Incompleteness Theorem
This is a placeholder for a future post: In the last week I’ve run across Gödel’s Incompleteness Theorem in the comments section here, in my own reading (Douglas Hofstadter, also Jay Garfield and Graham Priest on Nagarjuna), and then this morning, listening to A Briefer History of Time on audiobook.
Suffice it to say, this calls for some kind of post, since the discussion of limits, singularities, boundaries etc. are germane to what I’ve been writing about here and the theorem keeps rearing its ugly head(s).
From the Goedel Society web page:
“In 1930/1931 Gödel proved his most famous result, the incompleteness theorem. It states that any not too weak formal theory, in particular any reasonable formalization of number theory, cannot prove everything that is true, i.e. such theories are necessarily incomplete.”
This, however, doesn’t mean that just anything goes (i.e., that you can state any two contradictory conclusions from a theory and justify their existence together as being coherent).
I’ll point you to the Stanford Encyclopedia page on Goedel (my intuition says ignore Wikipedia) for starters, and come back to this on Monday or the weekend sometime.
August 10th, 2007 at 7:17 pm
Quine draws explicitly upon Gödel’s critique of Carnap’s project. This is where I picked him up and I borrow Quine’s synthesis whenever I teach on Logical Positivism because Gödel (I think) ends many discussions about the possibilities of perfected logical systems. I think it just drives us where we were all going in the first place: logic will not get us to the real world. All attempts have failed and even Carnap supposedly gave up.
August 10th, 2007 at 7:46 pm
Thanks, Dru. Now I see more clearly what you’ve been getting at in some of your comments. I wonder why this wasn’t emphasized more at UMSL. Could just be the courses I took. It’s mind-boggling stuff–plus the BHoT that I’m listening to ties my gray matter in knots. (Better than getting knots from the traffic out here though!)
August 10th, 2007 at 7:55 pm
I had Paul Roth for epistemology, so we were thoroughly drenched in Quine and Epistemology Naturalized (which included the Carnap, Gödel, et al build up to the ‘epistemological revolution’ as Roth called it).
What’s BHoT again?
August 10th, 2007 at 8:42 pm
Sorry, “Briefer History of Time” by Stephen Hawking. It’s a good overview. Gödel came up when talking about some of the paradoxical consequences of Einstein’s General Theory of Relativity concerning time travel.
(And yes, that’s what I get for taking the epistemology comp without the course… I had it, sort of, with Streeter, but as part of a larger seminar on action, mind and epistemology. Some Quine in philosophy of science, but I’m trying to fill in gaps this summer.)