Incompleteness, Omega, Chaitin, AIT

  1. Philosophy Forum
  2. » Logic
  3. » Incompleteness, Omega, Chaitin, AIT

Get Email Updates Email this Topic Print this Page

Reply Fri 7 May, 2010 07:59 pm
Chaitin proved incompleteness in a different way, from a Turing angle. His book "Meta-math" is great. Especially for anyone who has done even the simplest programming, and understands the concept of algorithm. This video opens in the midst of the description of the number Omega.
YouTube - Gregory Chaitin Lecture Carnegie-Mellon University 2000 Pt 6

---------- Post added 05-07-2010 at 09:32 PM ----------

YouTube - Gregory Chaitin Lecture Carnegie-Mellon University 2000 Pt 3
YouTube - Gregory Chaitin Lecture Carnegie-Mellon University 2000 Pt 4
YouTube - Gregory Chaitin Lecture Carnegie-Mellon University 2000 Pt 5
 
ughaibu
 
Reply Fri 7 May, 2010 09:28 pm
@Reconstructo,
Chaitin is something of a charlatan and self inflater, 'Godel, Turing and ME!!', kind of thing. I think his results are interesting and offer easy access to some consequences of randomness, on the other hand, van Lambalgen is pretty hard on him: http://staff.science.uva.nl/~michiell/docs/JSL89.pdf
 
Reconstructo
 
Reply Sat 8 May, 2010 04:56 pm
@ughaibu,
ughaibu;161522 wrote:
Chaitin is something of a charlatan and self inflater, 'Godel, Turing and ME!!', kind of thing. I think his results are interesting and offer easy access to some consequences of randomness, on the other hand, van Lambalgen is pretty hard on him: http://staff.science.uva.nl/~michiell/docs/JSL89.pdf


Thanks for the link. I like Chaitin, tho. I love programming, though, so maybe his angle is just something that clicks for me. AIT is as simple and intuitive as the coordinate plane. Personally, I like simple. I love mathematics for finding the hidden structure in the apparently haphazard. And so does Chaitin, as he makes clear in his book. It's ironic that he applies this desire for clarity to randomness, incompleteness.

For me, the essence of calculus, for instance, is more beautiful than chipping away at difficult integrals. On the other hand, I do enjoy tackling difficult algorithms. I also lean more toward pure math than applied math. For me, there's a strong aesthetic philosophical attraction to math.

What value for x gives us the maximum value for this expression: "the x root of x?" As you may already know the answer is e. I find this quite fascinating. It seems that it must be related somehow that the fact that e^x is its own derivative, but I can't yet process how. (I find e more fascinating than pi.)

Anyway, comprehension as compression is beautiful, simple, sensible, and this is actually more important to me than Omega...but Omega is interesting. I'm digging into Turing now. For me, incompleteness has never been terribly important. Zeno's paradox, 0!, and other little questionables have always hinted that math had a shaky foundation. What fascinates me about incompleteness is its connection to philosophy.

The Kronecker/Cantor divide is also fascinating. I'd like to read more about K, but information is scarce.
 
TuringEquivalent
 
Reply Sat 8 May, 2010 05:32 pm
@Reconstructo,
I think he is a moron, but one day, i have to rationalize my conviction.
 
Reconstructo
 
Reply Sat 8 May, 2010 10:04 pm
@Reconstructo,
I was just reading about how passionate and certain Hilbert was that his formalization program would succeed. Perhaps I am lucky to have come into the door from the realm of metaphor. I expected precision but not perfect closure. And precision is still there.

Incompleteness is in many ways a blessing. This ties in to Rorty, who insists on the impossibility of closure, and finds the good in that. This also reminds me of Dostoevsky's underground man, who insists that he would throw a wrench in the perfect machine, out of an all too human perversity. Closure is akin to death.

Zizek writes well on the so-called "death drive." The death drive is not simply a desire for death, in my reading of his reading. Instead, the death drive is a desire for immortality. This makes sense because life is flux, creation and its inseparable destruction. You have birth/death on one hand, and stasis on the other. The vampire myth is a great example of the death drive. Neither breeding nor dying. Reminds me of Daedalus in Ulysses, who hates water. Well, mathematicians seem to have a love affair with eternity. Except for mischief makers like Chaitin. I can only wonder about Godel, who starved himself to death because he was paranoid....Was he ambivalent about his proof? Was it a sort of suicide?

I can't imagine Chaitin going mad. Kronecker is written about as if he was quite the social butterfly. According to E.T. Bell, he could have been a great businessman. True, Kronecker was passionate about mathematics, and did his best to keep Cantor's ideas down. E.T. Bell writes about Kronecker and Weierstrass in a way that I won't try to describe. Let's just say he humanized them, made them one hilarious pair of argumentative friends.
 
