What am I supposed to be learning with long proofs of the main theorems in class?

It seems like this is exclusively how (most) people teach graduate/upper div math classes. They go through the proof of some big theorem, sometimes it might take two lectures. It's obviously important. But I honestly have no idea what I am supposed to be getting from this. It honestly seems useless to me. I know it isn't. I know it's a rite of passage to prove xyz theorem. But is this really worth sometimes two class periods?

I honestly don't know what I'm supposed to be learning. These proofs require unique logic a lot of the time, and this logic is difficult to transfer to other problems.

In lower div classes, it would be informal explanations of things followed by a lot of examples. I'm not saying that that's the way to teach functional analysis for example. But at least it made sense to me. I know what I was supposed to be learning.

I can go through all the details myself, I go to lecture for motivation/intuition/things you can't normally get from a textbook. The proof of Riesz representation theorem for example can be found in any text. Why can't I just read it on my own? It's much easier to read proofs like this on your own rather than in class.

• Sometimes you are just learning what a proof of a non-trivial (and non-computational) fact looks like. Most students will not read the proof on their own. Many of the remainder won't recognize what parts of the proof are important. Sometimes the lecturer just really likes the proof. – Adam Dec 7 '15 at 4:49
• Perhaps you should ask yourself what you expect to get from these graduate/upper div math classes.  Presumably, you expect to get an education.  What kind of education do you want?  What do you expect to do with it?  Why are you taking graduate/upper div math classes? – Scott Dec 7 '15 at 6:17
• @Scott That doesn't really answer my question, no. – user5108 Dec 7 '15 at 6:29
• Funny thing happened to me recently. I remember taking a topology course as a beginning grad student and being completely lost by the proof of Urysohn's lemma. I spent the rest of my Ph.d. studying things which do not involve a lot of point set topology. Sat in on a friend's class toward the end where prof was proving Urysohn's lemma, and the theorem and proof now seem completely obvious! The students in the class were all lost. So clearly I have gained something through the process of going through graduate school... – Steven Gubkin Dec 7 '15 at 20:26
• Sometimes (and it's more often than not), you have to understand the proof to understand the theorem. Why the assumptions are necessary? Where they are used? What would happen if we took them out or assumed something else? If you can't answer these questions (at least partially, it happens that these might be some very hard problems), your intuition is lacking (of course, that does not mean being able to answer is enough). – dtldarek Dec 8 '15 at 0:58

6 Answers

I agree with the sentiment in this question. I too often feel that lecturers go through a detailed proof because they think that everything must be proven pedantically to be able to use it. Sometimes unnecessarily complicated proofs are skipped, but not often enough to my taste.

But there is a point to proving these "big theorems". The proofs contain new ideas and those ideas will often become useful later, however unlikely that feels when you first encounter the ideas. The new clever ideas should be explained in lecture time, but the entirety of a technical proof may be too much. Reading a book or lecture notes with supporting exercise problems seems like a better way to go than a full lecture proof.

I hope lecturers would feel less of an urge to prove everything. What to skip and how to give students access to the proofs if they want them is another issue. I want to skip this question because the OP has the point of view of a student. It can be hard to persuade the teacher to change their ways; I will let it for you to judge whether or not that is reasonable.

As a student, if you feel that a proof given in the lectures is too much, try to look at it from a distance. Try to see the big picture instead of trying to understand all technical details. If the proof can also be found in a textbook or other source, there is little need to take notes. Ask the lecturer for intuition and explanation.

If you feel that the proof is way too hard for you and you don't get the big picture either, you can ignore it entirely. It's perfectly fine to occasionally take a theorem as a black box and just learn to use it (as opposed to learning how to prove it and understanding why it is true). If the situation keeps bugging you, you can go back to the result later and fill the gap. A year or two more of studying mathematics will make the proof more accessible to you. I see no reason to hurry to learn everything; it can deter you from making the progress you can make.

