Ian Hacking
University of Toronto, St. George
Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand.
Keywords philosophy of mathematics  Plato  Kant
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DOI 10.4314/sajpem.v30i1.64407
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The Mind, the Lab, and the Field: Three Kinds of Populations in Scientific Practice.Rasmus Grønfeldt Winther, Ryan Giordano, Michael D. Edge & Rasmus Nielsen - 2015 - Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences 52:12-21.
The Structure of Scientific Theories.Rasmus Grønfeldt Winther - 2015 - Stanford Encyclopedia of Philosophy.

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