What if the lecturer is not rigorous?

Question

I study mathematics as my major subject and theoretical physics and statistics as my minor subjects. I found that, sometimes in physics or statistics lectures, the lecturer makes mistakes, like forgetting to prove that a series converges, or computing multi-dimensional integrals by using only one path. Once I spent three weeks to find a correct reasoning why one particular series converges. Should I say anything about these mistakes to the lecturer?

Answer 1

In physics we deliberately do not prove the series converges, because we are not interested in teaching concepts like convergence. Physics courses are not intended to be mathematically rigorous. It just is not one of the goals.

Answer 2

Let me add a thought following the other post, which asserts that this is not a mistake, pedagogically. Take this as a given: you don't need to "call out" the professor for failing to teach "properly." It is, however, still the case that you personally are wanting to dig into the mathematical foundations of these concepts more deeply, and find it important to your comprehension.

That's great! You might learn something really interesting, and might set yourself on a path to become a person who makes scientific advances by attacking these sorts of questions.

Now, I would suggest approaching your professor from that perspective, instead of considering it a problem with their teaching. Ask if there are books or other resources that the professor would suggest where you can learn more about the proofs behind these assertions. If the professor doesn't have good suggestions for you, try looking in places like Physics.SE. If you can't find a satisfactorily rigorous proof, it may well be that it does not exist (unlikely, but it happens), and that may be an interesting opportunity!

Answer 3

Echoing parts of the other answers, and some of the comments: first, it is inaccurate to declare omissions such as "proof of convergence" a "mistake". There simply is no absolute obligation to verify that all parts of the mathematics work as a physicist expects for other reasons. Yes, you or I and others might want to see the proof, that is, mathematical causality, but this is simply not obligatory. (Conversely, we can prove things without direct physical manifestations or physical reasoning...)

In fact, "convergence" is merely a simple form of what one might want, and itself not obligatory (much less its proof). Indeed, I have read that Poincare discovered in the late 19th century that a series expansion of a solution to a differential equation used for many decades (successfully) in celestial mechanics did not converge. Not that its converge was difficult to prove, but that it definitely diverged. But/and people had been getting correct numerical outcomes. Well, it was an "asymptotic expansion", ... but/and such expansions are more delicate in some regards (e.g., term-wise differentiation) than convergent power series, and the mathematical details were not filled in for several decades.

Another example is P.G.M. Dirac's book on quantum mechanics, which used distributions and unbounded operators in manners that would not be justified for 20 years (in the work of L. Schwartz). I have read that J. von Neumann and others were considerably disturbed by the lack of "rigor", or even the pretense of it, which motivated them to try to provide such... Nevertheless, the predictive and explanatory power of Dirac's work was unquestionable, and it would have been ridiculous to have dismissed it because he couldn't provide proofs, or didn't care to.

As remarked above, it really does appear to be that hard-to-justify mathematics is fairly tolerable when it quasi-magically predicts physical details, or quasi-magically proves to be an accurate book-keeping or computational device for observable physical phenomena.

Yes, we should think very differently when/if we aim to "subvert" such mathematics to purely mathematical situations, where there may be no genuine physical phenomenon to observe and test. No, I do not have that physics-y intuition that suggests (to my perception) outrageous mathematical manipulations, so I myself definitely need either or both pithy examples and persuasive (!) proofs that assure me there's some "causality" beyond the literal tangible world. But, in fact, history suggests that much interesting mathematics has come from "outrageous" mathematical stunts by imaginative physicists, so such stuff is a good source!

And, yes, sometimes the purely mathematical justification for obviously-necessary mathematical tricks in physics is far more sophisticated than the immediate physical explanation/motivation/phenomenon. Sure, sometimes the mathematics is not hard, and simply omitted due to lack of interest. Sometimes the mathematics is profoundly difficult, or in fact impossible in a particular year with technical limitations of the time. That fact, that has appeared over and over, is philosophically and scientifically provocative in itself, in my opinion.

So, yes, I, too, have been disturbed by reading physics-y accounts that did (to my perception) crazy mathematical things. Long ago, I thought that this was a definite failing, and that rigor was required, and possible. By now I see that these situations are much more complicated than that, and that gauging any particular instance may be unexpectedly non-trivial!

