Timeline for What is the oldest open problem in mathematics?
Current License: CC BY-SA 4.0
24 events
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| Dec 31, 2022 at 12:30 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| May 11, 2014 at 20:14 | comment | added | Joël | @Todd We would absolutely not mind if you discard our comment (with this one, our two comments) in this thread. | |
| May 11, 2014 at 19:14 | comment | added | Todd Trimble | (Edited after 3 hours) I cleaned up by deleting some comments which in my opinion began to veer off-topic and cause tempers to rise, while trying to retain at least most of the core mathematical statements. A few hours have passed and I expect temperatures have lowered, but let's please stay strictly on-topic and strive to be courteous if you wish to say anything more -- thanks. | |
| May 11, 2014 at 18:25 | comment | added | user9072 | Your original answer in full read: "I was surprised not to find Zeno on this list. Giving a satisfactory mathematical account for resolving Zeno's paradox must surely be the oldest of all." What even is in your opinion the precise unresolved mathematical problem here? If you cannot pin this down, then surely somebody wondered how the starts move and so on before Zeno and surely there is some math there surely some of which can be considered as unresolved. | |
| May 11, 2014 at 16:07 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 11, 2014 at 15:53 | comment | added | Mikhail Katz | @Terry, OK, but nonetheless it cannot be dismissed as being of significance solely as far as physics is concerned. The fact that logicians of the caliber of Keisler are led to provide alternative accounts of it using a hyperreal continuum (based on the same foundations of ZFC) indicates that the paradoxes are richer than is often admitted. | |
| May 11, 2014 at 15:51 | comment | added | Terry Tao | I agree that Zeno's paradoxes have mathematical significance. I do not agree that they currently pose a mathematical open problem, which is the focus of the question under discussion. | |
| May 11, 2014 at 15:49 | comment | added | Mikhail Katz | @Terry, thanks for your comment. I was merely illustrating the mathematical significance of Zeno's paradoxes, a view you seem to share. This is not the place to argue for changing the said foundations, and I am not sure my professional qualifications as differential geometer allow me to argue in favor of changing foundations. The point is that some editors seem to believe that the said foundations provide ultimate answers, which is after all a hypothesis. | |
| May 11, 2014 at 15:48 | comment | added | Terry Tao | My point in the section containing the quote is that from a modern perspective, Zeno's paradoxes can be interpreted as a precursor to the modern theory of the continuum, by highlighting the need for a continuous axiom for the real numbers, as well as the need to specify higher order initial conditions in order for higher order equations of motion to be well posed. | |
| May 11, 2014 at 15:43 | comment | added | Terry Tao | The sentence of mine that you quote is only asserting that real analysis requires a continuous mathematical axiom, such as the Dedekind completeness axiom, in order to achieve a satisfactory theory. However, this axiom is certainly part of standard mathematical foundations, and I do not believe that any revision of these foundations is necessary in order to perform real analysis. | |
| May 11, 2014 at 15:02 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 11, 2014 at 14:27 | comment | added | Mikhail Katz | @Emil, I agree with what you wrote with the proviso that the passage from the problem to the model is not as mechanical as your answer tends to suggest. Often it is not clear what the right model is. Part of solving a math problem is determining what the best model is. To take an elementary example, one might think that the Extreme Value theorem is a well-defined topic whose truth value had been resolved long ago. Not so according to constructivists; see this recent article for a discussion. | |
| May 11, 2014 at 14:23 | comment | added | Emil Jeřábek | ... the paradox, this does not give an mathematical problem (which in any case would be new). It only shows that the physics problem to determine the appropriate model may not be resolved. In order for Zeno’s paradoxes to make the oldest problem of mathematics, you’d need a definite mathematical model of the situation proposed by the ancient Greeks, such that the mathematics of the paradoxes in this particular model is unsolved so far. I don’t see anything like that happening. | |
| May 11, 2014 at 14:17 | comment | added | Emil Jeřábek | Re rock: it splits into a physics problem to determine an adequate mathematical model of the physical reality in question (Newton’s laws, gravity, friction forces, ...), and a mathematical problem to figure out the trajectory in said model. Likewise for Zeno’s paradoxes, finding a mathematical model of the situation is a problem of physics, and working out the solution of the paradox in such a model is a problem of mathematics. Each model yields a different mathematical problem. In the recent literature you mentionn, if someone proposes a novel model of motion and immediately solves ... | |
| May 11, 2014 at 13:28 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 10, 2014 at 18:29 | comment | added | Mikhail Katz | @S.Carnahan, I don't think the mathematics can be separated from physics in this case. Furthermore, the view that Zeno's paradoxes have a mathematical aspect to them is a fairly common view in the literature. | |
| May 10, 2014 at 18:26 | comment | added | Mikhail Katz | @Emil, this is a problem of physics only to the extent that the trajectory of a rock thrown in the air is a problem of physics. Certainly the rock is a problem of both mathematics and physics. Furthermore, the Zeno paradoxes have a theoretical aspect about them (infinitely many steps, etc.) that would place it outside of the realm of physics as ordinarily conceived. | |
| May 9, 2014 at 9:57 | comment | added | Emil Jeřábek | I tend to think that the possibility of various ways to model mathematically the idea of motion make it an open problem of physics rather than mathematics. | |
| May 9, 2014 at 8:39 | comment | added | Mikhail Katz | @S.Carnahan, this is precisely my point. Since there are various ways to model mathematically the idea of motion, there is more than one possibility of accounting for the paradox. The uncertainty principle you mentioned is certainly a different way of accounting for it from the infinite series approach suggested by HJRW. The fact is that there are several recent mathematical papers providing yet a different account of the paradox. | |
| May 9, 2014 at 8:35 | comment | added | S. Carnahan♦ | The Wikipedia page does not suggest anywhere that the mathematical content of the paradox is unresolved. Any remaining problems seem to involve the question of how motion in the physical world should be described. Heisenberg's Uncertainty Principle strongly indicates that infinitely dividing the trajectory of a real-world moving object is not a meaningful thing to do. | |
| May 9, 2014 at 7:32 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 9, 2014 at 7:19 | comment | added | Mikhail Katz | @HJRW, whether or not this has been resolved is certainly a matter of dispute; for a discussion see e.g., the wiki page I linked. | |
| May 8, 2014 at 15:39 | comment | added | HJRW | OK, I'll bite. Surely this is resolved by the notion of convergence of infinite series? | |
| May 8, 2014 at 15:12 | history | answered | Mikhail Katz | CC BY-SA 3.0 |