The Zoology of the MA Species - Aesthetics and the Immovable Object - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

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

Act II

5. Aesthetics and the Immovable Object

5.4. The Zoology of the MA Species

Okay. . . Anyways. . . Before the above exploration of the immovable object ex, we spent some time discussing the “four species.” In that discussion, we found that each member of the AM species could be written as fc(x) ≡ cx, and we later found that each such member of the AM species has a partner in the MA species that “undoes” it. That is, for any positive number c, we get two machines. First, there is a member of the AM species fc(x) ≡ cx. Second, we get a member of the MA species gc whose defining property is that it behaves like this:

As you may or may not have realized, the members of the MA species have a name in the standard textbooks: they are called “logarithms.” In those textbooks, we would see logc(x) written instead of what we have called gc(x), but they both refer to the exact same thing.

What we call them isn’t important. What is important is that, in yet another instance of backward pedagogy, we’re typically introduced to these topics before we’re exposed to calculus. It has somehow been decided that long before students are taught about calculus, they should learn about the number e, the machine ex, and logarithms. To top it all off, during this baffling discussion of logarithms, a particular logarithm is presented as “special” for some mysterious and unexplained reason. It goes by the name ln(x), and it is called the “natural logarithm,” or “log base e.” Students are then taught a series of incomprehensible incantations about these objects, in the hope that it might somehow prepare them for calculus. Unsurprisingly, this causes widespread confusion about just what these odd logarithm things are, what this number e is, or why mixing the former concept with the latter to obtain the function ln(x) is anything other than alchemy. Indeed, many graduate students and professors in non-mathematical fields will readily admit that they still have no idea what a logarithm is, despite the fact that they were once taught about them.

The source of this confusion is not the concepts themselves. As we’ve discovered in this chapter so far, not only did we need calculus in order to compute the number e, but the only reason we cared about this number in the first place was because of its relationship with the derivative! That is, we only care about the number e because the machine ex happens to be its own derivative, and we only care about ln(x), or loge(x), because it undoes the actions of ex. As such, if we don’t already know about derivatives, we have exactly no reason to care about e, or ex, or ln(x), or any of their properties. “No reason to care,” I would wager, is precisely how the majority of students feel when they are taught about these topics in high school. Can you blame them?

Having seen the conceptual origin of these odd things called “logarithms,” we still don’t know much about them, and we still don’t feel very comfortable with them. Although we managed to figure out what the members of the AM species look like (i.e., they are all number-to-the-x machines), we still have no idea what the members of the MA species look like, in the sense that we can’t describe them in terms of anything we know. This is a side effect of having defined the four species not by what they are but by how they behave. Whenever we define a mathematical object only by saying how it behaves, we are not allowed to be surprised if it is not immediately apparent what the object is, whatever that means.

This highlights a deeper reason why the topics of this chapter can be confusing to newcomers. Despite its simplicity, the strange dance of defining an object by its behavior is extremely foreign to the way human minds typically transact with the world. For the vast majority of human evolutionary history, human brains had exactly zero experience with this odd ritual of defining objects by their behavior, and to this day, our neural machinery does not come into the world expecting to deal with objects thus defined. During any stage of human evolutionary history, up to and including the present day, virtually all of the “things” encountered by humans have fallen into one of the following categories: humans, nonhuman animals, plants and fungi, invisible disease-causing microorganisms, human-made artifacts (like axes or computers), and inanimate features of the geography. In dealing with anything from any of these categories, it is always safe to assume that there exists a large set of preexisting facts about that thing of which you are unaware. In none of these cases is the set of all facts about the object in question discoverable from a simple principle that is postulated by the human doing the discovering. In our natural default setting, we are simply not accustomed to thinking this way.

As such, I expect that when students first hear about “logarithms,” they attribute a kind of hidden essence to these objects, and assume that there must be a world of (almost zoological) information about them that the professor has not revealed. In a sense, that’s true! But the odd fact is that allof their properties are implicit in their definition. While that’s true of any object in mathematics, it is more salient (and more intimidating) in the case of logarithms, because when students are first exposed to these objects, they are encountering this novel behavior-based style of definition directly for (almost) the first time: they are given a description only of how logarithms behave, not of “what they are.” That is, for familiar machines such as m(x) ≡ x2, the description on the right-hand side tells us about the inner workings of the machine, and how we can compute specific outputs given specific inputs. The definition of logarithms gives us no such clue about the inner workings of the machine, but only its behavior with respect to the operations of addition and multiplication: logarithms are whatever they have to be in order to satisfy the behavior log(xy) = log(x) + log(y).

