Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical exercise, I've gained $30$ kg, I've spent countless hours studying at night, constantly had sleep deprivation, I've lost my social life, and I got health problems. My grades are quite good, but I feel as though I'm wasting my life.
I love mathematics when it's done my way, but that's hardly ever. I would very much like my career to be centered around mathematics (topology, algebra or something similar). I want to really understand things and I want the proofs to be done in a (reasonably) rigorous way. I've been accused of being a formalist before, but I don't consider myself one at all. However, I am a perfectionist, I admit. For comparison, the answers of Theo, Arturo, Jim Belk, Mariano, etc. are absolutely rigorous enough for me. From my experience, $80$% or more mathematics in our school is done in a sketchy, "hmm, probably true" kind of way (just like reading cooking recipes), which bugs the hell out of me. Most classmates adapt to it. I for some reason can't. I don't understand things until I understand them (almost) completely. They learn "how one should do things", but less often do they ask themselves WHY is this correct. I have two friend physicists, who have the exact same problem. One is at the doctorate level, constantly frustrated, while the other abandoned physics altogether after getting a diploma. Apart from one $8$, he had a perfect record, all are $10$s. He said that he doesn't feel he understands physics well enough. From my experience, ALL his classmates understand less than he does, they just go with the flow and accept certain statements as true.
Also, my problem is having weak memory. I forget a lot. Having to study a different subject each day is killing me. If it were up to me, I'd change the way lectures are done. We'd study only ONE subject for a month or two, then have the exams, and the next month or two the next subject. Mixing it up has a terrible effect: I forget things, because I constantly change the topic. That's why I'm always behind schedual. For example, in the second year, during the school year, I understood almost nothing of multivariable calculus, because I had to simultaneously study abstract algebra, topology, computer programming, etc, and couldn't keep up. But then, I devoted the whole three months of summer and got through all $330$ pages of theory, understood it very good, including all the proofs ( which I was forgetting along the way), got a $9/10$ grade, and had absolutely no vacations (stayed at home), no free time, no sport, no nothing. It was complete crap.
- should I not read the proofs at all?
- should I try less hard, get worse grades and understanding, and 'have a life'?
- abandon the idea of doing math for a living?
- how can I spot the important/illustrative proofs, without studying them completely (it often happens, that I 'get the whole point' only after I've understood the whole proof)
I'm not gifted/bright at all, I'm completely average with a bad memory, but I do have interest in math, good grades, and a horrible lifestyle for the last few years. This question is directed at people who have a career doing mathematics. How did you manage to study everything on time, AND sufficiently rigorous, that you were able to understand it?
I often tend to be the only one to find serious issues in the proofs, in the formulations of theorems, and also in the worked out exercises at classes. Everyone else either understands everything/most, or doesn't understand and also doesn't care for possible issues. Often do I find holes in the proofs and that hypotheses are missing in the theorem. When I present them to the professor, he says that I'm right, and says that I'm very precise. How is this precise, when the theorem doesn't hold in it's current state. Are we even supposed to understand proofs. Are the proofs actually really just sketches? How on earth is one then supposed to be able to discover mathematical truths? Is the study of mathematics just one big joke and you're not supposed to take it too seriously?
I have a bunch of sports I like and used to do. Also, I had a perfectly good social life before, so you don't need to give advice regarding that. I don't socialize and do sport because digesting proofs and trying to understand the ideas behind it all eats up all my time. If I go hiking, it will take away $2$ days, one to actually walk + one to rest and regenerate. If I go train MMA, I won't be focused for the whole day. I can't just switch from boxing to diagram chasing in a moment. Also, I can't just study for half an hour. The way I study is I open the book, search up what I already know but forgot from the previous day, and then go from theorem to theorem, from proof to proof, correcting mistakes, adding clarifications, etc. etc. Also, I have a bad habit of having difficulty starting things, but when I do start 'my engine', I have difficulty stopping, especially if it's going good. That's why I unintentionally spend an hour or two before studying just doing the most irrelevant stuff, just to avoid study. This happens especially when I've had more math than I can shove down my throat, I have mental preparations to begin studying. But when my engine does start and studying goes well (proven a lot, understood a lot), it's hard for me to stop, so I often stay late at night, up to 4 a.m., 5 a.m., 6 a.m. When the day of the exam arrives, I don't go to sleep at all, and the night and day are reversed. I go to sleep at 13h, and wake at 21h... I know it's not good but I can't seem to break this habit. If I'm useless through the whole day, I feel a need (guilty conscience) to do at least smth. useful before I go to sleep. I know this isn't supposed to happen if one loves mathematics, but when it's 'forced upon you' what and how much and in what amount of time you have to study, you start being put off by math. It stops being enjoyment/fun and becomes hard work that just needs to be done.