Why mathematics needs engineering

TL;DR

This paper argues that mathematics benefits from engineering principles in its definitions and theorems, emphasizing the importance of careful design and consistency to improve understanding and foundational clarity.

Contribution

It introduces engineering-inspired guidelines for mathematical definitions and axiomatization, illustrated through relations, functions, and category theory.

Findings

Revised definitions improve clarity and consistency.

Engineering principles enhance mathematical foundational robustness.

Guidelines like the Halmos and Arnold principles aid in mathematical design.

Abstract

Engineering needs mathematics, but the converse is also increasingly evident. Indeed, mathematics is still recovering from the drawbacks of several "reforms". Encouraging is the revived interest in proofs indicated by various recent "introduction to proof"-type textbooks. Yet, many of these texts defeat their own purpose by self-conflicting definitions. Most affected are fundamental concepts such as relations and functions, despite flawless accounts 50 years ago. We take the viewpoint that definitions and theorems are tools for capturing, analyzing and understanding mathematical concepts and hence, like any tools, require diligent engineering. This is illustrated for relations and functions, their algebraic properties and their relation to category theory, with the "Halmos principle" for definitions and the "Arnold principle" for axiomatization as design guidelines.