Proof, Practice, and Progress
Dissertation, University of Toronto (Canada) (
2002)
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Abstract
This thesis presents an anti-realist account of mathematics as 'recreational', and argues that such a view can answer the central dilemma for the philosophy of mathematics as presented in Benacerraf's 'Mathematical Truth'. I argue that we should only be satisfied with a naturalistic solution to this dilemma, where I understand 'naturalism' minimally as requiring natural scientific explanations of our mathematical knowledge. In Chapter 2 I thus discuss several broadly naturalist attempts to understand mathematical statements as having standard truth conditions while not referring to non-natural objects, and argue that these attempts to grasp the first horn of Benacerraf's dilemma have failed to satisfy a minimal adequacy condition of accounting for ordinary mathematical practices. ;In order to ensure that my own account takes seriously the actual practices of mathematicians, I discuss in Chapters 3--5 various stages of mathematical activity. These chapters include my own case study of mathematical proof, in which I present my observations from two semesters participating in a research seminar at the Fields Institute for Research in Mathematical Sciences in Toronto. Chapter 6 returns to the argument for anti-realism; showing that the standard Quinean realist arguments from indispensability leave too much standard mathematical practice unaccounted for. Chapter 7 makes explicit the commitments of the anti-realist alternative I have argued for, including an explanation of the various things that could be meant by 'mathematical truth'. ;It is concluded that there is an important sense in which mathematical statements may be said to be true even though they are not true of any objects. However, if Benacerraf is right to insist that, for a predicate to be considered a genuine truth-predicate, its applicability conditions must be explained in terms of reference, then the radical conclusion of this thesis is that there is no such thing as mathematical truth