Forgetting Mathematics - 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

If you want to build a ship, don’t drum up people together to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.

—Antoine de Saint-Exupéry, Citadelle

1.1. Forgetting Mathematics

1.1.1Hello, World

Forget everything you’ve been told about math. Forget all those silly formulas you’ve ever been told to memorize. Make a little room in your head with clean white walls and no math. Without leaving that room, let’s reinvent mathematics for ourselves. Without the burden of teachers, without the burden of a classroom, without paying any attention to the thing called “mathematics” that has been handed down the generations to us, free of that ridiculous lie that the worst thing you can do is to be wrong. Only by doing this will we be able to understand anything.

I’m calling this chapter Ex Nihilo for two reasons. The first is to poke fun at the unnecessarily fancy terminology that shows up in all subjects, including math. We humans love to sound smart, and saying stuff in a different language (especially a dead language) makes it seem more important. Having said that, we can dispense with the Latin. The term Ex Nihilo means “out of nothing,” and I chose this as the name for the first chapter to emphasize that in this book, mathematics is ours. The term no longer refers to that thing you learned in school. We are pulling mathematics into existence, out of nothing.

I’ll assume that the language of addition and multiplication is familiar enough that you can speak it fluently. I don’t mean that it should be obvious how to calculate the square of 111111111 or how to find the square root of 12345678987654321 or anything crazy like that. In general, mathematicians don’t like to deal with numbers. All I mean is that I’ll assume you can convince yourself of basic things like the order of addition doesn’t matter. Same deal for multiplication. To say the same thing in more abbreviated form:

no matter what numbers (?) and (#) are.

To begin our journey into mathematics, we will not need to waste our time learning how to do boring things like calculating a specific number in decimal notation for . All we need to know is that the funny symbol refers to whichever number turns into 1 when we multiply it by 7. If you see something like , then don’t be fooled into thinking that there’s some mysterious thing called “division” that you have to learn random facts about. A symbol like is just an abbreviation for , which is just multiplication. What number does refer to? I have no idea, and you certainly don’t have to either. But we do know that it is whatever number turns into 15 when we multiply it by 72. That’s all.

Assuming that you get the basics of addition and multiplication, we’re going to take an utterly bizarre path through mathematics. After this first chapter, which largely consists of learning how to invent our own mathematical concepts, we’re going to jump straight into inventing calculus, and then use it to reinvent for ourselves all of the things that are usually thought of as prerequisites to calculus. By turning the subject on its head, we’ll discover that calculus — the art of the infinitely large and the infinitely small — can not only be invented before its so-called prerequisites, but that those “prerequisites” cannot be fully understood without calculus itself.

This approach also frees us from the need to memorize anything. Since we’ll never (intentionally) accept anything we have not created for ourselves, and since we can always look back at what we’ve already done, we find that mathematics — a field so often associated with memorization — actually requires less memorization than any other subject. While in other fields memorization may be unavoidable, in mathematics it is poison, and any mathematics teacher who makes you memorize something without apologizing for it on bended knee should be immediately teleported to the unemployment office and made to memorize the phone book.1 Mathematics is a beautiful discipline in which nothing ever needs to be memorized. It’s about time we started teaching it that way.

Author: Okay, that was too extreme. I didn’t really mean that. I just meant that memorization isn’t very helpful. But writing a book is an exciting experience, and I might occasionally get carried away. So try not to take my editorializing too seriously, okay? I mean, I’ve never written a book before, and I’m scared I might burn out along the way. I know I’ll never finish unless I make sure to enjoy the writing process. So if it’s not too much to ask, please try to tolerate my extraneous hyperbole. I promise it’s all in good fun. Anyways, let’s keep moving, dear Reader. Can I call you Reader?
Reader: Works for me.
Author: Great! You can call me Author. Or whatever you want. I’ll answer to any loud noise, really, so pick your favorite and let’s keep moving. I can’t believe I’m actually writing a book!

Our adventure will eventually lead us to some fairly “advanced” topics that typically aren’t taught until the latter half of a four-year bachelor’s degree in mathematics. We’ll see that this “advanced” stuff is really no different from the “basic” stuff, but at each stage the textbooks change the way they write things, just to confuse you.

