Timeline for Do there exist ideas which are true beyond axiom systems?
Current License: CC BY-SA 4.0
11 events
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| Jul 21, 2022 at 14:38 | comment | added | Mauro ALLEGRANZA | But yes, the sources of our mathematical theories and concepts must be human language and experience. We have some basic "intuitions" about collections of objects and parts of collections and we analyze and discuss them. At some point we find useful to try to manage them rigorously with mathematics. But then we have several related but different approaches: Set theory, Type theory, Mereology. | |
| Jul 21, 2022 at 10:54 | comment | added | Conifold | On the underlying intuitions and heuristic principles that feed ZFC set theory and its extensions see a nice expose by Maddy, Believing the axioms I and II. We cannot identify those truths "pre-emptively" because they are subtle, complex, only vaguely perceived until suitable conceptual apparatus is developed and correlated with prior knowledge, etc. Godel's results can be interpreted as implying that the process of formulating them is inexhaustible in principle. | |
| Jul 21, 2022 at 1:06 | comment | added | Double Knot | Indeed per your platonism penchant as expressed by Godel, there's fundamental truth out there than set theory which human mind may not be able to conceive exhaustively, so even ZFC turns out to be inconsistent it has nothing to do with math itself which under intuitionism it would be a much more serious problem as foundation of the whole math... | |
| Jul 20, 2022 at 19:56 | comment | added | David Gudeman | Any truth expressed in language contains implicit assumptions. An axiom systems is just an attempt to formalize some of those assumptions. | |
| Jul 20, 2022 at 17:34 | comment | added | Brian | I guess ideas relating to physical ideas we want to formalise through mathematics @Mauro ALLEGRANZA. For example the idea of having a set, taking its union intersection etc. I'm not quite sure about the idea of the dog. I'll think and reply | |
| Jul 20, 2022 at 13:25 | history | edited | Geoffrey Thomas♦ | CC BY-SA 4.0 |
'Do' replaces 'Does'
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| Jul 20, 2022 at 12:33 | comment | added | Mauro ALLEGRANZA | If the question is about "ideas", in what sense are you asserting that "ideas are true"? Is the idea of "dog" true? | |
| Jul 20, 2022 at 12:31 | comment | added | Brian | I understand but how does that relate to my question exactly ? @MauroALLEGRANZA | |
| Jul 20, 2022 at 12:06 | comment | added | Mauro ALLEGRANZA | In mathematics, we prove theorems starting from axioms and using logical rules. Outside mathematics, we have empirical fact whose knowledge is not based on "proof"; and we have other beliefs: art, society, religion. | |
| Jul 20, 2022 at 12:04 | history | edited | Mauro ALLEGRANZA |
edited tags
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| Jul 20, 2022 at 11:48 | history | asked | Brian | CC BY-SA 4.0 |