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?

And then you hear "we see this series diverges to infinity. Let's sum it". And in some situations, this is a perfectly reasonable thing to do, since you sum it in some sense that has physical interpretation. — Davidmh, Sep 23 '15 at 09:06
"forgetting to prove" something mathematical is not a "mistake" in physics. The only "proof" I am interested in as a physicist is experimental validation. — Calchas, Sep 23 '15 at 13:37
Note that for every student who wonders about the convergence of the series, there is a student who is completely baffled by all the math, and struggling to cope. This student needs a high level, intuitive overview, which is what the unrigorous lecturer is likely trying to provide. He's only got two hours. — Peter Bloem, Sep 23 '15 at 14:33
This is the main reason why I switched my major from engineering to pure mathematics - my math brain hated the hand-waving and inexactitude of the math done in science classes, and the rest of my brain doesn't love science enough to get over it. At least you can take it as a sign that the mathematical part of your thinking is on the right track. — Todd Wilcox, Sep 23 '15 at 15:37
Particularly in engineering, it's common to learn these concepts and their proofs rigorously in the math class. The physics lecturer then assumes that this knowledge is already well established. Often a physics course will have a math prerequisite for this reason. — , Sep 23 '15 at 16:12
"Once I spent three weeks to find a correct reasoning why one particular series converges." doesn't this answer it already? Are you studying convergence of series or physics? — Bakuriu, Sep 23 '15 at 18:32
These are not mistakes. Science lectures generally do less footwork on proving mathematical properties, for the sake of saving time and for the sake of actually focus on science. Most of this mathematics can be found in literature and has minor importance. — Greg, Sep 23 '15 at 23:16
If physics professors had to prove every little mathematical assertion they used, when would they have time for teaching physics? And what would be left for the mathematicians to do? — Kevin, Sep 24 '15 at 05:29
"Mathematicians are apt to regard phycisists as a somewhat lower category of being, who deal with knowledge that's somewhat certain, or at least fairly probable." ~ Dr Greg Moore, University of Toronto — Dɑvïd, Sep 24 '15 at 20:17
Is this really a "mistake"? Or would it be better categorized as an "omission"? — J.R., Sep 25 '15 at 10:10
This seems to confounding the lecture content with the course. Lectures are introductions to topics, and cannot be expected to cover everything you should know in complete detail. That's what are the extra reading, and working through exercises is for. For every hour of contact time (lectures and tutorials) I'm expect my students to be doing 2-3 hours of independent study. University isn't high school. — beldaz, Sep 25 '15 at 22:34
I like this question, I used to double major in mathematics and mechanical engineering . The frustration is even more in engineering classes than in physics or stats classes — Amr, Nov 26 '21 at 19:03

score 106 · Accepted Answer · answered Sep 22 '15 at 22:03

106

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.

