How to Invent a Mathematical Concept - Ex Nihilo - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

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

Act I

1. Ex Nihilo

1.2. How to Invent a Mathematical Concept

What I cannot create, I do not understand.

—Richard Feynman, from his blackboard at the time of his death

Before we invent calculus, we’ll need to know how to invent things in the first place. Specifically, we’ll need to know how to invent mathematical concepts. We’ll illustrate the creation process with two simple examples: the area of a rectangle and the steepness of a line.7 It doesn’t matter if you already know how to compute both of those things. Everyone has something to gain from a discussion of these issues, whether in their own understanding or in their teaching, because the invention process is so rarely discussed.

We’ll later find that these two concepts form the backbone of all of calculus. The latter is the basis of the “derivative,” and the former is the basis of the “integral.” These concepts are opposites, and the precise sense in which they’re opposites is described by the so-called “fundamental theorem of calculus.”

When we’re inventing mathematics from scratch, we always start with an intuitive, everyday human concept. The process of inventing a mathematical concept consists of attempting to translate that vague qualitative idea into a precise quantitative one. No one can really visualize anything in five dimensions, or seventeen dimensions, or infinitely many dimensions, so how do mathematicians define something like “curvature” in a way that allows them to talk about the curvature of higher-dimensional objects? How do human mathematicians arrive at their definitions, when these definitions are often so abstract that it seems as if one would have to be endowed with superhuman capacities of higher-dimensional intuition in order to “see” the truth?

This process is not as mysterious as it seems. Creation is simply translation, from qualitative to quantitative. Hopefully, explicit education in the creation process, at all levels of mathematics, will one day find its rightful place in the curriculum alongside lesser concerns like addition, multiplication, lines, planes, circles, logarithms, Sylow groups, fractals and chaos, the Hahn-Banach theorem, de Rham cohomology, sheaves, schemes, the Atiyah-Singer index theorem, the Yoneda embedding, topos theory, hyper inaccessible cardinals, reverse mathematics, the constructible universe, and everything else we teach students in mathematics from elementary school to the postdoctoral level. It is so much more important.

1.2.1Mining Our Minds: Inventing Area

In this section we’ll illustrate the essence of how to invent a mathematical concept by examining the concept of area, in the simplest possible case: the area of a rectangle. The fact that a rectangle with length and width w has area ℓw is a simple idea, and you almost certainly know it already. But try to forget it. Let’s imagine that we have no idea that the area of a rectangle is its length times its width.

Let’s assume that we know roughly what we mean by “area” in a nonmathematical sense. That is, we know it’s a word that describes how big a two-dimensional thing is, but we don’t know how to tie that concept to anything mathematical. At this point, we can use the abbreviation A to stand for area, and write pointless things like A =?, but that’s all we can do. However, based on our everyday non-mathematical concept, we know this for certain:

First thing our everyday concept tells us:

Whatever we mean by the “area” of a rectangle, it somehow depends on the rectangle’s length and width. If someone else has a definition of “area” that has absolutely nothing to do with length or width, then that’s fine, but it’s not what we mean by “area.”

Let’s make an abbreviation for all those words. We can express the above sentence in a highly abbreviated form by writing

A(, w) = ?

instead of simply writing A = ? like we did before. The new stuff inside the parentheses just says, “This somehow depends on the length and width, and I’ll abbreviate those as and w. I don’t know anything else.”

Notice that this abbreviation is similar to our abbreviation for machines from earlier. Either we could say “I’m not talking about machines, I’m just abbreviating,” or we could take the analogy with our machine abbreviations seriously and say, “Once we’ve really said what we mean by area in a precise way, it should be possible to build a machine that spits out the area of a rectangle when I feed it the length and the width. It’s this machine that I’m calling A.” Either of these interpretations will get us to the same place, so pick your favorite and let’s keep going.

Since we’re building the precise mathematical concept of area by starting with our intuitive everyday concept, we don’t have any numbers to start with. If we don’t have anything quantitative to start with, then we have to start with something qualitative. While there are no laws telling us what to do, we want to make sure our precise concept acts like our everyday concept. Toward that end, how’s this for another thing our everyday concept tells us:

Second thing our everyday concept tells us:

Whatever we mean by the “area” of a rectangle, if we double the width without changing the length, then we have two copies of the original rectangle, so the area should double. If someone else has a different definition of “area” that doesn’t behave like this, then that’s fine, but it’s not what we mean by “area.”

Figure 1.3: Whatever we...

Figure 1.3: Whatever we mean by “area,” if we double the width of a rectangle without changing the length, then the area should double.

In case that didn’t make sense, look at Figure 1.3. Our vague, intuitive, nonmathematical concept of area isn’t enough to tell us that the area of a rectangle is length times width, but it is enough to tell us that if we double the width, then the area should double (as long as we keep the length the same). We can abbreviate this idea by writing:

A(, 2w) = 2A(, w)

For the same reason, if we double the length without changing the width, the area should double too. We can abbreviate this as

A(2, w) = 2A(, w)

Even more, there’s nothing special about the word “double” in this sentence. If we triple the width, then we have three copies of the original thing, so the area should triple. Same for length, and same for quadrupling, or for multiplying by any other whole number. What about numbers other than whole numbers? Well, if we change the length from l to “one and a half l” (without changing the width) then we have one and a half copies of the original thing, so the area should be one and a half of the original. Clearly, whatever we mean by area, sentences like this capture our intuitive concept, no matter what the amount of magnification is. We can abbreviate this infinite bag of sentences all at once by writing

and

no matter what the number # is. But if that’s true, then we can trick the mathematics into telling us the area of a rectangle, by thinking of as · 1, and thinking of—

(A faint rumbling noise is heard in the distance.)

Aaah! What was that?!. . . Was that you?

Reader: Uhh. . . I don’t think so. I think that was on your end.

Author: You sure?

Reader: Yeah, pretty sure.

Author: Hmm. . . Okay, where were we? Right, equations 1.1 and 1.2 tell us that we can pull numbers outside the Area machine, no matter what those numbers are. But if that’s true, then there’s nothing preventing us from being tricky and pulling the lengths and widths themselves outside! They’re just numbers, after all. Since is the same as · 1 and w is the same as w · 1, we can sneakily use the two facts provided by equations 1.1 and 1.2 on the numbers and w themselves, like this:

Which says that the area of a rectangle is length times width. . . times some extra thing? What on earth is that A(1, 1) doing there?!

