most recent 30 from http://mathoverflow.net 2013-06-18T23:33:03Z http://mathoverflow.net/feeds/user/1902 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/13089/why-do-so-many-textbooks-have-so-much-technical-detail-and-so-little-enlightenmen Mike Benfield 2010-01-27T01:52:40Z 2012-09-23T08:12:11Z

<p>I think/hope this is okay for MO.</p> <p>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.</p> <p>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. </p> <p>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. </p> <p>So what I want to know is:</p> <blockquote> <p>Why do authors write books like this?</p> </blockquote> <p>And:</p> <blockquote> <p>How do others handle this situation?</p> </blockquote> <p>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? </p>

http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets/19164#19164 Mike Benfield 2010-03-24T00:12:54Z 2010-03-24T00:12:54Z

<p>This may be a naive answer, but for me the concept captured by a topological space is when a point is infinitely close to a set. This happens when the point is in the closure of the set (or, equivalently, when every neighborhood of the point intersects the set). The definition by open sets may obscure this. </p>

http://mathoverflow.net/questions/18820/how-do-you-motivate-a-precise-definition-to-a-student-without-much-proof-experien/18821#18821 Mike Benfield 2010-03-20T03:02:59Z 2010-03-20T03:02:59Z

<p>Cauchy published a "proof" that a convergent sequence of continuous functions converges to a continuous function, relying on a not-completely-rigorous idea of continuity. This is particularly notable given Cauchy's role in giving precise definitions here, and also given how easy it is to think of counterexamples. </p> <p>The discussion in the following link may be relevant: <a href="http://www.math.usma.edu/people/Rickey/hm/CalcNotes/CauchyWrgPr.pdf" rel="nofollow">http://www.math.usma.edu/people/Rickey/hm/CalcNotes/CauchyWrgPr.pdf</a></p>

http://mathoverflow.net/questions/14262/the-problem-of-infinity/14271#14271 Mike Benfield 2010-02-05T14:43:22Z 2010-02-05T15:34:02Z

<p>You say, "There are physically impossible things that can be spoken about in mathematics." When mathematics is used to model a physical system, it is always an approximation. So, the fact that the model is not a perfect match for the system (that is, it describes some "physically impossible things") is not a surprise. There's no need to get fancy to show examples: our space is not plain old Euclidean 3 dimensional space, but that model has been very useful. Or even something as basic as possible: using natural numbers to count stuff. Well, it's physically possible that half of one of the things you were counting could burn up, and it turns out the mathematical model you were using is inadequate to describe the system (because you can't talk about half an object). </p> <p>The fact that some mathematics is abstract but finds applications seems like a different issue. Some number theory is such an example, but I don't think anyone ever denied that integers were highly useful in modeling real systems (and thus, in some sense I guess, "real"); it's just that the questions people were asking about them seemed to not have practical uses. </p> <p>So I actually think there is less here than meets the eye. It's not really a big deal that we talk about infinite sets of things even though maybe there is no infinite collection of objects in reality, just like it's not really a big deal that we treat our space like it's continuous even though maybe it's actually discrete. </p>

http://mathoverflow.net/questions/7120/too-old-for-advanced-mathematics/7130#7130 Mike Benfield 2009-11-29T12:27:40Z 2009-11-29T12:27:40Z

<p>I started college in Jan 2007 when I was just about to turn 27. At the time I knew <em>nothing</em>: I didn't really remember trigonometry. I didn't know what it meant to raise a number to a negative power. I'm graduating this December (at 29; I was motivated enough by feeling like I was years behind schedule that I was able to finish this degree in 3 years), in the process of applying to graduate schools, in 3 graduate classes now. </p> <p>Yes, I do regret that I didn't do all this sooner. And honestly it still feels weird. Sometimes I go to class and it just seems so strange that I'm actually going to college. Still, overall, this is one of the best things I've ever done. I guess you have a couple years on me, but it doesn't matter. </p> <p>The unfortunate thing is that, at least at my school, you don't get to do much mathematics until your junior year, so you may be in for a long slog of general education classes you are not particularly interested in. (I actually think this is a <em>big</em> problem... but that's a whole different issue I guess.) </p> <p>Also, you say "I'm getting into material that is going to be very difficult to learn without structure or some kind of instruction." Maybe. But it's also possible you just haven't yet developed the skills/habits to learn that kind of thing. </p>

http://mathoverflow.net/questions/29297/undergraduate-approach-to-learning-math/32694#32694 Mike Benfield 2010-07-21T02:57:51Z 2010-07-21T02:57:51Z

There is one sense in which you DO have only one chance to do it right: putting together a strong profile for grad admissions. I'm sure there are many who need the advice you have given. But speaking as someone whose grad admissions did not go according to plan (despite what was considered a very strong profile at my undergrad institution), I wish someone had given me different advice when I was a freshman/sophomore. I wish someone had laid out a very ambitious study/course plan for me, told me which profs to get to know, and so on.

http://mathoverflow.net/questions/28158/a-learning-roadmap-request-from-high-school-to-mid-undergraduate-studies Mike Benfield 2010-06-14T22:10:17Z 2010-06-14T22:10:17Z

1. Some of the answers you have received are a little odd, IMO. I see some fairly difficult graduate level books being recommended for a high school student who knows some calculus. (Homological algebra? Really?) I honestly don't know what to make of this. 2. Generic advice that may or may not be something you need to hear: Don't get discouraged. Math is hard for everybody. Be persistent, but if you have a book you can't make progress on, don't feel the least bit of shame in turning to a more elementary treatment or going back to learn prerequisite topics or whatever it takes.

http://mathoverflow.net/questions/27881/who-is-the-last-mathematician-that-understood-all-of-mathematics/27882#27882 Mike Benfield 2010-06-12T01:42:42Z 2010-06-12T01:42:42Z

Somewhere I read an anecdote about von Neumann talking to some grad students who were stunned to realize that he didn't know anything about some rather simple concepts of topology. The point of the anecdote I think was in fact to illustrate that you don't have to know everything.

http://mathoverflow.net/questions/26878/what-are-your-favorite-calculus-books-for-math-majors-of-various-levels/27009#27009 Mike Benfield 2010-06-04T04:49:33Z 2010-06-04T04:49:33Z

I think it's great that Spivak doesn't talk about compactness or connectedness. Or, rather, he doesn't use those words. This kind of thing goes a long way towards making sure courses are not just about absorbing piles of definitions but have some content.

http://mathoverflow.net/questions/23525/how-does-an-academic-mathematician-educate-him-herself-about-job-opportunities-ou/23541#23541 Mike Benfield 2010-05-05T13:06:11Z 2010-05-05T13:06:11Z

Except for the &quot;women in science&quot; article, those are mainly about graduate school in the humanities, which (I have been told) really is considerably less likely to lead to a tenure track job than math or science. I also have problems with the methodology behind that job ranking.

http://mathoverflow.net/questions/20959/roots-of-polynomials-with-multiple-of-unity-coefficients-implies-algebraically-cl/20966#20966 Mike Benfield 2010-04-11T15:08:30Z 2010-04-11T15:08:30Z

I think this can be shown quite directly without Galois Theory.

http://mathoverflow.net/questions/20516/complex-torus-cn-versus-cn/20518#20518 Mike Benfield 2010-04-06T16:24:31Z 2010-04-06T16:24:31Z

I thought commutative groups were called Abelian because Abel showed that if the group associated to a polynomial is commutative, the polynomial is solvable by radicals.

http://mathoverflow.net/questions/19328/what-is-enough-to-conclude-that-something-is-a-cw-complex-part-ii Mike Benfield 2010-03-25T18:15:04Z 2010-03-25T18:15:04Z

But anyway, unless I'm misunderstanding something, the boundary of e has to be attached to stuff in X, and that does not happen in your construction, so it doesn't make a CW complex considering e as an open cell. Right?

http://mathoverflow.net/questions/19328/what-is-enough-to-conclude-that-something-is-a-cw-complex-part-ii Mike Benfield 2010-03-25T18:04:51Z 2010-03-25T18:04:51Z

It is a CW complex, but does it follow that it meets his additional criteria? It has to be a CW complex using the partition into cells he described.

http://mathoverflow.net/questions/18820/how-do-you-motivate-a-precise-definition-to-a-student-without-much-proof-experien/18821#18821 Mike Benfield 2010-03-22T09:51:06Z 2010-03-22T09:51:06Z

Yemon, I thought that Cauchy gave the epsilon-delta definition of a limit, but apparently this is not quite right; it was indeed Weierstrass. It seems that Cauchy gave a somewhat imprecise definition of a limit, but that when using this definition in proofs he used the epsilon-delta language (and maybe was the first to do so? I am unclear on this).

http://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion/18623#18623 Mike Benfield 2010-03-18T23:34:15Z 2010-03-18T23:34:15Z

I went through a similar period (see my comment on another answer). I think sometimes authors and instructors see all this as so obvious that it doesn't get explained to some students' satisfaction. It all worked itself out when I decided to stop thinking about it too much. dx is a very small change in x, and dy is the corresponding small change in y. Then dy/dx is the derivative. Not really, because we have to take limits, and if you want a rigorous interpretation of the notation obviously you'll need a different approach, but this helped me.

http://mathoverflow.net/questions/18593/what-are-the-worst-notations-in-your-opinion/18602#18602 Mike Benfield 2010-03-18T19:45:01Z 2010-03-18T19:45:01Z

In my first calculus class, a very big deal was made to make sure students understood that dy/dx was NOT a quotient; dy and dx had NO meaning whatsoever on their own, it was just notation. And then I got to differential equations, and on the first day the professor said &quot;Now multiply by dx.&quot; The other students seemed perfectly OK with this, but it confused the heck out of me for a while. Maybe I was the only one who had actually believed the calculus professor that dx had no independent meaning.

http://mathoverflow.net/questions/18522/how-should-i-visualise-rpn Mike Benfield 2010-03-17T23:42:18Z 2010-03-17T23:42:18Z

Have you just looked at the simple case to understand that attachment? Draw RP^1 as a circle with a dot on it. Then you take an open 2d ball, and you want to attach its boundary to the circle you've just drawn... but you have to do it in a way that you wrap the boundary all the way around the circle twice. Then you've got RP^2. In effect this is the same thing as taking a closed 2d ball, then taking the quotient identifying antipodal points of its boundary.

http://mathoverflow.net/questions/17888/whats-the-semantic-meaning-of-partial-derivate Mike Benfield 2010-03-11T19:23:36Z 2010-03-11T19:23:36Z

No one ever explained what a partial derivative meant in my first multivariable calculus class either, so I'm not at all surprised by this question. Seriously. &quot;Treat y as a constant and differentiate with respect to x&quot; was all we got. Yes, Yemon Choi, you should be quite worried.

http://mathoverflow.net/questions/14162/finite-versus-infinite-on-non-hausdorff-topologies Mike Benfield 2010-02-09T05:09:07Z 2010-02-09T05:09:07Z

Yeah, sorry, I clicked your name and a few clicks later I was here: <a href="http://qwiki.stanford.edu/wiki/Ian_Durham" rel="nofollow">qwiki.stanford.edu/wiki/Ian_Durham</a> It was too much of a weird/funny coincidence not to mention it.