How is that any different than simply saying they are computable to
arbitrary accuracy, in the Church-Turing sense.
Brent
On 6/17/2018 3:32 PM, Lawrence Crowell wrote:
The Schrödinger equation is integrable for completely unitary systems.
This hold for Hamiltonians that are complex or nonlinear, where
perturbation methods are often used. My thought is that all possible
quantum eigenvalues are computable with Diophantine equations, where
some Gödel numbering scheme from the solutions to the actual numbers
can exist.
LC
On Sunday, June 17, 2018 at 3:42:56 PM UTC-5, Jason wrote:
Hi Lawrence,
Is the evolution of states of the wave function computable? If so
then the result of MRDP implies it is Diophantine.
Jason
On Sunday, June 17, 2018, Lawrence Crowell
<[email protected] <javascript:>> wrote:
I have Matiyasevich's paper on the MRDP theorem. I have not as
yet read it. I have had this idea that a general scheme for
quantum eigenvalues could by Diophantine. This would then be a
sort of universal dovetailer of all possible physical states.
Unfortunately this is an area I have thought about some, but
as yet have never endeavored to pursue.
LC
On Saturday, June 16, 2018 at 7:18:41 PM UTC-5, Jason wrote:
In solving Hilbert's 10th problem
<https://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem> in
the negative, the work of Martin Davis, Yuri Matiyasevich,
Hilary Putnam and Julia Robinson culminated in 1970 with
the MRDP theorem
<https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich%27s_theorem>
which concludes:
*Every computably enumerable set has a representation as a
Diophantine equation
<https://en.wikipedia.org/wiki/Diophantine_equation> (an
equation involving only integer coefficients and variables).*
This shocked number theorists, because it meant simple
equations involving nothing more than a few integer
variables have the full power of Turing machines. In fact,
it was shown by Yuri Matiyasevich that a universal
Diophantine equation can be made with as few as 9 unknowns.
Some examples:
* k is even if there exists a solution to: k - 2x = 0
* k is a perfect square if there exists a solution to: k
- x^2 = 0
* k is a Fibonacci number if there exists a solution to:
k^4 - k^2*x^2 - x^4 - 1 = 0
* (k+2) is a prime number if there exists a solution to
the sum of: (these 14 equations
<http://mathworld.wolfram.com/PrimeDiophantineEquations.html>)
* k is a LISP program having output n, if the equation
having variables: k, n, x1, x2, x3 ... x20000 (a
polynomial having ~20,000 variables
<https://arxiv.org/pdf/math/0404335.pdf>) has a solution.
The universality of Diophantine equations means there are
polynomial equations that compute things quite surprising,
such as polynomials that have solutions of 0, IFF:
* One of the variables "k" is a valid MP3 file.
* One of the variables "k" is a JPEG image containing
the image of a cat (where the equation implements the
same computation as a neural network trained to
recognize images of cats)
* For two of the variables "y" and "x", "y" equals a
state of a chess board after deep blue makes a move
given a chess board with a state of "x".
* For two of the variables "y" and "x", "y" equals the
state of the Universal Dovetailer after performing "n"
steps of execution.
The last example seems to suggest to me, that pure
arithmetical truth, concerning the solutions to equations,
is identical to computation. That is to say, certain
mathematical statements carry with them (effectively)
Turing machines, and their executions.
Just as all solutions to the deep-blue implementing
equation is equivalent to the computations that Deep blue
makes when evaluating the board, and all solutions to the
cat recognizing equation are equivalent to the processing
done by the trained neural network, all solutions to the
LISP equation are equivalent to the execution of every
possible LISP program (including the UD).
Does this our conscious experience might be a direct
consequence of Diophantine equations?
Can Diophantine equations for a single set of parameters
model non-halting programs like the UD, or one must
consider the set of of all possible parameters?
Jason
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