We’re about to set out on an adventure into a beautiful world of necessary truth in which nothing is accidental. You may occasionally get discouraged (and it’ll probably be my fault). You may have to play around with ideas on your own to convince yourself that you understand them. You may have to think very hard, and you’ll have to try even harder not to be intimidated when you see symbols (abbreviations). But you won’t have to trust me. You won’t have to wonder what’s being hidden from you. And you won’t have to memorize anything. . . unless you want to. Here we go.

1.1.2“Function” is a Ridiculous Name

The term ‘function’ got into mathematics, I was told by Prof. K. O. May, due to a misinterpretation of a proper usage by Leibniz. Nevertheless, it has become a fundamental concept of mathematics and whatever it is called, it deserves better treatment. There is perhaps no better example in mathematical education of missed opportunities than in the treatment of functions.

—Preston C. Hammer, Standards and Mathematical Terminology

Machines do all sorts of things. A bread-maker is a machine that eats bread ingredients and spits out bread. An oven is a machine that will eat anything and spit out that same thing at a much hotter temperature. A computer program for adding one to a number can be thought of as a machine that eats a number and spits out one plus whatever number you put in. A baby is a machine that eats things and spits them out all covered in spit.

Figure 1.1: One of our machines.

Figure 1.1: One of our machines.

For whatever reason, mathematicians have decided to use the strange word “function” to describe machines that eat numbers and spit out other numbers. A much better name would be. . . just about anything. We’ll start by calling them “machines,” and then once we’re used to the idea, we’ll occasionally start calling them “functions,” but only occasionally.2 Let’s use the only tools we’ve got — addition and multiplication — to invent some machines that eat numbers and spit out numbers.

We’ll use certain nonstandard terms throughout the book, but I should emphasize that I don’t necessarily think my terminology is “better” than the standard terminology, and I’m certainly not arguing that other books should use it! The purpose of occasionally inventing our own terms is simply to remind ourselves that the mathematical universe we are creating is entirely our own. It’s a universe we’re building, from scratch, so we get to decide what to call things. But please don’t think that my purpose is to convert everyone to a new set of terms. The word “function” may not be the best, but it’s not all that bad once you get used to it.

1.The Most Boring Machine: If we feed it a number, it hands the same number back.

2.Add One Machine: If we feed it a number, it adds one to what it ate and spits out the result.

3.Times Two Machine: If we feed it a number, it multiplies that number by two and hands back the result.

4.Times Self Machine: If we feed it a number, it multiplies that number by itself and hands back the result.

It takes a lot of words to talk about these machines, so let’s invent some abbreviations. All of the symbols in every area of mathematics, however complicated they look, are just abbreviations for things we could be talking about in words, except we’re too lazy. Because they usually don’t tell you this, most people are really intimidated when they see a bunch of equations they don’t understand, but less intimidated when they see an abbreviation like DARPA or UNICEF or SCUBA.

But math is just lots and lots of abbreviations plus reasoning. We’ll be inventing a bunch of abbreviations on our journey, and it’s important that we invent abbreviations that remind us of what we’re talking about. For example, if you want to talk about a circle, two reasonable abbreviations would be C and . Some good abbreviations for a square would be S and . This is so obvious that you might wonder why I’m saying it, but when you look at a page full of equations and think “Ahhh! That’s scary!”, all you’re really looking at is a bunch of simple ideas in a highly abbreviated form. This is true for every part of mathematics: deconstructing the abbreviations is more than half the battle.

We want to talk about our machines using fewer words, so we need to invent some good abbreviations. What makes an abbreviation good? That’s for us to decide. Let’s look at some of our options. We could describe the Times Two Machine by saying:

If we feed it 3, it spits out 6.

If we feed it 50, it spits out 100.

If we feed it 1.001, it spits out 2.002.

And then we could just say what it does to every possible number. But that’s a crazy waste of time, and we’d never finish. We could say that whole infinite bag of sentences at once, simply by saying “If we feed it (stuff), it spits out 2 · (stuff),” where we’re choosing to remain agnostic about which specific number (stuff) is. We could abbreviate this idea even further by writing stuff 2 · stuff.

