Towards a general mathematical theory of experimental science

TL;DR

This paper develops a formal mathematical framework for scientific theories, showing that verifiability constrains theories to mathematical structures foundational in physics, thus unifying various physical theories under a common foundation.

Contribution

It introduces a formal framework for scientific theories based on verifiability, linking them to fundamental mathematical structures used in physics.

Findings

Verifiability constrains theories to specific mathematical structures.

Provides a formal foundation connecting scientific theories to manifold and measure theory.

Enables systematic incorporation of assumptions to guide theory development.

Abstract

We lay the groundwork for a formal framework that studies scientific theories and can serve as a unified foundation for the different theories within physics. We define a scientific theory as a set of verifiable statements, assertions that can be shown to be true with an experimental test in finite time. By studying the algebra of such objects, we show that verifiability already provides severe constraints. In particular, it requires that a set of physically distinguishable cases is naturally equipped with the mathematical structures (i.e. second-countable Kolmogorov topologies and $\sigma$-algebras) that form the foundation of manifold theory, differential geometry, measure theory, probability theory and all the major branches of mathematics currently used in physics. This gives a clear physical meaning to those mathematical structures and provides a strong justification for their use in science. Most importantly it provides a formal framework to incorporate additional assumptions and constrain the search space for new physical theories.