It turns out that equation 1.3 is trying to tell us about the concept of units. It’s telling us that we can figure out the area of any rectangle, but only once we’ve decided on the area of a single rectangle, the rectangle with a length of 1 and a width of 1 (or any other rectangle). If we’re measuring lengths in light-years, then we want A(1, 1) to be the area of one square light-year. If we’re measuring lengths in nanometers, then we want A(1, 1) to be the area of one square nanometer.

We usually get around this by forcing A(1, 1) to be 1, but that’s just for convenience. We could choose to represent A(1, 1) by the number 27 if we wanted, and then we’d have formula A(ℓ, w) = 27ℓw. That may look odd, but it wouldn’t be wrong in the least. Instead of forcing A(1, 1) to be 1 or some other number, we can interpret equation 1.3 in a different way, by rewriting it like this:

What this says is that we don’t have to talk about units (that is, we don’t have to decide what we want A(1, 1) to be), but we can no longer talk about the areas themselves. This interpretation tells us that something is equal to length times width, but it’s not an area. It’s a “ratio” of areas, or however many A(1, 1)’s you can fit into A(l, w).

After all that inventing, we see that the mathematics was really smarter than we were — not only did it try to tell us about the concept of units, but it automatically tells us how to convert areas from any system of units to any other (say, from nanometers to light-years). This is one of the many cases we’ll see throughout mathematics where inventing a concept for ourselves, even a simple one that we’re thoroughly familiar with already, can give us much more insight into the concept itself.

Even better, it’s not hard to convince ourselves that this same argument should work in any number of dimensions. Let’s imagine that we have a three-dimensional box-type thing, and let’s abbreviate its length, width, and height as , w, and h. For the same reasons as in the area example, if we double the height (say) without changing the length or width, then we’ve got two of the original box, so the volume should double. Just like before, there’s nothing special about the word “double,” and the same idea makes sense for any amount of magnification. Same for length and width. So in three dimensions, these three things should be true for any number #, not necessarily the same number in all three sentences:

V (#, w, h) = #V (, w, h)

V (, #w, h) = #V (, w, h)

V (, w, #h) = #V (, w, h)

Just like before, we can use these three ideas on the numbers , w, and h themselves, and write

V (, w, h) = ℓwh · V (1, 1, 1)

Now we can do something much more strange and interesting: we can start to say things about higher-dimensional spaces. If n is some large number, then we can’t picture anything in n-dimensional space. No one really can. At this point, we’re not even sure exactly what we mean by the phrase “n-dimensional space.” That’s fine! We’re completely free to say “Whatever I mean by n-dimensional space, and whatever I mean by the n-dimensional version of a rectangular box-type thing, they had better behave similarly enough to their cousins in two and three dimensions that we can make the same argument we just made. If they don’t behave like that, then that’s not what I mean by n-dimensional space right now.” Having said that, we can confidently write:

V (1, 2, . . ., n) = 12 · · · n · V (1, 1, . . ., 1)

where V stands for “whatever we want to call volume in an n-dimensional space” and we’ve decided not to give all the different directions their own quirky names anymore like we do in two and three dimensions. It’s easier to just abbreviate them all by , and then attach a different number to each one so we can tell them apart (hence: 1, 2, . . ., n).

Even though we can’t even begin to picture what we’re talking about, we can still use it to infer other things. For example, if all of the sides of this n-dimensional box-type thing are the same length (let’s call it l), then we’d have an n-dimensional cube-type thing. So if we force that ugly V(1, 1, . . ., 1) piece to be 1 (just for convenience), then we can infer that the “volume” of this higher-dimensional box is V = n. We can confidently say things about its “n-dimensional volume,” even though we can’t even begin to visualize what we’re saying!

So in summary, we saw that by thinking about our everyday concept of area, and abbreviating our thoughts in a way that expressed infinitely many sentences at once, our vague ideas turned out to force the area of a rectangle to be ℓwA(1, 1). We thus found not only the familiar “length times width” formula, but another piece we forgot to consider, though the mathematics was nice enough to remind us about it: the concept of units.

Now, we’ll use this simple invention to help us understand and visualize some of the so-called “laws of algebra” in a way that ensures that we’ll never have to memorize them ever again. Onward!

1.2.2How to Do Everything Wrong: A Sermon on the Folly of Memorization

But it required a few years before I perceived what a science teacher’s job really is. The goal should be, not to implant in the student’s mind every fact that the teacher knows now; but rather to implant a way of thinking that will enable the student, in the future, to learn in one year what the teacher learned in two years. Only in that way can we continue to advance from one generation to the next. As I came to realize this, my style in teaching changed from giving a smattering of dozens of isolated details, to analyzing only a few problems, but in some real depth.

—E. T. Jaynes, A Backward Look to the Future

One of the worst things about many early mathematics courses (at least the ones I had to take) is that the teachers seem to have somehow become convinced that the purpose of mathematics courses is to teach you facts about mathematics. I couldn’t disagree more. You might wonder what I think mathematics courses should be about, if not mathematics. This is important, so let’s say it once and for all, and put it in a box.8

Why write this in a box with such a grandiose name? Good question! Full disclosure: Sometimes when you’re writing a book (as I’ve learned since I started writing this one), it’s fun to insert the occasional homage to something you love. This box is an homage to my favorite textbook: E. T. Jaynes’s posthumously published magnum opus Probability Theory: The Logic of Science. Perhaps because he died before completing the book — but also because Jaynes was a fiery sort of guy — it’s full of Jaynes’s strangely heartwarming personal quirks, and various other things one rarely finds in textbooks. One such item is a box called “Emancipation Proclamation” in Appendix B. I always loved that section. Now I’m writing a book of my own, so I get to give Jaynes a well-deserved homage.

Declaration of Independence

The purpose of math courses is not to create students who know things about math. The purpose of math courses is to create students who know how to think.

We’d better be careful here, because at first the phrase “learning how to think” might conjure up an image of a beefy Stalin-esque man in a police uniform, holding a whip and shouting, “Think this way!” That’s not what I mean at all.

Mathematics is an entire world where nothing is accidental, and where the mind can train itself with an intensity and precision unmatched by any other subject. Moreover, in the course of training your mind, you’ll accidentally be learning the subject that just happens to describe everything about the world. It’s incredibly useful, but its practicality comes as a side effect of training the mind. Speaking of things that are important enough to write in boxes, here’s something else they never tell you:

Mathematics is NOT about

Lines, planes, functions, circles, any of the things you learn about in mathematics courses.

