The Splendors and Miseries of Heavisidisation

TL;DR

This paper explores the potential and challenges of representing scientific problems using iterated Heaviside functions, aiming to reformulate scientific theories within a constructive mathematical framework.

Contribution

It introduces a novel perspective on reformulating scientific problems through Heavisidisation, outlining initial steps towards this ambitious goal.

Findings

Identifies obstacles in expressing scientific answers as iterated Heaviside functions

Proposes a reformulation program for scientific theories using this representation

Highlights the potential for a constructive mathematics approach in natural sciences

Abstract

Machine Learning (ML) is applicable to scientific problems, i.e. to those which have a well defined answer, only if this answer can be brought to a peculiar form ${\cal G}: X\longrightarrow Z$ with ${\cal G}(\vec x)$ expressed as a combination of iterated Heaviside functions. At present it is far from obvious, if and when such representations exist, what are the obstacles and, if they are absent, what are the ways to convert the known formulas into this form. This gives rise to a program of reformulation of ordinary science in such terms -- which sounds like a strong enhancement of the constructive mathematics approach, only this time it concerns all natural sciences. We describe the first steps on this long way.