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maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?
I've also found these to be helpful: The Language and Grammar of Mathematics, from The Princeton Companion to Mathematics, by James Gowers: http://press.princeton.edu/chapters/gowers/gowers_I_2.pdf Reading Mathematics, by John Hamal Hubbard: http://www.math.cornell.edu/~hubbard/readingmath.pdf
I enjoy a mathematics textbook as much as the next person, but what annoys me is the lack of solutions to the problems in so many of the books I've skimmed outside of classes.
maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?
One approach is to look at a few different but overlapping primers on the same matter and do the exercises. If one treatment doesn't click, another might. And for well-known subjects, topics are often covered in roughly the same order. Maybe not for graduate classes, not sure, but linear algebra, set theory, group theory, analysis it's generally the case. Unless you can afford a tutor to teach it several different wa…
Recently I cracked open Principle of Analysis by Rudin. I got stuck pretty much straight away and looked it up - I saw a few suggestions for a bridge between high school maths and the rigor required by that book. I'm now reading another book that took inspiration from the previously mentioned book but gives a bit shallower a slope towards the rigor required.
maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?
"Does math have to be ambiguous or cryptically written? That doesn't sound right."
Well, no. Sometimes, the solution is just "get a better book" (although, see [-1]). However, even extremely well-written books will skip over certain details (which ones they elide depends on the target audience), for the simple reason that it
a) massively shortens the book
b) maintains the "flow" well, instead of stopping every few sentences to distract the reader with ideas not directly relevant to the argument
c) acts as a natural method to help people choose materials: if you're hitting things you can't understand or even parse every two lines, maybe you'd be better served by coming back to it a while later?
In other words, the skipped details and "why would that be the case?" are going to be there even in books that are considered masterpieces of mathematical writing, and everyone encounters them and learns to deal with them. So what do you do when you hit the inevitable "yeah, right, that's trivial, suuuuuure" next time? To paraphrase someone I can't place right now, you either go under it, over it, around it, and -- if all those fail -- then ($deity willing or no) you push right through it. (Exercise for the cough careful cough reader: what do I mean?)
Less faux-metaphorically, try
* skipping forward a little -- new ideas do not have to be learned and absorbed linearly; case in point: the next paragraph goes into detail on this :)
* checking out another book on the same topic
* asking a question on "MSE" / math.stackexchange.com (the /r/math subreddit is good too)
* checking out the extremely friendly MSE chatroom, which I credit with fixing a ton of my "I am just stupid" moments
* looking for lecture notes (shameless plug: [0]) online
* (failing everything above) spending a day walking around your house with your hands squeezing your head from the sides, trying to mimic the romantic picture of the tortured-genius lonewolf-mathematician-in-training and somehow trying to become more intelligent by increasing the density of your brain matter until you've triumphed over whatever puny margin-note was keeping you occupied
Also note that mathematical understanding and intuition always, always grow organically. You often end up understanding something "properly" years after you stop struggling with it, and it only gets better as you learn things and see how disparate ideas fit together or even become special cases of a general concept, or variations upon a basic theme. This is touched upon in Terry Tao's excellent career advice essays[3], which are aimed at students learning to make sense of the (mathematical) world: in particular, I'd recommend
* https://terrytao.wordpress.com/career-advice/there%E2%80%99s...
* https://terrytao.wordpress.com/career-advice/does-one-have-t...
Grothendieck talks about his journey (see [4]) himself, and his writing is beautiful (if also prone to creating a ton of hope in young people encountering this anecdote for the first time). It's profound, but not an authoritative account of what it means to do mathematics, by any means (just like Hardy's A Mathematician's Apology).
> Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.
Some of the stuff on this page[1] is very helpful, whatever your current level of knowledge. Ctrl-F "tendrils". You might find Thurston's essay "On Proof and Progress in Mathematics" (arXiv link: [2]) interesting too. Good luck!
[-1] There are some well-known books that students are almost always advised to come back to later (and there are probably more that they're advised to not bother with, ha).
[0]: http://refinator.herokuapp.com/
[1]: http://math.stanford.edu/~vakil/potentialstudents.html
[2]: https://arxiv.org/abs/math/9404236
[3]: https://terrytao.wordpress.com/career-advice/
[4]: https://quomodocumque.wordpress.com/2010/07/05/the-capacity-...
maybe i am just stupid, but I find that many texts in mathematics like to skip over details, or I get stuck because there is an ambiguity in the text and there is no one to ask about it.. What do we do in this kind of situation?
You're not stupid. At least, there is very little chance that the math you're trying to work on right now requires super-normal cognitive abilities of any sort: it's more about learning to be comfortable with the ambiguity or lack of comprehension that the process of learning entails. Here's a grab-bag of stuff that have helped me (and still do) lift myself out of "damn I'm dumb" situations such as the one you seem t…
I've also found these to be helpful: The Language and Grammar of Mathematics, from The Princeton Companion to Mathematics, by James Gowers: http://press.princeton.edu/chapters/gowers/gowers_I_2.pdf Reading Mathematics, by John Hamal Hubbard: http://www.math.cornell.edu/~hubbard/readingmath.pdf
I don't mean to be that person, but it's Timothy Gowers, not James.
Earlier quoted context omitted.
One approach is to look at a few different but overlapping primers on the same matter and do the exercises. If one treatment doesn't click, another might. And for well-known subjects, topics are often covered in roughly the same order. Maybe not for graduate classes, not sure, but linear algebra, set theory, group theory, analysis it's generally the case. Unless you can afford a tutor to teach it several different wa…
I can verify this approach works for me. The other thing I suggest is use "getting stuck" as a way to identify gaps - go look for books that fill those gaps. Recently I cracked open Principle of Analysis by Rudin. I got stuck pretty much straight away and looked it up - I saw a few suggestions for a bridge between high school maths and the rigor required by that book. I'm now reading another book that took inspiratio…
I'd like to see a list like this that included the field of mathematical logic. For whatever reason mathematical logic no longer seems to be a "popular" area of research, despite its deep connection to theoretical computer science. But there are distinction in study, as computer scientist tend not to go deeply into computability theory like a traditional mathematician would.