Ideas from 'Hilbert's Programme' by Georg Kreisel [1958], by Theme Structure
[found in 'Philosophy of Mathematics: readings (2nd)' (ed/tr Benacerraf/Putnam) [CUP 1983,052129648x]].
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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 nonmathematical concepts




Full Idea:
It is necessary to use nonmathematical 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.
