Author's introduction

Summary

It frequently happens in the history of thought that when a powerful new method emerges the study of those problems which can be dealt with by the new method advances rapidly and attracts the limelight, while the rest tends to be ignored or even forgotten, its study despised.

This situation seems to have arisen in our century in the Philosophy of Mathematics as a result of the dynamic development of metamathematics.

The subject matter of metamathematics is an abstraction of mathematics in which mathematical theories are replaced by formal systems, proofs by certain sequences of well-formed formulae, definitions by ‘abbreviatory devices’ which are ‘theoretically dispensable’ but ‘typographically convenient’. This abstraction was devised by Hilbert to provide a powerful technique for approaching some of the problems of the methodology of mathematics. At the same time there are problems which fall outside the range of metamathematical abstractions. Among these are all problems relating to informal (inhaltliche) mathematics and to its growth, and all problems relating to the situational logic of mathematical problem-solving.

I shall refer to the school of mathematical philosophy which tends to identify mathematics with its formal axiomatic abstraction (and the philosophy of mathematics with metamathematics) as the ‘formalist’ school. One of the clearest statements of the formalist position is to be found in Carnap [1937]. Carnap demands that (a) ‘philosophy is to be replaced by the logic of science …’, (b) ‘the logic of science is nothing other than the logical syntax of the language of science …’, (c) ‘metamathematics is the syntax of mathematical language’ (pp. xiii and 9). Or: philosophy of mathematics is to be replaced by metamathematics.

Formalism disconnects the history of mathematics from the philosophy of mathematics, since, according to the formalist concept of mathematics, there is no history of mathematics proper. Any formalist would basically agree with Russell's ‘romantically’ put but seriously meant remark, according to which Boole's Laws of Thought (1854) was ‘the first book ever written on mathematics’. Formalism denies the status of mathematics to most of what has been commonly understood to be mathematics, and can say nothing about its growth.