Mathematics IS about

Sentences that look like:

“If this is true, then that is true.”

Once we recognize this, we can see two things immediately. First, it’s obvious why training the mind in this way is useful, no matter what you are doing. Second, it’s obvious that mathematics courses are focused on exactly the wrong things.

Let’s look at a particular example of doing the wrong thing. In algebra courses, roomfuls of sleepy humans are told about something called the “FOIL” method, which is an abbreviation for “First, Outer, Inner, Last.” It’s a way to remember sentences like this:

(a + b)2 = a2 + 2ab + b2

or more generally,

(a + b)(c + d) = ac + ad + bc + bd

We can see immediately that the name “FOIL” is a way of helping you to remember a fact about mathematics, not a way of teaching you how to reinvent that fact for yourself whenever you need to. What on earth is the point of that? Most students can see what the point of that is better than the teachers can: there isn’t one. Let’s invent both of these facts in a way that guarantees that we’ll never have to remember them again.

Figure 1.4: This is...

Figure 1.4: This is basically all FOIL is saying.

If we take a piece of paper and draw a picture on it, drawing the picture doesn’t change the area of the piece of paper. This is true whether we draw a house or a dragon or anything else. So suppose we’re playing around with the ideas we’ve invented, and we come across something that looks like (a + b)2. We can think of that as the area of a square. Which square?

Well, if a square has length blah on each side, then its area is (blah)(blah), which can be abbreviated as (blah)2. So we can think of (a+b)2 as the area of a square whose sides are all a + b long. Let’s draw that square, and then let’s draw a picture on it. This is what we’re doing in Figure 1.4. The picture we’ll draw will look like a weird lopsided + sign, but it’s just two straight lines that divide up the sides into a piece that’s a long and a piece that’s b long. This will give us two ways of talking about the same thing. Since drawing all over the square didn’t change its area, we can see that

(a + b)2 = a2 + 2ab + b2

Now you never have to remember that formula ever again. Let’s see if this same type of argument also lets us invent this more complicated sentence:

(a + b)(c + d) = ac + ad + bc + bd

Any two numbers multiplied together, that is, anything of the general form (blah) · (blee), can be thought of as the area of a rectangle that is (blah) long in one direction and (blee) long in the other. Let’s draw a picture of this, where (blah) is (a+b) and (blee) is (c+d). The picture is in Figure 1.5. The picture says that the big rectangle’s area is just the area of all the small rectangles added up. So the gist of the picture can be expressed in abbreviated form by the sentence

(a + b)(c + d) = ac + ad + bc + bd

Figure 1.5: This is...

Figure 1.5: This is really all FOIL is saying.

Now you never have to remember that formula ever again. If you ever forget it, you can invent it for yourself. You shouldn’t even try to remember either of these formulas. In fact, you should probably try to forget both of them immediately! Every mathematics classroom should have this inscribed over the blackboard:

The First Commandment of Mathematics Education

A mathematics teacher should not urge students to remember, but to forget.

Since the goal is for you to be able to go through the same process of reasoning yourself, you should not try to memorize the steps in this argument, but rather to understand the argument well enough that if you ever forget either of these formulas (which you should), then you can reinvent them for yourself on the spot in a few seconds. When you do this, you’ll find that eventually you “memorize” things by accident, just because you understand them so well. The way to check whether you’ve been successful in this Zen-like process of “learning without remembering” is to see whether you can apply the same process of reasoning in new places.

Here’s the logic of it: if you can apply the same reasoning in places you’ve never seen, then it’s impossible for you to have just memorized the facts themselves. New contexts act like a sieve that filters out the possibility of memorization. Unfortunately, in an environment that punishes experimentation and failure (e.g., school), trying things out in new contexts becomes a source of anxiety rather than the intellectually gratifying play that it should be. Let’s ignore all that and just play.

Inventing Stuff

1.Above, we discussed the silly acronym “FOIL,” which stands for “First, Outer, Inner, Last.” This method of remembering a fact rather than understanding a process would lead us to come up with infinitely many different “methods,” all of which we would have to memorize. For example, it happens to be true that

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac

That’s a really ugly sentence, and no one in their right mind would want to memorize it. If our goal were just to remember it, rather than to learn general strategies of reasoning, then we might use the same approach as the guy who invented the acronym “FOIL” and call this the “LT.MT.RT.15.16.24.26.34.35” method.9 Let’s not do that. Instead, use the same strategy we used above (drawing a picture and staring at it) to invent this ugly expression for yourself. Hint: Draw a square and break up each side into three pieces, instead of two like we did.

This stands for “Left Two, Middle Two, Right Two, First and Fifth, First and Sixth, Second and Fourth, Second and Sixth, Third and Fourth, Third and Fifth” method, and it illustrates the absurdity that the “FOIL” mindset (memorizing facts) leads to if taken just one step further.

2.This time let’s play in three dimensions, and see if the same type of thinking works there. We really don’t want to have to remember an ugly sentence like

(a + b)3 = a3 + 3a2b + 3b2a + b3

So instead of remembering it, let’s invent it in the same way we did above: by drawing a picture and staring at it. Hint: Draw a cube and break up each edge into two pieces. It might help to stare at Figure 1.6. Also, you might want to try the one above this first, because this one can be a bit hard to visualize, so if you get stuck, it’s easy to get discouraged and think that you don’t understand the idea, even if you really do.

Figure 1.6: This picture...

Figure 1.6: This picture might help with invention #2.

Even though this kind of reasoning lets us invent things that other people told us to memorize, there are two things about it that are less than perfect. First, it’s not as simple as it could possibly be (I’ll say what I mean by this soon). Second, it’s basically useless on stuff like (a + b)4 or (a + b)100, because the human mind has trouble visualizing things in dimensions higher than three. It turns out that there’s a remedy for both of these things.

Instead of using the style of thinking we used above to shatter stuff like (a+b)4 into smaller pieces, let’s use something simpler. I know it might seem weird to solve a harder problem by using a simpler method, but this strategy turns out to work all over the place in mathematics. It’s really fortunate that things work out that way! Here’s the simpler method.

Say we’ve got a piece of paper, and imagine tearing it into two pieces however you want. Even if we don’t know either of the areas numerically, it’s clear that the area of the original piece is the area of both torn pieces added together. We can reinvent the “FOIL” method and all of its more complicated friends in any number of dimensions (whether we can picture what’s going on or not) just by applying this tearing idea over and over. Let’s write down the tearing idea in abbreviated form.

