Timeline for Taking a theorem as a definition and proving the original definition as a theorem
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2021 at 15:07 | comment | added | Aditya Guha Roy | I would like to add one more (which I essentially picked up from you :-) ): complex numbers are also a part of this list; they were initially meant to be the algebraic completion of reals, but this is not the usual definition of complex numbers we see these days and we only prove this as the Fundamental Theorem of Algebra. | |
| Sep 29, 2021 at 15:28 | comment | added | J. van Dobben de Bruyn | To add to you 5th example, also at some point there used to be a distinction between abstract $C^*$-algebras, concrete $C^*$-algebras (of operators), and $B^*$-algebras. Nowadays we know these to be equivalent, and call them all $C^*$-algebras. | |
| Sep 28, 2021 at 14:37 | comment | added | Timothy Chow | Perhaps Ostrowski's theorem is another example of this type. I remember taking a course from Goro Shimura in which he said he didn't think that Ostrowski's theorem was that important because there was no real need to have a rigorous proof that no other examples existed. I would be curious to know if Ostrowski's theorem has any applications. | |
| Sep 28, 2021 at 3:32 | comment | added | Terry Tao | That is the primary benefit to be sure, but there are some cases where knowing that a concrete representation exists can greatly simplify the proof of other useful theorems. Some examples are given at mathoverflow.net/questions/101061/proof-by-universal-receiver | |
| Sep 28, 2021 at 3:16 | comment | added | liuyao | IMO, these fundamental theorems don't serve as big a role as we often credit them. It's more a "peace of mind" that the abstract definition that supposedly generalizes the concrete examples doesn't give us more. (A counter-example is Lie groups, which do include non-matrix ones.) The benefit of the abstract definition is that it lets us see more clearly what is essential and what is subject to the particular presentation (which is useful for calculation). | |
| Sep 27, 2021 at 15:42 | history | edited | Terry Tao | CC BY-SA 4.0 |
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| S Sep 27, 2021 at 15:23 | history | answered | Terry Tao | CC BY-SA 4.0 | |
| S Sep 27, 2021 at 15:23 | history | made wiki | Post Made Community Wiki by Terry Tao |