Abstract

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Frege advocated the view that the senses of expressions constitute an objective realm to which our discourse can refer (what he called "indirect reference"). This can be used as the basis of a new answer to the question: what is the subject matter of pure mathematics? The 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 characterised as being the systematic study of senses, and in particular their logical inter-relations, regardless of whether or not they refer to anything. The train of thought leading to this conclusion has two main parts:

• pure mathematics is about entailment (what follows from given sets of axioms) rather than about truth (whether the axioms are true in the first place, or even whether they have reference at all);
  
  
• entailment is a property of sense - its validity can be ascertained regardless of whether the senses refer.
  

Sciences like physics and chemistry can be seen as making up the Science of Contingent Reference (SCR). Logicism can be described in the same way as the thesis that pure mathematics is the Science of Necessary Reference (SNR), while intuitionism can perhaps be described as seeing mathematics as being the Science of Alternative Reference (SAR), the alternative realm of reference being that perceived, via construction, by our mathematical intuition.

The virtue of the SS thesis is that it explains in a natural way the existence of applied mathematics. If pure mathematics is about some realm of reference distinct from the contingent, physical one, then it is something of a mystery why mathematics is so efficacious in our descriptions of the physical world. By contrast, it is inherent in the notion of sense that senses involve the possibility of reference. While we are doing pure mathematics we ignore this aspect, but if the senses turn out to refer, then our deepened understanding of their logical properties, gained by doing pure mathematics, enhances our ability to understand the phenomena referred to. Pure mathematics builds up a stock of explored senses which are available for subsequent referential discourse. The classic example of this happening was when Riemannian geometry, developed as an abstract possibility, was used by Einstein as the means of describing actual spacetime in General Relativity.

The SS thesis is related to the assertion that the axiom sets of pure mathematics are stipulations of the meaning of the otherwise undefined terms within them. This idea was attacked, from their different perspectives, by Frege and Brouwer, who argued, in effect, that mere stipulation can not establish the truth of axioms. With this we can agree. The SS thesis merely says that axioms establish the sense of certain terms within them; stipulation says nothing about reference in general and truth in particular.

The SS thesis does not answer many of the great questions concerning pure mathematics, but it does reframe them. This is illustrated by the following examples.

The fruitfulness of the SS thesis will depend to a large extent on the utility of this way of reframing the questions.

The SS thesis can be compared with other accounts of mathematics.

• The Science of Necessary Reference (SNR). Superficially mathematics is about the properties of certain necessarily existing objects: originally numbers and latterly sets. However when we examine why such objects must exist, this account loses its plausibility. More plausibly, mathematics is about the referents of certain higher order incomplete expressions constructed only from logical constants, whose reference is thereby guaranteed.
  
  
• The Science of Alternative Reference (SAR). Mathematics is here seen as being about an alternative realm of objects accessed through the operation of mathematical intuition. The possession of this intuition is part of the phenomenology of human experience. The puzzle then is why the study of the realm of physical objects should be expressed in terms of mathematics.
  
  
• Formalism. Mathematics is the study of the formal properties of certain symbol strings. These strings lack not only reference, but even sense. Mathematical physics has a referential periphery, where it comes into contact with the results of experiments, but has a purely formal core in which the symbols and their transformations do not refer to anything in the physical world. The SS thesis by contrast presents a picture of mathematics as consisting of statements expressing thoughts, and, in applied mathematics, being either true or false.

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