> On 17 Jun 2018, at 15:27, Jason Resch <jasonre...@gmail.com> wrote: > > 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 <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.
Yes, and that very statements has to be formalised to see what the machine can understand about this, which is very small for very elementary arithmetic but quite huge for the Gödel-Löbian machine. In that case a (halting) computation is just a number representing it, and the machine can distinguish the difference between that number representation and the computation represented, and the fact that it is emulated in virtue of the *truth* concerning the relation. Like it is true that 3^6 divides 2^5 * 3^6 *5^8 *7^2 *11^2 when we implement a register in arithmetic by the prime decomposition of the numbers. Here we translate (partially) that 6 belongs to the register (5, 6, 8, 2, 2). (You need also that 3^7 does not divide the number). The key point is in distinguish well the arithmetical truth from their representation in arithmetic. Bruno > > Jason > > > On Sat, Jun 16, 2018 at 7:18 PM, Jason Resch <jasonre...@gmail.com > <mailto:jasonre...@gmail.com>> 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 everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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