Online research in philosophy

Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege’s writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well informed overview provided by the first of its three chapters. Specialists will find the book an (...) indispensable reference and an invaluable source of insights and new results. Although Frege is widely regarded as the father of analytic philosophy, his work on the foundations of mathematics was for a long time rather peripheral to the ongoing research. The main reason for this is no doubt Russell’s discovery in 1901 that the paradox now bearing his name can be derived in Frege’s logical system. But recent decades have seen a huge surge of interest in Fregean approaches to the foundations of mathematics. (The work of George Boolos, Kit Fine, Bob Hale, Richard Heck, Stewart Shapiro, and Crispin Wright is singled out for particular attention in the present monograph.) A variety of consistent theories have been discovered that can be salvaged from Frege’s inconsistent system, and foundational and philosophical claims have been made on behalf of many of these theories. Burgess claims quite plausibly that the significance of any such modified Fregean theory will in large part depend on how much of ordinary mathematics it enables us to develop.1 His.. (shrink)

This is the verbatim manuscript of a paper which has circulated underground for close to thirty years, reaching a metethical conclusion close to J. L. Mackie’s by a somewhat different route.

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...) discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

History and Philosophy of Logic, Volume 32, Issue 2, Page 187-191, May 2011.

What is the simplest and most natural axiomatic replacement for the set-theoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? Several axiomatic systems, old (...) and new, are examined and evaluated as candidate answers to these questions, with results of Harvey Friedman playing a significant role in the examination. (shrink)

Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific theory of the world. Quine's (...) arguments against the analytic/synthetic distinction, even if fully accepted, still leave room for a notion of pragmatic analyticity sufficient for the indicated purpose. (shrink)

This paper re-examines the question of whether quirks of early human foetal development tell against the view (conceptionism) that we are human beings at conception. A zygote is capable of splitting to give rise to identical twins. Since the zygote cannot be identical with either human being it will become, it cannot already be a human being. Parallel concerns can be raised about chimeras in which two embryos fuse. I argue first that there are just two ways of dealing with (...) cases of fission and fusion and both seem to be available to the conceptionist. One is the Replacement View according to which objects cease to exist when they fission or fuse. The other is the Multiple Occupancy View – both twins may be present already in the zygote and both persist in a chimera. So, is the conceptionist position tenable after all? I argue that it is not. A zygote gives rise not only to a human being but also to a placenta – it cannot already be both a human being and a placenta. Neither approach to fission and fusion can help the conceptionist with this problem. But worse is in store. Both fission and fusion can occur before and after the development of the inner cell mass of the blastocyst – the entity which becomes the embryo proper. The idea that we become human beings with the arrival of the inner cell mass leads to bizarre results however we choose to accommodate fission and fusion. (shrink)

One textbook may introduce the real numbers in Cantor’s way, and another in Dedekind’s, and the mathematical community as a whole will be completely indifferent to the choice between the two. This sort of phenomenon was famously called to the attention of philosophers by Paul Benacerraf. It will be argued that structuralism in philosophy of mathematics is a mistake, a generalization of Benacerraf’s observation in the wrong direction, resulting from philosophers’ preoccupation with ontology.

A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.

Saul Kripke has made fundamental contributions to a variety of areas of logic, and his name is attached to a corresponding variety of objects and results. 1 For philosophers, by far the most important examples are ‘Kripke models’, which have been adopted as the standard type of models for modal and related non-classical logics. What follows is an elementary introduction to Kripke’s contributions in this area, intended to prepare the reader to tackle more formal treatments elsewhere.2 2. WHAT IS A (...) MODEL THEORY? Traditionally, a statement is regarded as logically valid if it is an instance of a logically valid form, where a form is regarded as logically valid if every instance is true. In modern logic, forms are represented by formulas involving letters and special symbols, and logicians seek therefore to define a notion of model and a notion of a formula’s truth in a model in such a way that every instance of a form will be true if and only if a formula representing that form is true in every model. Thus the unsurveyably vast range of instances can be replaced for purposes of logical evaluation by the range of models, which may be more tractable theoretically and perhaps practically. Consideration of the familiar case of classical sentential logic should make these ideas clear. Here a formula, say (p & q) ∨ ¬p ∨ ¬q, will be valid if for all statements P.. (shrink)

A revision of a sermon on the evils of calling model theory “semantics”, preached at Notre Dame on Saint Patrick’s Day, 2005. Provisional version: references remain to be added. To appear in Mathematics, Modality, and Models: Selected Philosophical Papers, coming from Cambridge University Press.

Dummett's case against platonism rests on arguments concerning the acquisition and manifestation of knowledge of meaning. Dummett's arguments are here criticized from a viewpoint less Davidsonian than Chomskian. Dummett's case against formalism is obscure because in its prescriptive considerations are not clearly separated from descriptive. Dummett's implicit value judgments are here made explicit and questioned. ?Combat Revisionism!? Chairman Mao.

