On 20 May 2014, at 22:03, Craig Weinberg wrote:
On Monday, May 19, 2014 2:40:54 AM UTC-4, Bruno Marchal wrote:
On 18 May 2014, at 21:37, Craig Weinberg wrote:
On Sunday, May 18, 2014 1:56:48 PM UTC-4, Bruno Marchal wrote:
On 18 May 2014, at 17:43, Craig Weinberg wrote:
Free Will Universe Model: Non-computability and its relationship
to the ‘hardware’ of our Universe
I saw his poster presentation at the TSC conference in Tucson and
thought it was pretty impressive. I'm not qualified to comment on
the math, but I don't see any obvious problems with his general
approach:
http://jamestagg.com/2014/04/26/free-will-universe-paper-text-pdf/
Some highlights:
Some Diophantine equations are easily solved
automatically, for example:
∃𝑥, ∃𝑦 𝑥² = 𝑦² , 𝑥 & 𝑦 ∈ ℤ
Any pair of integers will do, and a computer programmed
to step through all the possible solutions will find one
immediately at ‘1,1’. An analytical tool such as Mathematica,
Mathcad or Maple would also immediately give symbolic
solutions to this problem therefore these can be solved
mechanically. But, Hilbert did not ask if ‘some’ equations
could be solved, he asked if there was a general way to solve
any Diophantine equation.
...
Consequence
In 1995 Andrew Wiles – who had been secretly working on
Fermat’s ‘arbitrary equation’ since age eight – announced he
had found a proof. We now had the answers to both of our
questions: Fermat’s last theorem is provable (therefore
obviously decidable) and no algorithm could have found this
proof. This leads to a question; If no algorithm can have
found the proof what thought process did Wiles use to answer
the question: Put another way, Andrew Wiles can not be a
computer.
Also, he is the inventor of the LCD touchscreen, so that gives him
some credibility as well.
http://www.trustedreviews.com/news/i-never-expected-them-to-take-off-says-inventor-of-the-touchscreen-display
You will not convince Andrew Wiles or anyone with argument like that.
1) it is an open question if the use of non elementary means can be
eliminated from Wiles proof. Usually non elementary means are
eliminated after some time in Number theory, and there are
conjectures that this could be a case of general law.
2) machine can use non elementary means in searching proofs too.
Does computationalism necessarily include all that is done by what
we consider machines,
Only digital machines.
But how do you know the difference between what a digital machine
happens to do because of the way that it is implemented (machine +
sense + physics) rather than what follows from mechanism alone?
I don't know that.
I would know that, I would know that comp is true, which I don't know,
nor could I know.
or does computationalism have to be grounded, by definition, in
elementary means?
It does not, but always can, by Church Thesis.
Why doesn't it? Why isn't the same loose grounding afforded to
consciousness? I'm not really sure what would even constitute being
always allowed to be grounded in the elementary but not having to be.
Computationalism assumes that the relevant part of our bodies
(relevant for making our consciousness related to our probable
computations) is grounded in computations, a notion which is
elementary (definable in first order arithmetic language, and existing
provable in weak little segment of the arithmetical truth).
But our consciousness itself is not grounded on that tiny part of
arithmetic, nor is the behavior in general grounded in arithmetic,
even the whole of arithmetic. Even the machine's theology escapes
arithmetic.
For example, although G and G* are decidable, their first order
extension is as undecidable as they can be (qG is Pi_2, and qG* is
Pi_1 ... *in* the whole arithmetical truth as oracle. this means that
even using God as an oracle, you can't solve all general theological
question on the machines).
That is why the "ONE" of the machine (arithmetical truth) is already
overwhelmed by what emanates from it, the Noùs, which get bigger than
the ONE (which is rather natural given that is gives rise to the MANY,
but this is something that greeks could not seen (as they didn't
discover the universal languages and machines). Leibniz was close,
though.
When the complexity of machines grows, there is a threshold which
makes impossible for any machines or entities to confine the behavior
of the machine in any simple theories. That threshold is low, you get
it with any entities which is able to add and multiply.
