You have essentially made an argument for studying the historical development of a subject alongside the subject itself. I have no objection to doing this, and indeed, I try to do it myself as much as I can (the beautifully written historical notes are one of the things I love about the Bourbaki volumes). There’s no doubt that knowing the history of a concept can often help in developing a deeper understanding of the concept.

But that doesn’t undermine my contention that simplicity is to be valued more highly than familiarity. And note that the history of some subjects, such as quantum mechanics , can actually be a hindrance to understanding.

You’re right, I think, that our disagreement is about how much mathematical maturity to expect from students. I would simply argue that, regardless of the level they arrive with, they should be expected to reach the level of Lang (say) as quickly as possible.

It may be true, as a matter of fact, that no one actually begins the study of cohomology with Eilenberg-Steenrod, but perhaps they should. I certainly wouldn’t want to dismiss the idea out of hand. However, I’ll have to think about that before coming to any firm conclusion. The same goes for gauge theory–remember the example of quantum mechanics, after all.

The axioms for topology do not exist to summarize our pre-existing spatial intuition; they exist in order to allow us a glimpse of the deep relationships between our mundane universe and more exotic worlds. In this sense, the very beauty of mathematics resides in its (initial) “unintuitiveness”.

Of course I agree — this is exactly what I meant in my penultimate sentence. But pedagogically it makes no sense to approach this level of generality without first mastering the intuition that led to its study and formulation in the first place. No student — graduate or undergraduate — initiates a study of cohomology theories using the Eilenberg-Steenrod axioms before actually understanding more explicit instantiations of cohomology theories (de Rham, before anything else, usually) and how they can be related to make those axioms sensible. Neither should he or she read an introductory gauge theory text that fails to trace (even if briefly or only heuristically) the field’s development from fundamental insights about the manipulation of Maxwell’s equations. The axioms for topology indeed do not “exist to summarize our pre-existing spatial intuition”; but they exist precisely because someone first had to conceive of the right axioms — and it seems to me of utmost importance for students to learn how the greatest and most helpful abstractions have developed from messier, less general details. While the beauty of mathematics may consist in the elegant simplicity of its abstractions, it might as well be defunct if its practitioners don’t understand how to build new structures or develop new insights — in your words, to “glimpse the deep relationships between our mundane universe and more exotic worlds.” The power and vitality of mathematics resides in that very ability to “expand and develop.” Except for the most gifted and perspicacious among us, the cultivation of that ability usually requires learning about how others have formalized the great advances in the past.

Ultimately, I think we only disagree in what we understand to be the level of mathematical maturity of an average mathematics graduate student. I personally find it highly unlikely that most first- or even second-year PhD students have so thoroughly mastered a subject that they need, or even desire, a text like Lang’s — unless of course that subject happens to be one’s primary field of interest. But then again, there’s nothing incredibly hard about Lang’s text (in the sense of being beyond the abilities of any graduate student of pure mathematics) — except that it’s so dry and often unenjoyable to read. Perhaps the issue always devolves to one of taste, as it always has for me with the case of Spivak’s Calculus on Manifolds. Many of my peers exalt that compact little text; I’d rather have Munkres — no matter how much more verbose he may be — any day.reCAPTCHA WP Error:incorrect-captcha-sol

I do not disagree with you that “motivation” is important. What I would point out is that familiarity is not the only, nor necessarily the most important, source of motivation. We should not forget that simplicity is also a motivation, and for me it is a much more important kind. For example, the “motivation” behind the axioms for a topological space is the fact that there are only two of them (invoking nothing but elementary set theory), which nevertheless suffice to imply a whole host of “spatial” properties. Thus, for me, the ideal books on general topology–for all purposes, including a first course–are those of Kelley and Bourbaki

A certain type of person would complain about this type of presentation, on the grounds that the simple set-theoretic axioms seem “unmotivated”, having apparently nothing to do with space and continuity. Such a person, in my view, would be missing the whole point of the subject. Of course it’s “unintuitive” (at first) that such a theory could be developed from such premises; but the most important objective of mathematical activity is to expand and develop our intution–not to codify the limitations of our imagination. The axioms for topology do not exist to summarize our pre-existing spatial intuition; they exist in order to allow us a glimpse of the deep relationships between our mundane universe and more exotic worlds. In this sense, the very beauty of mathematics resides in its (initial) “unintuitiveness”.

This brings me to this divisive claim:

<i>The contrast I’m making is between two expository procedures: one that consists in presenting the definition of a group upfront, clearly, distinctly, and abstractly; and another that may, for instance, involve presenting a verbose discourse on a particular example of a group before giving the definition. (Note that in both cases, I assume that it is the general concept of a group that is of interest; but in the second case, the author is reluctant to explicitly discuss the general object without giving “preparation” that involves previously familiar objects.)</i>

The distance between these expository procedures and the pedagogical techniques they imply (which, it should be said, are not a true bifurcation; there’s clearly a gray area in between) divide mathematicians into two groups: those who enshrine the importance of motivating mathematical ideas, and those who have already intuited that motivation and are prepared to work with abstraction. As a pure math student, I of course value mathematics for the sake of mathematics; but one thing I hate and loudly decry is the idea that mathematics (say even at the graduate level) should be taught upfront — before any motivation, before any discourse about its development and the motivation of major ideas — in full-fledged abstraction. Mathematics is never envisaged or developed this way and should certainly never be presented this way to students. There’s a reason we call graduate students graduate students and not professional mathematicians.

That said, I share James’s disgust for those authors who obscure the big picture by disingenuously leaving parts out. One of the most helpful tools in understanding a major concept is to understand its limits or its difficulties (counterexamples, for example — no pun intended). Although I admit one cannot always give a thorough treatment of a subject, one should never forget to include in the exposition of a subject hints about its messier details or its location in the big picture.

To demonstrate the kind of mathematical exposition I consider ideal, I cite Munkres’s <i>Analysis on Manifolds</i> and John Lee’s <i>An Introduction to Smooth Manifolds</i> (from which it’s not hard to guess that my main interests lie in differential geometry and mathematical physics).

In the end, I believe this idea of working from a totally unmotivated, abstract definition of mathematical structures belies the human work that actually leads to its development — and stems from a tacit arrogance. Every mathematician — and even us lowly, helpless students — will admit the beauty in our work lies in its pure, Platonically ideal abstract structure. Only bad mathematicians will suggest they’ve never needed the more worldly concrete examples to help them better understand the implications of that structure.

http://cs.nyu.edu/~yap/book/alge/ftpSite/

from my perspective, this book answered all the questions about computational algebra i didn’t even know how to ask.

I think you’ll find that Serge was in fact a Bourbaki, at least that seems to be the general consensus (see http://en.wikipedia.org/wiki/Bourbaki for instance). This doesn’t take away from your point that Serge had his own style. I find Lang’s Algebra a very challenging and useful book. But I don’t think its good as a first graduate text on the subject. I’ve found this to be quite an interesting topic, thanks Walt!