Philosophical reflections on the foundations of mathematics

Erkenntnis 34 (2):187 - 209 (1991)
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Abstract

This article was written jointly by a philosopher and a mathematician. It has two aims: to acquaint mathematicians with some of the philosophical questions at the foundations of their subject and to familiarize philosophers with some of the answers to these questions which have recently been obtained by mathematicians. In particular, we argue that, if these recent findings are borne in mind, four different basic philosophical positions, logicism, formalism, platonism and intuitionism, if stated with some moderation, are in fact reconcilable, although with some reservations in the case of logicism, provided one adopts a nominalistic interpretation of Plato's ideal objects. This eclectic view has been asserted by Lambek and Scott (LS 1986) on fairly technical grounds, but the present argument is meant to be accessible to a wider audience and to provide some new insights.

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Citations of this work

Category theory.Jean-Pierre Marquis - 2008 - Stanford Encyclopedia of Philosophy.
What is the world of mathematics?J. Lambek - 2004 - Annals of Pure and Applied Logic 126 (1-3):149-158.

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References found in this work

Introduction to Metamathematics.Stephen Cole Kleene - 1952 - Princeton, NJ, USA: North Holland.
Natural Deduction: A Proof-Theoretical Study.Dag Prawitz - 1965 - Stockholm, Sweden: Dover Publications.
The Development of Logic.William Kneale & Martha Kneale - 1962 - Oxford, England: Clarendon Press.
The Development of Logic.W. C. Kneale - 1962 - Oxford University Press.
Mathematical Logic.Willard Van Orman Quine - 1940 - Cambridge: Harvard University Press.

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