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

1. Ex Nihilo

Interlude 1: Dilating Time

[I do not] carry such information in my mind since it is readily available in books. . . The value of a college education is not the learning of many facts but the training of the mind to think.

—Albert Einstein, on why he did not know the speed of sound, one of the questions included in the “Edison Test,” quoted in the New York Times on May 18, 1921

There’s a lot of things you never see, and you don’t know you don’t see ’em cuz you don’t see ’em. You gotta see something first to know you never saw it, then you see it and say, “Hey, I never saw that.” Too late, you just saw it!

—George Carlin, Doin’ It Again

Things You Never See

There are a lot of things you never see, and this interlude is devoted to some of them. We’ve all heard of Einstein. Here we’ll see how to invent for ourselves one of the many things he’s famous for — a mathematical description of how time actually works — using mathematics so simple that you’ll wonder why they didn’t tell you this years ago. In my view, the fact that this short mathematical argument exists but is not presented to every student at some stage of their education is one of the most telling signs that formal education, as it currently exists, has got its priorities all wrong. The simple derivation in the latter part of this interlude beautifully illustrates the strangeness of the universe and the excitement of science, and one can find it floating around mathematics and physics departments, shared between friends like a folk song or an epic poem, but never standardized as part of the body of knowledge that we teach to every member of our society. Yet for decades we’ve been teaching students everything they need to understand this argument, and then simply not bothering to show it to them. Why? Because special relativity is an “advanced” topic that doesn’t quite “belong” in an introductory physics class, nor does it seem to belong in the courses on Euclidean geometry in which students first learn the mathematics needed to understand the argument itself. The argument is homeless. So, lacking any proper home for this beautiful, intuition-shattering argument in the core curriculum, we instead attempt to inspire students about the mysteries of the physical world by focusing on the mathematical description of pendulums, projectiles, and how a ball rolls down a hill. Alright, enough editorializing. Time for some fun!

First, before we see how time really works, we’ll see an extremely short argument that makes it obvious why the formula for shortcut distances, usually called the “Pythagorean theorem,” is true. Second, we’ll see that this is the most complicated mathematical idea you’ll need to understand one of the main ideas from Einstein’s special theory of relativity: the fact that time slows down when you move. We’ll use the phrase “time slows down when you move” as an abbreviated description, but it isn’t exactly accurate. More precisely, whenever two objects (people included) aren’t moving at the same speed or in the same direction, they begin to observe each other’s “time” passing at different rates.¹³ As impossible as this idea sounds, it’s not just a theory, and it’s not just a fact about humans or clocks. It’s a fundamental fact about the structure of space and time, and it has received more experimental validation than virtually any other idea in all of science. By the end of this interlude, you’ll be able to completely understand the mathematical argument showing that this phenomenon of “time dilation” exists. However, you may come away feeling like you don’t understand it, because the conclusion of the argument is always surprising, no matter how well you’ve understood the argument itself. So even though our primate brains have trouble with the concept, the mathematics should make complete sense, and if it doesn’t, that’s my fault. Ready? Here we go.

It’s difficult to do justice to the idea with a brief verbal description, so if that didn’t make sense, don’t worry. It’ll make more sense soon, after we cover a bit of background.

Shortcut Distances

Not everything is horizontal or vertical. Things can be tilted in any direction. That’s unfortunate, because often the information we’re handed comes in the form of facts about two “perpendicular” directions — directions that could (in a vague sense) be thought of as horizontal and vertical. For example, “it’s 3 blocks east and 4 blocks north,” or “such-and-such is 100 meters tall and 200 meters away.” Suppose all we have is information about two such distances: one that we’ll call horizontal, and another we’ll call vertical. We can discuss this question by drawing a triangle with a horizontal side and a vertical side. To be clear, triangles aren’t the point, but this lets us discuss the question abstractly, while remaining agnostic about unimportant details. Let’s name the sides of this triangle a, b, and c (see Figure 1.9), and suppose we know the numbers a and b. Using only this information, can we figure out the length of the “shortcut distance,” c?