We can now show that all of the other “logarithm properties” of the standard curriculum follow for free, based on how we defined the MA species. At that point, we will know enough of their properties to unravel them with the Nostalgia Device, thus giving us a simple description of them as an infinite plus-times machine. In principle, this description will enable us to compute the logarithm of any number to arbitrary accuracy by using nothing but addition and multiplication.

5.4.1Something Else They Never Tell You

Let’s first choose some better abbreviations. We’ve been writing fc(x) ≡ cx for members of the AM species, and gc(x) for members of the MA species. Since the AM species are all “power machines,” let’s abbreviate them as pc(x) ≡ cx. Since the MA species are essentially the opposite of the power machines, let’s abbreviate them as qc(x), because the letter q looks like a backward p.

Okay, because of the way we defined the MA species (i.e., as opposites of the AM species), all facts about them should come in pairs. For each fact about the simpler AM species, we should be able to deduce a fact about the more mysterious MA species. Facts about AM machines are therefore a kind of currency with which we can purchase a greater understanding of the MA machines. So where do facts about the AM species come from? Well, since all such machines look like cx, all facts about the AM species must follow from the way we defined powers! Because we ourselves invented powers, everything we know about them follows from these two sentences:

and

where x, y, and s are any numbers. Now, because of the way we defined the MA species, we know that all of its members behave this way:

This is the “logarithm property” whose “opposite” is equation 5.19. Textbooks usually write the above sentence this way:

Is there a corresponding sentence about the MA species that is the “opposite” of equation 5.20? Let’s see. Let’s try to “undo” equation 5.20 by taking the “log base stuff” of both sides, that is, wrapping both sides in qs. We discovered earlier that MA machines undo their partner AM machines, which is to say qs(s#) = # for any number #. Using this idea, we can write

and by whacking the left side of this with equation 5.20, we can write

Now, this is completely true, and since it’s a fact about “logarithms” that we built using equation 5.20, it sort of qualifies as the “opposite” of equation 5.20 we were looking for. Given all that, we could simply stop here. However, the above equation is kind of ugly. If we want to get a sentence that is only talking about logarithms (i.e., the MA species), then it would be nice to kill the piece that looks like sx. How might we do this? Well, if equation 5.24 is true for all numbers x and y, then it has to be true when x = qs(z) for any particular number z. If we replace the x’s in 5.24 with xqs(z), then this will kill the sx piece, because by our tricky choice of abbreviations, we’ve made it true that sxsqs(z) = z, where the second equality comes from the fact that AM machines undo their partner MA machines. Using this tricky abbreviation, we can rewrite equation 5.24 in an equivalent but different-looking way, like this:

which says something much easier to interpret than equation 5.24, even though it’s exactly the same idea in disguise. All this says is that we can “bring powers outside of logarithms.” This is a fact about MA machines that we built using the fact about AM machines in equation 5.20. Yet again we see that facts about these two species come in pairs. Equation 5.25 is the “logarithm property” that textbooks usually write like this:

If the equation above looks at all scary and non-obvious, remember that it’s just saying the same thing as the much simpler-looking equation 5.20. We can continue to ask “why” until we hit rock bottom. “Okay,” you might say, “so equation 5.26 is true because of equation 5.20, but why is equation 5.20 true?” Good question! The simple answer is that equation 5.20 is true because we forced it to be true when we invented the idea of powers in Interlude 2! As usual in mathematics, if we continue asking “why” for long enough, we will eventually find that the answer to any question of the form “Why is such-and-such true?” is simply “Because of some decision we made earlier.”

Alright, so we’ve discovered a few things about these beasts. What else can we say? Well, if equation 5.21 is true for all x and y, then it has to be true when we choose to think of y as one over some other number. That is, if we think of y as , then because of the way we invented negative powers, we can rewrite equation 5.21 this way:

Just looking at the far left and far right, the above sentence says:

This is the “logarithm property” that textbooks usually write like this:

Is there any relationship between different members of the MA species? Well, remember that different members of the MA species are determined by which member of the AM species cx they “undo,” which is to say that they’re determined by which particular number c is. So we can rephrase the question as “Given two numbers a and b, is there any relationship between qa and qb?” If we write the obvious sentence x = x in the scary-looking form

and then wrap both sides in the function qa, we get

or equivalently,

This is the “logarithm property” that textbooks usually write like this:

