Mathematics as information compression via the matching and unification of patterns

TL;DR

This paper proposes a new perspective that views the foundations of mathematics as primarily about information compression through pattern matching and unification, supported by the SP Theory of Intelligence.

Contribution

It introduces ICMUP as a novel approach to understanding mathematical structures, linking mathematics, logic, and computing through a unified information compression framework.

Findings

Mathematical structures can be interpreted as variants of ICMUP

Logic and computing are understood as forms of ICMUP

Examples demonstrate the relation between mathematics and information compression

Abstract

This paper describes a novel perspective on the foundations of mathematics: how mathematics may be seen to be largely about 'information compression via the matching and unification of patterns' (ICMUP). ICMUP is itself a novel approach to information compression, couched in terms of non-mathematical primitives, as is necessary in any investigation of the foundations of mathematics. This new perspective on the foundations of mathematics has grown out of an extensive programme of research developing the "SP Theory of Intelligence" and its realisation in the "SP Computer Model", a system in which a generalised version of ICMUP -- the powerful concept of SP-multiple-alignment -- plays a central role. These ideas may be seen to be part of a "Big Picture" comprising six areas of interest, with information compression as a unifying theme. The paper describes the close relation between mathematics and information compression, and describes examples showing how variants of ICMUP may be seen in widely-used structures and operations in mathematics. Examples are also given to show how the mathematics-related disciplines of logic and computing may be understood as ICMUP. There are many potential benefits and applications of these ideas.