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)

En este articulo quiero discutir algunos temas centrales deI tratamiento fregeano de los contextos no extensionales. Limitaré mi discusión al análisis de oraciones de creencia y de la oratio obliqua. En la primera parte, voy a describir dos tipos de teoría dentro deI marco de la semántica de Frege. En particular, compararé y evaluaré los análisis de oraciones no extensionales de primer y segundo nivel que se pueden llevar a cabo en las teorías de ambos tipos. En la segunda parte, (...) examinaré en que medida se puede establecer una jerarquía infinita de sentidos indirectos. En la tercera parte, voy a examinar el principio fregeano de sustitutividad de expresiones coreferenciales salva veritate. Además, haré algunas observaciones críticas sobre artículos de Tyler Burge y Jaakko Hintikka relacionados con esta temática.In this article, I discuss some important aspects of Frege’s treatment of nonextensional contexts. I focus on the analysis of belief sentences and of oratio obliqua. In the first part, I deseribe two types of theory within Fregean semantics and assess the analyses of non-extensional sentences of first and of second level whieh can be carried out in the theories of both types. In the seeond part, I examine to what extent one can establish an infinite hierarehy of indirect senses. In the third and final part, I examine Frege’s principle of substitutivity salva veritate of coreferential terms. I conelude with critieal observationson articles of Tyler Burge and Jaakko Hintikka which deal with this principle. (shrink)

This essay challenges the widely accepted principle that a person is morally responsible for what he has done only if he could have done otherwise. The author considers situations in which there are sufficient conditions for a certain choice or action to be performed by someone, So that it is impossible for the person to choose or to do otherwise, But in which these conditions do not in any way bring it about that the person chooses or acts as he (...) does. In such situations the person may well be morally responsible for what he chooses or does despite his inability to choose or to do otherwise. Finally the author considers certain suggestions for revising the principle he rejects or for replacing it with a principle of an altogether different kind. (shrink)

It is my view that one essential difference between persons and other creatures is to be found in the structure of a person's will. Besides wanting and choosing and being moved to do this or that, men may also want to have certain desires and motives. They are capable of wanting to be different, in their preferences and purposes, from what they are. Many animals appear to have the capacity for what I shall call "first-order desires" or "desires of the (...) first order," which are simply desires to do or not to do one thing or another. No animal other than man, however, appears to have the capacity for reflective self-evaluation that is manifested in the formation of second-order desires. (shrink)

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.

In this essay, I critically discuss Dale Jacquette's new English translation of Frege's work Die Grundlagen der Arithmetik as well as his Introduction and Critical Commentary (Frege, G. 2007. The Foundations of Arithmetic. A Logical-Mathematical Investigation into the Concept of Number . Translated with an Introduction and Critical Commentary by Dale Jacquette. New York: Longman. xxxii + 112 pp.). I begin with a short assessment of Frege's book. In sections 2 and 3, I examine several claims that Jacquette makes in (...) his Introduction and Critical Commentary and put matters in the right perspective. In sections 4-7, I analyse errors and shortcomings in Jacquette's (and Austin's) translation(s) and show how they can be avoided. In this context, I consider several issues of interest for Frege's logic and philosophy of arithmetic. I conclude with general remarks. (shrink)

The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come. Contributors Paul Benacerraf, George Boolos, John P. Burgess, Charles S. Chihara, Michael Detlefsen, Michael Dummett, Hartry Field, Kit Fine, (...) Bob Hale, Richard G. Heck, Jnr., Geoffrey Hellman, Penelope Maddy, Karl-Georg Niebergall, Charles D. Parsons, Michael D. Resnik, Matthias Schirn, Stewart Shapiro, Peter Simons, W.W. Tait, Crispin Wright. (shrink)

