Hi Gareth,
Interesting post.
Usefulness and relevance as subjective and relative. Maybe we have overrated objectivity and the grail of the absolute. I think so. Objectivity is another one of those things that everyone seems to firmly believe is a good thing. At the same time, no one seems to have seriously considered alternatives to it, so I take their convictions about it with a grain of salt or two. I've noticed that there is little need for "objectivity" unless I'm trying to convince people who don't trust what I say. If I'm not trying to convince someone, objectivity isn't an issue. If they trust what I'm saying, objectivity isn't an issue. Couples don't usually start couples counseling just because they have issues. They bring "objective" third parties into the picture when trust becomes an issue. Would we have ever resorted to "objectivity" if we never lied to each other and never tried to make each other do what we don't want to do? Hypothetically speaking.
I actually didn't imply that philosophy, logic, math, etc., are pointless. Far from it. I just take issue with making them be-alls and end-alls. Math and science seem to have turned the corner in the 20th Century from noble human undertakings to become the closest things that we have to truly global religions. We believe in them, and our faith in them extends far beyond their capabilities. I take issue with any thinking that extends beyond what its foundations can support.
I found myself agreeing with you about Godel and his theorem. My understanding of the theorem is 2nd-hand, (tried reading that, too, but I ain't no mathematician,) so I defer to Ray Kurzweil's description, which jives with other descriptions I've read:
Early in the twentieth century mathematicians Alfred North Whitehead and
Bertrand Russell published their seminal work, Principia Mathematica, which
sought to determine axioms that could serve as the basis for all of mathematics.
However, they were unable to prove conclusively that an axiomatic system
that can generate the natural numbers (the positive integers or counting
numbers) would not give rise to contradictions. It was assumed that such a
proof would be found sooner or later, but in the 1930s a young Czech mathematician,
Kurt Godel, stunned the mathematical world by proving that within
such a system there inevitably exist propositions that can be neither proved
nor disproved. It was later shown that such unprovable propositions are as
common as provable ones. Godel's incompleteness theorem, which is fundamentally
a proof demonstrating that there are definite limits to what logic,
mathematics, and by extension computation can do, has been called the most
important in all mathematics, and its implications are still being debated.
The Singularity Is Near, p. 453
Then I got to thinking (always a risky proposition.) If we generalized math to any system of thought and then generously attributed the rigor of math to all systems of thought, (let's just say we're now in the year 5010 and can do such things,) we could have it one of two ways: 1) there are propositions in all of our systems of thought which can neither be proved or disproved or, 2) we limit our systems of thought to those propositions which can be proved or disproved, thus precluding the contradictions that Russell and Whitehead struggled to eliminate, and
thereby render the systems irrelevant (e.g., incapable of generating the natural numbers.) We can't have our cake and eat it, too.
So what is the difference between Godel's unprovable propositions and the notion of basic beliefs? see
http://en.wikipedia.org/wiki/Basic_belief Not much, I'd say.
I think that our obsession with getting rid of those pesky unprovable propositions reveals a couple of things. First, that we are really uncomfortable with the fact that our cognition has limitations, of which Godel's pesky propositions and basic beliefs are functions. Second, that we imbue artifacts of cognition with a sense of reality that takes them outside the domain of the cognitive. Philosophy is full of examples of logic that makes no sense unless you first attribute metaphysical reality to cognitive artifacts. Dualism is a great example. Just because we can't understand reality without thinking dualistically doesn't mean that reality is dualistic. I also think that another example is viewing mathematics as an attribute of reality rather than as a way of describing certain attributes of reality. There are always two things: what is and what we say about it, (unless we say nothing.) Somehow, after we say something about it, we are sorely tempted to believe that what we said is on an existential par with what we said it about, and we often go so far as to let what we said take precedence. Words don't transform into sticks and stones, even when we pick up sticks and stones.
So, back to Godel, I'd say that his theorem was very important for certain types of people, i.e., people who need that type of proof in order to accept the truth of something. Aside from them, Godel's proof was overkill. I'm glad for it, because it helps when I'm talking to the type of people who value such things. I'm thinking, for example, of people who require heavy math for what they are doing, and so, put a lot of stock in it. More interestingly, I'm also thinking about people who can't do the heavy math but who think that it's important that others who can do the math claim that Godel proved something. That brings me to the irony of modern epistemology.
We are so far past the point where any one person could master enough of what we need to know in order to do what we need to do that I just laugh when "rational" people object to the notion of faith. I laugh. Because I have read Kurzweil or the Encyclopedia Britannica or a zillion other sources and saw that everyone claims that Godel proved his incompleteness theorem, in what way did I conclude that the claim is true that is different than the way anyone concludes that gods exist or that life on earth was seeded by aliens? When I hear that Susskind won the bet with Hawking about black hole radiation, how do I know that he was right or that either one of them knows what they are talking about? Do I really have enough knowledge to judge?
We seem to think that phenomena like paranormal theories, conspiracy theories, and religions are functions of faults in the intelligence of the theorists and their adherents. Maybe they are rather indications of faults in our current versions of "rationality." When we have to take the words of so many people in order to supply our rationality with content, I'd argue that we are far more credulous and believing now than ever. And if our sanctioned, "rational" versions of knowledge were so compellingly comprehensive and satisfying, wouldn't the "irrational" versions start disappearing? I see a huge gap and have for a long time. Proponents of "rationality" just seem to find ways of dismissing the validity of the gap. I don't think that we'll ever get rid of religious crazies, for example, by telling them that the reasons that they turned to religion are all wrong. I think that maybe we can have an effect if we understand what they were looking for when they turned to religion and provide them with ways of understanding things that will satisfy them without making them crazy. And to listen to them, it's the very hubris of our "rationality" that is making them crazy, not their religions. Who is to say that they don't have a point?
As you can tell, I love this stuff. Please feel free to take me to task. Everything is a straw man just waiting to be knocked down. Truth is just the straw men that nobody managed to knock down.