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david petry wrote:
Patricia Shanahan wrote:
david petry wrote:
Jesse F. Hughes wrote:
All of the mathematics that has the potential to help us understand the
world in which we live falls within the scope of mathematics as the
science of phenomena observable in the world of computation.
So the statement, "The halting problem is not computable," does not help
us understand the world in which we live?
Well, if I told you that I have a short, simple, and efficient program
that will tell you whether or not a program will halt, as long as the
code is less than a zillion lines long, that would not contradict the
assertion that no general algorithm for the halting problem exists.
I won't believe you if you claim your halting problem decider is even a
little bit shorter than a zillion lines.
I'm not sure you get the point: the assertion that no general algorithm
for the halting problem exists does not by itself tell you anything at
all about the world we live in.
Yes and no. Obviously, blindly applying something proved using
abstractions such as infinite memory does not work. You need to think.
On the other hand, understanding and using abstractions can help us
understand the world we live in, or at least the world programmers work in.
It is because of ideas I learned in the abstract world of Turing
machines, that I am certain you don't have a short program that can
decide whether a significantly longer program halts on a given input.
Patricia
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