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]> 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'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 [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > -- 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