Suppose we’re inventing things and we get to a point where we’ve written down something that looks like (stuff) · (a + b), or maybe (a + b) · (stuff). They’re the same thing, so this argument works for either. Just like before, we can picture this as the area of a rectangle with two sides that are (stuff) long and two other sides that are (a + b) long. If we tore this rectangle right along the line in the middle-ish of Figure 1.7, then we’d have a piece with area a · (stuff), and another piece with area b · (stuff). The tearing doesn’t change the total area (because we’re not throwing any pieces away), so it has to be true that

(a + b) · (stuff) = a · (stuff) + b · (stuff)

Figure 1.7: The obvious...

Figure 1.7: The obvious law of tearing things: if you tear something into two pieces, then the area of the original thing is just the areas of the two pieces added together. To say the same thing in abbreviated form: (a + b) · (stuff) = a · (stuff) + b · (stuff). Textbooks usually call this fact the “distributive law.”

I’ll call this the obvious law of tearing things, but the name isn’t important. Call it whatever you want. Textbooks call it the “distributive law,” which sounds a bit pretentious, but that name makes sense too. After the next few paragraphs, we usually won’t need a name for the idea.

Just like the obvious law, all of the so-called “laws of algebra” can be thought of as abbreviations for simple visual ideas. For example, the fact that multiplication works the same both ways (i.e., a · b = b · a) just says that the area of a rectangle doesn’t change when we turn in on its side. That’s a really simple idea too, so they call it the “commutative law of multiplication” to scare you. But it just means we can switch the order of multiplication anywhere we want. In particular, we can use the obvious law even when (stuff) shows up on the left of (a + b), instead of on the right.

Now, if we wanted to sound like a textbook, we could have written c instead of (stuff) when we wrote down the obvious law. That would be okay too, but I wrote (stuff) to remind ourselves that the law is true no matter what (stuff) looks like. If (stuff) happens to be two things added together (or rather, if we choose to think of it that way), then we can replace the (stuff) with something like (c + d), and rewrite the obvious law like this:

(a + b) · (c + d) = a · (c + d) + b · (c + d)

But then using the obvious law again (on each of the pieces on the right), we get

(a + b) · (c + d) = ac + ad + bc + bd

which is just the expression that we invented earlier by drawing pictures. The above sentence is also just the “FOIL” method. But since we invented the “FOIL” method using the obvious law, we never have to remember it ever again. Ready, set, forget it forever!

As mundane as the obvious law seems, it turns out to offer us a window into higher dimensions. The visual way of thinking about (a + b)3 required us to picture a three-dimensional object (a cube), and we quickly noticed that this method didn’t really help us on (a+b)4 or any larger powers, because we can’t visualize four-dimensional objects. However! Even though we’re not interested in algebraic tedium like expanding (a+b)4 for its own sake, we might be interested in deeper questions like how to chop up a four-dimensional cube along each of its three-dimensional “surfaces,” none of which we can really picture due to the limitations of the human brain. But while we primates run into problems with the visual method, the obvious law has no such limitation. So if we felt like it, we could simply apply the obvious law to something like (a+b)4 repeatedly, and once we had completely unraveled it, the resulting (admittedly long) expression would give us a bit of insight into four-dimensional geometry. For example, the number of pieces in the resulting expression would be the number of different pieces that a four-dimensional cube would be carved into, if we sliced it along each of its three-dimensional faces. I have no idea how to picture what I just said, but it’s true! It has to be. Simply by using this mundane fact about tearing rectangles in two, we can coax the mathematics into telling us something that reaches far beyond the human brain’s powers of visualization.

1.2.3Unlearning Division / Forgetting Fractions

As we saw above, the small amount of mathematics we’ve invented so far turns out to be more than enough to reinvent many of the so-called “laws of algebra.” In the next few paragraphs, we’ll show how two more such laws follow naturally from what we’ve already done. Understanding where these “laws” come from will allow us to more comfortably forget the things we’ve been told about the strange nouns known as “fractions,” and the correspondingly strange verb “division.”

First, you may have been told in the past that we can (warning: jargon ahead) “cancel common factors from the numerator and denominator of a fraction.” That is, . However, in the universe we’ve invented thus far, there is no such thing as “division”: a symbol like is nothing more than an abbreviation for , which is just multiplication of one number by another. This may seem like cheating, because the symbol certainly looks like it involves division. But division is a concept imported from outside our universe. We’re simply using the symbol as an abbreviation for whatever number turns into 1 when we multiply it by 9. To put it another way, we are defining symbols like by their behavior, and treating whatever numerical values they might have as a secondary afterthought that we, personally, choose not to focus on. Defining these objects by their behavior also makes it simple to convince ourselves that sentences like are true. Here’s how.

We’ve convinced ourselves that the order of multiplication shouldn’t matter, and we also know that , for any number #. The following argument uses only these two ideas to show that the sentence has to be true. Notice that every equals sign below is ≡, except for one. The one that isn’t ≡ just involves switching the order of multiplication. Since we can think of that as turning a rectangle on its side (i.e., swapping its length and width without changing its area), I’ll write Turn above the equals sign where we use that fact. Here we go:

So the fancy-sounding “law” about “canceling” “common factors” from the “numerator” and “denominator” of a “fraction” really isn’t a law at all. Or maybe it is. Or maybe the term “law” isn’t really meaningful. Either way, it’s really just a consequence of the fact that (i) we decided when we began that the order of multiplication shouldn’t matter, and (ii) we’re using as an abbreviation for whichever number turns into 1 when we multiply it by stuff.

Here’s another quick one. At some point, we’ve probably all been told that we can “break fractions apart.” That is, someone told us that the sentence was true, probably without much justification. However, this is just the obvious law of tearing things in disguise. Let’s see why. Again, notice that all the equals signs in what follows are ≡, except for one. The one that isn’t ≡ is where we use the obvious law, so I’ll write Tear above that equals sign. Here we go:

So the ability to break fractions apart isn’t some special “law” about fractions. It really has nothing to do with fractions at all. It’s just the obvious law of tearing things, written in a slightly unusual way.

The point: Faced with a scary-looking sentence involving a lot of division, we can just rewrite it in the language of multiplication. Surprisingly often, this simple change of abbreviations will make things look a lot simpler, and it also lets us avoid having to memorize all sorts of quirky behaviors about fractions.

