Noisy Deductive Reasoning: How Humans Construct Math, and How Math Constructs Universes

TL;DR

This paper proposes a stochastic computational model of mathematical reasoning, explaining how mathematicians generate research, use heuristics, and consider multiple systems as isomorphic to possible universes, offering new insights into mathematical epistemology.

Contribution

It introduces a probabilistic framework for mathematical reasoning, integrating Bayesian and abductive approaches, and explaining phenomena like multiple proofs and isomorphic systems.

Findings

Mathematicians generate research programs probabilistically.

Bayesian models effectively describe mathematical heuristics.

Multiple proofs increase belief in propositions.

Abstract

We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.