I am not sure. If I tabooed “exist,” then my best guess is that ultrafinitists would argue that statements involving really big numbers are not meaningful. For example, they might argue that such statements are not verifiable in the real world. (Edit: as another example, as I mentioned in another comment, ultrafinitists might argue that you cannot count to really big numbers.)
For instance, I see that the anecdote-reporter refers to there being a series of points of a particular length, but I assume they don’t mean that in an intuitive, literal sense: there are certainly at least 2^100 Planck lengths between me and the other end of the room.
Yes, but just barely: 2^100 Planck lengths is something like 2 x 10^{-5} meters, so substitute 2^1000 Planck lengths, which is substantially larger than the diameter of the universe.
Seems weird to think that some of the possible configurations of bits on my 1.5TB hard drive don’t exist. Which ones? I hope none of the really good collections of pr0n are logically unreachable.
If that number does exist, then what about really big busy beaver numbers, like bb(2^10^13 )? They’re just a series of computations on hard drive contents. And that number is so close to infinity that we might as well just step from ultrafinitism to plain old finitism.
While I am not an ultrafinitist, I believe the idea is meant to be this: It is not valid to talk about those numbers, because there is no meaningful thing you can do with those numbers that can affect the real world. Therefore, the ultrafinitists say that it is not really logical to treat these numbers as “existing” as they can not affect the real world at all, and why say that something exists if it cannot affect anything at all?
Seems weird to think that some of the possible configurations of bits on my 1.5TB hard drive don’t exist.
Would you like to go through all of them just to be sure? How long do you think that will take you?
what about really big busy beaver numbers, like bb(2^10^13 )? They’re just a series of computations on hard drive contents.
Trying to actually compute a sufficiently large busy beaver number, you’ll run into the problem that there won’t be enough material in the observable universe to construct the corresponding Turing machines and/or that there won’t be enough usable energy to power them for the required lengths of time and/or that the heat death of the universe will occur before the required lengths of time. If there’s no physical way to go through the relevant computations, there’s no physical sense in which the relevant computations output a result.
It may not be possible to check all of them, but it certainly is possible to check one of them...any one of them. And whichever one you choose to check, you’ll find that it exists. So if you claim that some of the possible configurations don’t exist, you’re claiming they’d have to be among the ones you don’t choose to check. But wait, this implies that your choice of which one(s) to check somehow affects which ones exist. It sure would be spooky if that somehow turns out to be the case, which I doubt.
Exactly. And I could make my choice of which pr0n library to check—or which 1.5TB turing machine to run—dependent on 10^13 quantum coinflips; which, while it would take a while, seems physically realizable.
I am not sure. If I tabooed “exist,” then my best guess is that ultrafinitists would argue that statements involving really big numbers are not meaningful. For example, they might argue that such statements are not verifiable in the real world. (Edit: as another example, as I mentioned in another comment, ultrafinitists might argue that you cannot count to really big numbers.)
Yes, but just barely: 2^100 Planck lengths is something like 2 x 10^{-5} meters, so substitute 2^1000 Planck lengths, which is substantially larger than the diameter of the universe.
Seems weird to think that some of the possible configurations of bits on my 1.5TB hard drive don’t exist. Which ones? I hope none of the really good collections of pr0n are logically unreachable.
If that number does exist, then what about really big busy beaver numbers, like bb(2^10^13 )? They’re just a series of computations on hard drive contents. And that number is so close to infinity that we might as well just step from ultrafinitism to plain old finitism.
While I am not an ultrafinitist, I believe the idea is meant to be this: It is not valid to talk about those numbers, because there is no meaningful thing you can do with those numbers that can affect the real world. Therefore, the ultrafinitists say that it is not really logical to treat these numbers as “existing” as they can not affect the real world at all, and why say that something exists if it cannot affect anything at all?
This seems incredibly likely, doesn’t it? (As long as we are happy to bound ‘logically reachable’ to within the observable universe.)
Would you like to go through all of them just to be sure? How long do you think that will take you?
Trying to actually compute a sufficiently large busy beaver number, you’ll run into the problem that there won’t be enough material in the observable universe to construct the corresponding Turing machines and/or that there won’t be enough usable energy to power them for the required lengths of time and/or that the heat death of the universe will occur before the required lengths of time. If there’s no physical way to go through the relevant computations, there’s no physical sense in which the relevant computations output a result.
It may not be possible to check all of them, but it certainly is possible to check one of them...any one of them. And whichever one you choose to check, you’ll find that it exists. So if you claim that some of the possible configurations don’t exist, you’re claiming they’d have to be among the ones you don’t choose to check. But wait, this implies that your choice of which one(s) to check somehow affects which ones exist. It sure would be spooky if that somehow turns out to be the case, which I doubt.
Exactly. And I could make my choice of which pr0n library to check—or which 1.5TB turing machine to run—dependent on 10^13 quantum coinflips; which, while it would take a while, seems physically realizable.