Meta Math

April 13, 2006 3:32 AM

Gregory Chaitin's Meta Math! The Quest For Omega
"Okay, what I was able to find, or construct, is a funny area of pure mathematics where things are true for no reason, they're true by accident... It's a place where God plays dice with mathematical truth. It consists of mathematical facts which are so delicately balanced between being true or false that we're never going to know, and so you might as well toss a coin." From Paradoxes of Randomness.
"In my opinion, Omega suggests that even though maths and physics are different, perhaps they are not as different as most people think. To put it bluntly, if the incompleteness phenomenon discovered by Gödel in 1931 is really serious — and I believe that Turing's work and my own work suggest that incompleteness is much more serious than people think — then perhaps mathematics should be pursued somewhat more in the spirit of experimental science rather than always demanding proofs for everything." From Omega and why maths has no Theory Of Everythings.
[previously, see also, via]
posted by MetaMonkey (17 comments total)

This leads to my first result on the limits of axiomatic reasoning, namely that most numbers are uninteresting or random, but we can never be sure, we can never prove it, in individual cases.

I think Three Dog Night surely proved that one is the loneliest number that you'll ever do.
posted by three blind mice at 3:42 AM on April 13, 2006

Dammit, it's "math" not "maths". Otherwise this is some fascinating stuff that takes me back to my university days, but reminds me of why I hated the theoretical stuff so dang much. :)
posted by antifuse at 4:17 AM on April 13, 2006

Why isn't it "maths", since there are a number of different areas of mathematics. However, as an American, it doesn't sit well with me, it still makes sense.
posted by Goofyy at 4:22 AM on April 13, 2006

His theory of Omega is very interesting but the jump to empiricism at the end is strange, since he offers no arguments in support of it. His attitude seems to be 'some things can't be proved, so why bother trying to prove anything hard?'. He seems to think that the existence of Omega poisons the entire mathematical effort, but doesn't explain why.
posted by unSane at 4:25 AM on April 13, 2006

I say 'maths' not 'math' because it's short for 'mathematics'. We study 'mathematics', not 'mathematic'. But who cares if your variable is called X or Y?
posted by unSane at 4:26 AM on April 13, 2006

Maths is UK, math is American. In the immortal words of Van Morrison, It ain't why why why why why why why why why why, it just is.
posted by languagehat at 5:37 AM on April 13, 2006

Dammit, 'math' sounds wrong to me and 'maths' sounds wrong to you. Deal with it.

I think the fact that we're all discussing the word here means that the clever people who can understand this are all asleep.
posted by altolinguistic at 5:49 AM on April 13, 2006

I don't know that there's so much difference between today's math(s) world and the one he proposes. Recently I read the paper "Primes in P" (a proof of a polynomial-time primality test---most of it was over my head) and they give one result that is proven, and another better result that is proven if some particular conjecture is true:

We give a deterministic, O∼(log^(15/2) n) time algorithm for testing if a number is prime. Heuristically, our algorithm does better: under a widely believed conjecture on the density of Sophie Germain primes (primes p such that 2p + 1 is also prime), the algorithm takes only O∼(log^6 n) steps.

In my experience, this kind of thing is in fact common in mathematical papers.

In the PDF from the second link, there's this item (near the beginning of the conclusion) which I offer without further comment:

The topic of monotheism and polytheism is also germane here. I love the complexity and sophistication of South Indian vegetarian cuisine, and
my home is decorated with Indian sculpture and fabrics—and much more. I admired Peter Brook’s Mahabharata, which clearly brings out the immense philosophical depth of this epic.

But my intellectual personality is resolutely monotheistic. Why do I say that my personality is monotheistic? [...]

posted by jepler at 5:54 AM on April 13, 2006

my brain hurts!
posted by shmegegge at 7:16 AM on April 13, 2006

If you like this, don't hesitate to start digging into the CDMTCS research reports. Incredible stuff, fun mathematics.
posted by sonofsamiam at 7:31 AM on April 13, 2006

Neither the linked page nor the alternate page displays pi and the square root symbol properly. I see lowercase p and O with an umlaut on both. What am I missing? (Mozilla 1.7.2.)

posted by jfuller at 7:44 AM on April 13, 2006

His theory of Omega is very interesting but the jump to empiricism at the end is strange, since he offers no arguments in support of it. His attitude seems to be 'some things can't be proved, so why bother trying to prove anything hard?'. He seems to think that the existence of Omega poisons the
entire mathematical effort, but doesn't explain why.

What he means is, there are some mathematical problems that we will NEVER be able to prove- they are unprovable. He is not saying it is not worth trying to prove anything hard, he is saying some hard things are unprovable and we may just have to take them as a given.

For instance, in computer science, the question does P=NP is hugely important to many, many ideas. No one has been able to prove if P is or is not equalt to NP for the past 30 years. Even though it is not proven, it is widely assumed that P does not equal NP and many practical and important works have been based on this assumption. Chaitin is saying that we math could benefit from making some assumptions like this, especially if the problem ends up being unprovable.
posted by gus at 7:52 AM on April 13, 2006

Fun read. Reminds me of Hofstedter with the heavy math couched in the "I smoke pot too" lingo.
posted by Joseph Gurl at 7:57 AM on April 13, 2006

Uninteresting numbers, unprovable theories... really daunting beliefs this guy has.

Still interesting stuff. Definitely on the order of playful mathematics - trying to glean conceptual ideas rather than concrete proofs.
posted by destro at 8:31 AM on April 13, 2006

I spent a few days with Chatin back in 1996 or so, when he was developing a lot of this theory. If you have an undergraduate math background it's entirely accessible. A very constructive approach to logic and computability and quite interesting for it.
posted by Nelson at 11:44 AM on April 13, 2006

The sort of playful self-referential "paradoxes" he talks about are indeed fun, but the supposition that they have enough philosophical weight to provide the foundation for an ontology (or anti-ontology) cannot, I think, be taken for granted.

Many so-called paradoxes depend on trying to do things with language that language isn't necessarily equipped for. E.g., we can't just assume, uncritically, that "this statement is false" is the same sort of empirical, verifiable proposition about the world that "there's a coffee mug on my desk" is.

So many mathematicians (and quantum physicists) whose work borders on the philosophical/metaphysical keep poor Wittgenstein rolling perpetually in his grave . . .
posted by treepour at 1:34 PM on April 13, 2006

Then perhaps mathematics should be pursued somewhat more in the spirit of experimental science rather than always demanding proofs for everything.

Wow, this fits nicely with a pet theory of mine that mathematics is not Platonic, i.e. it is not "inherent in the universe" or "true whether or not anything exists" or "existing in a higher dimension" or something like that.

My hypothesis is that mathematics is the formalization of the naturally evolved capacity of human beings to abstract and categorize in order to better control their environment, i.e. the formalization of language itself.

In this interpretation, mathematics is only a tool -a beautiful and powerful one nonetheless- that enables humans to reason about their world. Therefore, the limits inherent in the world itself are made visible to us by their reflection in mathematics.

What do y'all think about this?
posted by koenie at 2:42 PM on April 17, 2006 [1 favorite]