Timeline for Why do mathematicians prefer one definition over the other when they both define the same concept?
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14 events
| when toggle format | what | by | license | comment | |
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| Oct 2, 2021 at 19:20 | comment | added | Timothy Chow | I asked my synthetic/analytic question as a separate MO question. | |
| Sep 28, 2021 at 11:32 | comment | added | Timothy Chow | @DavidRoberts Do people really use the words "synthetic" and "analytic" for those examples? I don't think I have ever seen the word "synthetic" used in the context of axioms for the reals, probability theory, or quantum mechanics (can't speak to HoTT about which I know very little). | |
| Sep 28, 2021 at 7:47 | comment | added | HJRW | @DavidRoberts: oh, I see. I shall be careful about those usages in future if my audience might include philosophers. When talking to mathematicians, I think the distinction is unlikely to cause confusion. | |
| Sep 28, 2021 at 5:42 | comment | added | David Roberts♦ | Reals as Dedekind complete ordered field vs a construction, HoTT = synthetic homotopy theory vs presented by topological spaces, probability as she is done using random variables vs specifying everything to be a measurable function (cf blog posts of Terry Tao on this), quantum theory as axiomatised by collection of desired properties vs explicit models (matrix mechanics or wavefunctions).... | |
| Sep 28, 2021 at 3:05 | comment | added | Timothy Chow | @DavidRoberts Now you've got me wondering. The analytic/synthetic distinction in philosophy basically started with Kant, didn't it? Descartes was talking about analytic geometry long before Kant was even born. And Kant thought that all of math was synthetic a priori. So I'm not sure I believe that "analytic" is a term that comes from philosophy, if you have in mind the traditional analytic/synthetic debate. Also, is there any other example of "synthetic X versus analytic X" in math besides X = geometry? I can't think of one offhand. | |
| Sep 28, 2021 at 0:13 | comment | added | David Roberts♦ | @HJRW except then there are things like constructive definitions that are designed to work in the absence of excluded middle, and this is an orthogonal concern. "Analytic" is a term that comes from philosophy. Either way, terminology is going to be overloaded. Saying "defined by a construction" is definitely preferable to saying "defined constructively", in this instance. | |
| Sep 27, 2021 at 13:17 | comment | added | HJRW | FWIW, “constructive” seems a much better word for this than “analytic”. Free groups provide examples of objects that can be defined either via a universal property or “constructively”, in this case as the group of reduced words on an alphabet and its inverses. Calling the latter definition “analytic” would be extremely weird. | |
| Sep 27, 2021 at 12:37 | comment | added | David Roberts♦ | It is totally clear what you mean, I'm just quibbling about the use of a word in scare quotes (that normally has a entirely different meaning), when there seems to be a perfectly good alternative! :-) | |
| Sep 27, 2021 at 12:28 | comment | added | David Roberts♦ | Yes, it's clear, but that's what I would call an analytic definition: building a model of some axioms out of some other stuff, rather than postulating properties and working with them (=synthetic definition). Uniqueness is a separate issue, cf synthetic plane geometry (w/o parallel axiom) vs building models using coordinates etc. | |
| Sep 27, 2021 at 11:32 | comment | added | Timothy Chow | @DavidRoberts Is the example of the real numbers not clear enough? One proves the existence of an ordered field with the least upper bound property by using Dedekind cuts or Cauchy sequences. | |
| Sep 27, 2021 at 6:52 | comment | added | David Roberts♦ | I'm not sure how one proves existence except via building a model, or else something like the compactness theorem (?). Certainly the discussion and example you give seems analytic to me, but if you mean something else, then fair enough... | |
| Sep 27, 2021 at 0:28 | comment | added | Timothy Chow | @DavidRoberts "Constructive" is not the best word, I agree, but I don't think I mean "analytic" either. I almost want to say that one definition focuses on uniqueness and the other focuses on existence. | |
| Sep 26, 2021 at 23:35 | comment | added | David Roberts♦ | Instead of constructive I guess you mean "analytic" (the opposite to synthetic) | |
| Sep 26, 2021 at 16:21 | history | answered | Timothy Chow | CC BY-SA 4.0 |