On 15 Oct 2015, at 00:48, Bruce Kellett wrote:
On 15/10/2015 2:10 am, Bruno Marchal wrote:
You talk like if I have claim knowing some truth. I do not. You are
doing philosophy of comp-theology. That belongs to the field of
philosophy of science, which is not my expertise. You cannot use
philosophy for making people doubting a logical argument.
..........
of course if you doubt the truth of RA axioms, then I can't explain.
Above you claim that you do not know some truth. So how do you know
the truth of the RA axioms?
By some truth, I meant some big metaphysical truth.
Then I do not *know* the truth of RA axioms, in any publicly
communicable way. I assume them, and that is not a problem, because we
all assume them when we do science.
Of course, you are just being cavalier with your use of the word
'truth'. Axioms are not things which can be said to be either true
of false, at best they are only useful and productive, or not.
They are true or false in the standard model of arithmetic, that is
the mathematical structure (N, +, *).
Even PA can define what is true for any formula with a bounded numer
of quantifiers. What PA cannot do is to define the truth for arbitrary
formula, by Tarski Theorem.
If arithmetic is false, Church-Turing thesis makes no more sense.
You will have difficulties in defining computable function from N
to N.
See above. Arithmetic is neither true nor false, it is only useful
or not, depending on the context.
That is too vague. What do you mean by "Arithmetic"? "true" or "false"
apply only to arithmetical sentences. Then "true" means "true in the
standard model". That can be defined in analysis or set theory (not in
any arithmetical theory).
All proof of negative results are argument from incredulity.
Proving ~p is the same as proving p -> f.
This is just nonsense. An argument from incredulity is an argument
that claims that the difficulty of believing a conclusion
(incredulity) is a valid reason for rejecting the argument.
Proofs are things that happen in formal systems --
Formal proofs. But in science we don't use formal proofs. We reason
informally about them.
In our case (step 8) the "incredulity" just show that if we keep
materialism we have to accept non Turing emulable components in the
(generalized) brain playing a necessary role for consciousness to
proceed. It shows that you need a creationist-like God of the argument
to save a metaphysical commitment, despite there is no evidence for
it. It is a religious move, in the pejorative sense of religious. You
can as well add that you need a god to sustain that primitive matter.
It is a God-of-the-gap, and it is used to not proceed in the
formulation of a problem.
starting from axioms and following pre-defined rules of inference.
So the proof of ~p is simply a sound demonstration that ~p follows
from the axioms according to the rules of inference.
~p is an abbreviation of (p -> f). It assumes at the metalevel that
you are incredule of f. (that is consistent).
It is not a 'proof' that p is false. As I must stress again, truth
and falsity are not words that can be applied within the context of
axiomatic systems.
It can be applied only there, when we do science about it. It is
called "model theory" or semantic.
As non-logician do not know much of model theory, I spare them with
using it too much. I can do that because everyone agree with the
elementary arithmetical truth (except when doing bad philosophy).
If you have a real doubt that 17 is prime, you can't proceed. But if
you agree with such proposition, you should not have any problem,
neither in UDA nor in the translation in arithmetic.
Bruno
Bruce
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