Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act II
5. Aesthetics and the Immovable Object
5.2. The Four Species
The Void contains an infinite swarm of things waiting to be invented, but the vast majority of them are sterile and uninteresting. Ludwig Wittgenstein said, “Whereof one cannot speak, thereof one must be silent,” and this is certainly true of mathematics. These ideas suggest that defining mathematical objects by how they behave is a surefire way to avoid the sterile parts of the Void. If we define an object by how it behaves, then we will always know what we can say about it, although we may not know what it “is.” In this section we will see several particularly striking examples of that principle. Given the central role played by addition and multiplication on our journey thus far, let’s define four species of machines purely by saying how they behave with respect to these two operations. At this point, the motivation for playing with these machines is purely aesthetic.
The Four Species:
The AA species is defined to be all machines that turn Addition into Addition:
The AM species is defined to be all machines that turn Addition into Multiplication:
The MA species is defined to be all machines that turn Multiplication into Addition:
The MM species is defined to be all machines that turn Multiplication into Multiplication:
In the definition of the four species, we force each sentence to be true for all numbers x and y. These are four especially elegant behaviors, but at the moment we have no idea what the members of each species look like. Maybe some of the species have no members. After all, it is possible to write down a sentence that cannot possibly be true, even though its impossibility may not be obvious from a quick glance at it. Let’s spend some time playing with these species, and see if we can figure out what each one looks like.
5.2.1The AA Species
Let’s imagine that we’ve trapped a member of the AA species. All we know is how it behaves. We have no idea what it looks like. Let’s see if we can figure out. The defining property of the AA species is that it behaves like this:
for all numbers x and y. The numbers x and y don’t refer to horizontal and vertical coordinates. They’re just abbreviations for any two numbers we feed the machine. Now, if the machine really behaves like this for all numbers x and y, then it has to behave this way when x and y are both zero. That is,
f(0) = f(0) + f(0) = 2f(0)
But if f(0) is whatever number is unchanged when we multiply it by 2, then f(0) has to be 0. This fact may not have been obvious from our definition of the AA species, but now we can see that the sentence f(0) = 0 was hiding inside that definition all along. Just to be clear, we really have no idea what to do in order to figure out what this beast looks like. We’re just playing around with it, using the only thing we know, equation 5.1.
What if we imagine that the number y is infinitely small? Abbreviating that number by dx, the definition of the AA species (equation 5.1) tells us that
This almost looks like a derivative. We know a few things about derivatives, but we don’t know much about the AA species, so let’s see if making this look more like a derivative will tell us anything helpful. If we move the piece f(x) over to the left side, and divide by dx, we’ll get
Nice! The left side is now just the definition of the derivative of f, so we can replace the stuff on the left with f′(x), like this:
The right side almost looks like a derivative too, but it’s missing something. Or maybe it isn’t. . . A minute ago, we figured out that all the members of the AA species had to have f(0) = 0. Since adding zero doesn’t change anything, we can write
It’s now easier to see that the right side is just the “rise over run” of two points that are infinitely close to each other: 0 and dx. Surprisingly, making this simple expression superficially more complicated (by throwing a bunch of zeros into it) actually made it easier for us to see how simple the expression was in the first place: the right side was just the derivative of f at the point x = 0. Using these ideas, we can rewrite the right side of equation 5.5 as f′(0), which turns it into
But this has to be true for any number x, which tells us that the steepness of f at one particular point (namely, x = 0) is the same as its steepness everywhere. So f has to be a straight line, and thus (using what we discovered in Chapter 1) the members of the AA species all have to look like f(x) =cx+b. Further, we already showed that this species’s members all spit out zero when we feed them zero, so we must have b = 0. Putting it all together, we can write:
The AA Species
The AA species was defined to be all machines that turn addition into addition:
We just figured out that all members of this species have to look like
f(x) = cx
for some number c.
So, we managed to figure out what the members of this species look like only by using information about how they behave. We didn’t really have any well-specified method for doing this. We just goofed around for awhile, feeding things to machines, and eventually noticed that all the members of this species had constant steepness. That told us that they were lines, which told us how to write them. Let’s see if we can do something similar for the other species.
