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.     

Consistency, Models, and Soundness

This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis.     

‘Metamathematics’ in Transition

In this paper, we trace the conceptual history of the term ‘metamathematics’ in the nineteenth century. It is well known that Hilbert introduced the term for his proof-theoretic enterprise in about 1922. But he was verifiably inspired by an earlier usage of the phrase in the 1870s. After outlining Hilbert's understanding of the term, we will explore the lines of inducement and elucidate the different meanings of ‘metamathematics’ in the final decades of the nineteenth century. Finally, we will investigate the earlier occurrences and come to the conclusion that the conceptual prehistory of the Hilbertian notion of metamathematics dates back to 1870, whereas the history of the word starts in 1799 at the latest.

What is this: metamathematics? It is something amazing everybody, […] since it makes the mind dizzy and withdraws thinking its sole fulcrum.1 ¹ Brunner 1898, p. 832: ‘Was ist das: Metamathematik? Es ist etwas, was einen Jeden in das äusserste Staunen versetzt, […] was das Gehirn schwindlig macht und dem Denken seinen einzigen Stütz- und Angelpunkt zu entziehen droht’.      

On a Paradox of Hilbert and Bernays

The paper is a discussion of a result of Hilbert and Bernays in their Grundlagen der Mathematik. Their interpretation of the result is similar to the standard intepretation of Tarskis Theorem. This and other interpretations are discussed and shown to be inadequate. Instead, it is argued, the result refutes certain versions of Meinongianism. In addition, it poses new problems for classical logic that are solved by dialetheism.     

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.     

Cantor on Frege's Foundations of Arithmetic: Cantor's 1885 Review of Frege's Die Grundlagen der Arithmetik

In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik. In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege ‘reckless’ for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell–Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning.     

Satisfying Predicates: Kleene's Proof of the Hilbert–Bernays Theorem

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of the arithmetization is still regarded as the standard one. It is highly compressed, however, and very difficult to read. My goals in this paper are expository: to present the basics of Kleene's arithmetization in a less compressed, more easily readable form, in a setting that highlights its relevance to issues in the philosophy of logic, especially to Quine's substitutional definition of logical truth, and to formulate the Hilbert–Bernays Theorem in a way that incorporates Kreisel's, Mostowski's, and Putnam's refinements of Kleene's result.     

Hilbert's Inferno: Time Travel for the Damned

Combining time travel with certain kinds of supertask, this paper proposes a novel model for Hell. Temporally‐closed spacetimes allow otherwise impossible opportunities for material kinds of damnation and reveal surprising limitations on metaphysical objections to Hell. Prima facie, eternal damnation requires either infinite amounts of time or time for the damned to speed‐up arbitrarily. However, spatiotemporally finite ‘time travel’ universes can host unending personal torment for infinitely many physical beings, while keeping fixed finite limits on rates of temporal passage. Such ‘Hilbert's Inferno’ spacetimes suggest neither materialism nor the finitude of time and space need forbid Hell. A material Hell can be spatiotemporally finite yet eternal for its inhabitants. Hilbert's Inferno also sheds light on Hell's location and accessibility, and shows that some spacetimes are intrinsically better suited to punishment than reward.     

Hume's Foundational Project in the Treatise

In the Introduction to the Treatise Hume very enthusiastically announces his project to provide a secure and solid foundation for the sciences by grounding them on his science of man. And Hume indicates in the Abstract that he carries out this project in the Treatise. But most interpreters do not believe that Hume's project comes to fruition. In this paper, I offer a general reading of what I call Hume's ‘foundational project’ in the Treatise, but I focus especially on Book 1. I argue that in Book 1 much of Hume's logic is put in the service of the other sciences, in particular, mathematics and natural philosophy. I concentrate on Hume's negative thesis that many of the ideas central to the sciences are ideas that we cannot form. For Hume, this negative thesis has implications for the sciences, as many of the texts I discuss make evident. I consider and criticize different proposals for understanding these implications: the Criterion of Meaning and the ‘Inconceivability Principle’. I introduce what I call Hume's ‘No Reason to Believe’ Principle, which I argue captures more adequately the link Hume envisions between his logic, in particular his examination of ideas, and the other sciences.     

Boolos and the Metamathematics of Quine's Definitions of Logical Truth and Consequence

The paper is concerned with Quine's substitutional account of logical truth. The critique of Quine's definition tends to focus on miscellaneous odds and ends, such as problems with identity. However, in an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level. In the article he shows that Quine's substitutional account of logical truth cannot be generalized properly to the general concept of logical consequence. The purpose of this paper is threefold: first, to introduce the reader to the metamathematics of Quine's substitutional definition of logical truth; second, to make Boolos' result accessible to a broader audience by giving a detailed and self-contained presentation of his proof; and, finally, to discuss some of the possible implications and how a defender of the Quinean concepts might react to the challenge posed by Boolos' result.     

