The n-Category Café

I find this topic really interesting, most likely because I have a bit of uncertainly about where I stand. Here’s an attempt, admittedly not fully cooked.

I’m pro-axiomatic in that I think, 1, the axiomatic method is very good when done properly. I’m neutral in that I think, 2, it is a mess when not done properly. And I’m anti in that I think, 3, number 2 is much more common than 1.

I would say that the group theory in a first course in algebra is an example of 1, and the ring theory is an example of 2. When teaching group theory, everything seems inevitable to me, but when teaching ring theory, I really feel like I’m repeating what Emil Artin happened to think of one afternoon and tell his class the next day. That could be because I know a lot more about (commutative) ring theory, but I doubt it. Are there any group theorists who would disown a large part of the standard first course?

Regarding 3, I think category theory, in its application to traditional mathematics, is really the savior of the axiomatic method, in that it greatly clarifies which abstractions are good and which are silly. I think of Artin and Noether as epitomizing the first, flawed approach, and Grothendieck as the champion of the new approach, though I have some quibbles at places with him as well. It’s too bad that his change in emphasis in commutative algebra hasn’t yet tricked down to the textbooks, even though it’s at least 55 years old and most of the textbooks were written much later. (Nothing against Artin and Noether, by the way. I wouldn’t be surprised if someone could make a good case that their approach was a necessary stepping stone.)

An interesting contrast to Emil Artin’s approach is Yitzhak Katznelson at Stanford. I attended a course he gave on Fourier analysis, in which he covered the board with rough sketches of convolution kernels and freely waved his hands while talking about pushing things out to infinity. He refused to write technical details on the board, insisting that for those one read his book, which wastes no space on the intuition. The book and lectures together actually make a great way to learn the subject.

Here’s that “Two Cultures of Mathematics” divide again.

“After stating the axioms and giving motivating examples to illustrate their meaning [emphasis added] he would then state some definitions and theorems.”

So, which is it:

(1) The purpose of stating the axioms and giving motivating examples [that enhance understanding and] to illustrate their meaning is to be able to solve problems?

– or –

(2) The purpose of stating the axioms and giving motivating examples to illustrate their meaning, so as to be able to solve problems, is to attain understanding?

It seems to matter BOTH how the teacher approaches the instruction, and how the teacher and students orient around the Two Cultures boundary.

I’ve never been through a Moore Method class myself, but I’ve read various testimonies. I’d like to record a few of them here.

First up is Paul Halmos who, as most readers will be aware, was widely regarded as a fine expositor and teacher. He raves about the Moore Method as follows (from the section “How to Teach” in his automathography “I Want to Be a Mathematician”):

One of the rules was that you mustn’t let anything wrong get past you – if the one who is presenting a proof makes a mistake, it’s your duty (and pleasant privilege?) to call attention to it, to supply a correction if you can, or, at the very least, to demand one.

The procedure quickly led to an ordering of the students by quality. Once that was established, Moore would call on the weakest student first. That had two effects: It stopped the course from turning into an uninterrupted series of lectures by the best student, and it made for a fierce competitive attitude in the class – nobody wanted to stay at the bottom. Moore encouraged competition. Do not read, do not collaborate – think, work by yourself, beat the other guy. Often a student who hadn’t yet found the proof of Theorem 11 would leave the room while someone else was presenting the proof of it – each student wanted to be able to give Moore his private solution, found without any help. Once, the story goes, a student was passing an empty classroom, and, through the open door, happened to catch sight of a figure drawn on a blackboard. The figure gave him the idea for a proof that had eluded him till then. Instead of being happy, the student became upset and angry, and disqualified himself from presenting the proof. That would have been cheating – he had outside help!

Since, as I have already said, I have not really seen Moore in action in a serious mathematical class, I cannot guarantee that the description of the method I just offered is accurate in detail – but it is, I have been told, correct in spirit. I tried the method, experimented with it, kept running into tactical problems, worked out modifications that seemed to suit the students and the courses I was faced with – and I became a convert.

Some say that the only possible effect of the Moore method is to produce research mathematicians, but I don’t agree. The Moore method is, I am convinced the right way to teach anything and everything. It produces students who can understand and use what they have learned. It does, to be sure, instill the research attitude in the student – the attitude of questioning everything and wanting to learn answers actively – but that’s a good thing in every human endeavor, not only in mathematical research. There is an old Chinese proverb that I learned from Moore himself:

I hear, I forget; I see, I remember. I do, I understand.

Okay. Next up is Mary Ellen Rudin, who was by her own words “a child of Moore” in all but the most literal sense. This is from the collection of interviews

• More Mathematical People: Contemporary Conversations, ed. D.J. Albers, G.L. Alexanderson, C. Reid; Harcourt Brace Jovanovich Inc., 1990

which I highly recommend – check out for instance the interview of Mac Lane.

(I’ve decided to give a rather long extract of the interview, roughly from page 287 to page 293, because I find it all very interesting. “MP” is “Mathematical People”. These passages start at the point where Rudin is entering as a freshman at the University of Texas, where Moore taught. Emphases are as in the original.)

Rudin: … So on the appropriate morning I went to the gymnasium to register for the things they had decided on. There was a mass of people, but there were very few people at the mathematics table so I was sent over there. The man who was sitting at the table was an old white-haired gentleman. He and I discussed all kinds of things for a long time. I now know the kinds of things he must have asked me. There would have been lots of sentences with if and then. I used if and then correctly. I also used and and or correctly from a mathematician’s standpoint. At the end of our conversation he signed me up for the courses that I had written on my little slip of paper. When I went to my math class the next day, I found that the professor was R.L. Moore – the same man who talked to me at the registration table.

MP: You met him literally on your first day on campus?

Rudin: On literally my first day on campus I met R.L. Moore. And literally on my first day I was selected by him.

MP: How large was the university at that time?

Rudin: Eleven thousand students.

MP: That’s still rather large for him to be sitting there in the gym and evaluating individual freshmen.

Rudin: He always did that. He was looking for students.

MP: Well, he got you.

Rudin: Yes. There was no one else in that first math class who was at all bright so far as I could tell. It was a trig class. The next semester I took analytic geometry, and R.L. Moore taught it. The next semester I took calculus, and R.L. Moore taught it. It never occurred to me that that was really peculiar. I was not very smart! But I was fully aware that in some way he was teaching for my benefit. He would call on other people first and let them fall on their faces. Then he would have me solve the problem for them. I was always the last person he would call on.

MP: Did this involve going up to the board?

Rudin: Yes. Everything was set up on the basis of proving theorems from axioms. It never occurred to me that there was any other kind of mathematics. At the end of the calculus course, for instance, I’m not sure I knew that the derivative of sin $x$ was cos $x$. But I could prove all sorts of theorems about continuity and differentiability and so on!

MP: In other words, you were getting an introductory analysis course.

Rudin: Exactly. And it was a first-rate course in introductory analysis. But it wasn’t until I taught calculus myself that I learned all those formulas. Now I find it really incredible!

…

MP: Did you have any mathematics teachers besides Moore?

Rudin: Well, yes, but not until my senior year. Then I had a course in algebra from somebody else and a course in differential equations from somebody else.

MP: So you had a course from Moore every semester?

Rudin: Every single semester during my entire career at the University of Texas. I’m a mathematician because Moore caught me and demanded I become a mathematician. He schooled me and pushed me at just the right rate. He always looked for people who had not been influenced by other mathematical experiences, and he caught me before I had been subjected to influence of any kind. I was pure, unadulterated. He almost never got anybody like that.

MP: You are a child of Moore.

Rudin: I’m a child of Moore. I was always conscious of being maneuvered by him. I hated being maneuvered. But part of his technique of teaching was to build your ability to withstand pressure from outside – pressure to give up mathematical research, pressure to change mathematical fields, pressure to achieve non-mathematical goals. So he maneuvered you in order to build up your ego. He built your confidence that you could do anything. No matter what mathematical problem you were faced with, you could do it. I have that total confidence to this day. You give me the definitions, and I’ll solve the problem. I’m a problem solver, primarily a counterexample discoverer. Part of that is a Moore thing, too. That is, he didn’t always give us correct theorems, at least half his statements were false. So we had to think about them as a research mathematician might. I still have this feeling that if a problem can be stated in a simple form that I can readily understand, then I should be able to solve it even if doing so involves building some complicated structure. Of course, I have had some failures. You can guess how often.

MP: But you’ve never failed in confidence?

Rudin: No, having failed 5,000 times doesn’t seem to make me any less confident. At least I don’t feel bound by any serious constraint or doubts about my ability.

MP: Tell us a little more about your graduate school experience.

Rudin: I had entered the university in 1941. That was the beginning of the war for the United States, and all Moore’s students went off to war except Bing, who had an old injury which didn’t allow him to go. Because of the speeded up wartime schedule, I had just three years as an undergraduate, but graduate school for me was fantastic. I started in December 1944, and by September 1945 the war was over and the men were back. So I started with R.D. Anderson, R.H. Bing, Ed Moise – let’s see, who have I left out? Ed Burgess was there, too. There were five of us.

MP: That’s quite a collection!

Rudin: We were a fantastic class. Each of us could eat the others up. Moore did this to you. He somehow built up your ego and your competitiveness. He was tremendously successful in that, partly because he selected people who naturally had those qualities he valued. He immediately separated Moise and Bing, who were further along than the rest of us. But still we were really together, and we have all been very close to each other for our entire careers. That is, we were a team. We were a team against Moore and we were a team against each other, but at the same time we were a team for each other. It was a very close family type of relationship. Actually in our group there was another, a sixth, whom we killed off right away. He was a very smart guy – I think he went into computer science eventually – but he wasn’t strong enough to compete with the rest of us. Moore always began with him and then let one of us show how to solve the problem correctly. And, boy, did this work badly for him! It builds your ego to be able to do a problem when somebody else can’t, but it destroys that person’s ego. I never liked that feature of Moore’s classes. Yet I participated in it. Very often in the undergraduate classes. I mean, I was the killer. He used me that way, and I was conscious of being used that way.

MP: You were Moore’s only woman student?

Rudin: Moore had several women students after me, and he had had two women students previously. The first, Anna Mullikin, wrote a fantastic thesis at Pennsylvania and then immediately went off to China as a missionary. The next was Harlan Miller. She later taught at Texas Women’s College and was very influential as a teacher and an administrator, but she never did any research mathematics after her thesis. He was tremendously disappointed on both of these women.

MP: How did Moore conduct a graduate class?

Rudin: He didn’t lecture about mathematics at all. He put definitions on the board and gave us theorems to prove. Most of the time we didn’t have the theorem that was supposed to be proved that day, and so we discussed whatever. We discussed life. And while we were doing that, he worked on us in various ways. He obviously worked on me – now that I think about it many years later – to make very sure that I would continue to do research after I got my degree. He viewed his two earlier women students as failures, and he didn’t hesitate to tell me about them in great detail so I would realize that he didn’t want to have another failure with a woman.

MP: He must have had some male failures, too.

Rudin: Oh, he had plenty of male failures. There’s no question.

MP: You were saying that you would talk about things other than mathematics in class. What sort of things?

Rudin: Moore would come in and stand at the board and sort of start the conversation. “Miss Estill, do you have anything to report today?” “Mr. Anderson, what do you have to say?” There were maybe three people in the class, always a small group.

MP: So if nobody had anything to report?

Rudin: Then we would start discussing something – it could be politics or anything. Ed Burgess, for instance, remembers the following incident, which I don’t remember at all. We were discussing locking doors. I said that I would never lock the door to my house unless my husband insisted. Ed says that Moore literally pounced on that, saying, “Husband! But, Miss Estill, I thought that you were going to be a mathematician.” Moore intended to elicit a response from me; but although he may have had his doubts, I never saw any contradiction in being both a housewife and a mathematician – of the two I was more driven to be a housewife. Now I don’t remember this particular incident, but I do remember that locking doors was a subject that we often discussed.

…

MP: Tell us about your thesis.

Rudin: It was one of Moore’s many unsolved problems. His technique was to feed all kinds of problems to us. He gave us lists of statements. Some were true, some were false, some he knew were true, some he knew were false, some were fairly easy to prove or disprove, others very hard. There would be all kinds of unsolved and solved problems in the same batch, and there was no way of distinguishing between them. We worked on whatever we jolly well pleased. I solved one of the unsolved problems. Actually I found a counterexample to a well-known conjecture. The technique I used is now called “Building a Pixley-Roy Space”. Two mathematicians named Pixley and Roy tried to read my thesis, which was written in Moore’s old-fashioned language and was not terribly well written besides, and gave a beautiful simplified description of the technique. At the time I wrote my thesis, I had never in my life seen a single mathematics paper!

MP: You had never read any mathematics papers?

Rudin: I told you that I was pure and unadulterated. I only knew the mathematics that Moore fed me. The mathematical language that he used was his own. I didn’t know the standard definition of compact; I didn’t know the correct definition of limit point. I didn’t know how mathematical words were used at all. Instead of open set, for instance, he used region. His language was completely different from the language of the mathematical literature. I didn’t know any other language.

MP: He told his students – at least so I have heard – “I don’t want you going to the library and reading papers.”

Rudin: I don’t remember ever being told that I shouldn’t read mathematics papers, but I was never tempted. It’s true, however, that he sometimes encouraged people to go out into the hall so that they would not hear a proof. I would not do such a thing. If somebody proved something first, he proved it first, and I would listen to it.

MP: Why do you think Moore did that?

Rudin: He wanted to build your independence – whether the other person’s proof was right or wrong. Of course if it was wrong, he’d be delighted to have you there because then you could discover that it was wrong.

MP: How would he know a proof was wrong in advance?

Rudin: First of all, he did have some inner sense. Second, he tried to get students to have the attitude that they didn’t want to listen to someone else’s proof. I rebelled against that, but there were people who did go out. I think that none of our group ever went out. That wasn’t our style. But when you read about Moore, you will read that he tried to get people to do that.

MP: He never referred in class to other people’s work?

Rudin: Never, never, never.

MP: You grew up in a strange world.

Rudin: In a strange, unreal world. Completely. I still dislike reading mathematical papers, and I learn any other way I possibly can. My first two or three papers were all written in Moore’s old-fashioned language.

MP: How did you feel about your mathematical education later?

Rudin: I really resented it, I admit. I felt cheated because, although I had a Ph.D., I had never really been to graduate school. I hadn’t learned any of the things that people ordinarily learn when they go to graduate school. I didn’t know any algebra, literally none. I didn’t know any topology. I didn’t know any analysis – I didn’t even know what an analytic function was. I had my confidence built, and my confidence was plenty strong. But when my students get their Ph.D.’s, they know everything I can get them to learn about what’s been done. Of course, they’re not always so confident as I was.

MP: Reading other people’s work is a great way to destroy one’s confidence.

Rudin: Maybe it is. At least that was Moore’s opinion.

MP: Weren’t there any departmental requirements at Texas? Any qualifying exams for the doctorate?

Rudin: He was the department. I took exams in philosophy, which was my graduate minor, but I never took an exam in mathematics in my life. Moore students were good in direct proportion to how fast they learned after they got out. I still feel seriously deprived by the shortage of things I learned. I resent that, I guess, but at the same time I’m conscious of how much – well, I wouldn’t have become a mathematician at all if it hadn’t been for R.L. Moore.

MP: How about finding a position after you got your Ph.D.?

Rudin: Getting a position was just like going to graduate school. I never applied for one. Moore simply told me that I’d be going to Duke the next year. He and J.M. Thomas, who was a professor at Duke, had been on a train trip together. Duke had a women’s college which was sort of pressuring them to hire a woman mathematician. So Moore told Thomas, “I’ve got the very best, and I’ll ship her to you next September.”

MP: What feelings toward Moore, as a person, did you develop over time?

Rudin: Oh, I had very warm, enthusiastic feelings for him, although I also had very negative feelings. I was conscious of both levels. I was aware that he was bigoted – he was – but I also was aware that he played the role of a bigot sometimes in order to get our reactions, maybe even to keep us from being bigots. I’m never really sure to what extent that was true. Moise, for instance, was a Jew. Moore always claimed that Jews were inferior. I was a woman. He alwys pointed out that his woman students were inferior. Moise and I both loved him dearly, and we knew that he supported us fantastically and did not think that we were inferior – in fact he thought that we were super special. On the other hand, he wanted us to be very confident of ourselves in what he undoubtedly viewed as a somewhat disadvantaged position. Now then, did he play the role of a bigot to elicit a response? I have no idea?

MP: His talking about his former women failures probably made you say to yourself, “I’m not going to be one of those!”

Rudin: I can’t say that I really ever identified with them. Something else Moore built into all of us was our responsibility to be part of the mathematical community – to take part in the American Mathematical Society, very strongly, and to take part in the Mathematical Association of America, even though it was not a research organization. He believed in going to meetings of professional organizations and participating in the meetings. That’s something that all of us have done more than our share of. Moore was president of the American Association for the Advancement of Science, and Moore, Whyburn, Wilder, and Bing were all presidents of the AMS as well as colloquium lecturers for the AMS and members of the National Academy of Sciences. Wilder, Bing, Moise, Young and Anderson were MAA presidents. All of us have served on endless committees for these organizations.

MP: What about editing responsibilities?

Rudin: He believed very strongly in doing that, too.

MP: Even though he never read?

Rudin: Oh yes, he considered publication absolutely vital. We should publish and be very involved, even if we shouldn’t read too much about what other people were doing.

MP: But he sent you off to Duke never having read anything?

Rudin: Right.

…

Scads of documents and videos regarding Moore and variations on his teaching methods (mostly from the point of view of the converted) can also be found here.

Moore’s style has sometimes been called “learning mathematics, Texas-style”. He himself was Texan down to the bone: able to fight with his fists and with firearms, upright and chivalrous, courteous, strong-willed, and an absolutely dominating presence in the classroom. Obviously an enormously charismatic figure to many who knew him.

I have several reasons for wanting to see motivation (as well as formalism) in Mathematics.

(0) I was raised by parents with degrees in English Literature, who attended many Broadway shows in the golden age of the Musical, living in an historically literary neighborhood. I am biased towards literary values, narrative, plot, and drama.

(1) I have a second-rate Mathematical mind, compared to many of my teachers and cohorts. Hence I demand motivation and explanation and worked examples in the same way that a student in Special Education requires adaptation and modification of the curriculum.

(2) My rejected pure math papers never complain about motivation, and often mention that the topic and presentation are interesting. I get rejected most often for lack of rigor, or insufficient proof. I’ve been a referee, too, and try to avoid bias, but improve the rewrites by the authors.

(3) Having taught Applied Math in colleges and universities, I never give more rigor than the students need. They do demand evidence that the math applies to their field of study, and makes som sort of sense.

(4) Having taught remedial elementary courses (Elementary Algebra for those who failed before once, twice, even thrice) I have found that engagement, variation in modality, differences in senory primacy, variation in learning style, may best be approached through sufficient motivation, clarity, connection to what they already know, and to their intuition and sensory awareness of the physical and social world.

(5) With less than 3 weeks remaining in my Student Teaching before I am approved for a California Secondary School Single Subject Mathematics Teaching Credential (just as California prepares to cut a bloody $5.3 x 10^9 from the education budget) I have become critical of theory for its own sake (in Math) and theory for its own sake (in Pedagogy). Pragmatically, what works for urban high school students at risk?

(6) When I teach the formulae for area and volume of spheres, I first have then take one each of the $60 worth of discounted golf balls, baseballs, softballs, volleyballs, basketballs, and soccer balls I have brought. They are asked how to find the radii. Eventually one student says “shoelace” and shows how to measure circumference with string and ruler. What is the area of one of the 2 congruent curvilinear pieces of leather of a baseball’s cover? They don’t know how. But they realize that if they know the total area, they need only divide by 2. What is the area and volume of Earth? Given the approximation that diameter = 8,000 miles, their orders of magnitude vary by orders of magnitude. But, collectively, the 9th and 10th graders determine that about 1,000,000 Earths would fit into the Sun.

I think that these general principals apply at higher levels than in High School. I think that motivation and narrative are essential, to all but the rara avis.