Classical logic -- Temporal logic -- Modal logic -- Conditional logic -- Relevantistic logic -- Intuitionistic logic.

The discovery of the note cards for Quine’s previously unpublished 1946 lecture on nominalism provides an obvious occasion for commenting on the differences between the issue of nominalism as Quine first publicized it to a wide philosophical audience and the issue of nominalism as debated among Quine’s successors today. Yet as I read and reread the text of Quine’s lecture, I found myself struck less by the differences between Quine’s position there and the positions of present-day writers than by differences (...) between Quine’s position there and the positions of Quine himself in later writings — and not his writings from many years later but his writings from the next few years, and especially one of his writings from the very next year, his notorious joint paper with Goodman. (shrink)

In this era when results of empirical scientific research are being appealed to all across philosophy, when we even find moral philosophers invoking the results of brain scans, many profess to practice "naturalized epistemology," or to be "epistemological naturalists." Such phrases derive from the title of a well-known essay by Quine,[1] but Paul Gregory's thesis in the work under review is that there is less connection than is usually assumed between Quine's variety of naturalized epistemology and what is today taken, (...) by opponents and proponents alike, to constitute epistemological naturalism. To put it bluntly, as Gregory does in the opening sentence of his introduction, Quine "has not been well understood." If there is less connection between the Quinian and other epistemological naturalisms than there has often been taken to be, on Gregory's account there is also much more connection between Quine's position on epistemology and his positions on other contentious issues. (shrink)

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Supecki and Bryll, is presented for the claim that the right demonstrability logic must be contained in S5, (...) and a more speculative argument for the claim that it does not include S4.2 is also presented. (shrink)

Adapated from talks at the UCLA Logic Center and the Pitt Philosophy of Science Series. Exposition of material from Fixing Frege, Chapter 2 (on predicative versions of Frege’s system) and from “Protocol Sentences for Lite Logicism” (on a form of mathematical instrumentalism), suggesting a connection. Provisional version: references remain to be added. To appear in Mathematics, Modality, and Models: Selected Philosophical Papers, coming from Cambridge University Press.

My contribution to the symposium on Goedel’s philosophy of mathematics at the spring 2006 Association for Symbolic Logic meeting in Montreal. Provisional version: references remain to be added. To appear in an ASL volume of proceedings of the Goedel sessions at that meeting.

1 Choice conjecture In axiomatizing nonclassical extensions of classical sentential logic one tries to make do, if one can, with adding to classical sentential logic a finite number of axiom schemes of the simplest kind and a finite number of inference rules of the simplest kind. The simplest kind of axiom scheme in effect states of a particular formula P that for any substitution of formulas for atoms the result of its application to P is to count (...) as an axiom. The simplest kind of onepremise inference rule in effect states of a particular pair of formulas P and Q that for any substitution of formulas for atoms, if the result of its application to P is a theorem, then the result of its application to Q is to count as a theorem; similarly for many-premise rules. Such are the schemes and rules of all the best-known modal and tense logics, for instance. Sometimes it is difficult to find such simple schemes and rules (though it is usually even more difficult to prove that none exist). In that case one may resort to less simple schemes or less simple rules. There is no generally recognized rigorous definition of "next simplest kind" of scheme. (In the case of schemes, one fact that makes a rigorous definition difficult is that, if the logic in question is axiomatizable at all, which is to say, if the set of formulas wanted as theorems is recursively enumerable, then by Craig’s trick one can always get a primitive recursive set of schemes of the simplest kind, even if one cannot get a finite set. Intuitively, some primitive recursive sets are much simpler than others, but it is difficult to reduce this intuition to a rigorous definition.) Neither is there any generally recognized definition of "next simplest kind" of rule, and hence there is no fully rigorous enunciation of the choice conjecture, the conjecture that schemes of the next simplest kind can always be avoided in favor of rules of the next simplest kind and vice versa. Nonetheless, there are cases where intuitively one does recognize that the schemes or rules in a given axiomatization are only slightly more complex than the simplest kind, including cases where one does have a choice between adopting slightly-more-complex-than-simplest schemes and adopting slightly-more-complex-than-simplest rules. In tense logic early examples of slightly more complex rules are found in [2] and [3]: there is one example of the embarrassed use of such rules in the former, and many examples of the enthusiastic use of such rules in the latter and its sequels. Accordingly the rules in question have come to be called "Gabbay-style" rules.. (shrink)

Hintikka and Sandu have recently claimed that Frege's notion of function was substantially narrower than that prevailing in real analysis today. In the present note, their textual evidence for this claim is examined in the light of relevant historical and biographical background and judged insufficient.

