The Price of Mathematical Scepticism

TL;DR

This paper explores the philosophical implications of doubting key mathematical principles, arguing that such skepticism should extend to the consistency of third-order arithmetic, emphasizing the alignment of beliefs about reality and mathematical axioms.

Contribution

It introduces a philosophical argument linking skepticism about the Continuum Hypothesis and Axiom of Choice to doubts about third-order arithmetic's consistency.

Findings

Doubting bivalence of the Continuum Hypothesis implies doubting third-order arithmetic consistency.

Skepticism about the Axiom of Choice aligns with doubts about mathematical consistency.

Mathematical beliefs are rooted in intuitions that are either fully accepted or rejected.

Abstract

This paper argues 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. 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.