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PLURALISM IN MATHEMATICS
, 2004
"... We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic. ..."
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We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.
QUINE’S PHILOSOPHY OF LOGIC AND MATHEMATICS
"... The last four words of my title may seem redundant, since virtually all Quine’s philosophical writings, early and late, pertain directly or indirectly to logic, mathematics, or both. My aim here will in fact be less ambitious and more realistic than my title would thus suggest. I will be concerned n ..."
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The last four words of my title may seem redundant, since virtually all Quine’s philosophical writings, early and late, pertain directly or indirectly to logic, mathematics, or both. My aim here will in fact be less ambitious and more realistic than my title would thus suggest. I will be concerned not with anything and everything that Quine has had to say as a philosopher and logician about logic and mathematics, but more specifically with Quine’s struggles as an avowed empiricist with the two main problems that logic and mathematics have traditionally posed for any philosophy that takes senseexperience to be the primary source of knowledge: first, the appearance that logical and mathematical knowledge are a priori or independent of any reliance on senseexperience; second, the appearance that the objects of mathematical knowledge are abstract and beyond the realm of senseexperience. I will take up the two issues in the order listed. Quine’s views on both were in large part developed in reaction to logical positivism, the dominant form of empiricism in the early years of his
I. The Received View
"... attempted (and failed) to establish phenomenalistic foundations for science and wielded the verificationist criterion of cognitive significance against traditional metaphysics, religion and values. This characterization of Carnap’s philosophy has come to us primarily through A. J. Ayer’s introductio ..."
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attempted (and failed) to establish phenomenalistic foundations for science and wielded the verificationist criterion of cognitive significance against traditional metaphysics, religion and values. This characterization of Carnap’s philosophy has come to us primarily through A. J. Ayer’s introduction of positivism to the Englishspeaking world in his Language, Truth and Logic1 and the preliminary sketches of positivistic doctrine with which many of W.V. Quine’s essays begin (and go on, inevitably, to repudiate). 2 It is now largely taken for granted that the various objections leveled at verificationism—that none of its many reformulations draws the intended line between meaningful science and meaningless metaphysics and that it is meaningless according to itselfare devastating. 3 As a result, Carnap’s work has been allotted a largely historical role, if a significant one: contemporary views are often identified and distinguished by what in his and the positivist’s account of philosophy, science, language, and values they reject. 1 Ayer 1952.