> I prefer the computables--I feel like the idea of a computable number is the formalized version of the idea that (non-integer) numbers in nature do not have definite values, but instead have processes by which you can overturn more digits with effort, and thus they can only be said to be "pretty equal" but never totally equal.
Consider an irrational number, where the fastest possible algorithm (given a standard Turing machine) to compute its nth digit requires (10^(10^100))*(n+1) operations. Now, by the standard definition, such a number is computable – but in practice, we will never be able to compute even its first digit. In practice, its first digit is just as unknowable to us as the first uncomputable digit of (any instantiation of) Chaitin's constant is.
"Computable" by the standard definition is unphysical, because it includes computations that require unphysically large amounts of time and space. Now, many uncomputable (by the standard definition) numbers can be computed by a hypercomputer (such as a Turing machine with an oracle, or a computer which can execute supertasks) – hypercomputers are unphysical, but "standard" computers (in the mathematical sense) are unphysical too. Why denounce one form of unphysicality but not another? It would seem to be more consistent to either embrace the unphysicality of mathematics and imbibe the psychedelic broth of the Cantorian paradise, or else reject unphysicality entirely and embrace ultrafinitism. The "infinity is verboten but unphysically large finitudes are A-okay" approach of (non-ultrafinitist) construcitivsm/intuitionism seems rather arbitrary.
> If a number can be defined in a language but you can't tell me if it's different from another number that I have in finite time, then it is not very interesting to me.
If a number can be defined in a language but I can't tell you whether it's different from another number that you have in a physical amount of time (let's say less than the age of the universe), is it interesting to you?
> Consider an irrational number, where the fastest possible algorithm (given a standard Turing machine) to compute its nth digit requires (10^(10^100))*(n+1) operations
There can be no such number. You can always make a Turing machine output initial digits faster by adding some more states to do so.
Yes, if you knew ahead of time that the first digit of some such number happened to be seven, you could just hardcode your program to start with "IF n = 0 THEN RETURN 7". But that's cheating...
Let's denote the nth prime by P(n). Now, consider the real number Q, where 0.0 < Q < 1.0, and the nth decimal digit (after the decimal point) of Q, Q(n), is given by Q(n) = P(10^(10^100) + n) mod 10. Q, as defined, is computable, but (unless we discover some mathematical "shortcut" for computing the last digit of a prime number), I doubt we are ever going to know what even its first digit is (well, we know it must be 1, 3, 7 or 9–but which of those four?). And even if some such "shortcut" were discovered, I'm sure someone could cook up another such number for which there is no known "shortcut".
How about this: let N(n) be any FNP-complete [FNP is to function problems what NP is to decision problems] computable endofunction of the natural numbers (there are many to choose from, pick any), then define:
Q(n) = P(N(10^(10^100) + n)) mod 10
You actually think we'll ever know the first digit of the real number so defined?
Well, I am glad to learn an even better notion for the concept I'm talking about! I'm just saying, I think "definable" is too loose, because it presupposes that I can even talk about real numbers in a definite way. "Computable" feels closer because it is like an idealized version of the notion, but I agree, it's too loose also.
The reason I've latched onto it is that I read about the idea that in constructive mathematics all functions are continuous, basically because you can't tell that they're not continuous (because to tell that would imply you could inspect infinite digits of function's inputs and tell that they lead to discontinuities in the outputs). I quite like that idea because it rhymes with the intuition that nature doesn't care about later digits of numbers either, and that properties of the real numbers like being uncountable, dense, etc cannot show up in any physical system and therefore aren't "real".
But I'm definitely looking for a better way of understanding this, so curious if you can suggest anything.
> I read about the idea that in constructive mathematics all functions are continuous
From what I understand, it is more that most systems of intuitionistic/constructive logic can't prove the existence of a discontinuous real function – but nor can they disprove it. They have models both in which such functions exist, and also models in which they don't.
Consider an irrational number, where the fastest possible algorithm (given a standard Turing machine) to compute its nth digit requires (10^(10^100))*(n+1) operations. Now, by the standard definition, such a number is computable – but in practice, we will never be able to compute even its first digit. In practice, its first digit is just as unknowable to us as the first uncomputable digit of (any instantiation of) Chaitin's constant is.
"Computable" by the standard definition is unphysical, because it includes computations that require unphysically large amounts of time and space. Now, many uncomputable (by the standard definition) numbers can be computed by a hypercomputer (such as a Turing machine with an oracle, or a computer which can execute supertasks) – hypercomputers are unphysical, but "standard" computers (in the mathematical sense) are unphysical too. Why denounce one form of unphysicality but not another? It would seem to be more consistent to either embrace the unphysicality of mathematics and imbibe the psychedelic broth of the Cantorian paradise, or else reject unphysicality entirely and embrace ultrafinitism. The "infinity is verboten but unphysically large finitudes are A-okay" approach of (non-ultrafinitist) construcitivsm/intuitionism seems rather arbitrary.
> If a number can be defined in a language but you can't tell me if it's different from another number that I have in finite time, then it is not very interesting to me.
If a number can be defined in a language but I can't tell you whether it's different from another number that you have in a physical amount of time (let's say less than the age of the universe), is it interesting to you?