I think/hope this is okay for MO.

I often find that textbooks provide very little in the way of motivation or context. As a simple example, consider group theory. Every textbook I have seen that talks about groups (including some very basic undergrad level books) presents them as abstract algebraic structures (while providing some examples, of course), then spends a few dozen pages proving theorems, and then maybe in some other section of the book covers some Galois Theory. This really irks me. Personally I find it very difficult to learn a topic with no motivation, partly just because it bores me to death. And of course it is historically backwards; groups arose as people tried to solve problems they were independently interested in. They didn't sit down and prove a pile of theorems about groups and then realize that groups had applications. It's also frustrating because I have to be completely passive; if I don't know what groups are for or why anyone cares about them, all I can do is sit and read as the book throws theorems at me.

This is true not just with sweeping big picture issues, but with smaller things too. I remember really struggling to figure out why it was supposed to matter so much which subgroups were closed under conjugation before finally realizing that the real issue was which subgroups can be kernels of homomorphisms, and the other thing is just a handy way to characterize them. So why not define normal subgroups that way, or at least throw in a sentence explaining that that's what we're really after? But no one does.

I've heard everyone from freshmen to Fields Medal recipients complain about this, so I know I'm not alone. And yet these kinds of textbooks seem to be the norm.

So what I want to know is:

Why do authors write books like this?


How do others handle this situation?

Do you just struggle through? Get a different book? Talk to people? (Talking to people isn't really an option for me until Fall...) Some people seem legitimately to be able to absorb mathematics quite well with no context at all. How?

closed as no longer relevant by Tom Leinster, Todd Trimble, Steven Gubkin, Yemon Choi, Qiaochu Yuan Sep 24 '12 at 3:11

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    I found myself thinking the exact same as you on multiple occasions. I find it most annoying in proofs to be honest: the solution is pulled out of a hat and then checked, with no insight about how the solution was found. It's as if someone was explaining how to come up with elliptic curves of high rank just by giving an example by Elkies and checking that it is of high rank. – Sam Derbyshire Jan 27 '10 at 2:01
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    Why did the comments go away? – Mariano Suárez-Álvarez Jan 27 '10 at 6:16
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    We decided that a discussion of the merits of Bourbaki was not a good use of the comments box (it had drifted far from the topic at hand), so we killed them. – Andy Putman Jan 27 '10 at 6:33
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    Two possible reasons why: 1) such books are easier to write 2) the books you want becomes very long, and , very often, surprisingly, more difficult to read. – kjetil b halvorsen Sep 22 '12 at 19:02
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    Most history textbooks? Where? Why are all questions and answers happening in the USA? – Yemon Choi Sep 23 '12 at 9:10

22 Answers 22

By now the advice I give to students in math courses, whether they are math majors or not, is the following:

a) The goal is to learn how to do mathematics, not to "know" it.

b) Nobody ever learned much about doing something from either lectures or textbooks. The standard examples I always give are basketball and carpentry. Why is mathematics any different?

c) Lectures and textbooks serve an extremely important purpose: They show you what you need to learn. From them you learn what you need to learn.

d) Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle". You have to do the work yourself, but you need someone else there to either help you over obstacles you can't get around despite a lot of effort or provide you with some critical knowledge (usually the right perspective but sometimes a clever trick) you are missing. Without the prior effort by the student, the knowledge supplied by a teacher has much less impact.

A substitute for a teacher like that is a working group of students who are all struggling through the same material. When I was a graduate student, we had a wonderful working seminar on Sunday mornings with bagels and cream cheese, where I learned a lot about differential geometry and Lie groups with my classmates.

ADDED: So how do you learn from a book? I can't speak for others, but I have never been able to read a math book forwards. I always read backwards. I always try to find a conclusion (a cool definition or theorem) that I really want to understand. Then I start working backwards and try to read the minimum possible to understand the desired conclusion. Also, I guess I have attention deficit disorder, because I rarely read straight through an entire proof or definition. I try to read the minimum possible that's enough to give me the idea of what's going on and then I try to fill the details myself. I'd rather spend my time writing my own definition or proof and doing my own calculations than reading what someone else wrote. The honest and embarrassing truth is that I fall asleep when I read math papers and books. What often happens is that as I'm trying to read someone else's proof I ask myself, "Why are they doing this in such a complicated way? Why couldn't you just....?" I then stop reading and try to do it the easier way. Occasionally, I actually succeed. More often, I develop a greater appreciation for the obstacles and become better motivated to read more.

WHAT'S THE POINT OF ALL THIS? I don't think the solution is changing how math books are written. I actually prefer them to be terse and to the point. I fully agree that students should know more about the background and motivation of what they are learning. It annoys me that math students learn about calculus without understanding its real purpose in life or that math graduate students learn symplectic geometry without knowing anything about Hamiltonian mechanics. But it's not clear to me that it is the job of a single textbook to provide all this context for a given subject. I do think that your average math book tries to cover too many different things. I think each math book should be relatively short and focus on one narrowly and clearly defined story. I believe if you do that, it will be easier to students to read more different math books.

