Frustated by the lack of mathematical rigour in physics topics

[spaghetti3451]

I have become so annoyed by the lack of mathematical detail in the physics textbooks. For example, to study Foundations of Quantum Mechanics, I had to spend hours and hours doing background reading from books, the net and Wikipedia until I got my way aroung vectors spaces and its nooks and crannies. I think I will have to do the same next year when I take the General Relativity course. This is becoming so annoying and actually I am beginning to think I should have studied maths instead of physics, so I could have known more mathematics and felt satisfied and then I could have specialised in theoretical physics.

This is actually making me wonder if I should self-study mathematics from the freshman upto senior level just so the mathematics is not so demotivating?

 

[Vanadium 50]

Are you complaining that the mathematics used in physics is not rigorous? Or are you complaining that the mathematics used in physics is not taught in physics textbooks?

 

[micromass]

Why not minor or double major in mathematics??

 

[madness]

Why not minor or double major in mathematics??

For me that was part of the frustration. Switching between rigorous mathematics and sloppy physics from one class to the next. Even so called mathematical physics courses are sloppy compared to courses taught in the maths department.

 

[spaghetti3451]

Are you complaining that the mathematics used in physics is not rigorous? Or are you complaining that the mathematics used in physics is not taught in physics textbooks?

Oh! I mean that the maths is not rigorous. But even if physicists only managed to use correct notation in every case (for example, f(x) is not a function; f is the function and f(x) is the output) and managed to show the representation of any mathemcatical expression in all its generality, then I guess the subject could have been more satisfying, because even though we would be still be using approximations in our calculations, at least we would know where the approximations have been made and why the approximations do not cause a huge difference between predictions and experimental observations.

But, there's this other point that vector spaces are an abstract concept and so it's must better to introduce the idea of vector spaces and all its properties by appealing to examples of functions, matrices, Euclidean vectors and other exotic objects and to show how their properties are all similar whenever a new property of a vector space is introduced. The end result is that we have to spend hours trying to convince ourselves of the validity of the statements so that we avoid taking the mathematical statements for granted.

 

[spaghetti3451]

For me that was part of the frustration. Switching between rigorous mathematics and sloppy physics from one class to the next. Even so called mathematical physics courses are sloppy compared to courses taught in the maths department.

Exactly! I can be satisfied with not knowing as much abt vector space as a maths undergrad, but at least i wish whatever little i knew i knew rigorously.

 

[micromass]

Exactly! I can be satisfied with not knowing as much abt vector space as a maths undergrad, but at least i wish whatever little i knew i knew rigorously.

The thing is that physicists use such advanced math at times that it becomes impossible to do everything rigorously. Ideally, you could study the math first and then do the physics, but then it would likely take a huge amount of time before you get to see anything interesting.

For example, in calculus you have seen that in some cases

$$\int_a^b\int_c^d f(x,y)dxdy=\int_c^d\int_a^b f(x,y)dydx$$

The proof for this result is quite annoying. So what is the benifit, for a physics major, to learn that proof?? It's so much easier to accept it as a given. Instead of getting into the details of the proof, one could spend his time better with the details of physics problems. After all, physics majors pay to learn physics, not annoying math proofs.

I mean, if the choice is "knowing much physics, but not knowing the rigorous math" and "knowing the rigorous math, but not much physics", it makes sense that physics educations go for the first option and that math educations go for the second option.

I guess that the only thing you can do about it, is self-study the math or take math courses in the math department.

 

[RoshanBBQ]

The thing is that physicists use such advanced math at times that it becomes impossible to do everything rigorously. Ideally, you could study the math first and then do the physics, but then it would likely take a huge amount of time before you get to see anything interesting.

For example, in calculus you have seen that in some cases

$$\int_a^b\int_c^d f(x,y)dxdy=\int_c^d\int_a^b f(x,y)dydx$$

The proof for this result is quite annoying. So what is the benifit, for a physics major, to learn that proof?? It's so much easier to accept it as a given. Instead of getting into the details of the proof, one could spend his time better with the details of physics problems. After all, physics majors pay to learn physics, not annoying math proofs.

