Extensions of the Finitist Point of View

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 ω -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.     

The Middle Wittgenstein’s Critique of Frege

ABSTRACT

This article aims to analyse Wittgenstein’s 1929–1932 notes concerning Frege’s critique of what is referred to as old formalism in the philosophy of mathematics. Wittgenstein disagreed with Frege’s critique and, in his notes, outlined his own assessment of formalism. First of all, he approvingly foregrounded its mathematics-game comparison and insistence that rules precede the meanings of expressions. In this article, I recount Frege’s critique of formalism and address Wittgenstein’s assessment of it to show that his remarks are not so much a critique of Frege as rather a defence of the formalist anti-metaphysical investment.     

Hilbert's ‘Verunglückter Beweis’, the first epsilon theorem,and consistency proofs

In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert's first epsilon theorem and a certain ‘general consistency result’ due to Bernays. An analysis of the form of this so-called ‘failed proof’ sheds further light on an interpretation of Hilbert's programme as an instrumentalist enterprise with the aim of showing that whenever a ‘real’ proposition can be proved by ‘ideal’ means, it can also be proved by ‘real’, finitary means.     

The Role of Structural Reasoning in the Genesis of Graph Theory

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ‘theory of quantity’ (‘Größenlehre’) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section 2, I first analyze Frege's use of the term ‘source of knowledge’ (‘Erkenntnisquelle’) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account.     

Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic

In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs'. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar problem'.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem.     

Aspekte der frege–hilbert-korrespondenz

In a letter to Frege of 29 December 1899, Hubert advances his formalist doctrineaccording to which consistency of an arbitrary set of mathematical sentences is a sufficient condition for its truth and for the existence of the concepts described by it. This paper discusses Frege’s analysis, as carried out in the context of the Frege-Hilbert correspondence, of the formalist approach in particular and the axiomatic method in general. We close with a speculation about Frege’s influence on Hilbert’s later work in foundations, which we consider to have been greater than previously assumed. This conjecture is based on a hitherto neglected revision of Hilbert’s talk Über den Zahlbegriff     

Hilbert,logicism, and mathematical existence

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).     

Wogegen wandte sich Husserl 1891?

A comprehensive and agreed-upon account of Husserl??s relation to Gottlob Frege does not yet exist. In this situation we encounter interpretations that allow systematic dogmas to reappear that should have long been vanquished??for instance, that the author of the Logical Investigations was not only decisively influenced by Frege, but also that he had already retracted his sharpest Frege-critique by 1891. The present essay contains a largely historical response to W. Künne??s new monograph on Frege that advocates such views. We will concentrate on a small remark that turns out to reference a defining moment for any understanding of Husserl??s early philosophy. We shall argue that Husserl??s supposed self-criticism does not turn on the critique that he had earlier leveled at Frege??s Grundlagen der Arithmetik; rather, it has to do exclusively with his own earlier systematic positions on the grounding of arithmetic. In this context, an important particular of Husserl??s Philosophie der Arithmetik takes center stage: this book is a mosaic composed from old and new insights, a fact that becomes most evident in the two distinct concepts of ??equivalence?? that are founded there, which reflects Husserl??s transition from a theory of arithmetic based on the concept of number to one based on the parallelism between proper and symbolic (improper) presentations. This change involves a long historical development that goes back to a tradition marked by the work of Bolzano, Lotze, Brentano, and Stumpf, and it is closely tied to the problem of how to distinguish between the sense and the object of an act. Systematic neglect of the historical background of the Frege?CHusserl relation has led to disputes over who owns the copyright to the sense/reference distinction, but it has obscured the very core of the original line of questioning.     

Wittgenstein’s challenge to enactivism

Many authors have identified a link between later Wittgenstein and enactivism. But few have also recognised how Wittgenstein may in fact challenge enactivist approaches. In this paper, I consider one such challenge. For example, Wittgenstein is well known for his discussion of seeing-as, most famously through his use of Jastrow’s ambiguous duck-rabbit picture. Seen one way, the picture looks like a duck. Seen another way, the picture looks like a rabbit. Drawing on some of Wittgenstein’s remarks about seeing-as, I show how Wittgenstein poses a challenge for proponents of Sensorimotor Enactivism, like O’Regan and Noë, namely to provide a sensorimotor framework within which seeing-as can be explained. I claim that if these proponents want to address this challenge, then they should endorse what I call Sensorimotor Identification, according to which visual experiences can be identified with what agents do.

     

Thought's Social Nature

Abstract: Wittgenstein, throughout his career, was deeply Fregean. Frege thought of thought as essentially social, in this sense: whatever I can think is what others could think, deny, debate, investigate. Such, for him, was one central part of judgement's objectivity. Another was that truths are discovered, not invented: what is true is so, whether recognised as such or not. (Later) Wittgenstein developed Frege's idea of thought as social compatibly with that second part. In this he exploits some further Fregean ideas: of a certain generality intrinsic to a thought; of lack of that generality in that which a thought represents as instancing some such generality. (I refer to this below as the ‘conceptual‐nonconceptual’ distinction.) Seeing Wittgenstein as thus building on Frege helps clarify (inter alia) his worries, in the Blue Book, and the Investigations, about meaning, intending, and understanding, and the point of the rule following discussion.     