  • @Deane I think from this and other posts you've made here,you were a lot better then most of us as students,Deane. I seriously doubt most of us could learn mathematics by reading any sophisticated text backwards.I DO agree with the spirit of what you're saying and (a)-(c) above. Active learning is the best teacher.Myself-I take extremely detailed notes I then convert to bite-size pieces on study cards WITHOUT MEMORIZATION.This last part is critical.I memorize definitions AND NOTHING ELSE. I try and reproduce everything else.Halmos' Dictum:"Don't just read it-FIGHT IT!" No better advice. – The Mathemagician May 28 '10 at 22:48
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    Andrew, I don't think I have any real disagreement with you. I probably overstated things when I said "backwards". The idea is to starting flipping through the book until you find something that actually seems interesting. If it's at the very beginning, then just start reading there. But sometimes I have to jump to somewhere a little farther ahead and then work backwards from there. And, to be honest, I doubt I've ever been able to read the entire contents of a really sophisticated math book. At best I learn fragments that I'm able to understand and I'm interested in. – Deane Yang May 29 '10 at 0:47
  • A minor comment @Deane: merely do as a goal is probably ultimately not the goal! Learning how to use (which admittedly is entangled with "do") mathematics (for aesthetic, pragmatic, theoretical, or any other purposes) might be a more valid goal, no? – Suvrit Apr 14 '14 at 22:39
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    Regarding a), I certainly agree that one of the major goals is to learn to do* mathematics, but regarding b) & c), I am led to remember Prof Tom Korner's essay "In Praise of Lectures". Roughly speaking, he argues that the real point of a lecture is for students to actually see mathematics be done . So a good lecturer actually does mathematics, rather than just lists what needs to be learnt. Thus e.g. the basketball example must be replaced by e.g reviewing videos of a basketball game, which can be instructive to someone who wants to play the sport well. (1/2) – Spencer Oct 12 '15 at 16:34
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    (Of course its easy to say that one should "do math" in lectures rather than just "tell math" but probably very difficult. Having been taught by Tom Korner, I found that he had almost, after many, many years of teaching, got this down: After stating some lemma, he would successfully put on the act of thinking through the proof on the spot by saying aloud the thoughts that any analyst would have when asked to prove the lemma and then turning those thoughts into the proof, in a way that seemed natural. (This also relates to fedja's comment elsewhere on this page)) – Spencer Oct 12 '15 at 16:41

Here are some words by Gromov that might be relevant.

This common and unfortunate fact of the lack of an adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to the binary nature of mathematical perception: either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud; moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which the idea appears nonobvious.

Source: M. Berger, Encounter with a geometer. II, Notices Amer. Math. Soc. 47 (2000), no. 3, 326--340.

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    This is so true! – Martin Brandenburg May 30 '10 at 9:59
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    I think it is true for some people. But if you're forced to teach a lot then you have to become sensitive to what the difficulty is -- either by remembering your own experience or by observing carefully the difficulties that others have. When I have this darkness-to-light experience, I almost always know what it was that I was missing before. Sometimes it was a key example, sometimes an interesting concrete problem that demands the abstract idea in question, etc. – gowers Jul 27 '10 at 17:45
  • The above quote describes a binary state of knowing. I would add that understanding or 'enlightenment' is more about the search. It's more of a gradient. Zero being not knowing something exists yet and 1 being expert knowledge. So 0.01 is learning to learn. Maybe 0.1 is asking here. The quality of the data is related more to where you search. – Eddie May 15 '17 at 4:55

I absolutely agree that this is a question worth asking. I have only recently come to realize that all of the abstract stuff I've been learning for the past few years, while interesting in its own right, has concrete applications in physics as well as in other branches of mathematics, none of which was ever mentioned to me in an abstract algebra course. For example, my understanding is that the origin of the term "torsion" to refer to elements of finite order in group theory comes from topology, where torsion in the integral homology of a compact surface tells you whether it's orientable or not (hence whether, when it is constructed by identifying edges of a polygon, the edges must be twisted to fit together or not). Isn't this a wonderful story? Why doesn't it get told until so much later?

For what it's worth, I solve this problem by getting a different book. For example, when I wanted to learn a little commutative algebra, I started out by reading Atiyah-Macdonald. But although A-M is a good and thorough reference in its own right, I didn't feel like I was getting enough geometric intuition. So I found first Eisenbud, and then Reid, both of which are great at discussing the geometric side of the story even if they are not necessarily as thorough as A-M.

As for the first question, I have always wanted to blame this trend on Bourbaki, but maybe the origin of this style comes from the group of people around Hilbert, Noether, Artin, etc. Let me quote from the end of Reid, where he discusses this trend:

The abstract axiomatic methods in algebra are simple and clean and powerful, and give essentially for nothing results that could previously only be obtained by complicated calculations. The idea that you can throw out all the old stuff that made up the bulk of the university math teaching and replace it with more modern material that had previously been considered much too advanced has an obvious appeal. The new syllabus in algebra (and other subjects) was rapidly established as the new orthodoxy, and algebraists were soon committed to the abstract approach.

The problems were slow to emerge. I discuss what I see as two interrelated drawbacks: the divorce of algebra from the rest of the math world, and the unsuitability of the purely abstract approach in teaching a general undergraduate audience. The first of these is purely a matter of opinion - I consider it regrettable and unhealthy that the algebra seminar seems to form a ghetto with its own internal language, attitudes, criterions for success and mechanisms for reproduction, and no visible interest in what the rest of the world is doing.

