Hi,
I really want to apology for my spelling. I will not correct my post
(I could add errors!), but I want to correct a statement I made:
On 18 Jul 2011, at 20:44, Bruno Marchal wrote:
Accepting what you can feel and see and test is the antithesis of
taking it for granted and the epitome of the scientific attitude.
That is Aristotle definition of reality (in modern vocabulary). But
the platonist defend the idea that what we feel, see and test, is
only number relation, and that the true reality, be it a universe or
a god, is what we try to extrapolate.
Of course this is a mistaken statement. Not all platonists are
pythagoreans. I thought writing this:
"But the platonists defend the idea that what we feel, see, and test
might be the shadow, or the border of something else, which might be
non physical.".
Plato knew the Pythagorean, and a part of the academia defended the
idea that the fundamental reality was mathematical. But other
Platonists, like Plato himself, were just more agnostic on this than
the so-called Mathematicians (notably Xeusippes). Plotinus show the
same agnosticism, despite its amazing enneads on the Numbers.
Best,
Bruno
We certainly don't see, feel, or test a *primitive* physical
universe. The existence of such a primitive physical reality is a
metaphysical proposition. We cannot test that. This follows directly
from the dream argument. That is what Plato will try to explain with
the cave.
The trouble with axiomatic methods is that they prove what you put
into them. They make no provision for what may loosely be called
"boundary conditions". Physics is successful because it doesn't
try to explain everything. Religions fall into dogma because they
do.
I don't criticize physics, but aristotelian physicalism. which is,
for many scientists, a sort of dogma.
Religion fall into dogma, because humans have perhaps not yet the
maturity to be able to doubt on fundamental question. To admit that
we don't know if there is a (primitive) physical universe.
Physicists use mathematics (in preference to other languages) in
order to be precise and to avoid self-contradiction.
That is the main error of the physicists. They confuse mathematics
with a language.
And the main error of mathematicians is they confuse proof with
truth.
That is unfair because all what I use here is the (big) discovery of
Gödel that arithmetical truth escapes all possible effective or
axiomatizable proof systems. So mathematicians are able to
distinguish mathematically, in many case, the difference between
proof and truth.
Only intuitionist confuse proof and truth, (like S4Grz!) but
classical mathematicians note that not only proof does not entail
truth, but that even in the case where proof entails truth, the
contrary remains false: truth does not entail proof.
The whole AUDA is based on the fact that arithmetical truth is
beyond all correct machines (proofs).
Let me comment a little part of your dialog with Jason. I comment
also Jason.
"True" is just a value that is preserved in the logical inference
from axioms to theorem. It's not the same as "real".
True is more than inference from axioms.
I think Jason said that. I agree. Truth is preserved in the
application of sound inference rules, but truth is far bigger than
anything you can access by inference rules and axioms. Arithmetical
truth is, compared to any machine, *very* big. The predicate truth
cannot even be made arithmetical.
For example, Godel's theorem is a statement about axiomatic
systems, it is not derived from axioms.
Well, the beauty is that Gödel's second incompleteness theorem is a
theorem of arithmetic. BDt -> Bf (or ~Bf -> ~B~Bf) is a theorem of
PA. It is the whole point of interviewing PA about itself. It can
prove its own Gödel's theorem. That is missed in Lucas, Penrose, and
many use of Gödel's theorem by anti-mechanist. Simple, but not so
simple, machine have tremendous power of introspection. Löbian one,
have, actually, maximal power of introspection.
Sure it is. It's a logical inference in a meta-theory.
Not at all. The second (deeper) theorem of Gödel, like the theorem
of Löb, is a theorem of Peano Arithmetic. The tedious part consists
in translating the "Bx" in arithmetic, but Gödel's succeeded
famously in the task (cf beweisbar ('x')).
G axiomatise all such metatheorem that a theory can prove about
itself, and G* formalize all the truth that the theory can prove +
that the theory cannot prove about itself. In that way, Solovay
closed the research in the modal propositional provability/
consistency logics, by finding their axiomatization, and this both
for the provable part of the machine (which contains BDt -> Bf), and
the non provable part (which contains typically Dt, DDt, DDDt, DBf,
DDBf, etc.).
Bruno
Brent
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