Can a Math Major Transition to a Physics PhD?

[homeomorphic]

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?

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.

 

[Group_Complex]

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.

That just means you have understood the problem, have not just read the solution etc.

Some definitions make very little sense until 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.

 

[deRham]

What homeomorphic might mean is the difference between being able to string through a linear algebra proof using familiar methods, and actually having a good idea of why you can and should expect that sort of result to hold, how it relates to the body of the theory, etc. It's a level that goes beyond just being able to complete the proof - it makes you a more active user of the theory, who can more likely use it in less familiar situations.

 

[Group_Complex]

What homeomorphic might mean is the difference between being able to string through a linear algebra proof using familiar methods, and actually having a good idea of why you can and should expect that sort of result to hold, how it relates to the body of the theory, etc. It's a level that goes beyond just being able to complete the proof - it makes you a more active user of the theory, who can more likely use it in less familiar situations.

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

 

[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 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.

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]

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.

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]

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

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]

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.

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'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 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.

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.

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.