Timeline for Are all mathematical theorems necessarily true?
Current License: CC BY-SA 2.5
31 events
| when toggle format | what | by | license | comment | |
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| Apr 14, 2019 at 19:30 | review | Close votes | |||
| Apr 15, 2019 at 9:31 | |||||
| Feb 13, 2017 at 23:53 | review | Close votes | |||
| Feb 14, 2017 at 1:33 | |||||
| Jan 29, 2010 at 19:34 | vote | accept | Paul Mickus | ||
| Jan 29, 2010 at 10:09 | answer | added | Neel Krishnaswami | timeline score: 2 | |
| Jan 29, 2010 at 9:02 | answer | added | Charles Stewart | timeline score: 2 | |
| Jan 28, 2010 at 22:43 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
Added math-philosophy tag
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| Jan 28, 2010 at 21:48 | comment | added | Dan Piponi | A sentence of ZF (say), being a sentence of first order logic, has no free variables. So it makes no sense to talk of truth-assignment does it? | |
| Jan 28, 2010 at 21:45 | answer | added | Joel David Hamkins | timeline score: 18 | |
| Jan 28, 2010 at 21:19 | comment | added | Paul Mickus | It's funny my original ambiguous question might be considered a simple mathematical question. Yet when I try to be more precise it falls into philosophy. :( | |
| Jan 28, 2010 at 21:15 | comment | added | Konrad Swanepoel | @some guy: your answer and François' reference to Solovay shows that there is some connection to mathematics here. But I guess it's going to be closed soon. | |
| Jan 28, 2010 at 21:14 | comment | added | Yemon Choi | @someguyonthestreet The question is whether or not the present question can be made into a precise one within modal logic or model theory, while still remaining the question which the original poster wanted answered. Formalizing a problem of philosophy in mathematics can have the side effect of defining away the original problem | |
| Jan 28, 2010 at 21:11 | comment | added | Konrad Swanepoel | @Paul: Perhaps the color of my shirt is necessary? Of course we don't think/believe/hope so, but who knows? It's just easier to imagine a different shirt color than it is to imagine that $2+2\neq 4$, but that could just be a failure of imagination. | |
| Jan 28, 2010 at 21:10 | comment | added | some guy on the street | very odd; a down-voted question with an up-voted answer, and lots of activity... @John Goodrick, the absence withing philosophy of a consensus on "possible worlds" doesn't encourage me to move a question over to the philosophers! If a question admits a precise interpretation and precise answer within modal logic or model theory, etc., then I think it's a decent mathoverflow question, too. | |
| Jan 28, 2010 at 21:08 | comment | added | Paul Mickus | John: Yeah you might be right. However many philosophical problems later became formalized in mathematical language. And surely asking a philosopher this question will only result in more philosophy! | |
| Jan 28, 2010 at 21:03 | comment | added | Paul Mickus | Oh very sorry some guy on the street :( Yeah maybe instead of possible worlds... we can think of them as accessible worlds. For instance a world that is like ours but some of the facts about it are changed. So maybe instead of wearing a blue shirt today I wear a white one. However if "2+2=4" was not true this world would be vastly different and inaccessible (impossible?). This perhaps makes it more clear what contingent and necessary mean. | |
| Jan 28, 2010 at 20:59 | comment | added | John Goodrick | Stalyn/Paul: I still think that mathoverflow is not the place for this question. The last I heard, there was no consensus in the philosophical community as to what exactly a "possible world" is, or whether this is even a coherent notion, or whether "necessity" really does mean "true in all possible worlds." Unless you mean to ask a precise question about first-order modal logic, then you'd be better off asking this question to philosophers. | |
| Jan 28, 2010 at 20:57 | comment | added | Konrad Swanepoel | @some guy on the street: it still deserves a +1. | |
| Jan 28, 2010 at 20:56 | comment | added | some guy on the street | oh dear! the question changed before I answered it! | |
| Jan 28, 2010 at 20:55 | answer | added | some guy on the street | timeline score: 1 | |
| Jan 28, 2010 at 20:53 | comment | added | Konrad Swanepoel | I'm deleting my answer, because it doesn't answer your rephrased question. So let's assume we are using classical logic. Then regarding the fundamental theorem of algebra, when written formally in for example the first order theory of fields, it has no free variables. So the fact that it follows from whatever axioms you use to fix the complex numbers, is a tautology. The theorem itself is then only a tautology if you give some interpretation to the axioms that will make them "necessarily true". To use your definition, you need to understand all possible worlds. What is possible? | |
| Jan 28, 2010 at 20:48 | comment | added | Paul Mickus | Aha yeah I agree possible worlds is still vague. Thank you very much for the article suggestion. | |
| Jan 28, 2010 at 20:44 | comment | added | François G. Dorais | I think your question is still a little vague from the strictly mathematical standpoint since it's unclear what the possible worlds might be. I think a reasonable answer to your question is given in Solovay's Provability interpretations of modal logic (Israel J. Math 25, 1976). However, I can't convince myself that this answer is correct in all possible worlds... | |
| Jan 28, 2010 at 20:32 | comment | added | Paul Mickus | Yes I have read Kripke... I think my question now rephrased makes more sense. It a mathematical question or at least of some mathematical importance. | |
| Jan 28, 2010 at 20:22 | history | edited | Paul Mickus | CC BY-SA 2.5 |
added 497 characters in body; edited title
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| Jan 28, 2010 at 20:13 | comment | added | John Goodrick | Stalyn: asking a bunch of mathematicians will probably not enlighten you on the question of "what is a necessary truth as opposed to a contingent truth?"; it would be much better to ask a local friendly philosophy student, or read Kripke's Naming and Necessity, or look on the Stanford Encyclopedia of Philosophy. | |
| Jan 28, 2010 at 19:54 | comment | added | Paul Mickus | There is no truth-value assignment for P where P or not P is not true. This is a tautology and necessarily true. But consider a more complex statement, for example the fundamental theorem of algebra. This does not seem to be a formal tautology as there exists truth-value assignments where the statement is false. But surely it seems to be necessarily true. Could the fundamental theorem of algebra be false in a world where it exists? | |
| Jan 28, 2010 at 19:42 | comment | added | Konrad Swanepoel | Is the statement "$P$ and not $P$" necessarily true for all statements $P$? | |
| Jan 28, 2010 at 19:38 | comment | added | Paul Mickus | Define 'necessarily true' as a the statement that is true in all possible worlds (where such a statement could exist). Maybe the questions is rather is all mathematical theorems necessarily true? | |
| Jan 28, 2010 at 19:33 | comment | added | Mariano Suárez-Álvarez | I would say that the question does not make sense as stated. True statements mathematical do not stop being true, so 'always true' is mostly a pleonasm, and 'necessarily true'... well, it would need at least context to become meaningful. | |
| Jan 28, 2010 at 19:28 | comment | added | Yemon Choi | I suspect that in proof theory, "tautology" might have a more restricted and precise meaning than the one you use here. Are you asking about the colloquial use of the word? Sometimes mathematicians say something is "merely a tautology", which might I guess be a misuse of language | |
| Jan 28, 2010 at 19:25 | history | asked | Paul Mickus | CC BY-SA 2.5 |