On 12/17/2018 11:02 AM, Bruno Marchal wrote:
On 17 Dec 2018, at 07:10, Brent Meeker <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 12/16/2018 9:42 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 10:27 PM Brent Meeker <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 12/16/2018 4:43 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker
<meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:
On 12/16/2018 2:04 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett
<bhkellet...@gmail.com <mailto:bhkellet...@gmail.com>> wrote:
On Mon, Dec 17, 2018 at 8:56 AM Jason Resch
<jasonre...@gmail.com <mailto:jasonre...@gmail.com>>
wrote:
On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
<meeke...@verizon.net
<mailto:meeke...@verizon.net>> 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.
Assuming Donald Trump is sound.
We don’t know what truth is, but we can believe that some formula are
true about our domain investigation. When I assume x + 0 = x, I ask
people if they agree with this, about the natural numbers.
Then a theory is sound, if the rule of inference preserves truth.
If a theory appears to be unsound, we put it in the trash, simply.
That happens sometimes, usually when theories manage too much big objects.
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.
By definition; soundness means that it reflect reality.
You're now messing with words. What does "reflect reality" mean? It
looks like an appeal to correspondence theory to truth. But that means
to know a theory is sound you need to know what is true.
Sound just means its theorems are tautologies, i.e. they are valid
inferences from the axioms.
Soundness implies consistency. But consistency does not imply
soundness. The robot describing the Venus of Milo in front of another
sculpture is consistent, but unsound.
All the machines I am talking about are supposed to be arithmetically
sound.
Meaning that what they "believe" are theorems. Not that they "believe"
everything that is true. In fact you have proven anything about what a
person might or might not believe. Your ideal machines do not include
any concept of acting on "beliefs" which is the real test of beliefs.
Brent
Bruno
Brent
Jason
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