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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.
A DEFENCE OF MATHEMATICAL PLURALISM
, 2004
"... We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context. ..."
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We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.
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.
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
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, 1999
"... SORITES) ISSN 11351349 Issue # 11. December 1999 Copyright © by SORITES and the authors MAIN AIN INTER NTERNET ET ACCESS CCESS: ..."
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SORITES) ISSN 11351349 Issue # 11. December 1999 Copyright © by SORITES and the authors MAIN AIN INTER NTERNET ET ACCESS CCESS:
MONTAGUE REDUCTION, CONFIRMATION, AND THE SYNTAXSEMANTICS RELATION
"... Abstract. Intertheoretic relations are an important topic in the philosophy of science. However, since their classical discussion by Ernest Nagel, such relations have mostly been restricted to relations between pairs of theories in the natural sciences. In this paper, we present a model of a new ty ..."
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Abstract. Intertheoretic relations are an important topic in the philosophy of science. However, since their classical discussion by Ernest Nagel, such relations have mostly been restricted to relations between pairs of theories in the natural sciences. In this paper, we present a model of a new type of intertheoretic relation, called Montague Reduction, which is assumed in Montague’s framework for the analysis and interpretation of natural language syntax. To motivate the adoption of our new model, we show that this model extends the scope of application of the Nagelian (or related) models, and that it shares the epistemological advantages of the Nagelian model. The latter is achieved in a Bayesian framework.
Carnap and Quine: TwentiethCentury Echoes of Kant and Hume
"... As a student at the University of Jena—where, in particular, he learned modern mathematical logic from Gottlob Frege—Rudolf Carnap was exposed early on to the Kantian view that the geometry of space is grounded in the pure form of our spatial intuition; and, as Carnap explains in his autobiography, ..."
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As a student at the University of Jena—where, in particular, he learned modern mathematical logic from Gottlob Frege—Rudolf Carnap was exposed early on to the Kantian view that the geometry of space is grounded in the pure form of our spatial intuition; and, as Carnap explains in his autobiography, he was initially strongly attracted by this view: I studied Kant’s philosophy with Bruno Bauch in Jena. In his seminar, the Critique of Pure Reason was discussed in detail for an entire year. I was strongly impressed by Kant’s conception that the geometrical structure of space is determined by the form of our intuition. The aftereffects of this influence were still noticeable in the chapter on the space of intuition in my dissertation, Der Raum. (1963a, 4) In particular, Carnap’s dissertation, completed—under Bauch—in 1921 and published in KantStudien in 1922, defends the view that the form of our pure intuition has only the infinitesimally Euclidean structure presupposed in Riemann’s theory of ndimensional manifolds (rather than a global threedimensional Euclidean