I think about mathematics differently when I'm 1) at my computer (with pen and paper), 2) while walking, 3) while in the shower, 4) while at a whiteboard (where I also don't have access to the internet). 5) when writing a document for others to read, |6 and 7) when writing a non-beamer talk, and when writing a beamer talk.
I was wondering how people feel about this (no specific question comes to mind)
Partly it seems strange that I can't use the knowledge that I think differently in these states to benefit my study. When you are writing up an undergraduate thesis, or a short note, you've already been through the primary learning stage. But it feels like only then do you really build a coherent story.
When writing a beamer talk you work out the essentials, and the 'apparent motivation' (which you may not care about lol)
I think we all agree that understanding/explaining for ourselves is not at the level of understanding/explaining/teaching others. The latter requires more thought and mastery.
For context a student I tutor has trouble answering basically anything without a pen and paper, and can reason things out pretty quickly when he has pen and paper - they asked me how to fix this, and I said that will go away with practice, which I later reflected may not be true (I can't remember if I ever had problems with that)
If I'm at the computer, I dedicate a much smaller amount of time trying to self-solve a problem than I would on a walk. If I can't work something out in like 20 min, I'll try googling around for a paper which answers the problem. So there's a difference in time/effort expended on independent work. I'll try to work things out explicitly, like find examples and such.
When walking I don't (usually) have a pen nor paper, and so explicit examples aren't possible, and I don't pull out my phone to google or anything, so I'll work more abstractly due to those constraints (and probably less precise…
Expanding on the whiteboard thing: I always feel that paper isn't big enough, and when I finish a page it feels like the information on that page isn't there any more on the next page (that may or may not make any sense), sort of like erasing a whiteboard. (For context the whiteboard I'm using is massive)
Yeah I relate to that whiteboard story. Boards (blackboards, for me) are excellent for jotting down the inherent nonlinearity of thought while doing math.
Pen and paper are oppressive in the sense that writing there always must be linear, it feels.
Like the fact that I think these false things, and don't catch them until I return to try to make them precise, means that they are 'true enough' to pass some low level filter, and there's some underlying actual math lurking nearby
I walk/pace around vigorously when I am unable to see a picture clearly. It would often be the case that it feels parts of the picture I want to draw are fazed out, and I can't concentrate enough to focus on what's happening in that hazy region of the picture while sitting down. I'll just start wondering about something else
@JoeShmo ? I just quoted the definition of 'himself' and it says 'male person or animal previously mentioned as the subject of the clause', and since you said person that rules out animal
I guess I feel like one should be able to use the knowledge that you reason differently in these different settings to ones advantage
It's not really practical to take a shower several times a day just for that state, or to go for lots of walks. You also can't really give the proper level of effort to preparing a literally useless beamer presentation (i.e. one you won't actually be presenting). Also one can't really put in proper effort to write up an article that won't be read
It makes me feel like you'd do your best work if you were required for whatever reason to give a talk every week, and to write up some notes every week, but there aren't enough people to care about you doing that unless you're a big mathematician already :P
I don't think I would call what I do in these various settings that different. I am extremely aware of how I think, and I don't think the way I think changes with context. During teaching, I try to con people into listening to my point of view, and so in the process of preparing talks try to figure out why my point of view is interesting.
Who cares if people want to listen to you or not, you can always write blogposts or something. I am currently writing two.
Just because I want to understand some things better
@BalarkaSen Do you feel massively advantaged to have pen and paper, or access to the internet when working on math? In some sense you could think of these as crutches, or obstructions to thinking freely (since you'll know if you're wrong very quickly)
My advisor looks at the roof and works out proofs on the spot, to statements that he's never heard before lol
@BalarkaSen Oh I don't, it's just that when you're forced to write a beamer presentation, and you know there's an audience, you will put in the effort to make it good. If you're just making it as some sort of study-aid, then it feels like more work than it's worth.
@BalarkaSen Well that's part of what I mean. Because you can be lazy when the internet is nearby, you often will be. But if you're on a walk, the statement feels like it may or may not be true, and that'll lead you to think other statements that seem to be implied, and that may yield some fruit
When sitting at the computer I may say "Don't look this up, it should be obvious if its true or not" and then I'll subconsciously make an assessment if its worth the time to work it out, whereas on a walk I may as well think about it until I return home (an hour)
I don't go outside to walk as much (maybe I should!). But very often I have to be my own sanity-check for the statements whose truth or falsity I care about.
I don't care if Japanese rings are Nagata. I will look that one up any day.
A rough sort of example is that when thinking about a stack as a pseudo functor to Grpoid, on a walk one might think of a groupoid as a particularly nice simplicial set. Or one may immediately replace the stack as a pseudo-functor with a pseudo-naturally equivalent strict 2-functor, and then 'take the underlying 1-functor' so you can compose it
And on that walk you would have no idea at all if replacing it in that way, and composing it is independent of making a choice, so you'd say whatever to that, and keep thinking about the consequences (even if they may be irrelevant because your thing doesn't actually work)
Like here's a rough process on a walk: 1) Stack is just a pseudo-functor X:(Sch/S)^op --> Grpd with nice descent 2) Well if I ignore the 2-category data on both sides, then I could try to take functors Grpd-->D where D is a nice category 3) Well I can't do that without replacing X with an actual functor somehow 4) Okay, we need it to be a strict pseudo-functor so that composition at least makes sense, we can do that, but it might mess everything up and make this pointless (we'll think about that later)
@BalarkaSen Well it's not even a functor because it has 'the data of a deformed composition law' (and it's between 2-categories)
@BalarkaSen Well it's not so fucked up. It takes objects, morphisms, 2-morphisms to objects, morphisms, 2-morphisms. It gives you the data of a 2-morphism between F(g)\circ F(f) and F(g\circ f), so deforms the composition law. You have genuine composition preservation for 2-morphisms
@BalarkaSen Well that's the thing, I remembered that you could do something like this (on the walk), but on the walk I had no idea what problems it would introduce, I actually still don't know what problems this introduces
If you think of the stack BG, which just assigns the groupoid of G-torsors to any S-scheme you input, and takes S-scheme morphisms to pullback functors, then for g\circ f:X-->Y-->Z you have isomorphic G-torsors f^*g^*P\cong (g\circ f)^*P, but non-equal, so strictification of this seems kinda weird
Nice meeting you, sleep well (or enjoy your boxing training?)
