” (Hilbert 1930, p. 1163).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 17

The most recent development of quantum mechanics—on the one hand, related to the works of Heisenberg, Born, and Jordan, on the other hand, to those of Schrödinger—has made it possible to com- prehend the whole ªeld of atomic phenomena from a unitary point of view, and to explain the most important observational results. It has also made it almost impossible to doubt the ªnding, in this theory, a network of ideas of as great importance as those of classical mechanics or electrodynamics. The great importance of quantum mechanics makes understanding its principles in the clearest and most general way an urgent necessity. (1927, p. 104) The paper elaborates Jordan and Dirac’s formulation of quantum theory, in the frame of Hilbert’s axiomatics. Once more, geometry is referred to as a model for axiomatization. The way leading to this theory is the following: one formulates certain physical requirements [physikalische Forderungen] concerning these probabilities, requirements that are plausible on the basis of our experiences and developments and which entail certain relations between these probabilities. Then one searches for a simple analytic machinery [analytische Apparat], in which quantities appear that satisfy exactly these relations. This analytic machinery and the quantities occurring in it receive a physical interpretation [physikalische Interpretation] on the basis of the physical requirements. The aim is to formulate the physical requirements in a way that is complete enough to determine the analytic machinery un- ambiguously. This way is then the way of axiomatizing, as this had been carried out in geometry for instance. (1927, p. 104; translated in Rédei p. 84) The main difªculty, however, lies in proceeding from the physical requirements—i.e., the probabilities associated with physical quantities— to the analytic machinery. The fundamental physical idea of the theory, the authors explain, comes from the replacement of the rigorous func- tional relations of the usual mechanics with probability relations. As to the physical meaning of these relations, only the formalism provides some suggestions. No ‘intuition’ guides the choice of the axioms. Therefore, the standard procedure for constructing an axiomatic system is somehow soft- ened.14 This helps understand, on the one hand, the caution of the authors who—“corresponding to the current status of the theory”—“do not want yet to establish a complete axiomatization,” but on the other hand, the re- 14. In his essay in this volume, Rédei speaks of an “opportunistic soft axiomatization.”

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 18 Hilbert’s Axiomatics as ‘Symbolic Form’?

liance of those same authors on the analytic machinery as bearer of physical meaning: What is univocally ªxed, however, is the analytic machinery, that— from a purely mathematical point of view—is not even susceptible to any variation. What can instead be modiªed, and presumably will be, is the physical interpretation, with which a certain freedom and arbitrariness still remain. Through axiomatization, concepts like probability and the like, previously a bit vague, lose their mys- tical character, and are implicitly deªned by the axioms. (1927, p. 106) What lesson can be drawn from such an arduous attempt to axiomatize? Does it emphasize the still uncertain and provisional status of the theory, in late twenties, or focus on axiomatics’ potential limits? How should physical meaning ºow from an axiomatic theory? In his 1927 lecture on Foundations of Mathematics, Hilbert maintains that the individual assumptions and laws of a physical theory have no meaning that can be immediately realized in intuition; in principle, it is not the propositions of physics taken in isolation, but only the axiomatic theory as a whole, that can be confronted with experience. To make it a universal requirement that each individual formula then be interpretable by itself is by no means reasonable; on the contrary, a theory by its very nature is such that we do not need to fall back upon intuition or meaning in the midst of some argument. What the physicist demands precisely of a theory is that particular propositions be derived from laws of nature or hypotheses solely by inferences, hence on the basis of a pure formula game, without extraneous considerations being adduced. (Hilbert 1927, p. 475; my emphasis) As explained in the 1927 joint paper, once a complete set of physical requirements allows the analytic machinery to be determined unambigu- ously, the ‘formalism’ should be kept separate, in as clear a way as possible, from intuitive or physical meaning. But quantum theory needs to take account of two completely different classes of objects, i.e., measurable, real values of physical quantities and operators which represent these quantities, and allow their values to be calculated. Consequently, insofar as one has to turn back to catch the essence of the physical meaning of the theory, the ‘natural’ order of the whole construction is blurred.15 This prompts Hilbert, Nordheim, and von Neumann to be wary of achieving a complete 15. For more details on this cf. Lacki (2000).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 19

axiomatization of quantum theory. It also serves as a warning for physics about the question of consistency. Over and over again Hilbert has stressed the need for a ‘direct’ consistency proof. In contrast to geometry or physics, purely theoretical ªelds of mathematics, such as number theory and set theory, do not have a method of reduction to another special domain of knowledge: nothing but logic can be invoked. The task then arises to “make the concept of the spe- ciªcally mathematical proof itself an object of investigation, just as the astronomer considers the movement of his position, the physicist studies the theory of his apparatus, and the philosopher criticizes reason itself” (Hilbert 1918, p. 1115). Hilbert presents the methodological approach with which the task should be undertaken in his Copenhagen (and Ham- burg) lecture: “The axiomatic method is and remains the indispensable tool, appropriate to our minds, for all exact research in any ªeld whatsoever” (1922, p. 1120); it is logically indisputable, and eliminates the naïve assumptions of axioms as fundamental truths. “But now it is a question of something even more important,” namely, to achieve “full clarity about the principles of inference in mathematics” by means of an appropriate axiomatization of logic itself. Now the “deepening of the foundations” must go beyond individual ªelds of knowledge, and penetrate into the ground of mathematical procedures. It was both Dedekind and Frege’s ‘logicism,’ on the one side, and Brower and Weyl’s ‘intuitionism,’ on the other side, that disposed Hilbert to the ‘formalism’ of his Beweistheorie. Hilbert’s formalism, however, is not a compromise between these two extremes of thought, but rather “a new general intellectual orientation” (Cassirer 1929, p. 379). Seeking a conclu- sive proof theory, Hilbert’s hope was to produce “the logical miracle that is grounded at the very essence of mathematics: the question of the inªnite would be made accessible to ªnite resolution, to resolution through ªnite processes” (1929, p. 385). Quoting from his Copenhagen lecture:

As we saw, abstract operation with general concept-scopes and contents has proved to be inadequate and uncertain. Instead, as a pre- condition for the application of logical inferences and for the activa- tion of logical operations, something must already be given in representation [in der Vorstellung]: certain extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something that cannot be reduced to something else. Because I take

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 20 Hilbert’s Axiomatics as ‘Symbolic Form’?

this standpoint, the objects [Gegenstände] of number theory are for me—in direct contrast to Dedekind and Frege—the signs themselves, whose shape [Gestalt] can be generally and certainly recognized by us—independently of space and time, of the special conditions of the production of the sign, and of insigniªcant differences in the ªnished product. The solid philosophical attitude that I think is required for the grounding of pure mathematics—as well as for all scientiªc thought, understanding, and communication—is this: In the beginning was the sign. (1922, pp. 1121–22) Cassirer traces Hilbert’s proof theory to that of Leibniz. The latter was intended to dispel Descartes’ skepticism about the certainty of the deductive method. If a mathematical proof is to be truly stringent, if it is to embody real force of conviction, it must be detached from the sphere of mere mnemonic certainty and raised above it. The succession of the steps of thought must be replaced by a pure simultaneity of synop- sis. This only symbolic thinking can achieve. For its very nature it does not operate with the thought contents themselves, but corre- lates a deªnite sign with each content of thought and through this correlation achieves a condensation which makes it possible to con- centrate all the links of a complex chain of proof in a single formula, and embrace them in one glance as an articulated whole. (Cassirer 1929, pp. 388–89) Hilbert’s emphasis on extra-logical discrete objects, namely signs or symbols, is motivated by the need to survey a proof in all its parts, to catch it as a ‘Gestalt’. Possible contradictions in which mathematical thought may get entangled no longer need an intricate discursive process to be discovered. They will result, instead, in an erroneous constellation of signs: “A proof is a ªgure, which we must be able to view as such” (Hilbert 1922, p. 1127). In this sense, Cassirer notices, “it is this basic idea of the Leibnizian characteristic which has been revived in Hilbert’s ‘formalization’ of the processes of logical and mathematical reasoning.” The further step of Hilbert’s axiomatics here involved is captured by Bernays (1922, p. 196): Just as he [Hilbert] had formerly stripped the basic relations and axioms of geometry of their intuitive content, he now eliminates the intellectual content of the inference from the proofs. Now the logical analysis of our mathematical thought demands “a strict formalization of the entire mathematical theory, inclusive of its

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 21

proofs, so that—following the example of the logical calculus—the mathematical inferences and deªnitions become a formal part of the ediªce of mathematics” (Hilbert 1922, p. 1123). Thus, the systems of formulas that represent mathematical proofs in a logical calculus are detached from their contents, and become the immediate object of study.16 Once that which is speciªcally mathematical is separate from everything contentual, we recognize—following Bernays—that the domain of the mathematical- abstract, into which the methods of mathematical thought translate all that is theoretically comprehensible, is not the domain of the contentual- logical [inhaltlich Logisches] but rather that of pure formalism. Here Hil- bert’s vision on mathematics and the axiomatic method reaches its highest point: “Mathematics turns out to be the general theory of formalism, and by understanding it as such, its universal meaning also becomes clear” (Bernays 1922, p. 196). This meaning of mathematics as a “general theory of forms,” writes Bernays, “has come to light in recent physics in a most splendid way, especially in Einstein’s gravitational theory.” It is no surprise that it was Hilbert, as noticed by Bernays, who ªrst set gravitational law in its simplest mathematical form. It is no surprise either, we may add, that Cassirer’s notion of ‘symbolic form’ has come to light through his work on Einstein’s theory of relativity. That the sciences, in particular, mathematics and the exact natural sciences furnish the criticism of knowledge with its essential material is scarcely questioned after Kant; but here [in the theory of relativity] this material is offered to philosophy in a form, which, even of itself, involves a certain epistemological interpretation and treatment. (Cassirer 1920, p. 355; my emphasis)

3. From ‘Axiomatic’ to ‘Formal’ Systems Gödel’s opening remarks, in his article On Formally Undecidable Propositions of Principia Mathematica and Related Systems I, allude to an evalution of Hilbert’s ‘axiomatics’: The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules....Onemight therefore conjecture that these axioms and rules of inference are sufªcient to decide any mathematical question that can at all be formally expressed in these systems. (Gödel 1931, p. 145) 16. In this perspective, Abrusci’s essay in this volume shows how Hilbert’s ideas will evolve in Gentzen’s Sequent Calculus and in more recent logical investigations.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 22 Hilbert’s Axiomatics as ‘Symbolic Form’?