At this point, it’s not remotely clear how to figure out the length of c if all we know is a and b. Since we have no idea of how to proceed, our only hope of making progress is to see if we can turn this hard problem into a problem that only involves things we’re familiar with. We haven’t invented much mathematics yet, so we’re not familiar with many things, but we do know the area of a rectangle. So a good first shot at the problem might be to see if we can build a rectangle out of several copies of the triangle above, and then maybe we’d be able to make a bit of progress (or maybe not, but it’s worth a try). Following this train of thought, the first thing I would personally think to do is to take two copies of the triangle in the above picture, and stick them together so we have a rectangle with a width of a and a height of b. Unfortunately, after a few moments of staring at the resulting picture, we’d still be confused, since this simplest way of building a rectangle doesn’t seem to tell us much about the shortcut distance. Luckily, the second-simplest way turns out to be more helpful, as shown in Figure 1.10.

Figure 1.9: This is not a caption.

Figure 1.10: Building a square inside a square, using four copies of our triangle and some empty space. This lets us talk about something we’re not familiar with (shortcut distances) in terms of something we are familiar with (the area of a square).

We’ve built a big square out of four copies of the original triangle, with the shortcut distances forming a square-shaped region of empty space in the middle. As in Chapter 1 when we invented the obvious law of tearing things, a surprising amount of knowledge can be squeezed out of the simple fact that drawing a picture on something doesn’t change its area. In this case, we’ve essentially drawn a tilted square inside a larger square. A square’s area is one of the few things we know at this point, so this trick lets us form sentences about the shortcut distance using our still very limited vocabulary. We can form such a sentence by talking about the area of the whole thing in two ways. The results are shown in Figure 1.11.

Figure 1.11: By writing the area of the whole thing in two different ways, we can invent a formula for shortcut distances, also called the “Pythagorean theorem” in textbooks.

On the one hand, the picture we’ve drawn is a big square whose length on each side is a + b, so its area is (a + b)². In Chapter 1, we convinced ourselves that (a+b)² = a²+2ab+b² by drawing a picture that made it obvious. That’s one way of describing the picture, but we can also describe it in another way. The area of the whole thing is just the area of the empty space (which is c²) plus the areas of all the triangles. We don’t know the area of a triangle, but if we imagine putting any two of the triangles next to each other (like we imagined in our first failed attempt at this problem), we’d have a rectangle with area ab. We have four triangles, so we can build two rectangles out of them. So by carving things up in this way, we can see that the area of the whole picture is also c²+2ab. We’ve described the same thing in two ways, so we can throw an equals sign between the two descriptions, like this: a² + 2ab + b² = c² + 2ab.

Now this next part is extremely important, so read carefully. The above mathematical sentence says that one thing is equal to another. If two things really are equal (i.e., identical), and we modify both in exactly the same way, then (although the two things will both change individually) they will still be identical to each other after the modifications. Two boxes with identical but unknown contents will continue to have identical contents if we perform an identical action to each. This is true no matter what that action is (e.g., “remove all the rocks,” or “add seven marbles,” or “count up the number of hats in each and double it”), as long as we agree that the two actions are identical. This is why we can now say (in the standard jargon) “subtract the term 2ab from both sides of the above equation.” Make sure you understand this. This is not a property of mathematics or of equations, and it’s not some mysterious “law” of “algebra.” It’s a simple fact about our everyday concept of two things being identical: identical modifications to identical objects must lead to identical results.¹⁴ If it doesn’t, then we cannot have been using the term “identical” very carefully. So, performing this modification, we arrive at this sentence:

An understanding of this simple fact and its consequences would allow us to skip a large proportion of a typical introductory course in algebra.

a² + b² = c²

This tells us how to talk about shortcut distances in terms of horizontal and vertical bits, so let’s call it the “formula for shortcut distances.” Textbooks usually call this the “Pythagorean theorem,” which sounds like some sort of magical sword, or a disease you catch from drinking unsanitary water.

The Fiction of Absolute Time

Events and developments, such as. . . the Copernican Revolution. . . occurred only because some thinkers either decided not to be bound by certain “obvious” methodological rules, or because they unwittingly broke them.

—Paul Feyerabend, Against Method

A few lines of reasoning can change the way we see the world.

—Steven E. Landsburg, The Armchair Economist

Get some popcorn, dear reader, and prepare yourself, because you’re about to see one of the most beautiful arguments in all of science. The conclusion isn’t easy for our primate brains to accept, so you won’t be able to understand it viscerally. No one can. But even the simple mathematics we’ve already invented offers us a way to circumvent and move beyond certain inherent limitations of the primate brain. This section will go a bit more quickly than we have been so far, but don’t worry. The derivation below is logically independent from the rest of the book, so even if you don’t understand anything in what follows, you won’t be behind when we move on to inventing calculus in Chapter 2. With that in mind, sit back and enjoy yourself. We’ll need three things to get where we’re going:

1.How far you go = (How fast you’re going)·(How long it takes), as long as your speed doesn’t change along the way. We all know this intuitively, but we can easily forget that we know it when it’s phrased in this abstract form. It’s just saying: (a) if you go 30 miles per hour for 3 hours, you will have gone 90 miles, and (b) there’s nothing special about the numbers we used in item (a). Let’s write this as d = st to stand for “(distance) equals (speed) times (time).”

2.The formula for shortcut distances that we invented above (i.e., the “Pythagorean theorem”).

3.A strange fact about light.

The strange fact about light is not a mathematical fact, but a physical fact, and it’s so completely ridiculous that you shouldn’t be surprised if it doesn’t make sense. The full absurdity of this fact about light is best expressed by seeing how it’s different from something we all know. So first, the thing we all know: if you throw a tennis ball at 100 miles per hour and then immediately (by some superpowers) run up behind it at 99 miles per hour, then it will look like the tennis ball is moving away from you at 1 mile an hour, at least until it falls to the ground. Not a mystery.

Here’s the seemingly impossible fact about light: if you “throw” some light (e.g., if you let a few light particles, or “photons,” out of a flashlight while you’re standing still) and then you immediately run up behind it at 99% of the speed of light, then it won’t look like the light is moving away from you at 1% of the speed of light! Rather, it will still look like the light is moving away from you at the full speed of light — the same speed it would have looked like it was moving away from you if you hadn’t bothered chasing after it.

If that seems impossible, good! That means you’re paying attention. Rather than trying to understand this fact by worrying about how it could possibly be true, we’ll try to understand it by playing a game similar to the one Einstein played in 1905. We’ll say, “Okay, this fact seems impossible, but the evidence suggests that it’s true, so how about we just ask ourselves: if it were true, what else would have to be true?”

To begin, let’s imagine a strange device that I’ll call a “light clock.” To build a light clock, just imagine holding two mirrors some small distance apart. Since light bounces off mirrors, this imaginary device will keep the light trapped, bouncing back and forth between them. As we all know, we can measure time in seconds, hours, days, or however else we want, so let’s choose to define our unit of time to be: however long it takes the light to bounce from one mirror to the other. We could give this amount of time a name, like “schmeconds” or something, but we don’t need to.

Figure 1.12: Our imaginary light clock consists of two mirrors, one raised a height h above the other, with a particle of light bouncing back and forth between them.

Now for some abbreviations. For strange historical reasons, people usually use the letter c to stand for the speed of light. Basically, c is the first letter of the Latin word for “swiftness,” and the speed of light is quite literally the fastest anything in our universe is able to go, so aside from the Latin, it sort of makes sense.

So, c will stand for the speed of light. Let’s use h to stand for the height difference between the two mirrors, and let’s use t_still to stand for the amount of time it takes for the light to go from one mirror to the other (we’ll see in a minute why we’re calling this t_still instead of just t). I’ll draw the light clock in Figure 1.12.

Now, at the beginning of this section, we convinced ourselves that (How far you go) = (How fast you’re going)·(How long it takes) as long as your speed isn’t changing. So using all the abbreviations we just defined, we can write h = ct_still, or to say the same thing in a different way:

Now, let’s imagine two people looking at the same light clock. We imagine that one of them is on a rocket moving horizontally, holding the light clock. The other person will be on the ground, watching the rocket and light clock fly by at some speed, which we’ll abbreviate as s. This is shown inFigure 1.13.

Okay, so the argument above where we wrote h = ct_still should describe what the guy on the rocket sees. You might be confused about why we’re using the word still to talk about this situation, since after all, the guy on the rocket is “moving.” We’re describing it this way because the guy on the rocket is not moving relative to the light clock, since he’s holding it, so it’s “still” compared to him. As we’ll discuss later, “moving” doesn’t really mean anything unless you say “moving relative to such and such.” Okay, now what will the guy on the ground see? Well, for him, the light trapped in the light clock will still be bouncing up and down, but it’s also moving past him horizontally, so the light particle will appear to be bouncing diagonally in a kind of sawtooth pattern, as shown in Figure 1.14.

Figure 1.13: Our light clock is on a rocket, moving past an observer on the ground at some speed s.

Recall that in the argument above (where we concluded h = ct_still) we were thinking about the time it takes for the light to bounce from one mirror to the other. Let’s do that again, this time from the point of view of the guy on the ground. We can write t_mov as an abbreviation for the amount of time it takes for the light to bounce from one mirror to the other, as seen by the guy on the ground. The subscript mov reminds us that this guy sees the light clock moving. Now, you might wonder why we would give two different names to this amount of time. After all, they’re clearly the same. But don’t be so confident! We already saw that light behaves in a very strange way, and Einstein took seriously the possibility that these times might not be the same. For now, let’s give them two different names just in case. If they actually are the same, we’ll discover that later. If they’re not, we’ll discover that too.

Now, just focusing on the light’s path from Figure 1.14, we can figure out the distance that the light travels in one “clock tick” as seen by the guy on the ground. This is shown in Figure 1.15. The vertical distance between the mirrors is still h, and the distance the light travels horizontally isst_mov, since the rocket is going at speed s and we’re thinking about what happens during a time t_mov.

Figure 1.14: Three snapshots of our light clock as the light moves up and down, as seen by the guy on the ground. From his perspective, the light is moving diagonally, since it is bouncing up and down between the mirrors, while the light clock itself is moving past him from left to right. As he watches the rocket go past, the light particles will trace out a kind of sawtooth pattern.

Here’s where we use the strange fact about light: that however fast you’re traveling, light always looks like it’s moving the same speed. Because of this, both our characters see the light traveling at the same speed c. But the guy on the ground sees the light traveling diagonally, and the distance that the light travels along the diagonal in a time period t_mov is still just “speed times time,” or ct_mov. This is weird. For example, if the light were any normal bouncy object bouncing back and forth between the mirrors, then the diagonal speed of the ball seen from the ground would be faster than the vertical speed of the ball as seen by the guy on the rocket. So we’ve “told the mathematics” that we’re assuming the strange fact about light. Now we can see what else has to be true as a result.

Here’s where we use the formula for shortcut distances. Since “horizontal” and “vertical” are perpendicular to each other, the picture in Figure 1.15 tells us:

h² + (st_mov)² = (ct_mov)²

We eventually want to compare the times t_still and t_mov, and we already have an expression for t_still from earlier, so let’s try to isolate t_mov in the above equation, and then maybe we can see if these two times are the same. If we want to isolate t_mov, it might help to throw everything involving t_mov onto one side of the equation, like this:

h² = (ct_mov)² − (st_mov)²

Figure 1.15: Drawing all the distances. Let’s think about the time it takes for the light to go from the bottom to the top, from the point of view of the guy on the ground. The vertical distance between the mirrors is still h. The horizontal distance traveled is st_mov, and the diagonal distance traveled is ct_mov, because of the strange fact about light from earlier.

