Version: v1, Published online: 1998

Retrieved September 21, 2020, from https://www.rep.routledge.com/articles/thematic/logicism/v-1

## 4. Extension to other mathematical domains

The extension of these results to the arithmetic of the rational, real and complex numbers offers no new fundamental difficulties, after the pioneering work that had already been done by mathematicians (see Numbers §7).

A difficulty does arise in the theory of functions: namely, the need – first brought to prominent notice by Zermelo (1904), who characterized it as a ‘logical principle’ – for what has come to be called the ‘axiom of choice’; this says, in effect, that if *f* is a function defined on a domain *A*, such that, for each *x* in *A*, f(x) is a non-empty set, then there is a function *g*, also defined on *A*, such that, for each *x* in *A*, g(x) is an element of f(x).

An interesting point arises when we come to geometry. Frege never extended the thesis of logicism to geometry; but, in a rather polemical exchange with Hilbert concerning the latter’s path-breaking work on the foundations of geometry (see Frege 1980), Frege actually came near to formulating the point of view from which this extension can – and, indeed, so far as logicism is viable at all, *should* – be made: namely, one regards geometric theory – or, rather, *a* geometric theory (for there are infinitely many) – as concerned with a species of structure (of any one of which there may be many exemplars), characterized – indeed, defined – by the ‘axioms’ of that geometry; the theorems deduced from those axioms are then demonstrated ‘by pure logic’ to hold of any exemplar of the species so defined. That is the view maintained by Whitehead and Russell in the projected treatment of geometry in their major work, Principia Mathematica (1910–13). And it is the view that had already been expressed, in effect, by Dedekind and still earlier by Riemann (see Stein 1988: 244, 251–3). What is also noteworthy is that Dedekind, in contrast to Frege and to Russell and Whitehead, adopts the same point of view concerning even the system of the whole numbers: that is, he characterizes the *structure* a system must have to serve as that of the whole numbers, in explicit preference to identifying those numbers with any ‘objects’ specified in other than those structural terms.

Stein, Howard. Extension to other mathematical domains. Logicism, 1998, doi:10.4324/9780415249126-Y061-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/logicism/v-1/sections/extension-to-other-mathematical-domains.

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