The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians (...) might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish. (shrink)

The paper comments on Dummett's Significance of Quine's Indeterminacy Thesis and discusses Quine's views on the translation of logical connectives. Some difficulties about the latter related to those raised by Morton (J. Phil. 70 (1973), 503–510) are considered. Quine seems here to be in a position considered by Dummett of not allowing a foreigner to be translated as conflicting with one's own firm theoretical commitment (in this case classical logic). But Dummett seems wrong in holding that entrenched theoretical statements (...) must be stimulus analytic. (shrink)

I comment on Feferman’s views on set theory, in particular criticizing a priori arguments claiming that the continuum hypothesis has no determinate truth value and commenting on his responses to my paper on his skepticism about set theory. I respond to criticisms of his of the structuralism that I have advocated and comment on his view of proof theory. On Koellner’s paper, I register little disagreement but note a difference of sympathy about views such as constructivism. On Tait’s paper, I (...) note that Kant gives more play to the notion of whole number than Tait seems to allow and that Kant’s conception of real numbers is unclear. Responding to Sieg’s paper, I note his emphasis on how much mathematics and its foundations changed from Kant’s time to that of Dedekind and Hilbert and mention my effort to find a limited role for an intuition distantly descended from Kant’s. (shrink)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the (...) concept of intuition and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

I consider different versions of a structuralist view of mathematical objects, according to which characteristic mathematical objects have no more of a 'nature' than is given by the basic relations of a structure in which they reside. My own version of such a view is non-eliminative in the sense that it does not lead to a programme for eliminating reference to mathematical objects. I reply to criticisms of non-eliminative structuralism recently advanced by Keränen and Hellman. In replying to the former, (...) I rely on a distinction between 'basic' and 'constructed' structures. A conclusion is that ideas from the metaphysical tradition can be misleading when applied to the objects of modern mathematics. (shrink)

The transcendental aesthetic -- Arithmetic and the categories -- Remarks on pure natural science -- Two studies in the reception of Kant's philosophy of arithmetic: postscript to part I -- Some remarks on Frege's conception of extension -- Postscript to essay 5 -- Frege's correspondence: postscript to essay 6 -- Brentano on judgment and truth -- Husserl and the linguistic turn.

This work is the long awaited sequel to the author’s classic Frege: Philosophy of Language. But it is not exactly what the author originally planned. He tells us that when he resumed work on the book in the summer of 1989, after a long interruption, he decided to start afresh. The resulting work followed a different plan from the original drafts. The reader does not know what was lost by their abandonment, but clearly much was gained: The present work may (...) be the most compactly organized of Dummett’s philosophical books, and its value as commentary on Frege is enhanced by its being organized as a study of Die Grundlagen der Arithmetik and selected parts of Grundgesetze. Moreover, since Dummett has been thinking and writing about Frege for forty years, we have a single, mature chronological stage of his thought. (shrink)

The paper undertakes to characterize Hao Wang's style, convictions, and method as a philosopher, centering on his most important philosophical work From Mathematics to Philosophy, 1974. The descriptive character of Wang's characteristic method is emphasized. Some specific achievements are discussed: his analyses of the concept of set, his discussion, in connection with setting forth Gödel's views, of minds and machines, and his concept of ‘analytic empiricism’ used to criticize Carnap and Quine. Wang's work as interpreter of Gödel's thought and the (...) importance of his writings as a source are also discussed. (shrink)

As the history of analytic philosophy is written, Gottlob Frege sits among the pantheon, one of the core creators of a novel way of philosophical thinking. It is a way of thinking that is notably infused with logical and semantic insights that are original to Frege. The source of these insights is well known. They arise in the context of logicism, Frege’s mathematical project that unfolded in a body of thought punctuated by three seminal works, Begriffsschrift of 1879, Die Grundlagen (...) der Arithmetik of 1884, and the two volumes of Grundgesetze der Arithmetik, published in 1893 and 1903. In these works, Frege sets out to show that arithmetic can be reduced to logic; more technically, that (a version of) the Peano-Dedekind axioms are derivable as theorems in a definitional extension of impredicative second-order logic. To achieve this goal, Frege introduced both a new logic and a new semantics; the conceptual apparatus that Frege brought to bear in these works, and in an accompanying series of classic essays, including most famously “On Sense and Reference,” has animated philosophical thought ever since. (shrink)

First published in 1990, this book consists of a detailed exposition of results of the theory of "interpretation" developed by G. Kreisel — the relative impenetrability of which gives the elucidation contained here great value for anyone seeking to understand his work. It contains more complex versions of the information obtained by Kreisel for number theory and clustering around the no-counter-example interpretation, for number-theorectic forumulae provide in ramified analysis. It also proves the omega-consistency of ramified analysis. The author also presents (...) proofs of Schütte’s cut-elimination theorems which are based on his consistency proofs and essentially contain them — these went further than any published work up to that point, helping to squeeze the maximum amount of information from these proofs. (shrink)

William Tait's standing in the philosophy of mathematics hardly needs to be argued for; for this reason the appearance of this collection is especially welcome. As noted in his Preface, the essays in this book ‘span the years 1981–2002’. The years given are evidently those of publication. One essay was not previously published in its present form, but it is a reworking of papers published during that period. The Introduction, one appendix, and some notes are new. Many of the essays (...) will be familiar to the readers of this journal; indeed two first appeared here.It should be no surprise to those who know Tait's work that this is a very rich collection, with contributions on a wide variety of issues, both systematic and historical. That poses a problem for a reviewer, because a serious discussion of even all the major issues would be beyond the scope of a review.Before the essays in this collection, Tait's publications were almost all in mathematical logic, especially proof theory. He was a leading actor in the work on the revised Hilbert program stimulated after the war by Georg Kreisel, Kurt Schütte, and Gaisi Takeuti. Tait thus has an experience in mathematical research that is rare among those whose careers have been institutionally in philosophy. In the 1970s Tait became disillusioned with the proof-theoretic program on which he had worked. However, he draws on that background in several of these essays, and he has continued to work on mathematical questions. In particular, the set-theoretic project reflected in essay 6 is as much mathematical as philosophical.It is not easy to characterize Tait's philosophical viewpoint briefly. Both the title of the book and some of its content mark him as a rationalist. His hero in the earlier history …. (shrink)

Kurt Gödel (1906-1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Gödel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Gödel's writings are among the topics covered. (...) Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible set"--Provided by publisher. (shrink)

Professor Flathman's main aim in this interesting paper is to set forth what we might call the “moderation thesis.” It holds that there may be occasions when the best thing to do, all things considered, is to violate a right – at least if the violation takes the form of what Flathman calls “civil encroachment” or “civil non-enforcement.” Moreover, it would be desirable, in a society whose practices include rights, for this belief to be generally accepted, so that those who (...) engaged with good reason in “civil encroachment” would receive the sympathetic support of the community. Let me begin by trying to explain the source of the problem to which this thesis is a response. As Flathman notes, rights have a discretionary character: if I have a right to do or to have something, it is up to me whether to do it or to have it. Having a right, I am entitled to make a choice. Since we cannot say, in advance, what choice I will make, we cannot say, in advance, what will be the state of the world after I make it. But since it seems plain that some states of the world are more desirable than others, it will always be possible that the consequences of my choice will be worse, on the whole, than the consequences of some other choice that I might have made but did not. To moderate rights is apparently to act in ways that redress or compensate for the undesirable results of their exercise. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.1 (2001) 59-79 [Access article in PDF] Money and Sovereignty in Early Modern France Jotham Parsons [The mint official] must above all seek integrity in the moneys, on which our features are imprinted and on which the general good depends. For what would be safe if our image were offended, and if that which a subject ought to venerate in his heart (...) were immediately violated by a sacrilegious hand? 1Early modern French commentators on money and monetary policy were fond of quoting this very late antique chancery formula; 2 it even appears in the epigraph to one unpublished commonplace book on the subject from the early seventeenth century. 3 Its popularity was no doubt due partly to the fact that classical literature contained few other theoretical statements on the practicalities of issuing money. 4 It also showed how deeply classical humanism had shaped economic thought, as it had done to all of French political culture. This was apparent as early as 1514 in Guillaume Budé's scholarly classic, De Asse et partibus ejus. 5 By 1600 these influences had ensconced themselves so firmly that economic writers found it nearly impossible to talk about money without a [End Page 59] full panoply of classical erudition. 6 In France the second half of the sixteenth century saw the rise of a specialized, expert, and increasingly systematic tradition of monetary thought. Despite its classical vocabulary, this economic literature was in many ways "modern" in its methods, ambitions, and subject. While developing an account which aspired to a scientific explanation of commercial society and of how government and money interacted, the French commentators came to a partial realization of how radically their own society differed from those which had gone before. In this respect the prevailing climate of thought about money influenced even a writer as non-technical as Michel de Montaigne.Traditionally, scholarly treatment of early modern French economic texts has focused on a relatively small range of issues and authors. One such issue, the origins of the "quantity theory of money" (with Jean Bodin as its hero), is notable for its sterility, since that concept was not and could not have been meaningful to contemporaries. 7 The origin and development of "mercantilism" or, more broadly, of "political economy" in general, has proven a more fruitful subject of inquiry. 8 Undeniably, the changing relationships among governments, merchants, and the economy were a major public issue and, in the seventeenth century, a major site of theoretical reflection on the nature of political power generally. Even so, an excessive concentration on mercantilism is dangerous, for however one defines it, it was not the first, and for a long time not the most significant, economic discourse in France. Rather, from the 1560s well into the seventeenth century, French discussions of economic issues centered on very concrete problems brought about by the government's obligation to issue and maintain a stable currency. 9 While one would search this literature in vain for the specific doctrines of modern economics, I argue that it did represent a real achievement in the early modern understanding of commerce and the state. Though they have not received their due from scholars, the writers who took part in these debates formed a large and influential school with close ties to the actual institutions of government. 10 The French monetary theorists did an enormous amount [End Page 60] to set the stage for the mercantilist era. They developed a detailed notion of monetary economics as a specialized discipline, and above all they placed money and economic policy in the context of a multi-national system of commercial states. In so doing, they sought to comprehend and control not just money but the modern state itself.Early modern money was perched uncomfortably between commodity and fiat. Bewildering varieties of coins circulated, composed of all sorts of alloys. The government assigned a face value for each coin which it authorized to circulate in the abstract accounting unit called the "money of account." 11 This value was almost always greater than... (shrink)