So, just by being agnostic about which number we were putting into a machine, we managed to collapse an infinite list of sentences down to a single sentence. Can we always do that? Well, probably not. We don’t know yet. But at this point we’ve decided to only think about machines that can be completely described in terms of addition and multiplication, and that’s what let us summarize an infinite number of sentences with just one. We can describe the rest of our machines in this abbreviated way too:

1.The Most Boring Machine: stuff stuff

2.Add One Machine: stuff stuff + 1

3.Times Two Machine: stuff 2 · stuff

4.Times Self Machine:3 stuff (stuff)2

We’re writing (stuff)2 as an abbreviation for (stuff) · (stuff). More generally, we’ll use the abbreviation (stuff)number to stand for “The thing you get when you multiply (stuff) by itself number-many times.” You’re not allowed to think “I don’t understand powers,” because at this point there’s nothing to understand. It’s just an abbreviation for multiplication.

In case that doesn’t make sense, here are some examples:

1.The Most Boring Machine:

3 3

1234 1234

2.Add One Machine:

3 4

1234 1235

3.Times Two Machine:

3 6

1000 2000

4.Times Self Machine:

2 4

3 9

10 100

Let’s try to abbreviate these machines as much as we possibly can without being ridiculous. By “being ridiculous,” I mean “losing information.” For example, we could abbreviate the entire collected works of Shakespeare by the symbol ♣, but that doesn’t help very much, because we can’t extract any of the information we’re abbreviating from the abbreviation.

How many abbreviations do we need to completely describe our machines? Well, we need to come up with names for (i) the machine itself, (ii) what we’re putting in, and (iii) what we’re getting out. Then we need to do one more thing: (iv) we need to describe how the machine works.

Let’s name the machines themselves with the letter M, so we don’t forget what we’re talking about. We might want to talk about more than one machine at a time, so let’s use the letter M with different hats to talk about different machines. We’ve been using the name stuffto refer to whatever we’re putting into the machine, but let’s abbreviate this even further and just write s (s stands for stuff). But now that we’ve got two abbreviations, we can build the third abbreviation out of the first two. This is a tricky idea, and I’ve never really heard anyone acknowledge that we’re doing it, let alone what an odd process it is, but this is where most of the confusion about “functions” comes from.

What do I mean by saying that we can build the third abbreviation out of the first two? Well, what name should we invent to talk about “the thing that the machine M spits out when I feed it some stuff s”? If we build the name for this using just the abbreviations M and s, then we don’t have to come up with any more abbreviations, and we’re using as few symbols as possible. So let’s call it M(s). Again, M(s) is the abbreviation we’re using to refer to “the thing that the machine M spits out when I feed it some stuff s.”

So we had to name three things, but instead we named two things, then we paused, looked around to see if anyone was watching, and sneakily used the two names we had already come up with as the “letters” to write down the third name. That’s a really weird idea, but it helps tremendously once we get used to it. If you’ve been confused about “functions” before, don’t worry. It’s all simple stuff about machines and abbreviations. They just don’t tell you that.

Okay, so we’ve got our three names, but we still haven’t described any particular machines in this new abbreviated language we invented. Let’s re-describe the four machines from earlier. I won’t list them in the same order. See if you can figure out which machine is which (i.e., which is the Add One Machine, which is the Times Self Machine, etc.)

The reason this intense abbreviation can be confusing is that, in one sense, we’re only describing the output, or what the machines spit out. Both sides of an equation4 a sentence like M(s) = s2 are talking about the thing that the machine M spits out. But in another sense, this sentence is talking about all three things at once: the machine itself, the stuff we put in, and the stuff we get out. Take another look at this crazy abbreviation:

The term “equation” causes most people to experience a discomforting combination of fear and boredom, a mixture of emotions that anyone familiar with the sympathetic nervous system might have thought impossible. So what is an “equation”? We’ve already talked about how mathematical symbols are just abbreviations for things that we could be describing with words. Against this background, “equations” are just sentences. Abbreviated ones. Once we realize that, the term “equation” doesn’t seem quite so bad. We’ll use both terms throughout the book.

M(s) = s2

We’re talking about the output on both sides, sure. But our abbreviation for the output — namely M(s) — is a weird hybrid that we built out of the two other abbreviations: the abbreviation for the machine itself, which was M, and the abbreviation for the stuff we put in, which was s. So the sentence M(s) = s2 has three abbreviations just on the left side. As if that weren’t enough, we then go on to describe the operation of the machine. The right side of this sentence, s2, is a description of the machine’s output, written in terms of the input.