Yet, as he continues, “It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned [Principia Mathematica and Zermelo-Fraenkel] relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.” Formal- ization allows proofs to be carried out by using few mechanical rules. This is not sufªcient, however, for preventing the construction of a true proposition concerning integers that cannot be proved within the formal system. The formal system cannot produce all true propositions, and therefore it is incomplete. Moreover, if the negation of the unprov- able proposition is added to the system, a consistent system containing false propositions can be attained. Gödel gave a preliminary an- nouncement of his ªrst incompleteness theorem at the “Conference of Epistemology and Exact Sciences,” held at Königsberg in the Fall of 1930, just the day before Hilbert gave his 1930 lecture mentioned above. Von Neumann, who was attending the conference to present Hilbert’s ‘formalism,’ promptly drew from Gödel’s words the conclusion that Hil- bert’s consistency goal was out of reach.17 As to the issue of the solvability of any “precisely formulated” mathematical question, Gödel would reªne Hilbert’s statement. According to Gödel (193?), his results do not necessarily imply a negative answer to the problem in its original formulation, “since the number-theoretic questions which are undecidable in a given formalism are always decidable by evident inferences not expressible in the given formalism.” The point is rather that “it is not possible to formalize mathematical evidence even in the domain of number theory, but the conviction about which Hilbert speaks remains entirely untouched” (193?, p. 164). What thwarts the transition from ‘evidence’ to formalism? The question demanded a mathematically satisfactory deªnition of the general concept of ‘formal system,’ and as Gödel emphasized, “the essential difªculty of this deªnition is the notion of mechanical procedure which comes in and which needs further speciªcation” (Gödel 193?, p. 166). Indeed, since mechanical procedures are applied to ªnite classes of expressions that can be mapped on integers, what has to be speciªed is the concept of a “calculable function of integers.” And the standard way of deªning calculable functions, Gödel noticed, is by recursion. Then, in 1951, he admits that “The most satisfactory way [. . .] is that of reducing the concept of ªnite procedure to that of a machine with a ªnite number of parts, as has been done by the British mathematician Turing” (Gödel 1951, pp. 304–5). How did a ‘Turing machine’ fulªll Gödel’s requirements for the general concept of a formal system?

17. For details cf. Sieg (2012).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 23

As Sieg’s detailed investigation has clariªed, Turing’s conceptual analysis of ‘mechanical’ calculability, “brings in human computers in a crucial way and exploits the limitations of their processing capacities, when proceeding mechanically” (2006, p. 201). But none of the restrictive conditions on the steps of a Turing machine would do for “arithmetically meaningful” steps. Gödel does not recognize “the genuinely distinctive character of Turing’s analysis, i.e., the move from arithmetically motivated calculations to general symbolic processes that underlie them.”18 Being computable in any formal system (containing arithmetic) endows the notion of general recursiveness (or Turing’s computability) with a character of ‘absoluteness’ which convinces Gödel of its great importance: “this importance is largely due to the fact that with this concept one has for the ªrst time succeeded in giving an absolute deªnition of an interesting epistemological notion, i.e., one not depending on the formalism chosen” (Gödel 1946, p. 84). By not depending on a particular formalism (a set of axioms or principles), the notion ought to grasp a certain characteristic feature common to all formalisms or, in other words, to grasp what makes a system ‘formal’. From Turing’s work, in the end, Gödel gained conªdence in “the mechanism that is connected with” any formal system: “A formal system can simply be deªned to be any mechanical procedure for producing formulas, called provable formulas” (Gödel 1964, pp. 369–70). The ‘deterministic’ layout of a Turing machine also satisªed Gödel’s requirement for a limited freedom in the activity of a mathematician: “If anything like creation exists at all in mathematics, then what any theorem does is exactly to restrict the freedom of creation” (Gödel 1951, p. 314). In Gödel’s view, mathematical concepts “form an objective reality of their own, which we cannot create or change, but only perceive and describe” (ibid., p. 320). Pondering over possible philosophical implications of his incompleteness theorems, he came to conjecture that: [I]f the human mind were equivalent to a ªnite machine, then objective mathematics not only would be incompletable in the sense of not being contained in any well-deªned axiomatic system, but moreover there would exist absolutely unsolvable problems. (1951, p. 310) The philosophical consequences for mathematics, which Gödel derived from his incompleteness theorems, appear not so much opposed to potential limitations of human mind as in favour of “some form or other of Pla- tonism or ‘realism’ as to the mathematical objects” (Gödel 1951, p. 311). 18. For a more extensive and accurate discussion cf. Sieg (2006).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 24 Hilbert’s Axiomatics as ‘Symbolic Form’?

