(Eliezer sometimes treats scope insensitivity as a simple arithmetical error that the brain commits, like in this quote: “the brain can’t successfully multiply by eight and get a larger quantity than it started with”. Considering that the brain has little trouble multiplying by eight in other contexts and the fact that scope insensitivity starts with numbers as low as 2, it seems more likely that it’s not an error but an adaptation, just like boredom.)
Arithmetic is a relatively late cognitive technology that doesn’t appear by its own (1). We can to a certain degree train ourselves to use exact numbers instead of approximate magnitudes in our reasoning, but that remains an imperfect art—witness the difficulty people have truly grasping numbers that are at all higher. An imprecise analog magnitude representation is the brain’s native way for representing numbers (2.pdf) 3), and while there is evidence about that analog system indeed being capable of multiplication (4), I’d be careful about making claims concerning what low-level systems we have no introspective access to can or cannot multiply.
(Especially since we do know plenty of cases where a particular system in the brain doesn’t share the capabilities other systems do—we might intuitively solve differential equations in order to predict a baseball’s flight path, but that doesn’t mean we can natively solve abstract equations in our head.)
we do know plenty of cases where a particular system in the brain doesn’t share the capabilities other systems do—we might intuitively solve differential equations in order to predict a baseball’s flight path, but that doesn’t mean we can natively solve abstract equations in our head.
Good point, but I’d go even further: we are not even solving differential equations in predicting a baseball’s flight path, but rather, pattern-matching it to typical falling objects. Though I frequently criticize RichardKennaway’s points about control systems, he is right that you actually need to know very little about the ball’s dynamics in order to catch it. You just need to maintain a few constant angles with the ball, which is how humans actually do it.
To the extent that “you” are solving a differential equation, the solution is represented in the motions of your body, not in any inference by your brain.
I’d be careful about making claims concerning what low-level systems we have no introspective access to can or cannot multiply.
I think I didn’t get my point across successfully here. I’m not making any claims about whether some low-level system can or can’t multiply, but instead saying that it’s not trying to multiply in the first place.
In other words, it’s likely wrong to believe that we wouldn’t have scope insensitivity, if only evolution could have come up with a way to make some subsystem do multiplication correctly. If that were the reason for scope insensitivity, then it would make sense to think of it as a simple arithmetical error.
Arithmetic is a relatively late cognitive technology that doesn’t appear by its own (1). We can to a certain degree train ourselves to use exact numbers instead of approximate magnitudes in our reasoning, but that remains an imperfect art—witness the difficulty people have truly grasping numbers that are at all higher. An imprecise analog magnitude representation is the brain’s native way for representing numbers (2.pdf) 3), and while there is evidence about that analog system indeed being capable of multiplication (4), I’d be careful about making claims concerning what low-level systems we have no introspective access to can or cannot multiply.
(Especially since we do know plenty of cases where a particular system in the brain doesn’t share the capabilities other systems do—we might intuitively solve differential equations in order to predict a baseball’s flight path, but that doesn’t mean we can natively solve abstract equations in our head.)
Good point, but I’d go even further: we are not even solving differential equations in predicting a baseball’s flight path, but rather, pattern-matching it to typical falling objects. Though I frequently criticize RichardKennaway’s points about control systems, he is right that you actually need to know very little about the ball’s dynamics in order to catch it. You just need to maintain a few constant angles with the ball, which is how humans actually do it.
To the extent that “you” are solving a differential equation, the solution is represented in the motions of your body, not in any inference by your brain.
Consider a related problem—how much dynamics do you have to know in order to make a 3-point shot in basketball?
Your link syntax (2) is broken; to fix it put backslashes before parentheses inside the URL, like this:
Thanks! Fixed.
I think I didn’t get my point across successfully here. I’m not making any claims about whether some low-level system can or can’t multiply, but instead saying that it’s not trying to multiply in the first place.
In other words, it’s likely wrong to believe that we wouldn’t have scope insensitivity, if only evolution could have come up with a way to make some subsystem do multiplication correctly. If that were the reason for scope insensitivity, then it would make sense to think of it as a simple arithmetical error.
Does that help make my point clearer?