would tend to render it totally useless.' some kind of natural selection process being effective for producing approximately valid algorithms' since 'the slightest 'mutation' of an algorithm. He is concerned about 'the 'paradox' of teleportation' whereby copies could be made of people, and thinks 'that Searle's argument has considerable force to it, even if it is not altogether conclusive.' He also finds it 'very difficult to believe.

'Penrose gives many reasons why he is uncomfortable with computer-based AI. Here's just a brief quote from that rebuttal: I believe Robin Hanson wrote up an excellent rebuttal to Penrose's highly speculative use of Godel's theorem ( link). And hence, Penrose rejects the plausibility of Strong A.I., pending the discovery of something like quantum gravitational effects in a brain (which he asserts we wouldn't be able to engineer or harness for the A.I. Kritchman and Raz give a cool argument that basically claims that it all hinges on what you mean by 'to know the day of the exam ahead of time.' It turns out that if you mean 'be able to prove the exam won't be tomorrow,' then Godel's theorem actually lets you escape the backward induction and hence the seemingly paradoxical set-up doesn't have to be paradoxical at all.Īlso, a very very important place where Godel's theorem was invoked is in Roger Penrose's book 'The Emperor's New Mind.' Penrose's main argument is that brains cannot be given a fully reductionist explanation in terms of currently understood physics because there's just something about a human mathematician that can somehow 'see' the consistency of the mathematician's own 'formal system' which ought to be prevented by Godel's theorem if our brains were just formal systems in the sense of Turing machines / Church-Turing thesis. There is a very naive 'resolution' to this paradox using backward induction. In it, they take the classic example of an exam that will be given next week, but you won't be able to know the day of the exam ahead of time (it's also often re-phrased in terms of an execution next week, but you won't know the day of the execution this is how it is described at Wikipedia). NP idea and what its ramifications would be on human mathematics if P happened to actually be equal to NP.Īnother recent paper that uses Godel's theorems in a very technical way to address a philosophical problem is 'The Surprise Examination Paradox and the Second Incompleteness Theorem' by Kritchman and Raz.

In particular there is a great (but not well-known) letter mentioned in it from Godel to Von Neumann in which Godel essentially anticipates the whole P vs. It covers a wide range of topics in philosophy that have been dramatically changed not just by computability but also by complexity theory. I think you might like to read a great recent paper by Scott Aaronson called 'Why Philosophers Should Care About Computational Complexity'.

Gödel's Theorem by Torkel Franzén