FeedShowKnowing.NET (Larry O'Brien)

Date: Sat, 18 May 2013 19:57:10 +0200

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A tangential post on an email list I subscribed to touched on a subject near and dear to my heart, but rather than go off-topic on that list, I thought I'd respond here:

[Roger Penrose] believes that the existence and human knowledge of the random real (uncalculable) numbers ... shows that computers will never be able to match human brains.... [M]athematicians know, ala K. Godel, mathematics is consistent even though they can't prove it.... Godel proved that mathematics can't prove its consistency (withour being inconsistent along the way). The fact that humans can make such a proof and a computer simply CANNOT is the R. Penrose basis for his belief. R. Penrose, also, has some biological arguments about axons to further bolster his belief.

This is a summation of the position that Roger Penrose laid out in a book called The Emperor's New Mind that was originally published in 1989. As luck would have it, I had just been hired as the Technical Editor of the premier magazine on artificial intelligence at the time, so not only did I have a chance to read the book closely, I had the privilege of being a conduit for some of the discussion regarding it (in those pre-Web days). (I almost pulled off a debate between Penrose, Searle, and Dennett, which would have been awesome.)

The OP slightly overstates Penrose's position on Godel's Proof: we know it is possible for a computer to construct Godel's Proof (10 Print "Theorem XI. Let k be any recursive consistent..."), the claim is that a computer cannot know Godel's Proof to be true. That mathematical certainty is a phenomenon that is not available to Turing machines (i.e., computers as we generally know them).

Penrose's claim is that mathematical certainty is a privileged phenomenon; that is, it's a real thing in the sense that mathematicians definitely experience and that it's something that we can be sure that a (Turing) machine cannot experience.

I've always felt Penrose's claim to be dubious.

Certainty, it seems to me, is just the phenomenon that arises from short-cutting previously-decided hard problems. Humans are very good at creating such short-cuts and their phenomenal impact is very strong (i.e., we feel them strongly). Penrose's mathematical certainty accords with mathematics, other people have certainty that accords with their religious beliefs.

Big deal.

Any machine operating in real time is going to massively rely on shortcuts, both built-in (e.g., objects persist when out of sight) and dynamic (e.g., "The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs").

When a mathematician looks at a proposition, they do not exhaustively explore it. Humans do not recurse deeply into a problem, they very quickly switch to meta-reasoning ("I assumed it was true and went down this path and I assumed it was false and went down that path. Now I see that the path's in front of me are similar to the path's I've already traveled, so instead of going down those paths, I'm certain of the proposition's validity.") I don't see why a computer cannot do the same thing, replacing "I'm certain" with "I'll create a shortcut." (Implementation note: When your stack gets so big, reason about the stack. And no, you can't define this with pure recursion. You have to have an arbitrary limit to the tactic. But you don't need many levels of meta-reasoning to be beyond human facility.)

Of course, all of us know there's a mile of difference between knowing of something and experiencing it: the phenomenon of reading a thermometer on a stove is fundamentally different than burning yourself. Similarly, a mathematician would say that "certainty" is not the same phenomenon as "pattern-match and pop the stack."

But this is a long-discussed problem: it's even called "the hard problem of consciousness." My problem with Penrose is that I just don't see what his arguments brings to the table beyond this existing problem. I don't see how he makes the hard problem any harder.

The easiest way to solve the problem is to say that subjective phenomena exist outside the realm of classical mechanics, that there is a "ghost in the machine." Penrose posited that nanoscale structures in the brain might be a channel by which quantum-mechanical mechanisms are amplified to become large enough to interact with neurons. In other words, our consciousness might be mechanical, but dependent on an unknown aspect of quantum mechanics.

Well, maybe.

I've never heard any evidence that nanoscale structures are important to consciousness. They exist and are complex, so to my evolutionist mind, they are probably not accidental (although they may be). But whether that function is structural or related to mental functioning is, I think, an open question.

In short, I've always felt that Penrose, while undoubtedly brilliant, didn't really advance the debate greatly.

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