The Nostalgia Device - On Circles and Giving Up - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Act II

4. On Circles and Giving Up

Interlude 4: The Nostalgia Device

The Void Is No Place for an Entity

Mathematics: I’VE HAD A TERRIBLE WEEK.

Reader: What happened?

Mathematics: WELL, IT ALL STARTED ON PAGE 177.

Reader: (Flips back) You weren’t even on page 177.

Mathematics: NO NO, I WASN’T HERE. I WAS AT HOME.

Author: Where do you live?

Mathematics: IN THE VOID, REMEMBER?

Author: Sure, sure. Sorry.

Mathematics: I THINK THAT’S THE PROBLEM. LIVING IN THE VOID. IT’S. . . HARD TO EXPLAIN. . .

Reader: Try. And don’t use so many parentheses this time.

Mathematics: WELL, IT’S. . . LIVING THERE, I’M FINDING IT DIFFICULT TO. . . TO FAMILIARIZE MYSELF WITH OR BECOME ACCUSTOMED TO OR FEEL AT HOME IN OR EASE INTO THOUGH NOT EASE IN THE SENSE OF FACILITY AND CERTAINLY NOT FACILITY IN THE SENSE OF A BUILDING OR ADAPT TO BUT NOT IN THE SENSE OF BIOLOGY OR HARMONIZE WITH MINUS THE MYSTICAL CONNOTATIONS OR ACCLIMATE TO BUT NOT IN THE SENSE OF ENVIRONMENT UNLESS BY ENVIRONMENT ONE MEANS “THE VOID” IN WHICH CASE I SUPPOSE (THAT)2’S TRUE BY DEFINITION OR COMPLY WITH MODULO THE TERM’S REGULATORY COLORING OR A WORD WHERE THE SYNTAX BEHAVES LIKE HABITUATE BUT WITH THE SEMANTICS OF A WORD MORE LIKE COMFORT OR GET USED TO OR LOVE OR ENDURE THIS. . . EXISTENCE.

Reader: It’s hard living in the Void because you exist now?

Mathematics: I SUPPOSE. . .

Reader: And because the Void doesn’t exist?

Mathematics: INDEED. . . OR, NOT IN THE EVERYDAY SENSE. . . I HAVEN’T EXISTED FOR LONG, BUT I ASSUMED I’D BE USED TO IT BY NOW. I’VE NEVER FELT THIS WAY BEFORE. THIS ODD MELANCHOLY. SO I THOUGHT I MIGHT FEEL A BIT BETTER IF I TALKED TO SOMEONE.

Author: Of course. Is that why you’re talking to us?

Mathematics: NO NO, THIS WAS ON PAGE 177. YOU TWO WERE BUSY. SO I CALLED NATURE TO ASK HER ADVICE. SHE DOESN’T LIVE IN THE VOID, BUT WE’RE OLD FRIENDS. AND SHE’S EXISTED LONGER THAN ANYONE I KNOW. BUT I DIDN’T QUITE UNDERSTAND HER SUGGESTION. . . IT WASN’T A SUGGESTION, REALLY. . . SHE SAID IT MIGHT BE A PROBLEM WITH MY HOUSE.

Reader: What?

Mathematics: SOMETHING ABOUT THE FOUNDATION.7 IT WASN’T BUILT TO HANDLE A CORPOREAL ENTITY. OR, NOT CORPOREAL. . . BUT AN ENTITY. I EXIST NOW! EVERYTHING WAS FINE BEFORE. BUT CREATION IMPLIES EXISTENCE! BY DEFINITION. AND IT’S ONLY GETTING WORSE.

Mathematics: OR THE FOUNDATIONS, I ALWAYS MIX THOSE UP. PLURALS ARE TRICKY.

Author: What are you gonna do?

Mathematics: WELL, NATURE SAID SHE HAS AN OLD FRIEND FROM THE VOID WHO’S SOME KIND OF EXPERT ON FOUNDATIONS, SO SHE GAVE HIM A CALL FOR ME.

