On Thu, Jun 6, 2013 at 10:02 AM, Joe Niederberger <email@example.com>wrote:
<< SNIP >>
> The only question that remains difficult, is whether humans do things > mathematical things with their minds that are non-computational in fact or > principle. Kirby has mentioned more than once a formula for pi from > Ramanujan that he says is evidence that the mind can do non-computable > tings. Others have pointed to Gödelian Incompleteness as serving the same > purpose. Roger Penrose wrote on book on it, "The Emporer's New Mind". > > So, those are a couple of people "taking issue". I find neither of them > conclusive. Do you have any issues to raise? > > > Cheers, > Joe N >
Having heard Penrose (Sir Roger) live on a few occasions, I will add a little more here on what I take to be his claim: that humans are sometimes a source of intelligence which no computational theory or practice currently has a way to explain much less duplicate.
However it's important to add that a formal proof may later be constructed that substantiates a human contribution i.e. proves that it's true. In the case of Ramanujan's formulas, I think there may be proofs here and there for some, whereas others have only been empirically verified.
However, be that as it may, he did not arrive at these formulas by a proving process. This was considered one of his weaknesses, which is another way of saying a cause of frustration to other mathematicians who were unable to gain any insight into how he achieved his results.
I believe this is quite a common phenomenon in mathematics: that a flash of insight puts a dot on the map, which is then incorporated only later (if ever) into a structure of proved theorems.
In terms of just practice, I don't find the claim controversial i.e. humans routinely do things which no algorithm can do given our current state of computational ability.
In terms of theory, the claim that humans *in principle* have access to a Platonic Realm that machines never will: that's harder to argue, but is what I take to be Penrose's position, push come to shove.