CONTENTS

1   INTRODUCTION

2   SCIENCE OF SENSE

3   APPLIED MATHEMATICS

4   AXIOMS AS DEFINITIONS

5   ALTERNATIVE THEORIES

6   CONCLUSIONS

2   SCIENCE OF SENSE

2.1 Mathematics and Entailment

The train of thought which leads to the SS thesis begins with the belief that there has been over the last two hundred years, a dramatic shift in the what counts as doing mathematics. This shift is primarily a change in the attitude of mathematicians to the axiomatic method. This is well illustrated by the following passage.

" ... Zermelo's version of the theory of sets differs from that of Russell and Whitehead in the role which it assigns to axioms. In Principia Mathematica the axioms are all supposed to be necessary truths, and the whole structure is conceived by its authors in the same way as Euclid conceived his Elements. That is to say, there is no suggestion that there could be any alternative, except possibly a system of the same content in which some of the present axioms appeared as theorems and some of the present theorems as axioms. ... Zermelo, no doubt, also wishes that the axioms in his set should appear plausible, but he presents them rather as a modern geometer might present a set of axioms for a non-Euclidean geometry. If they are consistent, they delimit a system for study and provide an implicit definition of the membership sign which is used as a primitive symbol in the formulation of them.

The notion of implicit definition is as old as Gergonne, but the importance which formal, as opposed to material, axiomatics has in the thought of modern mathematicians is largely due to the work of Hilbert."

The SS thesis is firmly in this Hilbertian tradition of formal, "as opposed to material", axiomatics, as is clear from the introductory sections of the first draft of my essay explaining the thesis. It is formalist in the Begriffsschrift sense of exposing the entire sense of linguistic entities in the structure of these entities. It is not formalist in the pejorative (to a realist, at least) sense of manipulating symbols without meaning. The notion of sense is brought in to save mathematical discourse, where one is not concerned with whether or not there is reference, from the charge of being meaningless. The formal axiomatics can become material if someone shows that the senses have a reference, either empirically, or in some a priori way (though there is a further step from showing that they refer to showing that they are true).

Modern pure mathematics is, according to the Hilbert picture, concerned with logical entailments. It posits sets of axioms and then explores the theorems which follow from them. Axiomatised mathematics (due to Euclid) was originally about truth. The axioms were intended as being obviously true, and the exploration of theorems was meant to discover and justify seriously non-obvious truths following from them. In this quest it was a stunning success. Modern pure mathematics, by contrast, can be dated from the discovery of non-Euclidean geometries. Thereafter the claim that axioms are true was abandoned. The truth, and indeed the reference, of the axioms was no longer of interest. All that mattered was correctly identifying what follows logically from the axioms. This does not mean that the truth of geometric axioms is unimportant, only that the study of this aspect of them has shifted from pure mathematics to physics.

It must be confessed right away that this is a deliberately selective account of the history of mathematics. It concentrates on geometry and ignores the other great theme in mathematics, namely arithmetic, the study of numbers. The most obvious way to interpret this part of mathematics (a part big enough to look pretty much like the whole) is that it refers to these abstract objects called numbers which exist - either objectively, before any mathematics has been done (Platonism), or once they have been constructed by mathematicians (constructivism). In parallel with the trend towards the Hilbert approach to axioms, there was the other great movement towards referring to abstract objects known as sets (or classes), among which the numbers are to be found.

The SS thesis was framed in terms of an analogy between the investigations carried out in mathematics, and those carried out in empirical science. Within this expanded sense of the word "science" the original usage, referring to empirical studies, can be characterised as the Science of Contingent Reference (SCR). Developing this idea further, Platonist mathematics could be said to aspire to the status of the Science of Necessary Reference (SNR). A possible way of characterising constructivist mathematics within the same framework is to call it a Science of Alternative Reference (SAR) - a vision of an autonomous alternative realm of reference, so radically different from classical (Fregean) reference that it requires an alternative form of logic. These rivals to the SS thesis as descriptions of what mathematicians are doing are discussed further in Section 5.

