A Finitist's Manifesto: Do we need to Reformulate the Foundations of Mathematics?

TL;DR

This paper questions the foundational assumptions of classical mathematics and computer science, especially regarding infinite objects and theoretical computation, urging a reconsideration of their conceptual basis.

Contribution

It critically examines the reliance on infinitary concepts in mathematics and explores the possibility of reformulating these foundations to address underlying issues.

Findings

Highlights issues with infinitary assumptions in mathematics

Questions the meaningfulness of 'existence' and 'in theory' concepts

Suggests the need for foundational reformulation

Abstract

There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the "existence'" of objects that contain an infinite number of elements, (ii) our ability, "in theory", to compute with an arbitrary level of precision, or (iii) our ability, "in theory", to compute for an arbitrarily large number of time steps. All of calculus relies on the notion of a limit. The monumental results of real and complex analysis rely on a seamless notion of the "continuum" of real numbers, which extends in the plane to the complex numbers and gives us, among other things, "rigorous" definitions of continuity, the derivative, various different integrals, as well as the fundamental theorems of calculus and of algebra -- the former of which says that the derivative and integral can be viewed as inverse operations, and the latter of which says that every polynomial over $\mathbb{C}$ has a complex root. This essay is an inquiry into whether there is any way to assign meaning to the notions of "existence" and "in theory'" in (i) to (iii) above.