I mean, if the choice is "knowing much physics, but not knowing the rigorous math" and "knowing the rigorous math, but not much physics", it makes sense that physics educations go for the first option and that math educations go for the second option.

I guess that the only thing you can do about it, is self-study the math or take math courses in the math department.

Mathematicians will detest this, but some things, with a solid intuitive understanding, just make sense. The thing you brought just makes sense if you use an intuitive understanding of integration as an infinite sum.

I believe the OP is not complaining about the absence of rigorous proofs. He is complaining about the absence of intuitive introduction to the topics. Or perhaps he's complaining about the book using mathematical concepts without explicitly defining their use. (So maybe his book had a vector space, used properties of a vector space, and never once said "Hey, this is a vector space." So it treated that fact like it was completely regular and not even worth stating.) I could be wrong, though.

 

[micromass]

Mathematicians will detest this, but some things, with a solid intuitive understanding, just make sense. The thing you brought just makes sense if you use an intuitive understanding of integration as an infinite sum.

Yes, it makes sense. Except that it's wrong some times. To discover when the formula is true, one needs the rigorous proof.

 

[RoshanBBQ]

Yes, it makes sense. Except that it's wrong some times. To discover when the formula is true, one needs the rigorous proof.

Well, I think this goes back to what the OP MIGHT be saying (I mean, he really needs to clarify). In this particular example, I don't think he would want a rigorous proof of this attribute. I think he would just want an intuitive description of how it makes sense alongside warning about when it might fail (combined with more intuitive understanding of why it might). I suspect in the vector spaces example, the book barely defined they were even using one, and only through a large amount of Googling and research (as he stated) did he even find out he needed to know this math.

 

[LBloom]

If you want to study physics and be a physics major do not expect your classes to be taught like math classes. Physicists and mathematicians do not think in the same way, even on the theoretical side. If physics classes spent time mathematically proving and being fastidious about mathematical notation they wouldn't get enough physics done! Be a double major and learn to jump between the styles. Its definitely worth it and I certainly enjoy it.

For example in Griffiths he gives some overview of linear algebra in the back of his QM book. He mentions that although it isn't hard to show that the e.vectors of a finite dimensional Hermitian matrix span the vector space, the proof used does not carry over to infinite dimensional cases. It's still true (I think...) but infinite dimensional matrices was beyond the scope of my proof based linear algebra class so is obviously beyond any quantum class.

My advice is to simply suck it up and deal with it for now. Both kinds of classes have their benefits and although physics classes aren't rigorous, they teach you to hone and work with your intuition. Don't underestimate its usefulness. The Standard Model, which is amazingly accurate, is a QFT with a gauge symmetry and guess what QFT is not mathematically rigorous! I mean progress has been made but the functional integral is notoriously illdefined. And developments in physics have spurred research in mathematics (especially in topology and knot theory) so its no longer this one way relationship where we physicists just borrow and abuse notation :)

 

[LBloom]

Also, if you stay just as a physics major, don't worry about trying to the mathematics in full. Like I said, the subtleties are not immediately relevant. You'll learn to pick up what you need to and when you get more advanced in your studies the mathematics will become more advanced and rigorous.

 

[atyy]

Rigour is the last thing you want in physics - first get the correct theory, then try to construct it rigourously. Even most of mathematics is that way, ideas first, rigour second - take numbers and the calculus, those came before ZFC. Logic is perhaps the one place where it's rigour first.

I mean progress has been made but the functional integral is notoriously illdefined. And developments in physics have spurred research in mathematics (especially in topology and knot theory) so its no longer this one way relationship where we physicists just borrow and abuse notation :)

Apparently it's still physicists who abuse notation. Turaev says in the Introduction of his book that the path integral approach to the Jones polynomial has still not been justified.

