CONTENTS

1   INTRODUCTION

2   SCIENCE OF SENSE

3   APPLIED MATHEMATICS

4   AXIOMS AS DEFINITIONS

5   ALTERNATIVE THEORIES

6   CONCLUSIONS

1   INTRODUCTION

Frege's separation of the previously unanalysed notion of meaning into sense and reference was a major advance in our understanding of what it is we do with language. Reference shows language reaching out beyond itself to talk about things in the world. (Among its capabilities is to reach out, and then back reflexively, to talk about itself, as if it were part of the world of the other.) It would have been easy for Frege to treat sense as something subjective - the product of our own mental activity as we strive to achieve reference. His choice of the word "thought" (Gedanke) to denote the sense of a sentence certainly has a strongly subjective flavour to it (at least in English).

Yet Frege insisted that sense is also objective. It is not the purpose of the present essay to defend this thesis, but to build upon it. It therefore suffices to give one example to illustrate the thesis and the thinking behind it. When, as realists, we say, "The moon is spherical", we intend this to be true independently of any person's grasping the thought. We claim that the sentence was true for a long time before there was anyone around to grasp the thought it expresses. The thought that the moon is spherical was true (and the thought that the moon is a two-dimensional disc was false) long before sense-graspers came on the scene. For thoughts to have this objective property, they must in turn be objective. Below it will be argued that logical implication relations are other objective features of thoughts.

If senses are objective, then they can in turn be referred to by our language. Frege identified indirect speech as the device used by our natural language to talk about thoughts. For a Fregean realist, the realm of sense, referred to in this way, is part of the greater objectivity. This realm is independent of (primary) reference to the extent that we can talk about senses whether or not they have reference. Indeed the question of whether or not given senses refer is an important part of our discourse about sense. However, as a whole the realm of sense is ontologically parasitic on the realm of primary reference. Sense depends upon the possibility of primary reference: if there were nothing primary to talk about, there would be no sense to talk about either.

This notion of a realm of sense gives us the possibility to develop a new account of mathematics. The central thesis of this essay is:

SS	Pure mathematics is to the realm of sense as science is to the realm of primary objective reference.

Pure mathematics is the systematic study of senses and their logical interrelationships. This thesis is independent of the claim that the realm of sense is objective, but taken together, they assert that pure mathematics is the study of an objectivity. Whatever its status, this subject matter was entirely unknown to us before Frege made his great voyages of discovery in the Philosophy of Thought.

Ironically, although the theory of mathematics developed here is based entirely on Fregean foundations, and although Frege's main objective in his philosophy was to give mathematics on a secure basis, the account given here is quite different from Frege's. For Frege arithmetic was the science of a type of logically necessary objects, the numbers. The class-forming operator was his way of generating these objects and linking them in to logic. It was also set him on the road to his final date with nemesis, in the shape of Russell's paradox. Frege's attitude to geometry can best be described as highly conservative.

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