Should a physicist learn math proofs?

[Chris Hillman]

Good problem solvers?

You can get away with a lot if you are a good problem solver (like straight A's in school), but it is those who really UNDERSTAND things that make the advances.

I think interested_learner was talking about students who are efficient at solving homework problems in lower division courses, but lest anyone conclude that, at the research level, "good problem solvers" (particularly in math) lack deep understanding, we should stress that most mathematicians will tell you that the good problem solvers (the ones who time after time resolve hard questions which baffle their peers) are the very rarest of aves. I'd say that good problem solvers (in this more exalted sense) may have the best understanding of all. Because they understand exactly as much is required to solve each problem, no less and (this is the perhaps the key) apparently no more.

 

[Chris Hillman]

Uh oh!

Hi, ZapperZ,

Einstein, by his own admission, was never good at math.

I think we agree on your real point here (that Einstein was held up by lack of a good textbook on Riemannian geometry c. 1913, which is generally held to be true), but I have to express horror to see this hoary old legend cropping up at PF.

First, Einstein never said "I am bad at math" (or do you have a verifiable citation?). At various times, in letters to friends like Grossman who knew the context, he expressed wry regret that he had not paid more attention in the Polytechnique to courses in mathematical subjects which by 1911 or so he was becoming aware would be of the greatest value to him. He made a self-deprecating joke about the so-called "Einstein summation convention" as his sole contribution to mathematics (hardly true, if you count the indirect influence of gtr on the development of differential geometry and other mathematical subjects). And so on. He said a lot of things which, taken out of context, can be twisted to fit almost any meaning.

Second, Einstein was actually a veritable master of those mathematical techniques which were in his toolkit, like power series.

Summing up: in his well known "scientific biography", Pais takes pains to debunk this myth, which other recent biographers (at least, the serious ones) have also quite properly discounted.

 

[ZapperZ]

Hi, ZapperZ,



I think we agree on your real point here (that Einstein was held up by lack of a good textbook on Riemannian geometry c. 1913, which is generally held to be true), but I have to express horror to see this hoary old legend cropping up at PF.

First, Einstein never said "I am bad at math" (or do you have a verifiable citation?). At various times, in letters to friends like Grossman who knew the context, he expressed wry regret that he had not paid more attention in the Polytechnique to courses in mathematical subjects which by 1911 or so he was becoming aware would be of the greatest value to him. He made a self-deprecating joke about the so-called "Einstein summation convention" as his sole contribution to mathematics (hardly true, if you count the indirect influence of gtr on the development of differential geometry and other mathematical subjects). And so on. He said a lot of things which, taken out of context, can be twisted to fit almost any meaning.

Second, Einstein was actually a veritable master of those mathematical techniques which were in his toolkit, like power series.

Summing up: in his well known "scientific biography", Pais takes pains to debunk this myth, which other recent biographers (at least, the serious ones) have also quite properly discounted.

I certainly did not mean to imply that his "poor math" is similar to someone not being able to do math. It is well-known that he is quite humble in terms of his ability, even with physics. However, from all the biographies that I have read, even from Pais, he had little inclination to study mathematics, at least for its own sake. So I can really understand when he seeked help in dealing with some of the most complex mathematics that only mathematicians would be an expert of. That is only logical. If one needs help, one would find someone who one thinks is truly an expert.

It certainly wasn't his lack of knowledge of the "proofs" of the mathematical tools being used that caused him to seek such assistance.

Zz.

 

[pivoxa15]

Hi, ZapperZ,
First, Einstein never said "I am bad at math" (or do you have a verifiable citation?).

Here are two interesting quotes by Einstein from http://en.wikiquote.org/wiki/Albert_Einstein

'I have no special talents. I am only passionately curious.'

So Einstein wasn't talented at maths. But didn't say that he was bad at it. This raises the question why didn't he say he was talented at physics? How can one be talented at physics? I can see how someone can be talented at maths or sport or English (i.e. linguistic talent) but how can one be talented at physics? Maybe talent can only be measured from subjects like maths and english. Other subjects like physics, chemistry, geography, psyhology, history are hybrids of those two. Physics leans heavily towards maths so talent in maths => good at physics. And maybe Talent in English => Good at history.

'I don't believe in mathematics.'

This quote might reinforce what ZapperZ has been saying - keep the (pure) mathematics to the mathematicians.

 

[ZapperZ]

Maybe talent can only be measured from subjects like maths and english. Other subjects like physics, chemistry, geography, psyhology, history are hybrids of those two. Physics leans heavily towards maths so talent in maths => good at physics.

