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1The foundational research of David Hilbert is typically read from a foundationalist perspective. Bringing the axiomatic method to perfection in proof theory, the German mathematician proposed a foundational program, the so-called “Hilbert’s program”, aimed at the final justification of mathematics, as well as of all the sciences in which mathematics played a prominent role. He also advanced a conception of scientific knowledge as consisting of two parts: i.e. the axioms of a theory that are the foundation, since they do not depend on other knowledge, and the theorems, which instead are derived, because they can be obtained from the first according to the principles of logic. Such a conception of knowledge has been questioned by the limitative theorems for formal mathematical systems, particularly by the incompleteness theorems discovered by Kurt Gödel in the early 1930s, which also showed the foundationalist aspirations of Hilbert’s program to be hopeless, at least its original version. Gödel’s discovery, in fact, showed that the hope of obtaining a final justification for mathematical sciences along the lines drawn by the program could never have been realized.
- 1 For a coherent foundationalist reading of Hilbert’s investigations, cf. Cellucci, Carlo, Filosofia (...)
- 2 Von Neumann was a universal mathematician capable, just like Hilbert, of deep methodological reflec (...)
2In this context, I will not discuss these readings of Hilbert’s research, since it is undeniable that it had a clear foundationalist character as well as clear foundationalist aspirations. Yet, a mere foundationalist reading of the German mathematician’s investigations – certainly capable of grasping the most visible motifs – tends to leave out other aspects equally present in them, i.e. aspects that I often call “pragmatic”, but which could also be read in utilitarian terms.1 I am talking about aspects related to the notion of success, which also reveal a methodological opportunism not only in science, but so too in axiomatics as it was practiced by both Hilbert and his School. These aspects can be read in utilitarian terms especially if one believes that utilitarianism is connected to a form of consequentialism not only in normative ethics, but in epistemology as well. Therefore, in the methodology of science, utilitarianism would share the opportunistic idea that no concept, principle or method that can be safely introduced within a science and can be useful for maintaining or extending its boundaries (i.e. successful as having positive consequences) should be banned from the realm of science. So, it is not by chance that, when Gödel’s results signaled the end of the foundationalist aspirations of Hilbert’s program, it was a mathematician who took part actively in the Hilbertian project of providing justification for all the mathematical sciences to propose, in an axiomatic key, an explicit opportunistic methodology of science based on the notion of success. I am referring to the “legendary” figure of John von Neumann, who also contributed significantly to the discovery and the assimilation of the incompleteness theorems.2
3In this article, I will focus on the aspects present both in Hilbert’s foundational investigations and in the methodology of science shaped by von Neumann after the discovery of the incompleteness theorems that could be read in utilitarian terms. Therefore, the first part of the article will be devoted to Hilbert (section 2), the other part to von Neumann (section 3).
- 3 For an image of the tortuous path of Hilbert’s research from Foundations of Geometry (1899) to Foun (...)
4In the last forty years, studies on Hilbert have shown the lengthy and dialectical evolution of the German mathematician’s foundational research.3 However, due to the fact that my task here is not strictly historiographic, I shall consider such research as divided into two moments, i.e. respectively before and after the proof theoretical turn in Hilbert’s research.
5This turn was announced in the celebrated talk given by the German mathematician in Zürich in 1917, Axiomatic Thought, in which he extensively considered the centrality of the axiomatic method for foundational research. In this discourse, he clearly affirmed the precise role of axiomatics in forcing the theoretician to touch upon epistemological questions that required a new field of research to be opened, i.e. proof theory. Among these questions, there were those related to the consistency of the axioms, those concerning the possibility of solving any problem clearly formulated, and those regarding the relation between formalisms and contents in the sciences:
- 4 Hilbert, David, «Axiomatisches Denken» (1918), in Gesammelte Abhandlungen (Berlin, Springer, 1935), (...)
All such questions of principle […] seem to me to form an important new field of research which remains to be developed. To conquer this field we must, I am persuaded, make the concept of specifically mathematical proof itself into 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.4
6So, in Hilbert’s foundational research I shall identify the aspects interpretable in utilitarian terms both before and after its proof theoretical turn. In the first case, the German mathematician was simply attempting to extend the axiomatic method, from which he had benefited in his investigations on the foundations of geometry, to any other mathematical science independently of whether it was pure or applied; while, in the second case, the same attempt came to be radicalized within a foundational program that aimed at the final justification of all the sciences that could be axiomatically reconstructed.
7Before considering the issues of success and opportunism, I would only like to highlight three elements of continuity between the two moments (besides those that could be read in utilitarian terms): i.e. the centrality given to the axiomatic method in investigating the foundations of a science, which never took second place in Hilbert’s thought; the necessity to face the issue of consistency, whenever a system of axioms has been found for a given body of knowledge; the need to pursue foundational research while keeping in mind that sciences cannot be deprived of their most valuable treasures, nor can they be prevented from further developments. Rather, sciences must be justified as they are ordinarily practiced, without imposing excessively rigid methodological restrictions.
- 5 Hilbert, David, «Über den Zahlbegriff», Jahresbericht der Deutschen Mathematiker-Vereinigung, 8 (19 (...)
- 6 With the new edition of Foundations of Geometry (19032), the axiomatic method was extended beyond t (...)
- 7 Reid, C., Hilbert, p.129.
8Hilbert officially entered into the debate on the foundations of the sciences with the publication of Foundations of Geometry (1899). In this work, the axiomatic method allowed the German mathematician to reconstruct Euclidean geometry in a perspicuous way, prefiguring analogous developments beyond the Euclidean domain, as well as beyond geometry itself. These developments were immediately pursued. The next year, presenting a system of axioms for real numbers in München, Hilbert made the following statement on the significance of the axiomatic method for foundational investigation: “for the final presentation and the complete logical grounding [Sicherung] of our knowledge the axiomatic method deserves the first rank”.5 Until the proof theoretical turn, therefore, he had been attempting to extend the axiomatic method to all the sciences in which mathematics played a prominent role and which required a deepening of their foundations.6 One of Hilbert’s assistants for physics, Paul Ewald, recalled the spirit of those years with the following words: “We have reformed mathematics, the next thing is to reform physics, and then we’ll go on to chemistry”.7 It was, indeed, a reform of the mathematical sciences under the flag of the axiomatic method.
9Now, when Hilbert’s investigations of the period are contextualized, bearing in mind the profound changes which many sciences were undergoing, as well as the methodological debates that inevitably arouse, one will notice some opportunistic hints, in his proposal of the axiomatic reform of the sciences, which can be read in utilitarian terms. In fact, the reform fundamentally aimed at justifying all the conceptual tools that in the practice of sciences had helped to preserve their results and, at the same time, had favored further developments. To be more precise on this point, I shall take as an example the case of analysis, to which Hilbert paid much attention after the proof theoretical turn of his foundational research.
- 8 On the transformations and expansions of mathematics during the 19th century, see for instance Klin (...)
10Analysis had gone through profound transformations and significant expansions throughout the 19th century, to the point that it had been extended to functions concerning domains of numbers far from experience (such as to functions of complex variables). Developments of this kind were favored by the conceptual renewal that had affected the discipline – similarly to many other branches of mathematics – and that led to the adoption of innovative techniques on which, however, there was no unanimous consensus. These techniques were deeply related to the use of infinitary concepts, principles or methods. And so, when paradoxes were found in analysis, even before the paradoxes of set theory, a methodological debate arose that involved the most important mathematicians of the time, discussing the exact techniques that had allowed the discipline to transform and expand significantly.8
11At the end of the 19th century, when Hilbert came on the scene, the methodological discussion was divided, generally, between two “parties”: the “critics” and the “liberals”, symbolically gathered around the figures, respectively, of Leopold Kronecker and Richard Dedekind. Each defended a distinct model of rigor in mathematics, which led them to take a position, either of refusal or of acceptance, with respect to the conceptual renewal which mathematics had undergone in the course of the century.
- 9 Weber, Heinrich, «Leopold Kronecker», Mathematische Annalen, 43 (1893), p.15.
12Without a doubt, it was Kronecker who defended the most restricting methodological positions. He belonged to the old generation of mathematicians who had inherited the image of science as based on rigid algorithmic procedures. The following motto best characterizes his position: “God created the integers; everything else is the work of man”.9 According to this view, only integers exist in a genuine sense and all other systems of numbers exist only if they can be constructed from integers through procedures that are finitary in character, i.e. algorithmic ones. Procedures of such a kind, however, had serious consequences both from an ontological and from a methodological point of view. Although he admitted the existence of rational numbers, for instance, Kronecker rejected the general notion of irrational numbers and accepted only the real numbers that could be constructed as the roots of algebraic equations. As for argumentative procedures, instead, the greater restriction that he imposed was linked to the use of the excluded middle, which had to be admitted in proofs only if limited to finite domains. He was also a cruel critic of the newly born set theory, which he despised as the most useless product of human imagination. All of these restrictions de facto prevented Kronecker from legitimating analysis as it had been developed over the preceding decades.
- 10 Cf. Sieg, Wilfried, «Relative Consistency and Accessible Domains», Synthese, 84 (1990), pp.261-266.
13Yet, in the same years in which Kronecker tried to impose his Diktat, a new generation of mathematicians was forming, which was fascinated by more liberal ideas and which did not disdain the advantages that a more abstract approach seemed to promise. Among those mathematicians was Hilbert. It was a generation pushed by Dedekind’s investigations (as well as by Georg Cantor’s) to take new directions, in order to return mathematics to the rigor that seemed to be put into question by the paradoxes in analysis and set theory.10
- 11 Dedekind, Richard, «Dedekind an Weber, 24 Januar 1888», in Gesammelte mathematische Werke, eds. Rob (...)
- 12 Cf. Sieg, Wilfried and Schlimm, Dirk, «Dedekind’s Analysis of Number: Systems and Axioms», Synthese(...)
