Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act III
N. New Is Old
N.4. Is That All?
N.4.1No
There’s plenty more that we could do in the world of multivariable calculus, but it remains true that there are really no fundamentally new ideas. Derivatives mean the same thing they’ve always meant, although they continue to be written in a variety of different-looking ways. Integrals mean the same thing they’ve always meant, and although in textbooks you might see several of them at once — ∫∫∫ instead of the familiar ∫ — this is just the old idea of ∫ three times in a row. So, rather than dwelling on all of the possible things we could do in this unfamiliar (but secretly familiar) world, let’s forge on into the infinite wilderness: calculus in infinite-dimensional spaces. In infinite-dimensional calculus, there is still a strong sense in which all of the new, scary concepts are simply the old, familiar concepts wearing different hats. However, the infinite wilderness has a rather different feel, and it springs from a beautiful unification of machines and vectors. This unifying idea enhances the beauty, elegance, and applicability of infinite-dimensional calculus quite a bit relative to finite-dimensional calculus and the semi-familiarity of this chapter. Conceptually (if not always historically), this unification of machines and vectors gives rise to a bouquet of ideas including Fourier analysis, Lagrangian mechanics, the idea of a function space, the max entropy formalism in probability theory, and the language in which quantum mechanics — our species’s deepest insight into the fundamental nature of reality — is expressed. We’ll discuss this fundamental idea in a short dialogue after the Reu—
(Mathematics wanders into the chapter.)
Mathematics: WHAT’S ALL THIS? WHY. . . WHAT. . . DID YOU TWO CREATE MORE OF ME WITHOUT ME?
(An awkward silence elapses.)
Author: Uh . . . no?
(Mathematics glances around at the nearby scenery.)
Mathematics: DID YOU???
Author: Maybe.
Mathematics: WHY DIDN’T YOU TELL ME? I WOULD HAVE—
Author: If it’s any consolation, the chapter isn’t even close to being finished. I mean, there’s so much to talk about on this subject, and I need to rewrite at least half of what I’ve written so far. It really doesn’t do the subject justi—
Mathematics: JUST FINISH THE CHAPTER!
Author: Would you mind waiting while we finish up?
Mathematics: I DON’T HAVE ALL DAY. HOW LONG IS THIS GOING TO TAKE?
Author: Well, we could go on for quite a while, but. . . hmm. . . actually, we’ve been talking about this for long enough. Here, I’ll write a Reunion right now and then we’ll be done with this chapter.
Mathematics: WRITE QUICKLY.
(Mathematics waits λP + (1 − λ)(imP) in an undefined location.)3
(Where λ ∈ [0, 1] and P ≡ patiently.)
Author: Alright, here we go. . .