In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that that what Frege calls faculty of intuition in his dissertation is probably meant to refer to a capacity of visualizing geometrical configurations structurally in a way which is essentially the same for most Western educated human beings. I further suggest that according to his Habilitationsschrift it is through spatial intuition that we (...) come to know the axioms of Euclidean geometry. In Sect. 3, I argue that Frege seems right in claiming in The Foundations, §14 that the synthetic nature of the Euclidean axioms follows from the fact that they are independent of one another and of the primitive laws of logic. If the former were dependent on the latter, they would be analytic in Frege’s sense of analyticity. But then they would not be independent of one another and due to their mutual provability would lose their status as axioms of Euclidean geometry, since according to Frege an axiom of a theory T is per definitionen unprovable in T. I further argue that only by invoking pure spatial intuition can Frege “explain” the epistemological status of the axioms of Euclidean geometry completely: their synthetic a priori nature. Finally, I argue that his view about independence in The Foundations, §14 seems to clash with his conception of independence in his mature period. In Sect. 4, I scrutinize Frege’s somewhat vague, but unduly neglected remarks in The Foundations, §26 on space, spatial intuition and the axioms of Euclidean geometry. I argue that for the sake of coherence Frege should have said unambiguously that space is objective, that it is independedent not only of our spatial intuition, but of our mental life altogether including our judgements about space, instead of encouraging the possible conjecture that in his view it contains an objective and a subjective component. I further argue that for Frege the objectivity of both space and the axioms of Euclidean geometry manifests itself in our universal and compulsory acknowledgement of the Euclidean axioms as true. I conclude Sect. 4 by arguing that there is a conflict between the subjectivity of our spatial intuitions as stressed in The Foundations, §26 and Frege’s thesis in his dissertation that the axioms of Euclidean geometry derive their validity from the nature of our faculty of intuition. To resolve this conflict, I propose that in the light of his avowed realism in The Foundations Frege could have replaced his early thesis by saying that although we come to know the axioms of Euclidean geometry through spatial intuition and are justified in acknowledging them as true on the basis of geometrical intuition, their truth is independent not only of the nature of our faculty of intuition and singular acts of intuition, but of our mental processes and activities in general, including the inner mental act of judging. In Sect. 5, I argue that Frege most likely did not adopt Kant’s method of acquiring geometrical knowledge via the ostensive construction of concepts in spatial intuition. In contrast to Kant, Frege holds that the axioms of three-dimensional Euclidean geometry express state of affairs about space obtaining independently of our spatial intuition. In Sect. 6, I conclude with a summarized assessment of Frege’s philosophy of geometry. (shrink)

In this article, I try to shed new light on Frege’s envisaged definitional introduction of real and complex numbers in _Die Grundlagen der Arithmetik_ (1884) and the status of cross-sortal identity claims with side glances at _Grundgesetze der Arithmetik_ (vol. I 1893, vol. II 1903). As far as I can see, this topic has not yet been discussed in the context of _Grundlagen_. I show why Frege’s strategy in the case of the projected definitions of real and complex numbers in (...) _Grundlagen_ is modelled on his definitional introduction of cardinal numbers in two steps, tentatively via a contextual definition and finally and definitively via an explicit definition. I argue that the strategy leaves a few important questions open, in particular one relating to the status of the envisioned abstraction principles for the real and complex numbers and another concerning the proper handling of cross-sortal identity claims. (shrink)

This 1988 volume is a collection of thirteen seminal essays on ethics, free will, and the philosophy of mind. The essays deal with such central topics as freedom of the will, moral responsibility, the concept of a person, the structure of the will, the nature of action, the constitution of the self, and the theory of personal ideals. By focusing on the distinctive nature of human freedom, Professor Frankfurt is able to explore fundamental problems of what it is to (...) be a person and of what one should care about in life. (shrink)

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. (shrink)

