I felt smart until grad school (in math), and after 6 years of it and not being able to graduate this year, I feel like a complete idiot. Not everyone feels stupid at the beginning, but almost everyone feels stupid at some point, once they get far enough because it keeps getting harder. But sometimes the people who felt stupid at the beginning can catch up to the others. Even when I didn't do well, I felt smart for a long time because I just felt like I was not that great at taking tests, but was pretty good at understanding things.
I had huge problems with calculus, not with the intuitive aspects but the lack of physical application / motivation for so many of the problems. The problems seemed to usually be just empty symbol-shuffling with no connection to any plausible situation.
Somehow, I don't have trouble with dumb calculus problems, even though I sympathize with your viewpoint. I just look at them as examples where you can compute things and test what you know about the intuitive concepts with calculation. Whether or not it's relevant to anything physical is not really the point. The idea of integration is applicable, so it makes sense to just get good at it by doing lots of examples and then when some example comes up in real life, once you are good at integration, you either work it out or have a computer do it. So, if the general idea of integrating stuff is applicable, I think that ought to be enough, rather than specific problems being applicable. As long as the calculations aren't needed in the further development of the theory, I'm happy.
I am told some people (math major "naturals") can limit the amount of disconnected facts they need to cram into their head by remembering the steps of the derivation and re-deriving as they go, but it seldom works for me.
I'm a math major natural. It's not necessarily the steps of the derivation. I just try to look at it in a way that makes it obvious that it's true. Often, this involves visualization of some sort. Once you get that feeling that it's obvious, a lot of times, it's easy to rederive the proof if you want.
I find the purely symbolic, unvisualizable "Bourbaki virus" approach to be a huge impediment in most math literature.
Yeah, it is pretty ridiculous, but I don't know if the problem is so acute in calculus. In differential equations, it can get pretty bad. But I don't think it's Bourbaki. There are TWO, not one source of bad math today (actually, there may be some overlap). One is Bourbaki-style which is really abstract and unmotivated. But the other source is that of the mathematicians who love ugly calculations, rather than abstraction. Both of these schools of thought HATE conceptual understanding and aspire to make math as ugly as they possibly can.
I have a sneaking suspicion that these mathematicians go to great pains to erase the method that they actually used to arrive at their conclusions and substitute more "elegant" or "rigorous" but less intuitive proofs.
I have a different sneaking suspicion. I think, in many cases, they don't know how to arrive at their conclusions and are just copying from other sources. Of course, they can follow the steps of their proofs, but in a lot of cases, I doubt they could have come up with the proofs on their own. So, probably the erasing when on a lot further back in the past (specifically, I mean, textbooks probably just copy other textbooks and papers).
There may be a reason why this erasing happens, and I don't think it's intentional. Speaking as someone who is trying to write down a fairly involved proof of a new theorem at the moment (the set-up and proof are about 20 pages, as of right now), I can say that I'm not confident that the theorem is true until it is written. Writing a proof down is essentially a way to convince yourself that your result is true. So, the primary purpose of writing a paper is deemed to be to establish the truth of a result, rather than communication. It's hard to worry about the exposition when you are worrying about whether the thing is correct. I hope my thesis will be reasonable clear conceptually, but it isn't optimized for teaching or learning the material. The main thing is to get it written down and checked for errors. With a finite amount of time at your disposal, you might not figure out the best way to convey your thought processes the first time you write something down.
Also, people just copy what they see other people doing. They see lots of other formal math papers and then they get the idea that that is what a math paper is "supposed" to be like. And then they write that way. Or they just get caught up in the efficiency of the abstract approach (doing things in generality means you don't have to prove the same things twice). Or they just aren't very imaginative, so they are unable to visualize anything, which cripples their ability to understand anything, as well as their ability to come up with anything understandable. I don't think they intentionally do it to make it obscure. Although Descartes is said to have done it intentionally.
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