 

[nucl34rgg]

Most of the math is rigorous. However, the purpose of an introductory textbook is not to present the technical details of the theory with full rigor. The purpose is to teach you how to do example calculations and understand the conceptual implications of the theory.

 

[AlephZero]

I'm still not quite sure what the OP is complaining about. If he/she tried to do a QM course without being told what the prerequiste math was, that's one problem. If he/she skipped the prerequisite math courses "because they were not rigorous enough", that's a different problem!

I'm with Micromass and Roshan on this one. I have a math degree but spent most of my working life in engineering companies. I once invented a neat numerical method but was concerned there was possibiltiy it would loop forever. The condition for that could be expressed in terms of the ranks of a sequence of submatrices of a matrix, and the conditions for selecting the next member of the sequence. It could also be described by a thought experiment about how to assemble a certain type of "rube goldberg" structure, using temporary supports in a particular way to stop the partly assembled object falling apart. There wasn't any point wasting time doing research into an obscure corner of linear algebra, when a straightforward physical argument answered the question!

 

[twofish-quant]

I have become so annoyed by the lack of mathematical detail in the physics textbooks.

That's because physicists aren't mathematicians. Physics textbooks focus on the physics and not the math.

This is becoming so annoying and actually I am beginning to think I should have studied maths instead of physics, so I could have known more mathematics and felt satisfied and then I could have specialised in theoretical physics.

That won't work. The point of theoretical physics is to describe nature. Sometimes you have to abuse math in order to do that. In most situations, having a rigourous proof turns out to be unnecessary of you have something that "just works."

 

[clope023]

I'm still not quite sure what the OP is complaining about.

In physics and engineering there is a tendency to be sloppy, unorganized, and unrigorous (ie unmethodological) with the mathematics. I've had many courses where formulas and equations are shot-gunned onto the white board with no justification and no derivation. If you care about being organized with your mathematics, this gets very frustrating and I can see why some people switch from physics/engineering to pure math.

 

[Pippi]

When I was studying electrical engineering in college, I stared at those CMOS equations and thought, "There is no rigor in physics."

 

[intwo]

If you're fluent in analysis, you can translate physics into mathematical terms. Alternatively, you can use textbooks that utilize the theorem-proof style. Check out Classical Mechanics by Gregory.

 

[Chunkysalsa]

I kinda understand the OP point but there's usually a large amount of material that needs to be covered and mathematical rigour is usually pretty low on the priority list. Most of the time it serves no purpose in a physical sense.

I'm an EE/Math dual major but I'm much farther ahead in my EE degree. Its nice when I learn something in a math class after I've seen it in engineering and the two sort of click together. Really helps in my understanding when I understand the point and the reason.

 

[yrreg]

I'm still not quite sure what the OP is complaining about. If he/she tried to do a QM course without being told what the prerequiste math was, that's one problem. If he/she skipped the prerequisite math courses "because they were not rigorous enough", that's a different problem!

I'm with Micromass and Roshan on this one. I have a math degree but spent most of my working life in engineering companies. I once invented a neat numerical method but was concerned there was possibiltiy it would loop forever. The condition for that could be expressed in terms of the ranks of a sequence of submatrices of a matrix, and the conditions for selecting the next member of the sequence. It could also be described by a thought experiment about how to assemble a certain type of "rube goldberg" structure, using temporary supports in a particular way to stop the partly assembled object falling apart. There wasn't any point wasting time doing research into an obscure corner of linear algebra, when a straightforward physical argument answered the question!

"There wasn't any point wasting time doing research into an obscure corner of linear algebra, when a straightforward physical argument answered the question!"

I was thinking all the time that physicists use mathematics to prove their theories, but with that statement above, it appears that "There wasn't any point wasting time doing research into an obscure corner of linear algebra, when a straightforward physical argument answered the question!"

