Frege: Philosophy of Mathematics, Misc

Frege believes (1) that his definition of number is (partly) arbitrary; (2) that it "makes" numbers of certain extensions; (3) that without such a definition we cannot even think or understand arithmetical propositions. this position is part of a view according to which mathematics in general involves the free construction of objects, their properties, and the very contents of mathematical propositions. frege tries to avoid excess subjectivism by the kantian device of treating alternative systems of arithmetic (e.g.) as different appearances (...) of a single realm of unstructured truths. (shrink)

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition (...) is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. (shrink)

This paper has three goals: (i) to show that the foundational program begun in the Begriffsschroft, and carried forward in the Grundlagen, represented Frege's attempt to establish the autonomy of arithmetic from geometry and kinematics; the cogency and coherence of 'intuitive' reasoning were not in question. (ii) To place Frege's logicism in the context of the nineteenth century tradition in mathematical analysis, and, in particular, to show how the modern concept of a function made it possible for Frege to pursue (...) the goal of autonomy within the framework of the system of second-order logic of the Begriffsschrift. (iii) To address certain criticisms of Frege by Parsons and Boolos, and thereby to clarify what was and was not achieved by the development, in Part III of the Begriffsschrift, of a fragment of the theory of relations. (shrink)

Antony Flew's ?A Strong Programme for the Sociology of Belief (Inquiry 25 {1982], 365?78) critically assesses the strong programme in the sociology of knowledge defended in David Bloor's Knowledge and Social Imagery. I argue that Flew's rejection of the epistemological relativism evident in Bloor's work begs the question against the relativist and ignores Bloor's focus on the social relativity of mathematical knowledge. Bloor attempts to establish such relativity via a sociological analysis of Frege's theory of number. But this analysis only (...) succeeds if the rejection of an explanatory theory entails that there are reasonable grounds for the rejection of the set of propositions which that theory was intended to explain. I argue against such an entailment, and thus against Bloor's attempt to relativize mathematical knowledge. (shrink)

Frege's analysis of Zahlangaben is expounded and evaluated.

Rather than reading Plato's philosophy of arithmetic ‘charitably’, it is better to try to explain its failure to generate any fruitful ideas. Prominent in the explanation is Plato's focus on predicates assigning cardinalities and on ‘groups’ falling under them. This focus left Plato unable to envisage the possibility, emerging in Dedekind and Frege but which arithmetic in Plato's time would not easily have suggested, of regarding numbers as objects essentially ranged in the structure of a progression.

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

In this review I briefly explain the most important points of each chapter of Dummett's book, and critically discuss some of them. Special attention is given to the criticisms of Crispin Wright's interpretation of Frege's Platonism, and also to Dummett's interpretation of the role(s) of the context principle in Frege's thought.

Frege’s argument against the ancient Greek conception of numbers as 'multitudes of units’ has been hailed as one of the most successful in his "Grundlagen". The aim of this paper is to show that despite Frege’s best efforts, the Euclidean conception remains a viable alternative to the Fregean conception of numbers by arguing that neither a dilemma argument Frege brings against the Euclidean conception nor a possible argument against it based on the truth of what is known as "Hume’s Principle" (...) succeeds. (shrink)

In his treatise Die Grundlagen der Arithmetik, Gottlob Frege tries to find a definition of number. First he rejects the idea that number could be a property of external objects. Then he comes with a suggestion that a numerical statement expresses a property of a concept, namely it indicates how many objects fall under the concept. Subsequently Frege rejects, or at least essentially modifies, also this definition, because in his view that a number cannot be a property – it should (...) be an object. In this article, I try to show that Frege’s first definition of number seems to be, despite his own opinion, much more promising than he supposed. I also argue that Frege’s argumentation against the empirical character of number is by no means convincing. (shrink)

The common thread running through the logicism of Frege, Dedekind, and Russell is their opposition to the Kantian thesis that our knowledge of arithmetic rests on spatio-temporal intuition. Our critical exposition of the view proceeds by tracing its answers to three fundamental questions: What is the basis for our knowledge of the infinity of the numbers? How is arithmetic applicable to reality? Why is reasoning by induction justified?

In this article, I compare Gottlob Frege's and Bernard Bolzano's rationalist conceptions of arithmetic. Each philosopher worked out a complicated system of propositions, all of which were set forth as true. The axioms, or basic truths, make up the foundations of the subject of arithmetic. Each member of the system which is not an axiom is related (objectively) to the axioms at the base. Even though this relation to the base may not yet be scientifically proven, the propositions of the (...) system include all of the truths of the science of arithmetic. I conclude the article by analyzing the respective views of Frege and Bolzano in the light of Gödel's first incompleteness theorem. (shrink)

The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides on Basic Laws of (...) Arithmetic (1893/1903) and "Thoughts" (1918). My goal is to clarify in which sense the Frege of Foundations and Basic Laws is a platonist concerning the natural numbers.1.. (shrink)

This paper defends the view that Frege’s reduction of arithmetic to logic would, if successful, have shown that arithmetical knowledge is analytic in essentially Kant’s sense. It is argued, as against Paul Benacerraf, that Frege’s apparent acceptance of multiple reductions is compatible with this epistemological thesis. The importance of this defense is that (a) it clarifies the role of proof, definition, and analysis in Frege’s logicist works; and (b) it demonstrates that the Fregean style of reduction is a valuable tool (...) for those who would investigate the nature of arithmetical knowledge. (shrink)

The essential aim of Professor P. Clark’s paper is to show that good reasons exist for opening the way for a revival of Frege’s Programme. Frege’s Programme tend to show that numbers are given to us by explaining the meanings of identity statements in which numbers words occur.

