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

N. New Is Old

Interlude N: (Mis)interpreted (Readings) (Re)interpreted

Misinterpretation

Author: I missed that. In the last sentence. Come again?

Reader: I didn’t say anything.

Author: I know. Mathematics. One more time?

(A cryptic silence elapses.)

Mathematics: I SAID, “IF I UNDERSTAND CORRECTLY, THESE ‘VECTOR’ THINGS ARE JUST MACHINES THEMSELVES, RIGHT?”. . . DID I MISUNDERSTAND SOMETHING?

Author: I think so. But don’t worry, it’s a tricky concept. A vector itself isn’t really like a machine at all. It’s just a list of numbers. Like, the vector v ≡ (3, 7, 4) is just a list of numbers, which we can think of as a point in three-dimensional space. It’s the point whose location, relative to the “origin,” is 3 units in the x direction, 7 units in the y direction, and 4 units in the z direction.

Mathematics: RIGHT, SO VECTORS ARE JUST MACHINES.

Reader: ???

Author: No no no. Why do you keep saying that?

Mathematics: I DON’T KNOW. I MUST BE MISUNDERSTANDING SOMETHING. EXPLAIN THE IDEA AGAIN.

Author: The vector v ≡ (3, 7, 4) is nothing like a function. . . er, I mean machine. It’s just three numbers. We can choose to think of it geometrically or we can choose not to. But as I’ve written it, it’s just a list with three items, which we can write as v₁ ≡ 3, v₂ ≡ 7, and v₃ ≡ 4.

Mathematics: I UNDERSTAND WHAT YOU’RE SAYING, BUT I’M NOT SURE YOU UNDERSTAND WHAT I’M SAYING. WE CAN STILL ABBREVIATE THINGS HOWEVER WE WANT, CORRECT?

Author: Of course. That’s basically the only “law” of mathematics: objects are only defined up to isomorphis. . . er, I mean, reabbreviation.

Mathematics: SO SUPPOSE I PREFERRED TO TALK ABOUT THIS VECTOR USING DIFFERENT ABBREVIATIONS. INSTEAD OF WRITING THIS:

SUPPOSE WE WERE TO WRITE THIS:

APART FROM THAT CHANGE OF ABBREVIATIONS, SUPPOSE MY VECTORS BEHAVE EXACTLY LIKE YOURS DO. WE CAN ADD THEM, MULTIPLY THEM BY NUMBERS, AND SO ON. THEN. . .

Author: (Wide-eyed) Hold on. . . I have a sense that this is going to be really important. Let me make another section.

Reinterpretation

Author: (To Mathematics) Go on, please!

Mathematics: NO, I THINK I WAS JUST CONFUSED. I ONLY HAD TIME TO SKIM THE CHAPTER. I WAS INCORRECTLY THINKING OF THE VECTOR

AS A MACHINE, WHICH, WHEN FED 1 SPITS OUT v(1) ≡ 3, WHEN FED 2 SPITS OUT v(2) ≡ 7, AND WHEN FED 3 SPITS OUT v(3) ≡ 4. GRANTED, IT WOULD BE A FUNNY TYPE OF MACHINE, BECAUSE USUALLY OUR MACHINES CAN EAT ANY NUMBER, WHEREAS THIS NEW MACHINE V WOULD ONLY BE ALLOWED TO EAT 1, 2, OR 3, SO ITS SET OF POSSIBLE FOOD WOULD BE THE SET {1, 2, 3} INSTEAD OF A CONTINUOUS SET OF NUMBERS. I BELIEVE THAT’S WHY I WAS CONFUSED.

Author: Wait. I don’t think you were confused. I was. Your argument means we have to acknowledge that vectors can be thought of as a particular type of machine. If we don’t acknowledge that, then we would accidentally be saying that we can’t abbreviate things however we want, because all your argument required was a minor reabbreviation.

Reader: So Mathematics wasn’t confused?

Author: No. And I think the same argument works in reverse. It almost has to. The whole argument was just a tiny abbreviation change. So, it’s not just that vectors can be thought of as machines. Machines can also be thought of as vectors.

Mathematics: HOW DO YOU MEAN?

Author: Well, take a machine like m(x) ≡ x² for instance. We can just think of this as a list of numbers, if we want to. I mean, instead of a list with a first slot, a second slot, a third slot, and so on, it would be a vector with a continuous infinitude of slots, one for each number.

Reader: Oh! I think I get the idea. So m(x) ≡ x² can be thought of as a vector that has the number 9 sitting inside the slot labeled 3, because m(3) ≡ 9. Do the slots have to be labeled by whole numbers?

Author: I don’t see why they would. For example, and , so m has the number sitting inside the slot labeled , and it has the number ² sitting inside the slot labelled .

Mathematics: AND JUST LIKE I SWITCHED FROM v_i TO v(i), CAN WE DO THE SAME IN REVERSE AND WRITE m₃ = 9 INSTEAD OF m(3) = 9?

Author: I don’t see why not! We’re not really doing anything. This is just a meaningless change of notation.

