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- #27

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This is wrong on so many levels. Do you actually have any experience with theoretical physics? This is generally the mathematical elitism I hear from undergraduate math majors who frankly, just dont understand what physics is about (or any science for that matter).

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You could be right about him having trouble with the lack of rigor, but he will still have to contend with the lack of practicality issue. Therefore, he will probably not be happy in pure math. Mathematical physics or applied math would be a better fit. Also, if physicists are too focused on calculations, then they are doing physics wrong. This isn't just a pesky math student thinking he knows better than the physicists. The fact is that calculations are very limited in what they can teach you. If you want to understand something, calculations aren't going to do the trick. And if you know enough about people like Feynman, you'll find they do always want to know the why behind the calculations. Feynman thought in a more visual, conceptual way than most physicists and that is why he was one of the best.I advise you not to leave mathematics. I had a similar crisis, mine being brought on by taking real analysis, complex analysis and Abstract algebra in my first year. However when I actually sat down to do physics I realised that Theoretical Physics seems to be more focused on calculations than understanding. I don't know how to convey what I mean by understanding, but let it suffice to say that a physicist is happy when his findings agree with experiment whereas a mathematician cares more for the reasoning and logic of the process. Physics, even Theoretical is a poor substitute for Mathematics in that respect.

It isn't inherent in physics. Yes, in practice, it may cause difficulties. When I took classical mechanics, I thought it was the ugliest thing I have ever seen. After several years of thinking about it on my own and finally understanding it conceptually, it is one of the most beautiful things I have seen. The truth is both communities are pretty messed up in the way they are doing things. I tried to run from physics to math to escape the conceptual shallowness, but I ended up going out of the frying pan, into the fire. There's no real refuge from it. If you want to be a real physicist or a real mathematician, you have to take matters into your own hands, either way.

I don't see why that matters. For all practical purposes, physics, as a subject, may as well be infinite.What is more Mathematics is infinite there will always be problems to solve, whereas Physics may only be finite. Yet even if we did have a fundemental "Theory of Everything", we will not be able to determine why this theory is the case and not something else. Theoretical Physics is a branch of mathematics which aims to determine a set of axioms for the physical world. I think most of the understanding in theoretical physics comes from the mathematics.

- #29

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Yes, I do have some experience with theoretical physics. I intended to become a theoretical physicist until I realised that mathematics held more understanding that physics. Proof in physics is experiment, proof in mathematics is reason. Is it no surprise that someone with a more abstract/theoretical perspective would prefer mathematics?This is wrong on so many levels. Do you actually have any experience with theoretical physics? This is generally the mathematical elitism I hear from undergraduate math majors who frankly, just dont understand what physics is about (or any science for that matter).

- #30

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What do you mean by conceptual? The only way to understand mathematics or physics is to do problems. Do you mean how certain proofs or methods are connected?You could be right about him having trouble with the lack of rigor, but he will still have to contend with the lack of practicality issue. Therefore, he will probably not be happy in pure math. Mathematical physics or applied math would be a better fit. Also, if physicists are too focused on calculations, then they are doing physics wrong. This isn't just a pesky math student thinking he knows better than the physicists. The fact is that calculations are very limited in what they can teach you. If you want to understand something, calculations aren't going to do the trick. And if you know enough about people like Feynman, you'll find they do always want to know the why behind the calculations. Feynman thought in a more visual, conceptual way than most physicists and that is why he was one of the best.

It isn't inherent in physics. Yes, in practice, it may cause difficulties. When I took classical mechanics, I thought it was the ugliest thing I have ever seen. After several years of thinking about it on my own and finally understanding it conceptually, it is one of the most beautiful things I have seen. The truth is both communities are pretty messed up in the way they are doing things. I tried to run from physics to math to escape the conceptual shallowness, but I ended up going out of the frying pan, into the fire. There's no real refuge from it. If you want to be a real physicist or a real mathematician, you have to take matters into your own hands, either way.

I don't see why that matters. For all practical purposes, physics, as a subject, may as well be infinite.

- #31

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No, it's more than just doing problems. The "just do the problems" approach is one of the wellsprings of the current conceptual disasters in both fields.What do you mean by conceptual? The only way to understand mathematics or physics is to do problems. Do you mean how certain proofs or methods are connected?

By conceptual, I mean being able to see that a theorem is obvious after internalizing the intuition behind the proof. That's quite a different thing from just following the logic of the proof. I also mean knowing the motivation for all the definitions. To feel like the subject is a part of you and you could have invented it yourself.

- #32

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That just means you have understood the problem, have not just read the solution etc.No, it's more than just doing problems. The "just do the problems" approach is one of the wellsprings of the current conceptual disasters in both fields.

By conceptual, I mean being able to see that a theorem is obvious after internalizing the intuition behind the proof. That's quite a different thing from just following the logic of the proof. I also mean knowing the motivation for all the definitions. To feel like the subject is a part of you and you could have invented it yourself.

Some definitions make very little sense untill you start trying to prove things, then it all falls into place. However I do not think mathematics lacks this at all, maybe this is just your perspective? I think it is a bit arrogant/delusional to claim there is a conceptual disaster in mathematics or physics, it's that way for a reason, it works.

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I see, but how is this lacking in the mathematical establishment? This seems to be the bread and butter of mathematics.

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- #36

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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.

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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.I think it is a bit arrogant/delusional to claim there is a conceptual disaster in mathematics or physics, it's that way for a reason, it works.

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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.

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

Zz.

Zz.

- #40

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

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.

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.

- #41

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

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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

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- #44

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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.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 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 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.

I'm not sure what they think.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.

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.

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