Reader: When’s he getting here?

Mathematics: I DON’T KNOW. THAT’S NOT UP TO ME. SPEAKING OF WHICH. . . MIND IF I ASK YOU TWO A FAVOR?

Reader: What is it?

Author: Anything.

Mathematics: DO YOU MIND BEING. . . AROUND? WHEN HE ARRIVES? ALL THIS IS NEW FOR ME. EXISTENCE. IT WOULD BE NICE TO HAVE SOME FRIENDS THERE.

Reader: Glad to be there.

Author: Of course. We promise.

Mathematics: WHAT’S A PROMISE?

Author: It’s a human thing. It’s where you say you will or won’t do something.

Mathematics: AND?

Author: And then the other person is supposed to believe that you will. . . or won’t.

Mathematics: HOW IS “I WILL DO X” DIFFERENT FROM “I PROMISE I WILL DO X”?

Author: The second one is more serious. The person’s supposed to believe you more.

Mathematics: I DON’T SEE HOW THAT PROVES ANYTHING.

Author: Okay, then how would you suggest we convince you? That we’ll be there, I mean.

Mathematics: WELL, THE COMMON PRACTICE IN THE VOID TO START BY DEFINING A FORMAL LANGUAGE. . .

(Mathematics defines a formal language.)

Mathematics: NOW YOU JUST SAY, IN THE FORMAL LANGUAGE: “AXIOM: DEAR MATHEMATICS, SUPPOSE THAT I WILL BE AT YOUR HOUSE TO OFFER MORAL SUPPORT WHEN THIS STRANGER ARRIVES.” WE WILL CALL THIS AXIOM S. THE S STANDS FOR STRANGER. . . OR SUPPORT. . . OR SUPPOSE. . . I HAVEN’T DECIDED YET.

(Author and Reader repeat the above incantation.)

Author: I don’t see how this is any better than a promise.

Mathematics: OF COURSE IT IS! NOW IF YOU DECIDE NOT TO SHOW UP, THEN YOU’VE VIOLATED AXIOM S BY PERFORMING ITS NEGATION, ¬S, SO YOUR FORMAL LANGUAGE IS INCONSISTENT. BEING INCONSISTENT, THE LANGUAGE CAN THEN PROVE ANYTHING, INCLUDING THE FACT THAT YOU’RE A FILTHY LIAR. . . AND A [NEGATIVE-ADJECTIVE] [NEGATIVE-NOUN], FOR ALL NEGATIVE ADJECTIVES AND NEGATIVE NOUNS. . . AND ALL THE EXPLETIVES, YOU’RE ALL THOSE TOO. IT’S MUCH BETTER THAN A PROMISE.

Author: . . .

Reader: . . .

(Narrator8 was tempted to point out that the above joke (or whatever that was) should have used the term “formal theory” in place of “formal language,” to guard against the possibility that anyone out there might be feeling pedantic. On the other hand, Narrator thought, the vernacular sense of the term “theory” comes with connotations that are likely to mislead a large proportion of readers (not to be confused with Readers; there is only one Reader). Finally, having weighed the pros and cons within the privacy of his own mind, he thought it best to simply remain silent and allow the conversation to proceed uninterrupted.)

(As he preferred to be called at this point in the book. But I digress. . .)

Reader: . . .

Author: . . .

Mathematics: . . . BUT ENOUGH ABOUT ME. HOW HAVE YOU TWO BEEN?

Nostalgia for Plus-Times Machines

Author: Pretty good.

Reader: A bit nostalgic for the old days.

Mathematics: HOW SO?

Reader: Well, back when everything in our universe was a plus-times machine, things were so much easier. Those machines with the new generalized powers weren’t so bad once we figured out how to deal with them, but in the last chapter we ran into these strange machines V and H that we couldn’t even write down a description of!

Author: They’re what the textbooks call “sine” and “cosine.”

