Question

Fixing the "I Must Prove This, Myself!" mindset

Answer 1

I've been trying to tell one young student that I have that professional mathematicians don't all understand every single detail of the proofs of every result they use. Plenty of people rely on modularity theorems in modern number theory, but Kevin Buzzard has publicly claimed he doesn't think there is anyone alive who knows the whole proof of Fermat's Last Theorem (which is a consequence of one such theorem). It's good to understand how the proofs go, but as for coming up with them yourself, that's somewhat unrealistic as you go deeper. You'll get to theorems that professional mathematicians spend serious amount of time coming up with proofs. The point is to not have to do all that work yourself, but try to learn from their insights. If you were studying analytic number theory, for instance, not trusting the Prime Number Theorem until you invented your own proof would be folly, seeing as it is sometimes seen as a landmark theorem of 19th century mathematics. Or being a geometer, and not availing oneself of the proof of the Calabi conjecture, or the Geometrisation conjecture, and not wanting to look at existing proofs, would lead to never moving past these results, since they won people Fields Medals.

Answer 2

You can always come up with some compromise.

For example, one compromise is to spend some amount of time trying to prove the statement, and if you don't make any significant progress and feel stuck, you can try to find one idea in the proof from reading a few lines and see if you can make some progress from that. This is just one example though, but it allows you to still get through the topics in some reasonable amount of time while giving you opportunities to come up with proofs on your own or at least partially solve them.

That is also another thing. Sometimes you cannot prove the whole statement on your own in a short period of time, but even partial progress towards a proof can often be a good achievement.

Answer 3

To *understand* proofs a strategy I found useful, even entertaining, was to try to come up with counter-examples. Not to the theorems, of course, but along the lines of "what can happen if I relax that condition/hypothesis?"

Answer 4

No I havn't struggled with this idealism. As soon as it became clear that i'd fall behind in a class for trying to come up with everything on my own, I did it at a level that was within my own pace.

Not sure how anyone in my measure theory class could just go all out verifying the whole theorem for every theorem covered before looking at it yourself, and still manage to keep up. They'd have to be extremely talented i guess.

Answer 5

I think part of what's going on here is that in introductory classes, most proofs do not require a radically new "idea" and so you internalize the idea that you can (and therefore should) prove it yourself. The point of presenting such arguments is more to get you practice with how proofs work than because the proofs are interesting in their own right. Reading some more advanced arguments where you really \*do\* need a new idea to make any progress should dispel the idea that you can reprove everything yourself, because \*nobody\* can do that.

Answer 6

Those statements are called theorems and lemmas for a reason - especially those with a name. Even genius mathematicians spend days or even years to prove those results (and no one before ever did that).  
For me, do I think I am as good as Gauss, Hilbert, Urysohn, … ? Absolutely not. So if these results are even hard for them to prove, how am I supposed to independently prove them? (Especially those that need some tricks to prove) I’ll just go ahead and read the book.

Answer 7

I started my undergrad degree in math at 28 years old after relearning the entirety of high school geometry and algebra via youtube video the previous year because I thought I wanted to major in business and knew they were gonna make me take college algebra to declare the major. 4ish years later I'm looking at a 4.0 in my first semester of grad school and talking to my advisor about switching to the PhD program out of the Masters program. One thing I did notice is that a lot of the "kids" (18-20 year olds lol) in my undergrad math classes had some kind of private tutoring or at the very least were in AP calculus in high school (FTR I barely passed Algebra 1 when I was 16 and my parents couldn't have afforded a private tutor even if I'd asked for one). In the end though we were all in the same classes learning the same new material. The only real difference between these kids that had always "liked" math and me was that I was much much less likely to get frustrated and complain when I didn't understand the material on the first or second pass through. The way I see it is if I don't get what's going on I generally just need to spend more time working thru whatever it is until I know what's going on.

Then again I also minored in history where the expectation in academic research is that you absolutely must understand your source material and previous scholarship on your chosen topic before you start delivering your own theories or making causal statements. I think this kind of "cite your sources" social science background probably also affects how I think about studying math especially at a graduate level.  Because of this I can understand people wanting to arrive at the end of proofs by figuring them out on their own but it makes more sense to me to go through what someone else has already done and understand it then put myself through the stress of trying to re invent the wheel. Coming up with my own interpretation is gonna be much easier if I actually understand stuff other people already did and I'm much less likely to come up with something that isn't correct.

Answer 8

I have that mindset, too. And hopefully I do maths now only for leisure, so I have no time constraint, other than what I decide.

Actually there are some drawbacks:
- It takes much more time to learn any theory.
- It can lead to a NIH (not invented here) syndrome where one likes one's proofs better, for no good reason.
- One's proofs may be wrong.

But I feel learning is stronger and more durable this way.

One algebra book states: "The usual way to compose exercises in a maths book is to take a more advanced course and arrange its theorems as exercises". So it shows that trying to demonstrate by oneself some classical results is part of the usual learning process.

I have been surprised, quite a few times, to come with some nice result which was not in the book, then to search it and to find a paper that proved the same result, often with the same proof. That's quite comforting, feeling to be on the right track. Of course  too bad for publishing, but oh well...

Answer 9

If you taught Einstein only basic arithmetic and not give him any books to read, he likely wouldn't develop anything further than basic calculus. Geniuses need to read too.

Answer 10

I get this. I had absolutely no interest in math in high school. And the little interest I did have, was to pursue other things such as engineering or sciences. And yet I found myself declaring a math major at the end of my sophomore year. Feynman once said “What I cannot create, I do not understand”. And I really took this to heart, and feel it deeply (it sounds like you do too). However, something we just need to accept to make great progress (unless we’re a prodigy like Feynman), we can’t literally reinvent everything. We can try to put our own spin on things here and there, and those are some of life’s delights. But the books written by mathematicians exist for a reason, take advantage! We have the nice ability to greatly accelerate our learning using these books. Although it’s not quite the same as completely coming up with it on your own, it’s still very satisfying to wrap your head around the concepts and results and proofs, and to be able to understand them well enough that you can reproduce the arguments on your own without referencing anything. And this is still a high bar to hit, it’s one I’ve accepted I should go for at this point, and I encourage you to strive for it too.

But anyway, you can do this. If anyone says you can’t do math unless you were planning to do it from a young age, fuck what they think, they’re wrong. Anyone can do it with hard work. Good luck.