This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.
Despite its centrality and its familiarity, W. V. Quine's dispute with Rudolf Carnap over the analytic/synthetic distinction has lacked a satisfactory analysis. The impasse is usually explained either by judging that Quine's arguments are in reality quite weak, or by concluding instead that Carnap was incapable of appreciating their strength. This is unsatisfactory, as is the fact that on these readings it is usually unclear why Quine's own position is not subject to some of the very same arguments. A satisfying (...) and surprising account is here presented that stiches together the puzzling pieces of this important philosophical exchange and that in turn leads to an explanation of why it is so difficult to say whether anything of substance is at stake. (shrink)
A reading is offered of Carl Hempel’s and Thomas Kuhn’s positions on, and disagreements about, rationality in science that relates these issues to the debate between W.V. Quine and Rudolf Carnap on the analytic/synthetic distinction.
The dream of a community of philosophers engaged in inquiry with shared standards of evidence and justification has long been with us. It has led some thinkers puzzled by our mathematical experience to look to mathematics for adjudication between competing views. I am skeptical of this approach and consider Skolem's philosophical uses of the Löwenheim-Skolem Theorem to exemplify it. I argue that these uses invariably beg the questions at issue. I say ?uses?, because I claim further that Skolem shifted his (...) position on the philosophical significance of the theorem as a result of a shift in his background beliefs. The nature of this shift and possible explanations for it are investigated. Ironically, Skolem's own case provides a historical example of the philosophical flexibility of his theorem. Our suspicion ought always to be aroused when a proof proves more than its means allow it. Something of this sort might be called ?a puffed-up proof?. Ludwig Wittgenstein, Remarks on the foundations of mathematics (revised edition), vol. 2, 21. (shrink)
Alexander George’s lucid interpretation of Hume’s “Of Miracles” provides fresh insights into this provocative text, explaining the concepts and claims involved. He also shows why Hume’s argument fails to engage with committed religious thought and why philosophical argumentation so often proves ineffective in shaking people’s deeply held beliefs.
Those inquiring into the nature of mind have long been interested in the foundations of mathematics, and conversely this branch of knowledge is distinctive in that our access to it is purely through thought. A better understanding of mathematical thought should clarify the conceptual foundations of mathematics, and a deeper grasp of the latter should in turn illuminate the powers of mind through which mathematics is made available to us. The link between conceptions of mind and of mathematics has been (...) a central theme running through the great competing philosophies of mathematics of the twentieth century, though each has refashioned the connection and its import in distinctive ways. The present collection will be of interest to students of both mathematics and of mind. Contents include: "Introduction" by Alexander George; "What is Mathematics About?" by Michael Dummett; "The Advantages of Honest Toil over Theft" by George Boolos; "The Law of Excluded Middle and the Axiom of Choice" by W.W. Tait; "Mechanical Procedures and Mathematical Experience" by Wilfried Sieg; "Mathematical Intuition and Objectivity" by Daniel Isaacson; "Intuition and Number" by Charles Parsons; and "Hilbert's Axiomatic Method and the Laws of Thought" by Michael Hallett. (shrink)
One effect of W. V. Quine’s assault on the analytic-synthetic distinction is pressure on the boundaries between mathematics and empirical science. Assumptions about reference and knowledge that are natural in the context of the empirical sciences have been exported to the case of mathematics. Problems then arise when we ask how, given the abstractness of mathematical entities, we can refer to them or know anything about them. For if abstractness entails causal impotence, and if reference and knowledge require causal intercourse, (...) then it seems a mystery how reference to, or knowledge of, abstract entities is at all possible. (shrink)
Few thinkers in the past three decades have exerted more influence on the philosophy of language than Quine, Dummett, and Chomsky. No investigation into the current state of philosophy of language can omit consideration of their views. Yet I believe that their work has often been seriously misinterpreted. I begin by trying to clear up some unfortunate and prevalent misunderstandings. In particular, I examine in detail the relationship between Quine's and Chomsky's thought and argue that rumors of their incommensurability have (...) been greatly exaggerated. I lay out the many affinities between their approaches and isolate the crucial junction at which they part company . I also reconstruct Dummett's arguments against truth-based theories of meaning and in favor of verificationist ones, and argue that some recent criticisms of these have rested on misunderstanding. During my exposition of Dummett's thought, various puzzling impasses are noted. I show that these are symptomatic of a deep tension within his view. Essentially, this consists in his desire to follow both Quine and Chomsky, to adopt simultaneously a Chomskyan position and a Quinean one on the very issue that I argued fundamentally divided these two thinkers. The resultant schizophrenia about the nature of linguistic knowledge is studied closely and suggestions for its cure are tendered. Its significance for the future of the philosophy of language is assessed. (shrink)
