The book begins with an overview that introduces the Theorem and the issues surrounding it, and explores how the essays that follow contribute to our understanding of those issues.

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)

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege's definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

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)

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.

Summary. This article develops two principal points. First, the so-called rivals of logical foundations, associated with Zermelo, Hilbert, and Brouwer, are here regarded as variants; specifically: to simplify, refine, resp. extend Frege’s scheme. Each of the variations is seen as a special case of a familiar strategy in the pursuit of knowledge. In particular, the extension provided by Brouwer’s intuitionistic logic concerns the class of propositions considered: about incompletely defined objects such as choice sequences. In contrast, Frege or, (...) for that matter, Aristotle thought that only propositions about precisely defined terms lent themselves to anything like logical theory.. This view of intuitionistic logic is fairly consistent with the mature Brouwer’s own research practice, but not at all with his rhetoric. Secondly, the article explains the silent majority’s scepticism about all variants of logical foundations, and, in note 7, sketches a genuine alternative which is almost explicit in the manifesto [#3] of Bourbaki with the aim of exploring the architecture of mathematics and our intuitive resonances to it. The alternative sees the weakness of logical foundations not in any logical defects, such as lack of precision, but in its sterility: stress on the logical elements of mathematics and mathematical reasoning draws attention away from discoveries of more significant features; more significant even for validity and reliability, though these matters are less prominent here than in logical foundations. The weakness in question is relevant beyond the present topic of foundations because this kind of weakness affects the whole analytic tradition in philosophy. Finally, for reasons explained there, an Appendix goes into some relations between the present paper and [17]. (shrink)

A brief, non-technical introduction to technical and philosophical aspects of Frege's philosophy of arithmetic. The exposition focuses on Frege's Theorem, which states that the axioms of arithmetic are provable, in second-order logic, from a single non-logical axiom, "Hume's Principle", which itself is: The number of Fs is the same as the number of Gs if, and only if, the Fs and Gs are in one-one correspondence.

Gottlob Frege's Grundgesetze der Arithmetik, or Basic Laws of Arithmetic, was intended to be his magnum opus, the book in which he would finally establish his logicist philosophy of arithmetic. But because of the disaster of Russell's Paradox, which undermined Frege's proofs, the more mathematical parts of the book have rarely been read. Richard G.

In recent work on Frege, one of the most salient issues has been whether he was prepared to make serious use of semantical notions such as reference and truth. I argue here Frege did make very serious use of semantical concepts. I argue, first, that Frege had reason to be interested in the question how the axioms and rules of his formal theory might be justified and, second, that he explicitly commits himself to offering a justification that (...) appeals to the notion of reference. I then discuss the justifications Frege offered, focusing on his discussion of inferences involving free variables, in section 17 of Grundgesetze, and his argument, in sections 29-32, that every well-formed expression of his formal language has a unique reference. (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...

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

The paper argues, as against Thau and Caplan, that the traditional interpretation that Frege abandoned his earlier views about identity and identity--statements is correct.

I examine Frege’s explanation of how Hilbert ought to have presented his proofs of the independence of the axioms of geometry: in terms of mappings between (what we would call) fully interpreted statements. This helps make sense of Frege’s objections to the notion of different interpretations, which many have found puzzling. (The paper is the text of a talk presented in October 1994.).

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that (...) Frege was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

Le problème de Frege-Geach est un problème qui se pose pour les théories selon lesquelles les jugements normatifs n’ont pas de contenu cognitif, mais expriment plutôt des états mentaux non cognitifs. Dans cet article, je présente le problème de Frege-Geach ; j’examine certaines stratégies existantes pour l’aborder dans sa forme traditionnelle ; et je me demande enfin si un problème de Frege-Geach se pose pour les raisons, et si l’usage des raisons peut mener à une solution. J’esquisse (...) une réponse positive à cette question. (shrink)

Mathematically, Truths have been said to be analytic. Leibniz tried to prove this in a way criticized by frege. The author states: "it is the purpose of this note to exhibit the full force of frege's criticism." frege also attempted to prove the same thing, But concludes the author, In his attempt, Has not "found universal acceptance among mathematical logicians." he finds the value in frege's analysis to be the fact of his attempt at proof and (...) the need for it. (staff). (shrink)

La paradoja del mentiroso lleva siglos atormentando la mente de filósofos y lógicos. El presente texto intenta una solución siguiendo la siguiente tesis: la paradoja no puede ser resuelta por la lógica dado que lo implicado en ella trasciende las fronteras de esa disciplina. Lo que trasciende estas fronteras (establecidas en gran medida por Frege, en tanto padre de la lógica moderna) es el propio ser humano como ser capaz de articular sentido. De ahí la necesidad de explorar cómo (...) la fenomenología aborda al ser humano para reconcebir también al lenguaje y al tiempo. Este giro será crucial para entender por qué el problema de dicha paradoja es la autorreferencialidad, lo que también conlleva a reinterpretar qué se entiende por niveles de lenguaje (sea en la forma de Tarski o de Kripke). (shrink)