• Thanks for the reply. I definitely agree that sometimes they DO contain certain important ideas. For example in model theory, we would often prove things by induction on the complexity of the formula. I used this in problem sets often. I often try to look at the big picture, but sometimes it's difficult without key ingredients and I feel like they can be missing, where the proof of the main theorem takes precedence. I of course won't try to persuade any prof, it's their class and they obviously know more than me (otherwise I wouldn't be taking their class). – user5108 Dec 6 '15 at 22:10
• @avid19, you are welcome! Your professor certainly knows more about the topic than you, but it does not imply that he knows more about how to teach mathematics effectively than you. Not all professors excel at teaching and making the material accessible to their students. I believe many of us know sad examples of this kind. – Joonas Ilmavirta Dec 6 '15 at 22:14
• +1 for "If the proof can also be found in a textbook or other source, there is little need to take notes. Ask the lecturer for intuition and explanation." Too often students are too busy writing and not busy enough engaging. – Aeryk Dec 7 '15 at 20:44
• "It's perfectly fine to occasionally take a theorem as a black box and just learn to use it (as opposed to learning how to prove it and understanding why it is true)." I like this a lot. – user5897 Apr 27 '16 at 0:48

Hmm apparently I will be the dissenter here. I think that long proofs taught in lectures are very much a good thing. This is particularly true for hard proofs. I will try and split the reasons why I think so into a couple of points.

• Hard proofs are no only hard to create but also hard to learn on your own.

Have you tried learning a complex proof on your own from the book? This is usually quite hard. It gets much harder when the proof is just in an article.

• When learning things on your own you often miss many details.

Quite often when you are reading a proof in a book you will miss places where the author performs a non-trivial step which might seem intuitive because they don't point it out. You can also miss steps which are actually in the book but may not be emphasized.

• Very few theorems you learn in upper div/lower grad level classes have proofs which are not eminently reproducible.

Pretty much every proof you will do in a class at this level will have methods and ideas which are the stock in trade of the field. Whether these are bounding methods for analysis, diagram chasing for algebra or c.c.c. proofs in set theory these methods come up time and again in the field. You can never see too many of them.

• When the professor explains a proof you can ask questions.

This in my mind is the biggest boon of these proofs. You can ask about the proof as it's being built. Any nagging doubt, any confusion or missed step can be clarified on the spot. Often you can ask questions which highlight misunderstanding or extend your knowledge of why and how a given method is used. This is never the same if you read the proof on your own and then ask questions later.

• Good professors will highlight the essential points of the proof presented.

This is another important thing. When reading a proof it is much harder to tell which part is just easy formula pushing and where the main idea of the proof is. A good professor can also explain where you might be able to relax or strengthen conditions and what results that can give.

All in all I think these reasons explain why teaching proofs in class is a good idea.

• So would you say, in summary, because the art of dissecting hard proofs is a valuable skill? – corsiKa Dec 7 '15 at 18:16
• @corsiKa Certainly that's one part of it. But to be perfectly honest unless you're a math genius you are getting a lot more out of the proof presentation then learning how to dissect proofs. I doubt the claim that it's easier to read Riesz representation theorem in a book then listen to it in class is true (and that's really an easyish theorem). In general any hard theorem in a book will have many parts of the proof omitted as they are too trivial to mention. This makes it hard to understand and tends to lead to the "oh sure" approach and then you miss essential points. – DRF Dec 7 '15 at 18:31
• +1 I would never have come close to understanding the Implicit Function Theorem without the wonderful professor I had going through it the way he did. I would add what to me is a very important point: Understanding the proof helps you understand the theorem. Even if nothing from the proof were useful, the theorems proved are building blocks for later math, and understanding them thoroughly is critical to using them effectively for later work. – Todd Wilcox Dec 7 '15 at 20:01
• "understanding them thoroughly is critical to using them effectively for later work." In my experience, not really. You couldn't possibly do research without taking some things for granted. Every paper I've read makes references to other papers. If I had to stop reading a paper every time there was a theorem from another paper, I would never complete reading a single paper. @ToddWilcox – user5108 Dec 8 '15 at 0:07
• @avid19 I personally strongly disagree with that approach even for research level math and I'm fairly certain many researchers do understand each theorem they use. But even if that weren't the case, we aren't talking about the cutting edge of research here. Implicit function theorem or Riesz representation theorem and any other theorems you are likely to learn at the level the OP is talking about are the absolutely basic theorems. You should be able to prove them any time anywhere. Heck I haven't done any linear algebra in years and I could prove Riesz's theorem from scratch. – DRF Dec 8 '15 at 5:35

This is a great question. Here are some thoughts on it.

