display all the ideas for this combination of texts
4 ideas
| 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) |
| 18518 | Infinite numbers are qualitatively different - they are not just very large numbers [Heil] |
| Full Idea: It is a mistake to think of an infinite number as a very large number. Infinite numbers differ qualitatively from finite numbers. | |
| From: John Heil (The Universe as We Find It [2012], 03.5) | |
| A reaction: He cites Dedekind's idea that a proper subset of an infinite number can match one-one with the number. Respectable numbers don't behave in this disgraceful fashion. This should be on the wall of every seminar on philosophy of 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. |
| 18500 | How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil] |
| Full Idea: I do not understand how structures could serve as truthmakers for mathematical truths, ...Mathematical truths are not true in virtue of any way the universe is. ...Mathematical truths hold, whatever ways the universe is. | |
| From: John Heil (The Universe as We Find It [2012], 08.08) | |
| A reaction: I like the idea of enquiring about truthmakers for mathematical truths (and my view is more empirical than Heil's), but I think it may be a misunderstanding to think that structures are intended as truthmakers. Mathematics just IS structures? |