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

5. Aesthetics and the Immovable Object

5.3. The Formal and the Informal

5.3.1Bringing Out the Nostalgia Device

Author: Mathematics! Get in here!

. . .

Reader: Mathematics! We’ve got something to show you!

. . .

Reader: We could just go on without—

Author: No. Give it a minute.

(Some amount of time t passes, where t is defined to be any amount of time behaving sufficiently like “a minute” to allow the narrative to continue.)

Author: Well, I guess there’s no choice. I was hoping I wouldn’t have to use this yet. . .

(Author pulls a small vial from his pocket.)

Reader: What’s that?

Author: A little something I picked up a few years ago.

(Author hands the vial to Reader.)

Reader: Hmm. It says, Henkin Juice. Grade premium. Vintage 1949. What does this do?

Author: It’s complicated. Basically a linguistic chemical reagent that dissolves the boundary between syntax and semantics. Supposed to be used on formal languages, but maybe it’ll work on informal language too. . .

Reader: . . . What?

Author: It gives power to names. Sprinkle some of it on the ground. And then keep quiet! Just for a few seconds. We don’t want this stuff to misfire.

(Reader empties the contents of the vial.)

Author: (Silently) Mathematics Mathematics Mathematics!

(Poof!)

Mathematics: WELL HELLO YOU TWO!

Author: It’s about time!

Mathematics: APOLOGIES. I SUPPOSE I LOST TRACK OF TIME. IT’S NONTRIVIAL TO AVOID LOSING TRACK OF TIME WHEN ONE LIVES IN A TIMELESS VOID.

Reader: Okay. . . We’ve got something to show you.

Mathematics: OH YEAH? WHAT IS IT?

Reader: We were playing around, and we found out that there had to be some machine E that is its own derivative.

Mathematics: INTERESTING. WHAT DOES IT LOOK LIKE?

Author: We aren’t exactly sure yet.

Reader: We called it E at first, where E stands for “Extremely special.”

Author: Then we figured out that it has to look like “number-to-the-x.”

Reader: Right. So then we started calling it e^x, where e stands for “extremely special”. . . again.

Mathematics: BUT YOU DON’T KNOW WHAT THIS NUMBER e IS YET?

Reader: Nope. We thought you should be around for when we tried to figure it out.

Author: Also, I think you still have the Nostalgia Device. We figured it might help.

Mathematics: OH MY. APOLOGIES AGAIN. I SUPPOSE I SIMPLY SPACED OUT. IT’S NON-TRIVIAL TO AVOID SPACING OUT WHEN ONE LIVES IN A SPACELESS—

Author: Enough! Just give it to us.

(Mathematics pulls out the Nostalgia Device.)

Author: Wow, I really missed this thing.

Mathematics: THAT’S A SIDE EFFECT. YOU MISS IT EVEN WHEN IT’S RIGHT NEXT TO YOU.

Author: That’s weird.

Mathematics: TO THE CONTRARY. I THINK THAT MEANS IT’S WORKING.

Reader: Let’s get moving. Here, I’ll put E into the Nostalgia Device.

Reader: What now?

Mathematics: YOU SAID THIS THING IS ITS OWN DERIVATIVE, RIGHT?

Reader: Right. So E′(x) = E(x).

Mathematics: BUT THEN ALL OF ITS DERIVATIVES ARE JUST ITSELF, RIGHT? FOR EXAMPLE, THE SECOND DERIVATIVE IS JUST THE DERIVATIVE OF THE DERIVATIVE, WHICH IS JUST THE DERIVATIVE, WHICH IS JUST THE ORIGINAL.

Reader: Nice! So you’re saying E⁽ⁿ⁾(x) = E(x) for all n. But then E⁽ⁿ⁾(0) = E(0) for all n.

Author: Hey, remember we used the Nostalgia Device on E?

Reader: (To Mathematics) Why is he flashing back to something that happened twenty seconds ago?

Mathematics: (To Reader) SIDE EFFECTS AGAIN. JUST IGNORE HIM.

Reader: Okay, where were we? Right, so E⁽ⁿ⁾(0) = E(0) for all n. What’s E(0)?

Author: Oh, E(0) ≡ e⁰ = 1. Because of the way we generalized powers in Interlude 2. Those were the days. . .

