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Julius Hamilton (May 24 2024 at 13:01):Curious if anyone experiences “having too many questions at the same time” and if any lessons or strategies were learned as one gained more experience in mathematics.
For example:
I want to explore some questions by drawing out a bunch of diagrams, but find drawing that many diagrams inefficient.
So I decide I want a convenient way to draw diagrams on my phone. I envision an application I could code where I can generate diagrams through functions, rather than manually specifying every object and arrow.
I realize such an application would be useful, but would require a lot of thinking to design effectively. It opens the question of what kinds of functions I should enable.
It makes me ask questions about the nature of diagrams in general. I want to know if there’s a systematic way to list all “diagram functions”, so I can be sure to discover any interesting ones I hadn’t thought of, not just a few obvious ones, like reversing the direction of the arrows, the pushout, a span, etc.
Then I find myself wanting to explore past questions I’ve had about using enumeration to list all examples of a class of things up to a finite value. In order to do this, I feel like I have unresolved questions in topics like logic and recursion I need to study first.
Just curious if other people experience this.
Ryan Wisnesky (May 24 2024 at 13:52):certainly 'the need to know' coupled with 'it takes year to understand' and 'the more you know, the more you know you don't know' sends people to grad school for long, long periods of time... fwiw, 'a desire to automate math' has been behind so much of the development of programming and computer science.
John Baez (May 24 2024 at 14:29):I feel a key reason I learned as much math as I did is that I didn't get sidetracked by trying to create software, etc.
Todd Trimble (May 24 2024 at 14:30):Then I find myself wanting to explore past questions I’ve had about using enumeration to list all examples of a class of things up to a finite value. In order to do this, I feel like I have unresolved questions in topics like logic and recursion I need to study first.
These things, such as finding all possible category structures on sets with at most five morphisms or some such, have a tendency to be insanely complicated and the number of them is huge, so in terms of learning category theory, I'm afraid that wouldn't be a great use of time.
Todd Trimble (May 24 2024 at 14:31):But on the bright side, for discussing limits and colimits of diagrams, it's nice that many questions are reducible to special cases of diagrams, and you are probably aware of most of them already. "All limits can be constructed from products and equalizers": if you can thoroughly describe exactly and in complete generality why that is, then that is a valuable insight. Same with "all limits can be constructed from a terminal object and wide pullbacks". A little further down that road, knowing for example that (in the category of sets) reflexive coequalizers commute with finite products, or that filtered colimits commute with finite limits -- all really good things to know. But it usually comes down to specific classes of diagrams, and the number of classes isn't exorbitant, I don't think.
Todd Trimble (May 24 2024 at 14:31):Of course, the list of "things to know" can be enormous and daunting. But for me, it usually comes down to "things I need to know", which is actually a different process. Typically, what I (organically) need to know will make itself known when I'm trying to tackle specific problems, not just trying to learn stuff in the abstract. But I won't know what those things are in advance.
John Baez (May 24 2024 at 14:39):It takes practice finding good problems, though. You've had a lot of experience, Todd, but a beginner can easily spend a lot of time on problems that are duds. The homework problems in a good book are more guaranteed to be helpful, though of course one can only develop the skill of making up good problems by trying to do it.
John Baez (May 24 2024 at 14:42):I'm very glad I went to college and grad school where I was forced to do a couple dozen proofs a week for homework - enough that I had to become a theorem-proving machine. I wouldn't had the discipline to do that on my own.
Todd Trimble (May 24 2024 at 15:17):Yes, all that is true. The point I really want to get across is that we think of all of category theory as ultimately applicable to things outside itself, not just theory of the sake of theory. Probably, for a lot of people coming to this Zulip, they have some potential applications in mind. In my own case, those applications are almost entirely to other mathematical topics, but anyway most people coming here probably already have a built-in need to know, a reason for wanting to learn category theory besides simple curiosity, and presumably a lot of that came up in some context where they thought that CT would have some practical application in their lives.
Todd Trimble (May 24 2024 at 15:17):So, the basic and simple idea I have is to nurture that and learn CT alongside whatever else you're trying to understand, symbiotically. It's a matter of grounding the experience of learning category theory. I can't help but think that makes the process much more fun, seeing what CT can really do for you. (Plus, of course, category theory is really, really cool and fascinating stuff.)
I agree: exercises in textbooks are a great testing ground. And I also agree that it takes a lot of discipline to do that all on one's own, as opposed to having to do it for a grade.
But I'd also add that learning what is a dud problem is also part of the process. :-)
David Egolf (May 24 2024 at 17:03):Having "too many questions" is a challenge for me, too! Here are a few things I've found helpful:
Matteo Capucci (he/him) (May 25 2024 at 06:56):In Julius' account of his experience I also see something I experienced many times (which might not be what they're talking about but here we go), which I feel like calling 'ADD-mode' based on what I know about that condition (which, ftr, I haven't bene diagnosed with).
Sometimes my mind just gets incessantly distracted in side matters during problem solving. The context-switch that naturally happens when one is solving a problem by iteratively going through the little tasks that compose it just hijacks my brain sometimes. These distractions range from checking social media to going through a rabbit-hole of references, to ponder a completely different problem related to a subtask.
