Timeline for Categories First Or Categories Last In Basic Algebra?
Current License: CC BY-SA 2.5
26 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 8, 2011 at 16:14 | comment | added | Todd Trimble | Thanks, Greg. I sort of want to ask what works of category theory you find silly -- and it's not because I disagree with you here -- I'm just curious what you find silly. Please feel free to write me at topological at musings dot gmail at com (interchanging at and dot). Best wishes - | |
| Nov 8, 2011 at 16:06 | comment | added | Greg Kuperberg | @Todd I can't really disagree with you. Yes, I was being silly. Yes, the more serious point is that you should learn category theory as you need it. My one specific defense of being silly here is that one can certainly find category theory work that has lost all sense of purpose and is itself silly. (Also a general defense of silliness isn't needed --- we're all entitled sometimes.) | |
| Nov 8, 2011 at 15:26 | comment | added | Todd Trimble | I've put off saying this for a long time, but with all due respect to Greg, his answer here is really silly. The significance of category theory is such that likening it to "a flavor", a taste sensation, is ridiculous. In the right hands, it gives ways of organizing and simplifying large tracts of mathematics, making the learning and deployment of mathematics more economical and effective. So much so that taking time to study it on its own terms can have huge benefits. Perhaps what you really mean to say (Greg) is that it should be learned on a need-to-know basis? That, I would agree with. | |
| Dec 21, 2010 at 18:45 | comment | added | The Mathemagician | I love garlic,personally. | |
| Oct 15, 2010 at 11:44 | comment | added | Harry Gindi | @Greg: I'm not sure if you've read this before, but Monique Hakim's thesis on (locally) ringed toposes and relative schemes is really a work of art. It starts out with a whole bunch of abstract nonsense, but by the first third of the book, you start to see some really beautiful geometry come right from the definitions. | |
| Oct 7, 2010 at 7:14 | comment | added | Greg Kuperberg | The issue with any book on category theory is not just that it's abstract. For instance, Hartshorne's GTM, Algebraic Geometry, is just as abstract as MacLane, but its impact is very different. The issue is that, as wonderful as category theory is for everyone else, it doesn't stand very well on its own. The best category theory is done for some purpose other than just category theory. So, MacLane is a fine book, I don't regret buying it at all. But as I suggested, I use it like a jar of garlic, only so much at one time. | |
| Oct 7, 2010 at 1:40 | comment | added | The Mathemagician | @ All CWM is a very well written,but overwhelming book.I'm very glad MacLane wrote 2 editions of it,but it's certainly not a book I look forward to reading at length.That's not really MacLane's fault,it's the nature of the subject.The level of sheer abstraction just makes one's eyes glaze over after 40-50 pages of trying. Adwodey's book is much more pleasant and very much in the same spirit-the student very much followed his teacher's footsteps in composing it,but he leads the reader up a much less steep slope of Mount Category. | |
| Oct 7, 2010 at 1:14 | comment | added | Andy Putman | @Steven : I don't mind it that much myself, though I have to admit that I've only used it as a reference. I can't imagine wanting to read an entire book on category theory! My opinion is that the tiny amount of category theory the average mathematician needs is best picked up along the way while learning something more exciting. | |
| Oct 6, 2010 at 2:11 | comment | added | Steven Gubkin | I actually really enjoyed Categories for the Working mathematician when I read it as an undergrad. I do not know why people think it is boring. | |
| Oct 6, 2010 at 1:16 | comment | added | The Mathemagician | @All I think the relectuance to fully accept category theory is the same reluctance the psychological community had with behaviorism:It seems very counterintuitive to some people to have a methodology where one does not "gets one's hands dirty"with an explicit object of study. | |
| Oct 6, 2010 at 0:17 | comment | added | Ryan Budney | @Tim Porter: I think people are making your point, just with a slightly different emphasis. Well, all except the Hob Goblin part. I don't think there's any fixation - categories are simply the focus of the thread. | |