answered Sep 22 '15 at 22:03

Anonymous Physicist

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Interesting. Why do you call such things as series if you don't use them like mathematicians? Why don't you create some new structure that fulfills your needs in particular situations? – studying Sep 22 '15 at 23:03
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You may have misunderstood. We use series because they get the right result. And we do care that they converge, we just don't futz around with the issue during class. We learned about series convergence in math class, and the serious theorists worry about it when writing papers (experimenters like myself generally don't have to because someone else already did), but class time is far to precious to waste on the matter unless the bounds of convergence are going to be important in this class of problems. – dmckee --- ex-moderator kitten Sep 23 '15 at 00:03
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Related on [physics.se]: http://physics.stackexchange.com/questions/27665/the-role-of-rigor – dmckee --- ex-moderator kitten Sep 23 '15 at 00:06
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This is the standard answer and I upvoted it. However I have always been profoundly frustrated by this and felt locked out of most of physics because of it. 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? – Pete L. Clark Sep 23 '15 at 01:16
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It is especially frustrating because it seems to be, as is hinted at in the comments, an issue of science pedagogy rather than science itself. If really no one knows that a series converges, that's a huge problem: e.g. a problem of this kind got Feynman, Schwinger and Tomonaga their Nobel Prize. All leading physicists I've met understand convergence of series just as well as I do; the difference is that this knowledge is kept mostly private in physics pedagogy. I don't think physics places any less of a premium on "true understanding" than mathematics, so this is a weird state of affairs. – Pete L. Clark Sep 23 '15 at 01:19
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@PeteL.Clark in the context of physics, "true understanding" occurs when theory predicts experiment. Whether the theory obeys some set of axioms is not very relevant. – Anonymous Physicist Sep 23 '15 at 02:46
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(I didn't mention "axioms", and I think convergence of series has little to do with them.) I think you're underplaying the use of math in physics: for hundreds of years, most or all physical theories have been written in the language of math, so if you don't understand the math then you don't really understand your theory. Just because it correctly predicts experiment so far doesn't mean that you will correctly apply it to future predictions. Isn't understanding the math indispensable for that? – Pete L. Clark Sep 23 '15 at 03:59
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If you're asking for complete rigor, how deep do you want to recurse? That could go all the way back to deriving the basic principles of math... At some point it becomes sufficient to say "we know this; it may or may not be trivially obvious; if you doubt it you're welcome to re-derive it on you own time but that's orthogonal to the point we're trying to teach now." – keshlam Sep 23 '15 at 04:52
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@PeteL.Clark And the physics department standard reply is that the math department is supposed to provide all those tools. The students can (and should) check that the claims made by their physics profs do indeed hold, but on their own time. (There's already not enough lecture time to bring undergrads past the year 1920 or so without the extra math.) Otherwise what good are all those math classes the physics majors are required to take? If the math department doesn't think this is their responsibility (as I've seen and is fair enough), then the blame falls on interdepartmental communication. – Sep 23 '15 at 06:35
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@Chris: Yes, students should learn about the theory of convergence of series in their math classes. That's not what the OP is saying: he's saying that series are being used without checking that they are convergent, and that he spent three weeks making this check himself. "Otherwise what good are all those math classes the physics majors are required to take?" I find the idea that a student who receives some undergraduate mathematical training should be completely self-sufficient in filling in mathematical details to be rather strange:... – Pete L. Clark Sep 23 '15 at 06:53
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@PeteL.Clark for understanding a particular part of physics it's sufficient to note "yes, this series converges and that integral is valid; the authors proved that but these details are out of scope for this class". If you're making a new statement, rigor is important; but the theory and practice on how to do that is supposed to be obtained during the math classes. – Peteris Sep 23 '15 at 06:55
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....I am a dozen years past a math PhD, and I'm not able to fill in all mathematical details unassisted. I have particular trouble when it comes to physics, and it was why I stopped taking physics classes many years ago. I think that one's mathematical training needs to be far in excess of what is used in order to be able to fill it in on the fly when studying physics. For instance, after getting my PhD I taught multivariable calculus, and I was pleased to find out that I was at 27 able to fill in the mathematical details behind physical explanations. I couldn't do this at age 17. – Pete L. Clark Sep 23 '15 at 06:57
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I didn't say this material should be covered during lecture time (and I think the OP's characterization of this as a "mistake" by his lecturer is inaccurate; by the standards of the respectable, intelligent community of physics educators, this is certainly not a mistake). Probably it should not be, most of the time. What frustrates me is that this material -- well known to many physicists -- is so thoroughly absent from physics textbooks. The knowledge is there but it is not being conveyed. This is a shame, and it turns off many mathematically minded people...I think unnecessarily. – Pete L. Clark Sep 23 '15 at 07:02
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@Peteris: " If you're making a new statement, rigor is important; but the theory and practice on how to do that is supposed to be obtained during the math classes." Infinite series are indeed taught in math classes. There one learns that in order to sensibly use a series it must be shown convergent (or possibly one should use a summability method...). Showing convergence need not be routine in any sense: the Riemann hypothesis can be stated in terms of series convergence. The OP spent three weeks on this. Since this isn't "new material", shouldn't there be resources available for him? – Pete L. Clark Sep 23 '15 at 07:06
