Loading... Gödel, Escher, Bach: An Eternal Golden Braid (original 1979; edition 1999)by Douglas R. Hofstadter
Work detailsGödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter (1979)
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Sign up for LibraryThing to find out whether you'll like this book. It's beginning to show its age, and the dialogs are less charming than I remember, but still it's quite combination of ideas skillfully blended. ( ) いや，かれの御手は広く開かれて，御心のままに，惜し・なく与えられる Never mind the Escher and Bach stuff; that’s just window dressing. This book is about Godel’s Theorem. And wow, what a book. Imagine a glorious future in which, by means of magic and genetic engineering, the human species is transformed into a better, smarter, faster, more beautiful, more creative, more moral, stronger, happier species, a more alive species. We make Elysium, then we live in the Elysium we’ve created. In this Arcadia, this Heaven, this Eden, this Platonic Form of the world animated and electrified by benevolent intelligence, you walk across grassy fields and you see the whole thing, The Dream: Everyone is wearing flowing white robes. (Why? Just because.) Over there athletic people engage in athletic contests, their goodnatured competition embodying grace, fluidity, and the confidence of a welldisciplined, healthy body. Over here, mathematicians use sticks to draw in the dirt on a river bank, proving astoundingly beautiful and useful new theorems. In another direction a young man or woman lounges, back against a tree, releasing sweet strains of melody into the air by means of some sort of elegant string instrument. Are you with me? Okay. In that universe, every nonfiction book is this good. What’s it about? It’s about, principally, Godel’s Theorem. The other stuff, at least in the first part (Escher, Bach, etc.), is just addons. Godel’s Theorem is often mischaracterized as “disproving all of mathematics!” or some similar nonsense. No. It says something about formal mathematical systems, systems of clearly stated axioms with clearly stated rules of inference for deriving implications of the axioms. The theorem essentially says that any formal system sophisticated enough to be used for number theory  reasoning about integers  either has internal inconsistencies or is unable to prove every truth in number theory. This does not “Undercut all of mathematics” or whatever. It simply means that a consistent formalistic approach to mathematics can never derive all mathematical truths. There are some truths that can only be proven in other ways. Indeed, Godel shows how to prove some of those truths by reasoning outside formal systems! To prove it, Godel had the profound insight that any formal system can be reinterpreted as a set of numbers and arithmetical operations on them, so formal number theory talks about itself! This is so cool. E.g., suppose your formal system has the symbol string x#@^&?G!y. (This might mean, say, “x is the largest number in the prime factorization of y.”) We also have a rule that allows us to derive x<=y (x is not greater than y) from the first string. But we also can interpret x as 5, # as 0, @ as 2, and so on, so the initial symbol string can be interpreted as a number. And so the second string is a number that we can derive from the first. So the rules of inference in this interpretation are arithmetic operations on numbers. Thus we can apply mathematical reasoning to the system and derive conclusions about the symbol strings it will generate and those that it won’t generate. A simplified analogy: Suppose that we can prove  by reasoning outside the formal system  that the system will never produce a string whose number is prime. What Godel proved, in this analogy, was there is always a symbol string that asserts “N is a prime number” (in the first interpretation) whose number was N (in the second interpretation). Thus, if the statement is true, the formal system will never prove it! (It is possible to verify that a welldesigned system will never "prove" a false statement, so you can avoid that problem.) In fact, no only do such truebutformallyunprovable statements exist, in any formal system complex enough to be useful, but an infinity of them exists! It was the idea of reinterpreting the symbols as numbers that was Godel’s real stroke of freakin’ genius. The theorem is based on that. Anyway: The next time someone tells you, “Godel’s Theorem proves that all mathematics is invalid,” or whatever, just give them a wedgie and move on. All it proves is that a certain approach to mathematics cannot prove everything. Which, unless you had unrealistic ambitions for it in the first place, is not that surprising. Notes whilst reading it. The first part on logic is a bit tough going in places. Take home message for me was: is order innate (normal) or depending on one's perspective? i.e. will supposed other worlds be able to make sense of a the record man sent into space. Will Bach be order for them too, or will they see order in Cage's creations? A bit dated as the author uses examples of records and jukeboxes to illustrate his stories. I get it; does a teenager? This is certainly the greatest book of popular science I've encountered or heard of yetit's accessible, engaging, playful, but also very deep and original in its analysis. It gives brief intellectual history where necessary, and repeats the same arguments in many surprisingly different ways for comprehension. But its ultimate ignorance/dismissal of the social world, assertion of objective meaning, and computational theory of mind force me to give it a lower rating. no reviews  add a review
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