That’s almost the definition of “thought experiment”. The snake eyes killer doomsday argument seems a bit fishy in that the expected number of victims is infinite: the chance that the nth group is kidnapped is (35/36) ^ (n – 1), so the chance that the nth group is the one that gets killed (not given that they were kidnapped) is 1/36 * (35/36)^(n-1). At this point it is customary to compare that number with the number of atoms in the observable universe, but frankly that seems like it would be belaboring the point. B := there is a counterexample to T How would you even do it? And Calculus III (multivariable calculus) is actually complex analysis made difficult. In any case, the “obvious” ones still seem to give a lot of people trouble, so it doesn’t seem to really be so all-or-nothing between obvious/worthless and opaque/worthless; there does seem to be that middle ground of difficult-but-understandable. Adams has some pretty good stuff – I liked his observations about “pivoting” and startup culture, for example – but for the most part he does come across as a decidedly weak-ass poor man’s SSC. What Gödel’s theorem does is it basically comes up with a statement of the form “S := ‘P does not know that S is true'”. So I’m more interested in things like the “Hydra problem in Peano arithmetic” example, where a seemingly straightforward problem turns out to be non-solvable given the axioms. Of course Scott’s book doesn’t make sense on the first reading. Or try to taboo “intelligence”. I would rate the Bohm interpretation and the transactional interpretation as still in the running. Just for the record, I don’t deny that Turing’s proof of the undecidability of the Halting Problem is legitimate. Geoff Pullum is awesome and I have no idea why I stopped reading Language Log. “the idea of emergence in the context of computation is problematic”. I think that I’ve already responded to this. Now, of course, “it seems unlikely” isn’t a valid proof. People come to me and ask: what is at the 13th place in this string, 0 or 1? If you can’t aprioritistically demonstrate that a computer system should have genuine semantics, because it us constructed to, and you also can’t demonstrate it empirically…it looks like you can’t demonstrate it. It lists all the countries of the world and their capitals, and is meant to be so comprehensive that a reader could use it to plot the most efficient journey from Timbuktu to Kalamazoo, taking into account tolls, weather, and levels of infrastructure development along the way. He understands the material backwards and forwards. You can look at Deep Blue, the Robbins conjecture, Google, most recently Watson – and say that’s not really AI. Other popular ideas about the universe being ‘digital’ and a ‘computer’ or composed of ‘cellular automata’ (aka Wolfram) or any theory where ‘information’ or ‘computation’ is taken to be fundamental I would also lump in the ‘wrong’ basket. In fact I pointed out that solving the halting problem must be at least as hard as finding a generic method for proving any theorem which can be expressed as a Turing machine which does a brute-force search for a counter-example. If they’ll all get released, you should be optimistic; if they’ll all get killed, you should be pessimistic. By any reasonable way of accounting just about all of them aren’t “teachable”. Completely agree about avoiding epsilon-delta whenever possible in first calculus courses. computational limited with randomness) verifier. How difficult the book is depends a lot on how much you already know, and I’d be hesitant to recommend it to anyone who hasn’t read Sipser. I hear a bunch of people telling me Bayesianism isn’t everything, it’s the only thing – and another bunch of people telling me it’s one useful tool in an entire bag of them. But eventually I realized that if I wanted to read Democritus the way it was supposed to be read, with full or even decent understanding, it would be a multi-year project, a page a day or worse, with my gains fading away a few days after I made them into a cloud of similar-looking formulae and three-letter abbreviations. This book is not that. Wang Tiles are interesting indeed; a nice example of a case where you might be given a seemingly straightforward puzzle without being aware that solving it would be equivalent to solving e.g. (Context – I’m seriously considering re-entering academia in the near future after screwing it up it hard the first time over a decade ago. Case2: Each new room is a random drawing. Well. What is it about bio-brains that make it ‘wet’, and the same thing happening in silicon not ‘wet’? I…I suddenly understand what the halting problem is. Of course it may take years of study for the subroutines to be installed into your brain via practice! Not magic. There are also intermediate Turing dregrees, i.e. And the Lord asked “Who told you this?” And Eve said “It was the serpent who bade me compute, for he told me if I did this I would be as God, knowing subgraph isomorphism and 3SAT.” Then the Lord cast them forth from the Garden, because He was Information Theoretic God and preventing people from screwing with complexity classes is like His entire shtick. Read honest and unbiased product reviews from our users. So our only method to outsmart all the clever algorithms, including those too clever for our brains, is to say – oh, this problem is so smart that it already knows something about any one of you. Aaronson tries to tie his own specialty, computational complexity theory, into all of this. I actually thought the same thing when I read it. Tegmark is right. So as a Bayesian, you should update *now*. I feel the main way your analogy breaks down is the “everybody would speculate maybe we finally found a real-world case” part. Aaronson goes one level deeper than most of the other popular science writers I know and speculates on why the laws of physics are the way they are. But all the other problems that have been proven non-computable are inevitably like this too, it’s just that their *rich structure* is not so obvious and we needed to work hard to uncover it. Wait a minute, this thought experiment isn’t about consciousness at all – its about intelligence/understanding which are pretty clearly continuums. Reality is that if I follow the procedure “for each apple in my basket, take out an apple from the barrel and put it in my basket”, then my basket now has twice as many apples. Now this kind of talk drives AI researchers up a wall. By the conditions above, P’ will run forever if it halts, or halt if it runs forever. If Hortas or Chenjesu (yes, fictional; take them as hypothetical evolved silicon-based sapients) were to realize they could simulate their semiconductor brains with organic chemical-based neurons, their following this line of reasoning would lead them to object that there’s no current and so it couldn’t possibly be conscious. number theory is applied math? Pity, because that’s exactly the kind of problem I was talking about. If you measure a microprocessors prefer,ance, it is what you measured it to be. was this a course in a cs department that in another school would be called “discrete math”? time travel. It’s still a high bar. I just wanted to make the point that the approach mentioned by Scott in his review of Scott’s book, of saying “if we could do X then it would allow us to do Y, therefore X must be at least as hard as Y” is often a more fruitful way of analysing a particular problem, than the Gödel/Turing approach which proves that any formal system must contain some undecidable theorems but doesn’t really give you any help in determining if a particular conjecture is likely to fall in that class. When you put it that way, maybe it’s not so hard to imagine this enormous Chinese-speaking entity that we’ve brought into being might have something we’d be prepared to call understanding or insight.