Against the “Research vs. Olympiads” Mantra

There’s a Mantra that you often hear in math contest discussions: “math olympiads are very different from math research”. (For known instances, see O’Neil, Tao, and more. More neutral stances: Monks, Xu.)

It’s true. And I wish people would stop saying it.

Every time I’ve heard the Mantra, it set off a little red siren in my head: something felt wrong. And I could never figure out quite why until last July. There was some (silly) forum discussion about how Allen Liu had done extraordinarily on math contests over the past year. Then someone says:

A: Darn, what math problem can he not do?!

B: I’ll go out on a limb and say that the answer to this is “most of the problems worth asking.” We’ll see where this stands in two years, at which point the answer will almost certainly change, but research \neq Olympiads.

Then it hit me.

Ping-pong vs. Tennis

Let’s try the following thought experiment. Consider a world-class ping-pong player, call her Sarah. She has a fan-base talking about her pr0 ping-pong skills. Then someone comes along as says:

Well, table tennis isn’t the same as tennis.

To which I and everyone else reasonable would say, “uh, so what?”. It’s true, but totally irrelevant; ping-pong and tennis are just not related. Maybe Sarah will be better than average at tennis, but there’s no reason to expect her to be world-class in that too.

And yet we say exactly the same thing for olympiads versus research. Someone wins the IMO, out pops the Mantra. Even if the Mantra is true when taken literally, it’s implicitly sending the message there’s something wrong with being good at contests and not good at research.

So now I ask: just what is wrong with that? To answer this question, I first need to answer: “what is math?”.

There’s been a trick played with this debate, and you can’t see it unless you taboo the word “math”. The word “math” can refer to a bunch of things, like:

  • Training for contest problems like USAMO/IMO, or
  • Learning undergraduate/graduate materials like algebra and analysis, or
  • Working on open problems and conjectures (“research”).

So here’s the trick. The research community managed to claim the name “math”, leaving only “math contests” for the olympiad community. Now the sentence

“Math contests should be relevant to math”

seems totally innocuous. But taboo the world “math”, and you get

“Olympiads should be relevant to research”

and then you notice something’s wrong. In other words, since “math” is a substring of “math contests”, it suddenly seems like the olympiads are subordinate to research. All because of an accident in naming.

Since when? Everyone agrees that olympiads and research are different things, but it does not then follow that “olympiads are useless”. Even if ping-pong is called “table tennis”, that doesn’t mean the top ping-pong players are somehow inferior to top tennis players. (And the scary thing is that in a world without the name “ping-pong”, I can imagine some people actually thinking so.)

I think for many students, olympiads do a lot of good, independent of any value to future math research. Math olympiads give high school students something interesting to work on, and even the training process for a contest such that the IMO carries valuable life lessons: it teaches you how to work hard even in the face of possible failure, and what it’s like to be competitive at an international level (i.e. what it’s like to become really good at something after years of hard work). The peer group that math contests give is also wonderful, and quite similar to the kind of people you’d meet at a top-tier university (and in some cases, they’re more or less the same people). And the problem solving ability you gain from math contests is indisputably helpful elsewhere in life. Consequently, I’m well on record as saying the biggest benefits of math contests have nothing to do with math.

There are also more mundane (but valid) reasons (they help get students out of the classroom, and other standard blurbs about STEM and so on). And as a matter of taste I also think contest problems are interesting and beautiful in their own right. You could even try to make more direct comparisons (for example, I’d guess the average arXiv paper in algebraic geometry gets less attention than the average IMO geometry problem), but that’s a point for another blog post entirely.

The Right and Virtuous Path

Which now leads me to what I think is a culture issue.

MOP alumni prior to maybe 2010 or so were classified into two groups. They would either go on to math research, which was somehow seen as the “right and virtuous path“, or they would defect to software/finance/applied math/etc. Somehow there is always this implicit, unspoken message that the smart MOPpers do math research and the dumb MOPpers drop out.

I’ll tell you how I realized why I didn’t like the Mantra: it’s because the only time I hear the Mantra is when someone is belittling olympiad medalists.

The Mantra says that the USA winning the IMO is no big deal. The Mantra says Allen Liu isn’t part of the “smart club” until he succeeds in research too. The Mantra says that the countless time and energy put into running each year’s MOP are a waste of time. The Mantra says that the students who eventually drop out of math research are “not actually good at math” and “just good at taking tests”. The Mantra even tells outsiders that they, too, can be great researchers, because olympiads are useless anyways.

The Mantra is math research’s recruiting slogan.

And I think this is harmful. The purpose of olympiads was never to produce more math researchers. If it’s really the case that olympiads and research are totally different, then we should expect relatively few olympiad students to go into research; yet in practice, a lot of them do. I think one could make a case that a lot of the past olympiad students are going into math research without realizing that they’re getting into something totally unrelated, just because the sign at the door said “math”. One could also make a case that it’s very harmful for those that don’t do research, or try research and then decide they don’t like it: suddenly these students don’t think they’re “good at math” any more, they’re not smart enough be a mathematician, etc.

But we need this kind of problem-solving skill and talent too much for it to all be spent on computing R(6,6). Richard Rusczyk’s take from Math Prize for Girls 2014 is:

When people ask me, am I disappointed when my students don’t go off and be mathematicians, my answer is I’d be very disappointed if they all did. We need people who can think about these complex problems and solve really hard problems they haven’t seen before everywhere. It’s not just in math, it’s not just in the sciences, it’s not just in medicine — I mean, what we’d give to get some of them in Congress!

Academia is a fine career, but there’s tons of other options out there: the research community may denounce those who switch out as failures, but I’m sure society will take them with open arms.

To close, I really like this (sarcastic) comment from Steven Karp (near bottom):

Contest math is inaccessible to over 90% of people as it is, and then we’re supposed to tell those that get it that even that isn’t real math? While we’re at it, let’s tell Vi Hart to stop making videos because they don’t accurately represent math research.

Addendums (response to comments)

Thanks first of all for the many long and thoughtful comments from everyone (both here, on Facebook, in private, and so on). It’s given me a lot to think about.

Here’s my responses to some of the points that were raised, which is necessarily incomplete because of the volume of discussion.

  1. To start off, it was suggested I should explicitly clarify: I do not mean to imply that people who didn’t do well on contests cannot do well in math research. So let me say that now.

  2. My favorite comment that I got was that in fact this whole post pattern matches with bravery debates.

    On one hand you have lots of olympiad students who actually FEEL BAD about winning medals because they “weren’t doing real math”. But on the other hand there are students whose parents tell them to not pursue math as a major or career because of low contest scores. These students (and their parents) would benefit a lot from the Mantra; so I concede that there are indeed good use cases of the Mantra (such as those that Anonymous Chicken, betaveros describe below) and in particular the Mantra is not intrinsically bad.

    Which of these use is the “common use” probably depends on which tribes you are part of (guess which one I see more?). It’s interesting in that in this case, the two sides actually agree on the basic fact (that contests and research are not so correlated).

  3. Some people point out that research is a career while contests aren’t. I am not convinced by this; I don’t think “is a career” is a good metric for measuring value to society, and can think of several examples of actual jobs that I think really should not exist (not saying any names). In addition, I think that if the general public understood what mathematicians actually do for a career, they just might be a little less willing to pay us.

    I think there’s an interesting discussion about whether contests / research are “valuable” or not, but I don’t think the answer is one-sided; this would warrant a whole different debate (and would derail the entire post if I tried to address it).

  4. Some people point out that training for olympiads yields diminishing returns (e.g. learning Muirhead and Schur is probably not useful for anything else). I guess this is true, but isn’t it true of almost anything? Maybe the point is supposed to be “olympiads aren’t everything”, which is agreeable (see below).

  5. The other favorite comment I got was from Another Chicken, who points out below that the olympiad tribe itself is elitist: they tend to wall themselves off from outsiders (I certainly do this), and undervalue anything that isn’t hard technical problems.

    I concede these are real problems with the olympiad community. Again, this could be a whole different blog post.

    But I think this comment missed the point of this post. It is probably fine (albeit patronizing) to encourage olympiad students to expand; but I have a big problem with framing it as “spend time on not-contests because research“. That’s the real issue with the Mantra: it is often used as a recruitment slogan, telling students that research is the next true test after the IMO has been conquered.

    Changing the Golden Metric from olympiads to research seems to just make the world more egotistic than it already is.

___ Students Have to Suffer

This will be old news to most of the readership of this blog, but I realize I’ve never written it down, so time to fix that.

Fill in the blank

Let’s begin by playing a game of “fill in the blank”. Suppose that today, the director of secondary education at your high school says:

“___ students just have to suffer.”

This is not a pleasant sentence. Fill in that blank with a gender, and you’d be fired tomorrow morning. Fill in that blank with an ethnic group, and you’d be fired in an hour. Fill in that blank with “special needs”, and you’d be be sued. Heck, forget “___ students”, replace that with “You”. Can you see someone’s career flashing before their eyes? How could you possibly get away with saying that about any group of students?

Those 500 hours

“Smart students just have to suffer.”
Director of Secondary Education at Fremont Unified School District

This happened to me. I haven’t told this story enough, so I will tell it some more.

When I was a senior in high school, I was enrolled in two classes and would thereafter run off to take graduate math at UC Berkeley. (Notes here.) This was fantastic and worked for a few weeks, so I got to learn real analysis and algebraic combinatorics from some nice professors.

Then the school district found out, and called me in for a meeting. The big guy shows up, and gives me this golden quote. I was then required to enroll in five classes a day, the minimum number of classes required for me to count towards the average daily attendance funding for my school district.

And that is why, for three periods a day, five days a week, I was forced to sit in the front office, saying “Hi, how may I help you?”.

(I didn’t even get paid! Could’ve asked for a cut of that ADA funding. It didn’t all go to waste though; I spent the time writing a book.)

Everywhere Else

Since I’ve had fun picking on my school district, I will now pick on the Department of Education.

“While challenging and improving the mathematical problem-solving skills of high-performing students are surely every-day objectives of those who teach such students, it is not a problem, relatively speaking, of major import in American education.”
Department of Education Reviewer

Oh boy.

The point is that the problem of neglecting gifted students isn’t at the level of individual teachers. It’s not a problem at the level of individual schools, or individual cities. This is a problem with national culture. The problem is that as a culture we think it’s okay to say a sentence like that.

Replace “high-performing” with any adjective you want. Any gender, any social class, any ethnic group, whatever, and you will get a backlash. But we’ve decided that it’s okay to mistreat the gifted students, because no one complains at that.

Maybe it’s too much to ask that schools do something special for top students. Can you at least not get in their way? Like not forcing students to be an office assistant for 500 hours to obtain ADA funding? Or more generally, how about just not forcing students to take classes which are clearly a waste of time for them?

Next Actions

So what can you do to change the national culture? As far as I can tell, this is mostly a lost clause. I wouldn’t bother trying.

The reason I wrote this post because I went through most of high school not really being aware of just how badly I was being mistreated. I’m really writing this for myself four years ago to point out that, man, us nerds really got the ugly end of the deal.

What you can do (and should) is make small local changes. You can persuade individual schools to make exceptions for a kid, and frequently individual teachers will do what they can to help a gifted student as well. Each individual student has good chance of finding a way around the big bureaucrats that rule the wastelands.

Ask a lot of people: if one administrator says no, ignore them and ask another one. Be prepared to hear “no” a lot, but keep waiting for the one or two crucial “yes” moments. If push comes to shove, switch schools, apply to college early, etc. Take the effort to get this one right. (See 56:30-60:00 of this for more on that.)

Dear past self, yell a little harder at the big guy when he comes, maybe you can save yourself 500 hours as an office aid.

Addendum: A Happy Story

In the comments, someone wrote the following:

Did your mistreatment as a gifted student hinder you in any significant way? … Where would you be today had the system not failed you?

I think it’s impossible to know. But here’s another story.

  • When I was in 7th grade, my school tried to force me to take pre-algebra. My mom begged the school teachers until they finally relented and let me take Algebra I. At the time, my 12-year-old self couldn’t have cared less: both classes were too easy for me, and I spent most of Algebra I playing Tetris on my TI-89.
  • Two years later this happened again: the school wanted to force me to take Algebra II. This time, my mom begged the teachers to let me take precalculus instead, which they eventually did. My 14-year-old self also couldn’t care less; both classes were too easy anyways, and I spent most of precalculus playing osu on my iPhone.
  • Two years later I was in Calculus BC, again bored to tears and in the last HS math class offered. That’s when my parents were able to persuade the school to let me take classes at UC Berkeley instead, since I had exhausted the HS math curriculum. I did very well in my first undergraduate classes, which then allowed me to take graduate classes for the rest of high school.

These professors were the ones that wrote my reference letters for college applications, which got me into all the top schools in the country (Berkeley, UCLA, MIT, Princeton, Stanford, Harvard). Without these reference letters I would certainly not have had as many options; winning the USAMO and making the IMO didn’t happen for me until the end of senior year.

But it wasn’t until I met the guy quoted above that I found out that I had unwittingly “broken district rules”, and technically shouldn’t have happened. (Belated thanks to those individuals who stuck out their necks for me!)

So here’s a surprisingly clear example of a near miss. Suppose that my mom had been more polite, or my school had been a little more firm, and any of the three events above didn’t occur. Not only would I have lost some college choices (potentially including MIT), I wouldn’t even know that this was the key event I could have changed.

[Bonus question: I estimate about 2% of USA high school students take the AMC. How would my life have changed if I had been in the other 98%?]

By analogy, if you ask me now what ways I’ve been affected, how am I to tell you? Without an Earth simulator I can’t point to which of the other 100 times I was mistreated hurt me the most. All I can do is point out that I (and many others) are being mistreated, which really should not be okay in the first place.

Against Hook-Length on USAMO 2016/2

A recent USAMO problem asked the contestant to prove that

\displaystyle  (k^2)! \cdot \prod_{j=0}^{k-1} \frac{j!}{(j+k)!}

is an integer for every {k \in \mathbb N}. Unfortunately, it appears that this is a special case of the so-called hook-length formula, applied to a {k \times k} Young tableau, and several students appealed to this fact without proof to produce one-line solutions lacking any substance. This has led to a major controversy about how such solutions should be graded, in particular whether they should receive the {7^-} treatment for “essentially correct solutions”, or the {0^+} treatment for “essentially not solved”.

In this post I want to argue that I think that these solutions deserve a score of {1}.

1. Disclaimers

However, before I do so, I would like to make some disclaimers:

  • This issue is apparently extremely polarized: everyone seems to strongly believe one side or the other.
  • This was an extremely poor choice of a USAMO problem, and so there is no “good” way to grade the HL solutions, only a least bad way. The correct solution to the dilemma is to not have used the problem at all. Yet here’s the bloodied patient, and here we are in the emergency room.
  • While I am a grader for the USAMO, I am one of many graders, and what I say in this post does not necessarily reflect the viewpoints of other USAMO graders or what score the problem will actually receive. In other words, this is my own view and not official in any way.

One last remark is that I do not consider the hook-length formula to be a “well-known” result like so many contestants seem to want to pretend it is. However, this results in the danger of what constitutes “well-known” or not. So in what follows I’ll pretend that the HL formula is about as well-known as, say, the Pascal or Zsigmondy theorem, even though I personally don’t think that this is the case.

One final disclaimer: I am unlikely to respond to further comments about an issue this polarized, since I have already spent many hours doing so, and I’ve done enough of my duty. So if I don’t respond to a comment of yours, please don’t take it personally.

2. Rule for citations

Here is the policy I use for citations when grading:

  • You can cite any named result as long as it does not trivialize the problem.
  • If the result trivializes the problem, then you are required to prove the result (or otherwise demonstrate you understand the proof) in order to use it.

This is what I’ve heard every time I have asked or answered this question; I have never heard anything to the contrary.

Some people apparently want to nit-pick about how “trivialize” is not objective. I think this is silly. If you really want a definition of “trivialize”, you can take “equivalent to the problem or a generalization of the problem” as a rule of thumb.

Clearly it follows from my rule above that the hook-length formula deserves {0^+} grading, so the remainder of the post is dedicated to justifying why I think this is the correct rule.

3. Subjective grading

I would rather have an accurate subjective criteria than a poor objective one.

In an ideal world, grading would be completely objective: a solution which solves the problem earns {7^-} points and a solution which does not solve the problem earns {0^+} points. But in practice this is of course not possible, unless we expect our contestants to write their solutions in a formal language like Coq. Since this is totally infeasible, we instead use informal proofs: students write their solutions in English and human graders assign a score based on whether the solution could in principle be compiled into a formal language proof.

What this means is that in grading, there are subjective decisions made all of the time. A good example of this is omitting steps. Suppose a student writes “case {B} is similar [to case {A}]”. Then the grader has to decide whether or not the student actually knows that the other case is similar or not, or is just bluffing. One one extreme, if {A} and {B} really are identical, then the grader would probably accept the claim. On the other extreme, if {A} and {B} have some substantial differences, then the grader would almost certainly reject the claim. This implies that there is some intermediate “gray area” which cannot be objectively defined.

Fortunately, most of these decisions are clear, thus USAMO grading appears externally to be mostly objective. In other words, accurate grading and objective grading are correlated (yay math!), but they are not exactly the same. All scores are ultimately discretion by human graders, and not some set of rigid guidelines.

Citations are a special case of this. By citing a theorem, a student is taking advantage of convention that a well-known proof to both the student and grader can be omitted from the write-up.

4. Citing the problem

In light of this I don’t think you should get points for citing the problem. I think we all agree that if a student writes “this is a special case of IMO Shortlist 1999 G8”, they shouldn’t get very many points.

The issue with citing HL in lieu of solving the problem is that the hook-length formula is very hard to prove, and it is not reasonable to do so in an olympiad setting. Consequently, it is near certain that these students have essentially zero understanding of the solution to the problem; they have not solved it, they have only named it.

Similarly, I think you should not earn points for trivializing a problem using Dirichlet, Prime Number Theorem, Zsigmondy, etc. This is also historically consistent with the way that grading has been done in the past (hi Palmer!).

5. Citing intermediate steps

Now consider the usage of difficult theorems such as Dirichlet, Prime Number Theorem, etc. on a solution in which they are merely an intermediate step rather than the entire problem. The consensus here is that citing these results is okay (though it is not unanimous; some super harsh graders also want to take off points here, but they are very few in my experience).

I think this is acceptable, because in this case the contestant has done 90% of the solution, and cannot do the remaining 10% but recognize it as well-known. In my eyes this is considered as essentially solving the problem, because the last missing bit is a standard fact. But somehow if you cannot do 100% of the problem, I don’t think that counts as solving the problem.

What I mean is there is a subjective dependence both on how much of the solution the student actually understands, and how accessible the result is. Unfortunately, the HL solutions earn the worst possible rank in both of the categories: the student understands 0% of the solution and moreover the result is completely inaccessible.

6. Common complaints

Here are the various complaints that people have made to me.

  • “HL is well-known.”
    Well, I don’t think it is, but in any case that’s not the point; you cannot cite a result which is (a) more general than the problem, and (b) to which you don’t understand the proof.

  • “Your criteria is subjective!”
    So what? I would rather have an accurate subjective criteria than a poor objective one.

  • “It’s the problem writer’s fault, so students should get {7}.”
    This is not an ethics issue. It is not appropriate to award points of sympathy on a serious contest such as the USAMO; score inflation is not going to help anyone.

  • “It’s elitist for the graders to decide what counts as trivialized.”
    That’s the grader’s job. Again I would rather have an accurate subjective criteria than a poor objective one. In practice I think the graders are very reasonable about decisions.

  • “I don’t think anyone disputes that the HL solution is a correct one, so certainly the math dictates a {7^-}.”
    I dispute it: I don’t think citing HL is a solution at all.

  • “Why do we let students use Pascal / Cauchy / etc?”
    Because these results are much more reasonable to prove, and the “one-line” solutions using Pascal and Cauchy are not completely trivial; the length of a solution is not the same thing as its difficulty. Of course this is ultimately subjective, but I would rather have an accurate subjective criteria than a poor objective one.

  • “HL solutions have substance, because we made an observation that the quantity is actually the number of ways to do something.”
    That’s why I wish to award {1} instead of {0}.

  • “Your rule isn’t written anywhere.”
    Unfortunately, none of the rules governing USAMO are written anywhere. I agree this is bad and the USAMO should publish some more clear description of the grading mechanics.

  • “The proof of the HLF isn’t even that complicated.”
    Are you joking me?

In summary, I don’t think it is appropriate to give full marks to a student who cannot solve a problem just because they can name it.

Things SPARC

I often get asked about what I learned from the SPARC summer camp. This is hard to describe and I never manage to give as a good of an answer as I want, so I want to take the time to write down something concrete now. For context: I attended SPARC in 2013 and 2014 and again as a counselor in 2015, so this post is long overdue (but better late than never).

(For those of you still in high school: applications for 2016 are now open, due March 1, 2016. The program is completely free including room/board and you don’t need rec letters, so there is no reason to not apply.)

The short version is that about 1/3 to 1/2 of the life skills I use on a regular basis are things I picked up from SPARC. (The rest came from some combination of math contests and living in college dorms.) On paper SPARC seems like a math or CS camp, but there is a strong emphasis on practicality in the sense that the instructors specifically want to teach you things that you can apply in life. So in addition to technical classes on Bayes’ theorem and the like, you’ll have classes on much “softer” topics like

  • Posture (literally about having good body posture)
  • Aversion factoring (e.g. understanding why I’m not exercising and fixing it)
  • Expanding comfort zones (with hands-on practice; my year I learned to climb trees)

and so on. This makes it hard to compare to other math camps like MOP (though if you insist on drawing a comparison, I think most MOP+SPARC students agree they learned more from SPARC).

Some more testimonials:

Now, here is my own list of concrete things which SPARC has taught me, in no particular order:

  • Being significantly more introspective / reflective about life. Example: realizing that some class/activity/etc. are not adding much value to life and dropping them.
  • Being interested in optimizing life in general; I now find it fun to think about how to be more productive and live life well the same way I like to think about hard math problems.
  • Thinking about thinking: things like mental models, cognitive biases, emotions, aversions, System 1 vs System 2,
  • Becoming very aggressive at conserving time. I’m much more willing to trade money for time, and actively asking whether I really need to do something, or if I can just axe it.
  • Using game theory concepts to think about the world. College tuition is expensive because this is the Nash equilibrium. Recognizing real-life situations which are well understood as games, like prisoner’s dilemma, chicken, etc.
  • Applying Bayes’ theorem and expected value to real life. Trying out X activity has constant cost but potentially large payoffs, hence large positive EV.
  • Being able to use probabilities in a meaningful way. Being able to tell the difference between being 90% confident and 70% confident in an event happening.
  • Actively buying O(n) returns for O(1) cost.
  • Being more willing to take less conventional paths, like essentially dropping out of high school to train for the IMO (I describe this in the first FAQ here). Another good example I haven’t done myself (yet) is taking a gap year.
  • Using Workflowy. It’s a big part of why I can think as clearly as I do. In context of SPARC, this is a special case of understanding the idea of working memory, which is also an idea I picked up from camp.
  • Writing a lot more. This is probably actually a consequence of the things above rather than something that I directly learned from SPARC; some combination of understanding working memory well, and being much more reflective.
  • Peer group and culture. This is a bigger one than people realize. In the same way that math contests establish a group of people where it’s cool to think about hard math problems, the SPARC network establishes a group of people with a culture of encouraging people to think about rationality. It’s very hard to be good at reflection in an isolated environment! SPARC lets you see how other people go about thinking about how to live life well and gives you other people to bounce ideas off of.

I’m sure there’s other things, but it’s hard for me to notice since it’s been so long since I had to live life pre-SPARC. And there’s some things that other people learned from SPARC that never stuck with me (lots of the social skills, for example). Much like your first time attending MOP, there will be more things to learn than you’ll actually be able to absorb. So the list above is only the things that I myself learned, and in fact I think the set of things you acquire from SPARC more or less molds to whichever particular things matter to you most.

Time is Money

At some point since arriving at college, when I was trying to decide whether something was worth the time, I started quantifying it in term of money (examples follow). This worked pretty well for me: money is easy to quantify, and the thought of spending money is something I’m naturally averse to.

The most noticeable effect was that it made me a lot more conscious of wasted time.

Roughly, I estimate that my time is worth about $30 an hour to me — I can get a $20/hour job without too much trouble, but I imagine that there would be things I’d rather do than a $20/hour job. And time is worth strictly more than the earned money — time cannot be saved, stored, or earned back.

With this figure, I got some pretty interesting translations:

  • “I was on Facebook for two hours today” is “I paid $60 to read my Facebook news feed”.
  • “I slept in until 1PM today” is “I paid $150 to avoid getting out of bed” (assuming one can wake up at 8AM and still function).
  • “I spent all of today procrastinating on work” is “I just wasted $300” (10 waking hours).

Things on the left side are things I overhear all the time at lunch in the freshman dining hall. I’ll bet the right side is something that no one would feel comfortable saying. We’re so used to spending time we forget how valuable it is!

And people think I’m weird for waking up early. Who wouldn’t wake up an extra couple hours early if they got paid $60 to do so? Or more negatively: would you still press snooze if it cost $60?

It’s even more fun to take this and scale up.

  • “I didn’t drop this class even though I didn’t like it” is “I passed up a refund for a $5,000 item I didn’t like”.
  • “I’m not sure what I got out of my last year at college” is “I bought a $100,000 item but I’m not sure what it was”.

Money is remarkably good at putting things in perspective.


I’ve recently come to believe that “deep conversations” are overrated. Here is why.


Human short term memory is pretty crummy. Here is an illustration from linguistics:

A man that a woman that a child that a bird that I heard saw knows loves

This is a well-formed English phrase. And yet parsing it is difficult, because you need a stack of size four. Four is a pretty big number.

And that’s after I’ve written the sentence down for you, so your eyes could scan it two or three times to try and parse it. Imagine if I instead said this sentence aloud.

Other examples include any object with some moderately complex structure:

Let ABC be a triangle and let AD, BE, CF be altitudes concurrent at the orthocenter H.

This is not a very complicated diagram, but it’s also very difficult to capture in your head unless you’ve seen it before — and again, that’s after I’ve written it down for your viewing pleasure.

Now imagine what you’re talking about isn’t just six lines and seven points, but “what do you think the point of college is?”, or “should a high school diploma be required to obtain a driver’s license?”, or “what is algebraic geometry about?” (all examples from my life, mind you). The answer to these questions is far more complex than the trivial examples I’ve given above. To try and talk about these things merely by voice seems fruitless.

It’s worth pointing out that you can get away with things that have a lot of breadth as long they do not have depth — loosely, as long as there are not too many dependencies. To give an example, the linguistics example is tricky because all the subjects and verbs depend on each other. The geometry diagram is tricky because the points are all tied together in a certain way. But I could read the first chapter of And Then I Thought I Was A Fish out loud, or tell you the story of the cute girl I met three summers ago, because the parts don’t depend (as much) on one another: at any point in a story you can remember the last couple sentences and still enjoy the story. But if I try to read you the first chapter of The Rising Sea: Foundations of Algebraic Geometry, it would make a good bedtime story only because you’d probably fall asleep.

It just strikes me as bizarre that people talk about “deep” issues without writing a single thing down. I think if you’re having lunch with a friend and discussing something like this, you ought to at least have a piece of paper out on the table where you can both jot down the main ideas of what’s been said. It doesn’t need to have every word because then you just get bloat, but still, at least get the key insights somewhere visible. (That’s what presentation slides and blackboards are for, right?)


The other strange thing is that in conversations, you have to process and respond in real-time. You can only spend as long thinking about a sentence as it takes for the next one to be said.

This is fine if I ask you a question such as “what is your birthday?”, because lookup queries are fast. It’s fine if I ask you “what did you think of X book you read?”, again because it is just a lookup query. Note that this is true even if you spent a long time reading and thinking about the book, because the computation was already done. It’s even fine if I ask “what is two plus five?” because it takes not very long to add.

But if I ask “what do you think about the war on drugs?” and you haven’t been thinking much about it, then the best answer you can give is “I don’t know”; because you can’t do a lookup query for an answer you haven’t computed yet.

Put another way, suppose someone asks me some complex question like “how do I get better at math contests?”, and I respond “do a lot of problems that are right above your ability”. One of two possible things just happened:

  1. I had already thought about this question, and this is a pre-computed answer, or
  2. I came up with this in the half second between the end of your question and the start of my response. (Though you can increase this time by pre-pending “uh”, “like”, “I think”.)

In other words, the fast nature of conversations prevents anything other than cache lookups and first impressions (or I suppose possibly a combination of both). And if the issue you’re talking about is sufficiently complex, first impressions are likely not so insightful. So if I sound really smart in a conversation, the only reason is that you’re asking questions I already pre-computed good answers for.

In other words, the best you can do from a typical conversation is learn what people’s existing ideas are. There isn’t a tractable way to generate new ideas from feedback, just because the time-scale involved is too small. Eliezer Yudkowsky makes a similar point in a Less Wrong Post:

If you want to sound deep, you can never say anything that is more than a single step of inferential distance away from your listener’s current mental state. That’s just the way it is.


I will now point out that both issues I mentioned have easy partial fixes. If I’m correct, then deep conversations can be substantially enhanced if we use paper or blackboard or anything else, and agree it is socially acceptable to take a minute to respond. Neither of these actions will completely alleviate their respective problems, but trust me when I say having 60 seconds to think is a world of difference compared to 2.

Both of these initially struck me as weird conclusions, but they do seem to make sense on closer inspection. In fact, I have actually seen both done in practice (albeit not simultaneously). So this means I have a way to test what I’ve written in this post now…


Apparently even people on Quora want to know why I transferred from Harvard to MIT. Since I’ve been asked this question way too many times, I guess I should give an answer, once and for all.

There were plenty of reasons (and anti-reasons). I should say some anti-reasons first to give due credit — the Harvard math department is fantastic, and Harvard gives you significantly more freedom than MIT to take whatever you want. These were the main reasons why transferring was a difficult decision, and in fact I’m only ~70% sure I might the right choice.

Ultimately, the main reason I transferred was due to the housing.

At MIT, you basically get to choose where you live. All the dorms, and even floors within dorms, are different: living on 3rd West versus living on 5th East might as well be going to different colleges. Even if for some bizarre reason you hate 90% of the students at MIT you can still have a fantastic social experience if you’re in a dorm you like.

This is not true at Harvard, which shoves you in dorms more or less at random. Specifically,

  • In freshman year, you are assigned a random dorm, and eat in a segregated dining hall (Annenberg) exclusively with freshman. All students are placed on a mandatory unlimited meal plan, I guess to discourage them from eating out.
  • After freshman year, you get a random House, and eat in a dining hall built into the House. There are restrictions that make it deliberately difficult to eat at other Houses.

The result of this random mixing is that (a) you only know people in your own year, and (b) zero dorm culture. Lounges are deserted, doors are shut, and people are unfindable — in fact I still don’t know the names of the students who lived next door to me. This a bigger deal than people give it credit for: students are busy and campus is large, so you don’t really see someone unless you share a class, live near them, or date them. For example, I rarely talked to James Tao, even though we’d known each other for three years beforehand and had plenty in common.

Put more harshly: “Harvard’s dominant typical social tone is superficial, inane, and too frequently alcohol-drenched to be interesting. It actively thwarts any attempts to escape this atmosphere, by assigning groups of students to dorms randomly — thus guaranteeing all students a more-or-less uniformly superficial, inane and alcohol-drenched experience.”

The problems I mentioned were worse for me specifically since I took exclusively upper-level math courses. My classmates were all upperclassmen who all already knew each other and ate/lived elsewhere. For my own meals, the typical Annenberg conversation was either classes or gossip, so I had little to say to the other freshman (if I talked about my classes I sounded like a showoff). I was often sitting alone in my room, which was great for learning category theory but not so much for my mood. I ended up moving in to an MIT dorm for a good chunk of the school year, where it was much easier to find people I could relate well to (because they all lived in one place).

At Harvard I was constantly isolated and bored. I got sick of it and left.

Diversity and Neg EMH

Those of you who know me personally will know how much I don’t like the word “diversity”, so for once let me give an argument in favor of it.

Efficient Market Hypothesis

One of the biggest take-aways I got from freshman year was something I like to call “neg EMH”, short for “negation of the Efficient Market Hypothesis”. This is a concept from economics which roughly says that the market self-regulates due to competition. For our purposes, we can of it more generally as saying

EMH: The world is big, and if you think you see something that everyone else misses you are probably wrong. If you are right, it doesn’t hurt to do a bit of due diligence anyway.

Its negation, then, would go something like

Neg EMH: The world isn’t that big; if you care, are thoughtful and intelligent, and have relevant expertise and skills, you shouldn’t be surprised to see something that no one else does, or do something that no one else can.

(These quotes are from a class at SPARC 2014.)

I think the biggest change I had freshman year was moving from the first point of view to the second point of view and even past it. It’s not merely possible that most people in the world are mistaken, it’s frequently the case. In fact, it’s frequently the case that most of the population is obviously mistaken. The first instance I saw of this was realizing how high school math education was (is) totally broken. Indeed there’s quite an elephant in the room:

You can’t actually be serious. Do people really think that knowing the Pythagorean Theorem will help in your daily life? I sure don’t, and I’m an aspiring mathematician . . . . It’s hilarious when you think about it. We’ve convinced millions of kids all over the country that they’re learning math because it’s useful in their lives, and they grudgingly believe it.

And ironically, the quote above is from an earlier blog post when I described how high school English was also broken. In both cases, people weren’t just wrong about something obscure, they were blatantly wrong about something which ought to have been patently obvious.


I realized that math was broken early in high school, and that writing was broken during about senior year, but at the time I dismissed it as just saying “high school is broken” and thought nothing more of it. It didn’t occur to me to think beyond that box.

Then I went to college and wasted a few hundred hours in one of the worst experiences of my life. It begin to dawn on me that perhaps there were more things that were broken than I thought. So I started looking harder, and I began to notice more and more inefficiencies. To give concrete examples:

  • In some (many) classes, people go to lectures to listen to a professor copy the notes he/she prepared onto a blackboard, then copy down the things on the blackboard into their notebooks. It’s almost like we never invented copy machines. [a]
  • Math textbooks are still written in a deadly boring, formal tone. To me, that’s just absurd. There are no jokes. There are few concrete examples. There is no “this is the standard example you should refer to”, “this is a surprising proof”, “this is just a routine calculation”. There’s rarely even “the main idea of the proof is…”. Instead, all you get is a sea of definitions, lemmas, theorems and proofs, which is often indecipherable.
    No one talks this way. No one thinks this way. Why are so many textbooks written this way?
  • Most people still use Microsoft Word (instead of Vim/LaTeX) and mouse-oriented operating systems and window managers (instead of tiling window managers, say). These are good examples of not buying O(n) returns at O(1) costs.

There are plenty more examples I have, but many are things I’m not really comfortable saying in public, so I’ll refrain from giving more examples [b].

Minority Rule

Now what does this have to do with diversity? Well, you might notice that most of the examples I gave had to do with math and college (and this becomes more true if you look at my full list). In fact, if you asked me what the two things I have the strongest feelings about are, I would say (i) math is usually taught poorly, especially at the lower levels, and (ii) college is an egregious waste of money [c].

What’s with that? Well, these happen to be the two things I think about the most.

This is a reflection of the totally obvious fact that if you spend lots of time thinking about something, you get to see things that most people don’t see [d]. I spend most of my time learning math, so I get to see when things are obviously broken [e]. I’d imagine someone who spends lot of time thinking about effective altruism (say) can probably see tons of inefficiencies there. And I had to spend lots of time thinking about the value of college because I’m applying for transfer. That’s why my views on the value of college are so strong.

This is the main value of talking to people with different specialties. It’s not merely that they see things differently; that’s tautologically true. The sinker is that smart people in different fields can often see that large portions of the population are blatantly wrong. It’s the word “blatantly” that’s important! I make fun of college all the time. I wonder who’s making fun of me.

A corollary: if most people believe X, but a smart “specialist” believes not X, the specialist is likely to be right, at least surprisingly often. (The “smart” condition cannot be dropped.) Put another way: an informed minority, perhaps even a single informed individual, is substantially better than an uninformed majority. That’s the power of neg EMH. So if I were looking for advice on picking colleges again, I’d first go find the kid who transferred out . . .

Go Do Good Things

Another corollary is that it’s easier to change the world than one might expect.

When I was younger, I used to think that “changing the world” was this glamorous thing that was near impossible, but that people attempted anyways for egotistic reasons; I explained to myself that stories of people succeeding were probably just survivor bias.

I still believe the survivor bias part, but I no longer think that most attempts are made by people trying to stroke their egos, but rather people who notice blatant market failures and feel compelled to act. In such a situation, it seems almost stupid not to take a shot. It’s not so much a feeling of “I can be the one to change the world” but rather “why on Earth has no one done this yet?”. That’s what I say all the time when I explain to people how I got the idea behind my geometry book, or any of my olympiad handouts for that matter. It is not Pride that changes the world, but Wrath.


[a] Of course, not all classes are like this, but many of them are, and I actively avoid classes which do this. As a rule of thumb, it’s easier to be personal in smaller classes, so taking higher-numbered classes over intro classes seems to be a good idea in general.

[b] Trust me, I would love to.

[c] Oops, that’s one of the things I was supposed to not say in public, although not one of the big ones. I’ll comment that I try very hard to only take classes that are worth my while. The fact that I actually learned math in high school is the only reason this is possible.

[d] Ironically, this sentence is an argument against individual diversity as follows: if you know a little bit about lots of things, then you don’t get the big “aha” moments in any of them.

[e] Not the math itself, of course, since in math we have proofs. (I imagine in other fields, you might notice things that are clearly wrong.) Though actually I’m told once you’ve delved enough into math research, you’ll can realize things that no one thought of before even though they should have. For example, Grothendieck on schemes…


In high school, I hated English class and thought it was a waste of time. Now I’m in college, and I still hate English class and think it’s a waste of time. (Nothing on my teachers, they were all nice people, and I hope they’re not reading this.)

However, I no longer think writing itself is a waste of time. Otherwise, I wouldn’t be blogging, even about math. This post explains why I changed my mind.

1. Guts

My impression is that teachers in high school got it all wrong.

In high school, students are told to learn algebra because “we all use math every day”. This is obviously false, and somehow the students eventually are led to believe it.

You can’t actually be serious. Do people really think that knowing the Pythagorean Theorem will help in your daily life? I sure don’t, and I’m an aspiring mathematician. (Tip: Even real mathematicians stopped doing Euclidean geometry ages go.) It’s hilarious when you think about it. We’ve convinced millions of kids all over the country that they’re learning math because it’s useful in their lives, and they grudgingly believe it.

The actual answer of why we teach math in schools is that it is supposed to teach students how to think. But even the teachers have lost sight of this. Most high school math teachers are now just interested in making sure their students can “do” certain classes of problems in a short time, where “do” here doesn’t refer to solving the problem but regurgitating the solution that’s already been presented. The process is so repetitive and artificial that in high school I wrote computer programs to do my homework for me, because all the “problems” were just the same thing with numbers changed. If you’re interested in just how far off math is, I encourage you to read Lockhart’s Lament.

How can this happen? I think the answer is that many high schoolers don’t really have the guts to think, “my math teachers don’t have a clue”, even though they like to joke about it. I have the guts to say this now because I know lots of math. And it’s amazing to know that millions and millions of people are just plain wrong about something I believe in.

But on to the topic of this post…

2. The world lied to me

I was always told that the purpose of English class was to learn to write. Why is this important? Because it was important to be able to communicate my ideas.

Dead wrong. Somehow the skill of being able to argue on the nature of love in Romeo and Juliet was going to help me when I was writing a paper on Evan’s Theorem years down the road? That’s what my parents said. It sounds absurd when I put it this way, but people believe it. (And let’s not forget the fact that theorems are named by last name…)

I claim that the situation is just like math. People are just being boneheads. As it turns out, the standard structure of an English essay is nothing more than a historical accident. Even the fact that essays are about literature is a historical accident. But that’s beyond the scope of what I have to say.

So what is the purpose of writing? It turns out that there is one, and that it has nothing to do with communication. It’s that writing clarifies thinking.

3. Writing lets you see everything

“I sometimes find, and I am sure you know the feeling, that I simply have too many thoughts and memories crammed into my mind…. At these times… I use the Pensieve. One simply siphons the excess thoughts from one’s mind, pours them into the basin, and examines them at one’s leisure.”

— Harry Potter and the Goblet of Fire

Here’s some advice to all of you still in doing math contests — start keeping track of the problems you solve.

There’s superficial reasons for doing this. A few days ago I was trying to write a handout on polynomials, and I was looking for some problems on irreducibility. I knew I had seen and done a bunch of these problems in the past, but of course like most people I hadn’t bothered to keep track of every problem I did, so I could only remember a few off my head. So I had to go through the painful process of looking through my old posts on the Art of Problem Solving forums, searching through old databases, mucking through pages of garbage looking for problems that I did ages ago that I could use for my handout. And all the time I was thinking, “man, I should have kept track of all the problems I did”.

But there are deeper reasons for this. As I started collating the problems and solutions into a list, I started noticing some themes in the solutions that I never noticed before. For example, basically every solution started with the line “Assume for contradiction that {f} is not irreducible and write {f = g \cdot h}”. And then from there, one of three things happened.

  • The problem would take the coefficients modulo some prime or prime power, and then deduce some things about {g} and {h}. Obviously this only worked on the problems with integer coefficients.
  • The problem would start looking at absolute values of the coefficients and try to achieve some bound that showed the polynomial had to reduce in a certain way.
  • If the problem had multiple variables, the solution would reduce to a case with just one-variable. This was always the case with problems that had complex coefficients as well.

You can’t really be serious — I’m only noticing this now? Here I was, already a retired contestant, looking at problems I had done long long ago and only realizing now there was a common theme. I had already done all the work by having done all the problems. The only difference was that I didn’t write anything down; as a result I could only look at one problem at a time.

Needless to say, I was very angry for the rest of the day.

4. External and Working Memory

Why does this happen? More profoundly, it turns out that humans have a finite working memory. You can only keep so many things in your head at once. That’s why it’s a stupid idea to not write down problems and (sketches of) solutions after you solve them and keep them somewhere you can look at.

I probably did at least 1000 olympiad problems over the course of my life. Did I manage to keep all the solutions in my head? Of course not. That’s why at the IMO in 2014, I didn’t try a maximality argument despite the {\sqrt n} in the problem. I think if I had kept better records I wouldn’t have missed this. How else do you get exactly {\sqrt n} in the lower bound? It’s not even an integer! Poof. There goes my neat 42.

I didn’t realize this wasn’t just a math thing until much later. I was talking about something along these lines during my interview for Harvard College; my interviewer was an artist. When I was talking about writing things down because I couldn’t keep them all in my head, he said something that surprised me — his easel was covered with sticky notes where he wrote down any ideas that occurred to him. He called it “external memory”, a term I still use now.

It’s actually obvious when you think about it. Why do people have to-do lists and calendars and reminders? Because you can’t keep track of everything in your head. You can try and might even get good at it, but you’ll never do as well as the old-fashioned pen and paper.

This isn’t just about “I need to remember to do {X} in exactly {Y} time”. There’s a reason we use blackboards during math lectures instead of just talking. The ideas in math are really, really hard, because math is only about ideas, and nothing else. If the professors didn’t write the steps on the board, no one would be able to keep more than two or three steps in their head at once. The difficulty is only compounded by the fact that math has its own notation. We didn’t develop this notation because we were bored. We developed notation because the ideas we’re trying to express are so complex that the English language can’t even express them. In other words, mathematicians were forced to create a whole new set of symbols just to write down their ideas.

5. An Imperfect Analogy to Teaching

But so far I haven’t really argued anything other than “if you want to remember something you better write it down”. There’s a difference between a to-do list and an exposition. One is just a collection of disconnected bullet points. The other needs to do more, it needs to explain.

The following quote is excerpted from Richard Rusczyk’s article “Learning Through Teaching” ).

You can’t just “kind of get it” or know it just well enough to get by on a test; teaching calls for complete understanding of the concept.

  • How do you know that?
  • When would you use that?
  • How could you come up with that in the first place?

If you can’t answer these questions for something you “know”, then you can’t teach it.

I knew this was true from my own experiences teaching, but it took me more time to realize that writing well is a similar skill. The difference is the medium: when you’re teaching in person, you get real-time feedback on whether what you said makes sense. You don’t get this live feedback when you’re writing, and so you need to be much more careful. Yet all the nuances of teaching are still there — distinguishing between details, main ideas, hardest steps; deciding what can be worked out from what other things, even deciding which things are worth including and which things should be omitted.

This all really started to become obvious to me when I started my olympiad geometry textbook. In senior year of high school, I decided that I had a good enough understanding of olympiad geometry to write a textbook on it. I felt like I could probably do better than all the existing resources; not as hard as it sounds, since to my knowledge there aren’t any dedicated books for olympiad geometry.

After I had around 200 pages written, I realized that I had gotten a lot better at geometry. There were lots of things that happened in the process of thinking about the best way to teach geometry.

  1. Most basically, I did in fact fill in gaps in my knowledge. For example, I studied projective transformations for the first time in order to write the corresponding section in my book. The ideas definitely clicked much faster when I was thinking about how to teach it.
  2. I made new connections. I realized for the first time that symmedians and harmonic quadrilaterals are actually the same concept; I discovered a lemma about directed angles that I wished I had known before; I found a new proof to Menelaus using an elegant strategy I had used on Monge’s Theorem. None of this would have happened from just doing problems.
  3. Most profoundly, I got a much better understanding for when to apply certain techniques. One of the main goals of my book was to make solutions natural — a reader should be able to understand where a solution came from. That meant that at every page I was constantly fighting to try and explain how I had thought up of something. This unending reflection was exhausting and reduced me to a rate of about one page written per hour\footnote{But conveniently, this process is something that just requires a laptop, not even paper and pencil. So I got a lot of pages written during office assistant.}. But it improved my own ability significantly.

Ultimately what this exemplifies is that trying to explain something lets you understand it better. And that’s in part because you can only manage so many things in your head at once. If you think keeping track of your appointments in your head is hard, try doing that with a complex argument. Can’t do it. Writing solves this problem.

6. Finding the Truth

But that’s not a perfect analogy. What I’ve presented above is a model where you have ideas in your head and you output them onto paper. This isn’t totally accurate, because as you write, something else can happen: the ideas can change.

I’ll draw an analogy from painting, again courtesy of Paul Graham.

The model of painting I used to have is that you would have something you want to draw, and then you sit down and draw it, then polish up the details. (That’s how I did all my high school art projects, anyways.) But this turns out to not be true: Countless paintings, when you look at them in x-rays, turn out to have limbs that have been moved or facial features that have been readjusted. I was surprised when I first read this. But it makes sense if you can think about it: how you can be sure what’s in your head is what you want if you can’t even see it yet?

I propose that writing does the same thing. I don’t start by thinking “these are the ideas and I will now write them down”. Rather, I just write my thoughts down, not sure where they’re going to end up. That’s how my geometry textbook actually got written. I didn’t start with a table of contents. I started by putting down ideas, finding the connections between them, noticing new things I hadn’t before. I created new sections on the fly as the need arose, added new things as I thought of them, and let the whole thing sort itself out with a simple \verb+\tableofcontents+. You can even think of the table of contents as a natural bucket sort — put down related ideas near each others, add section headers as needed, and bam, you have an outline of the main ideas. And I never know what this outline will look like until it’s actually been written.

By the same token, revising shouldn’t be the art of modifying the presentation of an idea to be more convincing. It should be the art of changing the idea itself to be closer to the truth, which will automatically make it more convincing. This is consistent with the Latin: the word “revise” literally means “see again”.

This is where high school and college essays get it really wrong. In a college essay, the goal is to “sell an idea” to the reader. If something in the essay looks unconvincing, you fix it by trickery: re-writing it in a way that it sounds more convincing without changing the underlying idea. The way you say something goes a long way in selling it. That’s what English class should have taught you. Sure, some teachers tell you to make concessions or counterarguments, but you’re doing this to try and pretend to be “honest”. You only write such things with an agenda in mind.

But since when are you always right? That’s absurd. The English class model is “I have a thesis that I know is right, and now I’m going to explain to the reader why”. But how can you know you’re right about a thesis before you’ve written it down? If the thesis and its accompanying argument is even remotely complex, it wouldn’t have been possible to sort through the whole thing in your head. Worse still, if the thesis is nontrivial, odds are that someone who is about as smart as you will disagree with you. And as Yan Zhang often reminds the SPARC attendees, you should really only expect to be right about half the time when you disagree with someone about as smart as you. If an essay is supposed to move you closer to the truth, and your original thesis is wrong half the time, do you scrap half your essays? Unfortunately, I don’t think you’d ever pass English class that way.

The culture that’s been instilled, where the goal of writing is to convince, is intellectually dishonest. I might even go to say it’s dangerous; I’ll have to think about that for a while. There are times when you do want to write to convince others (grant proposals, anyone?) but it seems highly unfortunate that this type of writing has become synonymous with writing as a whole.

7. Conclusion

So this post has a few main ideas. The main purpose of writing is not in fact communication, at least not if you’re interested in thinking well. Rather, the benefits (at least the ones I perceive) are

  • Writing serves as an external memory, letting you see all your ideas and their connections at once, rather than trying to keep them in your head.
  • Explaining the ideas forces you to think well about them, the same way that teaching something is only possible with a full understanding of the concept.
  • Writing is a way to move closer to the truth, rather than to convince someone what the truth is.

So now I’ll tell you how I actually wrote my geometry book, or this blog post, or any of my various olympiad articles. It starts because I have an idea — just a passing thought, like “this would be a good way to explain Masckhe’s Theorem”. Some time later I’ll another such thought which is related to the first. Then a third. My memory is especially bad, so pretty soon it bothers me so much that I have to write it down, because I’m starting to lose track. And as I write the first ideas down, I start noticing new ideas, so I add in these ideas, and then more new ideas start flooding in. There are so many things I want to say and I just keep writing them down. That’s how I ended up with a 400-page textbook written from what originally was just meant to be a short article. There were too many things to say that other people hadn’t said yet, and I just had to write them all down. The miraculous things is that these ideas naturally sorted themselves out. The bulleted main ideas I listed above weren’t things I realized until I looked at the resulting table of contents.

I’m sometimes told by people I respect that they like my writing. But I think this actually just translates to “I like the ideas in your writing”, and so I take it as a big compliment.