| Oct 5, 2010 at 19:55 | comment | added | The Mathemagician | @Andy I'm forbidden by moderator edict to reply to said person's opinions.But personally,I prefer Adowey's book to learn category theory followed by MacLane. | |
| Oct 5, 2010 at 19:14 | comment | added | Harry Gindi | @Andy: It appears that I have a soul after all (and that it can be crushed by reading Mac Lane's CWM). | |
| Oct 5, 2010 at 18:33 | comment | added | Andy Putman | @Harry : Is what I'm seeing actually true? Is Harry Gindi complaining that some part of mathematics is too abstract and boring? First BCnrd made an unnecessary technical assumption in another thread, and now this. Have I been magically teleported to Htrae or something? | |
| Oct 5, 2010 at 11:57 | comment | added | Harry Gindi | @Greg: I agree. I think the best way to learn category theory is to see it through Lang, Hartshorne, etc. first, then go back to something like Mac Lane to fill in details. I can't read Mac Lane all the way through. It's too boring. | |
| Oct 5, 2010 at 10:50 | comment | added | Greg Kuperberg | Just to be clear, I like garlic a lot. Just not by itself. So it's not glamorous like Gauss' analogies were. That's okay; everything about Gauss was wonderful, but he liked to be a bit pompous. I stand by my analogy. On one occasion, I read about half of MacLane's book, "Categories for the Working Mathematician". I liked it a lot, but I was left with the feeling that I had eaten too much garlic. This book has a zillion definitions and somehow isn't as deep as it first seems. | |
| Oct 5, 2010 at 7:36 | comment | added | Harry Gindi | @Tim: You shouldn't use caps for emphasis (although some people here do it, it is generally frowned upon). Instead use italics, which can be gotten by surrounding the text with asterisks or underscores. | |
| Oct 5, 2010 at 6:54 | comment | added | Tim Porter | @All continued. Use category theory freely without too much of a song and dance, then introduce is more formally as needs be FOR YOUR aims and objects in the course. | |
| Oct 5, 2010 at 6:46 | comment | added | Tim Porter | @All Why this fixation on category theory as if it is a Hob Goblin lurking to pounce on the unwary algebraist or topologicst? (I know many of you do not feel that for yourself but there is a reluctance to admit it.:-) with notable exceptions of course.) Why not use categorical arguments lightly when they make the proofs clearer in your opinion and avoid them when they require too much machinery. When teaching a course on knots and surfaces, so partially in topology but with some combinatorial group theory there as well, I talked about the product topology using the idea of a product. | |
| Oct 5, 2010 at 1:17 | comment | added | The Mathemagician | @All cont.: The "nuts-and-bolts" approach has the advantage of being explicit in the composition of the objects it postulates and why they have the properties they do follows directly and clearly from this construction.For example,there's no mystery what a function is if you understand what a Cartesian product is.The problem is the constructions get extremely complicated and difficult sometimes to accomplish explicitly.The "behavioral" approach has the advantage of simplicity and large scale organization-but that many objects remain "black boxes" that can ONLY be defined in terms of relations. | |
| Oct 5, 2010 at 1:10 | comment | added | The Mathemagician | @All I view category theory as a "top down" approach to the foundations of mathematics,whereareas traditional set theory is a "bottom up" approach. The best analogy I've been able to come up with draws from psychology/cognitive science: Category theory and set theory are as models of mathematical structures what mature Skinnerian behaviorism and nuerobiology are as models of human behavior.Nuerobiology and set theory construct the models from the nuts and bolts of thier domains while CT and SB ignore all that and focus only on the relations between its components as the only relevant aspects. | |
| Oct 5, 2010 at 0:05 | comment | added | Ryan Budney | Well, it's certainly not the Durian of mathematics. | |
| Oct 4, 2010 at 23:38 | comment | added | Terry Tao | Gauss called number theory the queen of mathematics. Calling category theory the garlic of mathematics somehow doesn't have the same level of glamour... | |
| Oct 4, 2010 at 22:10 | comment | added | Steven Gubkin | (I mean this literally) | |
| Oct 4, 2010 at 22:10 | comment | added | Steven Gubkin | I have been known to eat raw garlic, and this has not won me many friends. | |
| Oct 4, 2010 at 20:26 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |