Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker the decimal positions n! and 0 in all the other places. Someone might readily say that a number like this is unlikely to occur in any real context. Analogy. In 1956, Richard Friedberg and Albert Muchnik carried out a complex and artificial construction of a computation of intermediate degree for one notion of solvability degrees and logicians such as J. C. E. Dekker did similar constructions for other notions of degrees. History. In 1874, Georg Cantor constructed a specific enumeration of the countable set of algebraic reals and produced a transcendental number by having it differ in the i th digit from the i th member of his enumeration of the algebraic numbers. Again, someone could say that Cantor’s number isn’t a number that would naturally occur, that it is artificial, and that it depends in an essential way upon higher-order concepts such as treating an infinite enumeration of reals as a completed object. Analogy. In the 1960s, the logician Gerald Sacks and his colleagues streamlined the techniques of Friedberg and Muchnik to produce a bewildering variety of computations of intermediate degree. Yet all of their constructions remained somewhat artificial. History. But in 1873 Charles Hermite proved that the relatively non-artificial number e is transcendental, and in 1882 Ferdinand Lindemann proved that the downright gardenvariety number pi is transcendental as well. Analogy. Hasn’t happened yet ⎯ and may never happen. But my guess is that eventually someone will prove that a natural and familiar computation like CA Rule 30 is a non-universal computation that has an unsolvable halting problem. Weinberg’s Zinger And of course Turing machines and two-state CAs are, for instance, much slower than a PC. As the physicist Stephen Weinberg remarks, “This is why Dell and Compaq don’t sell Turing machines or rule 110 cellular automata.” Stephen Weinberg, “Is the Universe a Computer,” The New York Review of Books, October 24, 2002, pp. 43- 47. This was one of the many smug and negative reviews which greeted the publication of Wolfram’s A New Kind of Science. As John Walker remarked about the reviews, “Most of them were An Old Kind of Envy.” Something I found particularly wrong-headed in Weinberg’s review was his cavalier remark that we don’t really want to know the outcomes of complex processes like medium energy particle collisions. But I have to admit Weinberg got off some good zingers, as when he attributes Wolfram’s universal automatism to excessive time spent in programming, and compares his world-view to that of a carpenter who looks at the moon and wonders if it might be made of wood. Joke About Cosmic Fry’s Electronics If you do find your way to that Fry’s in the Sky, get me a slow-down ray! (Ignore this, I’m slap-happy, punch-drunk, on the ropes. I’ve been thinking too hard about this stuff for way too long.) Everyone’s Different It’s no accident that every single person in the world looks different. Even if two fertilized eggs were to have the same DNA, the biochemical differences between the cells’ p. 104
Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker cytoplasm would have an effect, as would the differing environments of their mothers’ wombs. Even identical twins are different, if you take a magnifying glass to them. Stimulants The stimulant issue is a scam, an irrelevant sideshow. A trick for suckers. Who ever got smarter by taking a pill? Television When I watch television, I feel that I really am seeing predictable computations. News, commercials, entertainment ⎯ it’s all so stereotyped, so lifeless, so utterly false. Nobody in real life acts like the people on TV, not even like the people on reality TV, for the reality TV shows have been carefully edited to remove any trace of gnarly originality. If something unpredictable every happens on TV, the U. S. Congress angrily decries the anomaly for months. Whoah there, querulous geezer, there’s more to society than TV. Quantum Computation and Predictability You know how sometimes you’re at a restaurant with five or six other people, and at the end of the meal everyone’s paying their share of the check, tossing hard-earned cash dollar bills down on the table. And then there’s always the one schmuck who loses control at the sight of the cash and grabs it all up and says they’ll pay with their credit card (so they can get air miles, put the meal on their expense account, avoid an auto teller fee, and come out ahead by pocketing most of the tip). To my way of thinking that’s what quantum computation is like, sitting at the table of computation theory, scooping up the hard cash of informed speculation, and claiming it can render any process predictable ⎯ if you’ll just trust its credit card. Gödel’s Second Incompleteness Theorem We can formalize the proof of Gödel’s First Incompleteness Theorem within the formal system F itself. To do this, we represent the sentence “F is consistent” by a sentence Con(F) of the form “ ‘0=1’ is not a theorem of F.” By a mind-breaking feat of jumping out of the system, Gödel showed how one can in turn carry out this proof within the formal system F itself, to establish as a theorem a statement of the form “if Con(F) then G f ” As a consequence, Gödel draws a Second Incompleteness Theorem. Gödel’s Second Incompleteness Theorem. If F is a consistent finitely given formal system as powerful as arithmetic, then the sentence Con(F) is undecidable for F. Chaitin’s Proof In order to describe Chaitin’s proof, I need another definition relating to Turing machines that use the two symbols 0 and 1. Recall that in the Chapter One we discussed the notion of adopting a fixed enumeration of these Turing machines so that for an integer e, T e is a Turing machine. We can view T e as a kind of name for a string n by writing T e (0) = n to mean that the computation T e (0) halts and that when T e (0) halts, the binary expression for p. 105
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Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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