display all the ideas for this combination of texts
3 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) |
| 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. |
| 9113 | Just as unity is not a property of a single thing, so numbers are not properties of many things [William of Ockham] |
| Full Idea: Number is nothing but the actual numbered things themselves. Hence just as unity is not an accident added to the thing which is one, so number is not an accident of the things which are numbered. | |
| From: William of Ockham (Summa totius logicae [1323], I.c.xliv) | |
| A reaction: [William does not necessarily agree with this view] It strikes me as a key point here that any account of the numbers had better work for 'one', though 'zero' might be treated differently. Some people seem to think unity is a property of things. |