A theorem statement is a sign of an idea that tends to be useful in the pattern of mathematical inquiry in a given subdomain. A good theorem adds clarity to a subject in the sense that it can often be used to answer questions that arise. This is really the main reason point in demarcating theorems…they point to ideas that explain things often where other ideas fail to explain or resolve.

One might take the extreme view (as Halmos once wrote…perhaps facetiously) that one should never read the proof of a theorem. One should try to prove it for oneself and then, after checking that it is true, just use it.

Another perspective is that a useful theorem is a signpost for a cluster of useful ideas. The proof of a major theorem by its very nature is a small world of ideas that were essential to the resolution of a problem. One might regard learning about the theorem as living in this world for a while. One can internalize the ideas and then look around (in the exercises perhaps) for further implications of these ideas. Since graduate courses should ideally be about teaching students how to think like a research mathematician, this process of revisiting and internalizing the world of a proof with the intent on exploring its implications and structure can be a valuable experience. From this perspective, the proof of a major theorem is a "mini course" in itself.

If the energy and time is not being taken by the professor to include the context for the result, though, you may be victim of someone trying to easily fill their teaching time so they can get back to their research. There is no easier way to fill a class period than stepping through a proof that could be routinely read in a book...

• +1 for "you may be victim of someone trying to easily fill their teaching time". Honestly, reading between the lines, I suspect this is exactly the OP's problem. While going through a proof of a basic theorem in class can be an excellent teaching tool in the hands of a good teacher, it's sadly also totally possible to spend two hours writing a proof on a blackboard without really teaching anything at all. – Ilmari Karonen Dec 8 '15 at 17:21

I view avid19's frustration as an argument for presenting proofs within some historical context. Few major theorems have been achieved without a struggle, often involving several mathematicians over an extended period of time. Of course there is rarely the freedom to sketch out these struggles within the time-constraints of a specific course. But even a nod toward that historical context and struggle would help motivate "long proofs of ... main theorems."

Here are two books, separated by 50 years, that illustrate the struggle, the first for Euler's $V-E+F=2$, the second Villani's Fields Medal nonlinear Landau-damping theorem:

(1) Imre Lakatos. Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press, 1976.

(Image from Wikipedia.)

(2) Cédric Villani, Birth of a Theorem: A Mathematical Adventure. Farrar, Straus and Giroux, 2015.

(Image from www.learningpersonalized.com.)

• How often do mathematics professors wear giant spider brooches as accessories? – JAB Dec 8 '15 at 17:02
• He's been called the Lady Gaga of mathematics. :-) – Joseph O'Rourke Dec 8 '15 at 17:50

In honesty, two class periods does not seem much to me. But on the other hand, I have a four year mathematics degree that I've rarely used in the last 25 or so years. That's my perspective. You'd think that I regret spending a month on wavelets, a year on iterative function systems, etc, when I'm not actively using it, but I don't.

Studying mathematics is like studying Zen. Don't resist it. Open your mind. You will grow.

On the other hand, if you can not cope with the current situation, free your self from it.

I once dropped out of a class in mathematical statistics because I thought it was a waste of time. Took a job, worked for half a year. Then sat down with my textbooks and finished the course on my own, because I needed it for my exam.

Of all the courses I took, guess which has been most useful to me?

No idea how long these class periods are in your country (the US, supposedly), but I guess of the order of magnitude of one hour. Then multiple class periods are an excessive amount of time. During my time at the university as an engineering student, I rarely experienced proofs taking longer than one hour, as far as I remember.

Anyway, the usefulness would be the following:

• getting the mathematical way of thinking, and being able to apply this knowledge for other proofs or problems
• understanding that things can be long and tedious. This is important. One of the mistakes I made initially, was assuming a proof or other type of answering strategy to a question was incorrect because it did not take me to the result fast enough. And while fast and elegant solutions are the preferred ones, sometimes things are just hard end tedious (maybe until some clever person finds a faster way). For me it was useful to understand this.
• You have no sense of the scale of mathematics. The final proof of Fermat's Last Theorem would take at least a whole semester for an advanced graduate student to go through. Mochizuki has a purported proof of the abc-conjecture that no one (well maybe someone is, but it's not clear) is trying to check and work through even though it's expected to lead to significant insights because no expert wants to spend the two years of full time work needed to understand the purported proof. – Alexander Woo Dec 9 '15 at 7:57