Gödel has proven the opposite: all scientific truths have to be constructed, a notion of unreachable truth cannot be consistently maintained. We had to change the semantics of mathematics into something that works, instead of claiming that reality is inconsistent with mathematics
-
-
I am not sure I follow. He proved in a consistent system there are unreachable truths. So we agreed to not always have to stay in the system. and so new proof techniques were ok once they made sense to us.
1 reply 0 retweets 1 like -
Replying to @iamreddave @eschudy
No, he and Turing have shown that the classical mathematical notion of truth was not self consistent and had to be changed. Values are only equivalent to functions if the function is actually computable.
1 reply 0 retweets 2 likes -
By classical notion of truth did people think that all truths would be provable inside the system? Because I see how they disproved that.
1 reply 0 retweets 2 likes -
Replying to @iamreddave @eschudy
Classical mathematics was "timeless", i.e. a statement could be considered true regardless of the cost of computing it. Gödel and Turing could show that this breaks the semantics of mathematics if a proof is not effectively computable (i.e. the cost to compute it is not finite).
1 reply 0 retweets 2 likes -
Replying to @Plinz @iamreddave
Is this true? Because you can have proofs that are never ending, thus the halting problem. Think printing hello in an endless loop - this is in fact an infinite proof.
2 replies 0 retweets 0 likes -
Replying to @eschudy @iamreddave
Proving is just a lossless compression of a statement into axioms. A compression algorithm that does not terminate cannot be claimed to perform a compression.
1 reply 0 retweets 0 likes -
Replying to @Plinz @iamreddave
Ok, then by that reasoning, a proof should always halt. And there should be no halting problem, other than predicting when it stops, rather than IF it stops.
1 reply 0 retweets 0 likes -
Replying to @eschudy @iamreddave
I suspect that you still don't see it? In classical math, Pi is a number with a certain numerical value that can be expressed with a function. In constructive (computational math), Pi is just a function. All values are just decorated integers.
1 reply 0 retweets 0 likes -
I think you could be right. What is a decorated integer? If you have a link thats great.
1 reply 0 retweets 0 likes
A decorated integer is an honest vector of bits that you dress up with little dots and e's so it looks almost like a real number.
Loading seems to be taking a while.
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.