Ian and the Limits of Rationality

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,