On 18 Jul 2011, at 19:14, meekerdb wrote:
On 7/18/2011 2:48 AM, Bruno Marchal wrote:
On 17 Jul 2011, at 20:28, meekerdb wrote:
On 7/17/2011 10:11 AM, Bruno Marchal wrote:
On 15 Jul 2011, at 18:41, meekerdb wrote:
On 7/15/2011 2:15 AM, Bruno Marchal wrote:
Numerology is poetry. Can be very cute, but should not be taken
too much seriously. Are you saying that you disagree with the
fact that math is about immaterial relation between non
material beings. Could you give me an explanation that 34 is
less than 36 by using a physics which does not presuppose
implicitly the numbers.
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Nice, indeed. We do agree that 34 is less than 36, and what that
means.
I am not sure your proof is physical thought. Physics has been
very useful to convey the idea, and I thank God for not having
made my computer crashed when reading your post, but I see you
only teleporting information. That fact that you are using the
physical reality to convey an idea does not make that idea
physical.
I was expecting a physical definition of the numbers.
Of course there is no physical definition of the numbers because
the usual definition includes the axiom of infinity.
You don't need the axiom of infinity for axiomatizing the numbers.
The axiom of infinity is typical for set theories, not natural
number theories. You need it to have OMEGA and others infinite
ordinals and cardinals.
As finite beings we can hypothesize infinities.
Yes, but we don't need this for numbers. On the contrary, the
induction axioms are limitation axioms to prevent the rising of
infinite numbers.
By thinking that I can understand your proof, you are
presupposing many things, including the numbers, and the way to
compare them.
On the contrary I think you (and Peano) conceived of numbers by
considering such such examples. The examples presuppose very
little - probably just the perceptual power the evolution endowed
us with.
That is provably impossible. No machine can infer numbers from
examples, without having them preprogrammed at the start. You need
the truth on number to make sense on any inference of any notion.
Nothing can be proven that is not implicit in the axioms and rules
of inference.
OK.
So I doubt the significance of this proven impossibility.
?
It means, contrary of the expectations of the logicist that the
natural numbers existence is not implicit in many logical system.
We cannot derive them from logic alone, nor from first order theories
of the real numbers, nor from most algebra, etc. So, if we want
natural numbers in the intended model of the theory, they have to be
postulated, implicitly (like in wave theory, set theory) or
explicitly, like in RA or PA.
So it is a funny answer, which did surprise me, but which avoids
the difficulty of defining what (finite) numbers are.
It *is* a theorem in logic, that we can't define them
"univocally" in first order logical system. We can define them in
second order logic, but this one use the intuition of number.
If you agree that physics is well described by QM, an explanation
of 34 < 36 should be a theorem in quantum physics,
I'm sure it is. If you add 34 electrons to 36 positrons you get
two positrons left over.
Physics is not an axiomatic system.
That is the main defect of physics. But things evolve. Without
making physics into an axiomatic, the whole intepretation problem
of the physical laws will remain sunday philosophy handwaving.
Physicists are just very naïve on what can be an interpretation.
The reason is they "religious" view of the universe. They take it
for granted, which is problematic, because that is not a scientific
attitude.
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.
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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