Setting: Chesterfield High, an unusual school in the suburbs of Ohio.
The teacher writes on the board:
2, 3, 5, 7, ...
How, he asks, do we complete this pattern?
Now a student might say that the next term is 12. When the teacher asks him why, he says, “I looked out the window and saw the number 12 bus go by.”
What’s wrong with this answer?
One thing you might say is that there’s a metarule, a rule about rules, and the metarule is: The only valid rules are ones that don’t involve anything specific about the classroom in which the question is asked. There aren’t any “indexical” rules, in the philosopher’s terminology.
So then the student says, fine, the next number in the series is 5. And this time, when you ask him why, he says it’s because it’s the fifth term in the series.
So then you come back at him and say, but the fourth term is 7; according to your rule, shouldn’t it be 4?
Why is your context bigger, or more important, than my lived experience of the number 12 bus?
And he replies, no, no, the rule is only that some of the terms correspond to the order in which they appear. One of those special terms is the fifth one.
So what about the others? Well (he says), the fourth term is 7, because you’re counting upward from the fifth in units of two. After that, we start counting down again and that’s why the third term is five, and the second term is three. The first term is 2 because that’s where we start, and that sets the size of those backwards steps.
You’re understandably frustrated, because the correct answer is that the next term is 11, and the reason is that this is the list of prime numbers, and the student is clearly intelligent enough to know.
“Ian,”—call him Ian—“Ian, look, you know that the correct answer is that they’re primes, and the next number in the series is 11. This is 10th-grade mathematics, it’s end-of-term review, and we did the whole unit on the prime numbers.”
Ian objects. I thought the rule couldn’t be indexical! What’s the difference between my first answer and yours—mine was about the bus, yours is about where Honors Math is at the end of March? Why is your context bigger, or more important, than my lived experience of the number 12 bus? Is it just that you’re more powerful than me? Is this something to do with standardized tests and the post-War meritocracy?
Let’s begin there.
Teacher: I don’t doubt, Ian, that you’re clever enough to come up with any extension of the pattern you like. And I admit that we do have a list of things we want you to know, that the prime number sequence is on that list, and you and I both agree it’s the best match from that list.
Practically, we need to evaluate your ability to remember what we want, and we like to make things a bit tricky because one day your boss will want you to read his mind in roughly the same way. Welcome to the real world.
But the real reason, if we were to get into it—and, honestly, I’d rather not—is that my rule is better than your rule because my rule is simpler. It’s short, it’s easy to tell someone. It’s elegant. And so if you were to somehow encounter the first few terms of this sequence in the real world—say, in the crash log for a computer program you wrote, or a list of cicada generation times—you’d do far better to think about whether it’s to do with primes, than with your elaborate construction.
Ian: Thanks for leveling with me. I appreciate your concern for my future employment. And it’s really useful to know that prime numbers might be a good heuristic one day.
I even agree with you that elegance is a good metarule. I know you don’t mean what’s fashionable, or pretty, that you mean something that’s sort of beyond any particular culture—beauty in the abstract, that’s just as good here as it will be when I’m 80 (god forbid), and as it will be when me or my descendants meet aliens on their voyage to the stars. I don’t doubt that you, or whoever will teach me next, will have a whole list of elegant things for me to learn.
But how do I know what’s elegant? I’m not saying elegance isn’t real. What I want to know is how to know “they’re all primes” is high in elegance. It can’t be that it’s only three words—I mean, it took me 12 years to learn enough to appreciate the rule, but I explained mine to you in a few seconds.
Teacher: If you really want to get into it, we can. The stakes are high, however. If you’re asking questions about completing a pattern, you’re looking under the hood at reason itself.
When we look out into the world, it’s natural to say that we see patterns: patterns in how people behave, or how nature works, even patterns that we reflect on in our own minds.
But that’s not quite the case. What we see is a field of possible patterns, possible ways to complete the series.
Imagine, and this is the standard example from the theory of rational explanation, a doctor. On the surface, it looks like a doctor is in the business of translating a patient’s symptoms into a diagnosis.
