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Single Idea 17698

[from 'On the Foundations of Logic and Arithmetic' by David Hilbert, in 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique ]

Full Idea

In the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used, for example in the notion of set, and to some extent also of number. Thus we turn in a circle, and a partly simultaneous development is required.

Gist of Idea

Logic already contains some arithmetic, so the two must be developed together

Source

David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.131)

Book Reference

'From Frege to Gödel 1879-1931', ed/tr. Heijenoort,Jean van [Harvard 1967], p.131


A Reaction

If the Axiom of Infinity is meant, it may be possible to purge the arithmetic from the logic. Then the challenge to derive arithmetic from it becomes rather tougher.