More, still, on the unbearable inevitability of discretization

Here is my answer to Martin's third post on why he opposes the semantic web idea. Basically, this is the famous J.R. Lucas argument where Kurt Gödel's incompleteness theorem is said to suggest that an intelligent machine query engine (in our case, the semantic web) cannot possibly be trusted because it does not have the ability to formally prove its answers, and also that it does not have the inductive capacity to infer knowledge from disparate facts, like the (ideal) human brain does.

Human minds and any form of AI are necessarily inconsistent, which does not mean that they are unable to produce new verifiable statements from other previously verified statements. Therefore, Gödel's theorem cannot be used to infer that machines are intrinsically inferior to the human mind from an epistemological standpoint. I will not further delve into the specifics of the rebuttal because the page has been turned a long time ago and for that, you can read the very efficient Gödel's Incompleteness Theorem is Not an Obstacle to Artificial Intelligence, by Jeff Makey.

Here is a funny little side note. I found out that Mr Lucas' summary for his own gödelian argument was published in Thruth, An International, Inter-Disciplinary Journal of Christian Thought! A little later, I stumbled upon his ontological argument for the existence of a unique, christian god where he does not even mentions Gödel's similar effort. By this, I want to point out, again, that the anti-AI arguments are often devised as religious apologetic tools, which is always something to be very suspicious of.

Gödel against AI? Eeeeeeeeeeeeeeeeeeeeasy on the attitude there, boy! ;-)

What's funny with Gödel is that his work seems to allow any interpretation possible. In the AI debate, opponents from both sides have used Gödel's theorems to support their cause. Unfortunately, and correct me if I am wrong, Gödel never pronounced himself specifically on the subject of AI. Although Gödel was profoundly religious, he was not at all into touchy-feely humans-are-special-because-they-have-emotions kind of discourse. For instance, he was a member of the Vienna Circle, a group of intellectuals whose most significant contribution to epistemology was logical positivism (sometimes called neopositivism or logical empiricism). Wikipedia's definition of logical positivism reads as follows :

Logical positivism (later referred to as logical empiricism) holds that philosophy should aspire to the same sort of rigor as science. This means it should be able to provide strict criteria for judging sentences true, false and meaningless. The most characteristic claim of logical positivism asserts that statements are meaningful only insofar as they are verifiable, and that statements can be verified only in two (exclusive) ways: empirical statements, including scientific theories, which are verified by experiment and evidence; and Analytic truth, statements which are true or false by definition, and so are also meaningful.

Note that nowhere does the concept of proof appear. Yet truth (and verifiability), meaning (and meaningfulness) are the central objects of the positivist theory. Deliberately omitting any reference to the proof can be explained by the general belief de rigueur then, in mathematical and scientific circles, that proving something (anything at all, not only statements in maths) was exclusively about inducing one theorem from another using the Peano axioms. Science as a whole was therefore thought as directly modelizable solely within (natural) number theory system. Ontologically, it is correct because it was their definition of science, and their definition of provability. However, and that's what Gödel opposed, they wrongly concluded that a statement was true if and only if it could be proven. Gödel's only purpose with his theorem was that some instinctively true statements were indeed verifiable, but were not provable. He has shown that verifiability is weaker than provability (and formal decidability), yet verifiability produces meaningful statements. The Vienna Circle has purposely not given a formal definition to verifiability because it acknowledged that new induction mechanisms (outside formal logic) can always be discovered and could legitimately be used to decide whether a given statement is true or false or even to prove it, according to a more flexible proof theory. Now that the meaning of provability has indeed somewhat changed a lot, Gödel's argument does not have the reach it once had. Furthermore, our acceptance of softer proofs gives legitimacy to the idea that we could devise induction engines capable of superior intelligence than the human mind.

Logical positivism (as well as falsifiability and other epistemological views) is meant as an answer to dualism, which basically is that (discretizable) science will never significantly grasp the nature of the mind and of reality in general. Dualism, a mere reformulation of animism, is what I have been complaining in the second part of my first discretization post, and yet it always seems fashionably appropriate to bring it up.

What to do from now ?

Before going any further, I want to notify the readers of this blog that this is the last post in the so-called discretization series. The reason is that I now want to go a little further than just writing one-liners on a subject I consider to be one of the few most consequent aspects of contemporary science. I'm thinking of a semantic web manifesto of some kind. For efficiency's sake, I would like to take the conversation outside of the blogosphere for the time it takes to gather the necessary knowledge to produce a panorama on the semantic engineering of the Web as relevant and exhaustive as possible. I would also help setup a Team Montreal to participate to the semantic web challenge. I would like to gather people from both the academic and private sector worlds. What do you guys think? I think I have some very interesting ideas to share and implement.

BTW, trackback seems to be running fine again so don't hesitate.