Knowing/Proving What We Cannot Know/Prove

It can be asked how we can know or prove what we can't know or prove. Epistemologists, for example, have come at this question from different angles.

Take David Lewis. He had a notion of “proper ignoring” (shown in his paper 'Elusive Knowledge' – 1996) in which he more or less says that some propositions either can't be known to be true (or false) or must be ignored in order to get on with those propositions we do need to know at a particular juncture. Or, in Wittgenstein's case, those things we know we can't know (or at least question) act as hinges on which the rest of our questioning and endeavours hang. (Having said that, this is more a question of not attempting to know rather than one of the impossibility of knowing.)

Kurt Godel

Kurt Godel did something similar with his theorems. However, instead of knowledge, we can now talk of proof.

Godel proved that certain statements within a system can't be proven even though they can still be taken to be true. Godel knew that we couldn't prove these statements. Still, as I said, such statements are still taken to be true; just not provably true.

Nonetheless, Godel's proof of a lack of proof did go against what certain mathematicians – or even the majority – believed at the time (e.g., the mathematical constructivists).

More recently, John Horgan quotes Gregory Chaitin (a mathematician and computer scientist at IBM) saying the following:

“Normally you assume that if people think something is true, it's true for a reason. In mathematics a reason is called a proof, and the job of a mathematician is to find the proof, the reasons, deductions from axioms or accepted principles.” (230)

However, Chaitin goes on to say:

“Now what if I found is mathematical truths that are true for no reason at all. They are true accidentally or at random....”

The constructivists certainly believed that truth - as well as numbers themselves - came along (as it were) with proof. A statement becomes true only when it's proved to be true. It can't be true for any other reason. So, as far as the constructivists were concerned, statements such as “mathematical truths that are true for no reason at all” simply don't make sense. It's almost equivalent to saying: “I think that Santa exists; though I can give no reason for believing that he exists.”

The same can be said of mathematical statements being true “accidentally or at random” - on a certain picture of maths, such things are quite literally impossible (or even meaningless).

Yet, of course, Godel proved otherwise. He proved that certain true statements can't be proved to be true; yet they're still true. The question remains: How did Godel know that these statements were true?

More specifically, is it possible for someone who doesn't know the maths to get his head around the the notion of unprovable - yet true - mathematical statements/equations? (It's here that Roger Penrose talks of 'seeing' their truth. Others speak of 'intuition'.)

Alan Turing

Whereas Godel was concerned with unprovable truths, Alan Turing discovered unsolvable problems. Ray Kurzweil (mentioned in the last post) writes:

“These are problems that are well defined with unique answers that can be shown to exist, but that we we can also prove can never be computed by any Turing machine – that is to say, by any machine...” (187)

Admittedly, this is a rough explanation of what Turing advanced.

First of all, it's hard to get a grip of what's meant by 'problems' here. I assume they're mathematical problems which the computer fails to prove. So here again computers are backing-up what Godel had already discovered in his “incompleteness theorem” of 1931. (Having said that, the Turing machine, at least in the Turing test, isn't all about mathematical questions or solutions.)

The words “unique answers that can be shown to exist, but that we can also prove can never be computed by any Turing machine” again show the Godelian angle. Put simply, it's about proving that X or (X's) can't be proved. You have a proof that X can't be prove.

Despite saying that, I still have a problem with the opening clause which says that “unique answers that can be shown to exist”. So those unique answers are known, though we also know that they can never be computed by any Turing machine. Can we know of unique answers if they haven't been computed? If the answer is 'yes', then how is that the case? Perhaps “showing” the existence of “unique answers” isn't the same as knowing (not even proving) the unique answers. Indeed the word “showing” (rather than “knowing” or “proving”) seems to hint at intuition or “mathematical insight” again.

Following on from some of the things which have just been said, it's also hard to understand what's meant by saying that “Turing showed that there as many unsolvable problems as solvable ones”. One's immediate reaction to a statement like that is: How could Turing have possibly known that? The only answer I give is the mathematical truism that because all infinities are equal (in that they're all, well, infinite), then if solvable mathematical problems are at least potentially infinite in number, then the same must also apply to unsolvable mathematical problems.*

*) Of course, according to Georg Cantor and others, infinite sets aren't all “equal” (i.e., they're "larger" or "smaller" than each other) even though they're all, well, infinite – they're only equal in, as it were, infiniteness.