Alright! Our inventing muscles still aren’t exercised enough yet, so we’ll look at one more example of how mathematical concepts are invented, and then we’ll end the chapter by summarizing some general principles about this mysterious process of moving from the qualitative to the quantitative.

1.2.4Of Arbitrariness and Necessity: Inventing Steepness

An old definition of the lecture method of classroom instruction: a process by which the contents of the textbook of the instructor are transferred to the notebook of the student without passing through the heads of either party.

—Darrell Huff, How to Lie with Statistics

When we first hear about the idea of “slope” in mathematics, they usually just tell us that it’s “rise over run,” briefly say what that means, and then start doing some examples. I never heard anyone explain why it’s not “run over rise” or “52 rise over 98 run,” and you probably didn’t either. Want to know why they didn’t tell us? Because we could define slope to be “run over rise” or “76 rise over 38 run” or any other number of crazy things! It all depends on how much of our vague, everyday concept of “steepness” we want to force our mathematical concept to have, how we choose to do this, and what we think sounds reasonable. What’s more, our choice of a formal definition often depends (more than anyone wants to admit) on subjective aesthetic preferences, i.e., what we think is pretty.

The goal of this section is to see why the above paragraph is true by inventing the concept of steepness, or, as they usually call it, “slope.” This one is a bit more involved than the invention of area, but don’t worry. In both cases, the process of invention follows essentially the same pattern.

We know what the word “steepness” means in an everyday, non-mathematical sense, and we want to use this everyday concept to build a precise mathematical one. We’ll focus on straight lines for the moment, just to make things easy on ourselves, and we’ll deal with curvy things when we invent calculus in Chapter 2 (essentially just by zooming in until they look straight). So in this section, whenever I talk about “a hill” or “a steep thing,” I’m talking about straight lines.

At this point, we can abbreviate steepness by the letter S, but we don’t know anything mathematical about it, so we don’t know how to write anything other than

S = ?

But how exactly does our everyday concept work? What properties does it have? What behaviors do we implicitly assign to “steepness” when we reason about the concept in a non-mathematical setting? Before we decide what to do mathematically, we need to explore our everyday concept in some more detail.

Suppose you wake up on another continent. There’s no one around. You don’t know your latitude, your longitude, or your altitude, even vaguely. You see a hill in the distance, so you decide to walk over to it to see what’s on the other side. When you’re walking up the hill, you find it to be fairly steep, so you think about turning back and looking for help in the other direction.

The above paragraph reveals something about our everyday concept that we all know intuitively, but which is so obvious that we usually don’t bother to mention it, though mentioning such things explicitly will be a huge help in moving from the qualitative to the quantitative. That is, even though you had no idea where you were when you climbed the hill, you still knew that it was steep. A steep thing is equally steep whether we climb it when we’re underground or when we’re inside of an airplane.

Another way of expressing the same idea is to say that steepness doesn’t depend on your vertical or horizontal position in isolation. The steepness of a hill isn’t an intrinsic property of the horizontal or vertical location where it is located. It is a property of the change in vertical location as we walk up the hill. But it’s not just a property of the change in vertical location. If you walk 10 miles along a not-so-steep sidewalk, you might end up 1000 feet higher in altitude than where you started, but going up 1000 feet would be next to impossible if you only had 10 horizontal feet to do it. So, based on our vague, qualitative, pre-mathematical concept of steepness, we know this:

First thing our everyday concept tells us:

Steepness only depends on changes in vertical location and changes in horizontal location, not on the locations themselves.

Let’s come up with some abbreviations for this. We could write the above sentence in abbreviated language by writing

S(h, v) = ?

The S still stands for “steepness,” the new symbols h and v stand for differences in horizontal and vertical location. For example, if you walk 20 feet along the ground and then climb a 10-foot tree, the h between where you started and where you stopped would be 20 feet, and the v would be 10 feet. Note that these abbreviations h and v only make sense once we’ve decided on two points: where to start and where to stop. So which two points were we talking about when we wrote S(h, v) = ? in the sentence above? We aren’t saying. We’re just playing an abbreviation game at this point. But the sentence S(h, v) = ? stands for the idea that steepness depends only on changes in horizontal location (h) and changes in vertical location (v), not on the locations themselves.

Now, since h and v are both quantities that compare two points, we need to choose two points on a hill before we can talk about the steepness of it, so (as far as we know at the moment) the steepness of a line might change depending on which two points we choose. But that doesn’t seem quite right, because straight lines are straight. At least in an everyday sense of the term, a straight line only has one steepness. It shouldn’t depend which points we choose. Let’s try to tell the mathematics about this intuition:

Second thing our everyday concept tells us:

Whatever we mean by “steepness,” a straight line should have the same steepness everywhere. If someone else has a definition of “steepness” that makes straight lines change steepness in the middle, that’s fine, but it’s not what we mean by “steepness.”

Okay, great! There’s something that’s definitely true about our everyday concept of steepness, and we want to force our mathematical concept of steepness to behave like this.

Figure 1.8 lets us visualize this idea. Since steepness is about differences, we need two points to compute it. Imagine that we look at two points on a line that are h apart horizontally and v apart vertically. Looking at two such points essentially leads us to look at the small triangle in the bottom left of Figure 1.8. Now, if we were to look at a different pair of points on that same line, then the steepness should be the same. For example, imagine we now look at two points on the same line that are a distance 2h apart horizontally (that is, twice the horizontal spacing of the two points we looked at originally). Well, it shouldn’t be too hard to see that since we’re on a straight line, the vertical distance will double too. That is, it will be 2v. (Make sure you see why this is true. Figure 1.8 should help.) But because of the “second thing our everyday concept tells us,” above, the steepness should be the same in both cases. We can tell the mathematics about this intuition of ours by writing the sentence:

S(h, v) = S(2h, 2v)

Now, notice that there’s nothing special about the number 2 in this argument. If we triple h, then the same reasoning would tell us that v triples, and the steepness stays the same because we’re still talking about the same line. We can repeat this argument for any whole number and get S(h, v) = S(#h,#v), where # is any whole number.

What’s more, the same idea should work when # isn’t a whole number. For example, if you cut h in half, then v gets cut in half, so . This is an extra fact about our intuitive idea of steepness, and it gets us one step closer to the precise definition we’re looking for. Let’s write it once and for all:

Figure 1.8: An illustration...

