For the last few weeks I’ve been working as a counselor at the PROMYS program. The program runs, among other things, a course in abstract algebra, which was a good opportunity for me to get annoyed at the way people normally introduce normal subgroups, which is via the following unmotivated
Definition: A subgroup of a group is normal if for all .
It is then proven that normal subgroups are precisely the kernels of surjective group homomorphisms . In other words, they are precisely the subgroups you can quotient by and get another group. This strikes me as backwards. The motivation to construct quotient groups should come first.
Today I’d like to present an alternate conceptual route to this definition starting from equivalence relations and quotients. This route also leads to ideals in rings and, among other things, highlights the special role of the existence of inverses in the theory of groups and rings (in the latter I mean additive inverses). The categorical setting for this discussion is the notion of a kernel pair and of an internal equivalence relation in a category, but for the sake of accessibility we will not use this language explicitly.
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Posted in remarks, tagged pedagogy on December 18, 2009 |
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I recently added two new pages to the blog: a bibliography for listing references I cite on multiple occasions, and a suggestions and requests page. The bibliography is likely to soon contain citations for at least some of the following books which have recently come into my possession:
- Introduction to the Theory of Computation, Sipser
- Lectures on Quantum Mechanics, Faddeev, Yakubovskii
- Representation Theory: a First Course, Fulton, Harris
- Conceptual Mathematics, Lawvere, Schanuel
- Concrete Mathematics: a Foundation for Computer Science, Graham, Knuth, Patashnik
I haven’t looked at 2 or 4 very closely yet, but so far I find 1, 3, and 5 to be among the best written textbooks I have ever read. Sipser’s book, in particular, strikes me as having found a perfect balance between brevity and clarity. His tone is conversational but finely polished, and I rather like his habit of summarizing the basic strategy of a proof before actually writing it down. Generally I am finding the book an absolute pleasure to read, which I can’t say for most of the math textbooks I’ve seen. You will likely see me blogging a little about languages and automata once I finish up my current series (right now I’m stuck on what should be a trivial proof).
Why don’t more mathematicians write like Sipser?
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John Ewing wrote up a nice collection of quotes from Paul Halmos for the Notices of the AMS; let’s meditate on his words.
The best notation is no notation; whenever possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that you are explaining the subject to a friend on a long walk in the woods, with no paper available; fall back on symbolism only when it is really necessary.
I’d have to agree. I see this as one of the strongest aspects of, for example, Terence Tao’s expository style. His latest post on relativization is a perfect example; Tao is a master at recognizing when technical details would obscure his exposition and when they are necessary. A related point:
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The problems from IMO 2009 are now available. I haven’t had much time to work on them, though.
There are two classical geometry problems, which I already know I won’t attempt. While I am well aware that classical geometry often requires a great deal of ingenuity, I am also aware of the existence of the field of automatic geometric theorem proving. On this subject I largely agree with Doron Zeilberger: it is more interesting to find an algorithm to prove classes of theorems than to prove individual theorems. The sooner we see areas like classical geometry as “low-level,” the sooner we can work on the really interesting “high-level” stuff. (Plus, I’m not very good at classical geometry.)
Zeilberger’s typical example of a type of theorem with a proof system is the addition or multiplication of very large numbers: it is more interesting to prove symbolically than it is to prove individual “theorems” such as . Zeilberger himself played a significant role in the creation of another proof system, but for the far less trivial case of hypergeometric identities (which includes binomial identities as a special case).
But so I can make my point concretely, I’d like to discuss some examples based on the types of problems most of us had to deal with in middle or high school.
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Bill Gasarch is right – writing technical posts is tiring! (I’ve been trying to finish the next GILA post for days.) So I’ll share some more thoughts instead. Today’s thought was triggered by David Corfield:
In the first of the above posts I mention Leo Corry’s idea that professional historians of mathematics now write a style of history very different from older styles, and those employed by mathematicians themselves. …
To my mind a key difference is the historians’ emphasis in their histories that things could have turned out very differently [emphasis mine], while the mathematicians tend to tell a story where we learn how the present has emerged out of the past, giving the impression that things were always going to turn out not very dissimilarly to the way they have, even if in retrospect the course was quite tortuous.
This in turn reminded me of something else Rota wrote about his Walter Mitty fantasies:
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Posted in remarks, tagged pedagogy on June 30, 2009 |
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When someone linked me to Ravi Vakil’s advice for potential graduate students, I was struck by the following passage:
…[M]athematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you’ll never get anywhere. Instead, you’ll have tendrils of knowledge extending far from your comfort zone [emphasis mine]. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning “forwards”. (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)
It’s great to hear this coming from an expert because this is exactly what I’ve been doing for the past year without realizing it. Without formally learning anything, I’ve begun extending tendrils into algebraic topology, category theory, and all sorts of subjects about which I still can’t say anything particularly intelligent. However, from my experience so far I have a tentative list of the benefits of this strategy:
- It becomes easier to recognize related concepts or constructions across different subjects, hence to tie them together.
- If you have a concept you don’t fully understand sitting in the back of your head, it may come to pass that once you learn the necessary tools to understand it you may re-derive the concept partially based on your memory. As Richard Feynman said, “what I cannot create, I do not understand.”
- Certain things become better motivated if you can say to yourself something like, “oh, I know why we’re learning about Theorem X; it’s an instance of Phenomenon Y which has lots of other nontrivial instances.” Here I’ll give an example: Pontryagin duality.
- You are naturally led to ask lots of questions, and questions are great. “This looks a lot like Theory Z,” you might ask your professor. “What’s the connection?”
The idea that constantly working outside your comfort zone is key to progress appears to me to be a general phenomenon; in two-player games and sports, for example, playing opponents who are better than you is a great way to improve.
What I’m curious about, though, is whether the undergraduate math curriculum explicitly encourages “tendril” behavior. Perhaps it’s just something every math major should be motivated to do independently, but I can’t help but think that Ravi’s advice, which I’ve never seen written down anywhere else, should be more widely acknowledged.
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(A more appropriate title for this post would probably be “I hate Bourbaki,” but I like it as is.)
I spend a lot of my free time reading research papers, usually in combinatorics; those tend to require the least background. Today I decided to read everything I could find written by one of the great champions of combinatorics, Gian-Carlo Rota, and in his philosophical writings I found the explicit declaration of an opinion I’ve held for some time now.
Consider the following passage from Syntax, Semantics, and the Problem of the Identity of Mathematical Objects:
The real line has been axiomatized in at least six different ways. Mathematicians are still looking for further axiomatizations of the real line, too many to support the justification of axiomatization by the claim that we axiomatize only in order to secure the validity of the theory.
Whatever the reasons, the variety of axiomatizations confirms beyond a doubt that the mathematician thinks of one real line, that is, the identity of the object is presupposed and in fact undoubted.
The mathematician’s search for further axiomatizations presupposes the certainty of the identity of the object, but recognizes that the properties of the object can never be completely revealed. The mathematician wants to find out what else the real line can be. He wants ever more perspectives on one and the same object, and the perspectives of mathematics are precisely the various axiomatizations, which lead to a variety of syntactic systems always interpreted as presenting the same object, that is, as having the same models.
Or the following passage, from Combinatorics, representation theory, and invariant theory: The story of a ménage à trois:
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