With classical computationalism, Gödel's theorem applies to us, and it
explains to us that the arithmetical truth, and the truth about
machines in general, can only be scratched by us.
You did not provide evidence that they cannot do that.
His evidence was the negative answer to Hilbert's 10th problem.
By using Church thesis. The proof consists in showing that the 10th
problem of Hilbert is Turing complete. Diophantine polynomials are
Turing universal. See below for an example of UD written as a system
of Diophantine equations (exponent are abbreviation here(*)
From what I've gathered so far, it seems like the proof shows that
the halting problem has a Diophantine representation, so that
because Church-Turing proves the halting problem is not computable,
then Hilbert's 10th problem of whether Diophantine equations can be
computed generally must be a no.
OK.
The fact that Wiles did prove a solution to FLT but could not have
done so using a general algorithm shows, according to Tagg, that
Wiles is not a Turing machine.
That is not correct.
1) As I said, it is an open question if Wiles proof can be made
elementary, in which case PA could fin it, given enough time.
2) But even if that was not the case, a machine can also use non
elementary means. No mathematicians would doubt that Wiles theorem can
be proved in or by ZF + kappa.
Another example: there are no algorithm to win a chess game in less
than one hour, but this does not mean that a machine cannot learn and
use excellent heuristics and become gifted in beating humans in less
than one hour.
Machines can search the arithmetical reality, which is richer than
themselves, and so, they can find new things, and find new things,
without having been able to predict them.
And you could'nt as a machine like ZF, or ZF + kappa, can prove
things with quite non elementary means.
What theory addresses the emergence of non elementary means?
Mathematical logic, theoretical computer science.
Does it explain where the emergence comes from, or just demonstrates
that it appears to emerge from an unknown property?
I would say that it explains where the emergence comes from. It comes
1) from the self representation ability of the universal numbers
relatively to the arithmetical reality.
2) from the degrees of consistency and closeness to the arithmetical
truth.
Maybe there is something about the implementation of those machines
which is introducing it rather than computational factors?
?
The non-elementary part may be from the inference of the
mathematician's consciousnesses, or physics of implementation rather
than the math.
Yes. Sure. That is exactly what I show to be precisely testable. If
non-comp is true, machine's theology, by including machine's physics,
will gives a tool to measure "our" degree of non (classical)
computationalism.
But to be honest, we know already that it has to be a strong form of
non-computationalism, as G and G* applies also to large class of
divine (non Turing emulable) self-referentially correct entities.
Bruno
Craig
Bruno
(*)
Nu = ((ZUY)^2 + U)^2 + Y
ELG^2 + Al = (B - XY)Q^2
Qu = B^(5^60)
La + Qu^4 = 1 + LaB^5
Th + 2Z = B^5
L = U + TTh
E = Y + MTh
N = Q^16
R = [G + EQ^3 + LQ^5 + (2(E - ZLa)(1 + XB^5 + G)^4 + LaB^5 + +
LaB^5Q^4)Q^4](N^2 -N)
+ [Q^3 -BL + L + ThLaQ^3 + (B^5 - 2)Q^5] (N^2 - 1)
P = 2W(S^2)(R^2)N^2
(P^2)K^2 - K^2 + 1 = Ta^2
4(c - KSN^2)^2 + Et = K^2
K = R + 1 + HP - H
A = (WN^2 + 1)RSN^2
C = 2R + 1 Ph
D = BW + CA -2C + 4AGa -5Ga
D^2 = (A^2 - 1)C^2 + 1
F^2 = (A^2 - 1)(I^2)C^4 + 1
(D + OF)^2 = ((A + F^2(D^2 - A^2))^2 - 1)(2R + 1 + JC)^2 + 1
(unknowns range on the non negative integers (= 0 included)
31 unknowns: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R,
S, T, W, Z, U, Y, Al, Ga, Et, Th, La, Ta, Ph, and two parameters:
Nu and X. The polynomial emulates the universal question "X is in
w_Nu", or "phi_Nu(X) stops".
Craig
Bruno
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