Ideas from 'Hilbert's Programme' by Georg Kreisel [1958], by Theme Structure

[found in 'Philosophy of Mathematics: readings (2nd)' (ed/tr Benacerraf/Putnam) [CUP 1983,0-521-29648-x]].

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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite

17809

Gödel showed that the syntactic approach to the infinite is of limited value

			Full Idea: Usually Gödel's incompleteness theorems are taken as showing a limitation on the syntactic approach to an understanding of the concept of infinity.
			From: Georg Kreisel (Hilbert's Programme [1958], 05)

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics

17810

The study of mathematical foundations needs new non-mathematical concepts

			Full Idea: It is necessary to use non-mathematical concepts, i.e. concepts lacking the precision which permit mathematical manipulation, for a significant approach to foundations. We currently have no concepts of this kind which we can take seriously.
			From: Georg Kreisel (Hilbert's Programme [1958], 06)
			A reaction: Music to the ears of any philosopher of mathematics, because it means they are not yet out of a job.

27. Natural Reality / C. Space / 3. Points in Space

17811

The natural conception of points ducks the problem of naming or constructing each point

			Full Idea: In analysis, the most natural conception of a point ignores the matter of naming the point, i.e. how the real number is represented or by what constructions the point is reached from given points.
			From: Georg Kreisel (Hilbert's Programme [1958], 13)
			A reaction: This problem has bothered me. There are formal ways of constructing real numbers, but they don't seem to result in a name for each one.