Dieses Buch blickt in eine bedeutende Epoche der Philosophie der Mathematik zurück, deren Strömungen die heutige Gestalt der Mathematik prägten. In der Wende vom 19. zum 20. Jahrhundert befand sich die Mathematik in einem fundamentalen Umbruch, der die Mathematiker dieser Zeit herausforderte. Sie mussten Stellung beziehen. Die Grundsätze und Wege der philosophischen Richtungen, die dieses Buch verständlich, kritisch und anerkennend beschreibt, wurden von Mathematikern formuliert. Eine Zeit gravierender Disharmonien begann, die bis in Streit und Feindschaften mündeten und zugleich faszinierende und (...) fruchtbare Ergebnisse hervorbrachten, mathematisch wie philosophisch. Es war ein aufregendes, intellektuelles Abenteuer zu Beginn des 20. Jahrhunderts auf einem außergewöhnlich scharfsinnigen und kreativen Niveau. Die Debatte über die unversöhnlichen Ansichten versiegte allmählich und inzwischen ist wieder relative Ruhe in die Gemeinde der Mathematiker eingekehrt. Zentrale philosophische Fragen aber, die damals die Protagonisten spalteten, sind nach wie vor unbeantwortet. Die Suche nach dem Wesen der Mathematik geht weiter und greift auf die Ideen dieser Kontroversen zurück. (shrink)
Ludwig Wittgenstein has a recognizable approach that he regularly pursues in his philosophical investigations. There is a problem that he often presses, a form of criticism that he often develops, against traditional pursuits of philosophy. It is surprisingly difficult to say clearly what this problem is. But it is worthwhile to try, for this criticism is not only a hallmark of his thought but is also closely connected to other central features of it, for instance, to his conceptions of language (...) and of the nature of philosophical investigation. These features can be properly understood only in concert with a correct view of his terms of criticism of traditional philosophy. In this essay, Alexander George articulates a problem Wittgenstein sees with philosophy, shows how it illuminates otherwise peculiar features of Wittgenstein’s investigations, and finally considers an interesting situation in which Wittgenstein’s goals might be thwarted. (shrink)
Intuitionism is occasionally advanced on the grounds that a classical understanding of mathematical discourse could not be acquired, given limitations of the experience available to the language learner. In this note, focusing on the acquisition of the universal quantifier, I argue that this route of attack against a classical construal results, at best, in a Pyrrhic victory. The conditions under which it is successful are such as to redound upon the tenability of intuitionism itself. Adjudication will not follow merely from (...) attending to the learner''s experience. The nature of the agent''s ability to engage in conceptual extrapolation from that experience must be considered as well. (And divergent views regarding this are likely to recapitulate the original disagreement.). (shrink)
The foundations of linguistics continue to generate philosophical debate. Jerrold Katz claims that the subject matter of linguistics consists of abstract objects and that, as a consequence, the discipline cannot be viewed as part of psychology. I respond by arguing (1) that Katz misinterprets work in the philosophy of mathematics which he believes sheds light on foundational questions in linguistics; (2) that he misunderstands aspects of Noam Chomsky's position, against whose conception of linguistics many of his claims are directed; (3) (...) that Katz fails to dispose of a much more plausible analysis, according to which linguistics remains an empirical inquiry in spite of its abstract subject matter; and, finally, (4) that his arguments against what he calls‘generativism’, appealing to the existence of an infinitely long grammatical sentence of English, are flawed. (shrink)
Does the past rationally bear on the future? David Hume argued that we lack good reason to think that it does. He insisted in particular that we lack — and forever will lack — anything like a demonstrative proof of such a rational bearing. A surprising mathematical result can be read as an invitation to reconsider Hume's confidence.
A rich tradition in philosophy takes truths about meaning to be wholly determined by how language is used; meanings do not guide use of language from behind the scenes, but instead are fixed by such use. Linguistic practice, on this conception, exhausts the facts to which the project of understanding another must be faithful. But how is linguistic practice to be characterized? No one has addressed this question more seriously than W. V. Quine, who sought for many years to formulate (...) a conception of use that makes sense of certain key features of meaning. The nature, development, and adequacy of his formulations are here explored. All are found to fall short of what he wanted to achieve. Donald Davidson has introduced significant variations on Quine's project. The resulting position is also examined, but likewise found to be problematic. Finally, a neo-Quinean conception is sketched, as are some of the problems such a view would have to surmount. (shrink)