Two Streams in the Philosophy of Mathematics: Rival Conceptions of Mathematical Proof
There are two streams in the philosophy of mathematics. Philosophers in the ‘mainstream’ regard formal logic as the principal tool for explaining why mathematics counts as knowledge. Mathematicians establish theorems using mathematical proofs; formal logic models these proofs and shows that they prove theorems (rather than merely persuade mathematicians). Some philosophers in the mainstream (such as Crispin Wright) maintain a qualified version of Frege’s view that mathematics simply is formal logic.
Philosophers in the mainstream offer a wide range of accounts of what mathematics is about (abstract Platonic objects, social constructs, empirical posits, convenient fictions, structures, patterns, etc.). But they all agree that formal logic does the lion’s share of explaining why mathematics is knowledge.
Philosophers in the alternative or ‘maverick’ stream reply that the fully specified and gap-free proofs posited in formal proof-theory play no role in mathematical practice, and nor could they. In any case, not all mathematical activity is theorem-proving.
In the decades since William Aspray and Philip Kitcher gathered the various alternatives to mainstream philosophy of mathematics under the term ‘maverick’, this stream has thickened from a trickle of occasional publications to a steady flow of activity, much of it in continental Europe. The time is ripe to create a forum where philosophers from these two streams can address each other’s arguments in detail.
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