This is a direct continuation of the previous post, so read that one first if you haven’t yet. In some sense this post is simpler than the previous, in that it uses simpler concepts and doesn’t involve understanding of the real number system. But it may be harder for many readers, because I’m asking you to imagine an alien race which does not understand certain things that you probably can’t remember a time when you didn’t understand. And it’s hard to imagine what it would be like not to know what we know.
It’s interesting, isn’t it, how people are much better at temporarily adding an unfamiliar concept to their working context than they are at temporarily subtracting a familiar one?
Let’s suppose now that, flushed with success over your breakthroughs in communication with the aliens of last time, you are put in charge of establishing a shared mathematical language with another, yet more unfamiliar, alien race.
Everything proceeds smoothly for a while, moving along the same lines as last time. You can identify the number 0 easily enough (the only number such that ) and then you can get 1 (the only number, other than 0, such that ). You make your way up through rational numbers.
But when you try to talk about infinite decimals, when you try to talk about the numbers that “fill in the spaces” on the number line, when you try to bring up numbers like , it’s evident that you’re not getting through. These aliens don’t seem to recognize “real numbers” in any form you know them.
Could it really be that these aliens only acknowledge rational numbers? How Pythagorean… Then you remember something you read on some anti-fruit blog and think: the square root of 2, surely they have a square root of 2.
So you ask, “Do you have any numbers such that .
Finally the alien perks up and says “Oh, yes, we can always solve equations like that. Bleep and bloop are the two numbers with that property.”
You: “I see, I call those numbers and .”
Alien: “Weird names.”
You: “If you say so. I wonder which is which?”
Alien: “I don’t understand.”
You: “Which number is positive, bleep or bloop?”
Alien: “I don’t undersetand ‘positive’.”
You: “Well, 1 is positive, and all the numbers we got by adding up 1’s are positive, and ratios of positive things are positive.”
Alien: “Hmm. Bleep and bloop aren’t ratios of integers, so that doesn’t seem to apply.”
You: “Okay then, which one is bigger than 1? Bleep or bloop?”
Alien: “Bigger? Like in size? These are numbers, not objects. I don’t see how something like physical size applies.”
You: “Bigger makes sense for numbers: is bigger than if is pos— wait, damn.”
Alien: “Oh. You were going to say ‘positive’ again. I still only understand positive for fractions. Bleep – 1 and bloop – 1 aren’t fractions.
You, getting frustrated: “Which is between 1 and 2?”
Alien: “I don’t understand ‘between’.”
You: “Which is nearer 7/5?”
Alien: ” ‘Nearer’? You keep talking about these things like they’re physical objects in physical places. I thought we were talking about numbers.”
And somewhere around here is where you give up, because the alien understanding of number just doesn’t seem to contain the right kind of structure to tell the difference between bleep and bloop.
(This isn’t as far-fetched as you might suppose. For the ancient Greeks, geometric measures such as length and area were not dealt with as numerical objects. It is perfectly possible—see the work of Hartshorne, for example—to develop a robust theory of geometry in which “lengths” and “numbers” are distinct categories of thing, so that applying words like “between” or “bigger than” to numbers wouldn’t come naturally. Or perhaps this alien race is incorporeal, existing as patterns of electricity or some such sci-fi archetype, so that geometry is not a practical concern. In such a situation the real number system might seem just as esoteric to them as, say, the 19-adic completion of does to you.)
The point is, if you can’t take for granted an understanding of the real numbers, or of ordering, or of topology, if all you have to work with are the rational numbers and basic operations, then identifying is the least of your worries. You can’t even tell apart from in such a situation.
The theme of the previous post, in a nutshell, is that the complex number system has an interesting symmetry relative to the real number system.
Today, then, we are getting an inkling that much more is going on…there are many “symmetries” (in some sense we’ll need to make precise eventually) of the algebraic number system (those numbers which are roots of integer polynomials) relative to the rational number system. In particular, there are enough symmetries to relate any square root with . (And indeed there are many many many more than are envisioned by such a simple example!)
Again, what does it mean that are bound by one of these symmetries? It means that any statement in the language of basic algebra and rational numbers which is true about is also true about and vice versa.
Recall the quadratic formula, which says that the roots of a quadratic equation are . Based on our talk with the alien, we could have predicted this. As algebraic objects relative to the rational numbers, are indistinguishable, so it was inevitable that if one of satisfies some polynomial equation, so must the other.
Also, these symmetries are not unrelated. There is no one single symmetry, for example, that relates every irrational square root to its opposite. Why not? Because, say, . If a symmetric negates and but preserves basic alegbra, then it can’t negate .
(In our alien example, this manifests as follows. There is no way to distinguish for the aliens between , nor between , nor between . But if we take any two of these ambiguities as resolved, then the third is automatically resolved.)
We are scratching at the surface of something big now. What are these symmetries? How do they relate to one another? Among the numbers that we know, which ones are related to each other by symmetries? What does that tell us about numbers and about solving equations? What, in short, does it all mean?
To answer that question (or, more honestly, to begin to answer that question, since even today’s best number theorists are far away from a full understanding of all these symmetries), we will need a better grasp on what “symmetry” actually means. I claimed last time that mathematics provides the right language for discussing symmetry, and it’s getting on time for me to back that up.
See you next week for an exploration of “symmetry” and “groups”, and why I think there’s a case that “group” is the most important mathematical concept you’ve never heard of.