Figure 1.8: An illustration of the second thing our everyday concept tells us about steepness. Whatever we mean by “steepness,” a straight line should have the same steepness everywhere. In particular, doubling the horizontal distance between two points also doubles the vertical distance, and we want the steepness to be the same in both cases. Or, in abbreviated form, S(h, v) = S(2h, 2v).

where # isn’t necessarily a whole number. This is really neat, and it tells us a lot about the possible things that we might mean by “steepness.” For example, the possible definition S(h, v) = h doesn’t work because it’s not true that h = #h no matter what numbers # and h are! For the same reason, the definition of steepness we’re seeking can’t be S(h, v) = hv, or S(h, v) = h + v, or S(h, v) = 33h42v99, or a lot of other things.

In fact, the longer we stare at equation 1.4, the more it becomes clear how powerful a statement it is. It almost looks like the number # is “canceling out” of both sides. If we play around for a while, we’ll be able to test a bunch of ideas, and make a list of the ones that work (i.e., the ones whose behavior makes equation 1.4 true). Here are a few that work. (Note: In the list below, the symbol just means “these are all definitions we could choose, but we haven’t chosen any of them yet.”)

After playing with this for a while, it becomes clear that any machine that depends only on (h/v) or (v/h) should work.10 That is, any machine whose description doesn’t contain h or v in isolation, but always contains both together in either the form (h/v) or (v/h). Why do we need this? Because it’s hard to see how on earth we could get any number to “cancel out,” like it has to in equation 1.4 otherwise. There might be some other way to get them to cancel, but we don’t care!

Since h/v = (v/h)−1, we could have written this sentence by saying “any machine that only depends on the quantity v/h,” but we haven’t talked about negative powers yet, so we’re not acting as if we know they exist. In our universe, they don’t yet.

1.2.5Not Your Grandfather’s Anarchy

Science is an essentially anarchic enterprise: theoretical anarchism is more humanitarian and more likely to encourage progress than its law-and-order alternatives. . . The only principle that does not inhibit progress is: anything goes.

—Paul Feyerabend, Against Method

It’s worth stepping aside for a moment and reflecting on what exactly we’re trying to do. Are we trying to dig deeper and deeper into our intuitive concept, extracting constraints until we can cut down the possibilities to just one? Not necessarily! The choice of “what we’re trying to do” is entirely up to us.

By repeatedly translating verbal ideas about our everyday concept of steepness into abbreviated form, we’ve decided that our mathematical concept of steepness has to (a) depend only on position changes v and h, not the positions themselves, and (b) depend only on and . But this still doesn’t tell us why the textbooks choose “rise over run” (that is, ) instead of any multiple of that, like or . Whether and when we should give up and just pick one is a philosophical problem that we have to deal with, and it illustrates a problem that shows up whenever we’re inventing a mathematical concept. There are two strategies at this point:

1.The Soldiering-On Approach. We could keep trying to prune down the imaginary bag of candidate definitions by (a) thinking of qualitative features of our concept of steepness, (b) abbreviating them, (c) mentally throwing out all the ones that don’t work, and (d) repeating this until we get to one and only one possible definition. Of course, once there was only one possibility left, we might not realize it just from looking at the list of requirements we’ve imposed, so we would have to convince ourselves that there really was only one surviving candidate once we got there. That would be nice, because then we’d know exactly where our definition came from, down to every last detail.

2.The Giving-Up Approach. Instead, we could decide that we’re tired of this process of mining our minds, trying to squeeze the last drop of content out of our intuitions. Maybe our everyday concept isn’t specific enough to uniquely determine one and only one definition. So, we could simply give up. “Look at it this way,” we could say. “I just wanted some definition of steepness that does everything I asked for, and I have several options, so I’m just going to pick one.” Who or what gets to determine which one we pick? We do, of course. We could just pick the remaining candidate we thought was “prettiest” or most elegant, by any standard. This happens in mathematics more than anyone would like to admit. We’re inventing this stuff ourselves. We can do whatever we want. We can conjure up entities ex nihilo and give them life by giving them names. If anarchy exists, this is it!

Okay, wait. That last sentence may have given you the impression that I’m saying there’s no such thing as “mathematical truth,” because we’re making all this stuff up. That’s definitely not what I meant to say. Anarchy in the usual sense refers to the absence of human laws, not the absence of physical laws.

In a state of anarchy, there are no “laws,” but you still can’t fly, because of the “law” of gravity. These are clearly two different concepts. Mathematics outside the confines of a classroom is anarchy in the first sense: we can do anything we want, but we can’t make anything be true.

We can choose to define things however we want, and we can choose to play with anything we want, but once we agree on what we’re talking about, we find that there is already a preexisting set of truths about our newly invented objects of study, and we have to discover those truths for ourselves.11

Note to people who know what the terms “Platonist” and “formalist” mean, and who interpreted this section as a defense of either of these views over the other: It isn’t.

In summary, at this point we could just decide to give up and choose “rise over run” as the definition we thought was prettiest, and go on to invent calculus. However, it is important to stress that we could also give up and choose “run over rise” (the upside-down version) or “42 times (rise over run) cubed” as the definition we thought was prettiest! If we then went on to develop calculus using one of these nonstandard definitions, all of our formulas would look slightly different than they do in the standard textbooks, but they would all be saying essentially the same thing as the standard versions.

1.2.6Onward! Just for Fun

The essence of mathematics lies entirely in its freedom.

—Georg Cantor, Gesammelte Abhandlungen

Now that we’ve had that discussion about what is arbitrary and what is necessary in mathematics, let’s go on and see what we would have to assume in order to make the standard definition of slope emerge as the one and only possibility left standing.

Let’s mine our minds some more, and ask whether our everyday concept of steepness tells us anything else about the properties we want our mathematical concept to have. So far, we have no reason to chose (rise over run) instead of (run over rise). However, while the second of these is a completely fine way to measure steepness, it has one odd property.

The candidate definition “run over rise” says that flat, horizontal things have infinite steepness, and completely vertical cliffs have zero steepness. That is, if the vertical distance between two points is zero (so v = 0), then becomes , which is infinite (or at least it sort of makes sense to say that it’s infinite, because , and the huge piece gets bigger as we make the tiny piece smaller). Also, if the horizontal distance between two points is zero (that is, h = 0) then “run over rise” is zero. That may not be wrong, but it isn’t quite how we usually think. Still, we’re the owners of this universe, so we’re allowed to impose the intuitive-sounding requirement that flat things have zero steepness. Let’s make it official.

