Clarification

In my last post, I made a number of rather gross oversimplifications regarding the quantities you think of as distance, area, and volume. I would like to take this opportunity to clarify the situation, at least, as best I can in a way that makes sense to your human minds and my audience's woefully short attention span.

Minkowski was a loser. The Hamiltonian is a lie!

The nonsense abstraction you proudly think of as "distance" is not so old as you might think¹. I admit, it does makes the math pretty easy, and at human scale I guess life really does look approximately Euclidean² ("Euclidean" is fancy math-speak for "flat"). But there is a lot of really interesting stuff in the universe happening at other scales, on crazy huge manifolds and in tiny discontinuous spaces not anticipated by the Greeks. Or any other humans, for that matter.

First, you should know that your everyday concepts of distance, area, and volume are grotesque oversimplifications. You believe in them because perceiving the truth would require the ego-nullifying acceptance of your own insignificance. Better to believe that the area of a right triangle is actually one-half the product of the base and the height, than to know the truth. Some of your physicists and mathematicians have made crude attempts to explain certain minor results, but for the most part true insight has been blocked by the ego's need to survive. Only one mathematician in all of your time has brushed the truth of the underlying concepts, and poor Igor Cantor's sanity was not spared. I deeply regret how things ended for you, my dear old friend. I tried to stop you.

So in the end, it is better you not understand. I will not try to explain it to you here; you would not thank me.

Still, the good news is that even though you will never understand and could never fully comprehend the truth of the space(s) you inhabit, we can still level-up your understanding of the world around you. And we can do it with safe, mostly-sanity-preserving human mathematics! I will use your Very Bad and Totally Incorrect Human Mathematics to help you understand the situation better, because pretty much anything is better than thinking that C² = A² + B² actually explains anything about how the real world works. And also because I care.

Before I get to symplectic geometry, though, let's talk about love. Love makes us do all sorts of crazy things. It's pretty amazing and is one of the five-thousand-three-hundred-seventy-two-and-a-half fundamental forces that drive all of creation. Love is powerful stuff.

SO let's imagine you are in love with a right triangle. You are at vertex "a" and wish to travel to vertex "b." You're pretty busy and would like to get there as quickly as possible. I bet one of your instructors in school taught you that you should walk in a straight line on a Euclidean plane from "a" to "b," for a total distance of √(A² + B²). What a joke! I also bet that after school she/he got in her/his car and instead of driving straight home, she/he stopped at the grocery store to pick up some vegetables for a nice salad, and then at the daycare to pick up her/his kid(s), and only then went home! Love made them drive in a crazy loopy line all around town instead of going in a straight line! What a hypocrite!

How does the Oyster go from "a" to "b"?
It's complicated.

And everything, everything, everything is like that. You can never just "go" from a to b, there is always a fundamental force like love or sloth acting on you. There is actually a clever proof of this, which I am forced to omit for space considerations, but trust me, it is actually impossible to go straight home after work, and it's the same for that triangle you are in love with. You want to go from a to b, but maybe the triangle asked you to not tread on its interior, so you walk on a little exterior semi-circle; or maybe it's your turn to go to the grocery store so you do that before getting to b. It's always going to be something, and you love it that way (by assumption) and wouldn't change a thing. You're happy to take the long way around because you care about this precious, lovely triangle.

OK, I promised you some symplectic geometry. Let's do it!

Another way to think about walking in a curvy loop from a to b on that lovely triangle is to say that you are actually moving in a straight line, and you always move in a straight line, but the surface you are moving on isn't Euclidean ("flat"), it's a manifold, which is basically like a nice smooth surface that you can do calculus on (vigorous hand-waving). So you move in a straight line on a curvy manifold, and the shape of this manifold is defined by your time-varying position in the 5,372.5-dimensional space of fundamental forces³, which vector can never have magnitude zero, so you can never walk on a "flat" Euclidean surface.