Mathematics: WHY ARE YOU TELLING ME THAT? I HAVEN’T READ THE TEXTBOOKS.

Author: Right. Sorry.

Reader: Anyways, we basically defined these two machines visually — using pictures. They were originally the names we gave to the unknown solutions of a problem we were trying to solve. But eventually we got stuck, so we just used the Molière strategy and pretended the names themselves were the answers.

Mathematics: THAT TRICK IS GREAT, ISN’T IT? HAVE YOU TWO FIGURED OUT WHAT IS YET?

Author: Nope, but the textbooks call it π.

Mathematics: AGAIN WITH THE TEXTBOOKS! WHO ARE YOU TALKING TO?

Author: Right. Sorry again.

Mathematics: ANYWAYS, THESE MACHINES V AND H ARE TROUBLING YOU?

Reader: Yeah. Somehow we managed to figure out their derivatives. We know V′ = H and H′ = −V, but we still can’t even write down a description of the machines themselves. It’s unsettling. I miss the old days. When everything was a plus-times machine, we could write down an actualdescription of anything in our universe. But since Chapter 3 ended, everything’s changed. I feel like I don’t really understand things as much anymore.

Mathematics: THAT’S TERRIBLE. ANY WAY I CAN HELP?

Reader: I don’t know. It would be nice if we could describe everything again. Like back when everything was a plus-times machine.

Mathematics: MAYBE EVERYTHING STILL IS. . .

Reader: No, you’re just trying to cheer me up.

Mathematics: NO REALLY. THE IDEA DOESN’T SEEM TOO OFF THE WALL. ARE YOU SURE EVERYTHING ISN’T A PLUS-TIMES MACHINE?

Reader: I guess we’re not really sure. . .

Mathematics: WELL, IT’S WORTH A TRY. ESPECIALLY IF THIS IS BOTHERING YOU TWO SO MUCH.

Reader: What do you mean “worth a try”?

Mathematics: LET’S JUST FORCE EVERYTHING TO BE HOW WE WANT IT TO BE. . . AND SEE WHAT HAPPENS.

Author: That’s crazy.

Mathematics: I KNOW! BUT LET’S TRY IT. LET’S IMAGINE WE’VE GOT SOME MACHINE. WE’LL REMAIN AGNOSTIC ABOUT WHICH ONE IT IS. LET’S JUST FORCE IT TO BE A PLUS-TIMES MACHINE, LIKE THIS:

Reader: How high does the sum go?

Mathematics: I DON’T KNOW. FOREVER.

Author: We defined plus-times machines to be finite sums. They couldn’t have infinitely many pieces.

Mathematics: WHY?

Author: I don’t know. The idea of an infinite description sort of. . . scares me.

Mathematics: NO NO, I’M NOT TALKING ABOUT AN INFINITE DESCRIPTION. BUT WE SHOULD BE ALLOWED TO SAY “AND SO ON.” WE’VE BEEN DOING IT ALL ALONG.

Author: What are you talking about?

Mathematics: LIKE THE WHOLE NUMBERS! THERE’S AN INFINITE NUMBER OF THEM. BUT IT DOESN’T TAKE AN INFINITE AMOUNT OF TIME TO TALK ABOUT THEM. WE JUST SAY:

0, 1, 2, . . . (AND SO ON)

Reader: Huh. . .

Author: I’ve never thought of it like that before.

Mathematics: CAN I CONTINUE?

Author: By all means.

Mathematics: SO WE’VE FORCED OUR ARBITRARY MACHINE TO LOOK HOW WE WANT IT TO. NOW WE’VE JUST GOT TO FIGURE OUT ALL THE NUMBERS #i WE USED TO DESCRIBE IT.

Author: How could we possibly do that? We know exactly nothing about this machine M.

Mathematics: OH. . . YES I SUPPOSE THE HOPE WAS A BIT MISGUIDED. . .

Reader: Well, we know what #0 is.

Author: Come again?