We’ve said the same thing in two ways: the M(s) on the left side is our name for the output, and the s2 on the right side is a description of the output. Since we’ve said the same thing in two ways, we throw an equals sign between them, and we’ve described this particular machine in a way that expresses infinitely many different sentences in a few symbols! It expresses infinitely many different sentences because it tells us: If you feed 2 to the machine M, it spits out 4. If you feed 3 to the machine M, it spits out 9. If you feed 4.976 to the machine M, it spits out whatever (4.976) · (4.976) is, and so on.

1.1.3Things We Rarely Hear

The idea of these machines is very simple. Like I mentioned before, they’re usually called “functions,” which is an odd name, and it doesn’t convey the idea very well. Not only is the word “function” a bit confusing at first, but the common abbreviations used to talk about functions can be pretty counterintuitive when we first encounter them. Here are some reasons such a simple idea can be so confusing:

1.They don’t always explain that we’re talking about machines.

2.They don’t always explain that everything we’re saying about these machines could be expressed in words, but we’re lazy (the good kind of lazy!), so we’re doing it in a highly abbreviated form.

3.They don’t always explain that we’re using the shortest abbreviations we can, or how we built a weird hybrid abbreviation out of two other abbreviations.

4.They don’t always distinguish between the name of a machine, M, and the name for its output, M(s). Sometimes books will talk about “the function f(x),” which isn’t really what they mean. To be fair, sometimes it’s useful to use our language incorrectly like this (it is our language after all, so we’re allowed to), but we’ll try not to do that until we’re much more familiar with the idea.

We rarely hear all this. A sizable proportion of textbooks and lectures just say that a function is “a rule that assigns one number to another number,” then they draw some graphs, pace back and forth a bit, and start writing stuff like f(x) = x2 a lot. For some (including myself when I was first exposed to the idea), this is a rather confusing conceptual leap.

I want to draw your attention to something puzzling in the previous sentence. Why do they write x? We wrote s instead of stuff because we got tired of writing out the whole word. But then what on earth is x an abbreviation for? Maybe it’s not an abbreviation for anything. There’s no law that all of the names we give things have to be abbreviations. Maybe the x is like Harry S. Truman’s middle name: it looks like an abbreviation, but it’s really not! Maybe the letter is the name itself. As it turns out, the letter x is an abbreviation for something. What? Let’s take a break and find out.

1.1.4The Unbearable Inertia of Human Conventions

Why do textbooks always use x? The answer is pretty funny.5 It’s actually a bastardized translation from Arabic. See, back in the old days, some Arabic mathematicians went through a train of thought similar to the one we’ve gone through here, and they decided to use the word “something” for the same reason that we used “stuff.” Perfectly reasonable. The idea is to always choose abbreviations that remind you of the thing you’re abbreviating, so that you don’t have to memorize anything. Up to this point, everything made sense. Then came the problem. The first letter of the word “something” in Arabic makes a sound similar to the sound “sh” in English. It turns out that the Spanish language has no “sh” sound, so when all of this Arabic mathematics was translated into Spanish, the Spanish translators chose the closest thing they could think of. This was the Greek letter “chi,” which makes a “ch” sound (as in Bach, not Cheerios). The letter chi looks like this:

χ

This explanation comes from a guy named Terry Moore, in his wonderful short talk “Why is ‘x’ the Unknown?” So credit goes to him.

Figure 1.2: Generally speaking...

Figure 1.2: Generally speaking, humans are slow to change things.

Look familiar? Later, as you might expect, this χ turned into the familiar letter x from the Latin alphabet. . . and this bastardized abbreviation continues to haunt our textbooks as the most common abbreviation for stuff.

The Arabic mathematicians were smart folks, and they chose their abbreviations well. They could do so because they were essentially in the same type of situation we are: in a world without much mathematics, inventing it as they go. Like them, we can always abbreviate things however we want. For example, consider the following two problems. Don’t bother doing them. Just stare at them for a few seconds.

1.Here’s a description of the f machine:

f(x) = x2 − (5 · x) + 17

What does the f machine spit out when we feed it the number 1?

2.Here’s a description of the machine:

What does the machine spit out when we feed it the number 1?

We don’t have to do either of these problems to see that they have the same answer (it’s 13, but that’s not the point). We’re describing the same machine, and we’re feeding it the same number in both cases, so we know they have the same answer, even if we didn’t bother to figure out what that answer is. Everyone knows that we can abbreviate things however we want. And yet when I’m explaining some piece of mathematics to someone, and I change abbreviations so that we can remember what we’re talking about, one of the most common things I hear is “Oh! I didn’t know we could do that!” It’s important that we practice changing abbreviations, because a lot of ideas in mathematics look scary and complicated when we use one set of abbreviations, but suddenly seem obvious when we use another. We’ll see some funny examples of this later.