Sections “Introduction: Hume’s Principle, Basic Law V and Cardinal Arithmetic” and “The Julius Caesar Problem in Grundlagen—A Brief Characterization” are peparatory. In Section “Analyticity”, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle, bearing in mind that with its analytic or non-analytic status the intended logical foundation of cardinal arithmetic stands or falls. Section “Thought Identity and Hume’s Principle” is concerned with the two criteria of thought identity that Frege states in 1906 and (...) their application to Hume’s Principle. In Section “The Nature ofion: A Critical Assessment of Grundlagen, §64”, I scrutinize Frege’s characterization of abstraction in Grundlagen, §64 and criticize in this context the currently widespread use of the terms “recarving” and “reconceptualization”. Section “Frege’s Proof of Hume’s Principle” is devoted to the formal details of Frege’s proof of Hume’s Principle. I begin by considering his proof sketch in Grundlagen and subsequently reconstruct in modern notation essential parts of the formal proof in Grundgesetze. In Section “Equinumerosity and Coextensiveness: Hume’s Principle and Basic Law V Again”, I discuss the criteria of identity embodied in Hume’s Principle and in Basic Law V, equinumerosity and coextensiveness. In Section “Julius Caesar and Cardinal Numbers—A Brief Comparison Between Grundlagen and Grundgesetze ”, I comment on the Julius Caesar problem arising from Hume’s Principle in Grundlagen and analyze the reasons for its absence in this form in Grundgesetze. I conclude with reflections on the introduction of the cardinals and the reals by abstraction in the context of Frege’s logicism. (shrink)

A clear, accessible exploration of how and why we love by prominent philosopher and bestselling author Harry Frankfurt In The Reasons of Love, leading moral philosopher and bestselling author Harry Frankfurt argues that the key to a fulfilled life is to pursue wholeheartedly what one cares about, that love is the most authoritative form of caring, and that the purest form of love is, in a complicated way, self-love. Through caring, we infuse the world with meaning. Caring provides (...) us with stable ambitions and concerns; it shapes the framework of aims and interests within which we lead our lives. Frankfurt goes on to explain that the most important form of caring is love, a nonvoluntary, disinterested concern for the flourishing of what is loved. And he contends that the purest form of love is self-love. This sounds perverse, but self-love—as distinct from self-indulgence—is at heart a disinterested concern for whatever it is that the person loves. The most elementary form of self-love is nothing more than the desire of a person to love. Insofar as this is true, self-love is simply a commitment to finding meaning in our lives. (shrink)

Name der Zeitschrift: Kant-Studien Jahrgang: 110 Heft: 1 Seiten: 74-125.

In this essay, I critically analyze Wittgenstein’s dispensation with “ = ” in a correct concept-script. I argue inter alia (a) that in the Tractatus the alleged pseudo-character of sentences containing “ = ” or = -sentences remains largely unexplained and propose how it could be explained; (b) that at least in some cases of replacing = -sentences with equivalent identity-sign free sentences the use of the notion of a translation seems inappropiate; (c) that in the Tractatus it remains unclear (...) how identity of the object as that which is expressed by identity of the sign should be understood specifically; (d) that there are = -sentences which have no obvious equivalent in Wittgenstein’s novel notation; (e) that Wittgenstein’s adherence to (non-relational) identity, although he dispenses with “ = ”, is probably motivated by his desire to ensure that the expressive power of an identity-sign free concept-script of first-order is on a par with standard first-order logic containing “ = ”. In the concluding section, I critically discuss some claims in Lampert and Säbel (The Review of Symbolic Logic, 14, 1–21, 2021) and defend Wehmeier’s account of pseudo-sentences in the Tractatus (2012) against the objections they raise. (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 (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)

Presents a theory of bullshit, how it differs from lying, how those who engage in it change the rules of conversation, and how indulgence in bullshit can alter a person's ability to tell the truth.