To read the rest of Reid's commentary you'll have to get the book, which I highly recommend doing anyway.

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    Related to this theme, I like this essay by Arnol'd: pauli.uni-muenster.de/~munsteg/arnold.html – Qiaochu Yuan Jan 27 '10 at 2:30
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    I think that assessment is really unfair. Before Bourbaki, Noether, Hilbert, Artin, etc., and the other early algebraists, the connections between mathematical subjects were so convoluted and difficult to follow that obvious connections weren't really being made. What the early algebraists did was reduce complicated mathematical objects to their fundamental properties and structures. While it's true that some geometric intuition is lost, in the process, it is a worthwhile trade. – Harry Gindi Jan 27 '10 at 2:35
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    I am told that the lectures of the elder Artin were extremely polished but given with a minimum of motivation. Later, when talking to him one-on-one, he would happily give all the motivation behind what he had just lectured about. The problem I see with imitating the abstract approach too thoroughly from a pedagogical point of view is that one has to strike a balance between the power of abstraction and its point, and I don't think most modern textbooks strike this balance successfully. – Qiaochu Yuan Jan 27 '10 at 3:00
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    @Qiaochu -- "I am told" = "I have read Rota's Indiscrete Thoughts"? :) Excellent book. – Pete L. Clark Jan 27 '10 at 4:17
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    OMFG! I'd always wondered where the word "torsion" came from! Upvoted. :) – Vectornaut Jan 27 '10 at 5:15

This is a consequence of the following fact:

One simply cannot communicate what one understands, but can only communicate what one knows.

This does not mean that it is impossible to provide motivation and/or context. But, ultimately, the fact kicks in.

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    I find it especially true when I teach analysis (at any level, from calculus to topics graduate courses). Forget about motivation or context, just try to pass directly the structure of the proofs as you see and remember them without routine technical details that you just add on the fly. You'll find it impossible. So, instead of a few ideas in their pure form that constitute the argument for you, you are forced to present tedious technicalities that do not really matter in the hope that the students will be able to find their way through them and recover the underlying ideas themselves. – fedja Jan 27 '10 at 4:04
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    So true! We call that the Monad tutorial fallacy, as explained in byorgey.wordpress.com/2009/01/12/… – Zsbán Ambrus Sep 23 '12 at 15:09

I also suffer from this problem -- I used to learn best from books, but in grad school, I'm having real trouble finding any book I can learn from in some subjects. There are a few reasons for this sad state of affairs that come to my mind. I'll list them first and expand on them below.

  1. Providing real enlightenment well is very, very hard, and requires a very intimate relationship with a subject.

  2. Different mathematicians need vastly different motivations for the same subject.

  3. Mathematics needs to age before it can be presented well.

  4. Good writing is not valued enough in the mathematical community.

The first of these is true to such a strong degree that it surprises me. Even for well-established subjects, like undergraduate mathematics, where there are a million mathematicians who know the subject very well, I find that all the really good books are written by the true titans of the field -- like Milnor, Serre, Kolmogorov, etc. They understand the underlying structure and logical order of the subject so well that it can be presented in a way that it basically motivates itself -- basically, they can explain math the way they discovered it, and it's beautiful. Every next theorem you read is obviously important, and if it isn't then the proof motivates it. The higher-level the subject, the fewer the number of people who are so intimate with it that they can do this. It's interesting how all the best books I know don't have explicit paragraphs providing the motivation - they don't need them. (Of course there are exceptions -- some amazing mathematicians are terrible writers, and there are people with exceptional writing ability, but the point stands).

Regarding the second point, different people want completely different things for motivation. The questions that pop into our heads when we read the theorems, the way we like to think, the kind of ideas we accept as interesting, important, etc., is different for all of us. For this reason, when people try to explicitly describe the motivation behind the subject they almost always fail to satisfy the majority of readers. Here, I'm thinking of books like Hatcher, Gullemin & Polluck, Spivak, etc., where some people find that they finally found the book that explains all the motivation perfectly, and others are surprised at the many paragraphs of text that dilute the math and make finding the results/proofs they want harder and reading slower. At the same time, the amount of effort each of these authors must have spent on organization of their book seems absolutely immense. For this reason, unless there are 50 books written on a subject, the chances that you will find a book that seems well-motivated for you are low.

The third reason is simple: it takes time for a new subject to stop being ugly, for people to iron out all the kinks, and to figure out some accepted good way to present it.

Finally, it seems to me that good writing, especially expository writing, is not particularly valued in the community, and is valued less now than it was before. Inventing new results seems to be the most respectable thing to do for a mathematician, teaching is second-best, and writing has the third place. People like Hatcher & co. seem to be rare, and I don't know of many modern titans of mathematics who write any books at all, especially on a level more elementary than their current research.

So what do we do? I think what algori said in his answer is the only way to go.

  • And some of the most prolific writers are terrible writers (Serge Lang, although I can't say I don't have some affection for some of his books..). – Harry Gindi Jan 27 '10 at 5:31
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    "...the chances that you will find a book that seems well-motivated for you are low." That really is an important point! And if it's a choice between a) a book with lots of motivation which makes no sense to me; and b) a book with no motivation, but which is consequently shorter, I'll pick (b) every time. – Matthew Daws Jan 27 '10 at 12:19
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    The fourth point is especially unfortunate. Shouldn't it be recognized in the mathematical community that a mathematician who produces an exemplary undergraduate textbook could influence the development of mathematics much more than a mathematician who proves a few obscure results, however brilliant? – Qiaochu Yuan Jan 27 '10 at 14:50
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    @Qiaochu I couldn't agree more.The student of John Milnor's who probably ended up making the single greatest impact on modern mathematics through his writings was one that published very little research;yet argueably had as great if not a greater influence them Milnor himself:Micheal Spivak.This however is the elephant in the room a research-based profession like ours refuses to acknowledge. – The Mathemagician May 28 '10 at 22:53
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    While I agree that good writing is undervalued, etc., I find the words about Michael Spivak pretty exaggerated ("single greatest impact on modern mathematics" -- really??). – Todd Trimble Dec 30 '12 at 13:21

To answer the question in title of the posting (here I am rephrasing what I learnt from philosophical writings by several great mathematicians; Vladimir Arnold and Andre Weil are two names that come to mind, but there surely are others who said something similar, although I can't give you a reference now): because mathematics is discovered in one way and written in a very different way. A mathematical theory may start with a general picture, vague and beautiful, and intriguing. Then it gradually begins to take shape and turn into definitions, lemmas, theorems and such. It may also start with a trivial example, but when one tries to understand what exactly happens in this example, one comes up with definitions, lemmas, theorems and such. But whichever way it starts, when one writes it down, however, only definitions and lemmas remain and the general picture is gone, and the example it all started with is banned to page 489 (or something like that). Why does this happen? This is the real question, more difficult than the original one, but for now let me concentrate on the practical aspects: what can be done about it?

Here is an answer that I found works for myself: try to study a mathematical theory the way it is discovered. Try to find someone who understands the general picture and talk to that person for some time. Try to get them to explain the general picture to you and to go through the first non-trivial example. Then you can spend weeks and even months struggling through the "Elements of XXX", but as you do that you'll find that this conversation you had was incredibly helpful. Even if you don't understand anything much during this conversation, later at some point you'll realize that it all fits into place and then you'll say "aha!". Unfortunately, books and papers aren't nearly as good. For some reason there are many people who explain things wonderfully in a conversation, but nevertheless feel obliged to produce a dreadfully tedious text when they write one. No names shall be named.

Here is another thought: when one is an undergraduate or a beginning graduate student, one usually doesn't yet have a picture of the world and as a result, one is able to learn any theory, no questions asked. Especially when it comes to preparing for an exam. This precious little time should be used to one's advantage. This is an opportunity to learn several languages (or points of view), which can be very helpful whatever one does in the future.

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    At the risk of being overly repetitive, I will repeat the advice of David Kazhdan as reported to me by one of his students: "You should know everything in this book, but don't read it!" – Deane Yang Jan 27 '10 at 16:09
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    In antiquity, knowledge was passed down by oral tradition. Gradually knowledge became what could be written down, rather than what could be passed on orally. – user2529 Apr 25 '10 at 14:28

This is a quote from a beautiful little book by D. Knuth called Surreal Numbers.

B: I wonder why this mathematics is so exciting now, when it was so dull in school. Do you remember old Professor Landau's lectures? I used to really hate that class: Theorem, proof, lemma, remark, theorem, proof, what a total drag.

A: Yes, I remember having a tough time staying awake. But look, wouldn't our beautiful discoveries be just about the same?

B: True. I've got this mad urge to get up before a class and present our results: Theorem, proof, lemma, remark. I'd make it so slick, nobody would be able to guess how we did it, and everyone would be so impressed.

A: Or bored.

B: Yes, there's that. I guess the excitement and the beauty comes in the discovery, not the hearing.

A: But it is beautiful. And I enjoyed hearing your discoveries at most as much as making my own. So what's the real difference?

B: I guess you're right at that. I was able to really appreciate what you did, because I had already been struggling with the same problem myself.

... and so on.

I believe that normal subgroups were first defined in the context of Galois theory (in particular, normal field extensions), by Galois. If one wants to abstract the situation slightly and see what kind of setting this is and why it makes normality important, I think the following is a fair representation:

If a group $G$ acts transitively on a set $X$, and $H$ is the stabilizer of $x \in X$, then $g H g^{-1}$ is the stabilizer of $g x$. Thus a normal subgroup has the property that it leaves one $x \in X$ invariant, then it leaves every $x \in X$ invariant.

Indeed, one could define a normal subgroup this way:

a subgroup $N \subset G$ is normal if and only if for every set $X$ on which $G$ acts transitively, $N$ fixes some $x \in X$ if and only if $N$ fixes every $x \in X$. (Proof: take $X = G/N$.)

This is not the same definition as being the kernel of a homomorphism, although of course it is equivalent.

What is my point? Mathematical ideas have many facets, often multiple origins, certainly multiple applications. This creates a difficulty when writing, because to focus on one point of view one necessarily casts other points of view into the shadows. Any author of a textbook has to walk a line between presenting motivation, perhaps by focusing on a certain nice view-point, and maintaining applicability and appropriate generality.

A related issue is that the example that will illuminate everything for one reader will seem obscure or even off-putting to another. When you lament the omission of a favourite piece of motivation from a textbook, bear in mind that the author may have found that this motivation doesn't work for a number of other students, and hence was not something they wanted to include.

The solution to this is to find texts that focus in directions that you are interested in.

Perhaps the ultimate solution is to move away from texts to reading research papers. If you find papers on topics or problems that you are interested in, you will hopefully have the motivation to read them. In doing so, you will then find yourself going back to earlier papers or texts books to understand the techniques that the author is using. But now all your study will have a focus and a context, and the whole experience will change.

  • I fixed an error that was happening with your LaTeX – Harry Gindi Feb 17 '10 at 21:26
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    I always appreciate it when a text makes a definition and immediately shows it to be equivalent to other natural concepts. For example, to define a normal subgroup in the usual way and then immediately to state and prove: "Theorem. The following are equivalent: (a) N is a normal subgroup of G; (b) N is the kernel of a homomorphism; (c) for every G-set X..." – Darsh Ranjan Mar 7 '10 at 0:43

Good question, but perhaps a little unfairly stated? With a topic like group theory, for example, it is true that, historically speaking, topics such as Galois theory played a crucial motivating role in the development of the theory, however, a posteriori, Galois theory is a more sophisticated topic than (elementary) group theory, and a student can profitably learn about groups as natural mathematical incarnations of symmetry, before he/she learns about Galois theory.

Therein lies, I think, a core issue: while explanation of the motivation behind a part of mathematics is very enlightening to those who have a rich enough background to appreciate it, it is not so clearly helpful to be given that motivation as one is first learning the subject: to be able to appreciate torsion as a phenomenon in the homology of manifolds, for example, requires considerably more sophistication than I would require of someone to explain (rigourously) what a finite (abelian) group was.

To put it another way, if I have thought hard about a piece of mathematics, and over time realized a good way to describe it, then it's not at all clear to me that telling you all the motivations I had, and the failed attempts I made, will ease your path to understanding what I have figured out, and therefore why should I burden you with all that baggage? The same verdict is I expect made more brutally by people who clean up the work of those that have come before them.

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    Good point. But rather than the difference between group theory with and without Galois theory I think the OP is complaining about the difference between a group as a group of automorphisms of an object and a group as a binary operation satisfying certain axioms. (I may have misrepresented his concerns in my own response.) – Qiaochu Yuan Jan 27 '10 at 14:53
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    I think that what people really want/need as motivation is not the actual history of the subject but an idealized history: how it should have been developed. For example, when learning topology, I think it's appropriate to start the rigorous development with open/closed sets (motivated perhaps by $\varepsilon-\delta$ proofs), rather than explaining what definitions people used before open sets. – Ilya Grigoriev Jan 27 '10 at 16:27
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    @Qiaochu: of course I agree, I just meant the more math someone knows, the easier it is for me to give motivation for a topic, but if someone knows relatively little, it's harder to know what contexts are helpful. The danger is perhaps that people react to this by writing only what is logically essential. @Ilya: I agree, but it's surely a subjective question to decide what the "idealized history" should be no? – Kevin McGerty Jan 27 '10 at 17:15

Bourbaki volumes are certainly not the sort of textbooks one puts into the hands of young students. but an advaced student, familiar with the most important classical disciplines and eager to move on, could provide himself with a sound and lasting foundation by studying Bourbaki. Bourbaki's method of going from general to specific is, of course, a bit dangerous for a beginner whose store of concrete problems is limited, since he could be led to believe generality is a goal for itself. But that is not Bourbaki's intention. For Bourbaki, a general concept is useful if applicable of more special problems and really saves time and effort.

-Cartan, "Nicolas Bourbaki and Contemporary Mathematics"

Bourbaki probably had some unintended influence on textbook writers, however, during the 20th century. More motivation, examples, applications, diagrams and illustrations, informal scholia to go with formal proofs, etc. than are found in the typical Bourbaki-inspired would be great. The "from the general to the specific" approach of bourbaki was adopted for specific, non-pedagogic reasons.

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    So many people tend to miss this point. There are so many points that become much clearer when one goes from general to specific. Take, for example, the intermediate value theorem. It is a very specific and unenlightening proof. However, if one proves it from the context of general topology, one ends up with the much stronger statement that connectedness is a topological invariant. – Harry Gindi Jan 27 '10 at 4:18
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    One could say that the IVT motivates the connectedness property. – Steve Huntsman Jan 27 '10 at 4:56
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    I'd say the notion of connectedness doesn't really need motivation! (The definition might, though.) – Sam Lichtenstein Jan 27 '10 at 5:12
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    Harry, topological invariance of connectedness is not "stronger than" the IVT. I would say that the main point of IVT is that intervals in ℝ are connected, however unenlightening that may be. – Jonas Meyer Jan 27 '10 at 5:36
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    Actually, getting accustomed to the formal definition of connectedness and manipulations with it, is IMHO an important part of learning point-set topology. If one starts looking at, say, connectedness of invertible groups in certain topological algebras wrt different topologies, then having precision and not just intuition becomes invaluable. – Yemon Choi Jan 27 '10 at 5:37

To play devil's advocate for a moment: sometimes, it is worth learning how to do some things in generality and abstraction early on in one's mathematical education. I'm not a group theorist, but sometimes there is merit in learning the abstract stuff and then seeing how it applies -- because then one sees just how much can be done "formally" or "naturally". That's not to say it should always be done that way round, or that the emphasis should be on terseness and "purity"; just that to dogmatically decry abstract formulations is IMHO no better than dogmatically disdaining examples.