Answer 4

Without detracting from the merit of the more philosophical answers, here's a simpler practical suggestion:

• Go see your professor after class or in reception hours.
• Tell them that since you're a Mathematics minor, you find the mathematical reasoning important to follow.
• Tell them that you sometimes cannot tell whether a step they makes is actually trivial, or might take a lot of time/effort to justify rigorously.
• Ask them that, when they are making a 'mathematical leap' (the second kind above), they tell the class specifically that they are doing so. For example "this step requires a proof, but it is a purely mathematical one which we will not delve into."

You can of course also ask them for a textbook with more mathematical rigor.

Answer 5

In theoretical physics, for various reasons, standards of mathematical rigor tend to be looser than they are in math. Individual physicists' preferences vary widely, however. In my experience, which seems to be echoed by Pete L. Clark's, many physicists tend to default to a looser standard of rigor while teaching, so your lecturers may or may not think about their course material at a much tighter level of rigor than the one they present it at.

You're definitely not alone in being frustrated by leaps of mathematical faith in physics lectures, and spending a great deal of time trying to fill them in. Here are some things I'd recommend doing to help deal with this, based on my own experience.

• Do try talking to your teacher outside of class about mathematical gaps that confused you. You may find that your teacher knows exactly how to fill them in, and simply omitted the details from their presentation in class.

• Do seek out other mathematically-minded people at your university, especially more experienced people, and talk to them about the things that confused you. As Pete L. Clark notes, many mathematically-minded physicists (and physicsy-spirited mathematicians!) have a private stash of rigorous insight into the less rigorous parts of a typical physics class, built up over years of experiences like yours. At some universities, the math department can be a gold mine of knowledge like this.

• As a corollary, do write down your own work when you fill in the gaps yourself! Someday, the three weeks you spent proving that series converges might save someone else three weeks of trouble.

• Do remember that not everything in physics has been formulated rigorously, and some topics are notoriously resistant to mathematical formalization. When you're confused by reasoning used in a physics class or the physics literature, it can be hard to tell whether you've encountered a small crack that can be paved over with a few hours of thought, an big gap that can be bridged using sophisticated techniques hidden in some corner of the math literature, or a gaping chasm that people have tried and failed to cross for decades. This is another reason talking to more experienced people can be helpful.

On the other hand, here are some things I'd recommend not doing.

• Don't think of gaps in mathematical reasoning as mistakes, especially when you're talking to other people about them. This doesn't match the way most physicists approach mathematical reasoning, and it can turn your conversations unpleasantly confrontational.

• If you've tried bringing your confusions to your teacher after class, and they've been consistently unable to help you, don't keep asking, especially if they seem annoyed by your problems. Your teacher may just prefer a looser standard of rigor than you, and there's nothing you can do about that. Seek out other sources of help instead.

• Don't ask about leaps of reasoning during class. If your teacher doesn't know how to fill them in, nothing is gained. If your teacher does know how to fill them in, that means they've made a conscious decision not to, so they might prefer to talk to you outside of class.

• Don't feel responsible for filling the mathematical gaps in your physics classes. In the comments here, people have said that "students can (and should) check that the claims made by their physics profs do indeed hold," and that "it's common to learn these concepts and their proofs rigorously in the math class." In my experience, those things just aren't true. You'll hit problems that you don't have the tools to resolve, and you'll hit problems that nobody has found the tools to resolve. Your confusion is not your fault.

• Don't feel like your teachers are responsible for filling the gaps either. They're just doing physics as physics is generally done, and sometimes as it has to be done.

• Don't spend too much time and energy trying to fill the gaps. Pancaking yourself against the far wall of the canyon a few times is okay, but at some point it's best just to walk away. You may come back later and discover that you've gained the tools and knowledge you need to get over, or that there's a bridge just a few miles away, or that getting over isn't likely to happen any time this century.

• But, with that said, don't stop looking for more rigorous and less confusing ways to understand physics. Efforts to shore up the mathematical foundations of physics have proven very worthwhile in the past, and I firmly believe that they'll keep proving worthwhile in the future. They may feel thankless, but they're not worthless, and I think they're great things to read about and think about when you have the time and energy to spare.

I hope at least some of this advice is helpful for you. If you ever bring your mathematical physics troubles to Math.SE, I hope I'll see your question, and I hope I'll have the time and the knowledge to help answer it.

Answer 6

I hope the OP won't be too offended if I say that this question seems to show an immature understanding of the meaning of rigor and the relationship between different academic disciplines. As an illustration, consider the following problem, which could be used as an exam question on a freshman physics or calculus exam.

