On 25 Oct 2015, at 10:39, Bruce Kellett wrote:
On 25/10/2015 6:12 pm, Pierz wrote:
It's hard to see how physics can be self-consistent without the a
priori existence of arithmetic.
Maybe it is because the self-consistency of physics is what makes
arithmetic possible.
That is too much fuzzy. To make it precise you might need to give the
theory "physics", and prove the consistency of arithmetic in it.
But by Gödel's incompleteness theorem we know once thing, which is
that to prove the consistency of arithmetic, you will need more than
arithmetic. But you don't need a primary physical universe to prove
that, nor anyone primary one actually.
Now, I have not yet heard of any physical theory which does not assume
arithmetic. The proof that photon and mass zero even use the well know
number theoretical truth that 1+2+3+4+... = -1/12, which is indeed
provable in Peano-Arithmetic.
To avoid confusion, we should distinguish:
1) the theory/machine, (a finite sequence of symbols, a machine)
2) the proofs/computations (finite sequences of symbols, the finite
working of a machine)
3) The model of the theory (a structure validating the theorems of the
theory, the beliefs of the machine)
4) Reality or parts of reality: when it happens the reality we are
interested in, and on which we bet on the existence, fits with a
model, partially or completely defined by the theory.
We might decide to forget about the "reality", and concentrate only on
the theories, and judge which one explains better the other one.
So, give me your theory "physics".
I am a scientist, and I hate wasteful pseudo-philosophy. I have
already given three different way to express "my" theory. (Classical
logic + Robinson Arithmetic, or SK-combinators + identity theory, or a
system of diophantine polynomials). The derivation of mind and matter
is independent of which one.
In any of such theory, I can define the universal numbers, prove their
existence, and prove the existence of all there finite computations,
and I can prove the discovery by richer universal numbers of the
existence of many non-finite computations as well. "Richer" can be
defined in term amount of induction axioms:
(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in
the arithmetical language (with "0, s, +, *),
In fact I recover that incompleteness forces, from the machine's point
of view to distinguish 5 main variants of truth:
1) truth itself
2) the part which is justifiable/comprehensible/intelligible
3) the part which is justifiable/comprehensible/intelligible and which
is also true
4) the part which is justifiable/comprehensible/intelligible and which
is also consistent
5) the part which is justifiable/comprehensible/intelligible and which
is also consistent, and true
For each machine, each such view defines a set of numbers with a
highly non trivial structure, and if you understand the UDA, and
Gödel's beweisbar, you get that 3, 4, and 5, give the physical modes,
making this testable.
If you accept the idea that the brain is Turing emulable, it is easier
to explain how numbers hallucinate in physical realities, than to
invoke some primary matter for which we have no evidence at all, and
which would do a selection (how?) of the many lives of the universal
number, or better the universal person associated to it, in arithmetic.
At first sight, it looks we get an inflation of possibilities, but
taking into account the self-referential correctness of the machine
gives very strong constraints
Though admittedly that is a different point to whether or not
physics is "emulated" in arithmetic.
True. But no-one has yet emulated any physics in arithmetic.
I insist often on this, but I guess it is a bit subtle. Neither
matter, nor consciousness are ever emulated in arithmetic. Only the
computations are emulated in arithmetic. Consciousness and matter are
defined by the first person points points of view of the (universal)
person supported by those computations, and are somehow defined by the
projections of all their histories. Your current first person point of
view is defined by *all* computations in arithmetic getting through
your cognitive states. That subtlity, when translated in arithmetic,
concerns the fact that, although we have :
p -> []p,
(recall that the propositional variable p are interpreted by the
arithmetical sigma_1 sentences).
and that the machine can prove or justifie it (the Löbian machine, not
RA!)
And we have also that
[]p -> p,
but luckily, for all this making sense, the machine cannot prove it
for all p.
G1* can prove p -> []p, and []p -> p, but G1 cannot.
Despite this equivalence, we have not that []<>p -> p, but we do have
p -> []<>p. making this playing some role on the graded quantizations.
(All this for those who remember previous explanations, or reaad my
papers, with the help of the right books).
Bruce, you need to study some books (if interested in the problems, of
course).
All the we have is mathematical accounts of discovered physical laws
--
Well, the laws of physics are given by mathematical relations,
involving many type of numbers, the natural numbers, the integers, the
rationals, the real numbers, the complex numbers, and the quaternion
entered QM to with Pauli matrices, and I doubt we will ever marry QM
and GR without the octonions (I bet), and with comp, it would be a
priori a miracle that all that works, unless there is a reason fro
that: get the right measure on the (relative: in 7 sense) sigma_1
(true) sentences. The seven sense are the fives senses above, except
that three of them split on G/G*, and so there are eight points of
view (minus truth, which is not a point of view).
arithmetic based in physics.
In term of theory, physics assumes arithmetic. And arithmetic does not
assume anything physical, or related to physical notions.
The advantage of starting from a simple theory arithmetic, or anything
Turing equivalent, is that we can prove in such theories (and all
their extensions) that any simpler theory can't get any universal
numbers.
I recall what is a universal number. We fix a universal system, say
LISP, we enumerate (without repetitions) the programs in LISP. We
enumerate correspondingly (with high repetitions) the partial
computable functions phi_i. And a number u is said universal, if for
all x and y phi_u(x,y) = phi_x(y). u imitates (run, executes) x on
y. We can define LISP in arithmetic, and that makes the notion easily
definable, but what we prove from that need some cautiousness (and
usually I use another slightly weaker definition, based on the
Recursively Enumarable set W_i. Given the importance of the
intensional nuance, some common notion of recursive equivalence can't
be used.
You need to study a bit of the theory of computability, and its
relations with arithmetic and mathematical logic.
I take it that you have a not wrong idea on the seven first steps. The
problem is in step 8, for you. But let us do a bit of the math to get
the technical point. (Hoping you are not here just to say it fits not
your religion/conception-of-reality).
Buy or try to find in the library, the books by Smullyan, and Boolos
1979, 1993.
My point is purely technical. Classical computationalism is testable
(by the usual physical means). And QM without collapse confirms its
most striking intuitive aspects (all computations, all sigma_1 truth)
and formally, it fits, up to now.
An recently I saw that Moderatus of Gades made the point even clearer
than Plotinus, until I realized that if you read the enneads in the
chronological order, a point can be made that Plotinus got the point
rather well.
What I say is not original conceptually, the neopythagoreans and the
neoplatonists saw, not this, but something quite close, I think.
Basically what any machine can find inward.
Bruno
Bruce
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