On 16 Jun 2015, at 01:41, meekerdb wrote:
On 6/15/2015 8:15 AM, Brian Tenneson wrote:
Therefore, perhaps proof of truth is an unattainable goal in math.
Perhaps proof of truth is an unattainable goal anywhere.
Math isn't concerned with true, it's only concerned with what
theorems follow from given axioms.
I disagree a lot with this. I'm afarid you confuse the tools, like a
theory, formal or informal, with the subject matter.
Traditionally the axioms are assumed to be true, but this concept of
true is no more than a marker like #t which marks a property
preserved under logical inference rules.
For a logician, we can roughly say that truth means satisfied by a
reality. But then logicians have studied the notion of "reality"
mathematically through semantic or model theory.
The other kind of true, as when we say "It's true that the Earth is
round." is a rough or approximate relation between a statement, "The
Earth is round." and some facts in the world which can in principle
be tested empirically. It's like truth in jury trials, we may
believe it beyond reasonable doubt, but we're never sure.
Same in math. We might say that arithmetical truth, and computer
science elementary basic truth, are, contrary to all other notion of
truth, very well understood and accepted.
So when you say truth is unattainable you need to distinguish the
different uses of the concept. I think it is possible to determine
that some things are true beyond a reasonable doubt.
Yes, like most if not all theorem in arithmetic. But even more when we
have an elementary proof (not making detour in analysis, like the use
of complex number in natural number theory).
Bruno
Brent
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