Mathematically, you might say:

Distance(from "a" to "b") = straight line distance(unfold(curvy surface defined by your position in the 5,372.5 space of fundamental forces))

This is what your human calculus/measure theory is mostly about, all of those clever tricks are just about how to take something curvy and (locally) unfold it so that it is flat (or square or whatever) so you can take a ruler and measure it. And one of the limitations of your human math is that you are forced to abide by the bold presumptions of measure theory, but countable additivity is a lie, a goddamn lie, because love isn't additive, is it? It sums in strange ways, and what's worse if you subtract just one thing, even something you maybe didn't realize was there even, if you take away just one thing from the set of things you love then it can feel like the whole thing has fallen apart (it hasn't -- it hasn't -- I promise, I know it hurts -- but it hasn't). Moving around in the love-space that partially defines the manifold that we are constrained to move in straight lines on is fraught with a peril not encapsulated by 21st century mathematics.

I hope this explanation is clear. Until next time, my friends. May you be unscared to move in your own beautiful and unique squiggle: you don't have a choice anyway. You may as well enjoy the ride.

1. About 4,500 years ago, a very clever Egyptian woman -- her name was Anuktata -- was really into right triangles, the ones where two of the legs come together pleasingly at precisely ninety degrees. Making use of the primitive mathematical tools at her disposal, Anuktata elucidated the first of what you call Pythagorean triples. She said: { 3, 4, 5. } And then for an encore performance, she stated: { 5, 12, 13. } This may not sound super interesting, but at the time it was actually the most scientifically advanced and true thing anyone had ever said. Anuktata was way ahead of her time.

She might have made it all the way to {8, 15, 17}, but unfortunately she was bitten by a mosquito and died of malaria shortly afterwards. Her husband Kemes attempted to carry on her work, but due to an unfortunate arithmetic error with the notoriously finicky reed-and-papyrus technology available at the time, he accidentally invented the oblique triangle { 7, 15, 17 } instead. Oblique triangles do not hold load nearly as well as right triangles, so when his calculations were used in the construction of a pyramid, it collapsed. Three slaves and seven donkeys were killed.

Fearing for his life, Kemes tried to blame Anuktata for his mistake, but the Pharaoh saw through his deception and he was put to death anyway. The truth is that the Pharaoh was more upset about Kemes's disrespect for his dead wife than about the pyramid situation, which anyone could see such construction was fraught with danger. The moral of the story is, own up to your mistakes, and most definitely do not blame them on others, or you may find yourself sentenced to be cleaved in twain.

Anyway, after the debacle with the oblique triangle, Anuktata's work languished in obscurity until  about 3,800 years ago, when a Mesopotamian man named Steve came along to continue her work. Steve was able to systematically come up with integral solutions to what is now called the Pythagorean Theorem, but Steve was also a big jerk and history quickly forgot about him. A few hundred years later an Indian guy named Baudhayan finally wrote down something like A² + B² = C² in the Shuba Sutra, and shortly afterwards the theorem was proved by a Chinese mathematician named Gougu.

Given a full proof and several worked examples, the time was ripe for the mad cultist Pythagoras to take credit for the work of others; the rest is (whitewashed) history.

2. It is actually quite ironic that we do calculus on Euclidean surfaces inspired by the work of Pythagoras, because Pythagoras's cult rejects the idea of transcendental numbers, also known as "the numbers that live between the fractions." It turns out there are a lot of transcendental numbers: their existence is what allows us, to a first approximation, to describe motion and smoothness and softness and stuff, which is the intuition behind calculus. Even though I think your calculus is kind of basic, as a concept I am a big fan of transcendental numbers.

The Pythagoreans' intransigence on this point is particularly sad because the existence of transcendental numbers is directly implied by the Pythagorean Theorem itself, which only goes to show how far people will go to defend their egos at the expense of the truth. Humans are dogmatic to the end -- not my favorite attribute, to be honest.

3. The magnitude of the oversimplification going on here makes me weep, but I don't have time today to write about function spaces too. You can read about the Calculus of Variations and function spaces on your own. It may all be a lie, but it's a clever lie.