In the most recent edition of Language, Truth and Logic , Professor A. J. Ayer still maintains that pure mathematics is analytic, being in fact merely a vast system of tautology. He is much more confident about this than are most contemporary professional mathematicians who have investigated the foundations of their subject. Following the breakdown of the efforts both of Frege and of Russell and Whitehead to derive pure mathematics from logic, i.e. to prove that the denial of any (...) one proposition of mathematics would necessarily be self-contradictory, Hilbert attempted to prove the more modest thesis that pure mathematics is consistent, i.e. that no two propositions of mathematics can contradict each other; but in 1931 Gödel discovered that even this thesis was undecidable according to the “rules of the game.” As Weyl has recently lamented, “From this history one thing should be clear: we are less certain than ever about the ultimate foundations of mathematics.” The sense in which Ayer uses the terms analytic and tautology implies also that in his view the activities of pure mathematicians lead to nothing new. It is true that he remarks that “there is a sense in which analytic propositions do give us new knowledge. They call attention to linguistic usages of which we might otherwise not be conscious, and they reveal unsuspected implications in our assertions and beliefs.” But, he continues, “we can also see that there is a sense in which they may be said to add nothing to our knowledge. For they may be said to tell us what we know already. ” This denial of novelty in mathematics is as typical of contemporary positivism as the prophecy of Comte that the composition of the stars would never be revealed to us and the objections of Mach to the atomic hypothesis were characteristic of nineteenth century positivism. Indeed, one wonders why the term positivism should have been appropriated by successive philosophers whose common outlook could be so much more fittingly described as negativism. (shrink)

Aldo Antonelli offers a novel view on abstraction principles in order to solve a traditional tension between different requirements: that the claims of science be taken at face value, even when involving putative reference to mathematical entities; and that referents of mathematical terms are identified and their possible relations to other objects specified. In his view, abstraction principles provide representatives for equivalence classes of second-order entities that are available provided the first- and second-order domains are in the equilibrium dictated by (...) the abstraction principles, and whose choice is otherwise unconstrained.entities are the referents of abstraction terms: such referents are to an extent indeterminate, but we can still quantify over them, predicate identity or non-identity, etc. Our knowledge of them is limited, but still substantial: we know whatever has to be true no matter how the representatives are chosen, i.e., what is true in all models of the corresponding abstraction principles. This view is backed up by an “austere” conception of universals, according to which these are first-order objects, i.e., ways of collecting first-order objects. Antonelli thus claims that second-order logic does not import any novel ontological commitment beyond ontology of first-order, naturalistically acceptable, objects. Moreover, in the case of arithmetic, even if Hume’s Principle is construed as described above, Frege’s Theorem goes through unaffected, since there is nothing in its proof that depends on an account of the “true nature” of numbers. A viable construal of logicism is then given by the combination of semantic nominalism and a naturalistic conception of abstraction. [Editors note]. (shrink)

In designing stratified models of human language, understanding notions of logical consequence and validity of inference require separating the aspects of meaning that vary between models from logical constants. Modelling meaning requires choices regarding the primitives, where the Fregean program is still offering us the fundamental insights on the role of truth, judgement, and grasping or sharing of thoughts.

Church has recently proposed a solution of the paradox of analysis as propounded by Langford in which Church makes use of Frege's distinction between the sense (Sinn) of a name and its denotation (Bedeutung). The main purpose of the present note. is to show that a, version of the paradox may be presented which is not directly solved by Church in his review but which, in turn, may be solved by using; another distinction of Frege-that between the ordinary (...) (gewihnlich) and the oblique (ungerade) use of a name. The second part of the note will be concerned with the relative merits of this second distinction of Frege and Quine's related distinction between purely designative occurrences of a name and occurrences which are not purely designative; they will be compared merely with respect to their effectiveness in removing or solving the paradox of analysis. (shrink)

In this paper, I mainly pursue the following two goals: on the one hand, I want to show how a central Fregean insight is tried to be captured within a two-dimensional strategy. On the other hand, I want to show that, in the light of Saul Kripke’s arguments against descriptivism, this strategy is faced with a fundamental problem. I proceed in four steps: in a first step, I bring together the passages that contain a central Fregean insight as a source (...) of inspiration for a two-dimensional reconstruction and that suggest a descriptivist reading of Frege’s view. In a second step, I shortly present Kripke’s threefold argumentation against two versions of descriptivism, which is also the basis for Kripke’s view that proper names are rigid designators. In a third step, I explain the basic idea of a Frege-inspired two-dimensional strategy which is used to reconcile theories of definite descriptions with theories of rigid designation and which is sometimes characterized as ‘neo-descriptivism’. Since the two-dimensional strategy is dependent on rigidified definite descriptions, I argue, in a fourth step, that, in the light of Kripke’s epistemological and semantic arguments, the two-dimensional strategy is problematic and untenable – though it is, nevertheless, motivated by Kripke’s modal argument. (shrink)