On Structural Features of the Implication Fragment of Frege’s <Emphasis Type="Italic">Grundgesetze</Emphasis>

We set out the implication fragment of Frege’s Grundgesetze, clarifying the implication rules and showing that this system extends Absolute Implication, or the implication fragment of Intuitionist logic. We set out a sequent calculus which naturally captures Frege’s implication proofs, and draw particular attention to the Cut-like features of his Hypothetical Syllogism rule.     

The End(s) of Philosophy: Rhetoric, Therapy and Wittgenstein's Pyrrhonism

In Culture and Value Wittgenstein remarks: ‘Thoughts that are at peace. That's what someone who philosophizes yearns for’. The desire for such conceptual tranquillity is a recurrent theme in Wittgenstein's work, and especially in his later ‘grammatical‐therapeutic’ philosophy. Some commentators (notably Rush Rhees and C. G. Luckhardt) have cautioned that emphasising this facet of Wittgenstein's work ‘trivialises’ philosophy – something which is at odds with Wittgenstein's own philosophical ‘seriousness’ (in particular his insistence that philosophy demands that one ‘Go the bloody hard way’). Drawing on a number of correlations between Wittgenstein's conception of philosophy and that of the Pyrrhonian Sceptics, in this paper I defend a strong ‘therapeutic’ reading of Wittgenstein, and show how this can be maintained without ‘trivialising’ philosophy.     

Peter Winch on ‘Aristotelian’ and ‘Socratic’ Reasoning

Peter Winch often returned to questions about the nature of logic. In the context of his work on Wittgenstein and political philosophy in the 1990s, Winch described a contrast between ‘Aristotelian’ and ‘Socratic’ reasoning. Aristotelian conceptions of reasoning, attributed to Frege and Russell, would see logic as a formal science and rationality as consistency with pre‐existent rules of inference. The Socratic conception, attributed to Wittgenstein, understands rational argument as a form of socially embedded dialogue that involves moral relationships and a dimension of depth. Rational persuasion may also involve use of persuasive images and examples.     

Is God His Essence? The Logical Structure of Aquinas’ Proofs for this Claim

In this article I consider whether Aquinas’ arguments for the claim that God is His essence are conclusive, and what was his purpose of upholding this thesis. I show his proofs from Summa Theologiae and Summa Contra Gentiles to be problematic and argue that the defense of Aquinas’ views on that matter suggested by certain remarks of P. T. Geach is flawed.     

Wittgenstein,Quasi-fideism,and Scepticism

In the discussion of certainties, or ‘hinges’, in Wittgenstein’s On Certainty some of the examples that Wittgenstein uses are religious ones. He remarks on how a child might be raised so that they ‘swallow down’ belief in God (§107) and in discussing the role of persuasion in disagreements he asks us to think of the case of missionaries converting natives (§612). In the past decade Duncan Pritchard has made a case for an account of the rationality of religious belief inspired by On Certainty which he calls ‘quasi-fideism’. Pritchard argues that religious beliefs are just like ordinary non-religious beliefs in presupposing fundamental arational commitments. However, Modesto Gómez-Alonso has recently argued that there are significant differences between the kinds of ‘hinges’ discussed in Wittgenstein’s On Certainty and religious beliefs such that we should expect an account of rationality in religion to be quite different to the account of rational practices and their foundations that we find in Wittgenstein’s work. Fundamental religious commitments are, as Wittgenstein said, in the foreground of the religious believer’s life whereas hinge commitments are said to be in the background. People are passionately committed to their religious beliefs but it is not at all clear that people are passionately committed to hinges such as that ‘I have two hands’. I argue here that although there are differences between religious beliefs and many of the hinge-commitments discussed in On Certainty religious beliefs are nonetheless hinge-like. Gómez-Alonso’s criticisms of Pritchard mischaracterise his views and something like Pritchard’s quasi-fideism is the correct account of the rationality of religious belief.

     

A Comfortable Sureness: Knowledge,Animality and Conceptual Investigations in Wittgenstein’s On Certainty

This essay analyses some remarks of Wittgenstein’s On Certainty in which Wittgenstein compares human behaviour to that of animals and says he wants to consider man as an animal. The essay’s main purpose is to show that these remarks are essentially understood as part and parcel of what Wittgenstein calls “conceptual investigations” and that, consequently, they give little support to On Certainty’s naturalistic interpretations. A second purpose of the essay is to show that Wittgenstein does not intend to combat the use of “I know” in contexts such as those evoked by Moore; rather he wants to draw attention to the different ways in which we say or can say “I know.”

     

On the Meaning of Hilbert's Consistency Problem (Paris, 1900)

The theory that ``consistency implies existence' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems.