Buy F&SF • Read F&SF • Contact F&SF • Advertise In F&SF • Blog • Forum 
November/December 2020

Science
Don't read this just before bedtime. It'll keep you awake. I warn you right up front that it's a puzzler, and it doesn't have a resolution. If you read enough of it to understand it, you'll be intrigued, as generations of mathematicians and philosophers before you have been, but unless you're more insightful than the lot of them, you won't figure it out either. But you'll try. So pick a time to read it when you don't mind being distracted for a while, and dive into one of the most interesting existential questions I've encountered in my life: Is mathematics real? The question doesn't even make sense at first. Of course it's real. We study it in school, and we use it to count change and to launch rockets to the Moon. Doesn't get much more real than that. Yet consider the number two. Can you imagine the number itself, without reference to anything? Not two apples, nor two elephants, nor a twodollar bill. Just two. I'm betting you can't. And if you can, then let's see you imagine the square root of two. It kind of looks like the very foundation of math—numbers—might not be real. They might not even be imaginary. What the heck are they?
That's just the beginning of the problem. Only it's not, really. To get to the beginning of the problem, we have to go all the way back to the beginning of the universe, to the Big Bang. Imagine that moment of creation, when all the matter and all the energy in the universe burst into being. Space itself burst into being, too, complete with all the rules and regulations that govern how subatomic particles formed and combined to create the physical universe we're familiar with. Rules that follow precise mathematical relationships. Where did the math come from? It was always there, some say. Math is a framework that's independent of physical reality. Two plus two equaled four even before the universe began. Yet if that's true, then something existed before the Big Bang. Even if that something was merely conceptual, it was still something. So if math predates the Big Bang, then the Big Bang wasn't really the beginning of everything.
It's just as difficult to imagine the opposite, though. If math didn't predate the Big Bang, then there was a time (even if it was just the moment when time itself was created) when two plus two didn't yet equal four. Utilitarians will say that without anything to count, there were no numbers to count with, so of course math didn't exist, but that leads to a chickenandegg conundrum. Did the number one come into being with the first particle? Did two come into being with the second particle? And so on? Then how in the world universe did 2.718 x 10^{123456} come into being? I just made that number up, but it's comfortably larger than the number of particles in the entire universe. And if it's not, then I can easily make up a bigger one that is. If numbers only come into being when there are particles enough to count with them, then how can I create a number bigger than the number of particles in the universe?
We haven't even gotten into the complicated bits yet. Let's consider addition. One plus one is two, except when we're adding raindrops. One raindrop plus another raindrop just makes a bigger raindrop. The distinction is semantic, but also very real. One rabbit plus another rabbit equals a half dozen rabbits if you give them a little time. So math isn't absolute without defining our terms, and we can whittle away at those terms as much as we want, splitting hares until one plus one equals roughly two trillion living cells for a pair of averagesize rabbits. It gets worse. Not only do we have to agree on our terminology, we also have to agree on where the math is performed. For instance, one of the basic rules of geometry, a form of math that deals with the shapes of things, says that the sum of the three interior angles of any triangle adds to 180 degrees. You can draw as many triangles as you want on a flat sheet of paper and that relationship will always be true. But draw a triangle on a pingpong ball and measure the angles, and you'll find that the sum is way higher. Why? Because the space you're working on is curved. And space is always curved. Einstein pretty much proved that—mathematically.
This is the point when people usually say, "Math describes reality; it doesn't define reality." Another way of putting it is "The map is not the territory," but that also depends upon the context. If you tell me to go to New York, I can put my finger on the U.S. map stapled to my wall and in a very real sense I have gone to New York. But in an equally real sense I haven't. If you say "Find the distance between New York and Los Angeles," the map would be the better tool for that job than actually travelling from one to the other. So it is with math: there are many, many things we can calculate to much greater precision than we can measure (pi, for instance)^{1}, so in many ways math is the territory and reality is the poor substitute. In fact, math is so much better at calculating reality than reality itself is, some mathematicians (and even some physicists) believe the universe is actually nothing but math. Evidence in favor of that theory includes the possibility that the universe we know could be nothing more than a vast simulation. Whether or not it actually is doesn't matter; the fact that it could be (if that is indeed a fact) proves that the physical universe could be nothing more than math.
