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Logicism

DOI
10.4324/9780415249126-Y061-1
DOI: 10.4324/9780415249126-Y061-1
Version: v1,  Published online: 1998
Retrieved June 21, 2021, from https://www.rep.routledge.com/articles/thematic/logicism/v-1

Article Summary

The term ‘logicism’ refers to the doctrine that mathematics is a part of (deductive) logic. It is often said that Gottlob Frege and Bertrand Russell were the first proponents of such a view; this is inaccurate, in that Frege did not make such a claim for all of mathematics. On the other hand, Richard Dedekind deserves to be mentioned among those who first expressed the conviction that arithmetic is a branch of logic.

The logicist claim has two parts: that our knowledge of mathematical theorems is grounded fully in logical demonstrations from basic truths of logic; and that the concepts involved in such theorems, and the objects whose existence they imply, are of a purely logical nature. Thus Frege maintained that arithmetic requires no assumptions besides those of logic; that the concept of number is a concept of pure logic; and that numbers themselves are, as he put it, logical objects.

This view of mathematics would not have been possible without a profound transformation of logic that occurred in the late nineteenth century – most especially through the work of Frege. Before that time, actual mathematical reasoning could not be carried out under the recognized logical forms of argument: this circumstance lent considerable plausibility to Immanuel Kant’s teaching that mathematical reasoning is not ‘purely discursive’, but relies upon ‘constructions’ grounded in intuition. The new logic, however, made it possible to represent standard mathematical reasoning in the form of purely logical derivations – as Frege, on the one hand, and Russell, in collaboration with Whitehead, on the other, undertook to show in detail.

It is now generally held that logicism has been undermined by two developments: first, the discovery that the principles assumed in Frege’s major work are inconsistent, and the more or less unsatisfying character (or so it is claimed) of the systems devised to remedy this defect; second, the epoch-making discovery by Kurt Gödel that the ‘logic’ that would be required for derivability of all mathematical truths can in principle not be ‘formalized’. Whether these considerations ‘refute’ logicism will be considered further below.