Suppose a child comes and asks you how to multiply 6,373 by 418. Suppose also that there was an ancient Mayan priest in the room with you. This Mayan priest overhears the child's question and barges in:
"The answer to your question is very simple. Go out and find yourself 418 baskets. Then, into each of these baskets, place 6,373 beans. Once you are done, pour out all the beans from all the baskets into a single pile, and then count them all up. The number that you have counted is the result of multiplying 6,373 by 418."
You, being a sane person with a finite lifespan, respond with the usual method that we all learn in school:
"Child, that's insane, don't listen to this priest. Here is what you do...
First, take the larger of the two numbers — which in this case is 6,373 — and write that on a piece of paper. Then, write the smaller number — which in this case is 418 — below it on the same piece of paper, and line the 8 in 418 up directly below the right-most 3 in 6,373. Finally, draw a line below the 418 — below this line is where we'll start writing our answer.
Now, take the right-most digit in 418 and multiply it to the right-most digit in 6,373. That results in 24, so write the 4 below the 8 (and below the line) and put the 2 on top of the 7 in 6,373, because we're 'carrying the 2.' Now multiply the 8 in 418 by the 7 in 6,373 and be sure to add the 2, giving 58. Write the 8 from the 58..."
I think you see the point.
Who Was Right?
In a vacuum, both of these answers are insane. The priest's method is completely impractical for any human being to actually utilize, and the second answer is convoluted and seemingly completely disconnected to the child's actual question.
However, while both answers are insane, both are also correct. The magic lies in being able to utilize the merits of both. The merit of the priest's answer is that it shows what the essence of the thing actually is. It truly shows "how to multiply" — i.e., what it means to multiply — these two numbers together. Our "school" approach has the merit of being useful — probably 100,000 to maybe 1,000,000 times faster than the priest's method.
Another merit of the Mayan priest's method is the fact that the child can go off immediately and start the process. It's a very simple — though not easy — task to perform. Our method, on the other hand, would probably require quite a lot of practice before we'd allow the kid to perform such calculations unsupervised.
Grad School vs. The Essence
This dichotomy between "the essence of a thing" and the efficient methods that we use to do the thing is pervasive in mathematics. We therefore need to be aware of it.
If we don't understand the essence of the thing, then all the fancy tricks and techniques that we wrap around that thing — no matter how useful or efficient — will be meaningless. Conversely, those tips and tricks become awesome when we do understand the essence, and we see that they allow us to do things that otherwise would be impossible or impractical.
The essence of a thing is usually very simple, often something an eight-year-old can understand. Calculus is a great example — the essence of the whole field is the single realization that "smooth" things can be viewed as limits of "not-smooth" things. This is a monumentally important realization, but it is not what makes calculus "difficult." What makes calculus difficult is the gigantic framework that we've built over centuries in order to precisely define words like "limit," and "smooth," as well as the complex set of tricks for explicitly computing many of the relevant quantities.
And this leads us to "Grad School" — which we're just using as a euphemism for school in general ("extreme" school, so to speak). Grad school is for learning the vast ecosystem of complex techniques for making "essences of things" concrete and calculable (and, in the case of actual grad school, hopefully contributing to that vast ecosystem). It is, in a sense, simply building a lot of scaffolding on top of the essence of the thing.
Keeping it all in your head
I had not heard of this distinction until Richard Feynman — the great physicist — brought up a similar analogy with Mayan priests long ago. But once you stumble upon this distinction, you see it everywhere. You also see some related phenomena.
Often, people get so good at the grad school stuff that they largely forget about the simple essence of things. This not only often kills their spirit towards whatever it is they've mastered, it also makes them pretty poor teachers.
Conversely, those who are the best at what they do are the ones who can — quickly and flexibly — move between different levels of complexity when describing or working on their topic of choice. Being able to freely move through the entire scaffolding — from the "essence" at the foundation to the most esoteric techniques up in the penthouses — shows true mastery of a subject, and often mastery of subjects that are adjacent as well. These are also often the best teachers — those who can meet their students wherever they are, and build together from there.
This is not easy, but most things aren't. It also certainly will not be the last of Feynman's philosophies towards pedagogy that we'll be discussing in this newsletter.