5.2.2The AM Species
Now let’s imagine we’ve trapped a member of the AM species. Just like before, all we know is how it behaves, not what it looks like. The AM species was defined by this 'margin-bottom:0cm;margin-bottom:.0001pt; text-align:center;line-height:normal'>
Let’s play around a bit. What if we set x and y to be zero? Then we get f(0) = f(0)f(0). This isn’t quite enough to determine what f(0) is, because both the number 1 and the number 0 behave this way (that is, 0 = 0 · 0 and 1 = 1 · 1). So let’s keep exploring. What if we set just one of the inputs to be zero (say, y = 0) and don’t do anything with the other one? Then we would get
f(x) = f(x)f(0)
Much better. This sentence tells us that f(0) = 1. Or at least it tells us that f(0) = 1, unless f(x) is the boring machine that always spits out zero, in which case the above sentence would still be true, but f(0) wouldn’t be 1. For the sake of argument, let’s imagine we’re working with a member of the AM species that doesn’t always spit out zero. Then the above argument tells us that f(0) = 1.
Well, we don’t really know what to do, but trying to make both sides look like a derivative helped us last time, so let’s try that. Actually, last time we were only dealing with addition, so it was easier to make both sides look like a derivative. Maybe we can play with derivatives, but not in the same way. Remember, we’re just using x and y to stand for two things we might feed the machine, not horizontal and vertical coordinates; we can change y without changing x, so 
, which means that 
. Let’s differentiate with respect to y, using this fact. Using the hammer for reabbreviation (the “chain rule”), we can write
f′(x + y) = f(x)f′(y)
where the prime stands for the derivative with respect to y. Can we conclude anything interesting about the machine f from this? Well, if we set x = 0, we get a sentence that doesn’t tell us anything, namely, f′(y) = 1 · f′(y). That’s not very helpful. Let’s try setting y = 0 instead. Then we get
f′(x) = f′(0)f(x)
Hey! This says that the members of the AM species are almost their own derivatives. They’re just their own derivatives multiplied by some fixed number. This is nothing like any machine we’ve seen before. For example, no plus-times machine can behave this way, because the derivative knocks the power of each term down by one: (xn)′ = nxn−1. If there is a particular member of the AM species that has f′(0) = 1, then it would be exactly its own derivative! Now, if we had a machine f that was its own derivative, then any constant multiple of that machine would be its own derivative too. That is, if f were its own derivative so that f′(x) = f(x), then any machine m(x) ≡ cf(x) would satisfy m′(x) = cf′(x) = cf(x) ≡ m(x), so m would be its own derivative as well. So we’ve got infinitely many machines that are their own derivatives — one for each number c — but only one of these guys is a member of the AM species. Why? Because we just saw above that AM machines possess the behavior f(0) = 1. So there’s only one extremely special machine that is both its own derivative and a member of the AM species. We have no idea what it looks like, but let’s call this particular machineE, to stand for “Extremely special.” So E is the only machine that satisfies
We still have no idea what the AM machines look like, but it would be nice to get a general idea, because we know they’re unlike anything we’ve seen before. What else can we do? All we have to work with is the fact that these beasts turn addition into multiplication. So if n is a whole number and f is an AM machine, then we can write
Interesting. . . The input went upstairs and became a power. And f(1) is just some number we don’t know. I wonder if this will be true for any number x. Earlier, we discovered that we can always approximate any number as well as we want by using numbers of the form 
, where n and m are whole numbers. So if we could show that this “the input goes upstairs” behavior was true for any number that looked like , then we’d be pretty convinced it was true for any number x. Suppose some number x can be written as 
, where n and m are whole numbers. Then we can pull the same kind of trick we just pulled, but in reverse:
where the symbol 
is where we used the definition of the AM species. So the above equation says that f(n) is equal to the thing on the far right. But just before this, we found that f(n) = f(1)n. We’ve described the same thing in two ways, so let’s combine these two descriptions to write
Figure 5.1: We discovered that all members of the AM species have to look like f(x) = cx, with a different member of the AM species for each different positive number c. This is what those machines look like.
What now? Well, we’d like to isolate the 
piece so we can get a better idea of what these AM machines look like. We can kill the mth power on the left by raising both sides to the 
power:
So, at this point we’re fairly convinced that members of the AM species all have the “input goes upstairs” behavior for any number x, since any number x can be approximated as closely as we want by numbers of the form 
. Since f(1) is just a number we don’t know, we may as well just write it as c, and then summarize what we’ve said by writing:
f(x) = cx
Let’s check our reasoning to make sure that machines of the form f(x) = cx are really members of the AM species. Well,
f(x + y) = cx + y = cxcy = f(x)f(y)
Perfect! So these cx machines really are all AM machines, no matter what positive number c is, and the behavior of these machines meshes very nicely with the way we defined powers! Earlier we discovered that there had to be some extremely special AM machine E(x) who was its own derivative, even before we had any idea that the AM machines all looked like cx. Now we know that we get a different AM machine for each positive number c. Let’s summarize everything we’ve discovered about this species so far in a box.