Mathematics: BUT THE FACT THAT THE TWO ABBREVIATIONS COULD END UP SO SIMILAR WAS POSSIBLE ONLY BECAUSE BOTH CONCEPTS NEED THE SAME TYPE OF INFORMATION TO SPECIFY THEM.

Author: I think so. I mean, instead of m(x) ≡ x², there’s no reason we can’t write m_x ≡ x².

Mathematics: SO VECTORS ARE MACHINES AND MACHINES ARE VECTORS?

Figure N.6: Up until now, we were thinking of a vector like (3, 7, 4) as a point in three-dimensional space. However, we can also think of it as a machine. This machine is far from omnivorous: it can’t eat anything but the numbers 1, 2, and 3. One advantage of picturing vectors this way is that our powers of visualization no longer break down once we move beyond three dimensions. Picturing seventeen dimensions simply requires us to picture seventeen vertical lines whose heights can vary independently. Picturing a point in a space with infinitely many dimensions simply requires us to picture the mundane “graph” of a mundane “function,” in two mundane dimensions.

Author: I think they have to be! After all, the two concepts behave in exactly the same way. We just didn’t recognize that at first. We were thinking of vectors with n slots as points in n-dimensional space, so we came up with abbreviations suited to that interpretation. Now even though there was nothing wrong with those abbreviations, they didn’t make it obvious that vectors were really just a type of machine, and vice versa. The two concepts are logically identical, but the notation we used didn’t make them psychologically identical, so we missed the underlying unity of the two ideas!

Mathematics: BUT WHICH ARE THEY REALLY? ARE MACHINES REALLY JUST VECTORS, OR ARE VECTORS REALLY JUST MACHINES?

Author: I’m not sure we have to decide. There’s no hidden essence inside these things that could somehow trick us. We’re inventing this stuff ourselves. It’s not as if something could act like a vector in every way but really secretly be a machine, or vice versa.

Reader: But isn’t this a bit arbitrary? I mean, a machine like m(x) ≡ x² seems like it’s really a machine, even though we might be able to think of it as a vector. Are you saying that it’s just as much a vector as it is a machine?

Author: Exactly! And if we aren’t willing to grant that it is, we’re actually being inconsistent. Since the beginning, the only “law” of mathematics in our universe has been: we can abbreviate things however we want. That implies that we can only define things by how they behave, not by what they are. And that must be why mathematicians are so obsessed with defining their objects axiomatically. In every branch of the field, the way they choose their definitions in the first place is always driven by this secret, underlying law: mathematical objects are not notation. The things we’re studying are not just squiggles on paper. If no particular choice of abbreviations is sacred, then the axiomatic approach and all that business about things being “only defined up to isomorphism” all follows naturally! It has to be that way. If it weren’t, then we would have violated the Only Law of mathematics: we would have assumed, at some point, however implicitly, that some set of abbreviations was inherently special.

Reader: Hey, uh, now that you’re done ranting, I have a question. Since vectors like (3, 7, 4) can be thought of as points in three-dimensional space. . . and since vectors with n slots can be thought of as points in an n-dimensional space. . . can we think of machines like m(x) ≡ x² as “points in an infinite-dimensional space”?

Author: I don’t see why not.

Mathematics: THIS IS BEAUTIFUL! CAN WE DO A CHAPTER ABOUT THIS?

Author: Of course! We’ll have to figure it out as we go.

Reader: Isn’t that what we’ve been doing all along?

Author: Hah! Yeah, I guess so. This’ll be fun! Come on, let’s go!

· · ·

(Author realizes something, and falls silent. He decides to stay behind and reread the interlude, while the others jump into the next chapter.)

· · ·

Author: . . .

Reader: . . . Hi!

Author: (Startled) Oh! I thought you had left. You’re not rereading this too, are you?

Reader: No. What’s going on?

Author: Nevermind. It’s nothing.

Reader: Seriously, what’s wrong?

Author: No no, it’s okay. . . It would just interrupt. . .

Reader: Interrupt what?

Author: Well, it’s just. . . there are no dialogues in the next chapter. . . and there are no chapters after that. . . so, I just realized. . . I’m not sure I’ll ever get to, you know. . . see you. . . again.

Reader: Oh.

Author: Yeah.

Reader: Why not just add a final section?

Author: I mean, I’ve been planning to. Not really a chapter or an interlude. More like a. . . chapterlude. But still, all my ideas for it are pretty weird. Who knows what it’ll be like when we get there. And either way, there’s no mathematics in it. That’s all in Chapter , right after this. So I just realized. . . even if we all get to see each other one more time. . .we won’t get to do this again. The mathematics. That was our last time.

Reader: Oh. . . I see.

Author: I hadn’t realized. . . until I finished writing the interlude just now.

Reader: Aren’t you still writing it?

Author: No. I just do this sometimes. Talk to you. This isn’t for the book.

Reader: What? Why?

Author: I don’t want to change the tone of the section. Your. . . their. . . the interpretation would be all wrong. Sometimes you just need to hide these things.

Reader: From who?

Author: From you! For the book! For the good of the narrative.

Reader: But what about. . . you know. . . full disclosure?

(Author sighs.)

Author: I’m gonna miss you so much when this book is over.