This is wonderful, because it tells us that we can ignore virtually all of the members of the MA species! Why? The piece loga(b) is just a number, independent of x, so equation 5.32 tells us that all members of the MA species are simply constant multiples of each other! Because of this, it is no longer worthwhile to continue talking about all of the members of the MA species. We can simply pick our favorite one, and then talk about that one. This amounts to picking our favorite “base.” We could choose 2 or 52 or 10 or 93.785 or anything else. However, we’ve developed quite a fascination with the immovable object ex, it being the only machine that is both its own derivative and a member of the AM species. As such, purely for aesthetic reasons, we will choose to talk only about the member of the MA species that is the opposite of the immovable object, namely, the machine qe(x). Our lone surviving MA machine qe(x) is what textbooks call the “natural logarithm,” or ln(x). Having happily jettisoned every logarithm except the one with base e, we can continue our journey with a much lighter load.

5.4.2Reabbreviation Hammer to the Rescue

Since there’s only one MA machine left, we don’t need the subscript on qe(x) anymore. From now on, let’s just call it q(x). Can we use what we know to differentiate the machine q(x)? Well, we don’t know the derivative of q(x), but we do know that

q(E(x)) ≡ q(ex) ≡ x

So maybe by writing the machine M(x) ≡ x in this complicated way, we can convince the mathematics to tell us the derivative of the so-called natural logarithm q(x). On the one hand, the derivative (with respect to x) of the stuff in the above equation is just 1, because it’s all just x. Let’s use the hammer for reabbreviation to try to figure out the derivative in another way. Let’s abbreviate ex as s, where s stands for stuff. Then we’ve got q(s) = x, and we want to figure out the derivative of q with respect to whatever variable is inside it. We can write

Hmm. . . what on earth is ? Well, we defined s to be shorthand for ex, so

Figure 5.2: Having discovered that all logarithms are multiples of each other, if we can graph one of them, then we’ll know what they all look like. Here we’ll picture the “base two” logarithm, because it’s easy to explain. We know 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 = 16, etc. This is just another way of saying log2(1) = 0, log2(2) = 1, log2(4) = 2, log2(8) = 3, log2(16) = 4, etc., as shown in the picture.

So the combined insight of the above two equations is . Throwing s over to the other side gives

Hey! This equation contains no mention of x, so even though we originally defined s to be shorthand for ex, we can forget about that now, and just treat it as a meaningless abbreviation. Equation 5.33 is saying, “The derivative of q with respect to whatever is inside it is one over whatever is inside it.” That is, we can replace the s with x if we want, because at this point it’s just a symbol. To emphasize this freedom to reabbreviate, let’s summarize what we learned from equation 5.33 in several equivalent but different-looking ways:

Or as the textbooks write it:

Hooray!

5.4.3Lying and Correcting Gets Us Further by Getting Us Nowhere

That was a fairly bizarre argument, and we may or may not trust the result. As often happens when we’re inventing mathematics for ourselves, we’ve made an argument and we’re not sure if it’s correct. Can we derive the same result in another way? Let’s try to use the definition of the derivative and see if we get the same answer. Just this once, let’s use the textbooks’ strange ln(x) notation, to make sure we don’t get too accustomed to any particular choice of abbreviations. Here we go.

At this point we’re stuck, but the place we got stuck looks strangely familiar. Recall that during the dialogue from earlier in this chapter, we discovered that

This looks almost like the spot where we just got stuck. The last line of equation 5.36 would be exactly the number e (thanks to 5.37), were it not for that obnoxious x sitting inside. Maybe we could get rid of it. Let’s try to massage the stuff at the end of equation 5.36 until it looks more like equation 5.37.

Okay, well there’s nothing special about the notation dx in equation 5.37. It just refers to an infinitely small number, and the only important thing is that the two dx’s in equation 5.37 are the same number. If dx is infinitely small, and x is not, then dx/x is also infinitely small. So if only the power in the last line of equation 5.36 were x/dx instead of 1/dx, then we could sneakily define the abbreviation

and use this to write

However, that would be lying, and lying changes the problem. But this line of argument suggests that the familiar trick of lying and correcting might help. Let’s do that! First, we lie by changing the power from 1/dx to x/dx. This makes the problem easier. However, we then have to correct for the lie by changing the power back to 1/dx. This entire process would have gotten us nowhere, except now we can see that rewriting 1/dx as x/(x · dx) might help us get past the point where we got stuck. Having rewritten the power, we can now reabbreviate the infinitely small number dx/x as dy in equation 5.36. Summarizing:

Now we can use this to break through the wall we hit earlier. I’ll throw numbers above equals signs to remind us where each step comes from. Starting from where we got stuck, we can write

Just like we hoped, we got the same answer as before. In both cases, we found that the derivative of the machine q (a.k.a. “natural log”) is 1/x. Having arrived at the same answer in two different ways, we become much more confident that this is indeed the correct answer.

