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.
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)
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)
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)
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)
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)