Third thing our everyday concept tells us:

Whatever we mean by “steepness,” a horizontal line should have zero steepness.

This rules out a lot more possibilities. For example, it rules out , it rules out , it rules out , and anything else that isn’t zero for horizontal things (i.e., when v = 0). This is great! Let’s list some possibilities that have survived all of our purges:

There are still infinitely many candidate definitions, but a lot of them are really weird. We could quit at this point and simply choose our favorite, but let’s keep going just to see how much we have to assume in order to arrive at the standard definition.

We still haven’t said much about how different hills relate to each other. For example, what does it mean to say that one hill is “twice as steep” as another? We haven’t really thought about that yet, so as of right now, there’s no correct answer. But we want the concept we invent to make senseto us, so let’s think about what we want “twice as steep” to mean. Suppose we’ve got two points, one higher up and to the right of the other, so that the line between them looks like a hill. Then imagine that we grab the higher point and move it up even further, until we’ve doubled the original vertical distance without changing the horizontal distance. That is, imagine we transform one hill into another by doubling the height of the original hill, without changing its horizontal width. Now, it makes a certain amount of sense to say that if two hills are equally wide horizontally but one is twice as tall as the other, then the steepness of the second one should be twice as big. As before, there’s no law that forces us to think about things this way, but all the other ways of thinking about it seem even worse: for example, it seems less reasonable to say that doubling the vertical distance should multiply the steepness by 72, because it’s not clear why we should prefer this rule over the infinite number of other possibilities. But the idea that doubling height should also double the steepness has a certain simplicity and elegance to it. Let’s make it official:

Fourth thing our everyday concept tells us:

Whatever we mean by “steepness,” if we double the height of a hill without making it longer horizontally, then its “steepness” should double.

How could we abbreviate this idea? Well, we’re doubling v without changing h, and we want that to force the steepness to double, so we could abbreviate it like this:

S(h, 2v) = 2S(h, v)

So, we’ve managed once again to perform the translation from qualitative to quantitative. Now, as usual, there’s nothing special about the 2 in the above argument. The idea we really wanted to convey was more general than that. For example, the same line of reasoning suggests that if you triple the vertical distance without changing the horizontal distance, then the steepness should triple. Let’s abbreviate this idea in a way that expresses infinitely many sentences at once, just like we have before:

S(h, #v) = #S(h, v)

Perfect. Now let’s look in our imaginary bag of candidate definitions and perform another purge. Which of the remaining possibilities can satisfy this requirement? Let’s try some. Well, if we look at the possible definition and we imagine doubling v like we did above, then we get

So we doubled the verticalness and the steepness quadrupled. That means we can throw this candidate definition away, because it doesn’t live up to the fourth thing our everyday concept tells us. Alright, so we just tested where # was 2, but what about when # is 3 or 5 or 119? Instead of testing each possible power individually (which would take an infinite amount of time), let’s test them all at once by remaining agnostic about which power we’re testing. By an argument just like the one above:

But all of this has got to be equal to 2S (h, v), or else it violates the fourth thing our everyday concept tells us. So in order for the steepness to double when we double the height, it has to be the case that 2# is just 2. But that’s only true when # is equal to 1, so this lets us throw out almost all of our remaining possibilities! Let’s make a list of some survivors.

Basically anything except “some number times rise over run” has been eliminated by one of the requirements we’ve listed earlier. We’re starting to see how much of the reasoning process the textbooks sweep under the rug when they just say “Slope is rise over run.” All the remaining definitions we can think of look like , so let’s see if our intuitive concept has any of its own opinions about what that (number) should be.

Imagine that gravity changes direction a bit. Then everything that used to be flat would be slightly tilted. If gravity changes its direction by 90 degrees, then stuff that’s now horizontal would be vertical, and vice versa. Now, if the direction of “up” changes by 90 degrees, then the steepness of everything is going to change. . . except one thing: a hill that’s halfway between vertical and horizontal. That is, a hill whose horizontal distance is the same as its vertical distance (a hill for which h = v) is going to be the only thing whose steepness doesn’t change when gravity changes like this. Any steepness definition of the form will assign this special hill a steepness of (number), because the special hill has the property v = h. So deciding what we want (number) to be is equivalent to deciding on the steepness of this special hill.

Let’s consider some possibilities. Suppose we decided that we want (number) to be 5. Then the special hill would have a steepness of 5 both before and after a gravity swap, but other hills would do much weirder things. A hill with v = 3 and h = 1 would have a steepness of 15 before a gravity swap and a steepness of after. There’s nothing wrong with that, but it seems pretty arbitrary, and the steepnesses before and after a gravity swap aren’t related to each other in a nice, visually appealing way. However, if for purely aesthetic reasons we choose to assign the special hill a steepness of 1, then other hills behave much more nicely. Then, a hill with a steepness of 3 before a gravity swap would have a steepness of after. A hill with steepness beforehand would have a steepness of after. That makes things look a lot simpler. If we make this choice, purely motivated by aesthetics, then we arrive at

as the one and only surviving possibility. Let’s write that down:

Fifth thing our everyday concept “tells” us, but not really:

There’s only one hill whose steepness is the same before and after a 90-degree gravity swap (i.e., switching v and h). For the sake of elegance and simplicity, we assign this hill a steepness of 1. This makes all the other hills act nicely under gravity swaps.

Now you know exactly how much they weren’t telling you in school. As is always the case when inventing a mathematical concept, the definition we finally arrived at was built from a strange blend of translation and aesthetics: some of our definition’s behaviors came from our desire to make it behave like our everyday concept, while others came from a desire to make the resulting definition as elegant and simple to deal with as possible, by our own human standards of elegance and simplicity.

1.2.7Summarizing the Inventing Binge in Words

We’ve covered the invention process in some detail, because it’s important to have at least a few simple examples of the process of inventing a mathematical concept spelled out fully and completely, making clear at every stage what we’re making up, what’s necessarily true as a consequence of what we’ve made up, and spelling out the reason behing every step. The invention process is fundamentally important to understand, so let’s summarize what we did, first in words, then by listing all the math we invented at once. To save space, we’ll abbreviate the phrase “or it’s not a good translation” by ONGT. All mathematical concepts are invented like this:

1.You start with an everyday concept that you want to formalize or generalize.12

Once we have invented more of mathematics, this set of “everyday concepts” that serves as the raw material for the creation process will come to include simpler mathematical concepts that we have invented earlier. This occurs, for example, when we generalize the basic concepts of calculus invented in Chapter 2 to a related set of concepts in a space of infinitely many dimensions, as discussed at the end of the book in Chapter . The more deeply we explore our mathematical universe, the less clear the distinction between everyday concepts and mathematical concepts becomes.

