Only in the same sense that you can talk about kittens by saying “Those furry things!” There’ll always be some ambiguity over whether you’re talking about kittens or lions, even though kittens are in fact furry and have all the properties that you can deduce to hold true of furry things.
Yes, and that’s OK. I suspect you can’t do qualitatively better than that (viz ambient set-theoretic universe for second-order logic), but it’s still possible (necessary?) to work under this apparent lack of absolute control over what it is you are dealing with. Even though (first order) PA doesn’t know what “integers” are, it’s still true that the statements it believes valid are true for “integers”, it’s useful that way (just as AIs or humans are useful for making the world better). It is a device that perceives some of the properties of the object we study, but not all, not enough to rebuild it completely. (Other devices can form similarly imperfect pictures of the object of study and its relationship with the device perceiving it, or of themselves perceiving this process, or of the object of study being affected by behavior of some of these devices.)
Likewise, we may fail to account for all worlds that we might be affecting by our decisions, but we mostly care about (or maybe rather have non-negligible consequentialist control over) “real world” (or worlds), whatever this is, and it’s true that our conclusions capture some truth about this “real world”, even if it’s genuinely impossible for us to ever know completely what it is. (We of course “know” plenty more than was ever understood, and it’s a big question how to communicate to a FAI what we do know.)
Not in the same sense at all. All of the numbers that you have ever physically encountered were nameable, definable, computable. Moreover they came to you with algorithms for verifying that one of them was equal to another.
Only in the same sense that you can talk about kittens by saying “Those furry things!” There’ll always be some ambiguity over whether you’re talking about kittens or lions, even though kittens are in fact furry and have all the properties that you can deduce to hold true of furry things.
Yes, and that’s OK. I suspect you can’t do qualitatively better than that (viz ambient set-theoretic universe for second-order logic), but it’s still possible (necessary?) to work under this apparent lack of absolute control over what it is you are dealing with. Even though (first order) PA doesn’t know what “integers” are, it’s still true that the statements it believes valid are true for “integers”, it’s useful that way (just as AIs or humans are useful for making the world better). It is a device that perceives some of the properties of the object we study, but not all, not enough to rebuild it completely. (Other devices can form similarly imperfect pictures of the object of study and its relationship with the device perceiving it, or of themselves perceiving this process, or of the object of study being affected by behavior of some of these devices.)
Likewise, we may fail to account for all worlds that we might be affecting by our decisions, but we mostly care about (or maybe rather have non-negligible consequentialist control over) “real world” (or worlds), whatever this is, and it’s true that our conclusions capture some truth about this “real world”, even if it’s genuinely impossible for us to ever know completely what it is. (We of course “know” plenty more than was ever understood, and it’s a big question how to communicate to a FAI what we do know.)
Not in the same sense at all. All of the numbers that you have ever physically encountered were nameable, definable, computable. Moreover they came to you with algorithms for verifying that one of them was equal to another.