Mr. Thomas Storer does not believe that a science of signs is of basic importance to philosophy, for philosophy, he holds, is not a formal or a factual science but an activity of clarifying meaning and building linguistic systems. I should like to defend the relevance to philosophy of a science of signs even when philosophy is so conceived, and then to question this conception of philosophy itself.

This is a commentary on Adrian Heathcote’s interesting paper ‘Hume’s Master Argument’. Heathcote contends that No-Ought-From-Is is primarily a logical thesis, a ban on Is/Ought inferences which Hume derives from the logic of Ockham. NOFI is thus a variation on what Heathcote calls ‘Hume’s Master Argument’, which he also deploys to prove that conclusions about the future (and therefore a-temporal generalizations) cannot be derived by reason from premises about the past, and that conclusions about external objects or other minds cannot (...) be derived by reason from premises about impressions. Heathcote raises an important question. Having (apparently) argued that our inductive inferences are not justified by reason, Hume puts them down to Custom, and seems to suggest that we OUGHT to indulge this propensity but NOt the superstitious propensities that lead to religious belief. (Query: Why is it right to indulge one non-rational propensity but not the others?) Finally Heathcote argues that just as there are valid, but not formally valid, arguments taking us from claims about inferential relations to claims about what we ought to believe, so there may be valid, but not formally valid, arguments taking us from factual claims about some situation to claims about what we ought to do. -/- I reply that Hume does indeed have a Master Argument and that it does rely on logical principles but not on the logic of Ockham which had been largely forgotten by Hume’s day. Instead Hume relies on the idea widely believed in the 18th Century and taught to Hume at Edinburgh by his logic Professor Colin Drummond, that in a logically valid argument the conclusion is contained in the premises. I reconstruct Hume’s Master Argument using this principle. I draw a careful distinction between two theses: 1) that we cannot get from non-moral premises to moral conclusions with the aid of logic alone and 2) that we cannot get from non-moral premises to moral conclusions with aid of analytic bridge principles. Hume believed the first but not the second. What then is the role of NOFI in the larger argument of the Treatise? To show that the truths of ethics cannot be derived via logic from self-evident truths of some other kind and thus that they are not demonstrable. How can we make sense of Hume’s apparent belief that it is sometimes right to transcend reason and sometimes not? In the case of Custom, we live in a world governed by causal regularities, and, in such a world, induction is in fact a fairly reliable belief-forming mechanism. Thus a suitably qualified spectator (one aware of the kind of world we live in) would tend to approve of indulging it, even if it cannot be justified by reason. However, our superstitious propensities are (and can be known to be) unreliable, since they produce different and inconsistent results in different people. Thus it is it is wrong (something a suitably qualified spectator would disapprove of) to indulge the faculty of Superstition. I also take issue with Heathcote’s penchant for valid, but not formally valid, inferences. I supply the missing premises for Heathcote’s Is/Ought inferences and argue that they are either not true or not necessary. (shrink)