Reader: Up above, in the description Mathematics wrote. If we’re forcing that to be true, then

M(0) = #0

Just plug in 0, and it kills all the pieces except the first one.

Author: Oh. . .

Mathematics: INTERESTING. . .

Reader: If we plug in 1, then maybe. . . oh, nevermind. I’m not sure we can figure out the other numbers.

Author: I liked your idea though. What if we take the derivative of M?

Reader: Why?

Author: I don’t know, but taking the derivative will knock each power down by one, so maybe we can use your trick to find #1. Here, let me try. If we take the derivative of what Mathematics wrote, we get

and then the same trick should work again. Just feed it zero and we get

M′(0) = #1

We can do it again too. Just take the derivative of the original description twice:

and then feed it zero:

M″(0) = 2#2

But we want to know what #2 is, so we can isolate that to get

Reader: Wait, does this mean we can describe anything in our universe again?

Mathematics: PERHAPS. I MEAN, WE BEGAN BY BEING AGNOSTIC ABOUT PRECISELY WHICH MACHINE WE WERE DESCRIBING, SO IN A SENSE OUR DESCRIPTION DESCRIBES AN ARBITRARY MACHINE. . . BUT TO DESCRIBE AN ARBITRARY MACHINE COMPLETELY, WE WOULD HAVE TO FIGURE OUT ALL OF THE NUMBERS IN ITS DESCRIPTION.

Reader: How would we do that?

Mathematics: IN A SENSE, WE ALREADY HAVE.

Author: How?

Mathematics: JUST FIGURE OUT #n WHILE REMAINING AGNOSTIC ABOUT n. THE SAME ARGUMENT SHOULD WORK. IF WE TAKE THE DERIVATIVE OF OUR ORIGINAL DESCRIPTION n TIMES, THEN WE’D GET RID OF ALL THE PIECES TO THE LEFT OF #nxn, SINCE EACH PIECE #kxk CAN ONLY SURVIVE k DERIVATIVES. THE FIRST DERIVATIVE KILLS THE #0 PIECE, THE SECOND DERIVATIVE KILLS THE #1x PIECE, AND SO ON. THE nth DERIVATIVE KILLS THE #n–1xn–1 PIECE, SO AFTER n DERIVATIVES, THE FIRST SURVIVOR IS #nxn.

Author: And the rest of the surviving pieces will have at least one x attached, so they’ll go away when we plug in zero.

Reader: We’re getting ahead of ourselves. What does the machine m(x) ≡ xn turn into when we differentiate it n times?

Mathematics: OH. I SUPPOSE WE DON’T KNOW. . . IF WE DIFFERENTIATE IT ONCE, THEN IT’S

m′(x) = nxn−1

Author: If we differentiate that again, then it’s

m″(x) = (n)(n − 1)xn−2

Reader: If we differentiate that one more time, then it’s

m′″(x) = (n)(n − 1)(n − 2)xn−3

Author: I think I see the pattern, but we need a new abbreviation. Let’s just write the nth derivative as m(n)(x). I put it in parentheses because it’s not a power, but I didn’t want to write a bunch of primes with “· · ·” in the middle. So taking n derivatives of xn will give us

m(n)(x) = (n)(n − 1)(n − 2) · · · (3)(2)(1)xnn

and the xn−n piece is just 1. So I guess the nth derivative of xn is just whatever number you get from multiplying all the whole numbers from n down to 1.

Mathematics: N!

Author: You want to call it n! . . . ?

Mathematics: NO NO, I MEANT TO SAY “NICE!” TO COMPLIMENT YOUR REASONING, BUT FOR SOME REASON WHEN I GOT TO THE LAST THREE LETTERS OF THE WORD. . . I JUST FROZE. . .

Reader: That’s a horrible joke.

Author: I know. But let’s call it n! anyway. It seems like a good abbreviation. That is:

n! ≡ (n)(n − 1)(n − 2) · · · (3)(2)(1)

For example

1! = 1

2! = (2)(1) = 2

3! = (3)(2)(1) = 6

4! = (4)(3)(2)(1) = 24

and so on.

