Timeline for What is the high-concept explanation on why real numbers are useful in number theory?
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| Nov 15, 2019 at 15:19 | comment | added | Kapil | Any proof in number theory that involves real numbers uses them to estimate a certain interval where some number lies. This estimate can always be weakened a bit to use only rational numbers. In fact, one could give a course in "Real Analysis" that uses only intervals and such intervals could be limited to those with rational boundaries. Thus, with a lot of effort, real numbers could be eliminated from the discussion entirely. Is it worth the effort? Perhaps, if one wants to make things "computationally effective". | |
| Nov 15, 2019 at 9:44 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Nov 15, 2019 at 1:35 | answer | added | Olivier | timeline score: 1 | |
| Nov 14, 2019 at 22:10 | answer | added | Timothy Chow | timeline score: 4 | |
| Jan 11, 2012 at 2:14 | vote | accept | Daniel Moskovich | ||
| Apr 17, 2011 at 23:29 | answer | added | John Stillwell | timeline score: 10 | |
| Apr 17, 2011 at 16:00 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
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| Apr 15, 2011 at 12:01 | comment | added | Daniel Moskovich | @Lubin: I'm arguing not that mathematics should be fragmented; rather that everything we do in mathematics should have a clearly defined/understood purpose, including the choice of tools used in a proof. | |
| Apr 15, 2011 at 2:26 | comment | added | Ramsey | Apropos the "utopian" comment/debate, I'm finding myself of two minds that I cannot seem to disentangle. As an example relevant to this thread, I'd have to say that I'm fine with the idea of solving a global problem by local means (eg looking at congruence to solve diophantine problems) but somehow less comfortable with the opposite (eg the Harris-Taylor proof of local langands). I really can't exactly characterize why. Perhaps a measure of canonicality? Or maybe "upstructuring" vs "downstructuring"? Can anyone help diagnose my affliction? | |
| Apr 15, 2011 at 1:02 | comment | added | Ryan Budney | Related thread: math.stackexchange.com/questions/5303/… | |
| Apr 14, 2011 at 23:24 | comment | added | user9072 | @Daniel Moskovich: I tried to give a selection of examples, some noninteresting to make a point and some perhaps interesting. Perhaps, I managed to include something you like; in any case, I found the task of doing so interesting. So thanks for the question. | |
| Apr 14, 2011 at 22:43 | comment | added | Daniel Moskovich | @unknown(google): ... where XYZ is "the dual/associated/whatever something isn't an integer, and it would be artificial to force it to be so". That would be one thing which would make me happy. | |
| Apr 14, 2011 at 22:42 | comment | added | Daniel Moskovich | @unknown(google): Your answer is excellent! I'm just a bit slow... Regarding your request, I don't know if this is the best example, but let's say Poincare duality in the simplicial world is most easily understood by passing to the more general setting of CW complexes, because the dual cell structure isn't a triangulation, and to triangulate would be artificial. So a "best" proof of Poincare duality for simplicial complexes would pass through CW complexes. I want to see something analogous for integers, in a clear simple context- it's most natural to adjoin reals because XYZ | |
| Apr 14, 2011 at 22:11 | comment | added | user9072 | May I ask that you provide some example of what you envision in any context whatsoever. That is an example of an application of (truly, so excluding what you called formal constants) real numbers wherever you like in which one can not more or less trivially poke holes by saying 'Look just successively approximate this. That it will be a total mess then, I don't mind.' That being said I will try one more update of my answer. | |
| Apr 14, 2011 at 21:22 | history | edited | Daniel Moskovich | CC BY-SA 3.0 |
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| Apr 14, 2011 at 19:38 | answer | added | Charles | timeline score: 9 | |
| Apr 14, 2011 at 19:33 | answer | added | anonymous | timeline score: 12 | |
| Apr 14, 2011 at 16:51 | answer | added | François Brunault | timeline score: 8 | |
| Apr 14, 2011 at 15:17 | answer | added | stankewicz | timeline score: 4 | |
| Apr 14, 2011 at 14:42 | answer | added | GH from MO | timeline score: 18 | |
| Apr 14, 2011 at 12:20 | answer | added | BSteinhurst | timeline score: 3 | |
| Apr 14, 2011 at 10:56 | comment | added | user9072 | @Quanta: what do you mean by 'in a finite way' precisely? Anyway, let $s(x)$ denote the number of natural numbers up to size $x$ not divisible by the square of a prime number (squaree free numbers). Then $s(x)/x$ converges to $6/\pi^2$. Also see Gerry Myerson's answer. | |
| Apr 14, 2011 at 10:05 | comment | added | Quanta | Can you please give some examples of real numbers being useful in number theory? Hopefully not algebraic numbers because they can be constructed in a finite way unlike the other trancendental real numbers.. | |
| Apr 14, 2011 at 9:39 | answer | added | François G. Dorais | timeline score: 80 | |
| Apr 14, 2011 at 9:37 | answer | added | user9072 | timeline score: 8 | |
| Apr 14, 2011 at 8:33 | comment | added | S. Carnahan♦ | What you suggest would be a utopia for people who don't want to learn new tools. We seem to have strong experimental evidence that fluency in one area is not a good way to make leaps in our ability to prove or understand mathematics. The question is somewhat related to the problem of whether tersely stated theorems should have short proofs - if the length of proofs were bounded by a computable function in the length of the theorem, we could check any mathematical statement in finite time. | |
| Apr 14, 2011 at 8:14 | answer | added | Sidney Raffer | timeline score: 19 | |
| Apr 14, 2011 at 5:26 | comment | added | Lubin | The real moral of the story is that there aren't separate ``worlds'' in mathematics, but rather a beautiful and often mysterious all-embracing unity. | |
| Apr 14, 2011 at 5:15 | answer | added | Ramsey | timeline score: 6 | |
| Apr 14, 2011 at 4:08 | answer | added | Gerry Myerson | timeline score: 8 | |
| Apr 14, 2011 at 3:30 | answer | added | Faisal | timeline score: 27 | |
| Apr 14, 2011 at 3:29 | comment | added | Daniel Moskovich | @Mariano: The hypothetical argument states possible existence of "correctly tooled" arguments, even ones which aren't known and won't be known for the next million years. | |
| Apr 14, 2011 at 3:13 | answer | added | Qiaochu Yuan | timeline score: 8 | |
| Apr 14, 2011 at 3:09 | comment | added | Mariano Suárez-Álvarez | Well, the hypothetical argument is pretty much against reality... doesn't that refute it by itself? One can also come up with «maybe partial differential equations are the wrong tool for the Poincaré conjecture, and maybe someone will come up with a purely topological argument and win a Fields^2 medal for it», and so on. The thing is, such alternative, correctly tooled arguments are slow in the coming! | |
| Apr 14, 2011 at 3:05 | comment | added | Qiaochu Yuan | I think Alon is disagreeing with the sentiment of the first clause of the first sentence rather than the entire first sentence. | |
| Apr 14, 2011 at 2:50 | comment | added | Daniel Moskovich | @Alon: "unless there were a good conceptual reason for the contrary". Are you arguing that statements and proofs should not live in the same world, and moreover, that the reasons for this should be unclear? | |
| Apr 14, 2011 at 2:45 | comment | added | Alon Amit | This is totally tangential but I just want to say I strongly disagree with the sentiment of the first sentence. If statements and proofs were typically "in the same world", math would be so dull. Dystopia. | |
| Apr 14, 2011 at 2:40 | history | asked | Daniel Moskovich | CC BY-SA 3.0 |