14Although he was almost the same age as Kronecker, Dedekind had quite innovative methodological ideas, which led him, for example, to use set theoretical notions in investigating the foundations of numerical systems. He abhorred any kind of restriction imposed by principle on the existence of certain objects, as well as on the legitimacy of certain methods. At the same time, he believed that the conceptual tools used by mathematicians, including the infinitary ones, should continue to be free creations of the mind. On January 24, 1888, Dedekind wrote the following words to Heinrich Weber: “We are a divine race and undoubtedly possess a creative power, not merely in material things (railways, telegraphs) but especially in things of the mind”.11 In this way, he demonstrated his admiration for the developments obtained by mathematics in the last decades. Before them, Dedekind had not intended to impose any preconceived limit, although he thought that reconsidering the notion of mathematical rigor was inevitable. He did so by practicing an abstract approach that aimed both at the axiomatic reconstruction of diverse numerical systems (for which he also defined the fundamental laws) and at the justification of his reconstructions by seeking consistency proofs that would protect from the danger of the paradoxes.12
- 13 I am referring to the talks Hilbert gave respectively in München, Paris and Heidelberg: cf. Hilbert (...)
15If one considers Hilbert’s early writings on the foundations of arithmetic (“arithmetic” understood in the broad sense that also includes analysis), then the background just outlined will immediately be recognized as their problematic horizon.13
- 14 Hilbert, D., «Über die Grundlagen der Logik und der Arithmetik», p.176. Engl. transl., p.131.
- 15 In the structure it is Dedekind’s approach, although, unlike him, Hilbert started to think at consi (...)
16By explicitly referring to the methodological debate in the 19th century, Hilbert saw the axiomatic method as the best way to free the diverse systems of numbers from the danger represented by the paradoxes, as well as to ground them safely. At the International Congress of Mathematicians in Heidelberg, in 1904, he expressed this point clearly: “It is my opinion that all the difficulties touched upon [i.e. the paradoxes] can be overcome and that we can provide a rigorous and completely satisfying foundation for the notion of number, and in fact by a method that I would call axiomatic”.14 At the same time, he acknowledged that, in order to accomplish this task, axiomatics first had to find an answer to the unavoidable problem of whether the axioms that can be identified for the diverse systems of numbers were free from contradictions.15 Hilbert had already expressed his views on the relevance of the consistency requirement a few years earlier, when in Paris he presented the twenty-three mathematical problems of the new century:
- 16 Hilbert, D., «Mathematische Probleme», p.299. Engl. transl., p.447. Translation slightly modified; (...)
When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. Upon closer consideration […] above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results.16
- 17 Cf. Hilbert, D., «Über die Grundlagen der Logik und der Arithmetik», pp.182-183. Engl. transl., pp. (...)
17It has been often stressed that in looking for systems of axioms for given bodies of knowledge and seeking consistency proofs for them, Hilbert was aiming at the final justification both of classical mathematics and of the sciences in which the discipline played a prominent role. It is undeniable that a clear foundationalist imprint, which during the years has become increasingly evident, is already present in the German mathematician’s early writings. However, to complete the picture, emphasis must be placed on the specific role played by consistency proofs in Hilbert’s axiomatic approach of these years. Protecting the axioms from the danger represented by the paradoxes, in fact, the proofs had to guarantee the use of the conceptual tools characterized by the axioms and, as a consequence, had to legitimate their introduction into mathematics, especially if they were necessary to not mutilate the discipline of the results already achieved, as well as of those achievable in the future. In other words, identifying in consistency the only, minimal criterion for the use of infinitary concepts, principles and methods, Hilbert de facto legitimated the freedom of the mathematicians’ mind in creating everything that (opportunistically) could be considered useful for preserving or extending the boundaries of a science. In Heidelberg, in particular, he spoke about the creative principle that constitutes the standard for the construction and further elaboration of mathematical thought and that, “in its freest use, justifies us in forming ever new notions, with the sole restriction that we avoid a contradiction”.17
18The convictions that no preconceived limits had to be imposed to the development of mathematical knowledge, and that everything that could be useful both to conserve or to expand our knowledge should be favored, were so strong for Hilbert, too, that they would remain unchanged when he developed his program in detail and made consistency the criterion for justifying entire formal theories axiomatically constructed.
- 18 From 1905 to 1917 Hilbert mainly paid attention to the extension of the axiomatic method to physics (...)
19Although systematically pursued from 1922 to 1931, the conceptual roots of Hilbert’s Program must be traced back to the period just considered, specifically that from 1900 to 1905. As clearly shown by the literature, the explicit formulation of the program had to wait for the careful assimilation by Hilbert and his School of Bertrand Russell’s and Alfred North Whitehead’s Principia mathematica (1910-1913), which provided the German mathematician with the logical framework necessary to technically develop the conceptual roots advanced more than a decade earlier.18 The assimilation occurred in conjunction with the proof theoretical turn of his foundational research.
20I shall return soon to the details of Hilbert’s program. At this point, I only would like to stress that, after the formulation of the program, both the German mathematician’s general approach and the problematic horizon of his reflection remained unchanged. Consider, for instance, the talk he gave in Copenhagen in 1922, in which he publicly presented the methodology of the program. On the one hand, Hilbert continued to think that in order to justify mathematics (and, in analogous way, the sciences) one must apply the axiomatic method and prove the consistency of the axioms:
The goal of finding a secure foundation for mathematics is also my own. I should like to regain for mathematics the old reputation for incontestable truth […] but I believe that this can be done while fully preserving its accomplishments. The method that I follow is none other than the axiomatic. […].
- 19 Hilbert, David, «Neubegründung der Mathematik. Erste Mitteilung» (1922), in Gesammelte Abhandlungen(...)
But now it is a question of something even more important. Precisely by means of the formulation which I believe I am able to give of the axiomatic method, we shall see how it leads us to full clarity about the principles of inference in mathematics. As I have already said, we can never be certain in advance of the consistency of our axioms if we do not have a special proof of it. Axiomatics therefore compel us take a stand on this difficult epistemological problem.19
21On the other hand, he continued to believe that this direction was the only one capable of neutralizing the “detractors” of science, who, in order to save science from the danger of the paradoxes, were ready to renounce many of its most relevant domains. When Hilbert came out with his program, there were no more mathematicians gathered around Kronecker, but there were those who took inspiration from Luitzen Brouwer’s intuitionism, among whom was one of Hilbert’s former mentees, i.e. Hermann Weyl. After some decades, therefore, the methodological debate had not substantially changed:
- 20 Hilbert, D., «Neubegründung der Mathematik», pp.159-160. Engl. transl., p.1119.
What Weyl and Brouwer do amounts in principle to following the erstwhile path of Kronecker: they seek to ground mathematics by throwing overboard all phenomena that make them uneasy and by establishing a dictatorship of prohibition à la Kronecker. But this means to dismember and mutilate our science, and if we follow such reformers, we run the danger of losing a large number of our most valuable treasures. Weyl and Brouwer calumniate the general concept of irrational number, of function, even of number-theoretic function, the Cantorian numbers of the higher number-classes, etc.; the proposition that among infinitely many integers there is always a smallest, and even the logical tertium non datur (for example, in the assertion: either there is only a finite number of prime numbers, or there are infinitely many) – all of these are examples of forbidden propositions or modes of inference. […]. No: Brouwer is not, as Weyl believes, the revolution, but only a repetition, with the old tools, of an attempted coup that, in its day, was undertaken with more dash, but nevertheless failed completely; and now […] this coup is doomed to fail [again].20
22Hilbert’s intent was to thwart the newly attempted coup, since through the program he strove to legitimate mathematics once and for all, precisely as it had come to be configured, namely, with all the infinitary concepts, principles or methods that had favored its development, and that could continue to do so in the future. From mathematics, such a program would have been extended to all the sciences in which the discipline played a prominent role.
23Now, let me briefly describe the way that Hilbert’s program aimed to justify mathematics and the sciences as they were effectively practiced.
- 21 Cf. Hilbert, D., «Neubegründung der Mathematik», pp.174-175. Engl. transl., pp. 1131-1132. Hilbert (...)
24Hilbert’s program for the justification of all the mathematical sciences was built around two fundamental ideas clearly presented in a talk he gave in Copenhagen in 1922.21
- 22 Hilbert usually talked about completeness in such an empirical sense. Something close to syntactic (...)
251) First of all, axiomatics needed to be practiced formally. This meant that for each discipline in need of justification, one had to find, through the identification of an appropriate formal language and a suitable logical calculus, a formal system capable of capturing its entire contents, including the infinitary ones. At the same time, the basic propositions (i.e. the “axioms”) of the system should be identified, in order to allow the formal derivation as “theorems” of all the remaining propositions. Clearly, a formal system of such a kind must satisfy some minimal requirements: e.g. the independence requirement, namely, that no superfluous axioms were in the system; the completeness requirement, i.e. that the axioms were sufficient for the entire logical development of the discipline; the consistency requirement, that is, a contradiction would never be derived formally from the axioms.22 Note that in this way the formal system would have captured the contents of the discipline as it was historically given, namely, with all of its usual concepts, principles and methods, including those whose legitimacy were called into doubt. Therefore, another idea collaborated with this in Hilbert’s program.
262) The consistency proof of the formal system, which was assigned the specific task of justifying the discipline, had to be obtained with finitary methods: i.e. methods whose legitimacy the detractors of science could never have doubted. This basically meant that the proof must have been obtained either indirectly, by showing that the consistency problem for a discipline could be reduced to the consistency problem for another, or directly, by effectively showing that within the discipline, one could not derive a proposition and together its contradictory one. In the former case, one usually speaks of relative consistency proofs, while in the latter, of absolute consistency proofs. Relative consistency proofs were mainly required for the formal systems of the sciences, while absolute consistency proofs were expected for the formal systems of the most fundamental disciplines of classical mathematics. While announcing the proof theoretical turn of his foundational investigation in Zürich, Hilbert was clear on this point:
We shall now examine […] the question concerning the consistency of the axioms. This question is obviously of the greatest importance, for the presence of a contradiction in a theory manifestly threatens the contents of the entire theory. […].
In accordance with this requirement I have proved the consistency of the axioms laid down in Grundlagen der Geometrie by showing that any contradiction in the consequences of geometrical axioms must necessarily appear in the arithmetic of the system of real numbers as well.
For the field of physical knowledge too, it is clearly sufficient to reduce the problem of internal consistency to the consistency of the arithmetical axioms. Thus, I showed the consistency of the axioms of the elementary theory of radiation […] presupposing in the process the consistency of analysis.
One may and should in some circumstances proceed similarly in the construction of a mathematical theory. […].
The problem of the consistency of the axiom system for the real numbers can likewise be reduced by the use of set-theoretic concepts to the same problem for the integers: this is the merit of the theories of the irrational numbers developed by Weierstrass and Dedekind.
- 23 Hilbert, D., «Axiomatisches Denken», pp.150,152-153. Engl. transl., pp.1111-1113.