Reader: Looks good! Then we can combine two things we just said to write the nth derivative of m(x) ≡ xn as

m(n)(x) = n!

This is getting a bit confusing. Let me write down everything we’ve done so I can make sure we know what we’re doing. . .

(Reader looks back at what we’ve done.)

We started off by hoping we could describe any machine like this:

If we differentiate this description n times, then all the pieces to the left of #nxn will go away, the piece #nxn will turn into #nn!, and all the pieces to the right of #nxn will still have at least one x attached, so they’ll get killed-off when we plug in zero. After all the dust settles, we’ll get:

M(n)(0) = #nn!

But since the point of all this was to figure out the numbers #n, what we really want is to rewrite it like this:

Author: Wow. . . Did we really just show what I think we showed?

Mathematics: I THINK SO! UNLESS WE MADE A MISTAKE, IT LOOKS LIKE WE JUST SHOWED THAT ANY MACHINE M CAN BE WRITTEN AS A PLUS-TIMES MACHINE LIKE THIS:

THAT’S AN UNPLEASANTLY LARGE DESCRIPTION. I’LL REWRITE IT LIKE THIS:

Author: Wait, that abbreviation has weird stuff in it. We never said what 0! is. Same for the zeroth derivative of something.

Reader: No no, Mathematics was just abbreviating the line above it. So we can just define 0! and the zeroth derivatives to be whatever they have to be to make the two sentences equal.

Author: How so?

Reader: Well if the two sentences are equal, then we have to define M(0) ≡ M. So the zeroth derivative of a machine is just the machine itself. And we have to define 0! ≡ 1. Make sense?

Mathematics: WORKS FOR ME.

Author: Okay, I’m fine with the abbreviations, but still. . . I’m a bit skeptical about all this. Sure, we wrote this down, but there’s no way this is going to work. It’s too good to be true.

Mathematics: DON’T YOU TWO HAVE THOSE MACHINES THAT WERE BOTHERING YOU?

Reader: You mean V and H?

Mathematics: WHY NOT TEST THIS NOSTALGIA DEVICE ON THEM?

Reader: Which device?

Mathematics: EQUATION 4.5. WE BUILT IT BECAUSE YOU WERE FEELING NOSTALGIC FOR THE OLD DAYS. WHEN WE KNEW HOW TO DESCRIBE EVERYTHING.

Reader: Oh, right.

Mathematics: LET’S TEST IT ON V AND H. HOW DID YOU TWO DEFINE THEM?

Reader: Well, if we have a straight line of length 1 tilted at angle α, we defined V (α) to be “however much of the line is in the vertical direction,” and H(α) was “however much of the line is in the horizontal direction.” So for example, if α = 0, then we’ve got a horizontal line, so H(0) = 1, and V(0) = 0. Aside from that and a few other specific examples, we have no idea how to get numbers for V and H if someone just hands us a random angle.

Mathematics: WHAT’S WITH THE α?

Reader: It reminded us of the word “angle.”

Mathematics: OH. I WASN’T THERE WHEN YOU DID THAT. MIND IF I JUST USE x?

Reader: Go ahead.

Mathematics: ALRIGHT, WELL OUR NOSTALGIA DEVICE TELLS US THAT

SO WE JUST NEED TO FIGURE OUT WHAT V (n)(0) IS FOR ALL n. HMM. . . I GUESS THIS IDEA ISN’T SO USEFUL AFTER ALL. HOW ARE WE GOING TO FIGURE OUT ALL OF V’S DERIVATIVES AT ZERO IF WE DON’T EVEN KNOW HOW TO DESCRIBE THE MACHINE ITSELF? I’M SORR—

Author: Wait! We kind of did figure out all of V’s derivatives. I mean, we know V′ = H and H′ = −V, so we know V″ = H′ = −V, and then we can just keep looping around, like this:

And moving from each step to the next is easy, because we can always just use V′ = H or H′ = −V to get from one derivative to the next. Even better, we know V and H at zero, so we can write

We get back to where we started after four derivatives, so things just keep looping around forever!