Now, since the order of multiplication doesn’t matter, it has to be the case that (ab)² ≡ abab = aabb ≡ a²b², no matter what a and b are. Toward the goal of isolating t_mov, let’s try to rewrite the above equation like this:

But then each piece on the right has a t_mov attached, so we can turn this into:

h² = (c² − s²(t_mov)²

Or to say the same thing another way:

Now remember earlier we found that , and the left side of the equation above almost has a piece in it that looks like . The trouble is that obnoxious −s². If that piece didn’t exist, then we’d have on the left side, which is just , so the two times would be equal. However, that s² is getting in our way. So let’s perform a sneaky mathematical trick: lying, and then correcting for the lie. Here’s the idea. We want to compare the times t_still and t_mov, because we feel so strongly that they should be the same. If they’re not, then that means “time” in the everyday sense doesn’t exist — an unsettling thought! We could compare these two times if only that −s² weren’t there. We can’t just get rid of the −s², because that would be lying, and then our conclusions wouldn’t be right. However, if we lie and then correct for the lie, then we’ll have a correct answer, so let’s do that. We want to rewrite equation 1.11 so that it looks like this:

Now, we have absolutely no idea what the symbols ♣ and ♠ are! Our job is to figure out what they have to be, in order to make that sentence true. Why would we do this? Well, if we can dream up some values for ♣ and ♠ that would make the sentence true, then we could turn the h²/c² bit in the above equation into a by using equation 1.10, which would let us compare the times, so then we could see how time really works. Our goal is make this sentence true:

c² (♣ − ♠) = (c² − s²)

but when we frame the problem like that, it’s not too hard. We want the symbol ♣ to turn into c² when we multiply it by c², so we can just choose ♣ to be 1. We want ♠ to turn into s² when we multiply it by c², so we can just choose ♠ to be s²/c², so that the c² on the bottom kills the c² on the top.

Most math books would avoid all this stuff about ♣ and ♠, and instead just say “factor out c².” We’ll do that too, once we’re comfortable with the idea. However, saying that at this stage might make it sound like “factoring” is something that we have to spend time learning. We don’t. While the end result of this whole process could be described as “factoring,” that isn’t a good description of the thought process we went through. What really happened is that we wanted something to be true (namely, we wanted there to be a c² on the bottom), so we lied in order to make it true (i.e., we just wrote a c² where we wanted it to be), and then we corrected for the lie so that we’d still get the right answer.

Even more importantly, the phrase “factor out a c²” makes it sound like there already has to be a c² inside. There doesn’t! If we ignore the concept of “factoring” and instead think about lying and correcting, then it’s perfectly clear that we can pull anything out of anything; we can pull a c out of (a+b), a term that doesn’t even have a c inside. How? Same logic as with ♣ and ♠ above. If we happen to want a c outside of (a+b), just follow that same logic, and you’ll end up rewriting it as . Okay, sorry for the sermon, but it wasn’t just a random change of topic. This stuff is fundamentally important, and I can think of no better time to mention it. Anyways, we’ve now figured out that

Taking the square root¹⁵ of both sides, and using the fact that t_still = h/c, we get

We haven’t talked in depth about square roots yet, although later we’ll see that they arise as part of a strange process whereby a formerly content-free abbreviation gains new life and becomes a genuine idea. If you didn’t understand the step where we took the square roots of both sides, don’t worry. We’ll talk about them soon. For now, we’re just using the symbol to stand for whatever (positive) number turns into stuff when you multiply it by itself. That is, stands for whichever number (?) makes the sentence (?)² = stuff true. You’re definitely not expected to know how to compute the square roots of any particular numbers. As long as you get the general idea, that’s more than enough for now.

