Computers can efficiently calculate anything that can possibly be calculated, predict anything that can be predicted, simulate anything that can be simulated, etc..., so they actually are kinda special in a way that previous things aren't.
The mechanisms of computation have allowed us to prove beyond a shadow of a doubt that there are true things that cannot be proven, and that there is such a thing as an unknowable quantity. It also turns out that predicting if certain particular computer programs will ever stop is equivalent to proving basically anything.
It's called the Church-Turing hypothesis for the former, and then there's the Gödel incompleteness theorem for the second, as well as the Busy Beavers for the second. it's pretty neat.
Computers can efficiently calculate anything that can possibly be calculated
This is pretty much the P=NP problem though, which hasn't been proven (and many believe it'll eventually be proven in the negative), with 'efficient' being defined as polynomial time.
If something can only ever be calculated in exponential (or worse) time, then a computer which can calculate it in exponential time I'd say is still efficient, so it would be true whether or not P=NP. When I say that a computer is efficient what I really mean is that it's not exponentially worse than any other possible way of arriving at the result. I should have worded it better, sorry.
Why? If there is a number that can't be computed, and there's no evidence that it's physically present, what's the contradiction here?
Computers are only limited in creating things that can be proven because creating something is a proof that it exists. This has a more formal meaning if you look at the Curry-Howard correspondence.
They exist in the mathematical sense that they are basically the solution to an equation. We know the equation, we know there is a solution, but we don't know what the solution is, and it has no physical relevance. Many mathematicians (constructivists) would say that this means that they don't exist until you can find a way of finding the exact solution.
For many of these numbers, we know that there cannot be any rigorous procedure by which they can be found, because if there was, there would be a logical contradiction.
Computers can efficiently calculate anything that can possibly be calculated, predict anything that can be predicted, simulate anything that can be simulated, etc..., so they actually are kinda special in a way that previous things aren't.
The mechanisms of computation have allowed us to prove beyond a shadow of a doubt that there are true things that cannot be proven, and that there is such a thing as an unknowable quantity. It also turns out that predicting if certain particular computer programs will ever stop is equivalent to proving basically anything.
It's called the Church-Turing hypothesis for the former, and then there's the Gödel incompleteness theorem for the second, as well as the Busy Beavers for the second. it's pretty neat.
This is pretty much the P=NP problem though, which hasn't been proven (and many believe it'll eventually be proven in the negative), with 'efficient' being defined as polynomial time.
If something can only ever be calculated in exponential (or worse) time, then a computer which can calculate it in exponential time I'd say is still efficient, so it would be true whether or not P=NP. When I say that a computer is efficient what I really mean is that it's not exponentially worse than any other possible way of arriving at the result. I should have worded it better, sorry.
Godel's a motherfucker for that :agony-immense:
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Why? If there is a number that can't be computed, and there's no evidence that it's physically present, what's the contradiction here?
Computers are only limited in creating things that can be proven because creating something is a proof that it exists. This has a more formal meaning if you look at the Curry-Howard correspondence.
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They exist in the mathematical sense that they are basically the solution to an equation. We know the equation, we know there is a solution, but we don't know what the solution is, and it has no physical relevance. Many mathematicians (constructivists) would say that this means that they don't exist until you can find a way of finding the exact solution.
For many of these numbers, we know that there cannot be any rigorous procedure by which they can be found, because if there was, there would be a logical contradiction.