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@PeteL.Clark I think it is not in the books because it is not in the normal way of thinking of a physicist. Since we are modelling reality, solutions usually exist, and are unique; convergence can be handwaved away noticing that the next term is orders of magnitude smaller... all the things that make a mathematician's teeth cringe of imprecission. There are cases where one has to be careful and check, but most of the time, we don't bother because the equations are well behaved enough (and of course, we get bitten when we assume what is false!). – Davidmh Sep 23 '15 at 09:15
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@PeteL.Clark There is only limited time in the curriculum, a text book publisher will only be happy with so many pages, and a student will only endure so many hours of instruction. If series convergence and a bunch of other mathematical principles go in, at the level of rigour a mathematics undergraduate would accept without sneering (I used to be one), there will be no time for anything interesting from physics. Your statement about "mathematically minded" students being put off cuts both ways; we don't want those without a taste for mathematical rigour put off from physics either. – Calchas Sep 23 '15 at 13:23
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@PeteL.Clark In my actual work as an experimental physicist, if I wanted to know whether a series converges, I would google it or ask Mathematica. I vaguely remember the underlying theory, but to be honest, it doesn't really matter to me. – Calchas Sep 23 '15 at 13:26
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@studying If we were interested in creating mathematical structures, we would be mathematicians. – Calchas Sep 23 '15 at 13:32
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@Calchas: Frankly I find this kind of argument very sad. In particular the claim that justifying the mathematics that one is doing would take up a prohibitive amount of space in the textbooks: on the one hand I see little evidence for that, and evidence against is the number of mathematically minded physicists who know exactly how to explain things but generally choose not to; on the other hand: physical (!!) textbooks and curricula determined by textbook publishers is so last millennium. There is plenty of space on the internet. – Pete L. Clark Sep 23 '15 at 14:43
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@PeteL.Clark Sorry to bring misery to your day. There is always a compulsory mathematics course covering the basics, the students generally prefer a concise and clear curriculum (and exam) rather than it being spread all over the internet, and the interested student is free to work through the mathematics in other topics if she likes and has the time. I don't see what it would add to the physical intuition to sit through dozens of proofs of convergence theorems. I would rather the students spent that time in the lab doing physics. – Calchas Sep 23 '15 at 14:49
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And the idea that making material available to population X could discourage population Y: come on, that's a terrible argument. It's totally antithetical to the way academia works. Finally, the argument that you are a professional X, that you don't personally use A in your work and therefore you discourage A from appearing in undergraduate instruction is also discouraging. I think the cultural incompatibility between the contemporary math and physics communities is not inevitable but rather a big missed opportunity. – Pete L. Clark Sep 23 '15 at 14:52
@Calchas: "I don't see what it would add to the physical intuition to sit through dozens of proofs of convergence theorems." I said that this material probably should not be covered in class most of the time. Your comment about the internet doesn't make much sense to me: many people in the world learn material from the internet; those that are taking undergraduate courses and otherwise. I've put over 2000 pages of material on the internet and I get correspondence from people all over the world about it. But when it comes from the courses I teach, fully reading the notes is not required. – Pete L. Clark Sep 23 '15 at 14:56
@PeteL.Clark Well in that case maybe we aren't disagreeing. :) – Calchas Sep 23 '15 at 14:59
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Comments are not for extended discussion; this conversation has been moved to chat. – aeismail Sep 23 '15 at 22:35
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You guys forget that many of these theorems are true only with an "under physical conditions". E.g we can talk about a Taylor series and the general assumption that which elements will be dominant, based on our physical intuition, while it can be very hard to mathematically prove (or may not even true) the same thing for a any mathematical function. – Greg Sep 23 '15 at 23:24
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I understand what you're saying, but there's a little nagging point here: "like forgetting to prove that a series converges, or computing multi-dimensional integrals by using only one path". From the wording, it sounds like the professor is simply skipping over these issues without mention. Shouldn't the professor at least mention what assumptions are being made and what shortcuts are being taken? This is important not just to math students, but also to science students. You must have an understanding of what you can and cannot use your tools for. Students that don't will misapply them. – jpmc26 Sep 25 '15 at 19:30
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@PeteL.Clark Could you tell me how much time it took you, until you realized that physics wasn't something for you ? I'm taking physics courses now, although math is my major and I have similar mixed feelings and concerns that you expressed here. – resu Sep 26 '15 at 13:54
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It'd be an interesting pedagogical experiment to make a mathematical supplement to some of the standard physics textbooks. Then a student who was bothered by these issues could have their questions answered easily in one place. – Noah Snyder Sep 27 '15 at 14:26
The issue in my oppinion is that the students learn that you don't need to check if the series is convergent, and that they should not care if it is... I see from time to time (which means way to often) physicist drawing the wrong conclusion because they think a series is convergent and it is not... What you are saying is that you guys are training the next generation this way, even if I am sure this is NOT your goal... – Nick S Sep 25 '19 at 20:39