A uniform rod with mass per unit length b is initially upright and at rest in a gravitational field g. At t=0, the rod is released. At a later time t, find the rate at which mass flows past a horizontal surface passing through the rod.

Those of us who are physicists or mathematicians can easily find "the" answer, which is bgt.

Now suppose we want to make this a little tougher so we can use it as an interview question for a potential TA. We state the question, but now we ask specifically for a high level of rigor in the answer.

If the field is math, a good answer might be something along the following lines. The solution of the problem involves a derivative. One way of defining a derivative is as a limit, and limits are in turn defined using epsilons and deltas. Here's a rigorous epsilon-delta proof that the limit we're talking about does converge.

Now suppose the field is physics. (I'm a physicist.) An example of a nice, rigorous answer would be one in which the interviewee explained why the observable we're talking about cannot possibly converge to the expression bgt. A sufficient argument for nonconvergence would be to point out that the rod is made of atoms, so the motion of mass across a horizontal line starts to look discrete once we get down to a certain scale. (An even nicer answer might focus on effects that might be more practically observable. For example, when the support of the rod is released, the disturbance travels outward through the rod at the speed of sound, not instantaneously.)

Both of these are rigorous approaches to knowledge, but they are different notions of rigor. One emphasizes the internal self-consistency of mathematics. The other emphasizes the careful consideration of how mathematical models relate to reality, which is more complicated.

Answer 7

I would like to answer based on my personal experience. During my undergraduate years, I used to double major in mechanical engineering and pure mathematics. I used to like my mechanical engineering classes until I first got exposed to real analysis and mathematical rigour. After that, I steadily became more frustrated and skeptical of the correctness of the methods used in my engineering classes that I stopped taking my engineering major seriously, skipping lots of classes, doing the bare minimum to get the grade etc.... Looking back at my younger years, I think my reaction was immature. I also wasn't lucky enough to meet someone older than me who has experienced this dilemma, that partially explains my immature reaction. Now let me answer your question:

Should I say anything about these mistakes to the lecturer?

In 99% of the cases, its not worth it. (The 1% case is if it happens that your instructor is aware of the rigorous mathematical foundation of his/her subject). What I encourage you to do instead is to try your best to understand what your instructor and textbook are saying and then axiomatize it or rigorously justify any claims. This is similar to how Cauchy, Weierstrass did their best to understand calculus in its non-rigorous form and were able to turn it to real analysis. This is a good mathematical exercise and might even help you understand your physics or stats classes better (However, make sure you have learnt enough mathematics like differential geometry and probability theory first so that you don't run into the problem of having to discover already known mathematics)

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I will reply to one of the comments by Pete L. Clark here, as my reply will be a good addition to my answer.

"... Ever since high school, things were fully explained in my mathematics courses and very erratically explained in my science courses. We care about convergence of series because divergent series may be absolutely meaningless. How can understanding why the procedures one is performing are meaningful not be part of science education?"

One has to note that there is a difference between cultures of pure mathematics and say engineering/science. The criterion of what counts as "understanding" or what counts as "full explanation" is different in these cultures. For a mathematician, what counts as full explanation is rigor which basically means a sequence of lines of proof in a formal system whose rules are clear enough to be checked by a computer, so that in principle you can write a computer program that distinguishes good arguments from flawed arguments. For an engineer, "understanding" is being able to reliably make engineering designs that work when implemented. I guess for a physicist, "understanding" would be having a semi formal system whose predictions match experiments. If some mathematically dubious infinite sum manipulations actually succeed in predicting the outcome of an experiment consistently, then that's enough for a physicist to count as "understanding". A mathematician however, might make a big deal of proving the jordan brouwer separation theorem even in the non pathological case (say smooth or piece wise linear). This might seem a pointless intellectual activity to non-mathematicians, but remember that thinking about these seemingly pointless questions is what opened the way to the field of topology and its more complex ideas which later on found applications to physics and computer science.

Answer 8

Generally, you better assess yourself well enough to call out a lecturer publicly, otherwise, there's plenty of opportunities to make corrections privately. That is generally the more politically-correct path. Beyond that, you should know that by choosing to address it publicly, you are (consciously or not) engaging in a battle of power. Such a battle can have positive or negative outcomes.

More specifically, I have questions for you: why would a lecturer have to prove that a series converges? Further, shouldn't multi-dimensional integrals always have the same answer regardless of the path? Otherwise, there's a deeper problem in the formulation of the expression (like including terms from a domain that doesn't belong there).