From a different perspective, however, it can be argued that if mathematics—as a “general theory of forms”—is not held hostage by ‘reality,’ either of physical objects or of Plato’s ideas, but is triggered by experience, it must depend upon the limits of the ‘stuff’ involved in its design. In this perspective, Hilbert (1930) observed that the most general and fundamental idea of the Kantian epistemology retains its signiªcance: namely, the philosophical problem of determining a certain intuitive, a priori outlook necessary for the genesis of our knowledge, and thereby of investigating the condition of the possibility of all conceptual knowledge and of every experience. I believe that in essence this has occurred in my investigations into the principles of mathematics. The a priori is nothing more and nothing less than a fundamental outlook, or the expression of certain indispensable preconditions of thought and experience. (1930, p. 1162) In the same perspective, the Turing machine provided mathematics with a scaffolding for building the “connecting bridges” between experience and thought. Turing’s model of computability, indeed, was achieved by investigating the condition of the possibility of performing a computation, i.e., through a scrutiny of the processing capacities required for the ‘agent’ involved. Considering the actions of a human computer, and focusing on the bare minimum needed to carry them out, it was possible to ªx a few simple mechanical operations on symbolic conªgurations indispensable for computing. These elementary operations on symbols were apt to be con- verted into a ªnite set of instructions for a “Turing machine”. The point to stress in the present context is that Turing’s limits on computability are bounded to the limitations of the computer involved. They are limits of observability and limits of memory: at each step of a computation, both the number of symbols, which a (human) computer can observe, and the number of the “states of mind” must be ªnite. If we permitted an inªnite number of symbols or states of mind, some of them would be arbitrarily close, hence they would be confused, whereas the symbols and the states of mind which take part in a computation must be “immediately recognizable,” hence distinguishable. Turing’s conceptual analysis might provide a perspicuous explanation of Hilbert’s “extralogical concrete objects” as a priori given signs to be manipulated according to rules. Quoting from Hilbert’s lecture On the Inªnite: If logical inference is reliable, it must be possible to survey these objects completely in all their parts, and the fact that they occur,

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 25

that they differ from one another, and that they follow each other, or are concatenated, is immediately given intuitively, together with the objects, as something that neither can be reduced to anything else nor requires reduction. This is the basic philosophical position that I consider requisite for mathematics and, in general, for all scientiªc thinking, understanding, and communication. (1926, p. 376) In giving a full expression to this basic philosophical position, a Turing machine shows the far reaches of a mathematical knowledge generated through research and manipulations of ‘recognizable’ symbols, transfor- mations of observable ‘states of things’. If ‘mental processes’ of some sort appear to be involved in Turing’s conceptual analysis, they must be read in conformity with Hilbert’s attitude: The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds. Thinking, it so happens, parallels speaking and writing: we form statements and place them one behind another. (Hilbert 1927, p. 475) In this perspective, for Hilbert, “the axiomatic method is basic to his anti- metaphysical account of the nature of mathematics” (Hallett 1990, p. 242). But Gödel’s later concern was about a philosophical error in Tu- ring’s argument: insofar as it shows that “mental procedure cannot go beyond mechanical procedures,” it would imply the same kind of limitation on the human mind. What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing, i.e. that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. [. . .] although at each stage the number and precision of the abstract terms at our disposal may be inªnite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge towards inªnity in the course of the application of the procedure. (Gödel 1972, p. 306) Gödel presumes that something similar may occur in the process of form- ing stronger and stronger axioms of inªnity in set theory. Our understanding of abstract concepts of this kind may involve a procedure not attain- able by ªnite means. Not unlike Gödel, Turing regards certain aspects of mathematical thought as needing a deeper understanding. Not unlike Tu- ring, Hilbert conjectured that even mathematical concepts that demand a

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 26 Hilbert’s Axiomatics as ‘Symbolic Form’?

great measure of abstraction should be understood by the application of ‘surveyable’ methods. The main purpose of his investigations into the foundations of mathematics was indeed “to endow mathematical method with the deªnite reliability.” The main challenge was to master the inªnite: [. . .] the deªnite clariªcation of the nature of the inªnite has become necessary, not merely for the special interests of the individual sciences, but rather for the honor of the human understanding itself. (Hilbert 1926, p. 370)

4. Ideal Elements The problem of the emergence and use of more and more abstract mathematical concepts brings us back to the roots of Hilbert’s axiomatic investigations. The distinctive abstractness of mathematical concepts in the second half of the 19th century has not solely stimulated Hilbert’s methodological sensibility, but has also posed a challenge for the Kantian philosophy of mathematics. Among the Neo-Kantian philosophers, Ernst Cassirer analyzed most carefully the formation and epistemological justi- ªcation of abstract mathematical concepts. What is of particular interest, in the present context, is that Cassirer’s turn from the philosophy of the Neo-Kantian Marburg School to his philosophy of Symbolic Forms matured through a methodological reºection inspired by Dedekind. What is more, Dedekind’s foundations of arithmetic, according to Jeremy Heis (2011), provided Cassirer with a modern version of the Kantian thesis that mathematics is rational cognition from a ‘construction of concepts’. Thus, if there is a kinship between Hilbert’s axiomatics and Cassirer’s philosophy of symbolic forms, it must be traced to Dedekind’s fatherly guidance. In Substance and Function, Cassirer argues that the development of sci- entiªc arithmetic in the last decades has shown the need to deduce the concept of numbers from purely logical premises. While geometry has been viewed as a science concerning empirical intuition, “all determina- tions of number are to be grounded, without any appeal to sensible objects or any dependence upon concrete measurable magnitudes, ‘by a ªnite system of simple steps of thought’” (1910, p. 35). He actually refers to Dedekind’s considerations:

If we scrutinize closely what is done in counting a set or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. (Dedekind 1888, p. 791)

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 27

As Cassirer underlines, Dedekind does not take “things” as given objects; rather they are terms of relations, and “as such can never be given in isolation but only in ideal community with each other.” The procedure of “rep- resenting” is also transformed. The question no longer concerns the ability to produce a conceptual copy of outer impressions, but to see a new necessary order among ideal operations and objects, namely, to produce a mapping: If in the consideration of a simply inªnite system N ordered by a mapping f we entirely neglect the special character of the elements, simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the ordering mapping f, then these elements are called natural numbers or ordinal numbers or simply numbers, and the base element 1 is called the base-number of the number-series N. With reference to this liberation of the elements from every other content (abstraction) we are justiªed in calling the numbers a free creation of the human mind. (Dedekind 1888, p. 809) Here, the critical analysis of the ‘abstraction theory’ carried out by Cassirer throughout Substance and Function sets its goal. Abstraction is not conceived, like in Aristotelian tradition, as a ‘negative’ process; it does not consist in selecting common properties from a plurality of things, while the remaining properties are neglected. The negation involved is func- tional for a ‘positive’ process, for what remains bears the relevant connecting structure. It would be naïve to confuse the object which comes to light with a ‘real’ thing, for its form can be recognized in its purity only after the relation from which it develops has been grasped. In this sense, abstraction has the character of a liberation: “it means logical concentration on the relational connection as such with rejection of all psychological cir- cumstances, that may force themselves into the subjective stream of pre- sentations, but which form no actual constitutive aspect of this connection” (1910, p. 39). A key of Cassirer’s reºection on the theory of concept formation lies in Dedekind’s deduction of the irrational number: The conceptual ‘being’ of the individual number disappears gradually and plainly in its peculiar conceptual ‘function’. On the ordi- nary interpretation, with which Dedekind’s deduction is at ªrst connected, although a certain number, given and at hand, produces a deªnite ‘cut’ in a system, none the less the process is ªnally reversed, for this production comes to be the necessary and sufªcient condition of our speaking of the existence of a number at all. The element cannot be separated by the relational complex, for it means

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 28 Hilbert’s Axiomatics as ‘Symbolic Form’?

nothing in itself aside from this complex, which it brings to expression, as it were, in a concentrated form. (Cassirer 1910, p. 61) What Dedekind claims for the existence of the irrational numbers is ex- panded by Cassirer so as to hold in general for the production of theoretical concepts. Whenever we have a system of conditions that can be realized in different contexts, we should be indifferent to the contexts and hold to the form of the system itself as an invariant, and then develop its laws deductively. “In this way we produce a new ‘objective’ form, whose structure is independent of all arbitrariness” (1910, p. 40). Through the attentive and detailed exploration of the means by which mathematics and physics give form to their concepts, Cassirer proceeds to sharpen his picture of symbolic forms. In this proceeding, Hilbert’s axiomatic plays a central role. The beginning of all mathematical concept formation is that thought, while not absolutely detaching itself from the intuitively given and intuitively representable, strives to liberate itself from the ºuid, indeterminate aspect of intuition...Itistheaxiomatic thinking of mathematics that ªrst posits the possible subjects for all genuinely mathematical statements. (1929, p. 400; p. 403) If Dedekind’s ‘structural deªnitions’ provided Cassirer with a modern version of the Kantian thesis that mathematics is rational cognition from a ‘construction of concepts,’ Hilbert’s ‘ªnitism’ reªned the Kantian ‘conditions’ for the construction of concepts. In the light of Turing’s limits on computability, Hilbert’s extralogical signs, intuitively given before all thought, should be traced neither to Kant’s a priori pure intuitions nor to Kant’s sensory intuitions, but to Kant’s conditions for ‘the possibility of experience’. They turn the latter into conditions for ‘cognitive acceptabil- ity’. Consequently, Hilbert’s consistency program can be understood “a way of strengthening the broader and more fundamental notion of accept- ability by trying to make it more tractable mathematically” (Hallett 1990, p. 240). Kant already taught—and indeed it is part and parcel of his doc- trine—that mathematics has at its disposal a content secured independently of all logic and hence can never be provided with a foundation by means of logic alone; that is why the efforts of Frege and Dedekind were bound to fail. Rather, as a condition for the use of logical inferences and the performance of logical operations, something must already be given to our faculty of representation [in der Vorstellung], certain extralogical concrete objects that are intuitively

