A small correction: there was an error in the definition I have for the
Fibonacci calculation. It should have been *(k^2 - kx - x^2)^2 - 1 = 0*
Below is some Python code that searches for every solution where k and x
are less than 100:
for k in range(100):
for x in range(100):
if pow(k*k - k*x - x*x, 2) - 1 == 0:
print k
It gives:
*$python main.py*
0
1
1
2
3
5
8
13
21
34
55
89
Which are all the Fibonacci numbers under 100.
Anyone interested may also execute or modify this example code online by
going here: http://tpcg.io/pA1E4q
John Clark often tells Bruno mathematical truth won't put Intel out of
business, but this case, (more than any other I have seen), leads me to
believe that mathematical truth does embody computation. No physical
computer is necessary for these computations to exist, only for us to
access it. Just as the existence of a solution to "x+x = 4" implies the
mathematical existence of 2, the existence of a solution to Chaitin's LISP
evaluating equation implies the mathematical existence of the computation
of of that LISP program.
Jason
On Sat, Jun 16, 2018 at 7:18 PM, Jason Resch <[email protected]> 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's_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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