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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.
>