# Mathematics without Numbers: Towards a Modal-Structural Interpretation

## Synopsis

Hellman here presents a detailed interpretation of mathematics as the investigation of "structural possibilities," as opposed to absolute, Platonic objects. After treating the natural numbers and analysis, he extends the approach to set theory, where he demonstrates how to dispense with a fixed universe of sets. Finally, he addresses problems of application to the physical world.

## Excerpt

. . . mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Bertrand Russell

THE idea that pure mathematics is concerned principally with the investigation of structures of various types in complete abstraction from the nature of individual objects making up those structures is not a novel one, and can be traced at least as far back as Dedekind classic essay, 'Was sind und was sollen die Zahlen?' (originally published in 1888). It represents a striking contrast with the Fregean pre- occupation with identifying "the objects" of particular branches of mathematics, and it seems to have lain behind Hilbert's refusal to accept Frege's point of view on such fundamental matters as the nature of mathematical definitions and axioms, mathematical existence, and truth.

With the rise of the comprehensive "logicist" systems of type theory and axiomatic set theory, however, the structuralist idea was either neglected in favour of some arbitrarily chosen relative interpretation of ordinary mathematics (number theory, analysis, etc.) within the comprehensive system, or else it was given metalinguistic lip-service through the apparatus of Tarskian model theory, carried out within set theory itself.

Despite the attractive unifying power of modern set theory, embedding the structuralist intuition within it has its disadvantages. Must we accept anything so powerful as, say, the Zermelo-Fraenkel axioms--categorically asserted as truths about Platonic objects--in order to carry out a structuralist interpretation of number theory or classical analysis? And what of set theory itself; can it not also be understood along structuralist lines, and would it not constitute a philosophical advance so to understand it? "It", after all, is actually a multitude of apparently conflicting systems. On the standard Platonist picture, at most one of them can correctly describe "the real world of sets". On a structuralist interpretation, there is at least the . . .

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