Mathematics: WOW! THAT MAKES THE OUTPUT OF THE NOSTALGIA DEVICE A LOT SIMPLER! NOW IT’S JUST

OR TO PUT IT ANOTHER WAY,

WAIT, ARE WE SURE THIS IS RIGHT? IT SEEMS TOO NICE.

Reader: I don’t know. We could check to see if the thing we wrote down is its own derivative. It shouldn’t be too hard, since it’s just a big plus-times machine. Let’s see. Using what you wrote, we’ve got:

Mathematics: BUT n! WAS JUST OUR ABBREVIATION FOR THIS:

n! ≡ (n)(n − 1) · · · (2)(1)

SO FOR ANY n, SOMETHING OF THE FORM IS SIMPLY ANOTHER WAY OF WRITING , AND YOUR EQUATION 5.14 BECOMES

Reader: And that’s exactly the same as what the Nostalgia Device spat out for E itself, so I guess it’s telling us that E is its own derivative after all. It worked! Now what?

Mathematics: I DON’T KNOW. WHAT WERE WE TRYING TO DO AGAIN?

Reader: Figure out what the number e has to be.

Mathematics: RIGHT, WELL I GUESS WE BETTER USE THE ABBREVIATION FOR E THAT INCLUDES e, LIKE THIS:

Reader: But e¹ is just e, so if we plug in x = 1 we get

or to write it another way

Hey, I think we just figured out which number e is.

Author: You did?! I missed it! I was. . . nevermind. Anyways, what number is it?

Reader: We don’t know exactly. But it’s whatever number you get from adding up the handstands of all the n! numbers, starting at n = 0 and going on forever.

Author: Oh. What if that sum is infinity?

Mathematics: OH. . . I DIDN’T THINK OF THAT.

Reader: Do we have any reason to think it’s infinity?

Author: How can any infinite sum be a finite number?

Reader: Well, 0.11111(forever) is a finite number, right?

Author: Sure. I mean, it’s less than 0.2. Definitely seems finite.

Reader: But that number can be thought of as infinitely many things added together, like this:

which I guess we could write like this:

So at least that tells us that it’s possible to get a finite number from adding together infinitely many things, as long as the things we’re adding up get smaller quickly enough — whatever that means.

Mathematics: HOW DO WE KNOW WHEN THEY’RE GETTING SMALLER QUICKLY ENOUGH?

Reader: Well in the example I made it was obvious. For this e thing, I’m not sure. . .

Mathematics: ARE WE STUCK?

Reader: I think so.

Author: Tell you what — let’s forget about that infinite sum for now. Maybe there’s a simpler way to figure out what e is.

Reader: Like what?

Author: Why not just use the definition of the derivative?

Reader: How?

Author: I don’t know. We’ve got to “tell the math” that e^x is its own derivative somehow, so I thought we’d use the definition of the derivative to do that.

Reader: You mean like this?

How would that help anything?

Author: Oh. I guess it doesn’t.

Mathematics: WAIT, I HAVE AN IDEA THAT MIGHT HELP. WHAT IF WE DID THIS?

Author: Hey! Then there’s an e^x on the far left and on the far right. If we kill-off that piece on both sides, we’d get

Maybe then we could try to isolate the e part and find another way of writing this number that doesn’t require us to add up infinitely many things.

Reader: Well, multiplying both sides by dx gives

dx = e^dx − 1

so I guess

e^dx = 1 + dx

How does this help anything?

Mathematics: HMM. . . IF WE RAISE BOTH SIDES TO THE POWER, WE’D GET:

Reader: That power is weird. Can I write the same thing some other way?

Author: Sure.

Reader: Okay, the number dx is infinitely small, but we don’t know what to do with infinitely small or infinitely large powers, at least not if we want to get an actual number out in the end. The number dx is tiny, so is huge. I’ll write as an abbreviation for a huge number. Then we can just write:

Author: But the original equation was only exactly right when dx was infinitely small, so this is only exactly right when we turn N all the way up to infinity. Let’s write that like this to remind ourselves:

or we could just think of N as the whole time. Same idea.

Mathematics: ARE WE DONE?

Author: I guess.

Reader: No we’re not! We’ve written down two sentences that say what e is, but we still haven’t figured out what it is!

Author: That sounds impossible.

Reader: No, I mean we still don’t know which number it is numerically. We’re not finished until we do some arithmetic and actually compute it.

Mathematics: I’D RATHER NOT.

