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Memories of Kurt Gödel
- cicero 9y agoThat's a beautiful story. I've been fascinated by Gödel since I read Gödel, Escher, Bach by Douglas Hofstadter back in the 1980s. I'm not a mathematician, so I can't plumb the depths of his work, but I gain a little more insight by reading articles like this one.
- ianai 9y agoI remember reading the proof of his incompleteness theorem (or the portion of it that I had time to between classes). It's surprisingly approachable. AFAIR, he came up with a way to encode proofs as symbol sequences. Proofs that actually exist have finite representations. He then proved the theorem with that somehow. (Granted, I read it over a decade ago...)
- Agentlien 9y agoYou're thinking of Gödel numbering which is an elegant technique used extensively throughout that proof. He uses it to map logical symbols, statements and entire proofs to natural numbers and then proceeds to prove things about those statements and proofs by referring to them via their Gödel numbers.
- deleted 9y ago[deleted]
- yequalsx 9y agoI know this post is aimed at laymen but this sentence is not correct. "A bit more precisely, the Incompleteness Theorem shows that human beings can never formulate a correct and complete description of the set of natural numbers, {0, 1, 2, 3, . . .}." The second order Peano Axioms are categorical and thus, up to isomorphism, the only model for this axiom system are the Natural numbers {0, 1, 2, 3, ...}. This is a complete system. We can't happen is a recursively enumerable axiomatic description of the Natural numbers that is complete. Another way to get a complete description of the Natural numbers is to take the collection of all true statements of the Natural numbers and make that our axiomatic system. It's just not a useful axiomatic system but it is a complete description of the Natural numbers.
- monet 9y agoThanks for pointing this out. Two questions: (1) Are there any resources you'd recommend reading to learn more about this? (2) If you are familiar, could you restate the Incompleteness theorem in your own words? Just looking to learn more here :)
- gfody 9y agothere's a tiny book called Godel's Proof that is both short and easy to read. I picked it up in my quest to read everything Hofstadter (he wrote the foreword) and thought it made it pretty straight forward. any axiomatic system that's flexible/powerful enough to make some sort of statement about itself will succumb to this method for constructing a paradox - and so all such systems are either inconsistent (paradox is possible) or incomplete (lacks sufficient flexibility to make a statement about itself).
- posterboy 9y agoso by contraction, self-reference is paradox? That's certainly true for the liar's paradox. But that shouldn't be mistaken for the actual statement, that some unprovable propositions follow from incomplete axioms and that the axioms can't be completed.
- nicky0 9y agoYou're on a dangerous road. I asked these questions once, got a couple of books. Decided I needed to enrol on a couple of courses... 7 years later I had completed a math degree. :)
- smallhands 9y agoreally i will love love to hear your story.i have feeling that there is more to it ! wow , i am impressed.please what is your email?
- jimhefferon 9y agoLots of reading are available to technical people who are willing. People elsewhere in this thread have mentioned Nagel and Newman, and also Smullyan. Here are two more: 1) Smith's Introduction to Godel's Theorems http://www.logicmatters.net/igt/ http://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy. 2) Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse http://www.ams.org/notices/200703/rev-raatikainen.pdf http://www.ams.org/notices/200703/rev-raatikainen.pdf is enlightening in a different way. I'll also mention a favorite author around here http://www.scottaaronson.com/blog/?p=710 http://www.scottaaronson.com/blog/?p=710 .
- beautifulfreak 9y agoIt reminds me of this article in the New Yorker about Gödel's and Einstein's talks. http://www.newyorker.com/magazine/2005/02/28/time-bandits-2 http://www.newyorker.com/magazine/2005/02/28/time-bandits-2 "Although other members of the institute found the gloomy logician baffling and unapproachable, Einstein told people that he went to his office 'just to have the privilege of walking home with Kurt Gödel.'"
- galaxyLogic 9y agoI wonder what he would have come up with had he had access to computers
- djmips 9y agoHe was right there at the Institute for Advanced Study (IAS) when they built the seminal IAS machine whose plans were distributed widely to help germinate computing around the world.