Your "logic" makes no sense. You're arguing that if you're good at making screw drivers and hammers, then you must also be good at building a rocking chair or a house. What does a talent at making the tools used to build something automatically means that the person must also have the talent at buiding that something?

I have seen many people who are "talented" in math who have zero inclination to do physics, and I have seen many people who I consider to be some of the most brightest minds in physics who simply have no patience to study mathematics. This alone falsifies what you are claiming.

Zz.

 

[complexPHILOSOPHY]

Here are two interesting quotes by Einstein from http://en.wikiquote.org/wiki/Albert_Einstein

'I have no special talents. I am only passionately curious.'

I believe this quote is describing his passion and curious drive to discover and explore a universe, that was built for us to understand. While I am sure he was a modest person, I don't think he doubted his talents. I could be wrong, though.

So Einstein wasn't talented at maths.

I don't think the quotation makes this statement even approximately true. He perhaps lacked the fundamental knowledge and understanding that a mathematician might possess, however, I am quite certain he had a talent for maths considering his supposid lack of understanding.

But didn't say that he was bad at it. This raises the question why didn't he say he was talented at physics? How can one be talented at physics?

I believe that Einstein was a staunch determinist who presupposed the existence of some great cosmic architect of reality (which I think he referred to as "The Old One", but I might be incorrect), who essentially designed the universe with elegant and precise mathematical laws to propel and engineer it's evolution. As a determinist, one assumes they are simply the byproduct of chemical and physical reactions, coaleascing to produce a conscious and sentient human being possessing the properties of self-awareness, cognition, sapience, etc. Through genetic programming, your biochemical and physical architecture is determined and generated while your subsequent interactions with the environment fabricates your model of reality. All of your talents are byproducts of the laws of the universe so you should not be praised for them, however, some hard determinists also reject the notion of responsibility for 'negative' actions as their actions are not truly free (however, this is of philosophical debate even between determinists). I would assume he never claimed he was talented because of this philosophy.

I can see how someone can be talented at maths or sport or English (i.e. linguistic talent) but how can one be talented at physics? Maybe talent can only be measured from subjects like maths and english. Other subjects like physics, chemistry, geography, psyhology, history are hybrids of those two. Physics leans heavily towards maths so talent in maths => good at physics. And maybe Talent in English => Good at history.

I think one's talent in physics can emerge from a lot of factors as well as any other subject such as history or mathematics and that you can't really deconstruct it into a false dichotomy like that. While possessing a fundamental and rich understanding of mathematics might help an individual to excel in the mathematical aspects of phyics, it does not necessary follow to assume that their intuition of how the physical universe operates will emerge out of it. Also, while it might help a historian to understand and describe historical events, accounts and recollections more eloquently and vividly, I think merely possessing the curiosity to learn and remember history, will certainly help someone gain a deeper understanding of history than grasping the proper mechanics of grammar and spelling.

I think Feynman had something accurate to say on the subject (as he usually does):

"So, ultimately, in order to understand nature it may be necessary to have a deeper understanding of mathematical relationships. But the real reason is that the subject is enjoyable, and although we humans cut nature up in different ways, and we have different courses in different departments, such compartmentalization is really artificial, and we should take our intellectual pleasures where we find them." - Feynman

 

[pivoxa15]

Your "logic" makes no sense. You're arguing that if you're good at making screw drivers and hammers, then you must also be good at building a rocking chair or a house. What does a talent at making the tools used to build something automatically means that the person must also have the talent at buiding that something?

I never said that. Although you might be correct if you are trying to use it as a metaphor. Maybe I should have added 'have a better chance being' in front of physics and history.

so
talent in maths => 'have a better chance being' good at physics. And maybe Talent in English => 'have a better chance being' Good at history.The big question remains what is talent in physics or talent in history?
Wikidictionary has for the relavant entries...
Talent
1. A marked ability or skill
2. The potential or factual ability to perform a skill better than most people.

Lets use 2 because we will include people who hasn't done any physics or are students of physics. In the quotations, Einstein wasn't using any of the two definitions because if he had, he would at least have said he was talented at physics. His standards were probably super high, too high even for himself.

So 'talent' can be 'have a better chance beeing good'
'good' can be 'better than most people'.

Lets be more specific in order to suit the different disciplines.

talent in maths + curiosity for nature => talent at physics
talent in english + curiosity for the past => talent at history

It is possibly easier to measure talent in maths and english so we use these two disciplines to measure talent in other subjects. However, the arrow is only one way so one could be talented at physics without talent in maths for example.

I have seen many people who are "talented" in math who have zero inclination to do physics,Zz.

Obviously, one must be motivated to do something in order to do well. I know someone who was talented in math (showed brilliance in junior high and got a scholarship to a private school) but only passed senior high maths and did mediocre in other subjects. He wasn't a 'study' person so wasn't motivated to do maths. In fact he hated it as he got older. So there are even people who are talented at maths but show zero inclination to do maths.

and I have seen many people who I consider to be some of the most brightest minds in physics who simply have no patience to study mathematics. This alone falsifies what you are claiming.
Zz.

That can happen. The arrows only point one way.
I have modified my claim a bit as shown above.

 

[Edgardo]

Hello andytoh,

I found some websites about mathematics in theoretical physics. I thought you may find them interesting, although they do not deal with the question how important proofs areOW to BECOME a GOOD THEORETICAL PHYSICIST
http://www.phys.uu.nl/~thooft/theorist.html

How Much Mathematics Does a Theoretical Physicist Need to Know?
Dr. Dave Morrison, KITP & Duke University
http://online.itp.ucsb.edu/online/colloq/morrison1/

How to Learn Math and Physics,
John Baez
http://math.ucr.edu/home/baez/books.html

Mathematically rigorous physics
http://en.wikipedia.org/wiki/Mathematical_physics

A basic curriculum for Quantum Gravity
Christine C. Dantas
http://www.geocities.com/christinedantas/basic-curriculum-for-quantum-gravity.htmlI think the importance of mathematical proofs depends on what kind of physicist you want to become i.e. a theoretical physicist or a mathematical physicist as described in the wikipedia article. Are you more interested in calculations/simulations and the comparison with experiment or the mathematical framework for physical theories?

If proving theorems is too time consuming you could look up the proofs in the books, I don't think that you have to prove every theorem by yourself.You could also ask the folks at the "Beyond the Standard Model" subforum, who work on quantum gravity and quantum field theory.

 

[dextercioby]

I'm a mathematics specialist with interest in general relativity, and would later like to learn quantum field theory and superstring theory. Of course this requires learning mountains of mathematics that I haven't even learned yet because I spend 80% of my studies doing math proofs.

Doing math proofs and learning the proofs of all imporant theorems is key for a mathematician, but how much is it for the general relativist or the quantum field theorist?

It depends on what you're trying to do. Let's say you want to learn using a textbook. This book contains mathematical notions which, if not defined in the book itself, should have been familiar to the reader from other sources/previous university courses,... The whole point is that learning means understanding each word, phrase, definition, theorem, etc. the author puts on paper. It also means that you have to be able to "fill in the gaps" the author deliberately leaves through some words such as:"it's easy to show that ...", "From (3.5) one can show that...". That's all you have to do. Learn mathematical notions, be able to understand proofs and not "learn" them.

I'm pretty sure no one will ever ask you to prove the spectral theorem for self-adjoint linear operators on a Hilbert space, simply because the proof for that theorem has already been given and is well known (among the mathematicians/mathematical physicists) and one can simply give reference to a book on functional analysis or quantum mechanics.

The real challange occurs when you have to discover new mathematics, i.e. new theorems and obviously proofs to them.

Daniel.

 

[andytoh]

The foundation of the mathematical tool? No. Not knowing the tool, yes. I never had to go back to the "proof" or "derivation" of the tool in all my years of doing physics. All I need to know is (i) what is the tool and (ii) how do I use it correctly.

So Zapper, what are some example mathematical tools/topics that you did not know when tackling a physics problem in your physics research?

 

[symbolipoint]

So Zapper, what are some example mathematical tools/topics that you did not know when tackling a physics problem in your physics research?

An implication is perceivable in that one, and really ZapperZ should answer, but I have my own comment:

A physicist can use whatever mathematical tools that he knows. If a tool exists which the physicist does not know, then either he will not use it; or another physicist who does know the tool will use it.

Really, studying Mathematics and studying Physics are two different things. The first is for examing the inner-workings of the "tool"; the second is for using the "tool". The physicist only needs to understand his Mathematical tools well enough to know how to use them and in which situations to use them.

symbolipoint

 

[ZapperZ]

So Zapper, what are some example mathematical tools/topics that you did not know when tackling a physics problem in your physics research?

I can't remember if I had any. I have had to go back to my mathematics text to do a refresher on how to use the tools, but I know exactly what I was looking for and simply wanted to know or remember how to use it correctly. There wasn't a single case where I had to go back to the "proofs" from ground zero.

Zz.

 

[complexPHILOSOPHY]

ZapperZ, you are primarily an experimentalist, correct? Would your experience potentially apply to someone wanting to do theoretical or mathematical physics, or would they probably need to really know the framework of the mathematics?

If that question didn't make sense, I apologize. I am trying to understand the real differences between the two (other than the obvious ones).

Thanks,
cP

 

[ZapperZ]

ZapperZ, you are primarily an experimentalist, correct? Would your experience potentially apply to someone wanting to do theoretical or mathematical physics, or would they probably need to really know the framework of the mathematics?

If that question didn't make sense, I apologize. I am trying to understand the real differences between the two (other than the obvious ones).

Thanks,
cP

I have talked to several theorists, both when I was at Brookhaven, and here at Argonne. Again, while they do use a lot of mathematics, my impression was that they don't really have to rederive those mathematics from the very beginning. I would also like to point out that in many theoretical papers, I have no recall seeing anything in which the fundamental aspect of that mathematics comes into play, i.e. where the "proofs" actually is part of the issue of the physics being presented.

You need to know the tools, and how to use those tools correctly. This is true no matter if you're a theorist or an experimentalist. You don't, however, in most cases need to know how to make those tools. Of course there are exceptions to the case, but the question that was asked was not about "exceptions" was it?

Zz.

 

[complexPHILOSOPHY]

I have talked to several theorists, both when I was at Brookhaven, and here at Argonne. Again, while they do use a lot of mathematics, my impression was that they don't really have to rederive those mathematics from the very beginning. I would also like to point out that in many theoretical papers, I have no recall seeing anything in which the fundamental aspect of that mathematics comes into play, i.e. where the "proofs" actually is part of the issue of the physics being presented.

You need to know the tools, and how to use those tools correctly. This is true no matter if you're a theorist or an experimentalist. You don't, however, in most cases need to know how to make those tools. Of course there are exceptions to the case, but the question that was asked was not about "exceptions" was it?

Zz.

No, my friend, you certainly answered my question.

Although, I would love to hear the 'exceptions' you are referring to, it puts things into perspective.

Thanks for taking so much of your time to help us understand what we need to do for the future.

-cP

 

[pivoxa15]

You need to know the tools, and how to use those tools correctly. This is true no matter if you're a theorist or an experimentalist. You don't, however, in most cases need to know how to make those tools. Of course there are exceptions to the case, but the question that was asked was not about "exceptions" was it?

Zz.

But for me, knowing the tools and knowing how to use it is not easy when the mathematics get complicated. It seem the only way to fully know the tools and know how to use it is by knowing how the tools work from first principles. i.e in second year QM, they start solving the equations of the Hydrogen atom with some fancy mathematics like Legedrel polynomials and I felt i didn't know what was going on. Even though I could use it on face value and do some calculations (i.e differentiations) to solve some basic problems. Without knowing the maths (i.e knowing the mathematics ground up) I felt I didn't understand the physics either although it was QM which makes things even more fuzzier.

 

[^_^physicist]

I am similar with pivoxa15, in that although I can use the tool, and in many cases figure out which tool to use without much refereance, I do not feel confindent in the rule unless I have either seen where the tool itself was proved (and can then I proceed to also go through the proof myself). And it isn't because I don't trust the mathematicians that came up with the tools; it is just a personal thing. I don't "know" the "tool" until I have seen where it comes froml, and have had a chance to fittle with the mathematical conclusions gained from it.

Take for instance, taking the derivitive of some function; I have no problem actually preforming the task, but I could not form a concept of how differenation worked or even how I it could be of any real use to physics, until I took an advanced calculus/intro to real anylsis course, which had nothing to do with physics. I guess my brain is just wired that way, but hey I go with what makes it easiest for myself to learn the tool

Granted, from a professional prespective (which I am just gauging a guess), the concept of working through the proof for the usage of a tool could, (and I would guess in many cases would) become quite cumbersome.

Being that I am still a student, what I may state could be utter nonesense; however, I feel the discouragement of learning proofs does a diservace to physics students; as it both artifically distances physics from mathematics and discourages the development of possible new uses of "old tools."

Of course that's just my revised 2-cents on the subject.

 

[ZapperZ]

But for me, knowing the tools and knowing how to use it is not easy when the mathematics get complicated. It seem the only way to fully know the tools and know how to use it is by knowing how the tools work from first principles. i.e in second year QM, they start solving the equations of the Hydrogen atom with some fancy mathematics like Legedrel polynomials and I felt i didn't know what was going on. Even though I could use it on face value and do some calculations (i.e differentiations) to solve some basic problems. Without knowing the maths (i.e knowing the mathematics ground up) I felt I didn't understand the physics either although it was QM which makes things even more fuzzier.

There seems to be some confusion here in terms of "knowing mathematics" and "using in the workings of a typical physicist". I was tackling the latter.

The FACT that you have to take math classes as an undergrad means that you have to know how some of these mathematical idea came from. There's a pedagogical reason for that. It allows you to have a flavor of how such things came into existence. No one here, and certainly not me, would tell you not to study such a thing.

But the ORIGINAL question, if you recall, wasn't this! It is on whether, someone who is a physicist and have gone through years of education (and necessary studying), would need to know mathematical proofs to be able to perform his/her job as a physicist. I believe that I should not have to repeat everything I have said here already in answering that question.

Zz.

 

[pivoxa15]

There seems to be some confusion here in terms of "knowing mathematics" and "using in the workings of a typical physicist". I was tackling the latter.

The FACT that you have to take math classes as an undergrad means that you have to know how some of these mathematical idea came from. There's a pedagogical reason for that. It allows you to have a flavor of how such things came into existence. No one here, and certainly not me, would tell you not to study such a thing.

But the ORIGINAL question, if you recall, wasn't this! It is on whether, someone who is a physicist and have gone through years of education (and necessary studying), would need to know mathematical proofs to be able to perform his/her job as a physicist. I believe that I should not have to repeat everything I have said here already in answering that question.

Zz.

Good point, I should get back to study...

 

[Werg22]

It depends, I would say. Some physicist are mathematically inclined while others are "chemically" inclined.

 

[pivoxa15]

It depends, I would say. Some physicist are mathematically inclined while others are "chemically" inclined.

"chemically" as in more experimental or "chemically" as in more hand waving? Although the two go hand in hand at times because the real world is so complex.

 

[Werg22]

Go figure . I, for one, get very bothered by the idea of using a mathematical concept without knowing it's substance, which includes proofs. It's the satisfaction of mastery, I guess.

 

[pivoxa15]

Go figure . I, for one, get very bothered by the idea of using a mathematical concept without knowing it's substance, which includes proofs. It's the satisfaction of mastery, I guess.

Same as me, that is why I am leaning towards the mathematician road although I find nature extremely fascinating and exciting as well.

 

[tim_lou]

Well, in general, a physicist probably wouldn't care about something like the proof of a derivative using delta and epsilon... Just like in my mechanics class. When we see differential equations, he usually says... let's try a solution of the form... bla bla bla... while in my calc 4 class, my prof. goes over operators and such and derive the solution from scratch instead of "trying" solutions.
My guess is many physicist just get a sense (a justification) of what is probably right about the tools they use.

 

[mathwonk]

absolutely, a physicist who does not learn math proofs, his belly button falls off. and chikldren laugh at him when he walks down the street with his best girl.

 

[complexPHILOSOPHY]

absolutely, a physicist who does not learn math proofs, his belly button falls off. and chikldren laugh at him when he walks down the street with his best girl.

Fantastic! That sounds like something you say when you are tripping. Or atleast something i'd say.

I love you.

 

[Moridin]

The mathematics part of physics shouldn't be just to learn how input the right values and find the magical answer. Some level of understanding is crucial, even though it might not be full proofs, but a general or brief knowledge of the area in terms of derivation. Does a rock climber need to understand his gear to be a successful rock climber? Not really, but if he does, it might come in handy.

 

[andytoh]

We've gotten a lot of mixed opinions. I believe the following is the MINIMUM mathematical rigour requirement of all physicists who use mathematical tools on a regular basis:

A physicist needs not study the proofs of mathematical theorems that he uses, but he MUST be able to (better yet, actually do it) prove the basic properties of each mathematical tool that he uses.

For example, if a physicist uses Lie groups, he must be able to prove that GL(n,R), C-{0}, products of Lie groups, etc... are indeed Lie groups. If he uses homotopic functions, he must know how to prove that homotopic functions form equivalence classes, that compositions of homotopic functions are homotopic, that the fundamental group is in fact a group under composition of homotopy equivalence classes, etc...

These elementary results are not difficult and by being able to prove them, the physicist will get a stronger feel for what the mathematical tool really is and how it works. This is the minimum proving requirement for physicists in my opinion, and such basic proving skills will make the physicist better in his usage of the mathematical tools.

 

[mathwonk]

complex philosophy, i get the impression we may be kindred spirits. or perhaps that you are holed up in a conservative religious school where rampant lunacy is outlawed. hang in there buddy, there is fun to be had in math land. as to tripping, this is not recommended by artificial means, math provides many outlandishly delightful journeys.