2.You typically have some idea of what you want the concept to do in simple, familiar cases. These simple cases form the basis for your decision about which behaviors you want your new concept to have in cases that are less familiar.

Examples: Whatever “area” means, the area of a rectangle should double if you double its length, ONGT. Whatever “steepness” means, the steepness of a straight line should be the same number everywhere, ONGT. Here’s one we didn’t do: Whatever “curvature” means, the curvature of a circle or a sphere should be the same number everywhere, and the curvature of straight lines or flat planes should be zero, ONGT.

3.You force your mathematical concept to behave like your intuitive concept in these simple cases, and sometimes in straightforward generalizations from these simple cases.

Examples: I can’t even begin to picture a five-dimensional cube, but its “five-dimensional volume” should be 12345, ONGT. I can’t picture a ten-dimensional sphere, but its curvature should be the same number everywhere, ONGT. I can’t picture the fifty-two-dimensional version of a “line” or “plane” or whatever, but its curvature should be zero, ONGT.

4.Sometimes you find that all of your vague, qualitative requirements, when written in abbreviated, symbolic language, completely determine a precise mathematical concept.

5.Sometimes, all the intuitive requirements you want to impose may not be enough to single out a single mathematical definition. That’s okay! In these cases, mathematicians usually just look into the imaginary bag of candidate definitions that do everything they want, and pick the one they think is prettiest or most elegant. You may be surprised to see these ill-defined aesthetic concepts inserting themselves into mathematics. Don’t be.

1.2.8Summarizing the Inventing Binge in Abbreviations

Finally, let’s summarize the inventing binge in symbolic form, to remind us what we did.

Inventing Area

Based on our everyday concept of area, we forced the following two properties to be true of the corresponding mathematical concept, in the specific case of a rectangle:

We then found that this forces the area of a rectangle to be

A(, w) = ℓw

which is the formula they throw at us in math classes.

Inventing Steepness

Based on our everyday concept of steepness, we forced the following five properties to be true of our mathematical version of the concept, in the specific case of a straight line:

1.Steepness depends only on changes in vertical location and changes in horizontal location, not on the locations themselves.

2. for any number #.

3.We wanted the steepness of a horizontal line to be zero, so .

4.If you double the vertical distance of a hill without changing the horizontal distance, this should double the steepness. Also, this property should hold not just for doubling, but for any amount of magnification, so for any number #.

5.When h = v, we chose for purely aesthetic reasons.

We then found that these five requirements together force the steepness of a line to be

which is the formula they throw at us in math classes.

1.2.9Using Our Invention as a Springboard

The above discussion might have given you the impression that mathematics is just one big inventing binge and we’re not actually discovering anything. In the section “Not Your Grandfather’s Anarchy,” I tried to explain why this isn’t the case, but let’s look at a concrete example. We said what we meant by “slope.” Now we’ll find that by doing so, we have conjured up a world of truth that is independent of us. This world contains truths that we didn’t “put in” explicitly, and which may not be obvious to us, but which nevertheless follow from what we’ve done.

At some point in the past, you may have been told that the “formula” for a line is f(x) = ax + b, as if this fact were so simple that it should be self-evident. Notice that we never used this formula in the above discussion, even though we talked about lines the whole time. At least to me, it is neither obvious that machines of the form f(x) = ax + b happen to be lines when we draw them, nor that all lines (except vertical ones) can be represented by machines of that form.

Instead of just accepting the above statement about lines, let’s invent it for ourselves. That is, we already invented the concept of steepness, so now let’s show that the definition we invented actually forces lines to be described by machines that look like f(x) = ax + b.

Let’s assume that lines can be described by some machine M(x), but we don’t want to assume that it looks like ax + b, because that’s not obvious to us. Instead, we’ll just force the objects we’re describing by the word “line” to have constant steepness. We already made this assumption in the course of inventing steepness earlier, where we called it “the second thing our everyday concept tells us.” Let’s tell the mathematics that we’re assuming this. Suppose x and are any two numbers. No matter what x and are, if the machine M describes a line, then we want it to be true that

where the symbol # stands for “some fixed number that doesn’t depend on x or .” Okay, that was an awful lot of symbols. We did that so as not to take too many steps at once, but the main message of the stuff above is simply this:

We’re forcing the steepness to be a fixed number # everywhere because we want to talk about a line, and equation 1.6 is essentially just our way of “telling the mathematics” that. But now, since we forced equation 1.6 to be true for all x and , it has to be true when . There’s nothing special about , and we could have chosen any other number for , or for x. We’re just doing this because we’re playing around, and equation 1.6 looks a bit simpler when we look at the special case in which is zero. Alright, so when happens to be 0, equation 1.6 says:

no matter what number x is. Since we’re remaining agnostic about what x is, that term M(x) in the top left of equation 1.7 is a complete description of our machine! If we could isolate that piece, we’d have succeeded in going from our vague qualitative idea that the steepness of a line should be the same everywhere (part of the definition of steepness we invented earlier) to a precise way of writing down what a line is in symbols! Let’s try to isolate that description of our machine that’s hiding in the top left. Since the two sides of equation 1.7 are equal to each other, they’ll still be equal if we multiply them both by the same thing. So if we multiply both sides of equation 1.7 by x, and use the fact that , then we see that equation 1.7 is saying the same thing as the sentence M(x) − M(0) = #x. But that’s just another way of saying

People who like jargon call the number M(0) the “y-intercept,” but we can just think of it as whatever number our machine M spits out when we feed it 0. The symbols # and M(0) are both just abbreviations for numbers we don’t know (or if you prefer, numbers whose specific values we’re choosing to remain agnostic about), so we could write the same sentence this way:

which is the “textbook” equation for a line. We invented this ourselves, so from now on, we own it.

We did a lot of things in this chapter! Let’s remind ourselves what we did. I guess we could just call the next section “Summary.” But our own universe deserves a few of its own terms. We’re bringing everything we created back together in one place, so how about. . .

(Author thinks for a moment.)