On 11/19/2014 9:00 AM, Bruno Marchal wrote:
I would think the obvious way to parse what Bruno has said here is "science
cannot show that something is correct".
Is that right, Bruno?
Yes.
Of course empirical tests are better at showing a theory is wrong than
showing
it's right, which is Popper's observation.
Indeed.
I'm curious as to how you define correctness properly?
I can't do it for myself, nor can any machine do it for herself. But a
"sufficiently strong" machine can do it for a lesser strong machine. You
can define
arithmetical truth and PA's correctness in the set theory ZF for example.
In that
case "correctness" is defined in the manner of Tarski: p is correct if it
is the
case that p is satisfied by this or that mathematical structure, (for RA
and PA,
you can use the usual (N,+, *) structure, and with computationalism, that
arithmetical truth (not definable in arithmetic) is enough).
This sounds like a description of which mathematical theories suggest the existence of
higher more-correct selves.
Not more correct, but knowing much more things. ZF knows that PA is consistent, and ZF
knows much more than PA about arithmetic, although of course we still don't know if ZF
knows the truth or the falsity of Riemann hypothesis, but few doubt that ZF has any
doubt about it.
So in summary "correct" just means logically consistent under some set of axioms. PA
correctness is relative to ZF's axioms. Which tells me that that mathematics cannot show
something is correct either.
Brent
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