Should a physicist learn math proofs?

[andytoh]

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? Is it worthwhile for them (and you) to study the proofs of the mathematical theorems that you use in your relativity studies? Do you just memorize the mathematical theorems, accept the results, and just make sure you know how to use them to solve your relativity problems? Or do you actually take the time to learn the proofs of the theorems and understand why the theorems are true. Do you make sure that you can derive the curvature tensor instead of just know how to use it? Do you make sure that you understand why the paracompactness of the manifolds you use in your relativity problems is the necessary condition for the existence of a partition of unity? Or you don't care and just use the results?

I ask this because I spend so much time doing math proofs that I cannot learn fast enough the enormous amount of math (and physics) I need to tackle, say, superstrings and supergravity. Yet, I don't feel right accepting new mathematical definitions and theorems without understanding why the theorems are true.

 

[cesiumfrog]

Philosophically...

A physicist only cares about one thing: how well it matches experiment. Doesn't even matter if there is absolutely no mathematical justification (yet), and it is frequently said that advances in mathematics are driven by physics.

 

[pivoxa15]

I'd say unless you have a super intuition (whatever that means) than the more maths you learn, the easier and faster you will learn your physics. I'd say spending all that time on maths is worth your while. Memorising results in maths will limit you at physics, not a good long time approach.

 

[andytoh]

Philosophically...

A physicist only cares about one thing: how well it matches experiment. Doesn't even matter if there is absolutely no mathematical justification (yet), and it is frequently said that advances in mathematics are driven by physics.

I'd say unless you have a super intuition (whatever that means) than the more maths you learn, the easier and faster you will learn your physics. I'd say spending all that time on maths is worth your while. Memorising results in maths will limit you at physics, not a good long time approach.

These are two completely contradictory answers.

I guess the answer would depend on what type of physicist you are (I remember my nuclear physics professor admitting to his entire class that his mathematics is very rudimentary because he didn't need to know more than 2nd year mathematics). So let me narrow it down to the deeply mathematical branches of physics, like general relativity, quantum mechanics, quantum field theory, superstring theory etc... that depends more mathematical knowledge (but not necessarily more mathematical proving skills) than the average math student.

On the one hand, memorizing all the mathematical definitions and results (relating to, say, superstring theory) will enable you to learn superstring theory faster and solve problems. But by not knowing why the mathematical theorems are true will hamper you from developing new theory (perhaps this is why Einstein contributed so little to his own theory of relativity?).

But on the other hand, to learn the proofs of all the theorems and mathematical results takes so much damn time that you will be much older than you'd want to be when you finally reach the frontiers of the mathematical physics you strive to reach. For example, when self-studying general relativity, it took me many months to learn fully all the relevant mathematics (e.g. I didn't just want to know what a tensor space was, but wanted to prove that a tensor space was in fact a vector space and an algebra, etc...), while in that same time I could have finished the textbook if I did not go more deeply into the mathematics than the textbook presented.

 

[Ki Man]

it would be better to study it now then to be under-prepared later on if you might need it someday

 

[neutrino]

Last time I heard, string theory had very little physics in it. :rofl:

*We now return to our regularly-scheduled set of serious and helpful replies*

 

[andytoh]

So a pure mathematician can do research in string theory, without any physics under his belt?

 

[neutrino]

So a pure mathematician can do research in string theory, without any physics under his belt?

Sorry. You shouldn't be taking that comment too seriously. Wait for experts here to help you.

 

[chroot]

Generally speaking, physicists don't really care to prove mathematical theorems. The mathematics they use depends on proofs, of course, but they're perfectly content to let other people do the proofs. Physicists need to know that their mathematical tools are sound, but they don't necessarily need the capability of proving it themselves.

- Warren

 

[AlephZero]

Re the "two contradictory answers", remenber there are two (often contradictory) sort of physicists, experimentalists and theorists. Experimentalists need the math skills to make sure their apparatus works without self destructing - which is often more like doing engineering than proving theorems.

 

[Chris Hillman]

Advances in mathematics driven by physics?

Hi, cesiumfrog,

A physicist only cares about one thing: how well it matches experiment. Doesn't even matter if there is absolutely no mathematical justification (yet), and it is frequently said that advances in mathematics are driven by physics.

I hope you didn't mean to claim that all or even most advances in mathematics are driven by physics, for I would strongly demur from any such claim. However, I imagine that most mathematicians would agree that many advances in mathematics have been driven by technical demands from physics. Actually, the usual interaction is more along the lines of mutual feedback; gravitation-related examples include:

1. Cartanian geometry was inspired to a great extent by the desire to generalize techniques needed to handle gtr and was promptly applied by Einstein and Cartan to propose a new gravitation theory.

2. the Sobolev-Schwartz theory of distributions was inspired to a great extent by Heaviside's work, and has inspired Colombeau's algebra which is applied in gtr to the theory of colliding plane wave (CPW) solutions.

3. Noether's work on variational symmetries and conserved quantities was motivated in part by Hilbert's reaction to Einstein's struggle toward gtr, and returned the favor by proving applicable in gtr.

No doubt there are many other examples, these are just off the top of my head.

 

[Chris Hillman]

Is it worthwhile for them (and you) to study the proofs of the mathematical theorems that you use in your relativity studies? Do you just memorize the mathematical theorems, accept the results, and just make sure you know how to use them to solve your relativity problems? Or do you actually take the time to learn the proofs of the theorems and understand why the theorems are true.

Did you ask your professors for advice on what to focus on when learning mathematics? I imagine that the response would be that you should certainly try to understand and remember the gist of proofs and logical relations rather than the statements of theorems.

Do you make sure that you can derive the curvature tensor instead of just know how to use it?

I really shouldn't try to speak for gravitation physicists, but I would be astonished if any offered any other answer than "of course!"

Do you make sure that you understand why the paracompactness of the manifolds you use in your relativity problems is the necessary condition for the existence of a partition of unity? Or you don't care and just use the results?

Hmm... well it depends upon precisely what you wind up doing. Partitions of unity are certainly very important in modern mathematics for all kinds of reasons, and therefore important in physics too.

I ask this because I spend so much time doing math proofs that I cannot learn fast enough the enormous amount of math (and physics) I need to tackle, say, superstrings and supergravity. Yet, I don't feel right accepting new mathematical definitions and theorems without understanding why the theorems are true.

Right, you probably should insist on understanding.

I probably won't make any friends by pointing out the obvious: the fields you mentioned are highly competitive and use a lot of high level mathematics. Learning faster and retaining more surely than your peers would be an encouraging sign that such an area might be well suited to your talents. Conversely...

If you look at the Millenium problems, many of these have nothing to do with superstrings and can be completely stated in much more elementary mathematical language. Most would say that creativity is essential for attacking hard problems, not a huge store of background knowledge.

If you look at enough reviews of research papers, I think you'll see that most papers, even most good papers, are not trying to scale such Olympian heights. I think you can have a good career in math or physics without knowing much about all the stuff bandied about in string theory, you just should study other problems. If your reaction is that these problems can't be as interesting as "fundamental physics", I demur. My experience has always been that if you learn enough about any topic which has ever fascinated even one true born scholar, you too will agree that this topic is quite fascinating enough to spend a lifetime thinking about.

Many observers think that string theory and such like are vastly overpopulated. Even some of the wiser proponents of string theory have been heard to express concern about this "bandwagon effect".

I yak a lot about gtr, mostly because I happen to have taught myself a fair amount about this subject, tend to try to answer questions about things I know a lot about, and people tend to ask many questions about gtr. Now I doubt one could find a more enthusiastic fan of gtr than myself, but from time to time to I try to point out that there is a very very wide world out there, and there are many topics which IMHO are equally fascinating, but which throw up fewer stumbling blocks at the outset of study.

 

[interested_learner]

My take on all this is that you should learn whatever you are learning at the most fundamental level possible (given the time you have and the quantity of work). This is true whether you are a mathematician, phyisicist or an engineer (as I am). 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.

 

[^_^physicist]

My question would be: Does it hurt to know how to work with and perform proofs? I can't see a reason why this would hurt, then again I am just a student, so perhaps my experience limits me; however, proofs, from my limited experience, is often a means of practicing your ability to approach something with a decent amount of logic and, in some cases, a fair amount of creativity. It might just be my opinion, but I could see these talents becoming valuable assets for a physicist.

Additionally, I could imagine a situation where writing a proofs might be a useful aid for a theorist, as the work to in the mathematics that relates to the physics may give new insights.

But your questions, of course, is whether or not a physicist needs mathematical proofs to do their job and to understand advanced topics- My best guess, based on discussion with professors, is not really. A physicist, in general does not need a means of working with, nor writing proofs.

 

[Gib Z]

My Opinion is that it is worthwhile. Knowing how to prove these results may help you understand how you incorporate this into the physical results. Werner Heisenberg has to reinvent a branch of mathematics for his formulation of Quantum Mechanics, which he later realized was already invented, Matrices. Knowing how to prove things will help you in your aprroach to inventing mathematics, which is what physicists need to do.

 

[mathwonk]

i am a mathematician. my opinino is that everyone in science will want to know why the things people claim to be true, really are true. this involves knowing the argument.

in a sense, there is no such thing as a proof. there is only an arguemnt. i.e. a theoirem has hypotheses, without which it does not hold. so one will apply theorems incorrectly if he does not know the hypotheses. i.e. roughly in mathematics, one cannot prove B is true, one can only argue that if A is true, then B is true too.

moreover they are based on underlying unstated assumptions, which should also be understood, if the result is to be used reliably.

theorems in calculus e.g. are based on assumptions about numbers that are apparently false for atomic physics, hence discoverers of phenomena in quanta had to break free of restrictive ideas about the infinitely small.

it is always unhealthy to take someone elses conclusions for granted withoiut knowing his argument for believing them to be true. if his argument does not convince you, then you may wish to discard his conclusions as well.

so i feel that if you do not know at least the idea of the proofs of mathematical theorems, you may not know when they can be used and when they should be disregarded.

also in physics, things which are claimed to actually be true, are usually arguable conclusions derived from very restricted and controversial observations. so logical argument also plays a role here, and it helps to be familiar with this.

even if you read feynman you may come away thinking quanta are bullets which do not know which window to pass through on a house, when the "real" data are much less visible.

i suggest that anyone who wants to understand what he is being told, should ask why the person stating it believes it himself. i.e. what is the arguement?

 

[pivoxa15]

i suggest that anyone who wants to understand what he is being told, should ask why the person stating it believes it himself. i.e. what is the arguement?

That's excellent advice. It should happen in whatever discussion one finds oneself in. They emphasise this heavily in the Arts faculty as things there can get very contraversial. I remember one academic from the Philosophy of Science department said something like, 'Whatever you believe is unimportant, the important thing is your reasons for what you believe.'

 

[ZapperZ]

i am a mathematician. my opinino is that everyone in science will want to know why the things people claim to be true, really are true. this involves knowing the argument.

in a sense, there is no such thing as a proof. there is only an arguemnt. i.e. a theoirem has hypotheses, without which it does not hold. so one will apply theorems incorrectly if he does not know the hypotheses. i.e. roughly in mathematics, one cannot prove B is true, one can only argue that if A is true, then B is true too.

moreover they are based on underlying unstated assumptions, which should also be understood, if the result is to be used reliably.

theorems in calculus e.g. are based on assumptions about numbers that are apparently false for atomic physics, hence discoverers of phenomena in quanta had to break free of restrictive ideas about the infinitely small.

it is always unhealthy to take someone elses conclusions for granted withoiut knowing his argument for believing them to be true. if his argument does not convince you, then you may wish to discard his conclusions as well.

so i feel that if you do not know at least the idea of the proofs of mathematical theorems, you may not know when they can be used and when they should be disregarded.

also in physics, things which are claimed to actually be true, are usually arguable conclusions derived from very restricted and controversial observations. so logical argument also plays a role here, and it helps to be familiar with this.

even if you read feynman you may come away thinking quanta are bullets which do not know which window to pass through on a house, when the "real" data are much less visible.

i suggest that anyone who wants to understand what he is being told, should ask why the person stating it believes it himself. i.e. what is the arguement?

I don't quite agree with this, and I'm speaking from the point of view of a practicing physicist.

I look at mathematics as the "tools" that I use to solve things. Mathematicians may not like that, but that is the reality for many physicists. We just don't have time to do "proofs". That is like a carpenter spending time studying how his tools were built. All he (or she) cares about is how to use that tool correctly and effectively. That is why we study how to use them, the same way we make physics majors study mathematics, and mathematical physics. We make sure these tools are used correctly and effectively.

At some point, we HAVE to trust that mathematicians who have studied these tools have done it correctly. If not, what would be the function of mathematicians anyway if we have to do it ourselves? It is the same way with physicists. At some point, you have to trust us that we have formulated a valid description of something for you to use. If you insist on understanding the very fundamental description of the band structure of a semiconductor before you put your life and the lives of your family on an airplane, then you'll never fly.

Do physicists need math? Damn right, and according to Mary Boas in her text, sometime physics majors need more math than math majors. But are we required to study proofs of all the math principles? Nope! The majority of functioning physicists in this world are living proofs of my assertion here.

Now, is there a division within the field of physics where the "tools" become the focus of study? Sure! That's why we have a branch of study called "mathematical physics". This area of study focuses on formulating the tools that are needed in physics. The same way that a carpenter can make suggestions to the tool manufacturer on some of the stuff he/she might need, or improvements to existing tools that can make his/her job more efficient, physicists too have specific needs in solving a problem that can be fed back to those handling the mathematics of that problem.

But this doesn't mean that ALL physicists do that (I certainly don't). Considering that most of us are fully occupied with our task at hand especially if we are experimentalists, we simply do not have the patience nor the time to pay attention to mathematical proofs. We simply hope that mathematicians have looked at them deeply enough to give us an official verdict on the validity of various mathematics and how to use them correctly. So far, it has worked very well.

Zz.

 

[andytoh]

We just don't have time to do "proofs". ... If you insist on understanding the very fundamental description of the band structure of a semiconductor before you put your life and the lives of your family on an airplane, then you'll never fly.

I guess this is the bottom line here. How much we want to accomplish vs. how much we want to understand, within the limited time that we have. Let me try to formulate what I want to say regarding this point...

 

[ZapperZ]

I guess this is the bottom line here. How much we want to accomplish vs. how much we want to understand, within the limited time that we have. Let me try to formulate what I want to say regarding this point...

But you don't have to "understand" how a screwdriver was manufactured to be able to use it correctly, do you? You also don't have to understand Landau's Fermi Liquid Theory to accept and use Ohm's law in your electronics, do you?

I have to "understand" certain things. I don't have to understand the proofs of the mathematics that I use. The only thing I need to know is how to use it correctly. Just because I lack the total understanding of how the mathematics principle came into being does not mean that the physics in which I worked on isn't sound. All I need to know is that mathematicians have declared that this mathematics is sound. Then I'll use it.

Zz.

 

[andytoh]

Just because I lack the total understanding of how the mathematics principle came into being does not mean that the physics in which I worked on isn't sound.

Zapper, have you ever been stuck in a physics problem, and later on you realized that the solution required using the mathematical tools from first principles and so you weren't able to solve the problem because you didn't know the foundation of the mathematical tool to begin with?

 

[ZapperZ]

Zapper, have you ever been stuck in a physics problem, and later on you realized that the solution required using the mathematical tools from first principles and so you weren't able to solve the problem because you didn't know the foundation of the mathematical tool to begin with?

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. When I need to integrate across a pole, I don't have to know (even though I do) the logical and historical origin of the theory of residues to be able to use it correctly. Open a textbook on Complex Analysis, and compare that to, let's say, the same topic in Arfken and Boas texts. You'll notice a more compact introduction to that topic, with the latter bypassing most of the derivation and lemmas and corrilary etc...etc. They try to get as quickly as possible on how to use it correctly to solve problems. And as a physicist, that is all I care about.

Zz.

 

[andytoh]

A mathematician and a physicist go to a bar. They see a very pretty lady sitting all by herself. The bartender provides them with a newly discovered theorem that would allow them to pick her up successfully.

The mathematician says: "Wow! But before I use the theorem, let me work out the proof."

The physicis says: "Cool. You do that. I'll just use the theorem right now and take her home."

 

[EmilK]

I agree that physicists does not necessarily have to know the proofs for mathematical theorems to function in that particular profession. However, it will not harm people to review it if they are interested in it or simply wishes to. Whether that experience will actually lead to becoming a better physicist in terms of problem solving and lateral thinking can be discussed. Knowledge only adds, never subtracts, but to what extent?

 

[ZapperZ]

I agree that physicists does not necessarily have to know the proofs for mathematical theorems to function in that particular profession. However, it will not harm people to review it if they are interested in it or simply wishes to. Whether that experience will actually lead to becoming a better physicist in terms of problem solving and lateral thinking can be discussed. Knowledge only adds, never subtracts, but to what extent?

But I think this isn't the issue. I mean, we all can pursue whatever we want that interest us. What I disagree with is that somehow, physicist should learn all of these proofs in mathematics, or else their work will be either suspect, or not sound, or they can't function as a physicist. A small percentage of physicists, maybe. But in general, based on my observations and based on what *I* do, the answer is clearly no.

I have zero inclination to study mathematics. I also have zero inclination to study computer programming. Yet, I studied both not for the sake of mathematics or programming, but for the sake of using them to do physics. My goal is to accomplish what I want to do, and these are the tools and the means to achieve my goal. Until there is evidence that my de-emphasis on studying the proofs and derivations of these mathematical tools somehow is affecting my profession, I will continue to assert they are not necessary for a typical physicist. Study them all you want if you like doing this, but don't confuse liking it with it being necessary.

Zz.

 

[mathwonk]

i like andytoh's example. it illustrates why physicists often have more flexibility and imagination than mathematicians.

i greatly admire the intuition of physicists that let them free wheel to a certain extent, or at least function, without the baggage of mathematicians.

maybe rather than a knowledge of proof, a physicist needs to get that sense of when something can be used reliably, that wonderful "physical intuition" that guides them.

I don't know how to get that. We mathematicians are well aware that often physicists are able to predict the true state of things far before matheamticians can "verify" it.

riemann's discoveries in complex analysis are sometimes said to have been inspired by observations in electricity, anmd the theory of quantum cohomology was apparently inspired by Wittens conjectures on gravity. Ferynman integrals serve as another deep and rich source of problems for mathematicians.

If it were not for ophysicists and their rich source of plausible conjectures about the world and its behavior, mathematicians would often be left exporibng somewhat sterile questions. So we depend on you guys maybe more than you do on us!

 

[andytoh]

Einstein had problems formulating general relativity because he didn't know about tensors. Fortunately Ricci had forumated tensors years before but for Einstein, just learning tensors as a tool (e.g. knowing its transformation laws without understanding why), was not enough. He, and other physicists, had to devote a lot of time in studying the foundation of tensors to formulate general relativity, and general relativity is true physics isn't it?

 

[mathwonk]

here is a trivial but ignorant example or question: mathematicians can "prove" that pi is an ikrrational, number. yet I rpesume that to a physicits this is a meaningless statement, since physics does not permit the level of rpecision needed for this statement.

i.e. mathematicians assume the existence of a perfect dieal circle, which does not exist in the world, and we assume the number line has infinite precision, and forms an ideal connected continuum, which also makes no sense in real life.

thus I assume that the conceopt of itrrational numbers makes no sense to a physicist. hence the concept of an actual absolute maximum of a continuous function, or even of a continuous function, can not mean the same to a physicist as it does to a mathematician.

i,e, i think the conclusions of amthematiocal theoresm refer to an ideal world that does not exist.

of course the phyisicists have not told me their views on this. I have much to learn no doubt on this score and many others, and I am sure there is more than one opinion among them too on such matters.

 

[ZapperZ]

Einstein had problems formulating general relativity because he didn't know about tensors. Fortunately Ricci had forumated tensors years before but for Einstein, just learning tensors as a tool (e.g. knowing its transformation laws without understanding why), was not enough. He, and other physicists, had to devote a lot of time in studying the foundation of tensors to formulate general relativity, and general relativity is true physics isn't it?

Really? Einstein, by his own admission, was never good at math. And I am still convinced that it is the lacking of tools was the stumbling block. Did he have to learn, or find someone who knows more about tensors? Sure! But did he have to need to know the full formulation from the ground up? I don't think so.

Do you know all there is to know about Fermi Liquid Theory and how the band structure in the semiconductor running in your computer to be able to type that response just now? Would such an insight be useful in you formulating a reply to what I just wrote?

Zz.

 

[mathwonk]

what i mean is a mathematician will tell you that two numbers are distinctly unequal if they agree only to a trillion decimal places. i presume physicists are less picky.

it always haunts me to tell students that a continuous graph that goes from below the x-axis to above it must cross somewhere, when the chalk line i am drawing in fact has big gaps in it, and may actually miss the chalk x axis.

do they believe me when the evidence of their eyes is the opposite? and do they even know what I mean? when i say equal i definitely do not mean almost.

I.e. in math, if f(a) = .0000000000000000000001, and f(b) = 0, we say b is a root, but not a.

I think in real life a would often suffice as a root. In fact in real life one cannot even say f(b) = 0, because no amount of accuracy would suffice to actually check this completely.


once you decide to let "equals" mean "agrees to a billion decimal places", all the theorems of calculus go out the window. there are no maxima, no guarantee that a polynomial of degree n has only n roots, etc...

thewn it seems to me you have to shift over to finite math, which is much harder.

if these comments seem abysmally ignorant, well there you have my unfortunate situation.

 

[ZapperZ]

Is there a point to these "jokes" of yours? In case you didn't know, we do have a science jokes thread in the General Discussion forum. Please use that and confine these forums to discussion that you can back with actual examples, not made up ones.

Zz.

 

[andytoh]

I thought long and hard, and I cannot think of an example where a physicist (who's job does not require coming up with new mathematical physics theory) needs to know math proofs.

I will now try to think of example where a physicist (who's job includes finding new physics theory) needs to know math proofs.

 

[ZapperZ]

I will now try to think of example where a physicist (who's job includes finding new physics theory) needs to know math proofs.

Look at Bob Laughlin's Nobel Prize winning paper on the fractional quantum hall effect. That is certainly NEW physics there. Now see where he needed to know "math proofs" to come up with what is now known as the Laughlin quasiparticles.

Zz.

 

[andytoh]

How about this example from high school physics: If there is no air friction, a well known physics result is that there are generally two initial angles for a projectile to land a certain distance. If the physicist does not know the mathematical proof for this (i.e. why the sin of supplementary angles are equal), then he will fail to realize the physics result that at there is no second angle corresponding to 45 degrees. I know this example is a bit lousy, but there must be more advanced physics theory analagous to this.

 

[ZapperZ]

How about this example from high school physics: If there is no air friction, a well known physics result is that there are generally two initial angles for a projectile to land a certain distance. If the physicist does not know the mathematical proof for this (i.e. why the sin of supplementary angles are equal), then he will fail to realize the physics result that at there is no second angle corresponding to 45 degrees. I know this example is a bit lousy, but there must be more advanced physics theory analagous to this.

Sorry, but this is a "proof" of a PHYSICS, not a proof of "mathematics" that is being used. You need to look at proof of "algebra" and "trig" to support your argument, and nowhere in there did one has to invoke such a thing. All we do is USE the principles of algebra and trig to derive that physical description.

So what you have done instead is to prove MY point.

Zz.