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By A. R. D. Mathias, H. Rogers

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As PROGRESS IN THE PHILOSOPHY OF MATHEMATICS 49 far as I know there is at present no reliable evidence on this subject and the theoretical models that have been produced are highly hypothetical and, in any case, are not sufficiently specific to distinguish details. The material produced by the work on artificial intelligence, in particular in connection with theorem proving by machine, leads to a picture which is, most probably, still far removed from actuality. I believe that when the empirical discoveries that are necessary here are made, this will lead to further advances concerning some of the topics in class IIa that have been mentioned and also to unexpected new developments.

At first sight, each of the four topics has its own character which is generically distinct from that of the others. However, upon further inspection, we notice that they all go beyond the consideration of mathematics as an abstract universe. As far as the definition of the notion cardinal number is concerned this is not immediately obvious since we might regard it as a problem of abstract set theory. However, as such it is not a problem in the philosophy of mathematics at all. Rather, its great philosophical importance is due to its immediate relation to the external world, at least as far as our perception of concrete (and hence, finite) collections is concerned.

One hundred years ago, none of the specifically mathematical philosophies that are of central importance in our time had as yet emerged. There was no intuitionism and no predicativism, no logicism and no formalism. Traces of constructivism can in fact be found already in Euclid but the kind of abstract mathematics that was to provoke a constructivistic reaction was only beginning to take shape in 1870. And not only the various constructivistic schools of thought but also logicism and formalism owe their existence largely to this development.

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