The AM Species
The AM species was defined to be all machines that turn addition into multiplication:
We just figured out that all members of this species have to look like
f(x) = cx
where c is some number.
The Immovable Object
We also figured out that there has to be some extremely special AM machine that is its own derivative. We named this machine E, to stand for “Extremely special.” This machine is the “immovable object” of differentiation. Combining this discovery with our new knowledge of the AM species, we can say: There must be some extremely special number e for which the AM machine
E(x) ≡ ex
is its own derivative. We have no idea what this number is at the moment, but it has to exist.
5.2.3The MA species
We can immediately see that the MA species is, in a sense, the “opposite” of the AM species, so maybe both species will turn out to be related to this mysterious number e. Let’s see. We defined the AM species to be the set of all machines that behave like this:
and the MA species was defined to be the set of all machines that behave like this:
where we’re using different letters to describe the machines so that we don’t confuse them. Let’s suppose that g is an MA machine and f is an AM machine. Then even though we have no idea what the MA machines look like, we can write
But this is just saying that if we build a big machine by gluing an AM machine’s output tube to an MA machine’s input tube, then the whole thing is an AA machine! That is, if we define h(x) ≡ g(f(x)), then we just showed that the machine h behaves like this:
h(x + y) = h(x) + h(y)
So h is an AA machine. Notice that we never specifically asked for these three ideas of ours to be related in such an elegant way, but nevertheless this fact was hiding inside our definitions all along, as a necessary consequence of them. What now? Well, we can profit from the fruits of our labor with the AA species. That is, we discovered earlier that any AA machine looks like ax for some number a. Since we just discovered that h has to be an AA machine, this lets us write
h(x) ≡ g(f(x)) = ax
where a is some number. Which number? We don’t know, but feeding 1 to the above equation tells us that g(f(1)) = a, so we could also write
Interesting. . . This says that if we feed any AM machine to any MA machine, then they almost cancel each other out. If g(f(1)) were equal to 1, then we would have g(f(x)) = x. So f and g would perform opposite actions, leaving us with exactly what we put in. We’re getting closer to saying what we mean by AM and MA being “opposites.”
We discovered above that the AM machines all have to look like f(x) = cx, where c was just an abbreviation for f(1). We get a different AM machine for each different value of c, so let’s add a subscript to our abbreviations, and write fc(x) ≡ cx. We’re doing this so that we can more easily talk about particular members of the AM species, and so that we don’t confuse one AM machine with another. Now in these new abbreviations, equation 5.11 says that for any particular AM machine fc, we can write
So now we can say that if g(c) = 1, then fc and g cancel each other out. Just like we had a different AM machine for each number c, we’ve now got a different MA machine for each number c. So even though we have no idea what the MA machines look like, we can still recognize that each member of the MA species has a partner in the AM species: the partner who shares the same value of c.
Because of this partnership, let’s now write a subscript on g to refer to a particular MA machine, just like we did with the AM machines a moment ago. That is, gc is our abbreviation for whichever MA machine happens to satisfy gc(c) = 1. For example, when c = 2 we obtain the AM machinef2(x) ≡ 2x, and its partner g2 is whichever MA machine satisfies g2(2) = 1. Having discovered all these things about the partnership between the two species and about the mysterious machine that happens to be its own derivative, let’s write down everything we know about both species in a box, and spend a moment feeling all smug and content with ourselves.
The Partnership Between the AM and MA Species:
The AM species was defined to be all the machines that turn addition into multiplication:
We found that any AM machine could be written as
fc(x) ≡ cx
The MA species was defined to be all the machines that turn multiplication into addition:
We have no idea what these machines “look like,” in that we can’t describe them in terms of anything we know. However, we do know that each AM machine has a partner MA machine that undoes its actions. The MA machine gc is defined to be whichever machine makes this sentence true:
Even though we still don’t know which number e makes the machine E(x) ≡ ex be its own derivative, we can talk about the partner of this machine by writing ge(fe(x)) ≡ x, or if we prefer, writing ge(ex) = x. We don’t know anything about this machine, except that it is “special by association.” We only care about it because it is the opposite of the immovable object of differentiation: the machine that’s unchanged when we take its derivative.
5.2.4The MM Species. . . Soon
We’ve still got one more species to explore: MM. However, the discussion is getting a bit abstract, and we need to make sure we’re comfortable with what we’ve done, so we’ll come back and look at the MM species later. Returning to the present, we’ve just discovered a bunch of interesting things about the partnership between AM and MA, as well as the special machine that manages to be its own derivative. So, while that’s still fresh in our minds, let’s take a break and explore our immovable object a bit more informally.