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 29

[anschaulich] present as immediate experience prior to all thought. (Hilbert 1926, p. 376) Wilfried Sieg (in this volume) distinguishes two Ways of Hilbert’s Axi- omatics: the ªrst, named the existential axiomatic approach, grew on Dedekind’s logicism, the second, named the formal axiomatic method, was inºuenced by Kronecker’s ªnitism. Furthermore, to grasp the relation between Dedekind and Hilbert, Sieg observes that “two synonymies should be kept in mind: Hilbert’s axioms are Dedekind’s conditions of a structural deªnition; Hilbert’s models of axioms are Dedekind’s systems falling under the corresponding structural deªnition” (2013, p. 14).19 Turning to Cassirer, if Dedekind’s logicism and Hilbert’s existential axiomatics moti- vate a ‘critical’ revision of the Kantian theory of mathematical knowledge, the ultimate result of his Philosophy of Symbolic Forms matches the ‘spirit’ of Hilbert’s formal axiomatics. Rather than contrasting the two ways of axiomatics, Cassirer’s close scrutiny of mathematical concepts, in particular of the ‘ideal elements,’ encourages the envisioning of an ‘ideal’ path connecting the two: The pendulum of mathematical thinking swings, as it were, with a twofold movement: toward the relation and toward the object. This thinking continuously dissolves all being into pure relations; but on the other hand it always keeps uniting a totality of relations into the concept of one being. (1929, p. 399) In his talk On the Inªnite, Hilbert used the method of ideal elements to provide a clear explanation of his theory of mathematical proofs. The way in which ideal propositions can be generated from contentual, ªnitary propositions is already shown, he noticed, by elementary mathematics going beyond the intuitive number theory. In algebra, expressions composed of let- ters allow the contentual propositions of number theory to be formalized by means of themselves: “Where we had propositions concerning numer- als, we now have formulas, which themselves are concrete objects that in their turn are considered by our perceptual intuition, and the derivation of one formula from another in accordance with certain rules takes the place of the number-theoretic proof based on content” (1926, p. 379). The strat- egy is to generalize this process and regard mathematics as “an inventory of formulas.” The aim is to ensure that logical operations can be applied to the ideal propositions, as they are applied to the ªnitary propositions. This, however, cannot be done in a contentual way for the ideal propositions (the formulas), because they have no meaning in themselves. What is 19. Cf. also the introductory essay to Sieg (2013).

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 30 Hilbert’s Axiomatics as ‘Symbolic Form’?

needed is to formalize the logical operations and also the mathematical proofs themselves. For accomplishing this task, Hilbert observed that the same “preestablished harmony” that had presented Einstein with the general calculus of invariants for his theory of gravitation, presented now the logical calculus. But since the formulas of the logical calculus are also ideal propositions, even the logical signs need to be divested of all meaning: [. . .] hence contentual inference is replaced by manipulation of signs [äußeres Handeln] according to rules, and in this way the full transition from a naïve to a formal treatment is now accomplished, on the one hand, for the axioms themselves, which originally were naïvely taken to be fundamental truths but in modern axiomatics had already for a long time been regarded as merely establishing certain interrelations between notions, and, on the other for the logical calculus, which originally was to be only another language. (Hilbert 1926, p. 381) ‘Axioms’ are certain formulas that serve as building blocks for the formal ediªce of mathematics, while a ‘mathematical proof’ is “an array that must be given as such to our perceptual intuition.” Thus, even while illustrating his conception of mathematics as an inventory of formulas, Hilbert appealed to “intuitive understanding.” His call for ideal elements was intended to help understand how the inªnite conquers “a well-justiªed place in our thinking and plays the role of an indispensable notion.” If axioms themselves, i.e., the building blocks of mathematical science, are no longer fundamental truths but instruments for establishing connections, the distinctive character of mathematics is not retained in its objects, but in its method. Consequently, the new ideal elements are no less reliable than the old contents. In Cassirer’s opinion, the ‘ideality’ of the new concepts does not begin with them, is brought about by them. When a deep structural connection is established, the critical analysis must arrange the new objects as a necessary systematic development of the old contents. Through the method of ideal elements, mathematics is not led astray, but only guided to take one step further along the original stream of its concept formation. Hilbert notices that the method is already used in the elementary geometry of the plane: There the points and straight lines of the plane are initially the only real, actually existing objects. The axiom of connection, among others, holds for them: for any two points there is always one and only one straight line that goes through them. From this it follows that the two straight lines intersect each other in one point