A good doctor, however, inverts the process: He plays out each disease in turn, seeing how well the symptoms match the story. If a patient had a respiratory virus, for example, it might make some of his symptoms very likely indeed, but others he sees would require an unlikely coincidence. A different disease would easily produce all of the symptoms—but given who the patient is, he would be far much less likely to contract it at all.
The doctor contemplates two different ways to complete the pattern. In one case, the respiratory virus, he completes the pattern in a natural fashion—he counts off symptoms, although not all of them match very well. In the other case, he matches reality perfectly—but the pattern is an unusual one, more like your jumbled rule from before. The rational doctor’s job is to weigh the benefits of matching with the plausibility of the underlying template.
Ian: This all seems pretty contrived. You have your doctor imagining different diseases, but there are plenty of places where that doesn’t work. When I’m doing translations in Honors Latin, I don’t try a whole bunch of sentences in English, translate them to Latin, and see which one fits the best. I don’t reverse the natural direction. We haven’t even tried English-to-Latin translation yet, it’s much harder.
Teacher: Bear with me. In the doctor’s case, the two possible patterns are pretty easy to weigh against each other: He has a sense of how common the two diseases are, and it turns out that there’s an optimal procedure for how to combine these facts with the relative match to the symptoms at hand.
Even in your case, the case that started this whole discussion, the answer is pretty easy. Given “2, 3, 5, 7,” and the fact that I’m the one presenting it, and the context you’re in, if your goal is to figure out what’s in my head, the prime number pattern is far more likely than the rules you gave, or any other people have found.
Ian: We’ve been over this. That’s not what I’m after.
Teacher: Yes. You’re not interested in reverse-engineering the system. You want to know the truth—you want these methods to give you true answers, not useful ones.
If I need priors to choose between patterns, I can’t learn the priors themselves in the same way!
Teacher: So I’ll give you a new word: “prior.” A prior is a rule for how to judge a pattern before the evidence comes in. What you’re really asking about is how we get the right priors.
Having good priors is why Latin-to-English is easier than English-to-Latin. You know a lot about what English sentences look like, because of how you grew up—subjects and verbs and objects in familiar orders. You can spot a few key vocabulary words in the Latin, crossing off sentences before you even entertain them. It’s iterative, subconscious, and you don’t make the leap all at once.
Ian: Fair enough. If I think about it, I may even do a little conscious diagnosis, too, at the end. If I have two translations I’m deciding between, I might ask how they’d look in Latin.
Where do these priors come from?
Teacher: Practically speaking, the answer is that we learn them. If you spend enough time in high school, you learn to prefer patterns that are in the curriculum, and if a doctor spends enough time in practice he learns the right preconceptions—
Ian: No, no, I’m sorry to interrupt, but I can see where this is going and I don’t agree. If I need priors to choose between patterns, I can’t learn the priors themselves in the same way! It’s just regress; how can I learn the high school testing pattern without having certain preferences that prefer that pattern over, say, the Illuminati Conspiracy Pattern that tells me you and the rest of the faculty are lizard people who pass messages to me through the bus lines. If you contradict me, I can just say you’re wrong —I’ve done that to other teachers, you know.
Teacher: I do. If you trace the regress back far enough, our priors come from evolution. Ancestors with the wrong priors die out; our priors get better; we become more rational.
Ian: What does death have to do with rationality? Evolution doesn’t want you to know the truth. It wants you to believe the thing that will bring you more children. If evolution can get you to have more children by thinking beautiful women are actually shadows of a divine order, then a lot of people are going to walk around completely convinced of that, with these kinds of lizard-people priors that are constantly confirmed.
Obeying my biology might make me happy, but it won’t tell me the truth, and it won’t get me to the stars. Changing my biology is no better: drugs, brain hacking, it’s just more priors, and how can I judge between them?
Tell me about beauty. Beauty is a way out.
Teacher: That’s correct. It’s a value, but it’s a universal one—we call it beauty here in Chesterfield High, but other cultures might have called it something else, and that beauty, or elegance, or whatever it is, is “a thing.” It’s a real, measurable quantity, something that can guide us beyond our biological priors. It’s Occam’s Razor, preferring the simple, avoiding unnecessary complications. Some people even call it the “Universal Prior.”
