Ideas from 'On the Foundations of Logic and Arithmetic' by David Hilbert [1904], by Theme Structure

[found in 'From Frege to Gödel 1879-1931' (ed/tr Heijenoort,Jean van) [Harvard 1967,0-674-32449-8]].

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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
The existence of an arbitrarily large number refutes the idea that numbers come from experience
                        Full Idea: The standpoint of pure experience seems to me to be refuted by the objection that the existence, possible or actual, of an arbitrarily large number can never be derived through experience, that is, through experiment.
                        From: David Hilbert (On the Foundations of Logic and Arithmetic [1904], p.130)
                        A reaction: Alternatively, empiricism refutes infinite numbers! No modern mathematician will accept that, but you wonder in what sense the proposed entities qualify as 'numbers'.
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Logic already contains some arithmetic, so the two must be developed together
                        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.
                        From: David Hilbert (On the Foundations of Logic and Arithmetic [1904], 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.