This essay discusses the unity of mathematics by comparing the philosophies of Mill and Frege. While Mill is remembered as a progressive social thinker, his contributions to the development of logic are less widely heralded. In contrast, Frege made important and lasting contributions to the development of logic while his social thought, what little is known of it, was very conservative. Two theses are presented in the paper. The first is that in order to pursue Mill’s progressive sociopolitical project, one (...) must embrace Frege’s distinction between logic and psychology. The second thesis is that in order to pursue Frege’s project of accounting for the unity of mathematics, we must understand mathematics as a human activity and consider the role that history and psychology play in the growth of mathematics. (shrink)

Cette these divisee en trois parties, comporte deux caracteristiques essentielles: ; Elle comporte une teneure mathematique: des progres accomplis entre 1870 et 1914 par les mathematiciens au sein des savoirs que sont l'analyse infinitesimale, l'algebre, la geometrie etc. ont conferre une place sui generis a l'algebre. Sa redefinition va integrer des objets quelconques et non plus seulement les simples techniques du calcul litteral. Aussi certains mathematiciens observerent un desinteret pour ces calculs au profit d'une reflexion theorique autour du mot d'ordre (...) de Dirichlet de "substituer les idees au calcul" . L'enjeu consistait a donner un fondement sur au concept de nombre. ; Historiquement, c'est entre 1891 et 1906 que Frege et Husserl ont entretenu un echange epistolaire sur la question. Pour Husserl, eleve de Brentano, ce fondement est a trouver dans le sujet connaissant, pole constitutif de tous les ordres du savoir. Sa Philosophie de l'Arithmetique en donne les contours. Critique et oppose a cette base psychologiste, Frege va presenter son approche qui privilegie la logique. Dans les Fondements de l'arithmetique , Frege estime que l'arithmetique entiere peut etre reduite aux lois de logique generale. Cette reduction de l'arithmetique a la logique donna naissance a la tradition logiciste qui fut mortellement ebranlee par le paradoxe de Russell et inaugura la crise des fondements de type ensembliste. ;Aux alentours de 1894, Husserl abandonna le psychologisme. Faut-il y voir l'influence de Frege? Cette question est debatue dans la 2e partie de la these par Mohanty et Dummett. Ils ne s'entendent ni sur la date de l'abandon par Husserl du psychologisme ni sur l'influence que Frege aurait pu exercer sur lui. ;La troisieme partie est un recours a Wittgenstein. Elle porte sur ce qu'on peut percevoir comme esquisse d'une solution qui passe par le renoncement de la philosophie a fonder les objets mathematiques qui ne sont que des conventions, brisant ainsi 2000 ans de domination platoniste. D'aucuns taxent la position de Wittgenstein de strict finitisme. Si cela s'averait juste, n'y-a-t-il pas la un lien possible entre Wittgenstein et le constructivisme de Kronecker? (shrink)

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so that (...) nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

Discusses Frege's formal definitions and characterizations of infinite and finite sets. Speculates that Frege might have discovered the "oddity" in Dedekind's famous proof that all infinite sets are Dedekind infinite and, in doing so, stumbled across an axiom of countable choice.

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. (...) Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle. (shrink)

Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a (...) model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak version of uniformization (which does not involve knowledge of Jensen's fine structure theory). The axioms of the Grundgesetze are examples of abstraction principles, and the other primary aim of this paper is to articulate a sufficient condition for the consistency of abstraction principles with limited amounts of comprehension. As an application, we resolve an analogue of the joint consistency problem in the predicative setting. (shrink)

Frege proposed that sentences like ‘The number of planets is eight’ be analysed as identity statements in which the number words refer to numbers. Recently, Friederike Moltmann argued that, pace Frege, such sentences be analysed as so-called specificational sentences in which the number words have the same non-referring semantic function as the number word ‘eight’ in ‘There are eight planets’. The aim of this paper is two-fold. First, I argue that Moltmann fails to show that such sentences should be analysed (...) as specificational sentences. Second, I show that even if they are to be analysed in this way, Moltmann’s proposed specificational analysis is unsatisfactory. (shrink)

Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...) proves, without any use of arithmetical induction, that Frege's definition is extensionally correct and so answers the circularity objections. (shrink)

In the Grundlagen , Frege offers eight main arguments, together with a series of more minor supporting arguments, against Mill’s view that numbers are “properties of external things”. This paper reviews all eight of these arguments, arguing that none are conclusive.

In his "Grundgesetze", Frege hints that prior to his theory that cardinal numbers are objects he had an "almost completed" manuscript on cardinals. Taking this early theory to have been an account of cardinals as second-level functions, this paper works out the significance of the fact that Frege's cardinal numbers is a theory of concept-correlates. Frege held that, where n > 2, there is a one—one correlation between each n-level function and an n—1 level function, and a one—one correlation between (...) each first-level function and an object. Applied to cardinals, the correlation offers new answers to some perplexing features of Frege's philosophy. It is shown that within Frege's concept-script, a generalized form of Hume's Principle is equivalent to Russell's Principle ofion — a principle Russell employed to demonstrate the inadequacy of definition by abstraction. Accordingly, Frege's rejection of definition of cardinal number by Hume's Principle parallels Russell's objection to definition by abstraction. Frege's correlation thesis reveals that he has a way of meeting the structuralist challenge that it is arithmetic, and not a privileged progression of objects, that matters to the finite cardinals. (shrink)