Timeline for Are there applications of category theory to countable sets?
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| Feb 24, 2023 at 11:01 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
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| Jul 1, 2010 at 3:54 | comment | added | KConrad | Andrew, you seem to be hung up on category theory vs. set theory. They're not incompatible, as Andy wrote, and your instructors (undergraduate?) were working implicitly within ZFC since they wanted to teach you something about the rest of math, so you shouldn't blame them. You're new to diagram chasing, and we all were at one point. You'll get used to it; everyone else did. The point is not to treat the integers or other objects in number theory in isolation, but in a category where the universal qualities of interest are revealed. Nobody does category theory only on the natural numbers. | |
| Jul 1, 2010 at 2:50 | comment | added | The Mathemagician | @KConrad I'm well aware of the applications in algebra and topology and I've begun a serious study of MacLane's classic.I guess I've never really been comfortable with diagram chasing since I was originally trained by old fashioned mathematicians who used naive set theory to formulate everything,even algebra. I finally got somewhat used to using it for very large systems,like the category of compact topological spaces and Abelian groups-but for objects like the natural numbers,it seems a little strange. | |
| Jul 1, 2010 at 2:48 | answer | added | Boyarsky | timeline score: 12 | |
| Jul 1, 2010 at 2:44 | comment | added | Andy Putman | One confusion that might be lurking here (and this is only a guess, so I might be totally off base) is your opposition between category theory on one hand and ZFC style set theory on the other. The two are not incompatible! While some category theorists promote it as an alternate set of foundations to ZFC set theory, I don't think that's what most people mean when they talk about using it in number theory, alg geometry, etc. Rather, it is an extremely expressive language for phenomena that reside entirely within standard, ZFC mathematics (indeed, often entirely in the finite world!) | |
| Jul 1, 2010 at 2:34 | comment | added | KConrad | Andrew, there are important constructions in number theory, algebraic geometry, and elsewhere that are impossible to make sense of without category theory, so it's not a wrecking ball. Keep in mind that number theory is concerned with structures related to the integers, not only the integers themselves (kind of like real analysis is not the study of real numbers). Please ask your friend for some suggested books or articles that can inform you about the themes which interest him, so you can get a better idea of the hard questions that he would like to study. | |
| Jul 1, 2010 at 2:03 | answer | added | Joel David Hamkins | timeline score: 10 | |
| Jul 1, 2010 at 1:58 | history | made wiki | Post Made Community Wiki by The Mathemagician | ||
| Jul 1, 2010 at 1:58 | comment | added | The Mathemagician | @KConrad The finite groups-particularly the Abelian ones-are a very good example indeed. It's just that the use of such concepts seem to be tantamount to swatting flies with a wrecking ball.But of course,just because you CAN approach the integers via set theoretic constructions doesn't mean you HAVE to,does it? | |
| Jul 1, 2010 at 1:19 | comment | added | Gerry Myerson | Some find it puzzling that the logarithm has any applications in number theory - but, it does. The countability of the primes has not discouraged the zeta function from having deep things to say about them, even though one must go well beyond the countable and indeed beyond the real to grok zeta. | |
| Jul 1, 2010 at 1:16 | comment | added | KConrad | Andrew, almost any functorial construction you can imagine is useful in number theory. Just open up a book on the subject at the graduate level. And look at finite groups: they're finite but still homological concepts are quite useful there (group cohomology?). Lebesgue said once that if you put everything on an equal footing, without choosing among everything that is exact, then you'd hardly think of many useful concepts. You have to treat the natural numbers as something more than just a countable set to get somewhere in understanding them. | |
| Jul 1, 2010 at 0:36 | comment | added | Yemon Choi | This seems a bit like asking why/how exotic algebraic constructions can shed light on, say, Diophantine problems. My own very paltry understanding, when it comes to the use of categorical methods, is that the category theory really plays an important role in understanding the hidden structures that lie underneath seemingly elementary statements and questions in number theory. So I think your last paragraph, while an understandable and natural POV, is somewhat missing the point. | |
| Jul 1, 2010 at 0:10 | history | asked | The Mathemagician | CC BY-SA 2.5 |