Comments: Goedelian Blogging

Did you know that you can make a good living as a blogger just by making fun of the crazy shit Republicans say?

You guys are working too hard.

Posted by SteveB at July 23, 2009 04:48 PM

I'll second the recommendation of Gödel, Escher, Bach.

Posted by Cloud at July 23, 2009 04:52 PM

Rob will never write this comment.

Posted by Rob Payne at July 23, 2009 05:08 PM

Commenter SunMesa made an interesting parallel between the completeness of ethical codes and Goedel.

Having been thus called out, I feel compelled to comment, if only briefly. I will shortly be away from internet access for some days, but I look forward to reading the discussion when I return. These are some deep waters.

What I am very skeptical about is any attempt to use "Goedel" outside mathematics. I've read statements to the effect that sociology cannot be proven to be consistent. That's utter, embarrassing nonsense. The Incompleteness Theorems concern only formal systems and are meaningless in any other context.

I am similarly skeptical, although I am not well prepared to delineate what lies "outside" mathematics. My reference to Godel and incompleteness was sparked by the "i.e." in your Counterpunch essay equivalencing an ethical code with a universal decision procedure. The latter, I would contend, must incorporate a formal system, and moreover, one sufficiently general that it can generate mathematical proofs.

Goedel's result is an exotic byproduct of self-reference, which is among the most important concepts from the last 100 years.

I did read Godel, Escher, Bach some 30 years ago, along with some some additional arcane material concerning the incompleteness theorem that, shall we say, I skimmed over.

I have often heard Godel's result characterized as a 'trick' depending on self-reference. Has this actually been established? To put it more explicitly, consider Proposition A: "All unprovable truths in (sufficiently general) formal systems are self-referential".

It strikes me that A is possibly true, possibly self-referential, and possibly unprovable, or some combination thereof.

At this point I would need a stiff drink to proceed further, and need to finish packing. Good luck, all.

Posted by SunMesa at July 23, 2009 06:37 PM

SteveB: I wish I could say I enjoy blogging about politics, but I don't. The ugliness of the people in politics disgusts me. But math, music, philosophy, humor, yes I really love that stuff...

Rob: You wouldn't do this to us, would you? So I have to assume that Jon impersonated you.

SM: Self-reference is a tricky thing, right, because I can hide it very well. But if by that you mean, are all unprovable sentences obviously self-referential, in the sense that they look like P(t), where t refers to P(t), then the answer is clearly no. There's nothing self-referential about the unsolvability of Diophantine equations, or the Continuum Hypothesis, etc.

I believe that diagonalization is always needed in the proof, however. Certainly there's a lot of that in the proofs of the 2 examples I mentioned. Of course, maybe one can argue that self-reference and diagonalization are related. I guess they are. But the point is that they need not appear explicitly in the statement. They can be hidden in the proofs.

I've always viewed Goedel's proof as a nice twist on Cantor's work. I don't understand why Cantor doesn't get more credit. They were two madmen, but Cantor is the one who built the madhouse.

Happy packing (and a good trip, wherever you're going).

Posted by Bernard Chazelle at July 23, 2009 07:26 PM

Bernard,
Naw, Jonathan has much better taste than that. Besides, I’m a habitual liar and that’s the truth.

Seriously, your essay on torture is outstanding. I couldn’t agree more with your conclusion.

Posted by Rob Payne at July 23, 2009 11:34 PM

"That life is worth living is the most necessary of assumptions and, were it not assumed, the most impossible of conclusions."
-George Santayana

Posted by I do not recommend this site at July 23, 2009 11:37 PM

Hey, Chazelle, I take off my shoes to count over ten. Go back to politics.

Posted by Rosemary Molloy at July 24, 2009 06:56 AM

This is an amusing post, Bernard, but I object to your description of the proofs as not hard. When I was in eleventh grade, my math teacher loaned me GEB, and after I read it the first thing I did was to get a translated copy of On Undecidable Propositions. I had no prior experience with formal logic and found it very difficult reading. I eventually understood most, but not all of the proof. Four years later, in college, I took a course in recursion theory, and read the paper again. Of course this time, it made much more sense, much more easily, but I still wouldn't characterise it as "not hard." The string of thirty-odd definitions on the way to Theorem 6 is definitely nontrivial, and further, as far as I know no one before Goedel had thought of coding first-order formulae as naturals and performing operations on them. It is hard. That's why there are a number of books trying to help people understand it.

Also, trying to use the Incompleteness Theorems to prove something unrelated to formal systems is annoying, and also amateurish. (I'm not saying this is what SunMesa was doing; s/he was merely analogizing.) This desire to stretch math further than it can go in describing vague philosophical concepts makes me root for a return to natural-language philosophy in spite of my own fondness for formal logic.

Posted by Save the Oocytes at July 25, 2009 11:54 AM

bernard,
please, give us a break. the posts on bach and bird and math and whatever aren't good reading and they're off topic. they detract from atr. move 'em to bernardsmusicandmusings.blg and let's keep atr to more political topics.

Posted by jerry at July 28, 2009 01:57 AM

The mathematical is political.

Posted by Solar Hero at July 28, 2009 02:16 PM