Then again, I'm someone who liked Banach's contraction mapping principle as an undergraduate, and didn't care much for solving differential equations; so my bias is obvious and undeniable ;)

  • Couldn't agree more! – Vipul Naik Mar 26 '10 at 13:52

To further Yemon Choi's thread, consider two historically popular algebraic topology textbooks. Currently Hatcher's book is very popular. Beforehand Spanier was quite popular. Spanier is in a sense more terse and to-the-point. But it also erases much of the context that you get from Hatcher's book. I was the TA for Hatcher's algebraic topology class a couple times at Cornell and remember some students having trouble dealing with the richness of the context in the book. Some questions in Hatcher's book present you with a picture and ask you to argue a certain pictured loop isn't null-homotopic. For a student used to dry set-theoretic rigour, this can be a major and uncomfortable leap.

I'm not saying that Spanier is in any way a better book, but by providing a rich layer of context you're giving students a lot more to learn. If they're ready, great. But if they're not, it can be a problem. Everyone deals with those issues in different ways. Sometimes you teach less technical material and give more context (like an undergrad differential geometry of curves and surfaces in R^3 type course) and sometimes you head for the big machine and maybe sacrifice context for later -- let the students "add up" the context when they can. Many undergraduate measure theory courses operate this way.

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    Switzer is appropriate for very few beginning students. – Tyler Lawson Jan 27 '10 at 5:44
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    Proposing Switzer as a textbook for beginning students (I think Ryan is talking about undergraduates even...) is quite close to trolling. – Mariano Suárez-Álvarez Jan 27 '10 at 6:08
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    You could certainly try, but this isn't the place for that, Harry. Especially since there are plenty of commentators here who have background in the terminology of categories and they don't agree with you. So the point seems pretty moot to me. Keep in mind that by-and-large, students first exposure to category theory is in an algebraic topology class. You can always ramp-up how efficient a book is by ramping-up the prerequisites, but putting category theory before algebraic topology is ahistorical. I would have found it pretty boring, too. – Ryan Budney Jan 27 '10 at 17:10
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    The problem with Switzer is not the category theory. Rather, it does things in far too much generality for a first course and is overburdened with things a beginner will find confusing (ie bordism theory, the Adams spectral sequence, cohomology operations in generalized homology theories, etc.). These are all topics I love, but I can't imagine wanting to teach them in a first course. The goal in teaching is not to show off how "smart" the teacher is, but rather to TEACH. Math-as-ego-trip has no place in advising students. – Andy Putman Jan 27 '10 at 17:18
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    Earlier you were making a more general assertion about anyone who was "comfortable with categories", not just your own experience. I'm not sure I see that Switzer does what you claim: "explained [homotopy groups] in one fell swoop". Perhaps you mean define, not explain? All the theorems about homotopy groups require proofs and Switzer doesn't have any shortcuts as far as I can see (I just found a Russian translation of Switzer, strangely enough, and am browsing through it). Here I'd argue category language doesn't really contribute much to the discussion. – Ryan Budney Jan 27 '10 at 18:39

It is interesting that we often also see the opposite complaint... For example: Here is this monster thousand-page calculus textbook. But see this old text by Courant: it covers the same material in 200 pages, just has less fluff. (And, of course, much of what they call "fluff" is what others call "motivation and context".)

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    That's taking this discussion to a logical extreme, I suppose. But I'd say Courant covers far more than that any 1000-page calculus text I know of, and has more interesting examples. But these books are targeting far more divergent audiences. – Ryan Budney Jan 27 '10 at 18:49
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    Although Courant is probably not suitable for today's freshmen, I would like to see people try to write terser textbooks. The new "say the same thing 4 different ways including color fonts and pictures" textbooks are just total overstimulation. They're trying to replace the role of lectures. – Deane Yang May 29 '10 at 0:52
  • Today, textbooks could be shorter, and then much example material, colour pictures and son on could be in web material – kjetil b halvorsen Sep 22 '12 at 20:14

Authors of mathematics have to make a lot of tradeoffs. Ideally, you want a book that is well motivated, has easy proofs, gives you a good intuition for working in an area yourself, covers lot of material etc. These are usually conflicting goals.

If you want to motivate a problem historically you are pretty much limited to using historical tools. So you proof a lot of theorems in general topology using transfinite induction and the well-ordering theorem instead of applying Zorn's lemma. This makes things obviously harder to read for people uzsed to the modern toolkit. The proofs are likely to be longer and it is harder to cover much material.

The intuition behind a result that is the easiest for a beginner, may not be the same intuition useful in actually working in an area. For the latter, you think in terms of big, abstract concepts.

Also, it is clearly not the case that a proof that is easier for a beginner is also easier for someone more advanced. The proof for the beginner may use elementary techniques but a lot of computation. For someone more advanced, the computation is confusing noise. A proof that relates to an idea already seen in other contexts would be much simpler.

There are books that are bad for every ausience at every stage of learning, but no book is perfekt for everyone at every stage of learning.

I apologize if this topic has been discussed to death so far. Many of the posts above are absolutely correct in saying that mathematicians all learn math in different ways. Some are fine slogging through swamps of technical details, and some prefer to learn the "bigger-picture" intuition before trying to understand proofs. Many fall somewhere in the middle.

I find it extremely helpful to have two sources when learning mathematics: one technical result/proof driven text and another more intuition and example oriented source. The latter doesn't need to be a book; indeed, as the thread author noted, many subjects lack such a book. However, more experienced mathematicians in the field tend to be able to provide a considerable amount of motivation for whatever you are learning. As an example, I learned differential topology from Gullemin & Pollack (motivation) and Lee's Smooth Manifolds book (details).

Also, if you want an example of a book that provides a ton of motivation and almost no detail (which, I think, is extremely rare in a math book), you should look at Thurston's Three-Dimensional Geometry and Topology.

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    Thurston's book is also the best exception I know to my claim that the most amazing mathematicians don't write books any more. It's a very strange book, but I find it quite inspiring. – Ilya Grigoriev Jan 27 '10 at 21:15
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    I agree with all of this, but especially the mention of Thurston's book (it was only a bound set of notes distributed by the Princeton math department when I used it). What beautiful and inspiring stuff. – Deane Yang Jan 28 '10 at 0:57
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    Thurston's book is a deep,awesome,infuriating read and it HAS to be on the must read list of any mathematics or physics graduate student beyond the first year courses. – The Mathemagician May 28 '10 at 22:40
  • BMann, what's the correspondence? 'Motivation' is for 'Example' and 'Details' is for 'Proof'? – BCLC Nov 17 at 10:18

Books are expensive, and a book that can be used in many different problems is more useful than one that focuses exclusively on one. That is why nice stories of the adventures of mathematics are harder to sell than dry theoretical expositions.

A story of solving a problem or proving a theorem is likely to be more entertaining and easier to follow and to remember even if the solution involves a lot of difficult mathematics. But each each story can hold just a small amount of theory, and once you know the stories, the story book becomes useless.

Dry theoretical expositions find their way into our own stories, when we consult them in order to find a solution for one of our problems. We are more likely to buy such books, because they are so much more useful to us in reality. Beyond that it is all economics: writers of mathematical texts develop a dry theoretical style, because that is what their readers demand.

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    I think this is an important point. The story (i.e., motivation) behind a piece of mathematics is very important but usually easy to remember. What is hard to remember are the rigorous details. So we want a book to remind us of the hard details. A book that is too wordy is in the long run just something that is too big and heavy to keep around. Now that we have the web, maybe that's a better place to maintain and organize the stories? – Deane Yang Jan 28 '10 at 13:46

I agree that sometimes authors present a concept simply because it's a standard example in the subject, but then spend a single page on it and just move on to other things. One example that comes to mind is a particular text on undergraduate real analysis which introduced Fourier Series in a few pages and then had a single sloppy exercise related to applications to PDEs. I'm not saying the book should have dedicated a chapter to PDEs, but one ugly exercise seems like a travesty and makes you scratch your head about why you're wasting your time on this stuff. I don't expect incredibly motivated concepts in graduate texts on the same subject simply because by then I should have already been motivated enough to study onwards.

However, motivation for what you're doing is one of those dangerous phrases in mathematics. For the more difficult and abstract stuff out there, it's not always straightforward to communicate the direct usefulness of an idea. Just because I tell you a result is incredibly useful in say, the sciences, does that make all the difference? When I learned the Radon-Nikodym theorem in real analysis, I could not for the life of me see a genuinely useful application of it, until I came to the formal definition of conditional expectation in probability. In short, the proof of existence and uniqueness of conditional expectation is by the abstract nonsense argument of the Radon-Nikodym theorem. I certainly think it would have been quite nice if somebody told me in my real analysis class why we were learning the Radon-Nikodym theorem, but at the same time I don't think I would have been ready to learn the substantial amount of probability to really understand what the heck the formal definition of conditional expectation is (let alone why it's useful!).

In the end, you're going to need to find a textbook which suites your needs. Each person has their own style for absorbing the material they need. Some people love the straightforward definition - theorem - proof approach while others like to see a section on "applications" after every idea presented (I personally fall into the latter category). If you want to learn the nitty-gritty version of complex analysis, you pick up Complex Analysis by Ahlfors. If you want to learn complex analysis from an engineering point of view, you pick up Complex Analysis For Engineers. It's up to you which applications you want to see, so supplement your knowledge accordingly. Plus, much of the time I don't come to appreciate a textbook until I've read it all the way through. If you're curious about "applications" of what you're learning, try going ahead 20-30 pages, and hopefully the author will have started subjects which apply what you have learned.

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    I sympathize with the idea here, but I don't know if it really justifies that approach. If it's truly not possible at a certain stage to convey to students why the R-N Thm is interesting or useful, why teach it to them? I realize this attitude would lead to a very different approach in education, but I'm not sure it's bad. It seems similar to what some here are advocating for personal study (like Deane Yang's reading books backwards). – Mike Benfield Feb 3 '10 at 1:34

I think it is just another instance of Sturgeon's law "90% of everything is crud". (Google for details.)

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    The question is not about the existence of crud but its nature, which I suspect is specific to math textbooks; the crud of science textbooks, for example, probably looks very different. – Qiaochu Yuan Jan 27 '10 at 4:08
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    Still, I do not understand why this answer got so many negvotes. It was a really nice quote. – Anweshi Jan 27 '10 at 22:02
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    +1 because this law is something one easily forgets in such debates. It's important to keep in mind, when talking about good & bad, that some part of everything is just bad because nobody tried to make it good. And then the other part (to fill up to 90%) is what is discussed here. – Konrad Voelkel Feb 7 '10 at 13:40
  • Equivalent to Pareto Principle? – BCLC Nov 17 at 10:16

I have noticed a similar trend in a different setting: highly technical parts of computer science, in particular POPL-style approaches to programming languages, and ISSAC-style symbolic computation. But there also arises a solution, of sorts: people's proceedings papers are precise, often dry, and full of details. The good presentations of the same material at a conference will typically involve a lot slides for motivation, the big picture, worked examples that give the general idea, and so on.

In other words, the proceedings paper alone is dry and only cursorily motivated, while the talk slides (on their own) could be seen as fluffy and imprecise. And yet, if you take both together, they give an absolutely fantastic view of the results. There is thus an increasing trend for computer scientists in these disciplines to post both their paper and their slides on their web page -- because each gives very different aspects of their actual contribution.

I like this style. Is there a way this could be transposed to mathematics?

  • This style already exists in mathematics. I've been to plenty of math talks for which I've seen the corresponding journal articles which fit your description perfectly. – Mark Meckes Feb 18 '10 at 15:05
  • But do the speakers post their talk slides alongside the journal article on their home pages? The logical next step would be to have archival sites that make it the 'usual process' to submit (slides, paper) as a pair. – Jacques Carette Feb 18 '10 at 19:09
  • Some speakers do that, but not many (I don't). You're probably right that it would be better if we did so more regularly. One (minor) difficulty with doing literally what you describe is that mathematicians tend to give talks that don't correspond to just one paper. – Mark Meckes Feb 20 '10 at 12:19
  • Also, many mathematicians don't use slides at all. – Darsh Ranjan Mar 7 '10 at 0:46

I hope no one will object my raising this question from the dead...

One point which has been alluded to by Tracer Tong but which is worth emphasizing is that it is sometimes very difficult to justify the usefulness of a fundamental concept without starting a whole new book. Just saying "This gets very important later on" may satisfy the lecturer/writer who knows what he is talking about but will leave the student with an aftertaste of argument by authority.

This happens most often with exercises : it is very tempting for the author to take an example or a theorem from a more advanced corner of his subject and strip it down of its fancy apparel.

I'll list a few examples of mathematical concepts I encountered in this way "before their times" and came out with the first impression that those were silly and unmotivated - and changed my mind when I learned about them in a more thorough manner :

  • Hyperbolic geometry (!!)
  • p-adic numbers (!!!)
  • Dirichlet series
  • Milnor K-theory

I don't know the best option here... It is nice to see glimpses of more exciting subjects, but sometimes it is more a way to satisfy the (quite natural) inclination of the teacher for what lays further down the road.

I agree with the sentiment of the original post, but I have also seen people perfectly happy and willing to plow through pages of technical details. I think their drive is to learn theory X, because big names say its important(nothing wrong with that just doesnt work well for me). So ultimately its a matter of what is your goal in mathematics and what is your personality.

Instead of arguing "why", we should try to exchange the missing motivation using the wonderful new tools we are privileged to have in 21st century (like MO, although not sure if MO staff would frown upon flood of questions like "what is the idea behind this definition".)

Also, consider checking out this thread I started out of my own frustration with the lack of motivation. By reading two of the books suggested in that thread, I can testify that the examples and motivation are out there, you just have to find the right authors. books well-motivated with explicit examples

Motivation is especially important in beginners, for instance in sophmore and junior undergraduate courses. A student who has seen three or four well-motivated steps to an abstraction approach would, I expect, be better prepared for a course that goes straight to it.

That said however, I just finished two weeks of historical motivation for my Theory of Computation course and they were impatient with it. So some of how best to teach depends on what learners bring to it.

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    I believe in a minimalistic approach. Try to figure out exactly what you want the students to be capable of doing by the end of the course. Provide the absolute minimum of every aspect (motivation, definitions, proofs, etc.) needed to achieve your goal. Too much of anything drags the course down. And of course try to make the students do as much of the work as possible. – Deane Yang Jan 28 '10 at 1:00
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    Too many professors use that as an excuse to not teach,Deane. "And,of course,try to make the students do as much of the work as possible." Does that benifit THEM or US,Deane? I wonder......... – The Mathemagician Apr 8 '10 at 5:23
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    Andrew, here I disagree with you. You mentioned "active learning" in another more recent comment. This is for me critical. A professor who presents overly complete beautiful crystal clear lectures is not necessarily doing the students any favors. It's better to give crystal clear incomplete lectures (gaps carefully chosen) and make the students finish the work. – Deane Yang May 29 '10 at 0:58

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