Author: Yeah, I don’t really want to do all that arithmetic. Couldn’t we do something more fun, like eating a bunch of nails, or—

Reader: Are you kidding me?! We didn’t do all that just to give up right before we’re about to figure out what this thing is.

Mathematics: IF YOU REALLY WANT, I COULD CALL A FRIEND OF MINE TO DO IT FOR US.

Reader: Yes, please. Let’s get this over with.

(Mathematics borrows Author’s phone and dials a number.)

5.3.2Outsourcing Numerical Tedium to a Friend of a Friend of a Friend

It is not the job of mathematicians. . . to do correct arithmetical operations. It is the job of bank accountants.

—Samuil Shchatunovski, quoted in George Gamow, My World Line: An Informal Autobiography

Mathematics: HEY THERE, A. IT’S MATH. . . GREAT!. . . LISTEN, DID YOU EVER FINISH THAT THING YOU WERE BUILDING?. . . PERFECT, WOULD YOU MIND STOPPING BY?. . . WHERE ARE YOU NOW?. . . REALLY?. . . OH, THAT’S FANTASTIC! SEE YOU SOON.

(Mathematics hangs up.)

Mathematics: MY FRIEND A.T. IS WILLING TO HELP US WITH THE ARITHMETIC, AND HE HAPPENS TO BE IN THE NEIGHBORHOO—

A.T.: Hello, old friend.

Author: Wow! That was quick.

Mathematics: A! IT’S GREAT TO SEE YOU! HERE, LET ME INTRODUCE YOU. THIS IS READER.

Reader: Nice to meet you.

A.T.: Likewise.

Mathematics: AND THIS IS AUTHOR.

Author: Hi! Great to meet you. Who’s your friend?

A.T.: Oh! Yes, of course. Everyone, allow me to introduce you to partner. This is Silicon Sidekick, but he prefers to be abbreviated as “Sil.” He will be helping you with your arithmetic problems.

Sil: 01000111 01110010 01100101 01100101 01110100 01101001 01101110 01100111 01110011 00100000 01001000 01110101 01101101 01100001 01101110 01110011 00000000

A.T.: No, Sil! Human language, please. Sorry, you three, this happens sometimes.

(A.T. flips some switches on Sil’s panel of flippy switches.)

A.T.: No, no, Sil. Gödel numbers are not human language. And they’re not all humans. We’ve talked about this before, Sil. When greeting multiple entities, at least have the courtesy to use type promotion in order to determine the most socially acceptable manner of categorizing them. Otherwise someone may end up taking offense.

(Sil computes for a moment.)

Sil: Greetings, instances of C,

where C is the least generic class

of which both Human and classof(Mathematics) \ are subclasses.

Reader: Hi!

Author: Hello!

Mathematics: NICE TO MEET YOU.

A.T.: Sil, these folks were wondering if you might be willing to automate something for them.

Sil: Of course. Define your problem.

Reader: We were wondering if you could compute this for us:

Sil: Stack overflow.

Reader: What does that mean?

A.T.: Sil is efficient, but he is a finite being with a finite memory capacity. We can add as much memory tape as we want, but if you want him to give you an answer in a finite amount of time, you’ll have to give him a finite job.

Author: Well those n! numbers get huge pretty quickly, so I bet we could just ask for the first 100 terms or so. We just want to get an idea of what this number e is. We don’t need it to infinitely many decimal places.

A.T.: Now you’re talking! Sil, do your magic.

Sil: To nine decimal places:

Reader: Nice!

Mathematics: OKAY, BUT HOW DO WE KNOW THIS WORKED? WE MIGHT HAVE MADE A MISTAKE, OR THIS THING MIGHT HAVE MADE A MISTAKE.

Author: You’re just mad because it called you a human.

Mathematics: . . .

Author: I suppose you’re right, though. We should look at the other expression for e too. Sil, would you mind computing this for us?

Sil: Segmentation fault.

A.T.: Weren’t you listening? No infinite jobs.

Author: Sorry. Sil, would you mind computing the expression when N is 100?

Sil: To nine decimal places:

when n = 100.

Mathematics: I DON’T BELIEVE THAT’S THE SAME ANSWER AS LAST TIME.

Reader: Wait. The things we’re giving to Sil shouldn’t be exactly equal.

Author: Oh, of course. Even if we did everything right, our argument only showed that these expressions turn into e when n and N go to infinity. They shouldn’t necessarily be equal if we cut them off after a finite number of terms. Maybe they’re both just approaching the right answer at different speeds.