score 78 · Answer 2 · answered Sep 23 '15 at 02:41

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!

For lists of physics books geared toward mathematicians there is this question on Math.SE (advertisement: where one can even find an answer of mine :-) ). — Massimo Ortolano, Sep 24 '15 at 05:41

score 31 · Answer 3 · answered Sep 23 '15 at 20:41

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!

paul: This is a thoughtful answer. First of all, you're right that divergent series are much more useful in physics than they are in most branches of mathematics. To my mind, this is an argument that physicists need to pay more attention to these issues than most mathematicians. It is true that asymptotic expansions can be used successfully even when the mathematical formalism is not known. It is also true that they can be used unsuccessfully, and there are examples of both... — Pete L. Clark, Sep 23 '15 at 22:29
...You're also right that sometimes the "rigorous" (not my favorite word...) mathematical justification of something which is being used by physicists is very sophisticated, that sometimes no one knows how to mathematically justify what is being physically used, and that this is not necessarily a bad thing: in fact quasi-magical physical predictions are one of the payoffs of cultural relations between mathematics and physics. I would like to see more that, not less. What I don't believe is that "it doesn't matter" whether physical theories have a solid mathematical grounding. — Pete L. Clark, Sep 23 '15 at 22:33
....Isn't that idea some kind of clubhouse handshake that older undergraduate physics students teach younger undergraduate physics students, a certain kind of machismo? And now some people here are repeating this. But of course it matters: brilliant people, some called "mathematicians" and some called "physicists" have worked very hard to put physical theories on a firm mathematical grounding. And yet there is little to no trace of these efforts in undergraduate physics. That's what I object to: not everything is known, but some things are. Those that are should be told. — Pete L. Clark, Sep 23 '15 at 22:37
@PeteL.Clark, yes, of course, incidentally, there is some exclusionary riff (human sociology) that confounds everything. E.g., rationalizing limitations or failings as virtue. :) Undergrad and graduate-student perceptions of things (in my experience) tend to drift into caricature/over-simplification/machismo... if only because they're adolescent (nevermind the historical/evolutionary sense of adulthood at age 13, etc...) So, we have bowdlerization + clubhouse/insider-trading denial. — paul garrett, Sep 23 '15 at 23:15
@PeteL.Clark This question of mine is somewhat off-topic, but I'm asking you now, because this question here is the perfect opportunity to do so and I always wanted ask a knowledge-able, more older (than me, that is) mathematician this: Have you heard of or do you know any people that also started out as mathematicians having the same frustrations/complaints with physics that you mentioned in different comments to answers to this question (I am a mathematician doing a PhD right now and have absolutely the same qualms you describe) but succeeded in overcoming their "need for rigor" and [...] — l7ll7, Sep 25 '15 at 19:49
[...] now do significant work in physics (physics so theoretical that it's almost mathematics doesn't count!) ? And, if it's not to personal to ask (it it is, please ignore it), how did your choice not to study more physics feel long afterwards, later on: Were you happy to have quit something where you had the impression of being "locked out" or to you still feel (as I occasionally do, since I have already stopped some time ago, early in my career, studying physics) a slight frustration that this was one castle of knowledge that always forbade you entrance ? — l7ll7, Sep 25 '15 at 19:57
@PeteL.Clark I second user10324 with his inquiry! It would be great if you could answer his questions, even only some of them. I'm also a math student that does not come to gripes with physics! — MyCatsHat, Sep 26 '15 at 13:46
@PeteL.Clark Yes, please, please give us struggling mathematicians an answer (it is a blessing for my tormented soul to see that other mathematicians also have problems with physics). And please see also my comments on the other answer.... — resu, Sep 26 '15 at 13:56

einpoklum · Answer 4 · 2021-11-26T21:20:47.257

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.

score 12 · Answer 5 · edited Apr 13 '17 at 12:49

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.

This is a great answer. – Pete L. Clark Sep 26 '15 at 23:04 — Pete L. Clark, Sep 26 '15 at 23:04

score 10 · Answer 6 · answered Sep 24 '15 at 00:44

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.

Perhaps you run in different mathematical circles than I do, but I don't think I've ever met a mathematician who would consider the definition of a derivative to be a useful part of a rigorous solution to the problem you describe. Rigor in mathematical physics isn't about epsilons and deltas for their own sake; it's about saying very clearly what you're trying to calculate, and arguing very convincingly that what you're doing is going to calculate it. — Vectornaut, Sep 26 '15 at 19:02
If you haven't done so already, I would encourage you to poll some mathematical physicists around you about what they'd consider a "rigorous" answer to the problem you posed. I, for one, would be curious about the results. — Vectornaut, Sep 26 '15 at 19:03
@Vectornaut: I think you're missing the point. The point is not that the rigorous underpinnings are nontrivial to practitioners of either field. They are trivial to both, and that's why I chose to dramatize the story by making it a job interview for a TA position -- so that we would have some motivation to ask someone to consider the trivial foundational issues. The point is that a practitioner of discipline A, using the correct notion of rigor for A, says that foo converges to bar. Meanwhile the person in field B says that foo does not converge to bar. They're both right. — , Sep 27 '15 at 01:39
But, in your story, the practitioner of math is not using the correct notion of rigor for math. That's what I'm trying to say: if you want your story to communicate the idea it's supposed to communicate, you should rewrite it so that the practitioner of each discipline is using the correct notion of rigor for that discipline. — Vectornaut, Sep 27 '15 at 18:26

Amr · Answer 7 · 2021-11-26T19:50:04.673

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)

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.

@Pete L.Clark I replied to one of your comments here. – Amr Nov 26 '21 at 19:51 — Amr, Nov 26 '21 at 19:51

score -2 · Answer 8 · answered Sep 23 '15 at 21:22

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).

What if the lecturer is not rigorous?

8 Answers8