Combining Philosophers

All the ideas for Anaxarchus, Georg Kreisel and Paolo Mancosu

unexpand these ideas     |    start again     |     specify just one area for these philosophers

5 ideas

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 [Kreisel]
     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 [Kreisel]
     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.
9. Objects / D. Essence of Objects / 7. Essence and Necessity / b. Essence not necessities
13166 Essences are no use in mathematics, if all mathematical truths are necessary [Mancosu]
     Full Idea: Essences and essential properties do not seem to be useful in mathematical contexts, since all mathematical truths are regarded as necessary (though Kit Fine distinguishes between essential and necessary properties).
     From: Paolo Mancosu (Explanation in Mathematics [2008], §6.1)
     A reaction: I take the proviso in brackets to be crucial. This represents a distortion of notion of an essence. There is a world of difference between the central facts about the nature of a square and the peripheral inferences derivable from it.
13. Knowledge Criteria / D. Scepticism / 1. Scepticism
3061 Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius]
     Full Idea: Anaxarchus said that he was not even sure that he knew nothing.
     From: report of Anaxarchus (fragments/reports [c.340 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.10.1
27. Natural Reality / C. Space / 3. Points in Space
17811 The natural conception of points ducks the problem of naming or constructing each point [Kreisel]
     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.