BS in math to PhD in physics?

deRham

I don't think he was claiming it is lacking (and had a different point of view entirely), but I could be mistaken. I don't think this is lacking, like you say.

homeomorphic

I am claiming it is LACKING. That is not at all the same thing as saying that it is entirely ABSENT. It is there. It's just that there's not enough of it.

Doing the problems sometimes makes it fall into place, but only sometimes.

This is not just my view. This is also Vladimir Arnold's view:

http://pauli.uni-muenster.de/~munsteg/arnold.html

Vladimir Arnold was a great mathematician.

I don't completely agree with that essay, but I agree with the general thrust of what he was trying to say there.

homeomorphic

I don't think it's arrogant when you see something that was beautiful obscured and destoyed, to just point that out. And believe me I have seen beauty, and I've seen it destroyed. This is not a delusion. This is video tape of the statue being destroyed. I don't just say things. I speak from what I've seen.

deRham

Maybe we are speaking of different parts of the establishment - is the article not more a criticism of how textbooks introduce mathematics while obscuring a lot of what goes behind the definitions?

I of course think there is more to mathematics than solving problems - it is also understanding the solutions on deeper and deeper levels. Many times, without attempting one solution, despite it being less than ideal in terms of shedding light, nothing gets somewhere, but we don't stop there.

Aside from some bad teachers, I feel a lot of mathematicians are quite interested in not obscuring the ideas behind the definitions.

If this is missing the point of the article, I would be curious to know more - it does seem an interesting one.

ZapperZ

Isn't this already deviating too far from the original question of this thread?

Zz.

homeomorphic

I don't think he was only talking about textbooks.

Being interested in not obscuring the ideas is not a sufficient condition to insure that the ideas are not actually obscured. That requires that you have a clear understanding of the motivation yourself. I am convinced that many mathematicians are not fully aware of the gaps in their understanding. In some cases, it may be that they believe in very formal writing, so that they are almost intentionally obscuring the ideas, not for the sake of obscuring the ideas, but in order to conform to a rigidly formal writing style. But, in other cases, people just don't seem to care about understanding the motivation because, presumably, they are not aware of its existence. Over and over again, I was not satisfied with what I was taught, I intuitively sensed there was a more reasonable way to do it, and, magically, I found it, either by looking around for different books or thinking about it on my own.

homeomorphic

It's somewhat relevant. The point to take away from this is that I don't think you can ever feel "safe" from the sorts of problems that the original poster brought up. If you have a strong feeling that you want to do things this way or that way, it's hard not to be disappointed at some point. So, if you want to do it, you have to take the good with the bad, I think.

deRham

Sure, it wasn't only textbooks, but my impression was he was talking about things meant to introduce a topic (at whatever level). You yourself mentioned that some books or sources or people seemed to convey things better, and I wonder if to some degree, this just means there need to be various resources out there, as opposed to a single best way.

I'm a fan of the style of introducing the intuition first, and then writing the formal proof next, so it's clear where everything is coming from.

You're definitely right that a lot of very poor motivation and communication happens in parts of the mathematics community, although I've gotten somewhat convinced that a significant portion agrees with the sentiment that things aren't ideal and need to improve.

deRham

@Zz - the question of the thread brings up a number of things, so while I share your sentiment that there's a large amount of information swimming around in unorganized form, the question of whether to switch from one discipline to another involves addressing misunderstandings or insights about each of those disciplines. Particularly given one suggestion for avoiding the apparent difficulties of switching involves studying some kind of applied math, and considering exactly how much similarity to formal mathematics he wants to see in his future studies, a lot of the things that spun off have at least some relevance.

homeomorphic

It would be great if there were 5 great ways. But the way things are now, for some topics you can find one or 2 great ways if you are lucky, but in many topics, you find 10 pretty bad ways, and you have to cobble together what you can find from the 10 bad ways, in addition to a Herculean personal effort to get a reasonable understanding.



I also like that style. Not everyone does. I just want my style to be easily available on the internet, all in one place. Which is what I plan to do. Then, other people can do whatever they want, but students who think like I do won't have to be tortured by it because they'll know where to look.


I'm not sure what they think.

Part of it is just a communication problem. A lot of stuff isn't written down. It's "folklore" intuition. Written communication is kind of secondary. But part of it is a thinking problem. After a while, sometimes the oral "folklore" is forgotten and all that is left is the written stuff, which is too formal.