In only two cases is this method of reduction to another special domain of knowledge clearly not available, namely, when it is a matter of the axioms for the integers themselves, and when it is a matter of the foundation of set theory; for here there is no other discipline besides logic which it would be possible to invoke.23
- 24 For a detailed account of absolute consistency proofs both in Hilbert’s program and School see Zach (...)
27Of course, when Hilbert arrived at the formulation of his program, he had already prefigured specific techniques of obtaining finitist consistency proofs in both directions. Relative consistency proofs would have gone through the exhibition of models for axioms (thanks to techniques that had been known for decades), while, for absolute consistency proofs, he excogitated new techniques. Specifically, these techniques tried to eliminate the infinitary (or “ideal”) elements from the formal systems, in order to prove that a finitary (or “real”) formula such as 0 ≠ 0 could not be formally derived from the axioms. In fact, if a contradiction was formally derivable from the axioms, any other formula would have been derivable from them (due to the pseudo-Scoto law); therefore, for the consistency proof of formal systems, it was sufficient to prove the impossibility of deriving a finitary (false) formula in a finite number of logical steps. This result would have been simpler to obtain, once each formal system was reduced to its finitist part, through an adequate elimination of the ideal elements present in it.24
- 25 The German mathematician explicitly recalled such a reductive strategy in Hilbert, D., «Über das Un (...)
- 26 Bernays, Paul, «Über Hilberts Gedanken zur Grundlegung der Arithmetik», Jahresbericht der Deutschen (...)
28Now, careful consideration of the entirety of Hilbert’s program along the lines just sketched will reveal a strategy that, in principle, aimed at the final justification of all the mathematical sciences in a unitary way. For any given scientific discipline, one had to axiomatically construct a formal system and, at the same time, had to prove finitistically the consistency of such a system either in an absolute or in a relative sense. In this way, the ordinary mathematical and scientific practice (with all its concepts, principles and methods) would have been fully justified. As a whole, Hilbert’s program also showed a strategy of justification that was reductive in character: using relative consistency proofs, the program de facto reduced the problem of justification of the sciences to that of pure mathematics, and reduced the problem of justification of pure mathematics to absolute consistency proofs to be pursued in finitist mathematics, i.e. the finitist part of classical mathematics. Therefore, it was a strategy that led progressively to the problem of justification of all the mathematical sciences within the most fundamental part of pure mathematics.25 This procedure explains why Paul Bernays, who assisted Hilbert in his proof theoretical investigations, said: “The great advantage of Hilbert’s procedure rests precisely on the fact that the problems and difficulties themselves in the grounding of mathematics are transferred from the epistemologico-philosophical domain into the domain of what is properly mathematical”.26
- 27 Cf. Peckhaus, Volker, «The Pragmatism of Hilbert’s Programme», Synthese, 137 (2003), in part. p.148
- 28 I have discussed the dynamical procedure of Hilbert’s axiomatic method in Formica, Giambattista, «O (...)
29A further remark must be added here. The final justification of all the mathematical sciences was the aspiration of Hilbert’s program: i.e. a sort of regulative idea, in the Kantian sense, that guided the German mathematician and his School in their foundational research.27 Therefore, this attempt must not be viewed too dogmatically. Hilbert conceived foundational research – as well as the axiomatic method28 – as a very dynamical procedure and, when consistency proofs were not available, he was even ready to adopt more opportunistic criteria of justification. As a consequence, Hilbert’s program could also be read in utilitarian terms, as it will be explained below.
- 29 Hilbert, D., «Über das Unendliche», p.190. Engl. transl., p.302.
30I have just said that Hilbert’s program could be read in utilitarian terms. The reason for such a judgement is quite straightforward, given the context of Hilbert’s foundational research I have outline above: the finitist standpoint, which ideally would have led via consistency proofs to the final (axiomatic) justification of the mathematical sciences, was mainly adopted by the German mathematician to legitimate the infinitary standpoint in the ordinary practice of the sciences. In a talk given in Münster, On the Infinite, Hilbert was very explicit on this point: “The right to operate with the infinite can be secured only by means of the finite”.29 This statement basically meant that, for him, no concept, principle or method freely created by the theoretician’s mind, which was also useful for maintaining or extending the boundaries of a science, should be banned from the realm of science, especially if the introduction of that tool could be guaranteed by a finitist consistency proof of the formal system that can be axiomatically constructed for the science. Therefore, one can reasonably say that Hilbert’s program, once realized, would have justified a sort of opportunism in the ordinary practice of mathematical sciences.
- 30 Zermelo, Ernst, «Neuer Beweis für die Möglichkeit einer Wohlordung», Mathematische Annalen, 65 (190 (...)
31For example, consider the way in which the German mathematician defended the axiom of choice in the writings of this period. It is an infinitary principle, profoundly debated in mathematics, as is well known. In the axiomatization of set theory proposed by Ernest Zermelo in 1908, the principle appeared among the axioms of the discipline; a few years earlier, in fact, Hilbert’s mentee, thanks to the principle, had demonstrated a significant intermediate result (i.e. the well-ordering theorem for the real numbers) that was necessary for the proof of the continuum hypothesis. In defending the use he had made of the principle, not only did Zermelo recall its great fertility, as well as its widespread presence in mathematics, but he also proposed to adopt for it a methodological rule that had a clear utilitarian flavor. As he stated, “principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all”.30 In that statement, he meant to say that if science required it, in principle, any tool had to be accepted as viable for scientific inquiry. Now, in his writings of the period, Hilbert defended a very similar position. Viewing the principle of choice as the source of all the infinitary concepts, principles and methods in use in mathematics, he believed that in no way was the mathematician to renounce it; in fact, he would habitually employ the principle in order to formulate the transfinite axioms necessary for his formal systems that captured the higher domains of mathematics, e.g. analysis. At the same time, however, with the request of finitist consistency proofs for the formal systems that he constructed, Hilbert believed that through the program, he would have fully legitimated the principle of choice both in the ordinary practice of mathematics and in the construction of those formal systems:
- 31 Hilbert, D., «Die logischen Grundlagen der Mathematik», pp.178-179. Engl. transl., p.1137.
Let us recall the axiom of choice in set theory, which Zermelo was the first to formulate, and on the basis of which he gave his ingenious proof of the well-ordering of the continuum. The objections that were raised against this proof – and against the developments in set theory that are bound up with it – were essentially directed against the axiom of choice. And even today, most people probably believe that the admissibility of the axiom of choice is dubious, but that the other modes of inference that occur in set theory in general (and in Zermelo’s proof in particular) are not objectionable to the same extent. I believe this opinion is false. Instead, logical analysis of the sort studied in my proof theory shows that the essential thought underlying the principle of choice is a general logical principle which is necessary and indispensable even for the most elementary rudiments of mathematical inference. If we make these rudiments secure, we simultaneously establish the principle of choice: my proof theory does both.31
32Hilbert’s program could be read in utilitarian terms, however, not only because once realized, it would have justified a sort of opportunism in the ordinary practice of the sciences. It could also be read as utilitarian because it approved the recourse to opportunistic criteria (beyond finitist consistency proofs) in justifying the introduction of a concept, principle or method useful for maintaining or extending the boundaries of a science. Therefore, when the final justification of a particular tool was not available, the program also allowed for justification in opportunistic terms. This second sense of “utilitarianism” in Hilbert’s program is very explicit, for instance, in the following passage of the talk in Münster, where the German mathematician explicitly legitimated success as an alternative criterion of justification:
- 32 Hilbert, D., «Über das Unendliche», pp.162-163. Engl. transl., p.370.
[I]n recent times we come upon statements like this: even if we could introduce a notion safely (that is, without generating contradictions) and if this were demonstrated, we would still not have established that we are justified in introducing the notion. […]. No, if justifying a procedure means anything more than proving its consistency, it can only mean determining whether the procedure is successful in fulfilling its purpose. Indeed, success is necessary; here, too, it is the highest tribunal, to which everyone submits.32
- 33 Hilbert, D., «Über das Unendliche», p.180. Engl. transl., p.384.
33As a criterion of justification, success could be even extended to entire theories, from concepts, principles and methods, according to Hilbert’s reflection from this period. In Münster, in fact, he explicitly stated that “[t]he final test of every new theory is its success in answering preexistent questions that the theory was not specifically created to answer. By their fruits ye shall know them – that applies also to theories”.33
- 34 Cf. e.g. Hilbert, D., «Probleme der Grundlegung der Mathematik».
- 35 Consider Hilbert’s reaction to Zermelo’s axiomatization of set theory. It is well known that there (...)
34Phrases of such a kind are particularly astonishing, especially if one considers that with the proof theoretical turn the foundationalist aspirations of Hilbert’s research became more pronounced.34 However, it seems clear to me that the German mathematician was favorable to a sort of opportunism not only in the scientific investigation, but also in his foundational research, when, for instance, consistency proofs, which would have guaranteed final justifications, were not available.35
35Therefore, in Hilbert’s foundational research, success was not a criterion of justification to be demonized, or to be simply replaced by stronger epistemological criteria. Rather, it was integrated with appropriate consistency proofs and, when necessary, could even replace this kind of proofs in order to justify the recourse to particular tools, as well as the formulation of specific theories. Like an underground river, success flowed below the foundationalist front of Hilbert’s program, ready to emerge at the surface as soon as the occasion would have required.
36Both in Hilbert’s consideration and in that of his School, success was a criterion with a clear opportunistic flavor that, when necessary, could perform specific epistemological functions, such as justifying not only particular tools but also entire theories. As I shall now seek to show, for one mathematician at work in the School at the time, success would become the key to preserving Hilbert’s axiomatic standpoint after Gödel’s incompleteness theorems, i.e. after the theorems showed the program to be hopeless.
- 36 Von Neumann, John, «Die formalistische Grundlegung der Mathematik», Erkenntnis, 2 (1931), pp.116-12 (...)
37In Hilbert’s School, several young but already prominent mathematicians, including Paul Bernays, Wilhelm Ackerman and John von Neumann, were at work and to one of them was assigned the task of presenting the program to the Second Conference on the Epistemology of the Exact Sciences (Königsberg, September 5-7, 1930). At that moment, the end of Hilbert’s program was just around the corner: at the conference, two days after von Neumann had presented the guidelines of the program,36 Gödel announced an early version of his first incompleteness theorem, during a roundtable on the foundations of mathematics. Soon, this theorem led to a second that clearly showed the hope of the German mathematician for justifying the mathematical sciences once and for all to be untenable.