Equation 1.12 may look complicated, but let’s first ignore most of the complexity and just mention the most important part of it: the times t_mov and t_still are not the same unless s is zero! This tells us that whenever two objects are moving at different speeds, then their light clocks desynchronize, and start “ticking” at different rates. We can rephrase equation 1.12 by throwing all the time-related stuff over to one side (i.e., dividing both sides by t_still). The only reason we might feel like doing this is because then the right side would only depend on the speed s. Of course it also depends on the speed of light c, but that’s just a number that never changes (this is that strange fact about light from earlier). However, the speed s is something that we can change. This lets us visualize this strange time-slowing phenomenon a bit better, which is what we’re doing in Figure 1.16. The figure tells us how the quantity t_mov/t_still changes as we change s, the speed of the rocket. We can think of that quantity as telling us how many times bigger t_mov is than t_still. The larger this quantity becomes, the more our everyday concept of time breaks down.

Now, it turns out that equation 1.12 isn’t just a fact about light clocks, or even about clocks in general. It’s a fact about the fundamental structure of space and time, and it has been experimentally tested too many times to count since Einstein discovered it in 1905. Why don’t we notice this effect in our everyday lives? That is, if you and I are hanging out, and then I drive to the store and back, we don’t tend to think that we’ve lived through two genuinely different amounts of time. However, as a glance at Figure 1.16 shows, the times we experience are equal when we’re moving at the same speed with respect to each other, and they’re extremely close to being equal when we’re moving at speeds that are small compared to the speed of light.

Figure 1.16: Visualizing time dilation. The horizontal axis is speed, and the vertical axis is the quantity t_mov/t_still, which tells us how much bigger t_mov is than t_still (i.e., how much our everyday concept of time has broken down). In our everyday life, we feel like time is a universal thing, which is to say we think t_mov = t_still, or equivalently t_mov/t_still = 1. This is the horizontal line in the figure. The curvy line is reality: when you are moving relative to someone, their time appears to be moving more slowly. For speeds that are small compared to the speed of light, our everyday concept of time is very close to being correct, but it increasingly breaks down as this relative speed more closely approaches the speed of light (roughly 300 million meters per second).

But even this tiny quantitative difference, as insignificant as it may be in our daily lives, requires a large qualitative change in how we think about the universe. The world we’re used to, in which there is a single absolute notion of time, is simply a helpful approximation: a lie that happens to be useful, as long as we’re not moving too fast relative to the objects around us. But as useful as our everyday concept of time may be, it is a startlingly poor description of the fundamental nature of reality.

Even worse, it turns out that we’re not completely justified in using the subscripts _mov and _still when we wrote t_mov and t_still. More careful consideration of this issue shows that as long as both guys are moving with some fixed speed and direction (i.e., neither one of them is speeding up, slowing down, or changing direction), then we’re not justified in saying that either of them is “still.” We’re used to using words like “moving” and “still” because we live on a gigantic rock covered in air, and whenever we’re on or near the surface of the Earth (i.e., pretty much always), there’s a special frame of reference that appears to be “not moving” — namely, standing still with respect to the Earth. However, this frame of reference isn’t really “still” in any universal sense, and if we imagine two people floating past each other in outer space, the issue becomes more clear. Each person might think it’s the other one who is really moving, and he who is standing still. Or he might think that he’s still and the other guy is moving. Or he might think that they’re both moving. All of these ways of thinking would be equally right and equally wrong.

The more we consider arguments like the one we just made, the more we see that it makes no sense to say that my real speed is such-and-such. It only makes sense to say that my speed is such-and-such compared to some arbitrary other thing that I’m defining to be “still.” Because of this, the conclusions of the above argument are much weirder than they might have seemed at first. In our light clock example, it’s not as if person A observes person B’s time to be slowed down, and person B observes person A’s time to be sped up, so that everyone can agree. The reality is far stranger than that. They would each see the other person’s time slowed down, and — as long as neither person changed speed or direction — neither of them would be wrong! Are you wondering who would be older if one of two twins left Earth on a near-light-speed rocket ship while the other stayed home, and then they eventually met back up and compared watches? Good! Look up the twin paradox. The universe is crazy. Let’s learn some more.