The Price of Mathematical Scepticism

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Conference Proceeding

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I argue that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. Thus scepticism about mathematical reality comes at a price.

Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, choice and consistency should all be aligned.

A key part of the talk is the “bivalence questionnaire”, designed to help us think about our intuitions and beliefs. By examining the different ways to answer it, we are led to a spectrum of positions, dubbed ultrafinitism, finitism, countabilism, sequentialism, particularism and totalism. Though they are very different, each maintains the belief alignment that I advocate.

Based on a paper in Philosophia Mathematica:


This presentation was part of the Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics in spring 2024.