Philosophical Analysis in the Twentieth Century by Scott Soames reminds me of nothing so much as Lectures on Literature by Vladimir Nabokov. Both are works that arose immediately out of the needs of undergraduate teaching, yet each manages to say much of significance to knowledgeable professionals. Each indirectly provides an outline of the history of its field, through a presentation of selected major works, taken in chronological order and including items that are generally recognized as marking decisive turning points. Yet (...) neither Soames’s work nor Nabokov’s is a history in any conventional sense, both being immediately disqualified from that category by the general absence of coverage of minor and middling works and writers. The emphasis is pedagogical rather than historiographical: the emphasis is on introducing the student to the field through very close examination of the limited number of key texts selected for inclusion. The author’s distinctive personality is also apparent in both works. Each writer has a favorite theme he repeatedly sounds: for Soames, the danger of conflating the analytic, the a priori, and the necessary; for Nabokov, the philistinism of expecting an uplifting “message” from works of literary art. Each also includes some quirky, individual selections: The Right and the Good, The Strange Case of Dr. Jekyll and Mr. Hyde. Few others would have taken R. L. Stevenson to be up there with Dickens, Flaubert, and Proust, or W. D. Ross with Russell, Wittgenstein, and Quine. Each also sets aside for separate treatment elsewhere a major body of work one might have expected to be covered. Nabokov reserves Russian literature for a companion volume, while Soames gives only slight coverage to what he describes as “work in logic, the foundations of logic, and the application of logical techniques to the study of language” — a category that in practice turns out to include the bulk of the relevant material (by such writers as Frege, Carnap, and Tarski) that published originally in German without simultaneous English translation.. (shrink)

This long-awaited volume is a must-read for anyone with a serious interest in\nphilosophy of mathematics. The book falls into two parts, with the primary focus of\nthe first on ontology and structuralism, and the second on intuition and\nepistemology, though with many links between them. The style throughout involves\nunhurried examination from several points of view of each issue addressed, before\nreaching a guarded conclusion. A wealth of material is set before the reader along\nthe way, but a reviewer wishing to summarize the author’s views (...) crisply will be\nfrustrated. The chapter-by-chapter survey below conveys at best a very incomplete\nand imperfect impression of the work’s virtues, and even of its contents, falling\nshort even of supplying a full menu for the banquet of food for thought that Parsons\nserves up to his readers. (shrink)

This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in ...

The species of indeterminist tense logic called Peircean by A. N. Prior is proved to be recursively decidable.

Foundational work in mathematics by some of the other participants in the symposium helps towards answering the question whether a heterodox mathematics could in principle be used as successfully as is orthodox mathematics in scientific applications. This question is turn, it will be argued, is relevant to the question how far current science is the way it is because the world is the way it is, and how far because we are the way we are, which is a central question, (...) if not the central question, of philosophy of science. (shrink)

Earlier results on climinating numerical objects from physical theories are extended to results on eliminating geometrical objects.

It is shown that for invariance under the action of special groups the statements "Every invariant PCA is decomposable into (1 invariant Borel sets" and "Every pair of invariant PCA is reducible by a pair of invariant PCA sets" are independent of the axioms of set theory.

Recently it has become almost the received wisdom in certain quarters that Kripke models are appropriate only for something like metaphysical modalities, and not for logical modalities. Here the line of thought leading to Kripke models, and reasons why they are no less appropriate for logical than for other modalities, are explained. It is also indicated where the fallacy in the argument leading to the contrary conclusion lies. The lessons learned are then applied to the question of the status of (...) the formula. (shrink)

The consequences for the theory of sets of points of the assumption of sets of sets of points, sets of sets of sets of points, and so on, are surveyed, as more generally are the differences among the geometric theories of points, of finite point-sets, of point-sets, of point-set-sets, and of sets of all ranks.

Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel’s incompleteness theorems, but also a large number of optional topics, from Turing’s theory of computability to Ramsey’s theorem. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive functions, a (...) traditional stumbling block for students on the way to the Godel incompleteness theorems. (shrink)

We show it is consistent with ZFC that the Hanf number of Ellentuck's Souslin logic should be exactly $\beth_{\omega_2}$.

It has become, however, one of the commonest catchwords of modern political science. Especially is it so used and abused by French, English and American ...

Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege’s writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well informed overview provided by the first of its three chapters. Specialists will find the book an (...) indispensable reference and an invaluable source of insights and new results. (shrink)

John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail. I am grateful to Philip Welch, two anonymous referees, and especially Ignacio Jané for written comments on earlier versions of this paper, which have led (...) to substantial improvements. Thanks also to the participants in a discussion group at the University of Bristol, where an earlier version was presented. (shrink)

In the final chapter of their book A Subject With No Object, John Burgess and Gideon Rosen raise the question of the value of the nominalistic reconstructions of mathematics that have been put forward in recent years, asking specifically what this body of work is good for. The authors conclude that these reconstructions are all inferior to current versions of mathematics (or science) and make no advances in science. This paper investigates the reasoning that led to such a (...) negative appraisal, and it produces a rebuttal to this reasoning. I am grateful to the following mathematicians who were kind enough to provide me with their thoughts about nonstandard analysis: Martin Davis, Laura Chihara, Ted Chihara, Steve Galovich, Bonnie Gold, and especially Roger Simons, whose comments about an earlier version of this paper were very helpful. Thanks also go to two referees for their useful suggestions and criticisms of an earlier version of this paper. (shrink)

This paper addresses John Burgess's answer to the ‘Benacerraf Problem’: How could we come justifiably to believe anything implying that there are numbers, given that it does not make sense to ascribe location or causal powers to numbers? Burgess responds that we should look at how mathematicians come to accept: There are prime numbers greater than 1010 That, according to Burgess, is how one can come justifiably to believe something implying that there are numbers. This paper (...) investigates what lies behind Burgess's answer and ends up as a rebuttal to Burgess's reasoning. (shrink)

John Burgess (Burgess, 2004) combines plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail.

There is a recent and growing trend in philosophy that involves deferring to the claims of certain disciplines outside of philosophy, such as mathematics, the natural sciences, and linguistics. According to this trend— deferentialism , as we will call it—certain disciplines outside of philosophy make claims that have a decisive bearing on philosophical disputes, where those claims are more epistemically justified than any philosophical considerations just because those claims are made by those disciplines. Deferentialists believe that certain longstanding philosophical problems (...) can be swiftly and decisively dispatched by appeal to disciplines other than philosophy. In this paper we will argue that such an attitude of uncritical deference to any non-philosophical discipline is badly misguided. With reference to the work of John Burgess and David Lewis, we consider deference to mathematics. We show that deference to mathematics is implausible and that main arguments for it fail. With reference to the work of Michael Blome-Tillmann, we consider deference to linguistics. We show that his arguments appealing to deference to linguistics are unsuccessful. We then show that naturalism does not entail deferentialism and that naturalistic considerations even motivate some anti-deferentialist views. Finally, we set out deferentialism’s failings and present our own anti-deferentialist approach to philosophical inquiry. (shrink)

Hermeneutic fictionalism about mathematics maintains that mathematics is not committed to the existence of abstract objects such as numbers. Mathematical sentences are true, but they should not be construed literally. Numbers are just fictions in terms of which we can conveniently describe things which exist. The paper defends Stephen Yablo’s hermeneutic fictionalism against an objection proposed by John Burgess and Gideon Rosen. The objection, directed against all forms of nominalism, goes as follows. Nominalism can take either a hermeneutic (...) form and claim that mathematics, when rightly understood, is not committed to the existence of abstract objects, or a revolutionary form and claim that mathematics is to be understood literally but is false. The hermeneutic version is said to be untenable because there is no philosophically unbiased linguistic argument to show that mathematics should not be understood literally. Against this I argue that it is wrong to demand that hermeneutic fictionalism should be established solely on the basis of linguistic evidence. In addition, there are reasons to think that hermeneutic fictionalism cannot even be defeated by linguistic arguments alone. (shrink)

Machine generated contents note: Part I. General: 1. The Gödel editorial project: a synopsis Solomon Feferman; 2. Future tasks for Gödel scholars John W. Dawson, Jr., and Cheryl A. Dawson; Part II. Proof Theory: 3. Kurt Gödel and the metamathematical tradition Jeremy Avigad; 4. Only two letters: the correspondence between Herbrand and Gödel Wilfried Sieg; 5. Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counter-example interpretation W. W. Tait; 6. Gödel on intuition and on Hilbert's finitism W. (...) W. Tait; 7. The Gödel hierarchy and reverse mathematics Stephen G. Simpson; 8. On the outside looking in: a caution about conservativeness John P. Burgess; Part III. Set Theory: 9. Gödel and set theory Akihiro Kanamori; 10. Generalizations of Gödel's universe of constructible sets Sy-David Friedman; 11. On the question of absolute undecidability Peter Koellner; Part IV. Philosophy of Mathematics: 12. What did Gödel believe and when did he believe it? Martin Davis; 13. On Gödel's way in: the influence of Rudolf Carnap Warren Goldfarb; 14. Gödel and Carnap Steve Awodey and A. W. Carus; 15. On the philosophical development of Kurt Gödel Mark van Atten and Juliette Kennedy; 16. Platonism and mathematical intuition in Kurt Gödel's thought Charles Parsons; 17. Gödel's conceptual realism Donald A. Martin. (shrink)

Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti-platonist argument proposed by Hartry Field avoids both horns of (...) their dilemma. (shrink)