In the OB post tautologies have to be empirically observed somehow, Eliezer writes about waking up one day and discovering all sorts of evidence that 2+2=3. This wouldn’t be evidence that 2+2=3 in Peano arithmetic, it would be evidence that Peano arithmetic just doesn’t apply for some reason. In my down-voted comment, I was just giving an example of how there can be different kinds of arithmetic if you are willing to be flexible about what arithmetic is. (If you are not willing to be flexible, then you are not willing to allow the observation that 2+2=3 as an observation about arithmetic, because this is not possibly true in standard arithmetic. Well, the observations are possible but you’d have to account for it as some kind of grand delusion.) My point is that 2+2=4 in Peano Arithmetic independent of observation, but observation tells you if Peano arithmetic applies or not.
This wouldn’t be evidence that 2+2=3 in Peano arithmetic, it would be evidence that Peano arithmetic just doesn’t apply for some reason.
Exactly.
My point is that 2+2=4 in Peano Arithmetic independent of observation, but observation tells you if Peano arithmetic applies or not.
It is worth emphasizing that to claim that “2+2=4 in Peano Arithmetic independent of observation” is not to claim that our knowledge of this fact about Peano Arithmetic is independent of observation. (The former claim is about our map of the territory; the latter is about our map of our map of the territory.)
It is worth emphasizing that to claim that “2+2=4 in Peano Arithmetic independent of observation” is not to claim that our knowledge of this fact about Peano Arithmetic is independent of observation. (The former claim is about our map of the territory; the latter is about our map of our map of the territory.)
Could you elaborate? It sounds to me like the former claim is about the territory, and the latter is just hard for me to parse.
I’ll emphasize with the following analogy: you need to observe the sun to know of it. However, you can nevertheless be certain—as certain as you are of anything at all—that the sun exists independently of observation. You need to define the Peano axioms and observe the deductions that lead to the tautologies to know of them, but they are mathematically true independent of your observation.
In the OB post tautologies have to be empirically observed somehow, Eliezer writes about waking up one day and discovering all sorts of evidence that 2+2=3. This wouldn’t be evidence that 2+2=3 in Peano arithmetic, it would be evidence that Peano arithmetic just doesn’t apply for some reason. In my down-voted comment, I was just giving an example of how there can be different kinds of arithmetic if you are willing to be flexible about what arithmetic is. (If you are not willing to be flexible, then you are not willing to allow the observation that 2+2=3 as an observation about arithmetic, because this is not possibly true in standard arithmetic. Well, the observations are possible but you’d have to account for it as some kind of grand delusion.) My point is that 2+2=4 in Peano Arithmetic independent of observation, but observation tells you if Peano arithmetic applies or not.
Exactly.
It is worth emphasizing that to claim that “2+2=4 in Peano Arithmetic independent of observation” is not to claim that our knowledge of this fact about Peano Arithmetic is independent of observation. (The former claim is about our map of the territory; the latter is about our map of our map of the territory.)
Could you elaborate? It sounds to me like the former claim is about the territory, and the latter is just hard for me to parse.
I’ll emphasize with the following analogy: you need to observe the sun to know of it. However, you can nevertheless be certain—as certain as you are of anything at all—that the sun exists independently of observation. You need to define the Peano axioms and observe the deductions that lead to the tautologies to know of them, but they are mathematically true independent of your observation.