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 31

at most. The proposition that any two straight lines intersect each other in some point, however, does not hold; rather, the two lines can be parallel. But, as is well known, the introduction of ideal elements, namely, points at inªnity and a line at inªnity, renders the proposition according to which two straight lines always intersect each other in one and only one point universally valid. (1926, p. 372) Then the ideal elements do not merely achieve a logical justiªcation be- side the others; by disclosing an underlying connecting structure, they secure the meaning of the whole system to a deeper ground. Here Cassirer points out a suggestive reversal of the epistemological issue that no longer lies in reducing the new to the old, but rather in drawing the meaning of the old from the new: [. . .] we should say that in the ideal elements mathematical thinking ªrst achieves the actual goal of its concept formation and comes to the critical understanding of what this concept formation is and of what it can do. Even if we assume that the ratio essendi of the ideal elements lies in the realm of the old ones, still the ratio cognoscendi of these must be sought in the ideal factors. (1929, p. 393) In this passage the peculiar form of mathematical knowledge clearly emerges from its ‘twofold’ method: on the one hand, it proceeds by intro- ducing new ideal elements, and on the other hand, by reªning the connecting structure from the new standpoint. An increasing logical compre- hension can be appreciated in all of those ªelds where ideal elements play a signiªcant role. A case in point is given by the discovery of the ‘imagi- nary’ in mathematics and the various attempts that have been made to jus- tify it logically. A new form of reasoning propagates itself from the theory of algebraic equations, through projective geometry, and physics. Indeed, the ‘vanishing point’ of perspective drawing acts already as an ideal element in making the ‘point at inªnity’ an object of mathematical thinking. It does not belong to the image, nor to the ‘real’ thing: it means nothing in itself aside from the act of thought which reaches through the given and beyond it. The actual intellectual miracle is that this process of reaching through, which already determined its beginning, never ªnds an end but is repeated over and over, always at a higher level. It is this alone that prevents mathematics from freezing into an aggregate of mere analytical propositions and degenerating into an empty tau- tology. The basis of the self-contained unity of the mathematical

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 32 Hilbert’s Axiomatics as ‘Symbolic Form’?

method is that the original creative function to which it owes its beginning never comes to rest but continues to operate in ever new forms, and in this operation proves itself to be one and the same, an indestructible totality. (Cassirer 1929, p. 403)

References Bernays, Paul. (1922) 1998. “Hilbert’s Signiªcance for the Philosophy of Mathematics.” Pp. 189–197 in From Brouwer to Hilbert. Edited & translated by Paolo Mancosu. Oxford: Oxford University Press. Boltzmann, Ludwig. (1897) 1974. “On the Indispensability of Atomism in Natural Science.” Pp. 41–53 in Theoretical Physics and Philosophical Problems. Edited by Brian Mc Guinness. Translated by Paul Foulkes. Dordrecht: Reidel. Boltzmann, Ludwig. (1900) 1974. “On the Principles of Mechanics.” Pp. 129–152 in Theoretical Physics and Philosophical Problems. Edited by Brian Mc Guinness. Translated by Paul Foulkes. Dordrecht: Reidel. Born, Max. 1922. “Hilbert und die Physik.” Die Naturwissenschaften 10:88–93. Cassirer, Ernst (1910) 1923. Substanzbegriff und Funktionbegriff. Berlin: Bruno Cassirer. Translated by William Swabey and Marie Swabey in Substance and Function & Einstein’s Theory of Relativity. Chicago: Open Court. Cassirer, Ernst (1920) 1923. Zur Einstein’schen Relativitätstheorie. Translated by William Swabey and Marie Swabey in Substance and Function & Ein- stein’s Theory of Relativity. Chicago: Open Court. Cassirer, Ernst. (1923–29) 1957. Philosophie der Symbolischen Formen. Three volumes. Translated by Charles W. Hendel, Ralph Manheim, and Wil- liam H. Woglom as The Philosophy of Symbolic Forms. New Haven: Yale University Press. Corry, Leo. 1999. “Hilbert and Physics (1900–1915).” Pp. 145–188 in The Symbolic Universe. Geometry and Physics 1890–1930. Edited by Jeremy Gray. Oxford: Oxford University Press. Dedekind, Richard J. W. (1872) 1996. “Continuity and Irrational Num- bers.” Pp. 765–779 in Ewald (1996). Translated by W. W. Beman. Dedekind, Richard J. W. (1888) 1996. “Was sind und was sollen die Zahen?” Pp. 787–833 in Ewald (1996). Translated by W. W. Beman. Ewald, William B. (ed.) 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics Vol. 2. Oxford: Oxford University Press. Frege, Gottlob. 1980. Philosophical and Mathematical Correspondence. Ox- ford: Basil Blackwell. Gödel, Kurt. (1931) 1986. “On Formally Undecidable Propositions of