1.1.5The Different Faces of Equality

There is another widespread problem with standard mathematical notation that causes tremendous unnecessary confusion to newcomers. That is the need to use different-looking versions of the equals symbol to remind ourselves why things are true.

When we use the normal equals symbol = in this book, we will mean the same thing that all mathematics books mean: A = B means that A and B refer to the same thing, even though they might look different. Therefore, the symbol = just tells you that something is true, but it doesn’t tell youwhy it’s true. We can do better by occasionally using different-looking symbols. In the rest of the book, these three symbols

will all mean the same thing. They all mean “the things on either side of me are the same,” but they’re different ways of reminding us why those two things are the same.

By far the most common alternative version of the equals symbol that I’ll use is ≡, and it says that two things are equal because of some abbreviation we’re using. A few examples will illustrate what I mean. One of the cases where the symbol ≡ will show up is whenever we’re defining something. For example, in the above discussion when we wrote M(s) = s2, we really could have written M(s) ≡ s2. I only used = because we hadn’t talked about ≡ yet. The ≡ symbol in the above sentence says “M(s) and s2 are the same thing, but not because of some mathematics that you missed. We’re just using M(s) as an abbreviation for s2 until we say otherwise.”

Now, using the ≡ symbol for definitions isn’t unique to this book. Lots of books do that.6 However, in an attempt to get the most explanatory bang for our notational buck, we’ll use this symbol in a slightly more general way. We will use ≡ in any equality that is true simply because of some abbreviation that we’re using, and not because of any mathematics that you missed. Just to choose a completely contrived example that refers to absolutely nothing, I might say something like this: Using the fact that M(s) ≡ s2, we can write

Ironically, this seems to be more common in advanced books than it is in introductions, where it’s most needed.

Just to stress the point, you should be able to understand the above pile of symbols even if you had never heard of addition, multiplication, or numbers! Since it involves ≡, all it is really saying is that the thing on the left and the thing on the right are equal because of some abbreviation we’re using, and not because of some mathematics that you missed. As such, whenever you see this kind of equals sign, you’re not allowed to be intimidated. There’s nothing to be scared about, because equations with ≡ aren’t really saying anything. However, we’ll see throughout the book how helpful it can be to change back and forth between different abbreviations, so it’s worth having a special kind of “equals” to remind ourselves when that’s all we’re doing.

Another way of using equals shows up whenever we’re forcing something to be true, and seeing what happens as a result. This is the version of equals that people are using when they say something like “Set yadda yadda equal to zero.” This is a strange concept, so it’s worth looking at a simple example. When a textbook insists that you “solve x = x2 for x,” it’s not always clear what that means. The equals sign is clearly being used in an odd way here. First, the sentence x = x2 isn’t even true, at least not in general. After all, if the sentence x = x2 were always true, then 2 would equal 4, and 10 would equal 100, and so on. Here’s the idea:

What they say: Solve x = x2 for x.

What they mean: Figure out which particular stuff makes the sentence (stuff) = (stuff) · (stuff) true. Ignore all the stuff that makes it false.

Since this meaning of equals is so different from ≡, we’ll write it a different way. How about this:

To reiterate, all of these different versions of “equals” mean the same thing as the the normal = symbol. The new ones just remind us of why something is true. Even if distinguishing between these different kinds of equals seems unnecessary now, we’ll see soon how much easier it makes things.

Attention Reader! This is important! Whatever you do, please don’t agonize about learning exactly when you should use which type of equals symbol! And if any teachers are reading this, please, for the love of mathematics, do not assign exercises where you make people determine whether = or ≡ or should be used in some equation or another. This isn’t a pedantic distinction we’re making because of a compulsive overattention to irrelevant details. It’s just a quick and easy way to remind ourselves why something is true. For the same reasons, I’ll also occasionally throw a number above an equals sign, like this:

What that means is “Blah = Blee because of equation 3.” By doing this, each equation can become a way to check and see if you understand an idea, but only if you want to. That is, you can try to figure out why something is true on your own, but whenever you get sick of that, the equals symbol tells you where you can look to find the reason why. I always wished more textbooks would do this. But enough about notation. Time for creation!