Ian: It sounds a little religious. At least it’s not indexical. I’ll take a universal principle over a specific one. And I confess it has a rather attractive, esoteric feel.
This prior, it has to do with succinctness, like a poem—how swiftly the rule can be conveyed. My rule is complicated, yours is not, so yours is more likely.
Teacher: Yes. We do as the great physicists did: equate beauty and simplicity, and judge the latter. When we ask which rule is simpler in the most general and abstract fashion possible, we’re talking about something called Kolmogorov Complexity. Every rule has a Kolmogorov Complexity, the length of the rule stated in the most efficient fashion possible. The smaller the complexity, the more beautiful and preferable the rule.
Ian: — but —
Teacher: And I know where you’re going; you’re going to give all sorts of objections to why your rule is actually simpler than mine.
Beauty is a way out.
Ian: Yes. And unless you can show me why, I’ll have to assume that this Kolmogorov Complexity is just another ideology, something that’s made up to tilt me in one direction over another. Maybe it’s beauty, but it’s basically the same as getting hot for girls. Or boys.
Teacher: Don’t get personal. It’s not a trick. Kolmogorov Complexity—let me put a lot of mathematics aside for a moment—is absolutely real. Every pattern has a simplicity, which corresponds to its Kolmogorov Complexity.
Ian: So let’s settle the original question. Is there an app that tells us the Kolmogorov Complexity of our rules?
Teacher: No, there’s no app.
Ian: Why not?
Teacher: Because even though Kolmogorov Complexity is real, and every rule has one, it’s not knowable. It’s not measurable.
Ian: That’s nonsense. It might be hard to calculate, but how can it be impossible? Why can’t I just work it out?
Teacher: We usually show that with a proof by contradiction. If you tell me that you have a calculation method, I’ll show you how it must—for an unknowable set of patterns —give the wrong answer. And I’ll even show you that you can’t get close. If you think you have a way to approximate it, I’ll show you how that method must be wrong as well.
Ian: So there’s a universal prior, but we’ll never know it? You’re using reason to tell me that rationality is unfounded, that there are these unknowable edges. That’s insane.
Teacher: A lot of things have edges. In the case of reason, it’s a very crinkly one, hard to spot, and easy to wander back and forth across. It’s quite beautiful, really.
Ian: You don’t understand. I’m 16 years old. I’m in high school. Half the people around me are morons, half of them are sex-crazed, and half sold out for status. A plurality are all three. We haven’t even gotten into how I feel about my body. The whole System has traumatized me, and I’m full of harmful—irrational—beliefs.
Rationality is my ticket out. The only reason I can trust you is that you seem rational enough to talk to. But now you’re telling me that rationality is just a layer on top of the System—it’s just as irrational as the people I’m trying to escape. I don’t know which is worse: being duped by someone else’s priors, or being a biological machine.
Teacher: Don’t go too far. You’re a smart kid—you can iterate faster than most. You can match patterns better. Evolution set you up well. You’ll get better at predicting the consequences of your actions, and better at adapting your environment to your will. Rationality is systematized winning.
Ian: It’s not winning I’m worried about. It’s my mind. Maybe it’s silly, maybe it’s a fetish, but I want to know the truth. It’s the principle of the thing. Wanting to know the truth got me this far, but now the only option you’ve given me is believing in something I can’t see. If I know it at all, it can’t be through rational, scientific calculation. There’s some kind of extra-rational process I have to engage in—but what’s beyond the edge of reason?
Teacher: Many things. Dreams, intuition, transcendence, love, ascending the ladder, repetition and the leap of faith, philosophy itself ...
Ian: ... delusion, fairy tales, fascism!
Teacher: Childhood’s end.
Simon DeDeo is a professor of social and decision science at Carnegie Mellon University, and external faculty at the Santa Fe Institute. In November, he’ll be leading a public seminar on the future of intelligence through the New Centre for Research and Practice.
Lead image: Sergey Nivens / Shutterstock