- 37 Von Neumann, John, «Von Neumann to Gödel. Berlin, 29 November 1930», with the original German in Gö (...)
38In a letter to Gödel in late 1930, von Neumann promptly recognized that the second incompleteness theorem meant the end for Hilbert’s program: “I think that your result has negatively solved the foundational question: there is no rigorous justification for classical mathematics. What sense to attribute to our hope, according to which it is de facto consistent, I do not know – but in my view that does not change the completed fact”.37
39It was clear to von Neumann, however, that Gödel’s results did not only affect the possibility of a final justification of classical mathematics, but also a justification of all of the mathematical sciences, in virtue of the reductive character of Hilbert’s program. The reason can be clearly explained. Gödel’s second incompleteness theorem established a fundamental fact: the fact that the consistency of a formal system, which is able to capture arithmetic, cannot be proved by means formalizable in the system itself. This proposition basically meant the impossibility of finitistically proving the consistency of arithmetic, because being finitist mathematics part of arithmetic would be formalizable within it. Therefore, given the reductive character of the program, the result would inevitably have affected classical mathematics as a whole, as well as the other mathematical sciences. The impossibility of proving finitistically the consistency of arithmetic implied, in fact, the impossibility of reaching, via (relative) consistency proofs, a final justification for classical mathematics, as well as for all the sciences in which mathematics played a prominent role. As a consequence, Gödel’s second incompleteness theorem had a very dramatic impact on the final justification of the mathematical sciences as it was envisioned in Hilbert’s program.
- 38 I am referring in particular to articles which von Neumann wrote in conjunction with the public cha (...)
40Von Neumann pondered this impact in his later foundational reflections, when he came out with a highly opportunistic methodology of the mathematical sciences, which, given its opportunism, could be read in utilitarian terms. Now, I shall focus on these reflections, which are found in few general articles written by the Hungarian mathematician in the last part of his life.38 These writings show a careful reconsideration of the concept of rigor in the mathematical sciences, as well as von Neumann’s attempt to save Hilbert’s axiomatic approach from the impact of Gödel’s incompleteness theorems. In particular, these reflections were founded on three pillars: i.e. the importance of not renouncing the axiomatic method while investigating the foundations of the sciences; the need to replace the consistency requirement (once an adequate system of axioms has been found for a given scientific field) with another one, more appropriate to the epistemological situation determined by the incompleteness theorems; the idea that the foundational research still had to aim at the justification of the sciences as they were de facto practiced, without posing methodological restrictions that would mutilate the sciences or make further developments impossible. Regarding the replacement of the consistency requirement, von Neumann arrived at attributing the key epistemological role to success.
- 39 Cf. Von Neumann, John, «Tribute to Dr. Gödel» (1951), in Foundations of Mathematics: Symposium Pape (...)
41In 1951, von Neumann had the chance to share his view on the relevance of the incompleteness theorems for the foundations of the sciences with a wider audience. As president of the American Mathematical Society, he presented Gödel’s results during the ceremony for the Albert Einstein Award, with which the scholar was honored in that year. He talked about Gödel’s contribution to modern logic as “a landmark which will remain visible far in space and time”,39 and, referring in particular to the incompleteness theorems, he added:
Gödel was the first man to demonstrate that certain mathematical theorems can neither be proved nor disproved with the accepted, rigorous methods of mathematics. In other words, he demonstrated the existence of undecidable mathematical propositions. He proved furthermore that a very important specific proposition belonged to this class of undecidable problems: The question as to whether mathematics is free of inner contradictions. The result is remarkable in its quasi-paradoxical “self-denial”: It will never be possible to acquire with mathematical means the certainty that mathematics does not contain contradictions. It must be emphasized that the important point is that this is not a philosophical principle or a plausible intellectual attitude, but the result of a rigorous mathematical proof of an extremely sophisticated kind.
- 40 Von Neumann, J., «Tribute to Dr. Gödel», pp.IX-X.
[…]. Gödel actually proved this theorem, not with respect to mathematics only, but for all systems which permit a formalization, that is a rigorous and exhaustive description, in terms of modern logic: For no such system can its freedom from inner contradiction be demonstrated with the means of system itself.40
42This passage unequivocally shows von Neumann’s judgment on the relevance of the incompleteness theorems for the foundations not only of classical mathematics, but also of all the mathematical sciences that can be approached by the means of modern logic. According to the Hungarian mathematician, in fact, besides having ushered in the end of Hilbert’s program, Gödel’s results showed unquestionably the impossibility of proving the absolute certainty of ordinary mathematical and scientific practice. In particular, these results called for a careful reconsideration of the notion of mathematical rigor, which was imported from mathematics into the sciences. It is impressive to note how many times the noun “rigor”, or the adjective “rigorous”, or the adverb “rigorously”, comes up within von Neumann’s reflections on the relevance of the incompleteness theorems.
- 41 Bear in mind that for him – as well as it was for Hilbert – the most “vitally characteristic fact” (...)
43It is clear that, for von Neumann, the theorems called for a reconsideration of the notion of rigor in mathematics and in the sciences, particularly considering the debate on the foundations of mathematics up to the discovery of the results by Gödel. This thought was displayed in philosophical articles such as The Mathematician (1947).41
- 42 Cf. Von Neumann, J., «The Mathematician», pp.5-6.
- 43 Von Neumann, J., «The Mathematician», p.5.
- 44 Von Neumann, J., «The Mathematician», p.5.
- 45 Von Neumann, J., «The Mathematician», p.6.
44Von Neumann identified three essential moments in the debate on the foundations of mathematics.42 The first was the discovery of certain difficulties in the most advanced branches of modern mathematics. The Hungarian mathematician usually referred to set theory, but analogous things can be said if one takes the example of analysis. The discovery of the difficulties de facto opened the foundational debate, according to von Neumann, and with it the ordinary practice of mathematics was put into question. As a consequence, a second moment arrived, since a reform of classical mathematics became necessary. In this context, according to von Neumann, two kinds of reform were advanced. One was led by personalities such as Russell, Brouwer and Weyl, who imposed such radical restrictions on the ordinary practice of the discipline that “a good fifty per cent of modern mathematics, in its most vital – and up to then unquestioned – parts, especially in analysis, were also affected by this “purge””.43 The other reform was led by Hilbert alone, who advanced an ingenious justification of classical mathematics by showing through indubitable means that the ordinary procedures of mathematical practice could be saved from contradictions. Von Neumann recalled the aspirations of Hilbert’s program with these words: “Had this scheme worked, it would have provided a most remarkable justification of classical mathematics”.44 However, a third moment came when, after “about a decade of attempts to carry out this program, Gödel produced a most remarkable result”,45 which showed Hilbert’s program as being essentially hopeless.
45For von Neumann, each of these three moments determined a significant change in the way theoreticians looked at the concept of rigor in mathematics, and the specific meaning of incompleteness theorems for such a concept – as well as for the evolution of the foundational debate – should be taken in particular consideration:
- 46 Von Neumann, J., «The Mathematician», p.6.
I have told the story of this controversy in such detail, because I think that it constitutes the best caution against taking the immovable rigor of mathematics too much for granted. […] I hope that the above three examples illustrate […] that it is hardly possible to believe in the existence of an absolute, immutable concept of mathematical rigor, dissociated from all human experience.46
46Now, if one compares von Neumann’s understanding of mathematical rigor in his later foundational reflections with the concept of rigor implied by Hilbert’s reductive program for the foundations of the sciences, one can note a relevant change in direction made by the Hungarian mathematician with respect to Hilbert. As I have said, the change was specifically required by the particular epistemological situation introduced by the incompleteness theorems. Hilbert’s program, in fact, since it had transferred the epistemological problem of the justification of the sciences into a domain that was purely mathematical (to recall Bernays’ insight), attempted to extend the concept of rigor of mathematics to all the sciences in which the discipline played a prominent role. Now, because Gödel’s results showed the impossibility of such an extension, it seemed reasonable for the Hungarian mathematician to make the opposite move, i.e. reconsidering the rigor of mathematics in light of the rigor proper to the sciences. Therefore, von Neumann affirmed that rigor of mathematics could not be dissociated from all human experience. The soundness of the thesis was supported by a simple historical fact:
- 47 Von Neumann, J., «The Mathematician», p.7. A similar reconstruction of the debate on the foundation (...)
The main hope of a justification of classical mathematics […] being gone, most mathematicians decided to use that system anyway. After all, classical mathematics was producing results which were both elegant and useful, and, even though one could never again be absolutely certain of its reliability, it stood on at least as sound a foundation as, for example, the existence of the electron. Hence, if one was willing to accept the sciences, one might as well accept the classical system of mathematics. […]. At present the controversy about the “foundations” is certainly not closed, but it seems most unlikely that the system should be abandoned by any but a small minority.47
47So, even though the attempt of justifying classical mathematics (and, through it, the sciences) once and for all was providing to be futile, many theoreticians were not ready to abandon the system. For von Neumann, this resistance unequivocally showed that the concept of rigor, in mathematics as well as in the sciences, satisfied more flexible criteria than those proposed by the defenders of foundationalist views of science. No concept, principle or method that could be shown to be useful for the growth of knowledge had to be abandoned, but, rather, had to be promoted in a new and appropriate methodological approach. Therefore, the time had come for him to present an opportunistic methodology of science, i.e. a methodology more adherent to the real practice of mathematics and the sciences. To von Neumann, in fact, science seemed highly opportunistic in its method.
- 48 That after Gödel’s results von Neumann did not leave the practice of the axiomatic method has been (...)
48Before considering these aspects of von Neumann’s later foundational reflections, one point must not be neglected, as it is the key to correctly interpreting von Neumann’s opportunistic methodology. Although Gödel’s results made him consider Hilbert’s program for the justification of the mathematical sciences untenable, he never questioned that the axiomatic method was the most fruitful method for deepening the foundations of any discipline. For him, the method still guaranteed the possibility of foundational research, without stripping the sciences of their most valuable treasures, nor limiting their further developments. Hilbert’s way towards the foundations required only reconsideration in light of the epistemological situation created by the incompleteness theorems. So, rendering explicit all of the pragmatic aspects already present in the German mathematician’s investigations, the axiomatic method needed to be coherently pursued in opportunistic terms, with a more flexible criterion for the justification of scientific theories axiomatically reconstructed. Von Neumann identified this criterion in success, which was not alien to Hilbert’s School, as I have already shown.48
49I have said that von Neumann never questioned, even after Gödel’s results, the fruitfulness of the axiomatic method for foundational research. Let me add some remarks on the point, since in von Neumann’s later reflection one will hardly find an explicit defense of the method, as one can easily find, instead, in Hilbert’s writings. Yet, von Neumann’s best defense of the axiomatic method was to continue practicing it throughout his life, to the point that Paul Halmos, one of his collaborators, identified in the practice of the method the secret of von Neumann’s success as a scientist:
What made von Neumann great? Was it the extraordinary rapidity with which he could understand and think and the unusual memory that retained everything he had once thought through? No. These qualities, however impressive they might have been, are ephemeral; they will have no more effect on the mathematics and the mathematicians of the future than the prowess of an athlete of a hundred years ago has on the sport of today.
