Mathematics and the real world

D.F.M Strauss

doi:10.4102/koers.v65i1.466

Abstract

In this article the initial discussion of the untenability of the distinction between “pure” and “applied" mathematics is followed by looking at alternative approaches regarding the relationship between mathematics and the “real world” - with intuitionism and Platonism representing the two opposite positions. The notions of infinity as well as the totality character of spatial continuity (and its implied infinite divisibility) turned out to occupy a central position in this context. In the final section brief attention is given - against the background of some perspectives on the history of mathematics - to an alternative approach in which both the uniqueness and the mutual irreducibility of number and space are conjectured.

Highlights

Mathematics acquired a distinctive and respectable status first and foremost owing to its apparently exact mode of thought and its ability to achieve rigorous results based upon sound assumptions
The belief that mathematics reveals something about the “real world” received its first and lasting impetus from the Platonic conception of the world of ideal forms in which the mathematical e/de occupied an intermediate position
Smart points out that according to Cassirer the main purpose of the critical study of the history of mathematics “is to illustrate and confirm the special thesis that ordinal number is logically prior to cardinal number, and, more generally, that mathematics may be defined, in Leibnizian fashion, as the science of order”. This brings us to the alternative approach of intuitionism - keeping in mind that Kurt Gódel’s discovery in 1931 forced Hilbert and Bernays to revert to intuitionistic methods in their “meta-mathe matics”

Summary

Introduction

Mathematics acquired a distinctive and respectable status first and foremost owing to its apparently exact mode of thought and its ability to achieve rigorous results based upon sound assumptions. The other option is to affirm a different “abstract” order of “existence” for mathematical “objects” and “structures” In this fashion modern mathe matics is often described as the science of “formal systems” (cf Kórner, 1972:124ff.) Paul Bernays (1976) specifies the term “formal” by equating it with “mathematical abstraction”, indicating to him that one only considers the structural elements of an object (i.e. the way in which an object is composed out of parts).. Smart points out that according to Cassirer the main purpose of the critical study of the history of mathematics “is to illustrate and confirm the special thesis that ordinal number is logically prior to cardinal number, and, more generally, that mathematics may be defined, in Leibnizian fashion, as the science of order” (my emphasis - DFMS; Smart, 1958:245) This brings us to the alternative approach of intuitionism - keeping in mind that Kurt Gódel’s discovery in 1931 (at an age of 25) forced Hilbert and Bernays to revert to intuitionistic methods in their “meta-mathe matics” (cf Hilbert & Bernays, 1934 and 1939)

Does the intuitionism of Brouwer bring mathematics closer to “reality”?

The conflict between platonism and intuitionism

An example of the amazing link between mathematics and the “real world”

The real world inherently displays a “mathematical side”

The history of mathematics explored two opposing reductions:

Literature