"Science et méthode" de Henri Poincaré. Mathématicien, physicien et philosophe français (1854-1912).

The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.

An introduction to the developments of nonstandard models. Beginning with Godel's incompleteness theorem, it covers the prime models, cofinal extensions, and extensions, Gaifman's construction of a definable type, Tennenbaum's theorem and Friedman's theorem on indicators, ending with a chapter on recursive saturation and resplendency.

This comprehensive volume gives a panorama of the best current work in this lively field, through twenty specially written essays by the leading figures in the field. All essays deal with foundational issues, from the nature of mathematical knowledge and mathematical existence to logical consequence, abstraction, and the notions of set and natural number. The contributors also represent and criticize a variety of prominent approaches to the philosophy of mathematics, including platonism, realism, nomalism, constructivism, and formalism.

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell’s Paradox being derivable in it.This system is, except for minor differ...

Gottlob Frege famously rejects the methodology for consistency and independence proofs offered by David Hilbert in the latter's Foundations of Geometry. The present essay defends against recent criticism the view that this rejection turns on Frege's understanding of logical entailment, on which the entailment relation is sensitive to the contents of non-logical terminology. The goals are (a) to clarify further Frege's understanding of logic and of the role of conceptual analysis in logical investigation, and (b) to point out the extent (...) to which his understanding of logic differs importantly from that of the model-theoretic tradition that grows out of Hilbert's work. (shrink)

In this paper, we explore Fregean metatheory, what Frege called the New Science. The New Science arises in the context of Frege’s debate with Hilbert over independence proofs in geometry and we begin by considering their dispute. We propose that Frege’s critique rests on his view that language is a set of propositions, each immutably equipped with a truth value (as determined by the thought it expresses), so to Frege it was inconceivable that axioms could even be considered to be (...) other than true. Because of his adherence to this view, Frege was precluded from the sort of metatheoretical considerations that were available to Hilbert; but from this, we shall argue, it does not follow that Frege was blocked from metatheory in toto. Indeed, Frege suggests in Die Grundlagen der Geometrie a metatheoretical method for establishing independence proofs in the context of the New Science. Frege had reservations about the method, however, primarily because of the apparent need to stipulate the logical terms, those terms that must be held invariant to obtain such proofs. We argue that Frege’s skepticism on this score is not warranted, by showing that within the New Science a characterization of logical truth and logical constant can be obtained by a suitable adaptation of the permutation argument Frege employs in indicating how to prove independence. This establishes a foundation for Frege’s metatheoretical method of which he himself was unsure, and allows us to obtain a clearer understanding of Frege’s conception of logic, especially in relation to contemporary conceptions. (shrink)

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to Brouwer, Weyl, Bernays and Hilbert, and the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

Not only can the influence of Gottlob Frege be found in contemporary work in logic, the philosophy of mathematics, and the philosophy of language, but his projects—and the very terminology he employed in pursuing those projects—are still current in contemporary philosophy. This is undoubtedly why it seems so reasonable to assume that we can read Frege' s writings as if he were one of us, speaking to our philosophical concerns in our language. In Joan Weiner's view, however, Frege's words can (...) be accurately interpreted only if we set that assumption aside. Weiner here offers a challenging new approach to the philosophy of this central figure in analytic philosophy. Weiner finds in Frege's corpus, from Begriffsschrift on, a unified project of remarkable ambition to which each of the writings in that corpus makes a distinct contribution—a project whose motivation she brings to life through a careful reading of his Foundations of Arithmetic. The Frege that Weiner brings into clear view is very different from the familiar figure. Far from having originated one of the standard positions on the nature of reference, Frege turns out not to have had positive doctrines on anything like what contemporary philosophers mean by "reference." Far from having served as a standard-bearer for those who take the realists' side of contemporary disputes with anti-realists, Frege turns out to have had no stake in either side of the controversy. Through Weiner's lens, Frege emerges as a thinker who has principled reasons for challenging the very assumptions and motivations that animate philosophers to dispute these doctrines. This lucidly written and accessible book will generate controversy among all readers with an interest in epistemology, philosophy of language, history of philosophy, and the philosophy of mathematics. (shrink)

Frege is celebrated as an arch-Platonist and arch-realist. He is renowned for claiming that truths of arithmetic are eternally true and independent of us, our judgments and our thoughts; that there is a third realm containing nonphysical objects that are not ideas. Until recently, there were few attempts to explicate these renowned claims, for most philosophers thought the clarity of Frege's prose rendered explication unnecessary. But the last ten years have seen the publication of several revisionist interpretations of Frege's writings (...) — interpretations on which these claims receive a very different reading. In Frege on Knowing the Third Realm, Tyler Burge attempts to undermine this trend. Burge argues that Frege is the very Platonist most have thought him — that revisionist interpretations of Frege's Platonism, mine among them, run afoul of the words on Frege's pages. This paper is a response to Burge's criticisms. I argue that my interpretation is more faithful than Burge's to Frege's texts. (shrink)

In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory T in monadic second-order logic, consisting of axiom V and 1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck and T (...) prove the existence of infinitely many non-logical objects (T deriving,moreover, the nonexistence of the value-range concept). Some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed. Finally, I explore the relation of T toCantor's theorem which is somewhat surprising. (shrink)

I am not a deflationist. I believe that truth and falsity are substantial. The truth of a proposition consists in its having a constructive proof, or truthmaker. The falsity of a proposition consists in its having a constructive disproof, or falsitymaker. Such proofs and disproofs will need to be given modulo acceptable premisses. The choice of these premisses will depend on the discourse in question.

