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 -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