In this paper, we attempt to show that a weak version of Hilbert's metamathematics is compatible with Gödel's Incompleteness Theorems by employing only what are clearly natural prov‐ ability predicates. Defining first “T proves the consistency of a theory S indirectly in one step”, we subsequently prove “PA proves its own consistency indirectly in one step” and sketch the proof for “If S is a recursively enumerable extension of , S proves its own consistency indirectly in one step”. The formalizations (...) of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs. (shrink)

This beautifully written book by one of the world's leading moral philosophers argues that the key to a fulfilled life is to pursue wholeheartedly what one cares about, that love is the most authoritative form of caring, and that the purest form of love is, in a complicated way, self-love. Harry Frankfurt writes that it is through caring that we infuse the world with meaning. Caring provides us with stable ambitions and concerns; it shapes the framework of aims and (...) interests within which we lead our lives. The most basic and essential question for a person to raise about the conduct of his or her life is not what he or she should care about but what, in fact, he or she cannot help caring about. The most important form of caring, Frankfurt writes, is love, a nonvoluntary, disinterested concern for the flourishing of what is loved. Love is so important because meaningful practical reasoning must be grounded in ends that we do not seek only to attain other ends, and because it is in loving that we become bound to final ends desired for their own sakes. Frankfurt argues that the purest form of love is self-love. This sounds perverse, but self-love--as distinct from self-indulgence--is at heart a disinterested concern for whatever it is that the person loves. The most elementary form of self-love is nothing more than the desire of a person to love. Insofar as this is true, self-love is simply a commitment to finding meaning in our lives. (shrink)

In this article, I try to shed some new light onGrundgesetze§10, §29–§31 with special emphasis on Frege’s criteria and proof of referentiality and his treatment of the semantics of canonical value-range names. I begin by arguing against the claim, recently defended by several Frege scholars, that the first-order domain inGrundgesetzeis restricted to value-ranges, but conclude that there is an irresolvable tension in Frege’s view. The tension has a direct impact on the semantics of the concept-script, not least on the semantics (...) of value-range names. I further argue that despite first appearances truth-value names play a distinguished role as semantic “target names” for “test names” in the criteria of referentiality and do not figure themselves as “test names” regarding referentiality. Accordingly, I show in detail that Frege’s attempt to demonstrate that by virtue of his stipulations “regular” value-range names have indeed been endowed with a unique reference, can plausibly be regarded as a direct application of the context principle. In a subsequent section, I turn to some special issues involved in §10. §10 is closely intertwined with §31 and in my and Richard Heck’s view would have been better positioned between §30 and §31. In a first step, I discuss the piecemeal strategy which Frege applies when he attempts to bestow a unique reference on value-range names in §3, §10–§12. In a second step, I critically analyze his tentative, but predictably unsuccessful proposal to identify all objects whatsoever, including those already clad in the garb of value-ranges, with their unit classes. In conclusion, I present two arguments for my claim that Frege’s identification of the True and the False with their unit classes in §10 is illicit even if both the permutation argument and the identifiability thesis that he states in §10 are regarded as formally sound. The first argument is set out from the point of view of the syntax of his formal language. It suggests though that a reorganization of the exposition of the concept-script would have solved at least one of the problems to which the twin stipulations in §10 give rise. The second argument rests on semantic considerations. If it is sound, it may call into question, if not undermine the legitimacy of the twin stipulations. (shrink)

One of the most influential of contemporary philosophers, Harry Frankfurt has made major contributions to the philosophy of action, moral psychology, and the study of Descartes. This collection of essays complements an earlier collection published by Cambridge, The Importance of What We Care About. Some of the essays develop lines of thought found in the earlier volume. They deal in general with foundational metaphysical and epistemological issues concerning Descartes, moral philosophy, and philosophical anthropology. Some bear upon topics in political (...) philosophy and religion. (shrink)