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...) independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

Nine of the papers collected here derive directly from a conference organized by Schirn in Munich in 1991. Seven others, three of them reprinted, have been intelligently chosen to complement the original nine. The collection has no overarching theme, nor is it dominated by any particular approach to Frege’s thought. It is “a mixed selection”, and aims to reflect “the prevailing tendency in current Frege scholarship”. The influence of Dreben is less in evidence than one might expect, but otherwise the (...) collection meets that broad brief pretty well. Some of the collection’s best papers—for example, those by Burge, Boolos, Dummett, Heck, and Terence Parsons—are essential reading. Inevitably, not all the others have that standing. But, excepting only an over-long essay by the editor, each deserves its place. (shrink)

This book is available either individually, or as part of the specially-priced Arguments of the Philosphers Collection.

In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege’s attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume’s Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck’s arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege's attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik Frege hardly could have construed Hume's Principle as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck's arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

We have seen that despite Feferman's results Gödel's second theorem vitiates the use of Hilbert-type epistemological programs and consistency proofs as a response to mathematical skepticism. Thus consistency proofs fail to have the philosophical significance often attributed to them.This does not mean that consistency proofs are of no interest to philosophers. We know that a ‘non-pathological’ consistency proof for a system S will use methods which are not available in S. When S is as strong a system as we are (...) willing to entertain seriously then a consistency proof for it will yield no epistemological gain. But in other cases philosophers might argue that the proof uses methods which are merely different rather than stronger than those available in the system in question. This claim has been made, for example, in the case of the constructive consistency proofs for elementary number theory. Similar philosophical investigations can be made on relative consistency proofs, since these differ from each other in the principles they employ. For example, most relative consistency proofs can be carried out within elementary number theory, but without using the theory of real numbers, no one has been able to prove the consistency of Quine's ML relative to that of his NF.What about the consistency of all mathematics or of some strong system for set theory? How do we answer the skeptic? Since here a convincing proof is not possible, we have established that the skeptic demands too much. We cannot be certain that our axioms are free from contradiction and must treat them as hypotheses which may be abandoned or modified in the face of further mathematical experience. This attitude is taken by many foundational workers who also go on to voice opinions about thelikelihood that various systems are consistent. Since these opinions are variously supported by appeals to the clarity of the mathematical concept formalized, the existence or non-existence of ‘weird’ models for the system and actual empirical experience with the system, this is surely a fruitful area for philosophical research. (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)

Parsons has given a (nonconstructive) proof that the first-order fragment of the system of Frege's Grundgesetze is consistent. Here a constructive proof of the same result is presented.

As is well-known, the formal system in which Frege works in his Grundgesetze der Arithmetik is formally inconsistent, Russell's Paradox being derivable in it.This system is, except for minor differences, full second-order logic, augmented by a single non-logical axiom, Frege's Axiom V. It has been known for some time now that the first-order fragment of the theory is consistent. The present paper establishes that both the simple and the ramified predicative second-order fragments are consistent, and that Robinson arithmetic, Q, is (...) relatively interpretable in the simple predicative fragment. The philosophical significance of the result is discussed. (shrink)

ZusammenfassungP. Bernays hat darauf hingewiesen, dass man, um die Widerspruchs freiheit der klassischen Zahlentheorie zu beweisen, den Hilbertschen flniter Standpunkt dadurch erweitern muss, dass man neben den auf Symbole sich beziehenden kombinatorischen Begriffen gewisse abstrakte Begriffe zulässt, Die abstrakten Begriffe, die bisher für diesen Zweck verwendet wurden, sinc die der konstruktiven Ordinalzahltheorie und die der intuitionistischer. Logik. Es wird gezeigt, dass man statt deesen den Begriff einer berechenbaren Funktion endlichen einfachen Typs über den natürlichen Zahler benutzen kann, wobei keine anderen (...) Konstruktionsverfahren für solche Funktionen nötig sind, als einfache Rekursion nach einer Zahlvariablen und Einsetzung von Funktionen ineinander .P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary stand‐point by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other. (shrink)

Michael Dummett; Discussions: Chairman's Address: Basic Law V*, Proceedings of the Aristotelian Society, Volume 94, Issue 1, 1 June 1994, Pages 243–252, https:/.