- 49 Halmos, Paul, «The Legend of John von Neumann», The American Mathematical Monthly, 80, 4 (1973), p. (...)
The “axiomatic method” is sometimes mentioned as the secret of von Neumann's success. In his hands it was not pedantry but perception; he got to the root of the matter by concentrating on the basic properties (axioms) from which all else follows. The method, at the same time, revealed to him the steps to follow to get from the foundations to the applications. He knew his own strengths and he admired, perhaps envied, people who had the complementary qualities, the flashes of irrational intuition that sometimes change the direction of scientific progress. For von Neumann it seemed to be impossible to be unclear in thought or in expression. His insights were illuminating and his statements were precise.49
- 50 For general accounts on the three contributions see, respectively, Ferreirós, José, Labyrinth of Th (...)
50Among the greatest scientific achievements of von Neumann’s life, there are three axiomatizations obtained respectively before, during and after the discovery of the incompleteness theorems by Gödel: in chronological order, set theory during the 1920s, quantum mechanics early in the 1930s, and economics early in the 1940s.50 In relation to the last axiomatization, in one of his last talks in Washington D.C., The Impact of Recent Developments in Science on the Economy and Economics (1956), von Neumann recalled the difficulties he had encountered when approaching economics as an exact science:
With regard to the application of the scientific method in economics, it is important to see which difficulties are real and which are only apparent. It is frequently said that economics is not penetrable by rigorous scientific analysis, because one cannot experiment freely. […]. Experimentation is a convenient tool, but large bodies of science have been developed without it.
[…]. What seems to be exceedingly difficult in economics is the definition of categories. If you want to know the effect of the production of coal on the general price level, the difficulty is not so much to determine the price level or to determine how much coal has been produced, but to tell how you want to define the level and whether you mean coal, all fuels, or something in between. In other words, it is always in the conceptual area that the lack of exactness lies. Now all science started like this, and economics, as a science, is only a few hundred years old. The natural sciences were more than a millennium old when the first really important progress was made.
- 51 Von Neumann, John, «The Impact of Recent Developments in Science on the Economy and Economics» (195 (...)
[…]. I think it is in the lack of quite sharply defined concepts that the main difficulty lies, and not in any intrinsic difference between the fields of economics and other sciences.51
- 52 Von Neumann, John and Morgenstern, Oskar, Theory of Games and Economic Behavior (Princeton, NJ, Pri (...)
51Von Neumann soon recognized, when approaching economics with mathematical methods late in the 1930s, that the concept of game would have been the best way to represent economic phenomena, since a wide variety of economic processes could be described with the language of games. As a consequence, he realized that, in order to overcome the lack of clarity in economics, one had to begin by making the notion of game more precise; he also realized that this operation could be done marvelously through the axiomatic method. In fact, after few years of collaboration with Oskar Morgenstern, the two scholars published a landmark book, Theory of Games and Economic Behavior (1944), in which they were able, thanks to the axiomatization of game theory, to define certain basic concepts of economics in terms of games, such as utility or rational behavior. Characterizing the general concept of game in the second chapter of the book, they explicitly stated: “This definition should be viewed primarily in the spirit of the modern axiomatic method”.52
- 53 Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, p.8.
52The way that von Neumann and Morgenstern applied the method was mainly opportunistic and the authors did not hide this fact: “Let it be said at once that the standpoint of the present book […] will be mainly opportunistic”.53 What they meant by opportunism in this context is reaffirmed and clarified to a certain degree in a later passage:
A choice of axioms is not a purely objective task. It is usually expected to achieve some definite aim – some specific theorem or theorems are to be derivable from the axioms – and to this extent the problem is exact and objective. But beyond this there are always other important desiderata of a less exact nature: The axioms should not be too numerous, their system is to be as simple and transparent as possible, and each axiom should have an immediate intuitive meaning by which its appropriateness may be judged directly. In a situation like ours this last requirement is particularly vital, in spite of its vagueness: we want to make an intuitive concept amenable to mathematical treatment and to see as clearly as possible what hypothesis this requires.
- 54 Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, p.25.
[…]. The further heuristic, and even esthetic desiderata, indicated above, do not determine a unique way of finding this axiomatic treatment.54
53Since the choice of a system of axioms is never univocally determined, but, instead, a scientific field can be captured equally by different systems of axioms, the necessary expectation of a chosen system (satisfying reasonable esthetic criteria) is the capacity to allow the achievement of definite aims, above all, that of obtaining particular results from the axioms. On this issue, however, von Neumann returned to his focus on an opportunistic methodology of science.
- 55 It has been argued that, in contrast to Hilbert who had a formal view of the axiomatic method (with (...)
54I shall now consider the methodology of science developed by von Neumann after the discovery of Gödel’s incompleteness theorems. By now it should already be clear that this methodology was committed to the axiomatic method and that the Hungarian mathematician believed that the method could be practiced in an opportunistic fashion.55
- 56 Von Neumann, J., «Method in the Physical Sciences», p.492.
55It should also be noted that, according to von Neumann, this methodology was not a way (i.e. a way different from another) to do science, but it was the way to do science as science, that is, the way in which science may effectively proceed. For him, in fact, science was mainly an opportunistic endeavor, as he argued in Method in the Physical Sciences (1955): “Not only for the sake of argument but also because I really believe it, I shall defend the thesis that the method in question [i.e. the method of science] is primarily opportunistic – also that outside the science, few people appreciate how utterly opportunistic it is”.56
- 57 Von Neumann, J., «Method in the Physical Sciences», p.492.
56In this perspective, science neither explained nor interpreted, according to von Neumann; rather, it constructed mathematical models – ideally with the help of the axiomatic method – in order to describe the phenomena of a given scientific area. The genuine sense of “description” here can be grasped in all its nuances, if one takes into account that, for the Hungarian mathematician, describing also meant predicting. Therefore, from the models of science, one had to expect both a good description of phenomena, as well as the capacity to make correct predictions in the area. Of course, von Neumann was well aware that models had to satisfy certain esthetic criteria, such as simplicity, which he judged as “largely a matter of historical background, of previous conditioning, of antecedents, of customary procedures”.57 However, he also believed that the satisfaction of the criteria had to be considered in function of what is described and predicted by the models, that is, it should not be dissociated from scientific practice. In fact, theoreticians usually prefer less simple models that capture more as opposed to more simple models that capture less.
57Therefore, since scientific models are mainly used to describe and predict, since their construction is hopefully guided by the axiomatic method, and since the choice of a particular system of axioms is never a purely objective task, it is crucial for von Neumann’s methodology of science to present a solution to the epistemological problem of justification of mathematical models, a problem left open by the discovery of the incompleteness theorems, which proved the prospective solution of Hilbert to be unfeasible.
- 58 Von Neumann, J., «Method in the Physical Sciences», p.492.
- 59 Hilbert, D., «Über das Unendliche», p.180. Engl. transl., p.384.
58As I have already anticipated, von Neumann identified success as the criterion to justify any scientific model axiomatically constructed: “The justification of such a mathematical construct is solely and precisely that it is expected to work – that is, correctly to describe [and predict] phenomena from a reasonably wide area”.58 Therefore, not only were the sciences an opportunistic endeavor for him, but, furthermore, those sciences had to be justified by opportunistic criteria, i.e. by success. “By their fruits ye shall know them – that applies also to theories”,59 as Hilbert would have said.
59However, especially when compared to his mentor, the Hungarian mathematician did not abandon the criterion to a sort of vagueness, but, rather, tried to characterize it more precisely. In fact, the capacity of a scientific model to describe and predict had to be judged, according to him, under three different aspects: 1) the number of phenomena to which the model could be applied; 2) their heterogeneity; 3) the possibility of extending the model to phenomena of unexpected areas. As von Neumann said in Method in the Physical Sciences:
The ability to describe – or predict – correctly is important in such a model, but it needs not be decisive per se. […]. Of course, it must be correct. However, as I mentioned above, it is considered very important that the material which has been correctly described or predicted should be heterogeneous. […].
If possible, the confirmation should not all stem from one area alone. In this sense, it is considered particularly significant to find confirmations in area which were not in mind of anyone who invented the theory. […].
- 60 Von Neumann, J., «Method in the Physical Sciences», p.593.
There are also other aspects of the matter which must be kept in mind. It is important that the phenomena which were correctly described should vary considerably, not only qualitatively, but also in their quantitative aspects.60
- 61 Von Neumann, J., «Method in the Physical Sciences», p.498.
60Fundamentally, the success of a mathematical model axiomatically constructed lay, for the Hungarian mathematician, in the power of the model to describe and predict, as well as in its capacity to be formally adaptable in view of correct extensions: “It is a matter of accepting that theory which shows greater formal adaptability for a correct extension”.61 Under different perspectives, both features enlighten the unifying and predictive power of the model. In The Mathematician, von Neumann explicitly pointed to this power:
- 62 Von Neumann, J., «The Mathematician», p.7.
[T]he criterion of success for such a [scientific] theory is simply whether it can, by a simple and elegant classifying and correlating scheme, cover very many phenomena, which without this scheme would seem complicated and heterogeneous, and whether the scheme even covers phenomena which were not considered or even not known at the time when the scheme was evolved. (The two latter statements express, of course, the unifying and the predicting power of a theory).62
- 63 Cf. Formica, G., «Von Neumann’s Methodology of Science», pp.488-492 and 496.
61Clearly, in these foundational reflections, von Neumann was developing his ideas on science in a coherent and opportunistic fashion, rooting them deeply in the mode of Hilbert's School. In particular, he was attributing to success a key epistemological role in justifying not only concepts, principles or methods that were useful for maintaining or extending the boundary of a science, but also entire scientific theories axiomatically constructed. In fact, in his opportunistic methodology, success fundamentally arrived to play the same role performed by consistency proofs in Hilbert’s program.63 The philosophical outcome of such a move is clear, i.e. that opportunism in the ordinary practice of science can be justified only by itself.
62I have said nothing about the role played by success in the justification of concepts, principles or methods useful to science. In this regard, it is sufficient to recall two short passages present in two unpublished letters by von Neumann. The first one is from a letter written to Roger Godement on May 5, 1950, in which the Hungarian mathematician discussed a non-constructivistic technique for solving problems related to the Lebesgue measure in analysis proposed by his colleague. In discussing the technique, von Neumann expressed his convictions about the possibility of violating constructivistic restrictions through the employment of non-constructivistic tools:
- 64 Von Neumann, John, «To Godement, May 5, 1950», in The Papers of John von Neumann (Washington D.C., (...)
Of course, I have no inhibitions about violating any of these strictures in a good cause. When there is a technical necessity I would certainly use the axiom of choice, any aleph, and any degree of impredicativity – but I don’t see that these are to be desired per se. In summa, it would seem to me that we are dealing here with fluctuations in mathematical style which it is still a little early to assess. I do think, of course, that it is most desirable to explore possibilities of the technique that you favor, and the only way to do it is to apply it and to try to displace by it everything that is displaceable, and to see where one gains anything by doing this. In this sense, I am very much in agreement with the change in technique that you advocate.64
63Here, success is related to the employment of non-constructivistic mathematical concepts or principles. In principle, such employment could not be inhibited, but must be assessed on the basis of the results they are allowed to achieve. In one phrase, von Neumann affirmed that no tool useful to the extension of the boundary of knowledge should be avoided in the realm of science.
- 65 Von Neumann’s principal critique of Wang was the following. The manuscript contained consistency pr (...)
64The second passage is from a letter that the Hungarian mathematician wrote to Hao Wang on January 21, 1949, in which he shared his impressions of a manuscript on consistency proofs in elementary number theory and constructivist set theory prepared by his colleague. Although the missive contains some critique (von Neumann did not consider significant per se the proof strategies adopted by Wang),65 he concluded the following consideration:
- 66 Von Neumann, John, «To Wang, January 21, 1949», in The Papers of John von Neumann (Washington D.C., (...)
I feel that the significance of a consistency proof for “elementary number theory and constructivistic set theory”, as you outline it, would be not so much “per se” but rather in the possibility of extending its method in some sense as indicated above [i.e. to impredicative systems]. The interest of a result of the kind, which is the subject-matter of your “Elementary consistency proofs, etc.” paper, would, therefore, depend on the success of subsequent applications and extensions of its method in such a sense. This interest should, therefore, only be judged by these last mentioned criteria, and only when a basis for such an assessment has been created by some success in the indicated direction.66
65The passage is about the proof methods to use in mathematics. Similarly to the discourse made about concepts and principles, in this context von Neumann affirmed that no procedure useful for expanding the contents of a discipline in science should be avoided, since the success of the procedure will be judged by its capacity to expand these contents.
66There is no doubt that the two passages can be placed seamlessly in the context of von Neumann’s opportunistic methodology, which represents the proper background for understanding not only their meaning, but also their attitude towards science, i.e. an attitude that is both conservative and extensive with respect to scientific research. In fact, von Neumann’s opportunism primarily sought to avoid the mutilation of science, both of the results already obtained and of those which could be reached in the future.
67Therefore, once carefully considered, von Neumann’s later methodological reflections, on the one hand, inherit all of the pragmatic aspects present in Hilbert’s foundational investigations, which, in my view, can be read in utilitarian terms; on the other hand, the methodological reflections complete these aspects by placing them in an explicit and coherent framework of opportunism.
68There seems to be a clear evolutionary line that, passing through the incompleteness theorems, can be traced from Hilbert’s foundational investigations to von Neumann’s later methodological reflections. Contrary to what one might think, however, following such a line does not show only how the need arose, in the light of Gödel’s results, to put aside the foundationalist aspirations of Hilbert’s program for mathematical sciences, but also shows the necessity to place all of the pragmatic aspects already present in the German mathematician’s foundational research in a coherent opportunistic methodology of science. This is exactly what von Neumann did in his later methodological reflections, after having recognized the dramatic impact of the incompleteness theorems on Hilbert’s program. The pragmatic aspects present in Hilbert’s foundational investigations orbited around the notion of success and can be read in utilitarian terms as soon as one interprets them as a form of consequentialism, so that in scientific methodology no concept, principle or method (even no entire theory) having positive consequences, i.e. safely introducible within a science and useful for maintaining or extending its boundaries, should be banned from the realm of science. Now, if one considers that this kind of opportunism, present in both Hilbert’s foundational research and von Neumann’s methodological reflection, was for both scholars the way to propose an axiomatic reform of the sciences tied to their real practice, then utilitarian aspects (broadly conceived) can be found, according to the thought of the two mathematicians, not only in the practice of the axiomatic method by them, but in the same practice of science as well.
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Hilbert, David, «Mathematische Probleme» (1900), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, pp.290-329. Engl. transl. «Mathematical Problems», by Mary Winston Newson, Bulletin of the American Mathematical Society, 8 (1902), pp.437-479.
Hilbert, David, «Über die Grundlagen der Logik und der Arithmetik», in Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg von 8. bis 13. August 1904, ed. Adolf Krazer (Leipzig, Teubner, 1905), pp.174-185. Engl. transl. «On the Foundations of Logic and Arithmetic», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), pp.130-138.
Hilbert, David, «Axiomatisches Denken» (1918), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, pp.146-156. Engl. transl. «Axiomatic Thought», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, pp.1107-1115.
Hilbert, David, «Neubegründung der Mathematik. Erste Mitteilung» (1922), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, pp.157-177. Engl. transl. «The New Grounding of Mathematics. First Report», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, pp.1117-1134.
Hilbert, David, «Die logischen Grundlagen der Mathematik» (1923), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, pp.151-165. Engl. transl. «The logical Foundations of Mathematics», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, pp.1136-1148.
Hilbert, David, «Über das Unendliche», Mathematische Annalen, 95 (1926), pp.160-190. Engl. transl. «On the Infinite», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), pp.369-392.
Hilbert, David, «Die Grundlagen der Mathematik», Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6 (1928), pp.65-85. Engl. transl. «The Foundations of Mathematics», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), pp.464-479.
Hilbert, David, «Probleme der Grundlegung der Mathematik», Mathematische Annalen, 102 (1929), pp.1-9. Engl. transl. «Problems of the Grounding of Mathematics», in From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, ed. Paolo Mancosu (New York and Oxford, Oxford University Press, 1998), pp.227-233.
Hilbert, David and Bernays, Paul, Grundlagen der Mathematik (1934 and 1939) (Berlin and Heidelberg, Springer, 1968 and 1970), 2 vols.
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Kline, Morris, Mathematical Thought from Ancient to Modern Times (New York and Oxford, Oxford University Press, 1971), vol. 3.
Mancosu, Paolo, «The Russellian Influence on Hilbert and His School», Synthese, 137 (2003), pp.59-101.
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Moore, Gregory, «The Emergence of First-Order Logic», in History and Philosophy of Modern Mathematics, eds. William Aspray and Philip Kitcher (Minneapolis, University of Minnesota Press, 1988), pp.95-135.
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Moore, Gregory, «Hilbert on the Infinite: The Role of Set Theory in the Evolution of Hilbert’s Thought», Historia Mathematica, 29 (2002), pp.40-64.
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Rédei, Miklós, «John von Neumann on Mathematical and Axiomatic Physics», in The Role of Mathematics in Physical Sciences: Interdisciplinary and Philosophical Aspects, eds. Giovanni Boniolo, Paolo Budinich and Majda Trobok (Dordrecht, Springer, 2005), pp.43-54.
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Notes
For a coherent foundationalist reading of Hilbert’s investigations, cf. Cellucci, Carlo, Filosofia e matematica (Roma and Bari, Laterza, 2002), chapp.1-11. Soon the English reader will also refer to Cellucci, Carlo, The Making of Mathematics: Heuristic Philosophy of Mathematics (forthcoming), chap.2. Having in mind Hilbert’s studies in the last decades, I considered the reading in Formica, Giambattista, Da Hilbert a von Neumann. La svolta pragmatica nell’assiomatica (Roma, Carocci, 2013), part I.
Von Neumann was a universal mathematician capable, just like Hilbert, of deep methodological reflections. For the general profiles of the two mathematicians, cf. Reid, Constance, Hilbert (Berlin and Heidelberg, Springer, 1970) and Israel, Giorgio and Millán Gasca, Ana, The World as a Mathematical Game: John von Neumann and the Twenty Century Science (Basel, Birkhäuser, 2008). On von Neumann and the incompleteness theorems, besides Formica, G., Da Hilbert a von Neumann, part II, see also Formica, Giambattista, «Von Neumann’s Methodology of Science: From Incompleteness Theorems to Later Foundational Reflections», Perspectives on Science, 18, 4 (2010), pp.480-499 and Formica, Giambattista, «John von Neumann’s Discovery of the 2nd Incompleteness Theorem» (forthcoming).
For an image of the tortuous path of Hilbert’s research from Foundations of Geometry (1899) to Foundations of Mathematics (1934 and 1939), refer in particular to Sieg, Wilfried, «Hilbert’s Programs: 1917-1922», The Bulletin of Symbolic Logic, 5 (1999), pp.1-44; Moore, Gregory, «Hilbert on the Infinite: The Role of Set Theory in the Evolution of Hilbert’s Thought», Historia Mathematica, 29 (2002), pp.40-64; Mancosu, Paolo, «The Russellian Influence on Hilbert and His School», Synthese, 137 (2003), pp.59-101; Zach, Richard, «The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert’s Program», Synthese, 137 (2003), pp.211-259; Ferreirós, José, «Hilbert, Logicism and Mathematical Existence», Synthese, 170 (2009), pp.33-70.
Hilbert, David, «Axiomatisches Denken» (1918), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, p.155. Engl. transl. «Axiomatic Thought», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, p.1115.
Hilbert, David, «Über den Zahlbegriff», Jahresbericht der Deutschen Mathematiker-Vereinigung, 8 (1900), p.181. Engl. transl. «On the Concept of Number», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, p.1093. Italics not mine.
