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. Theorems are not "truths" except in the conditional sense that
it is true that they follow from the axioms and the rules of inference.
Brent
Jason
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