CONTENTS

1   INTRODUCTION

2   SCIENCE OF SENSE

3   APPLIED MATHEMATICS

4   AXIOMS AS DEFINITIONS

5   ALTERNATIVE THEORIES

6   CONCLUSIONS

3   APPLIED MATHEMATICS

An advantage I claim for the theory of pure mathematics as the science of sense is that is leads to a natural account of the activity of applied mathematics, and even of why there should be such a subject in the first place. The fact that mathematics is efficacious in helping us with our descriptions of the world is often presented as a puzzle. Why should the timeless, idealised patterns of the realm of mathematics so often find an echo in the mutable, messy world of our physical experience? This question retains its force whether one believes that the mathematical realm exists as an objective, Platonic reality, or that it is a creation of the human mind. The discovery of mathematical regularities in the world is often accorded a mystical significance, being treated as evidence that behind the unsightly mess and mutability there are eternal crystalline structures or musical harmonies. These may be the higher reality of the world, or even the only true reality, connected to our vale of tears by the mysterious rule "as above, so below".

When mathematics is seen as the science of sense all this melts away. There is a ready made relationship between mathematics and the world, namely the relationship between sense and reference that is at the heart of Fregean semantics. Of course mathematics can be about the world, because that is ultimately what senses are for. The connection can be expressed as the thesis:

There is not a separate set of features which bear the label "mathematical", rather mathematics is a particular way of looking at what ever set of features might be in the world. Whenever we look at features with sufficient Begriffsschrift rigour we end up doing mathematics.

Originally, as discussed above, all mathematics was applied, that is, about the world and carried out as an aid to our living in the world. Arithmetic was about countable objects, while geometry was about spatial relations. Mathematics grew up as people tried to probe more deeply into the logical structure of these concepts, and eventually studying these structures for their own sake, as well as for counting things or measuring space. Mathematics became thereby pure, though it was still referential. With the discovery on non-Euclidean geometries, pure mathematics gained its autonomy. Mathematicians finally realised they could cut their links with the real world (except the residual link, expressed by the PR thesis), and begin to define their own new senses, either by imagining other ways the world might be, or purely formally, by adding, removing or modifying axioms. We no longer needed to rely on the world to supply us with the senses to study. The human imagination was given permission to explore the vastly wider realm of possibilities of reference.

By doing this, pure mathematicians began to build up a warehouse of thoroughly articulated senses which was available for physicists as they explored radically new and unexpected corners of the world. The classic case occurred when Einstein tried to reconcile gravitation with Lorentzian relativity - all the machinery of Riemannian geometry was waiting there for him to take advantage of, having been developed with no thought of its eventual reference. Later as physicists began to penetrate the world of the quantum, the machinery of Hilbert space was there in the warehouse, ready-made to describe the wavefunctions which are a key feature of this new physics.

This does not mean that the traffic is purely one way. The old process by which pure mathematics feeds off applied mathematics for its subject matter still continues. As we engage empirically with the contingent world, and then struggle to understand the results, we continue to grasp dimly new thoughts, and our attempts to make them more precise can generate new mathematics, which can then be taken up by pure mathematicians as objects of study in their own right. The mature process is a continuing dialectic between the free exploration of the human imagination, and the challenges presented to it by the contingent world.

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