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 Perspectives on Science 33

Principia Mathematica and Related Systems I.” Pp. 144–195 in Collected Works Vol. 1. Gödel, Kurt. (193?) 1995. “Undecidable Diophantine Propositions.” Pp. 164–175 in Collected Works Vol. 3. Gödel, Kurt. (1946) 1990. “Remarks before the Princeton bicentennial conference on problems in mathematics.” Pp. 150–153 in Collected Works Vol. 2. Gödel, Kurt. (1951) 1995. “Some Basic Theorems on the Foundations of Mathematics and Their Implications.” Pp. 304–323 in Collected Works Vol. 3. Gödel, Kurt. (1964) 1986. “Postscriptum to ‘On Undecidable Proposi- tions of Formal Mathematical Systems’ [1934].” Pp. 369–371 in Col- lected Works Vol. 1. Gödel, Kurt. (1972) 1990. “Some Remarks on the Undecidability Re- sults.” Pp. 305–306 in Collected Works Vol. 2. Gödel, Kurt. 1986–1995. Collected Works. Vols. 1–3. Edited by Salomon Feferman, John W. Dawson, Warren Goldfarb, Stephen C. Kleene, Gregory H. Moore, Charles Parsons, Robert M. Solovay, and Jean van Heijenoort. Oxford: Oxford University Press. Grey, Jeremy. 1999. “Geometry-formalisms and intuitions.” Pp. 58–83 in The Symbolic Universe. Geometry and Physics 1890–1930. Edited by Jeremy Gray. Oxford: Oxford University Press. Hallett, Michael. 1990. “Physicalism, Reductionism & Hilbert.” Pp. 183–257 in Physicalism in Mathematics. Edited by A. D. Irvine. Dordrecht: Kluwer Academic Publishers. Hertz, Heinrich. (1894) 1956. Die Prinzipien der Mechanik, Leipzig. Trans- lated as The Principles of Mechanics, New York: Dover Heis, Jeremy. 2011. “Ernst Cassirer’s Neo-Kantian Philosophy of Geome- try.” British Journal for the History of Philosophy 19(4):759–794. Hilbert, David. (1899) 1902. Grundlagen der Geometrie (Festschrift Zur Feier der Enthüllung des Gouss-Weber-Denkmals in Göttingen). Leipzig: Teubner. Translated by E. J. Townsend as The Foundations of Geometry. Chicago: Open Court. Hilbert, David. (1900a) 1996. “On the Concept of Number.” Pp. 1092– 1095 in Ewald (1996). Translated by William C. Ewald. Hilbert, David. (1900b) 1996. “From Mathematical Problems.” Pp. 1096–1105 in Ewald (1996). Translated by Mary Winston Newson. Hilbert, David. (1918) 1996. “Axiomatic Thought.” Pp. 1105–1115 in Ewald (1996). Translated by William Ewald. Hilbert, David. (1922) 1996. “The New Grounding of Mathematics.” Pp. 1115–1134 in Ewald (1996). Translated by William Ewald.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021 34 Hilbert’s Axiomatics as ‘Symbolic Form’?

Hilbert, David. (1926) 1967. “On the Inªnite.” Pp. 367–392 in van Heijenoort (1967). Translated by Stefan Bauer-Mengelberg. Hilbert, David. (1927) 1967. “The Foundations of Mathematics.” Pp. 464–479 in van Heijenoort (1967). Translated by S. Bauer- Mengelberg and D. Føllesdal. Hilbert, David. (1930) 1996. “Logic and the Knowledge of Nature.” Pp. 1157–1165 in Ewald (1996). Translated by William Ewald. Hilbert, David and Stephan Cohn-Vossen (1932), 1952. Anschauliche Geometrie, Göttingen. Translated by P. Nemenyi as Geometry and the Imagination. New York: Chelsea Pub. Co. Hilbert, David, Nordheim, Lothar, and John von Neumann. (1927) 1961. “Über die Grundlagen der Quantenmechanik.” Pp. 104–133 in Col- lected Works of John von Neumann. Vol. I. Edited by Abraham H. Taub. Oxford: Pergamon Press. Husserl, Edmund. (1900) 1970. Logische Untersuchungen. Translated by J. N. Findlay as Logical Investigations. London & New York: Routledge. Lacki, Jan. 2000. “The Early Axiomatizations of Quantum Mechanics: Jordan, von Neumann and the Continuation of Hilbert’s Program.” Ar- chive for History of Exact Sciences 54. Pasch, Moritz. 1882. Vorlesungen über neuere Geometrie. Leipzig: Teubner. Pieri, Mario. 1895. “Sui principi che reggiono la geometria di posizione.” Atti Accademia Torino. 30:54–108. Reid, Constance. 1996. Hilbert. New York: Springer-Verlag. Sauer, Tilman, and Ulrich Majer (eds). 2009. David Hilbert’s Lectures on the Foundations of Physics 1915–1927. Berlin: Springer. Sieg, Wilfried. 2006. “Gödel on Computability.” Philosophia Mathematica 14:189–207. Sieg, Wilfried. (2012) 2013. “In the Shadow of Incompleteness: Hilbert and Gentzen.” Pp. 155–192 in Hilbert’s Programs and Beyond, Oxford: Oxford University Press. Sieg, Wilfried. 2013. Hilbert’s Programs and Beyond, Oxford: Oxford Uni- versity Press. Stillwell, John. 2005. The Four Pillars of Geometry. New York: Springer. Turing, Alan. (1936–37) 2004. “On Computable Numbers, with an Ap- plication to the Entscheidungsproblem.” Pp. 58–90 in The Essential Turing. Edited by Jack Copeland. Oxford: Clarendon Press. van Heijenoort, Jean (ed.). 1967. From Frege to Gödel. Cambridge Mass.: Harvard University Press. Weyl, Hermann. (1932) 2009. “The Open World.” Pp. 34–82 in Mind and Nature. Edited by Peter Pesic. Princeton: Princeton University Press.

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/POSC_e_00116 by guest on 27 September 2021