With the new edition of Foundations of Geometry (19032), the axiomatic method was extended beyond the Euclidean domain. Contextually, the German mathematician started to approach axiomatically arithmetic and analysis, as well as some domains of contemporary physics. Few years later, a Hilbert’s mentee axiomatized set theory (cf. Zermelo, Ernst, «Untersuchungen über die Grundlagen der Mengenlehre I», Mathematische Annalen, 65 (1908), pp.261-281. Engl. transl. «Investigations in the Foundations of Set Theory I», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), pp.200-215). On Hilbert’s investigations on geometry and physics see respectively Toepell, Michael, Über die Entstehung von David Hilberts “Grundlagen der Geometrie” (Göttingen, Vandenhoeck & Ruprecht, 1986) and Corry, Leo, David Hilbert and the Axiomatization of Physics: From Grundlagen der Geometrie to Grundlagen der Physik (Dordrecht, Springer, 2004), while for those devoted to arithmetic and analysis refer to the historical essays collected in Sieg, Wilfried, Hilbert’s Programs and Beyond (Oxford and New York, Oxford University Press, 2013). For Zermelo’s axiomatization of set theory, cf. Mancosu, Paolo, Zach, Richard and Badesa, Calixto, «The Development of Mathematical Logic from Russell to Tarski, 1900-1935», in The Development of Modern Logic, ed. Leila Haaparanta (New York and Oxford, Oxford University Press, 2009), chap.3.
Reid, C., Hilbert, p.129.
On the transformations and expansions of mathematics during the 19th century, see for instance Kline, Morris, Mathematical Thought from Ancient to Modern Times (New York and Oxford, Oxford University Press, 1971), vol. 3, chapp.XL-XLIII.
Weber, Heinrich, «Leopold Kronecker», Mathematische Annalen, 43 (1893), p.15.
Cf. Sieg, Wilfried, «Relative Consistency and Accessible Domains», Synthese, 84 (1990), pp.261-266.
Dedekind, Richard, «Dedekind an Weber, 24 Januar 1888», in Gesammelte mathematische Werke, eds. Robert Fricke, Emmy Noether and Öystein Ore (Braunschweig, Vieweg, 1932), Bd. 3, p.489. Engl. transl. «Letter to Heinrich Weber (24 January 1888)», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, p.835.
Cf. Sieg, Wilfried and Schlimm, Dirk, «Dedekind’s Analysis of Number: Systems and Axioms», Synthese, 147 (2005), pp.121-170 and Sieg, Wilfried and Schlimm, Dirk, «Dedekind’s Abstract Concepts: Models and Mappings», Philosophia Mathematica, 25 (2017), pp.292-317.
I am referring to the talks Hilbert gave respectively in München, Paris and Heidelberg: cf. Hilbert, D., «Über den Zahlbegriff»; Hilbert, David, «Mathematische Probleme» (1900), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, pp.290-329. Engl. transl. «Mathematical Problems», by Mary Winston Newson, Bulletin of the American Mathematical Society, 8 (1902), pp.437-479; Hilbert, David, «Über die Grundlagen der Logik und der Arithmetik», in Verhandlungen des Dritten Internationalen Mathematiker-Kongresses in Heidelberg von 8. bis 13. August 1904, ed. Adolf Krazer (Leipzig, Teubner, 1905), pp.174-185. Engl. transl. «On the Foundations of Logic and Arithmetic», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), pp.130-138.
Hilbert, D., «Über die Grundlagen der Logik und der Arithmetik», p.176. Engl. transl., p.131.
In the structure it is Dedekind’s approach, although, unlike him, Hilbert started to think at consistency proofs in syntactical terms. On consistency proofs in Hilbert’s early period, as well as on a comparison with Dedekind, cf. Sieg, Wilfried, «Beyond Hilbert’s Reach?», in Reading Natural Philosophy of Science and Mathematics, ed. David B. Malament (Chicago, IL, Open Court, 2002), in part. pp.366-369 and Ferreirós, J., «Hilbert, Logicism and Mathematical Existence», in part. pp.59-63.
Hilbert, D., «Mathematische Probleme», p.299. Engl. transl., p.447. Translation slightly modified; italics not mine.
Cf. Hilbert, D., «Über die Grundlagen der Logik und der Arithmetik», pp.182-183. Engl. transl., pp.135-136.
From 1905 to 1917 Hilbert mainly paid attention to the extension of the axiomatic method to physics, although almost every year he devoted a lectures course to the foundations of mathematics. For further details see Sieg, W., «Hilbert’s Programs: 1917-1922» and Sieg, Wilfried, «Hilbert’s Proof Theory», in Handbook of the History of Logic, eds. Dov M. Gabbay and John Woods (Amsterdam and Boston, Elsevier and North Holland, 2009), vol. 5, pp.321-384, as well as Moore, Gregory, «The Emergence of First-Order Logic», in History and Philosophy of Modern Mathematics, eds. William Aspray and Philip Kitcher (Minneapolis, University of Minnesota Press, 1988), pp.95-135, Zach, Richard, «Completeness before Post: Bernays, Hilbert, and the Development of Propositional Logic», The Bulletin of Symbolic Logic, 5 (1999), pp.331-366 and Mancosu, P., «The Russellian Influence on Hilbert and His School».
Hilbert, David, «Neubegründung der Mathematik. Erste Mitteilung» (1922), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, pp.160-161. Engl. transl. «The New Grounding of Mathematics. First Report», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, pp.1119-1120.
Hilbert, D., «Neubegründung der Mathematik», pp.159-160. Engl. transl., p.1119.
Cf. Hilbert, D., «Neubegründung der Mathematik», pp.174-175. Engl. transl., pp. 1131-1132. Hilbert returned on the two fundamental ideas also in Leipzig, Münster and Hamburg: cf. Hilbert, David, «Die logischen Grundlagen der Mathematik» (1923), in Gesammelte Abhandlungen (Berlin, Springer, 1935), Bd. 3, pp.179-180. Engl. transl. «The logical Foundations of Mathematics», in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, ed. William B. Ewald (Oxford and New York, Oxford University Press, 1996), vol. 2, pp.1137-1138; Hilbert, David, «Über das Unendliche», Mathematische Annalen, 95 (1926), pp.176-179. Engl. transl. «On the Infinite», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), pp.381-383; Hilbert, David, «Die Grundlagen der Mathematik», Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6 (1928), pp.66-74. Engl. transl. «The Foundations of Mathematics», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), pp.465-471.
Hilbert usually talked about completeness in such an empirical sense. Something close to syntactic completeness, as nowadays we understand it, can be found late in his writings: e.g. in Hilbert, David, «Probleme der Grundlegung der Mathematik», Mathematische Annalen, 102 (1929), pp.1-9. Engl. transl. «Problems of the Grounding of Mathematics», in From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, ed. Paolo Mancosu (New York and Oxford, Oxford University Press, 1998), pp.227-233. The independence of the axioms is usually understood by Hilbert in analogous terms, as checking out the impossibility of deriving each axiom from the others. However, absolute independence proofs were conceived by Hilbert in model theoretical terms: i.e. searching for models that satisfy all the axioms except one and, if it happens, considering that axiom as independent. Even though in Hilbert’s early writings we find consistency proofs conceived in semantical terms (i.e. through the exhibition of a model for the axioms), very soon – e.g. already in Hilbert, D., «Mathematische Probleme» – he began to think of the possibility of obtaining consistency proofs in syntactical terms, as the impossibility of deriving two contradictory sentences from the same system of axioms.
Hilbert, D., «Axiomatisches Denken», pp.150,152-153. Engl. transl., pp.1111-1113.
For a detailed account of absolute consistency proofs both in Hilbert’s program and School see Zach, R., «The Practice of Finitism». A sketch of the idea behind these proofs can be found in Hilbert, D., «Die logischen Grundlagen der Mathematik»; while for the general argument I have reported, see e.g. Hilbert, D., «Die Grundlagen der Mathematik», pp.73-74. Engl. transl., p.471.
The German mathematician explicitly recalled such a reductive strategy in Hilbert, D., «Über das Unendliche», pp.179-180. Engl. transl., pp.383-384 and Hilbert, D., «Die Grundlagen der Mathematik», pp.74-75. Engl. transl., p.472.
Bernays, Paul, «Über Hilberts Gedanken zur Grundlegung der Arithmetik», Jahresbericht der Deutschen Mathematiker-Vereinigung, 31 (1922), p.19. Engl. transl. «On Hilbert’s Thoughts Concerning the Grounding of Arithmetic», in From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, ed. Paolo Mancosu (New York and Oxford, Oxford University Press, 1998), pp.221-222.
Cf. Peckhaus, Volker, «The Pragmatism of Hilbert’s Programme», Synthese, 137 (2003), in part. p.148.
I have discussed the dynamical procedure of Hilbert’s axiomatic method in Formica, Giambattista, «On the Procedural Character of Hilbert’s Axiomatic Method», Quaestio, 19 (2019), pp.223-246. Reproducing that reading here will bring me too far.
Hilbert, D., «Über das Unendliche», p.190. Engl. transl., p.302.
Zermelo, Ernst, «Neuer Beweis für die Möglichkeit einer Wohlordung», Mathematische Annalen, 65 (1908), p.115. Engl. transl. «A New Proof of the Possibility of a Well-Ordering», in From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, ed. Jan van Heijenoort (Cambridge, MA, Harvard University Press, 1967), p.189. On the principle of choice, with a special attention to Zermelo’s work, see Moore, Gregory, Zermelo’s Axiom of Choice: Its Origins, Development, and Influence (New York, Springer, 1982).
Hilbert, D., «Die logischen Grundlagen der Mathematik», pp.178-179. Engl. transl., p.1137.
Hilbert, D., «Über das Unendliche», pp.162-163. Engl. transl., p.370.
Hilbert, D., «Über das Unendliche», p.180. Engl. transl., p.384.
Cf. e.g. Hilbert, D., «Probleme der Grundlegung der Mathematik».
