On 12/16/2018 9:42 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 10:27 PM Brent Meeker <[email protected]
<mailto:[email protected]>> wrote:
On 12/16/2018 4:43 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker
<[email protected] <mailto:[email protected]>> wrote:
On 12/16/2018 2:04 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett
<[email protected] <mailto:[email protected]>> wrote:
On Mon, Dec 17, 2018 at 8:56 AM Jason Resch
<[email protected] <mailto:[email protected]>> wrote:
On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
<[email protected] <mailto:[email protected]>>
wrote:
But a system that is consistent can also prove a
statement that is false:
axiom 1: Trump is a genius.
axiom 2: Trump is stable.
theorem: Trump is a stable genius.
So how is this different from flawed physical theories?
Physical theories do not claim to prove theorems - they
are not systems of axioms and theorems. Attempts to
recast physics in this form have always failed.
Physical theories claim to describe models of reality. You
can have a fully consistent physical theory that
nevertheless fails to accurately describe the physical
world, or is an incomplete description of the physical
world. Likewise, you can have an axiomatic system that is
consistent, but fails to accurately describe the integers,
or is less complete than we would like.
But it still has theorems. And no matter what the theory is,
even if it describes the integers (another mathematical
abstraction), it will fail to describe other things.
ISTM that the usefulness of mathematics is that it's
identical with its theories...it's not intended to describe
something else.
A useful set of axioms (a mathematical theory, if you will) will
accurately describe arithmetical truth. E.g., it will provide us
the means to determine the behavior of a large number of Turing
machines, or whether or not a given equation has a solution. The
world of mathematical truth is what we are trying to describe.
We want to know whether there is a biggest twin prime or not, for
example. There either is or isn't a biggest twin prime. Our
theories will either succeed or fail to include such truths as
theorems.
This is begging the question. You taking one piece of mathematics,
arithmetic, and using it as a theory describing another piece of
mathematics, Turing machines. And then you're calling a successful
description "true". But all you're showing is that one contains
the other.
I'm not following here.
Theorems are not "truths" except in the conditional sense that it
is true that they follow from the axioms and the rules of inference.
I agree a theorem is not the same as a truth. Truth is independent of
some statement being provable in some system.
OK.
Truth is objective. If a system of axioms is sound and consistent,
then a theorem in that system is a truth.
No, c.f. Donald Trump.
But we can never be sure that system is sound and consistent (just
like we can never know if our physical theories reflect the physical
reality they attempt to capture).
But sometimes we can be sure that our theory does not reflect reality,
even if it is sound and consistent.
Brent
Jason
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