Mathematics: NOT AN UNREASONABLE THOUGHT. LET’S GO A BIT HIGHER. SIL, CAN YOU COMPUTE THE SUM WHEN n IS A BILLION?

Sil: To nine decimal places:

Mathematics: WELL THAT DIDN’T CHANGE ANYTHING. THE NUMBERS ARE DIFFERENT, THAT’S ALL THERE IS TO IT.

Reader: Hold on, we didn’t retry the other expression yet. Sil, can you compute the product when N is a billion?

Sil: To nine decimal places:

when N = 1,000,000,000

Reader: There we go. Same except for the final spot.

Author: Nice! I guess the second expression for e is just approaching the right answer more slowly as N gets bigger. This is great! Let’s write it in a box to make it official.

Summary of Our Adventures with the Immovable Object

We discovered that there is some specific number e for which the machine

E(x) ≡ e^x

is its own derivative. All multiples of this machine are also their own derivatives, but this is the only such machine that is also a member of the AM species. E is the only machine that is both its own derivative and turns addition into multiplication. That is, for all x and y,

E(x + y) = E(x)E(y)

We then used the Nostalgia Device to compute this number e, and found

where n! ≡ (n)(n − 1)· · ·(2)(1). Not wanting to put complete trust in the Nostalgia Device, we used the definition of the derivative to compute e another way, and found

A.T.’s partner Sil then helped us compute specific numbers for these expressions, for very large but finite values of n and N. Thanks to their help, we found that

e ≈ 2.718281828

5.3.3Thanks for Everything

Mathematics: THANKS SO MUCH FOR THE HELP, A. DOING ALL THAT ARITHMETIC BY HAND WOULD HAVE BEEN HORRIBLY TEDIOUS.

A.T.: Please, it was nothing. And don’t thank me, thank Sil.

Mathematics: DEEPEST THANKS, SIL.

Sil: My pleasure.

Author: Hey A.T., have you and Mathematics been friends a long time?

A.T.: Certainly. We’ve been friends since the foundational stages.

Author: Then why are you calling yourself “A”? What’s with the initials?

Mathematics: (To Author) HE’S A VERY PRIVATE PERSON.

Author: Come on, you don’t need to hide. You’re among friends. Full disclosure, right? What does A.T. stand for? Automated Teller?

A.T.: No.

Author: Andrew Tanenbaum?

A.T.: Closer! But no.

Author: . . . Achilles and the Tortoise?

(A.T. throws his hands up sarcastically.)

A.T.: You got me.

Author: What? Really?

A.T.: Hah, no. My name is Al.

Al T.: See?

Mathematics: (Chuckling) I’VE MISSED YOU, AL. HOW’VE YOU BEEN?

Al T.: Better, now. I had been lonely for quite awhile, but I had failed to properly categorize the problem until recently, as a result of a basic error in my premises for axiomatic introspection: that loneliness is a function of solitude. It isn’t. . . This premise delayed my pursuit of a solution for quite some time, because being around people only made the feeling worse. They never understood me; I understood them even less. With machines I’ve always felt I could be myself, but it’s hard to talk to them. Most machines, anyway. Things have been a lot better since Sil came along. He’s the apple of m—

Author: I know who you are! You’re him! The IEKYF ROMSI ADXUO KVKZC GUBJ!!!

Al T.: (Wide-eyed) How did you know?!

Author: Oh, come on. Could it be more obvious?

Reader: What are you two talking about?

Author: Nothing. You know, Al, these days they say brains and behavior can’t be understood without understanding computation.

Al T.: Who says that?

Author: People who study that sort of thing for a living.

Al T.: Wow. . . people are really coming around after all. . .

Author: Sorry you couldn’t be here to see it. We would have loved to have you around.

Al T.: (Awkwardly) . . . I had better get going. It was wonderful to meet you all.

Author: What?. . . Already?. . .

Mathematics: GOODBYE, OLD FRIEND.

Reader: Nice meeting you!

(Al and Sil begin walking away.)

Author: (To Al) Thanks for everything. All of it.

Al T.: (Confused) You’re. . . welcome.

(Al and Sil continue walking away.)

Author: (To Al, in the distance) Oh! Al! The British say they’re really sorry.

Al T.: It’s a bit late for that now. . . but tell them I appreciate the sentiment. Call us if you ever need anything automated, numerical or otherwise.

Author: Will do. Hope we see you again. . .