It can be hard to deal with this, when my brain is in 'ADD-mode' it eats productive time from my day. In my experience, handling it requires some discipline, lots of patience and lots of self-compassion. I learnt that often I guess distracted like so if the task I'm supposed to be doing doesn't have a really clear path and thus 'I get bored' while doing it, looking for the next interesting thing. I also learnt that limiting exposition to distractions can help a lot, so I try to work on a whiteboard/notebook if I want to avoid getting lost in side quests. Finally, I learnt that sometimes it's actually good to get distracted like this since it leads to serendipitous realizations, discoveries or just learning, and in a field like category theory is very hard to learn a completely useless fact.
John Baez (May 25 2024 at 07:18):I guess it may help to explicitly decide whether you are "solving a problem" or "roaming around looking for interesting ideas and trying to learn stuff". I've always deliberately spent a lot of time just looking around for fun stuff, and writing about fun stuff that I've just learned. But this is really different from "solving a problem". I try not to fool myself into thinking that having fun counts as work. I think it's better to let fun just be fun.
For example, right now I'm spending about 5 hours on weekdays working with a team of people trying to develop category theory and software for agent-based models. That's solving a problem, and it's a lot of work. By the end of each week I'm exhausted. To keep from getting drained it seems I need to spend an almost equal amount of time each day doing whatever I want: learning algebraic geometry and other bits of math, helping people work on a problem involving octonions, watching music theory videos, chatting here on Zulip and on Mathstodon, working out at the gym, etc.
Eric M Downes (May 25 2024 at 08:11):My answer is a question: "porque no los dos?" What is your long term goal? Can you go about your short term goal in a way that has a high dot product with the long term?
It also matters how you learn... if I could just work though a text from cover to cover without needing a main quest to orient and motivate me, I would certainly know more category theory. So maybe the side quests are just my way of coping with an unruly mind! But even still, its the only one I have!
What Julius is suggesting (an automated category drawer that can handle -cells and standard completions), if done right, could actually teach him a ton about category theory. In this case, I'd second what David said: look at quiv.er first before reinventing the wheel, and always scope with care, but generally building tools can be in service to larger goals. Especially if its what part of your brain already really wants to do.
Also: remain flexible, and checkin with awareness of the bigger picture every so often. Bilbo getting lost and winning a ring in a riddle game with a damp hermit turned out to be the main quest, and all that stuff with Smaug was ultimately trifling in comparison. More people have used TeX than have read the Art of Computer Programming (and the latter is no joke), and while I personally find Wolfram's pet projects uninteresting, Mathematica has had a huge impact. From a distance, I think Mike Shulman might be a master at building tools that synergize with (what I guess are) his long term interests. And I'd wager John Baez's pedagogical "roaming around looking for fun stuff" serves a highly synergistic effect as well, even if that wasn't planned!
Julius Hamilton (May 25 2024 at 18:16):Thank you everyone. This has no doubt given me some new ideas for how to approach the study of mathematics.
I’d like to ask a follow-up question.
I assume that even a very experienced mathematician can find themself in unfamiliar intellectual territory, since there are so many fields of mathematics.
The less familiar I am with a field, the more I struggle to articulate my intuitions about an idea I have, or communicate a question I have.
How would an experienced mathematician and/or researcher approach the situation of trying to formulate a question in a field while they still know very little about it?
John Baez (May 25 2024 at 21:34):I think I count as experienced, even over the hill. When I'm just learning a field I have tons of questions about it, and how it connects to other subjects, many quite vague. How and whether I ask them depends a lot on whom I'm talking to.
In public forums I usually try my damnedest to crystallize my confusion into precise, well-formed questions that use terminology in correct ways. (You can see my MathOverflow questions here.) I do this to avoid embarrassing myself, but also to get good mathematicians engaged. Experts are a lot more interested in talking to people who put in the effort to ask good questions. And putting in the work is good for me in other ways too. Very often when working to formulate a good question I read articles, books, Wikipedia and the nLab... and about half the time I figure out the answer or realize the question was misguided before even asking it!
In public forums I rarely ask the questions that puzzle me the most, because they are about mysterious patterns I sense but can't quite put into words. Instead, I try to ask more specific, well-formed questions whose answers may give me useful clues to the murkier puzzles I'm really concerned with.
At the opposite extreme, I have some math friends who I talk to, like James Dolan, who know how I think from years of conversation, and don't mind touchy-feely questions where I'm struggling to put vague intuitions into words. This usually works better in actual conversation rather than email, because it takes a lot of rapid back-and-forth to clarify things.
Valeria de Paiva (May 25 2024 at 22:17):In public forums I rarely ask the questions that puzzle me the most, because they are about mysterious patterns I sense but can't quite put into words. Instead, I try to ask more specific, well-formed questions whose answers may give me useful clues to the murkier puzzles I'm really concerned with.
once again, very well formulated, thanks!
I wish I was more like that, but lots of times I cannot resist and just ask the non-specific questions that do not elicit good answers.
John Baez (May 25 2024 at 22:29):Do you think it works out badly? Is it actually bad in some way, or does it just not help as much as you want?
David Egolf (May 26 2024 at 19:07):A related thought, on the topic of trying to focus one's efforts when learning math... I think there are (at least) two ways to put a preorder on learning resources:
I find it can be tempting to try and learn math by starting with "smaller" learning resources under preorder (1). That is, it's tempting to try and begin towards the "start" of mathematics, by first studying the most foundational of concepts or structures that underlie or support all the others.
However, in my experience it's often more practical to learn math by using "smaller" learning resources under preorder (2). That is, I find I learn more efficiently when I work from resources that I find relatively easy to understand, even if they are not discussing the most foundational or most "deep" concepts.
(As a side comment, I find it can be fun to try and explicitly draw the preorder (2) for some learning resources of interest.)