So, what is the "straightforward physical argument [that] answered the [what] question!"? if it is not founded upon really rigorous mathematics which is traditionally defined as an exact science?

Now, I can see that with the kind of mathematics employed by physicists, they can understand each other when they talk obvious nonsense about nothing is the origin or cause of the universe.

And let me not mention what freedom they enjoy with bringing in fudge factors to make their math calculations jibe with laboratory data.

So, physicists are not the sacred cows they present themselves to be but all clay idols: only the general public don't know it, because they have been fraudulently gypped to always keep a profound reverential silence when physicists speak, like Hawking saying, because of gravity God is not needed for the universe to come to existence, or something like nothing is the cause of the universe and not God, because there is vacuum fluctuation.




Yrreg

 

[Delong]

I have more classes in math than physics but I am way more interested in the physics. The lack of rigor doesn't concern, learning a lot of information in short time demands that one be flexible in how they learn. Although I appreciate math I could not just stick to math and ignore the world of forces and physics all around me and in me.

 

[Theorem.]

This is one of the main reasons I was lead to mathematics. I found the rigour to be quite attractive, and often elegant. Whereas unforunately it wasn't something that was given any emphasis in any of my physics courses. Some Universities offer a specialised dual program like 'Honours mathematical physics': the idea being you get a rigorous mathematics background. Unfortunately it didn't really carry over into the physics classes until maybe 4th year where MATHPHYS courses were being offered.

 

[WannabeNewton]

If you want mathematically rigorous physics then here: https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

and here: https://www.amazon.com/dp/0444515607/?tag=pfamazon01-20 (don't be fooled by the title-trust me )

 

[Theorem.]

If you want mathematically rigorous physics then here: https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Although this seems like a great resource that addresses rigor in physics, It is also only one book in one area of physics. Perhaps the OP is more concerned with the issue in general: How important is rigor in undergraduate learning? Should it be emphasized more? why aren't there more accessible undergraduate textbooks that share the same level of rigor as is taught in undergraduate mathematics?

I am not sure if this is what the OP was getting at but these are interesting issues in my mind. That being said my opinion is a little biased as a mathematician in training : ) I do however know that at my University many top notch undergraduate students in physics and mathematical physics find it to be an important issue and one that needs to be addressed.

 

[WannabeNewton]

I don't see any reason to include rigorous mathematics in undergraduate physics. I can't see it helping in any conceivable way; the hard part is the physics. I don't see rigorous mathematics making the understanding of the physics any easier. To give a good example, the book "A First Course in General Relativity" by B.Schutz has little to no rigorous point-set topology, differential topology, or Riemannian geometry in it but it still teaches the physics very well and in the end it is the physics of GR that is difficult; the mathematics of GR is trivial in comparison. Adding in the rigorous math might make some things more lucid but it is a HUGE time consuming task to teach rigorous math to undergraduate physics students on top of the (more important) physics that has to be taught. Of course this has never stopped physics students who want a rigorous math background from taking rigorous math classes regardless.

 

[Theorem.]

I don't see any reason to include rigorous mathematics in undergraduate physics. I can't see it helping in any conceivable way; the hard part is the physics. I don't see rigorous mathematics making the understanding of the physics any easier. To give a good example, the book "A First Course in General Relativity" by B.Schutz has little to no rigorous point-set topology, differential topology, or Riemannian geometry in it but it still teaches the physics very well and in the end it is the physics of GR that is difficult; the mathematics of GR is trivial in comparison. Adding in the rigorous math might make some things more lucid but it is a HUGE time consuming task to teach rigorous math to undergraduate physics students on top of the (more important) physics that has to be taught. Of course this has never stopped physics students who want a rigorous math background from taking rigorous math classes regardless.

Well it's definitely a complex issue. For the most part I agree with you here. For me I would have to say it's not so much about rigor as it is about creativity and problem solving. My physics classes did a poor job encouraging either. The first exposition i really had was with proof based mathematics, and I think that's why I am sometimes too quick to give "rigor" credit.

