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I frequently feel that the things I learn are not motivated properly. My understanding is that this is a common enough problem among students that it gets its own acronym: "What can you do with this?" (WCYDWT)

Here is a sample question and answer pair that I would have liked a few months ago:

Me: I have just learned Lagrange's theorem. WCYDWT?

Math.SE: The first proof in the wikipedia article about Euler's theorem is mostly based on Lagrange's theorem.

Would such a question have been on topic? I suspect it will almost always be a community wiki, except for very specific concepts.

I can ask this exact question (or another one) if we would like a trial run.

To clarify a bit:

  1. I come from a programming background, and have always attributed my success (or lack thereof) to the three virtues. Is it impatient of me to demand results immediately, and arrogant to expect to understand the reasoning? Probably, but I'm not convinced that this is a character flaw.
  2. I was inspired by Joriki's question, which wondered how to help people who ask questions beyond their expertise. I don't know the answer, but I do know that if the only goals in math are attainable only after years of study, you'll end up with a bunch of people who try to cut corners, or give up on math entirely. So I thought this might be one way of finding "intermediate" goals.

Anyway, it sounds like the community is unanimously against the acronym, but at worst neutral about the idea. So if I run into a time when I would like to ask a question like this, I think I will try it out.

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    $\begingroup$ I object to anyone using the acronym...! $\endgroup$
    – Mariano Suárez-Álvarez
    Jul 28, 2011 at 2:17
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    $\begingroup$ I don't know this acronym. And if I did, I might pretend that I didn't so that it wouldn't spread... $\endgroup$
    – davidlowryduda Mod
    Jul 28, 2011 at 2:41
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    $\begingroup$ Before you ask such a question you should first do web searches (e.g. in Google Books & Scholar) to uncover the common applications. Then, you should include these in your query. That will help to spark the imagination of the answerers, and to avoid duplicate effort. Seeing how a theorem is applied in many different texts helps to better appreciate its role in the whole theory. $\endgroup$
    – Bill Dubuque
    Jul 28, 2011 at 3:19
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    $\begingroup$ @Bill: Your comment probably applies to all questions, not only those of the kind the poster is asking about. $\endgroup$
    – Michael Hardy
    Jul 28, 2011 at 19:08

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I think such questions can be valuable (nod in Qiaochu's direction), but they can also be somewhat annoying (nod in Willie's direction). If you just learned about, say, Lagrange's Theorem, then isn't it a bit early to be demanding/asking to know "What is it good for?" How about waiting a bit and seeing what you do with groups in the near future?

If, after more study and with the benefit of hindsight you look back at what you learned, then it might be the time to ask things like "Why did we prove this? Is this important? Where does it show up? I've already done a whole bunch of stuff and it doesn't seem to come up much..." For a example of such a question, there's this question about Cayley's Theorem.

In short: what really rubs me the wrong way about the question you posit is the "I just learned about this" part. I'm tempted to answer "Have some patience, grasshopper."

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    $\begingroup$ I think that just as soon as you learn a new theorem is exactly the right time to start asking what it's good for...I just think that there should be a period where you ask yourself this question. In general, I wish undergraduates were more proactive (I used to hate that word; my how time flies) about their learning. $\endgroup$
    – Pete L. Clark
    Jul 28, 2011 at 3:02
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    $\begingroup$ I don't mean to sound harsh, but some of these questions seem to indicate a lack of contemplation on the part of the student. For instance, when you start studying group theory you study subgroups and you see that in general it's pretty hard to figure out what the subgroups of a given group are. Any help you can get on this seems evidently relevant, and Lagrange's Theorem is the most absolutely basic result along these lines. $\endgroup$
    – Pete L. Clark
    Jul 28, 2011 at 3:05
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    $\begingroup$ Ideally there will be some hw problems driving home some of the many consequences of this result -- e.g. if you have a subgroup with more than half as many elements as the total number of elements of a finite group, then by Lagrange's Theorem...In any case, one can certainly flip through the text and see where Theorem X.Y is used in Section Z.W where Z > X. If the theorem never gets used again, then it might be a good idea to ask what it's doing there. $\endgroup$
    – Pete L. Clark
    Jul 28, 2011 at 3:07
  • $\begingroup$ @Theo: if you can tell me a word with the same denotation as proactive and lacking its "yuck factor", I will gladly use it instead. $\endgroup$
    – Pete L. Clark
    Jul 28, 2011 at 3:12
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    $\begingroup$ @Pete: Certainly, see a theorem, wonder "What can it do for me?" But coming to MSE to ask it does seem to be a lack of contemplation; more, it is too easy to read it as a whining "What is this good for? Why am I being asked to know it?" than an honest query.... And reading the rest of your comments, it seems to me we are more or less on the same page. $\endgroup$
    – Arturo Magidin
    Jul 28, 2011 at 3:49
  • $\begingroup$ $@$Arturo: agreed. @Theo: thanks! $\endgroup$
    – Pete L. Clark
    Jul 28, 2011 at 6:37
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    $\begingroup$ this is specifically addressed in the don't ask section of the faq under "specifically, there is no actual problem to be solved". I am also reminded of a particular Airplane! quote. $\endgroup$
    – Jeff Atwood
    Jul 28, 2011 at 8:21
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    $\begingroup$ @Pete: Dear Pete, I also find myself using "proactive", and have a similary reaction to yours. I think that often "active" is a fine substitute, although it doesn't have quite the same anticipatory sense. Regards, $\endgroup$
    – Matt E
    Jul 28, 2011 at 11:33
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    $\begingroup$ Disagree completely. If someone is reading from a math book, what one could do with a theorem should ideally have been made apparent before the book got to the theorem. The theorem should be introduced once the reader has recognized the burning need to have it in order to do something. $\endgroup$
    – user13618
    Aug 6, 2011 at 4:16
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    $\begingroup$ @Ben: Who are you disagreeing completely with? $\endgroup$
    – Arturo Magidin
    Aug 6, 2011 at 4:20
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A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, "What do I get by learning these things?" So Euclid called a slave and said "Give him three pence, since he must make a gain out of what he learns." - Stobaeus, Extracts