SONYA & RASKÓLNIKOV – TOWARDS AN UNDERSTANDING OF THE ORIGIN OF BARTH'S DOCTRINE OF SIN AND GRACE1

This paper examines the origin of Barth's understanding of sin and grace in his reading of Dostoevsky in 1915. It is essentially the theological portrait of Sonya & Raskólnikov (Crime & Punishment) that regrounds Barth's understanding of sin and grace in an orthodox forensic model, which in turn develops into the mature doctrine we see in Die Kirchliche Dogmatik IV. The young Barth is exposed to many influences in his move away from nineteenth‐century neo‐Protestant liberal theology (characterized by a sociological‐humanistic model of sin). Mediated by his theological colleague Eduard Thurneysen, Dostoevsky is one such influence amongst many. Barth's reading has a profound effect on him: sin becomes defined by and in relation to God –eritis sicut deus. This sublapsarian perspective can then be discerned in his seminal paper ‘Die Gerechtigkeit Gottes’, delivered within months of his reading of Crime & Punishment, particularly in the Dostoevsky motif of the Tower of Babel (this reading occurs five to seven years prior to the generally accepted period of the influence of Dostoevsky). Barth's understanding then develops through his study of Romans (Der Römerbrief ) and by rediscovering a traditional approach in the Reformed Confessions in the 1920s; however, it is his reading of Crime and Punishment that initiates this model of sin and grace.     

FIRST‐ORDER LOGICAL VALIDITY AND THE HILBERT‐BERNAYS THEOREM

What we call the Hilbert‐Bernays (HB) Theorem establishes that for any satisfiable first‐order quantificational schema S, there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S. Our goals here are, first, to explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first‐order logical validity of a schema in terms of predicate substitution; second, to clarify the theorem by sketching an accessible and illuminating new proof of it; and, third, to explain how Quine's substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians.     

The lives of Mary Foote: painter and Jungian

Mary Foote (1872‐1968) was a successful early twentieth century American artist who suddenly closed her New York studio in 1926 to go to Zurich to study with Jung. There she joined his ‘Interpretation of Visions’ seminars (1930‐1934), which she recorded and edited. This work won Jung's praise and his friendship, but all too often Foote was seen merely as a secretary or background figure. Deirdre Bair's biography of Jung suggested that Foote's life and work deserved fuller study, if only to rebalance our view of Jung's early women followers. This paper takes up that work to ask how Foote's early life and career led to her important work in preserving and describing Jung's earliest attempts to apply his theories to clinical practice.     

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.     

Skolem and the löwenheim-skolem theorem: a case study of the philosophical significance of mathematical results

The dream of a community of philosophers engaged in inquiry with shared standards of evidence and justification has long been with us. It has led some thinkers puzzled by our mathematical experience to look to mathematics for adjudication between competing views. I am skeptical of this approach and consider Skolem's philosophical uses of the Löwenheim-Skolem Theorem to exemplify it. I argue that these uses invariably beg the questions at issue. I say ‘uses’, because I claim further that Skolem shifted his position on the philosophical significance of the theorem as a result of a shift in his background beliefs. The nature of this shift and possible explanations for it are investigated. Ironically, Skolem's own case provides a historical example of the philosophical flexibility of his theorem.

Our suspicion ought always to be aroused when a proof proves more than its means allow it. Something of this sort might be called ‘a puffed-up proof’.

Ludwig Wittgenstein, Remarks on the foundations of mathematics (revised edition), vol. 2, 21.     

Mourning the dead,mourning the disappeared: The enigma of the absent–presence

Freud's interest in the impact of death on the living goes back further than Mourning and Melancholia (1917e, [1915]). In Totem and Taboo (1912–13) Freud noted the ambivalence of the emotions we experience in relation to the dead. In this paper, I focus on Mourning and Melancholia as a landmark in the understanding of both the normal and psychopathological aspects of mourning and depressive processes in human beings. Mourning and Melancholia bridges Freud's first and second topographic theories of the psychic apparatus and constitutes for many authors the foundation of his theory of internal object relations. With this psychoanalytic understanding of mourning as a framework, I discuss ‘special mourning processes,’ such as the those confronted by psychoanalysts in Argentina when treating the relatives of thousands of people who were ‘disappeared’ by the military dictatorship in the 1970s; they are ‘special’ in the sense that the external reality [which] constitutes the starting point of the psychic mourning process, as described by Freud, is absent. I argue that the ‘absent–presence’ of the body as an enigmatic message initiates a special mourning process that bears certain characteristics of, and is isomorphic to, Laplanche's seduction theory.     

Do You Have the Heart to Come to Faith? A Look at Anti‐Climacus' Reading of Matthew 11.6

In Practice in Christianity, Søren Kierkegaard's pseudonym, Anti‐Climacus enters into an extended engagement with Matthew 11.6, ‘Blessed is he who takes no offense at me’. In so doing, he comes to an understanding that ‘the possibility of offense’ characterises the ‘crossroad’ at which one either comes to faith in Christ's revelation or rejects it. Such a choice, as he is well aware, cannot be made from a neutral standpoint, and so he is led to propose that it is ‘the thoughts of the heart’ (i.e. a person's disposition) that constitute the pivotal factor in determining whether or not God will reconcile a person into the Christian faith. In this paper, I discuss Anti‐Climacus' interpretation of Mt. 11.6 and consider his reasons for interpreting a person's predisposition as being so decisive for faith.