Lockhart's Lament

 Lockhart's lament response

To start off, I wholly agree with the overall sentiment behind this essay. I really do think that the way math is taught in high schools is detrimental to students' creativity, inspiration, and even study skills and problem-solving skills. When I was in high school, I didn't particularly enjoy math class. It was only in university that I decided that math was interesting. The first two things I remember being genuinely curious about in university math were,

(1) dy/dx is NOT a fraction but BEHAVES like one.. how strange! I made it my mission to understand WHY this is true. And after learning about differential forms, non-standard analysis, and the abstract algebraic definition of differentials, it makes MORE sense to me, but it's still a little mysterious. I think this is an interesting fundamental concept in advanced math -- an object that behaves "like" something, but isn't technically "that thing." 

(2) In a first-year computer science proofs class, the professor did a formal proof that mathematical induction is a valid proof technique. This was fascinating to me, because I was skeptical about mathematical induction at first. This led me down a whole rabbithole about ZF, the axiom of choice, continuum hypothesis, nonstandard axioms, and eventually I learned that foundations / set theory is one of my favourite areas of math. 

Two areas I disagree with Lockhart:

(1) On page 3, he states "there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic as mathematics." On this I wholly disagree. Sure, maybe Lockhart is showing his dramatic flair, but I'd hardly call a discipline "radical" or "subversive" when the majority of its research grants are funded by the military. I'd also argue that there are many things more poetic or psychedelic than math, like poetry or psychedelic art…

(2) On page 4, he says “[math is] fascinating, it’s fun, and it’s free!” I’m not sure if I quite agree. Sure, it’s technically free to do mathematics. All that’s required is your mind, and maybe some way to write things down. People have discovered novel mathematics in very desolate situations – Leray came up with the definition of a spectral sequence, a fundamental tool in algebraic topology, while he was a prisoner of war!

All that said, however, I think there are at least two senses in which mathematics is not free. One is that, unless one is Ramanujan, you need help from other people in order to make significant progress in understanding mathematics. I always maintain that mathematics is a semi-oral tradition. Sure, it’s possible to learn some math from books, but much of the real understanding, the trade secrets, is conveyed orally in classrooms or universities. And even if one is Ramanujan, he still had math class. So a person could pretend to be “Good Will Hunting,” and take out advanced math books from the library, but I think that without university, tutoring, or some other form of oral instruction, one is probably not going to learn much from the books. And oral instruction for math, beyond public school, is not free. 

Another sense in which I think math is not free is if we consider the costs to society. In our highly complex modern industrial society, social roles are so stratified and differentiated that we don’t think much of someone who doesn’t produce anything of value, but in most of history there would need to be very good reason to allow someone to enjoy the benefits of society (food, shelter, community), while spending most of their time thinking about abstract things that are not directly useful for the society. In my opinion, the reason that “professional mathematician who thinks about very abstract things all day that do not immediately provide value to the society” is a real job is a quirk of the military industrial complex more than anything else. The abstract math might provide tangible value to the development of technology in the future, and even if it doesn’t, it acts as costly signalling – by paying an intelligent, motivated person to think of useless things all day, like “Morava p-local sphere spectra,” the government is signaling to rival states that they have so much money and scholars they can afford to fund research that seems to provide no real value to the state.