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One thing to bear in mind is that most axiomatic mathematics is a distillation of concepts and points-of-view developed in the solution of more traditional and concrete problems. (I am thinking e.g. about group theory as arising from various research threads in number theory, geometry, and (especially) the theory of equations, commutative algebras as arising from parts of algebraic number theory and algebraic geometry, and measure and integration theory and functional analysis as arising from foundational questions in classical real analysis.)

Thus one way to try to develop motivation for some of theorems is to learn the history behind them and the subject that they are part of. Asking about this history seems to me to be a legitimate MSE question (and something that is not particularly emphasized in most texts, or indeed in most wikipedia entries).

Personally, I am also happy with questions/answers that ask for/provide motivations and explanations for the development of concepts which combine contemporary considerations and points-of-view with a historically informed discussion, but my impression is that questions elicting this kind of answer (not literally historical, but more historico-motivational) may be more controversial.

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I am generally a fan of questions like this, but I think it's important to keep in mind that the value of a piece of mathematics is not necessarily always to be found in its applications. Sometimes a piece of mathematics is valuable because it introduces a new way of thinking, or because it suggests interesting questions, etc. and I think focusing on applications detracts from these alternate modes of appreciating mathematics.

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  • $\begingroup$ But surely "it introduces a new way of thinking: ..." or "it suggests questions like ..." would be answers to "WCYDWT?"? $\endgroup$
    – Charles
    Sep 30, 2011 at 0:39
  • $\begingroup$ @Charles: Not, I think, in the sense that most people mean it. $\endgroup$
    – Qiaochu Yuan
    Sep 30, 2011 at 1:10
  • $\begingroup$ So you think that if you posted either of those as an answer to a "What can you do with this?" question, you wouldn't get upvoted? $\endgroup$
    – Charles
    Sep 30, 2011 at 4:40
  • $\begingroup$ @Charles: no, I just don't think that it's an answer to the question as stated. But most people don't ask the questions they mean to ask anyway, so it doesn't matter too much. $\endgroup$
    – Qiaochu Yuan
    Sep 30, 2011 at 12:05
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I am personally not a big fan of such questions. At least the way it is phrased, it seems like something out of an encyclopaedia entry. I feel that a good question for Math.SE should general be one that can be clearly stated as a mathematical question and one that admits an actual answer.

That said, if you do choose to ask those kinds of questions, make sure to flag it for moderator attention as they are by definition questions that should be Community Wiki.

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  • $\begingroup$ Is there another phrasing that would be preferable? Basically, there are a large number of theorems that could possibly be put into a textbook, so there must be some reason why theorem X was chosen instead of all others. I want to know what that reason is. $\endgroup$
    – Xodarap
    Jul 28, 2011 at 2:37
  • $\begingroup$ @Xodarap: well, as far as Lagrange's theorem goes, it is just a very basic ingredient in many arguments in group theory and not having it around would be enormously inconvenient. More generally it highlights that divisibility plays a big role in finite group theory, a theme further elaborated on by, for example, the Sylow theorems. $\endgroup$
    – Qiaochu Yuan
    Jul 28, 2011 at 2:51
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    $\begingroup$ @qia this is specifically addressed in the don't ask section of the faq under "specifically, there is no actual problem to be solved". I am also reminded of a particular Airplane! quote. $\endgroup$
    – Jeff Atwood
    Jul 28, 2011 at 8:18
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In general, I feel like "What can you do with this?" is a useful and commonly asked question. However, it is a question that many people can answer for themselves as they continue learning. Instead, a better question might be: Why is this notion more powerful than some other previously known idea? Why was this idea developed, and what was the motivation behind it? Why should I care?

I believe that this type of question is valuable because it asks people to look back on what they know about a subject, summarize their experiences and extract the "big ideas"; that type of thinking is often not seen in textbooks. With that in mind, I think that this type of question is only meaningful when applied to broader ideas than specific theorems. For example, I don't think a question about Lagrange's Theorem would be interesting, but I just read this question about Lebesgue integration and I enjoyed reading the answers.

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My feeling is that these questions can be good or bad, and they are better if there is some evidence that the questioner has put some serious thought into it already. I can't give a concrete rule, but I will shamelessly use my question about Lagrange inversion as an example of what I hope is "doing it right".

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