This paper engages the question ‘Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?’ within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof and reception of Gödel’s Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be, but also in which (...) Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be. (shrink)

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in (...) order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. (shrink)

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics duality. Whereas in classical algebraic geometry (...) a certain kind of rings can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A. As we shall argue, important dualities in different areas of mathematics can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding domains. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification (...) of classical mathematics. Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis. Work on the program progressed significantly in the 1920s with contributions from logicians such as Paul Bernays, Wilhelm Ackermann, John von Neumann, and Jacques Herbrand. It was also a great influence on Kurt Gödel, whose work on the incompleteness theorems were motivated by Hilbert's Program. Gödel's work is generally taken to show that Hilbert's Program cannot be carried out. It has nevertheless continued to be an influential position in the philosophy of mathematics, and, starting with the work of Gerhard Gentzen in the 1930s, work on so-called Relativized Hilbert Programs have been central to the development of proof theory. (shrink)

The essays in this volume concern the points of intersection between analytic philosophy and the philosophy of the exact sciences. More precisely, it concern connections between knowledge in mathematics and the exact sciences, on the one hand, and the conceptual foundations of knowledge in general. Its guiding idea is that, in contemporary philosophy of science, there are profound problems of theoretical interpretation-- problems that transcend both the methodological concerns of general philosophy of science, and the technical concerns of philosophers of (...) particular sciences. A fruitful approach to these problems combines the study of scientific detail with the kind of conceptual analysis that is characteristic of the modern analytic tradition. Such an approach is shared by these contributors: some primarily known as analytic philosophers, some as philosophers of science, but all deeply aware that the problems of analysis and interpretation link these fields together. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

This is a survey of Gödel's perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program, using his published and unpublished articles and lectures as well as the correspondence between Bernays and Gödel on these matters. There is also an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end.

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...) revival of infinitesimals (Laugwitz and Robinson—non-standard analysis) and—based upon category theory—the rise of smooth infinitesimal analysis and differential geometry. The spatial whole-parts relation is irreducible (Russell) and correlated with the spatial order of simultaneity. The human imaginative capacities are connected to the characterization of points and lines (Euclid) and to the views of Aristotle (the irreducibility of the continuity of a line to its points), which remained in force until the ninetieth century. Although Bolzano once more launched an attempt to arithmetize continuity, it appears as if Weierstrass, Cantor and Dedekind finally succeeded in bringing this ideal to its completion. Their views are assessed by analyzing the contradiction present in Grünbaum’s attempt to explain the continuum as an aggregate of unextended elements (degenerate intervals). Alternatively a line-stretch is characterized as a one-dimensional spatial subject, given at once in its totality (as a whole) and delimited by two points—but it is neither a breadthless length nor the (shortest) distance between two points. The overall aim of this analysis is to account for the uniqueness of discreteness and continuity by highlighting their mutual interconnections exemplified in the nature of a line as a one-dimensional spatial subject, while acknowledging that points are merely spatial objects which are always dependent upon an extended spatial subject. Instead of attempting to reduce continuity to discreteness or discreteness to continuity, a third alternative is explored: accept the irreducibility of number and space and then proceed by analyzing their unbreakable coherence. The argument may be seen as exploring some implications of the view of John Bell, namely that the “continuous is an autonomous notion, not explicable in terms of the discrete.” Bell points out that initially Brouwer, in his dissertation of 1907, “regards continuity and discreteness as complementary notions, neither of which is reducible to each other.”. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)

Since the discovery of the paradoxes of Zeno, the problem of infinity was dominated by the meaning of endlessness—a view also adhered to by Herman Dooyeweerd. Since Aristotle, philosophers and mathematicians distinguished between the potential infinite and the actual infinite. The main aim of this article is to highlight the strengths and limitations of Dooyeweerd’s philosophy for an understanding of the foundations of mathematics, including Dooyeweerd’s quasi-substantial view of the natural numbers and his view of the other types of numbers (...) as functions of natural numbers. Dooyeweerd’s rejection of the actual infinite is turned upside down by the exploring of an alternative perspective on the interrelations between number and space in support of the idea of infinite totalities, or infinite wholes. No other trend has succeeded in justifying the mathematical use of the actual infinite on the basis of an analysis of the intermodal coherence between number and space. (shrink)

While claiming that diagrams can only be admitted as a method of strict proof if the underlying axioms are precisely known and explicitly spelled out, Hilbert praised Minkowski’s Geometry of Numbers and his diagram-based reasoning as a specimen of an arithmetical theory operating “rigorously” with geometrical concepts and signs. In this connection, in the first phase of his foundational views on the axiomatic method, Hilbert also held that diagrams are to be thought of as “drawn formulas”, and formulas as “written (...) diagrams”, thus suggesting that the former encapsulate propositional information which can be extracted and translated into formulas. In the case of Minkowski diagrams, local geometrical axioms were actually being produced, starting with the diagrams, by a process that was both constrained and fostered by the requirement, brought about by the axiomatic method itself, that geometry ought to be made independent of analysis. This paper aims at making a twofold point. On the one hand, it shows that Minkowski’s diagrammatic methods in number theory prompted Hilbert’s axiomatic investigations into the notion of a straight line as the shortest distance between two points, which start from his earlier work focused on the role of the triangle inequality property in the foundations of geometry, and lead up to his formulation of the 1900 Fourth Problem. On the other hand, it purports to make clear how Hilbert’s assessment of Minkowski’s diagram-based reasoning in number theory both raises and illuminates conceptual compatibility concerns that were crucial to his philosophy of mathematics. (shrink)

Wilfred Sieg. Relative Consistency and Accesible Domains.

Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...) to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. (shrink)

On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über dasUnendlicheand is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist justification of classical mathematics and the role his proof theory was to play in it. But what has become of the ambitious program for securing all of mathematics, once (...) and for all? What of proof theory, the very subject Hilbert invented to carry out his program? The Hilbertian ambition reached out too far: in its original form, the program was refuted by Gödel's Incompleteness Theorems. And even allowing more than finitist means in metamathematics, the Hilbertian expectations for proof theory have not been realized: a constructive consistency proof for second-order arithmetic is still out of reach. Nevertheless, remarkable progress has been made. Two separate, but complementary directions of research have led to surprising insights: classical analysis can be formally developed in conservative extensions of elementary number theory; relative consistency proofs can be given by constructive means for impredicative parts of second order arithmetic. The mathematical and metamathematical developments have been accompanied by sustained philosophical reflections on the foundations of mathematics. This indicates briefly the main themes of the contributions to the symposium; in my introductory remarks I want to give a very schematic perspective, that is partly historical and partly systematic. (shrink)

In early major works, Cassirer and Schlick differently recast traditional doctrines of the concept and of the relation of concept to intuitive content along the lines of recent epistemological discussions within the exact sciences. In this, they attempted to refashion epistemology by incorporating as its basic principle the notion of functional coordination, the theoretical sciences' own methodological tool for dispensing with the imprecise and unreliable guide of intuitive evidence. Examining their respective reconstructions of the theory of knowledge provides an axis (...) of comparison along which to locate Cassirer's Neo-Kantianism and Schlick's pre-positivist empiricism, and an immediate background of contrast to the subsequent rise of logical empiricism. For in the absence of intuition all our knowledge is without objects and therefore remains entirely empty. (Kant A62/B87) And how awkward is the human mind in diving the nature of things when forsaken by the analogy of what we see and touch directly? (Boltzmann 1895). (shrink)

It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style (...) axiomatical systems are closed systems, a reproachposed by Carlo Cellucci. In the second section the question isdiscussed whether Hilbert's pragmatism in foundational issuescomes close to an a-philosophical ``naturalism in mathematics'' assuggested by Penelope Maddy. The answer is ``no'', because forHilbert philosophy had its specific tasks in the general projectto found mathematics. This is illuminated in the concludingsection giving further evidence for Hilbert's foundationalapriorism by discussing his ``axiom of the existence of mind'' andrelating it to the ``one and only axiom'' of the German algebraistof logic, Ernst Schröder, postulating the inherence of signs onthe paper. (shrink)

During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as continuity. (...) Epistemology of mathematics today only remembers the distinction, forgetting where they agreed, in this manner not only destroying the unity of the perceptual and conceptual but also forgetting what could be gained from Aristotelian demonstrative science. (shrink)

Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would (...) be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ?Die Widerspruchsfreiheit der reinen Zahlentheorie? (1936). As to Hilbert and Bernays 1939, we end on a ?critical? note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism. (shrink)

L'histoire des fondements des mathématiques du xxe siècle montre qu'il nous faut réviser la signification des notions philosophiques traditionnelles comme « évidence », « existence », « expérience » ou « rationalité ». On expose comment le logicien Paul Bernays, familier des conceptions de la philosophie de Jacob Friedrich Fries et de Léonard Nelson, donne aux résultats techniques une interprétation philosophique dont il s'inspire de plus en plus - à partir du milieu du siècle - de la « philosophie ouverte (...) » de Ferdinand Gonseth. Bien avant Thomas Kuhn, Bernays conçoit la révision envisagée non pas simplement comme une question de vérité ou de fausseté, mais comme un problème exigeant l'introduction d'un nouveau système conceptuel. Cependant, l'affaiblissement du concept de vérité est pour lui une conséquence épistémologique provenant de la théorie de la connaissance adoptée et non une conséquence méthodologique issue d'un changement de paradigme et donc d'une incommensurabilité. (shrink)

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new (...) analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom). (shrink)

David Hilbert's finitistic standpoint is a conception of elementary number theory designed to answer the intuitionist doubts regarding the security and certainty of mathematics. Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s. The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received (...) little attention, and on a number of philosophical reconstructions of the viewpoint (in particular, by Hand, Kitcher, and Tait). (shrink)

There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...) refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper.. (shrink)