5.4.4The Nostalgia Device Simplifies Life Yet Again

Even though we’ve discovered that the derivative of q(x) is 1/x, we still don’t know how to write down a description of q(x) in terms of anything simpler. That is, we have no idea how to compute a specific number for q(3) or q(72). This is the same predicament we found ourselves in during Chapter 4, when we didn’t know how to describe the machines V and H (what the textbooks call “sine” and “cosine”) in any way except by drawing pictures. In that case, we eventually discovered that our Nostalgia Device allowed us to write V and H as plus-times machines with an infinite number of pieces, which greatly eased our worries about them. Armed with the plus-times machine expansions of V and H, we knew that we could calculate specific numbers for them if we ever needed to, just by doing a bunch of addition and multiplication.

Maybe we could use the Nostalgia Device to get an idea of what this “natural logarithm” machine q(x) looks like and how we might be able to compute specific numbers for q(9) or q(42) or whatever. It might not work, but it’s worth a shot. Feeding q to the Nostalgia Device, it tells us that

Hmm. . . this isn’t going to work. We know that q’s derivative is 1/x and that’s going to be infinite at x = 0. Why does q get all weird at zero? Well, the machine q was defined by its behavior q(ex) = x. But ex is always positive: it gets smaller and smaller for large negative values of x, and only approaches zero as x goes to −∞. Based on this reasoning, it should be the case that q(0) = −∞, so maybe we shouldn’t use the Nostalgia Device on q directly, because q goes a bit crazy at zero.

What if we tried using the Nostalgia Device on the machine q(1+x) instead? This is the same machine, just shifted a little bit, and it’s much more well-behaved when x = 0, since q(1 + x) = q(1) = 0. It seems like thinking about the machine this way might make things easier. Okay, can we figure out all the derivatives of this machine at x = 0? Well, they’re just the derivatives of the original machine q at x = 1, so we need q(n)(1) for all n. Let’s list a few of q’s derivatives and see if we notice a pattern:

q′(x) = x−1

q″(x) = −x−2

q′″(x) = 2x−3

q(4)(x) = −(3)(2)x−4

q(5)(x) = (4)(3)(2)x−5

Hey, this is pretty simple! We just have to keep using the familiar pattern (x#)′ = #x#−1 that we discovered in Chapter 3. The powers are negative, so the sign changes every time we take a derivative. The power on the nth derivative will be negative n, and the number out front will be (n − 1)!, so we see the pattern. We can write everything we just said in shorthand like this:

q(n)(x) = (−1)n+1(n − 1)! xn

Okay, so we’ve got all the derivatives of q(x). What about the derivatives of q(x + 1)? Well, using the hammer for reabbreviation, we see that the derivatives of Q(x) ≡ q(x + 1) are just the derivatives of q(x) with x + 1 plugged in, because . Great! So we can write

Q(n)(x) = (−1)n+1(n − 1)! (x + 1)n

when n ≥ 1. All we need in order to use the Nostalgia Device is q(n)(1), which is another way of writing Q(n)(0). Using the above equation, these numbers are just:

for n ≥ 1. When n = 0, we have Q(n)(0) = Q(0) ≡ q(1) = 0. Now we can finally apply the Nostalgia Device to get:

The (n − 1)! in the above equation 5.41 will cancel against part of the n! pieces on the bottom, because n! = n · (n − 1)!. This leaves us with:

which is just an abbreviation for

Since q(x) blows up at zero, q(x + 1) will blow up at −1, so it’s not clear that we can trust this expression for all x. However, for now at least, it gives us a much more concrete way of thinking about the MA machines. The above equation gives us a way of thinking about the so-called “natural logarithm” q as an infinite plus-times machine. Since all of the members of the MA species are just constant multiples of q, we therefore have a way of describing “logarithms” (MA machines) without just saying “they’re whatever undoes the number-to-a-power machines cx.”