Galileo bought into that theory, famously saying that "Mathematics is the language in which God has written the universe." Yet a simple calculation called "the threebody problem" throws cold water on that notion. The threebody problem applies to orbital motion. I taught you how to calculate an orbit in the May/June 2019 issue, but I only used two bodies. I carefully avoided introducing a third body into the equation because adding a third body makes it unsolveable. Not just difficult, but unsolveable. The motions of three bodies in orbit are what's called "chaotic," in that tiny perturbations add up over time to completely disrupt any calculations we can make. It takes a very long time in some cases, but we can't predict with infinite accuracy how their orbits will evolve over time. If math were the language of the universe, that probably wouldn't be the case. Except maybe we just haven't discovered the right math yet.
On the other hand, math does a fine job of predicting real things that we haven't discovered yet. Einstein's general relativity was a mathematical construct, which has later proven out in experiment after experiment. Quantum theory is the same. It has famously predicted a multitude of elementary particles that were later discovered through experiment. The Fibonacci sequence of numbers, wherein each number is the sum of the previous two (1,1,2,3,5,8,13...) was known for centuries before we realized that snails and sunflowers have been using it all along to plan their spirals. Wait, what? Plants use math? Yep. Maybe not intentionally, but those spirals don't happen by accident.
Math can address some even stranger concepts. English majors (like me) might think they escaped mathematics by going into the study of literature and language, but that's not necessarily so. University of WisconsinEau Claire professor of mathematics, emeritus, Robert Andersen has proven that there are more adverbs than adjectives. How does he do that? We've already shown that you can't visualize a number without visualizing a number of somethings. So the counting numbers, at least, are modifiers. You have two apples, or five oranges. Apples and oranges are nouns. What part of speech modifies nouns? Adjectives. But you buy apples and oranges by the pound, so two pounds of apples is a different kind of two. It modifies "pounds" now, not apples. You might think there's no significant difference, until you realize that "pounds" is filling the same role that "two" was earlier. You can no more visualize a pound without it being a pound of something than you can visualize a number without it being a number of something, so "pound" is clearly modifying "apples." What part of speech modifies modifiers? Adverbs, of course. So numbers can be either adjectives or adverbs, depending on how they're used. Counting numbers are called natural numbers, and there are an infinite number of them. Measuring numbers are called real numbers, and there are an infinite number of them, too. But it can be (and has been) proven that there are more real numbers than natural numbers. (Yes, there are different sizes of infinity!) Therefore there are more adverbs than adjectives.
So is math real or imaginary? Is it invented or discovered? We don't know. The concept of "zero" was certainly discovered. As far as we can tell, it was first used about 300 BC as a placeholder. It was about 350 AD before it was really used as a number. It wasn't used for calculations until the 7th century in India or possibly the 8th century in the Middle East. It wasn't in common use in Europe for another 56 centuries. Think about that: The concept of "zero" was a newfangled idea when Isaac Newton was figuring out how gravity (and calculus) worked. Science fiction stories often portray first contact with aliens as happening through math. We transmit 1,2,3,5,7,9,11,13 at the aliens, and they transmit back 17, proving that they understand prime numbers—but they might just as easily scratch their glopiks in puzzlement because any dweezit knows 1 isn't prime (which it isn't under some definitions). There's no way to know which interpretation they'll use. Or maybe they won't have discovered (or invented) prime numbers, so they'll just think we can't count. Who knows? Nobody does. And in all probability, nobody ever will. Sleep tight.
^{1}
We've calculated pi out to some 31 trillion decimal places so far, but even if you start out with a circle as big as the known
universe it only takes 62 digits of pi to reach what's called the Planck length, which is the smallest unit of measure physically possible.
Jerry Oltion has been a science nut since he was old enough to spell "curious." He has written science fiction almost as long, and has done astronomy somewhat less. He writes a regular column on amateur telescope making for Sky & Telescope magazine, and spends many, many nights a year out under the stars.
 

To contact us, send an email to Fantasy & Science Fiction.
If you find any errors, typos or anything else worth mentioning,
please send it to sitemaster@fandsf.com.
Copyright © 1998–2020 Fantasy & Science Fiction All Rights Reserved Worldwide