EDIT: I should add, I do think that a somewhat more proof based mathematics background should be required for most physics classes and/or programs. Math and physics go hand in hand in many instances and the (sometimes rigorous) mathematical background can help students better understand what is going on in their physics classes. I recall taking E&M and having my prof "teach" vector calculus in 2 days, no student had yet been exposed to the topic and the result was horrible. It wasn't until I actually took the mathematics that developed vector calculus that I was able to reflect on what had really happened. Unfortunately most other students that took the class with me are still puzzled.

 

[WannabeNewton]

Well proof based math classes can certainly be very fun with regards to problem solving, that's for sure so I certainly agree with you there. Intro physics classes can also be of a same nature if they are rigorous enough from a physics standpoint (MIT's honors mechanics and honors EM as well as UChicago's honors classes for the same subjects come to mind).

 

[Theorem.]

Well proof based math classes can certainly be very fun with regards to problem solving, that's for sure so I certainly agree with you there. Intro physics classes can also be of a same nature if they are rigorous enough from a physics standpoint (MIT's honors mechanics and honors EM as well as UChicago's honors classes for the same subjects come to mind).

Hmm it would be interesting to give physics a second look just for fun through such classes. I have been completely absorbed in pure mathematics for so long- It would be an interesting experience for myself to give it a look.

 

[WannabeNewton]

Go for it! Who knows, maybe you'll switch back to physics :p

 

[Theorem.]

Go for it! Who knows, maybe you'll switch back to physics :p

Haha I doubt that I really love the stuff I am working on : ) but it could never hurt to learn

 

[George Jones]

How important is rigor in undergraduate learning?

Using rigourous mathematics to teach physics courses in: quantum mechanics would be insane; quantum field theory would be impossible.

Should it be emphasized more?

Possibly a little more; also more formality. For example, Dirac's quantum mechanics book is formal, but is not rigourous

why aren't there more accessible undergraduate textbooks that share the same level of rigor as is taught in undergraduate mathematics?

Because, while physics uses mathematics, physics isn't mathematics.

Adding in the rigorous math might make some things more lucid but it is a HUGE time consuming task to teach rigorous math to undergraduate physics students on top of the (more important) physics that has to be taught. Of course this has never stopped physics students who want a rigorous math background from taking rigorous math classes regardless.

Yes.

To understand the rigourous mathematics in such textbooks, physics students might have to take more pure mathematics than they want to, i.e., they are physics students, and have to take loads of physics courses. Either that, or develop the rigourous mathematics in the physics texts, in which case there would not be room in the book or time in course to cover the physics material that a typical physics department wants included in a typical physics course.

Don't get me wrong; I think such physics texts are both useful and interesting, but I don't think that they should be used as texts for standard physics courses.

I recall taking E&M and having my prof "teach" vector calculus in 2 days, no student had yet been exposed to the topic and the result was horrible.

I agree. Unfortunately, this happens all too often.

It wasn't until I actually took the mathematics that developed vector calculus that I was able to reflect on what had really happened. Unfortunately most other students that took the class with me are still puzzled.

Don't physics student have to take vector calculus (eventually)? If so, why did they remain puzzled?

 

[Theorem.]

Because, while physics uses mathematics, physics isn't mathematics.

I agree with you here up to a certain extent. In many instances understanding the mathematics can go hand in hand with understanding the physics. This can, at least in some cases, involve the development of the maths itself. I know that going through rigorous development really helped me understand what was actually going on in the math used, and consequentially the physics. Although, I am not saying that this should make rigour mandatory- simply stressing that it can have huge advantages (and as it has been mentioned, there are also some big negatives).

Don't physics student have to take vector calculus (eventually)? If so, why did they remain puzzled?

Yes I believe so, but I think by the time it was done much of the essence of the course was lost. It was definitely an odd ordering. Quite unfortunate! Part of that is, of course, the students responsibility.