Ron Maimon and Harvard’s Math 55A

Ron Maimon is one of the more intriguing online characters I’ve seen online.

For one thing, his marked disinterest in playing status games (characterized by e.g. false modesty) makes him stand out like a sore thumb. He’s been banned from multiple Q&A sites over the years, including Quora and Physics.SE, for his abrasive no-holds-barred CAPS LOCK LETTERS argumentative style. Peer pressure (including e.g. appeal to authority) doesn’t faze him; only substantive arguments do. He sometimes deliberately couches his counterarguments in the kind of language that would trip off all your heuristics for spotting crank arguments (John Baez is always a fun read) just to see whether the people he’s talking to would dismiss him just on the basis of style over substance. For instance, here’s how he describes himself on his physics.SE profile:

I have no PhD, I am almost entirely self taught. I like physics, but I think the professionals are, for the most part, completely incompetent. I have a lot of my own personal theories about physics which I like to spread online. I am unemployed and not by choice. Despite this, I consider myself to be the next Isaac Newton.

To say that “you need balls just to write that” would be to implicitly assume that what people think of you, considered separately from what they think of your ideas, matters: to Ron it’s trivially obvious that the truth-value of a claim is independent of the status of the person making the claim. Of course deciding whether claims would be worth investigating on the basis of accumulated status is a useful heuristic. But it’s exactly that: a heuristic. I honestly don’t think he’ll go far trying to convince people of his ideas simply because you need to play the status game to get people to listen to you first, but he’s a hard-core idealist about the scientific method, and where I’ve grown disillusioned because science doesn’t really work the way I thought it did he just refuses to give up and die (to bastardize Erdos’ use of the term: “die” for someone who’d “stopped doing math”, “left” for someone who’d died), and for that I take my hat off to him: all the goddamn respect in the world.

That said, here’s a quote by physics.SE user “Mad Scientist” on Ron:

Ron’s suspension was perfectly valid, he habitually ignores the rules and has announced a few times on different sites that he’ll never follow them. I was watching most of the stuff that happened when he was suspended, and he was just throwing a temper tantrum because one of the moderator candidates was someone he didn’t like.

Like I said, he’s interesting.

About substance: he’s one of the most impressive Q&A site users I’ve ever seen. Feynman talked about monster minds: he’s one of the few users I’ve ever seen who have impressed me purely by sustained quality of rapidly produced content. See for instance his output on Quora, or his user reputation on Physics.SE. To quote a response by physics graduate student Mark Eichenlaub (himself a very-high-reputation user on Quora and physics.SE) regarding a Less Wrong user’s snarky comment to Ron’s profile description above as a reason for dismissing Ron’s evaluation on Physics.SE of Eliezer Yudkowsky’s quantum mechanics sequence of posts (bonus: Scott Aaronson also makes an appearance!):

I’m pretty familiar with Ron Maimon, since I use Physics.Stackexchange heavily.

He seems to have other things going on in his life that prevent him from being accepted by the physics community at large, but in terms of pure knowledge of physics he’s really, really good. Every time I’ve read an answer from him that I’m competent to judge, it’s been right, or else if it has a mistake (which is rare) and someone points it out, he thanks them for noticing and corrects his answer.

When crackpots answer physics questions, they consistently steer away from the topic towards whatever their crackpot ideas are. Ron doesn’t do that. Crackpots tend to claim things that are pretty much known to be impossible, and display little depth of understanding or willingness to talk about anything other than their theories. Ron doesn’t do that. He also doesn’t claim that he’s being repressed by the physics establishment. He’ll call professional physicists idiots, but he doesn’t say that they’re trying to hide the truth or suppress his ideas. And when he sees a professional physicist who comes on the site and writes good answers, he generally treats them with respect. He leaves positive feedback on good answers of all sorts.

None of this fits in with being a crackpot.

He does get into fights with people about more advanced theoretical stuff that’s over my head. But when he talks about physics that I know, he does it extremely well, and I’ve learned a lot from him. He’s more knowledgeable and insightful than most professional physicists.

The stuff other users mentioned about his bible interests and his profile description is ad hominem.

See also for instance his answer on Physics.SE to “Why does kinetic energy increase quadratically and not linearly with speed?”, his answer to whether or not you should still read the Feynman lectures, etc.

Anyway, the main point of this overlong-digression-prefaced post is basically to point out his answer to a Quora user’s question: “What is it like to take Math 55 at Harvard?”, which I found really interesting because it basically trivializes Math 55.

Math 55 is reputedly one of the most difficult undergraduate courses in the country. There’s a Wikipedia page for it, and the Harvard Crimson has an article written on it in dramatic overtones. Enrollment has historically gone down to single digits, and USAMO finalists have dropped out because it was just too much.

The other answers were typical, although one (from Peter Dong’s song “Positive-Definite Non-Degenerate Symmetric Bilinear Forms”) was mildly amusing:

“Don’t ask me what the reason was I took Math 55.
I wonder now if anyone does get out of it alive.
As soon as I get into class, I’m fighting off a swarm
Of positive-definite non-degenerate symmetric bilinear forms.

Oh!  You stay up all night Tuesday working on them in your dorm.
You get the same topology however you transform.
You put ’em back together and you get your favorite norm.
Those positive-definite non-degenerate symmetric bilinear forms.”

Ron, in contrast, wrote this.

It’s not hard, if you know how to prove things coming in, but if you don’t already know proofs before you start, you just shouldn’t take it. You won’t learn how to prove things rigorously in the first two weeks before the first problem set is due. If you expect to learn the material from the class, don’t. Learn it a year or two before you go in, it will then be a breezy review with good peers, and it will introduce you to new stuff.

Because the class assumes familiarity with rigorous proofs, it mostly consisted of freshman from accelerated schools, who had exposure to proofs in high school. I was one of the few public school students, but I knew all the stuff from independent reading, so I was much much better prepared than the special school students. The class is simply another stupid method of social selection— take a certain fraction of the undergrads and give them special attention, and groom them for the Putnam (Harvard takes this seriously), and for a mathematics career. It’s a method of talent selection which is busted, like all other such methods.

If you take the class, for the sake of your TA, don’t write out rigorous proofs in full. Lots of students write out the solutions in lemma-theorem form, proving everything from rock bottom. I did this also. This makes your problem set ENORMOUS. You don’t need to prove the commutativity of integer addition. You should learn what the main idea of the proof is, and what can be taken for granted. This is not so easy to do in an undergraduate proof class, where nearly all the proofs are of obvious facts.

My complaint in hindsight is that the class didn’t sufficiently emphasize computational skills— you learn linear algebra without ever getting practice with row-reducing, or any other rote procedure. These are not conceptually difficult, but they are useful, and require practice, and this is more useful for undergrads than memorizing some specially selected route (as good as any other) through the rigorous development. I had personally already done some practical linear algebra, so it wasn’t a big deal for me, and I assumed everyone else was the same, but now I realize that’s not true. The other students did absolutely no mathematical reading at all before taking the class, and for them, it just wasn’t enough computational exercises. So there are often terrible gaps in the knowledge of the math-55 folks because they know abstract things without enough dirty computation. Also, they tend to become cocky from being selected as special, and this makes them useless. Perhaps I was saved by the fact that I wanted to be a physicist, so I didn’t care about the mathematicians, beyond poaching their methods and training my brain.

To learn how to prove things for the purposes of getting into the class and doing well, it is sufficient to become well acquainted with the material in a few standard rigorous undergraduate textbooks, I read Lang’s Calculus, Mukres topology, some books on General Relativity, and Dirac’s quantum mechanics, and this was far more than enough, it made the class boring, at least after the second problem set. The class only covers material that is standard undergraduate fare everywhere else, except rigorously. I cannot emphasize this enough, there is no magic, there is nothing in the syllabus that is beyond the standard  undergraduate multivector calculus, linear algebra, except of course,  you  need to prove everything. The only magic is in an occasional aside by the instructor, or a special topic.

The instructor my year was Noam Elkies, who has a wonderful insight into undergraduate teaching. He presented a strange introduction to Riemannian integration which develops finitely-additive measure theory instead of doing Riemann sums. It’s equivalent, and perhaps a little cleaner. In hindsight, I just wish they had gone straight to Lebesgue integration, there was no point in learning finitely additive measure separately. I also remember Koerner’s book on Fourier transforms being assigned, and I read that cover to cover, because it’s a great book. The lectures on Fejer’s proof and the FFT algorithm stick out in my mind as particularly insightful, I still have no problem writing an FFT routine when I need one. The rest is lost in my memory.

I took it in 1992, and I also TA’d it in 1993. While I have happy memories of the class when I took it, the TAing phase was difficult. I was a sophomore TAing 40 freshman! That was about double the number of students my year. And I had to take 3 undergrad classes plus 2 grad classes each semester that year, so my workload was approximately double that of a grad student— approximately 10 problems every week for math-55, meaning I had to write clean solutions for the problems, and I had to read 400 amateurish crappy enormously long proofs every week, in addition to doing 2 graduate problem sets, 2 undergrad problem sets, and a bunch of reading for whatever dippy core humanities course I was forced to take that semester. It was too much. The pay for an undergrad TA was also ridiculous, it was peanuts. But it was better than cleaning toilets, which is what I did my first year.

In the second semester, the class covered differential forms, while I introduced tensor analysis in section, to explain what these were, really. That was a mistake, the students didn’t like it, and they also didn’t appreciate that I would translate everything to tensor language, and then translate back to forms. But that was the only real collision between me and the instructor. The rest of the course was easy, because it was a subset of what Elkies covered.

I also remember making a mistake in one of my early sections— I said that a proof didn’t require choice, because I could see the construction more or less, but a bright student said “you are choosing a sequence”, and I said “oh yeah, I guess it does require choice”. Today, I would make the distinction between countable and uncountable choice, but at the time, I didn’t. Other than that, I remember having an easy time presenting proofs, because I had practiced presenting the proof in my head to learn the material.

TA’ing the problem sets meant that you have to find the mistakes in all of them. This took a long time. It made me lose sleep, and pull all-nighter after all-nighter. My social life disintegrated, and I think I went a little bit crazy. I would wander around Harvard Square at 4AM getting burgers at “The  Tasty” (now defunct), and making friends with homeless people, before  going back to my dormitory. But the students liked me, because I was close in age to them, I knew all the pitfalls of the class, I proved things well in section because I prepared well, and I actually read and understood each of their proofs, and commented on it. Also, I would make sure if there was an insightful original idea on one of their proofs, I would give more than full credit, so that you would get credit also for part of a problem you didn’t do, because you had an original idea somewhere else. The students appreciated this. I also explained the proofs from first principles, in a very rigorous way that I was really into back then. The students all said I was very helpful, and this was rewarding.

The one thing I learned from TA’ing that class was how to read crappy proofs very fast and find the mistake (if any), and this was a good skill to develop. This was probably the first time I acquired proficiency in quickly reading and evaluating mathematical proofs, from TAing, not from taking the class. Taking classes is useless for this.

I remember some problems from the first year, but only from one problem set, the first one in math55 proper. First, there was a superficially trivial problem regarding vector space duals that required the axiom of choice to solve in the infinite dimensional case. Elkies and the TA told me it didn’t require choice, but I kept on telling them that I thought it did, because whatever I tried without choice didn’t work. After hecktoring me a while, they realized it did require choice, so I got an undeserved reputation for being really smart. I talked to Dylan about this, and he told me why some people disbelieve choice, constructive principles and all that, although he tried to make it clear he wasn’t one of those people. This made a huge impression on me, I immediately embraced the constructive thing. I reevaluated the proof of the well-orderability of the reals, and realized it makes no sense. I read “constructive analysis”. I eventually got suspicious of all of classical mathematics by the time I took a grad real analysis course, and I gave up on math for another decade or so, before learning some logic. So you should make peace with the axiom of choice, and Cohen’s book “Set Theory and the Continuum Hypothesis” is really the only way to do so.

This problem set had 9 problems, all of which were good mathematical puzzles— they were genuine interesting things. They weren’t even graduate level stuff, but they were challenging. One of the easier ones I remember was to show that the dual of the vector space of eventully zero sequences of reals was bigger than the space itself. This I remember doing by finding an uncountable linearly independent set. There was another straightforward problem, which asked to calculate the  number of bases of an n-dimensional vector space over Z mod p, this was simple combinatorics, but it took me a while to figure out what was being asked (this was half the battle in the days before the internet). I did all the problems except for number 8, which stumped me. The problem asked to show that in Z mod 2 (the field with two elements) the diagonal of a symmetric matrix is in the span of the column vectors. The key idea was presented in lecture, but you had to take notes. It was a difficult problem for undergrads. I later figured out that a symmetric matrix in Z mod 2 is really an antisymmetric matrix also, that is the key idea. This was a nice problem, it was the last nice problem.

I remember being unhappy that I didn’t solve all the problems on that problem set. But then when it came back, the mean on that problem set was 2 out of 9, meaning 2 problems solved out of 9, and I had 8 out of 9, missing that stupid span Z mod 2 problem. Noam Elkies was told to tone it down in difficulty, and unfortunately, he did. The rest of the problem sets that year were loads of boring extremely straightforward standard exercizes, with an occasional good problem.


5 thoughts on “Ron Maimon and Harvard’s Math 55A

    • I think you mean “what an interesting pretense!”. But really, there is no pretense from me, as I do not claim any special insight beyond knowing this ridiculously trivial undergraduate mathematics, which me and every other mathematically trained person does. The class is pretty good as far as classes go, but it produces insanely cocky undergrads who believe they are some sort of supermen, because they passed an undergrad class. The main lesson of this answer (only half of which is included here) is that to get over this delusion, you need to try to produce real results instead of solving silly undergrad problems.

      Liked by 1 person

      • Hi there!

        While what I’ve written of you in this post (and probably several others, I forgot) basically amounts to “I think Ron’s a cool guy, here’s why”, I might’ve portrayed you in a disagreeable manner. Do tell if I’ve done so and I’ll take the post down immediately!


      • Will Fucking Hunting himself, came across you today on Quora. Much respect, you’re probably the most interesting and brilliant thinker I’ve come across online, ever. Thanks for existing, and fuck Quora’s policy.


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