Consider Hilbert’s reaction to Zermelo’s axiomatization of set theory. It is well known that there the scholar deferred the answer to the consistency problem for the system of axioms of set theory he presented: “I have not yet even been able to prove rigorously that my axioms are consistent, though this is certainly very essential; instead I have had to confine myself to pointing out now and then that the antinomies discovered so far vanish one and all if the principles here proposed are adopted as a basis” (Zermelo, E., «Untersuchungen über die Grundlagen der Mengenlehre I», p.262. Engl. transl., pp.200-201). Interestingly, also for this reason, Akihiro Kanamori defined “pragmatic” Zermelo’s axiomatization of set theory (cf. Kanamori, Akihiro, «Zermelo and Set Theory», The Bulletin of Symbolic Logic, 20 (2004), p.503). However, the choice of Zermelo to defer the answer to the problem of consistency of his axioms did not prevent Hilbert to see Zermelo’s axiomatization as the most brilliant example of a perfected elaboration of the axiomatic method. Moreover, it seems that the choice to defer that answer was precisely suggested to the scholar by Hilbert’s himself (cf. Moore, G., «Hilbert on the Infinite», p.57 and Peckhaus, Volker, «Pro and Contra Hilbert: Zermelo’s Set Theories», Philosophia Scientiae, 5 (2005), p.205).
Von Neumann, John, «Die formalistische Grundlegung der Mathematik», Erkenntnis, 2 (1931), pp.116-121. Engl. transl. «The Formalist Foundations of Mathematics», in Philosophy of Mathematics: Selected Readings, eds. Hilary Putnam and Paul Benacerraf (Englewood Cliffs, NJ, Prentice-Hall, 1964), pp.50-54.
Von Neumann, John, «Von Neumann to Gödel. Berlin, 29 November 1930», with the original German in Gödel, Kurt, Collected Works, eds. Solomon Feferman et alii (Oxford, Clarendon, 2003), vol. 5, pp.338-341. For details about the conference surrounding Gödel’s announcement, cf. Dawson, John W., Jr., Logical Dilemmas: The Life and Works of Kurt Gödel (Wellesley, MA, A K Peters, 1997), chap.4. I discuss the interactions between von Neumann and Gödel after the conference in Formica, G., «John von Neumann’s Discovery of the 2nd Incompleteness Theorem». For the discussion the two had about the end of Hilbert’s program, cf. Sieg, Wilfried, «Jacques Herbrand: Introductory Note», in Gödel, Kurt, Collected Works, eds. Solomon Feferman et alii (Oxford, Clarendon, 2003), vol. 5, pp.3-13; Sieg, Wilfried, «John von Neumann: Introductory Note», in Gödel, Kurt, Collected Works, eds. Solomon Feferman et alii (Oxford, Clarendon, 2003), vol. 5, pp.327-335; Sieg, Wilfried, «Only Two Letters: The Correspondence between Herbrand and Gödel», The Bulletin of Symbolic Logic, 11 (2005), pp.172-184.
I am referring in particular to articles which von Neumann wrote in conjunction with the public characterization of his figure as a scientist: e.g. Von Neumann, John, «The Mathematician» (1947), in Collected Works, ed. Abraham H. Taub (Pergamon, Oxford, 1961), vol. 1, pp.1-9; Von Neumann, John, «The Role of Mathematics in the Sciences and in Society» (1954), in Collected Works, ed. Abraham H. Taub (Pergamon, Oxford, 1961), vol. 6, pp.477-490; Von Neumann, John, «Method in the Physical Sciences» (1955), in Collected Works, ed. Abraham H. Taub (Pergamon, Oxford, 1961), vol. 6, pp.491-498.
Cf. Von Neumann, John, «Tribute to Dr. Gödel» (1951), in Foundations of Mathematics: Symposium Papers Commemorating the Sixtieth Birthday of Kurt Gödel, eds. Jack J. Bulloff, Thomas C. Holyoke and Samuel W. Hahn (New York, Springer, 1969), pp.IX-X.
Von Neumann, J., «Tribute to Dr. Gödel», pp.IX-X.
Bear in mind that for him – as well as it was for Hilbert – the most “vitally characteristic fact” about mathematics was its rather peculiar relation to the sciences (cf. Von Neumann, J., «The Mathematician», pp.1-2). For both the evolution of the former can be influenced by that of the latter, and vice versa.
Cf. Von Neumann, J., «The Mathematician», pp.5-6.
Von Neumann, J., «The Mathematician», p.5.
Von Neumann, J., «The Mathematician», p.5.
Von Neumann, J., «The Mathematician», p.6.
Von Neumann, J., «The Mathematician», p.6.
Von Neumann, J., «The Mathematician», p.7. A similar reconstruction of the debate on the foundations of mathematics can be found in Von Neumann, J., «The Role of Mathematics in the Sciences and in Society», pp.479-481. There, von Neumann explicitly identified himself as one of those mathematicians who had decided to not abandon the system of classical mathematics in the absence of a final justification: “This may sound odd, as well as a bad debasement of standards, but it was believed by a large group of people for whom I have some sympathy, for I am one of them” (p.481).
That after Gödel’s results von Neumann did not leave the practice of the axiomatic method has been sharply noted by all the scholars who considered his later foundational reflections: cf. Stöltzner, Michael, «Opportunistic Axiomatics – von Neumann on the Methodology of Mathematical Physics», in John von Neumann and the Foundations of Quantum Physics, eds. Miklós Rédei and Michael Stöltzner (Dordrecht, Kluwer, 2001), pp.35-62; Rédei, Miklós, «John von Neumann on Mathematical and Axiomatic Physics», in The Role of Mathematics in Physical Sciences: Interdisciplinary and Philosophical Aspects, eds. Giovanni Boniolo, Paolo Budinich and Majda Trobok (Dordrecht, Springer, 2005), pp.43-54; Rédei, Miklós and Stöltzner, Michael, «Soft Axiomatization: John von Neumann on Method and von Neumann’s Method in the Physical Sciences», in Intuition and the Axiomatic Method, eds. Emily Carson and Renate Huber (Dordrecht, Springer, 2006), pp.235-249; Israel, G. and Millán Gasca, A., The World as a Mathematical Game, chap.8. What has not been sufficiently highlighted by these scholars, however, is that he continued to practice the method plainly in Hilbert’s spirit. So, I tried to complete the picture in Formica, G., «Von Neumann’s Methodology of Science», considering how the spirit of Hilbert was present in von Neumann later foundational reflections.
Halmos, Paul, «The Legend of John von Neumann», The American Mathematical Monthly, 80, 4 (1973), p.394.
For general accounts on the three contributions see, respectively, Ferreirós, José, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (Basel, Birkhäuser, 1999), pp.365-393 and Murawski, Roman, «John von Neumann and Hilbert’s School of Foundations of Mathematics», Studies in Logic, Grammar and Rhetoric, 7 (2004), pp.37-55 for set theory; John von Neumann and the Foundations of Quantum Physics, eds. Miklós Rédei and Michael Stöltzner (Dordrecht, Kluwer, 2001) and Rédei, Miklós, «Mathematical Physics and Philosophy of Physics (with Special Consideration of John von Neumann’s Work)», in History of Philosophy of Science: New Trends and Perspectives, eds. Michael Heidelberger and Friedrich Stadler (Dordrecht, Kluwer, 2002), pp.239-243 for quantum mechanics; John von Neumann and Modern Economics, eds. Mohammed H. Dore, Sukhamoy Chakraborty and Richard Goodwin (Oxford, Clarendon, 1989) and Leonard, Robert, Von Neumann, Morgenstern, and the Creation of Game Theory: From Chess to Social Science, 1900-1960 (Cambridge and New York, Cambridge University Press, 2010) for economics.
Von Neumann, John, «The Impact of Recent Developments in Science on the Economy and Economics» (1956), in Collected Works, ed. Abraham H. Taub (Pergamon, Oxford, 1961), vol. 6, p.101.
Von Neumann, John and Morgenstern, Oskar, Theory of Games and Economic Behavior (Princeton, NJ, Princeton University Press, 1944), p.74.
Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, p.8.
Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, p.25.
It has been argued that, in contrast to Hilbert who had a formal view of the axiomatic method (within a foundationalist framework), von Neumann proposed a softer view of the method (within an opportunistic framework) in his later methodological reflections: cf. Stöltzner, M., «Opportunistic Axiomatics»; Rédei, M., «John von Neumann on Mathematical and Axiomatic Physics»; Rédei, M. and Stöltzner, M., «Soft Axiomatization». It is quite fair that in the two foundational researches there is a significant difference, determined by the discovery of incompleteness theorems; however, such a difference cannot be ascribed, according to me, to the way the two mathematicians looked at the axiomatic method. The method, in fact, seems to be conceived by both Hilbert and von Neumann in the same dynamic way, as well as a very flexible tool of inquiry, which, therefore, could be practiced (without changing in the character) within different methodological attitudes, either formally in a foundationalist framework or more softly in an explicit opportunistic framework. In «On the Procedural Character of Hilbert’s Axiomatic Method», I have described Hilbert’s axiomatic method in its procedural character: exactly the same procedure can be found applied in von Neumann’s later foundational achievements, such as that developed in the axiomatization of economics. See for instance Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior, pp.1-45 (in part., pp.6-8).
Von Neumann, J., «Method in the Physical Sciences», p.492.
Von Neumann, J., «Method in the Physical Sciences», p.492.
Von Neumann, J., «Method in the Physical Sciences», p.492.
Hilbert, D., «Über das Unendliche», p.180. Engl. transl., p.384.
Von Neumann, J., «Method in the Physical Sciences», p.593.
Von Neumann, J., «Method in the Physical Sciences», p.498.
Von Neumann, J., «The Mathematician», p.7.
Cf. Formica, G., «Von Neumann’s Methodology of Science», pp.488-492 and 496.
Von Neumann, John, «To Godement, May 5, 1950», in The Papers of John von Neumann (Washington D.C., Library of Congress, Manuscript Division), Box 3, Folder 19: General Correspondence – Godement, Roger, 1949-50, n.d., s.3/3.
Von Neumann’s principal critique of Wang was the following. The manuscript contained consistency proofs of predicative systems approached through predicative techniques; however, for the Hungarian mathematician, it would have been more interesting to use these techniques to approach consistency proofs of impredicative systems. Of course, such a strategy would have run up against the limits imposed by Gödel’s results, which, in contrast, would have forced one to reach, with predicative techniques, consistency proofs of impredicative systems relative to other impredicative systems. However, even if the former systems had been stronger than the latter ones, the result would have been interesting for von Neumann in any case.
Von Neumann, John, «To Wang, January 21, 1949», in The Papers of John von Neumann (Washington D.C., Library of Congress, Manuscript Division), Box 8, Folder 1: General Correspondence – “W” Miscellaneous, 1938-57, n.d. (1), s.2/5.
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