2.2 Entailment and Sense

The second step towards a new account of pure mathematics is the claim that logical entailment is a relationship between thoughts (the senses of sentences). The evidence for this is that the validity of arguments appears to survive the absence of reference. From

"Adam was the first man"

we can validly deduce (drawing also on the fact that the father must precede the son)

"Adam had no father"

even though we believe that "Adam" lacks a reference. In this case it might be argued that the validity of the argument depends on the pieces that have a reference, "first", "man" and "father". But suppose we define "gloop" as meaning the creatures on Mars which require three parents for the act of procreation: mother, father and para-father. Then from

"Quaam was the first gloop"

we validly deduce

"Quaam had no para-father" ,

even though neither "Quaam", "gloop" nor "para-father" has a reference.

The connection between sense and entailment is discussed by Dummett.

"Learning to use a statement of a given form involves, then, learning two things: the conditions under which one is justified in making the statement; and what constitutes acceptance of it, i.e. the consequences of accepting it."

There are two levels of grasping the sense of a sentence:

Level 1 understanding concerns the link to the primary world, while level 2 is concerned with connections within the realm of sense. Making this distinction does not mean that we can not use logical entailments to determine truth-value. But once we have done whatever is required to know how to determine truth-value, we have reached a set point in our engagement with the thought, and can be said to have grasped it, in the weaker sense. In general there then lies beyond that the long (possibly infinite) road of exploring its logical consequences.

It might be argued that since a valid argument is defined as one that always preserves truth, entailment only works with sentences which refer. But in fact all that is required is the possibility of reference (which was argued above to be the condition for there to be a realm of sense). A valid argument never takes one from a true to a false statement. If reference fails somewhere, then all bets are off. A similar move can be made with the related notion of analyticity. An analytic sentence is one which follows from no premises at all. Usually it is described as being always true. We can generalise this to cover senses without reference as well, by saying that if the sentence refers then it must be true. Another way of expressing this is to say that an analytic sentence is never false.

If pure mathematics is about entailment, and entailment is a property of sense, then pure mathematics is a study of sense. It is the systematic attempt to extend our second-level grasping of senses. Indeed, within pure mathematics we concentrate on the second level to the exclusion of the first; we are not interested in finding out what, if anything, the expressions refer to.

With this account we can make sense of Russell's dictum that in pure mathematics we do not know what we are talking about. This is true when we consider primary reference. But in the primary sense rather than not knowing what we are talking about, we do not care what we are talking about or even whether we are talking about anything. Looking at reference in the wider sense, which includes indirect reference, it becomes clear what we are talking about, namely the senses of the expressions and the logical relationships between them. As Dummett points out, it was Frege who discovered that logic was rich enough to make these relations, and the related notion of analyticity, into a non-trivial subject matter to investigate. We might fully grasp the sense of an expression at the first level, and yet still be utterly surprised by its logical consequences.

2.3 Mathematics as a Science

The SS thesis appeals to an analogy with what we usually mean by "science", namely SCR. An obvious disanalogy is that SCR has as an important component the experimental method. If one thinks about mathematical practice, however, some sort of analogy begins to reappear, between what experimental scientists do and the gedankenexperiments regularly carried out by mathematicians. In the latter we find processes like: "what happens if we put these axioms together, or add another axiom, or take one away". This is something like a chemical experiment, where we look for the reactions between chemicals. In mathematics the axioms may "react" together to produce a whole new range of new and exciting theorems. Or they may prove to be "immiscible", that is, unable jointly to produce a consistent axiom set.

There is also an analogy between realist SS and the interpretation of chemistry as a modal science. In chemistry, an entirely new compound is in a contingent sense the creation of the chemist, but ultimately the distinction between natural and artificial chemicals belongs to the natural history of the world, rather than to chemistry. The new compound was "already there" in the timeless laws of chemistry, and these are the subject matter which concerns the pure chemist. Similarly with the production of theorems in the Science of Sense. Who first proved a theorem, and when it was done are part of the history of mathematics, but the subject itself is concerned with the timeless realm of sense.

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