Mathematics: WELL, I SUPPOSE I CAN PICK UP WHERE I LEFT OFF. BEFORE WE GOT STUCK, I HAD JUST WRITTEN:

NOW, LOOKING AT WHAT YOU WROTE, IT’S CLEAR THAT ALL THE EVEN DERIVATIVES WILL BE ZERO, AND THE ODD DERIVATIVES JUST KEEP SWITCHING BACK AND FORTH BETWEEN 1 AND 1. SO IF I’M NOT MISTAKEN, THAT MEANS WE CAN WRITE:

Author: Oh. . .

Reader: I think that’s the description of the machine V that we gave up trying to find earlier! I bet we could use the same type of idea to find a description of H.

Mathematics: GIVE IT A SHOT!

(Reader plays around for a bit.)

Author: Hey! While Reader is working on H, I think I have an idea. I wonder if we could get a good approximate description of a machine by throwing away some of the terms in the infinite sum. Here, I’ll try with V. Let’s throw away everything except the first two terms in the description of V, and compare it to the graph of V we built in Chapter 4.

(Author draws Figures 4.16 and 4.17.)

Reader: I’m back! And I think it worked. I got that the machine H can be written like this:

Author: Wait, V and H were what the textbooks call “sine” and “cosine.” So we just figured out how to calculate sin(x) and cos(x) for any x without memorizing anything! So I guess now, for the first time, after all that. . . we actually know trigonometry.

Mathematics: WHAT’S TRIGONOMETRY?

Author: Nevermind.

Reader: Anyways, thanks for all the help, Mathematics.

Mathematics: ANY TIME. AND THANK YOU FOR STICKING AROUND. IT’S NICE TO HAVE SOMEONE TO TALK TO. . .

Author: My thoughts exactly. See you soon. Alright, time for the next chapter.

(Ahem.9)

(Narrator: Perhaps surprisingly (given what this “Author” fellow has told you thus far), the Nostalgia Device (called “Taylor series” in textbooks) is not guaranteed to work for any possible machine (though it works for most machines one encounters in practice). The question of when it works and when it doesn’t (or “convergence of Taylor series,” in textbook jargon) leads us down a deep rabbit hole into some extremely rich and interesting mathematics. It’s worth briefly mentioning what is perhaps the most surprising fact about the topic. That is, the conditions under which the Nostalgia Device works (or doesn’t) cannot be understood unless we are willing to think about complex numbers. Complex numbers are numbers of the form a + bi, where a and b are “real numbers” (the normal type of numbers with a decimal expansion like 9 or − 1.3 or 5.987654 . . .) and where i is the square root of −1. Like so many things we’ve encountered thus far, this number is defined by its behavior. The number i is defined to be whatever it has to be in order to satisfy the sentence i2 = 1. Long story short: It is impossible to fully understand the conditions under which the Nostalgia Device does or doesn’t work without admitting that complex numbers exist, even if we’re only concerned with machines whose input and output are real numbers. For a beautiful explanation of these ideas, there’s no substitute for Tristan Needham’s wonderfully nonstandard book Visual Complex Analysis.)

Figure 4.16: The dotted squiggle...

Figure 4.16: The dotted squiggle in the figure is V(x), also known as “sine” in the textbooks. The solid curve is , which is the machine we get from throwing away all but two terms in the Nostalgia Device’s description of V. As such, not only does the Nostalgia Device give us a way to describe the previously indescribable machines V and H, but throwing away all but a few terms in the description of these machines gives us an approximation that is very good near zero.

Figure 4.17: More examples of how adding more and more of the terms in the Nostalgia Device description of V(x) give us more and more accurate approximations of it, all of which eventually break down when we stray far from x = 0.