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)

In his paper `Finitism', W.W.~Tait maintained 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 argued that all finitist reasoning is essentially primitive recursive. In our 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 those $\Pi^0_1$-sentences that can be proved in. (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.

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)

Erratum to: Axiomathes DOI 10.1007/s10516-013-9222-7In the online publication, page 13, line 27, after the sentence “Hence, neo-logicism is doomed to failure.”, the following two sentences were missing:This argument was developed by Robert Trueman in a draft of his paper ‘Sham Names andion’. A revised version of this paper is forthcoming in Philosophia Mathematica under the tile ‘A Dilemma for Neo-Fregeanism’.

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 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)

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)

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)

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)

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.

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)

Having outlined a theory of bullshit and falsehood, Harry G. Frankfurt turns to what lies beyond them: the truth, a concept not as obvious as some might expect. Our culture's devotion to bullshit may seem much stronger than our apparently halfhearted attachment to truth. Some people won't even acknowledge "true" and "false" as meaningful categories, and even those who claim to love truth cause the rest of us to wonder whether they, too, aren't simply full of it. Practically speaking, (...) many of us deploy the truth only when absolutely necessary, often finding alternatives to be more saleable, and yet somehow civilization seems to be muddling along. But where are we headed? Is our fast and easy way with the facts actually crippling us? Or is it "all good"? Really, what's the use of truth, anyway? With the same leavening wit and commonsense wisdom that animates his pathbreaking work On Bullshit, Frankfurt encourages us to take another look at the truth: there may be something there that is perhaps too plain to notice but for which we have a mostly unacknowledged yet deep-seated passion. His book will have sentient beings across America asking, "The truth—why didn't I think of that?". (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)

Harry G. Frankfurt begins his inquiry by asking, “What is it about human beings that makes it possible for us to take ourselves seriously?” Based on The Tanner Lectures in Moral Philosophy, Taking Ourselves Seriously and Getting It Right delves into this provocative and original question. The author maintains that taking ourselves seriously presupposes an inward-directed, reflexive oversight that enables us to focus our attention directly upon ourselves, and “[it] means that we are not prepared to accept ourselves just (...) as we come. We want our thoughts, our feelings, our choices, and our behavior to make sense. We are not satisfied to think that our ideas are formed haphazardly, or that our actions are driven by transient and opaque impulses or by mindless decisions. We need to direct ourselves—or at any rate to believe that we are directing ourselves—in thoughtful conformity to stable and appropriate norms. We want to get things right.” The essays delineate two features that have a critical role to play in this: our rationality, and our ability to love. Frankfurt incisively explores the roles of reason and of love in our active lives, and considers the relation between these two motivating forces of our actions. The argument is that the authority of practical reason is less fundamental than the authority of love. Love, as the author defines it, is a volitional matter, that is, it consists in what we are actually committed to caring about. Frankfurt adds that “The object of love can be almost anything—a life, a quality of experience, a person, a group, a moral ideal, a nonmoral ideal, a tradition, whatever.” However, these objects and ideals are difficult to comprehend and often in conflict with each other. Moral principles play an important supporting role in this process as they help us develop and elucidate a vision that inspires our love. The first section of the book consists of the two lectures, which are entitled “Taking Ourselves Seriously” and “Getting It Right.” The second section consists of comments in response by Christine M. Korsgaard, Michael E. Bratman, and Meir Dan-Cohen. The book includes a preface by Debra Satz. (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)

From the author of the #1 New York Times bestseller On Bullshit, the case for worrying less about the rich and more about the poor Economic inequality is one of the most divisive issues of our time. Yet few would argue that inequality is a greater evil than poverty. The poor suffer because they don't have enough, not because others have more, and some have far too much. So why do many people appear to be more distressed by the rich (...) than by the poor? In this provocative book, the #1 New York Times bestselling author of On Bullshit presents a compelling and unsettling response to those who believe that the goal of social justice should be economic equality or less inequality. Harry Frankfurt, one of the most influential moral philosophers in the world, argues that we are morally obligated to eliminate poverty—not achieve equality or reduce inequality. Our focus should be on making sure everyone has a sufficient amount to live a decent life. To focus instead on inequality is distracting and alienating. At the same time, Frankfurt argues that the conjunction of vast wealth and poverty is offensive. If we dedicate ourselves to making sure everyone has enough, we may reduce inequality as a side effect. But it’s essential to see that the ultimate goal of justice is to end poverty, not inequality. A serious challenge to cherished beliefs on both the political left and right, On Inequality promises to have a profound impact on one of the great debates of our time. (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)

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)