- marktangotango 9y agoI wonder what is meant by this question. In a very naive sense of the question, the type of mathematics Gödel worked on (foundational?) wasn't really computation oriented or even formulaic, so the ability to perform lots of calculations very quickly probably wasn't of much use to him. Indeed, computers where available to him in later years. A more interesting interpretation of the question (to me) is what if he had access to computers used as mind amplification devices. For example; such as by using Mathmatica or Maple to explore and visualize theorems and things. I'd imagine the benefit of this activity for someone like Gödel would be "not much". Computations and simulation have inherit limitations; precision and rounding errors for scientific computation, and the fact they can only model what we can imagine for another. These people such as Gödel, Neumann, and their ilk, new this, they begat the era of computation we have today, and all the limitations that involved. Neumann in particular was famous for, when presented with your problem, would tell how to solve it. What's new today that they may not have foreseen is the vast level of internetworking and human communication that arose from ubiquitous presence of computers and networks. Something that strikes me about the present article is the fact that Gödel, keen to see Rucker before he knew him, was not so keen to converse with him after their first encounter. One might think Gödel was not too impressed with Rucker, maybe found him boring and dull for example.
- andyjohnson0 9y ago"What's new today that they may not have foreseen is the vast level of internetworking and human communication that arose from ubiquitous presence of computers and networks." In an alternate universe, Godel published his proofs in a paper on arXiv in 2002. However it was largely overlooked or dismissed as the work of a crank, so he went back to his day job at MSR. A casual reference to the paper in a comment on on Math Overflow five years later led to wider discussion and eventually to Scott Aaronson publicising it on his blog. Within a year the two theorems were accepted by the global mathematics community. Since then, numerous blogs, subreddits, and Facebook posts have challenged the legitimacy of the proofs or claimed prior credit. Speculation persists on some parts of the internet that GCHQ or the NSA knew of the theorems as early as the 1950s.
- vbuwivbiu 9y ago“The illusion of the passage of time arises from the confusing of the given with the real. Passage of time arises because we think of occupying different realities. In fact, we occupy only different givens. There is only one reality.” Is he saying that our brains exist over all time simultaneously but they "give" us a sequence of instants from which we perceive the illusion of passage of time ?
- gfody 9y agohe's not making a statement about brains, more the nature of reality. it's a perspective that's gaining traction lately with the dataist/reality-is-information line of reasoning.
- vbuwivbiu 9y agoah he's distinguishing between ontology and epistemology ? He's saying there's only one reality but our instantaneous measurements of it (the given - what we perceive with our brains) form the illusion of time
- nis10 9y agoAghhh who the ... picked the acid background of this page? cool
- arketyp 9y agoLooks like the Rock Paper Scissor cellular automaton: https://www.youtube.com/watch?v=lt9ihcg-bZc https://www.youtube.com/watch?v=lt9ihcg-bZc
- chx 9y agoGodel's incompleteness theorems are explored by many of Raymond Smullyan's books in a way that an inquiring high school student can understand it.
- westoncb 9y agoI'm surprised no one has mentioned anything about Rudy Rucker, whose blog this is from. Some credit him with authoring the first cyber punk novel, which was titled 'Software.' It's the first in his 'Ware Tetralogy' (Software, Wetware, Realware, Freeware) which I'd certainly recommend checking out.
- mirimir 9y agoYes, and I highly recommend Turing & Burroughs :)
- Synaesthesia 9y ago“Spaceland” was also a hoot. It’s his take on the classic “Flatland”.
- pge 9y agoFor further reading on Gödel, Rebecca Goldstein's Incompleteness is a great book. It is a biography of Gödel but also places his work in the context of the philosophical debates of the time.
- mirimir 9y agoSo does Gödel's incompleteness theorem apply to scientific knowledge? I mean, as I understand it, he proved that any useful logical system contains statements that can neither be proved true nor false. But does that apply to quantum mechanics, for example? Are there self-referential statements in QM?
- yaseer 9y agoIt applies to any formal system powerful enough to encode the properties of the natural numbers. Quantum Mechanics, as a formal system can encode the natural numbers, and thus can form a substrate for self referential Gödel encoding. The same principles would apply to a quantum system, and any other capable of encoding natural numbers.
- mirimir 9y agoWell, scientific hypotheses are tested through observations or, ideally, experiments. Maybe QM could encode arbitrary statements about natural numbers. But I can't imagine how one could test them through observations or experiments.
- kazagistar 9y agoThe question wasn't about reality, but about quantum physics as a formal system, which is very much a formal mathematical system that can exist separate of hypothesis testing and whatnot. The point is that as soon as we have any finte definition of a model for reality, incompleteness kicks in.
- fundabulousrIII 9y agoHe was quite insane. Dopamine is a hell of a drug.