On Fri, Jan 18, 2013 at 11:44 AM, George Corley <[log in to unmask]> wrote: > On Fri, Jan 18, 2013 at 4:29 AM, Mathieu Roy <[log in to unmask] > >wrote: > > > <<This reminds me of something I heard about for mathematics -- that we > > know > > for similar reasons that there are an infinite number of prime numbers > (you > > can always find a higher one), but no one has ever formally proven it.>> > > Well yes there is a proof. Let's say there is a finite number of primes. > > Take a list of all of them. Then multiply all of these prime together and > > let's call the result X. X+1 is another prime (because in order to be > > divisible by a prime - or any number other than 1 and X+1 - we would have > > had to add this prime to X, and 1 isn't a prime). X+1 wasn't on the list > > because it's bigger than all number on the list since we multiply them > and > > all of them were bigger than one. So we now have a prime number that > wasn't > > on the list of all prime numbers. Therefore the list did not have all > prime > > numbers. So there's an infinite amount of prime number. Reductio ad > > absurdum. > > > But my understanding was that that is not a formal mathematical proof (by > whatever rigid standard of logic that mathemeticians define a "proof"), but > Wikipedia disagrees with me, listing several different proofs of the > infinity of primes: http://en.wikipedia.org/wiki/Prime_number > > I dunno. This is not a math list, so I suppose this discussion doesn't > need to go any further. Maybe I'm thinking of some other obvious fact of > mathematics that can't be formally proven. > That's Goedel Incompleteness Theorem. Given a finite number of axioms and and derivation rules, you may always formulate a theorem that you may neither prove nor disprove using them. Back to linguistics, if you have a vocabulary and a grammar (which we may consider as a finite set of words and rules), there always exists a sentence for which a language processing machine won't be able to say whether it is grammatical or not. > <<That's hard to say. First, you would have to determine the lexicon you > > are generating from (and lexica of languages, I imagine, are arbitrarily > > large > > -- no doubt each individual's lexicon is finite, but it is unrealistic to > > expect to be able to catalog all words known by all speakers). My guess > > that generating a random string from any reasonably useful dictionary > will > > have a very low chance of providing a valid sentence, even allowing > > grammatical sentences that are semantically/pragmatically nonsense like > > "Colorless green ideas sleep furiously.">> > > Yes of course. I was thinking about using "Basic English". And I agree > that > > there's probably a very low chance that the sentence be valid. And maybe > > that would be more feasible to start with a lower number, such as 9 > instead > > of 15. > > > > I would go with 5. I'm not sure how long sentences tend to be in normal > speech, but I think that they're probably rather short. > > Still, I don't think you could calculate the probability with pure > mathematics. You'd need to generate a bunch of sentences, ask a native > speaker for grammaticality judgements, and do statistical analysis from > that. >