One of the most salient features of our culture is that there is so much bullshit. Everyone knows this. Each of us contributes his share. But we tend to take the situation for granted. Most people are rather confident of their ability to recognize bullshit and to avoid being taken in by it. So the phenomenon has not aroused much deliberate concern. We have no clear understanding of what bullshit is, why there is so much of it, or what functions (...) it serves. And we lack a conscientiously developed appreciation of what it means to us. In other words, as Harry Frankfurt writes, "we have no theory." Frankfurt, one of the world's most influential moral philosophers, attempts to build such a theory here. With his characteristic combination of philosophical acuity, psychological insight, and wry humor, Frankfurt proceeds by exploring how bullshit and the related concept of humbug are distinct from lying. He argues that bullshitters misrepresent themselves to their audience not as liars do, that is, by deliberately making false claims about what is true. In fact, bullshit need not be untrue at all. Rather, bullshitters seek to convey a certain impression of themselves without being concerned about whether anything at all is true. They quietly change the rules governing their end of the conversation so that claims about truth and falsity are irrelevant. Frankfurt concludes that although bullshit can take many innocent forms, excessive indulgence in it can eventually undermine the practitioner's capacity to tell the truth in a way that lying does not. Liars at least acknowledge that it matters what is true. By virtue of this, Frankfurt writes, bullshit is a greater enemy of the truth than lies are. (shrink)

In this article, I first analyze and assess the epistemological and semantic status of canonical value-range equations in the formal language of Frege’s Grundgesetze der Arithmetik. I subsequently scrutinize the relation between (a) his informal, metalinguistic stipulation in Grundgesetze I, Section 3, and (b) its formal counterpart, which is Basic Law V. One point I argue for is that the stipulation in Section 3 was designed not only to fix the references of value-range names, but that it was probably also (...) intended to play a regulative role with an eye to Basic Law V. I further intend to shed new light on the status of Frege’s identification of the truth-values with their unit classes in Grundgesetze I, Section 10 and determine its place and importance in his overall logicist project. In a subsequent section, I first discuss the hypothetical extendability of the logical system of Grundgesetze. In what follows, I analyze the incompleteness of that system—were it not inconsistent and hence trivially complete—by focusing on the role of the definite description operator and the axiom governing it (= Basic Law VI) as well as on the role of the application operator. The latter is defined by means of the former. I further comment on the redundancy of an axiom that could be designed to supplement Basic Law VI by incorporating the second clause of Frege’s two-part elucidation of the description operator. I argue that although it seems likely that Basic Law VI could have been shown to be dispensable in Grundgesetze, Frege would probably have been reluctant to dispense with it, despite his commitment to axiomatic parsimony. In my view, the reason is that he considers it essential that every primitive function-name of his formal language be governed and grounded by a basic law that is tailored to the nature and the role of the primitive name occurring at a key point in the law. Toward the end of the article, I propose solutions to two problems with Frege’s use of “=” in the concept-script sentences that result from his definitions involving the replacement of the double stroke of definition with the judgment-stroke. I conclude the article with a summary of the main results that I have achieved. (shrink)

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. (shrink)

Although the notion of logical object plays a key role in Frege's foundational project, it has hardly been analyzed in depth so far. I argue that Marco Ruffino's attempt to fill this gap by establishing a close link between Frege's treatment of expressions of the form ‘the concept F’ and the privileged status Frege assigns to extensions of concepts as logical objects is bound to fail. I argue, in particular, that Frege's principal motive for introducing extensions into his logical theory (...) is not to be able to make in-direct statements about concepts, but rather to define all numbers as logical objects of a fundamental kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Contrary to what Ruffino claims, reducibility to extensions cannot be the ‘ultimate criterion’ for Frege of what is to be regarded as a logical object. (shrink)

In his paper "Finitism", W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by him, (...) is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA. /// En su articulo "Finitism", W.W. Tait sostiene que la dificultad principal para quien quiere comprender la concepción hilbertiana de la matemática finitista es ésta: especificar el sentido de la demostrabilidad de enunciados generales sobre los números naturales sin presuponer totalidades infinitas. Además, Tait argumenta que todo razonamiento finitista es esencialmente primitivo recursivo. En este artículo tratamos de mostrar que su tesis "Las funciones finitistas son precisamente las funciones primitivas recursivas" es discutible y que otra, también defendida por él, resulta insostenible. La segunda tesis es que los teoremas finitistas son precisamente las clausuras universales de las ecuaciones que pueden demostrarse en PRA. (shrink)