Reconstructo
 
Reply Thu 27 May, 2010 02:17 pm
@Reconstructo,
I still think this is great stuff....
 
ughaibu
 
Reply Thu 27 May, 2010 02:34 pm
@Reconstructo,
Here's Solomonoff's autobiography: http://world.std.com/~rjs/barc97.pdf
 
ughaibu
 
Reply Thu 27 May, 2010 11:35 pm
@Reconstructo,
Reconstructo;161968 wrote:
I was just reading about how passionate and certain Hilbert was that his formalization program would succeed.
It's not dead yet: Hence Hilbert's goal of conclusively demonstrating that mathematics is trouble-free proves largely achievable.
Inconsistent Mathematics (Stanford Encyclopedia of Philosophy)
 
Reconstructo
 
Reply Thu 27 May, 2010 11:59 pm
@ughaibu,
ughaibu;169835 wrote:
It's not dead yet: Hence Hilbert's goal of conclusively demonstrating that mathematics is trouble-free proves largely achievable.
Inconsistent Mathematics (Stanford Encyclopedia of Philosophy)


Interesting. My time is limited tonight, so I must ask you: what in general are they up to? And have they narrowed the ambition of Hilbert to something achievable?

I should say that mathematics is obviously workable enough to put a man on the moon. What do you make of number? What is your stance on the issue? Is number invented, discovered, intuited, etc.? I think that the higher math is invented and adjusted till it works. But what about the foundation? What about arithmetic? What is a made of?
 
ughaibu
 
Reply Fri 28 May, 2010 05:08 am
@Reconstructo,
Reconstructo;169839 wrote:
what in general are they up to?
I dont know anything about it myself, but the idea seems to be that naive set theory can be supported by a paraconsistent logic. The article is very short, it'll only take you a few minutes to read. I haven't been able to locate online publications pages for the major players listed in the bibliography, and I'm too tired, today, to search for specific articles, but this review of Brady's book goes into some more detail: http://consequently.org/papers/universal.pdf
Reconstructo;169839 wrote:
What do you make of number? What is your stance on the issue? Is number invented, discovered, intuited, etc.? I think that the higher math is invented and adjusted till it works. But what about the foundation? What about arithmetic? What is a made of?
That's more questions than I have answers, I guess I'm a formalist.
 
Reconstructo
 
Reply Fri 28 May, 2010 01:42 pm
@ughaibu,
ughaibu;169868 wrote:
That's more questions than I have answers, I guess I'm a formalist.


Thanks for your direct answer. I very much respect formalism but I'm probably an intuitionist of some sort at heart. Some of this is related to my aesthetic response to mathematics. I love it because I experience it as intuitive. No doubt, not all of it is intuitive. I suggest though, that the core is intuitive. From there, internally consistent formalisms can be devised.

I'm not going to pretend to be a specialist on the matter. I cast my net fairly wide. I suspect that you are far more exposed to certain areas of mathematics, and likely to mathematics in general. I can claim that what I understand, I understand deeply, or so it seems to me. But I have focused on the philosophy/mathematics intersection. I think of some mathematics as absolute form, or in other words undiluted form. I'm a finitist all in all, and yet I love ideas like Cantor's. I like to see how far the finite concept can be pushed in continuity, infinity. I think that math, formal logic, and ordinary language are all founded upon a basic intuition of unity, of finiteness, boundedness, etc. Does any of this click for you? Does math strike you as a sort of perfect sculpture?
 
ughaibu
 
Reply Sat 29 May, 2010 07:51 am
@Reconstructo,
Reconstructo;170076 wrote:
I'm a finitist all in all
What's your definition of finiteness?
Reconstructo;170076 wrote:
Does math strike you as a sort of perfect sculpture?
No. Here's a fun read: http://philsci-archive.pitt.edu/archive/00001164/00/formfiz_preprint.pdf
 
Reconstructo
 
Reply Sat 29 May, 2010 03:14 pm
@ughaibu,
ughaibu;170347 wrote:
What's your definition of finiteness?No. Here's a fun read: http://philsci-archive.pitt.edu/archive/00001164/00/formfiz_preprint.pdf


I don't think we can truly comprehend the infinite. I think our concepts of the infinite are ultimate finite. But then one might have to define "truly comprehend."

My current theory is that math is founded on an intuition of unity, and that this unity is systematically iterated in many many ways. We can't calculate with transcendental numbers except in closed form, or so it seems to me. Consider the use of radians. 2/pi is an answer, right? But if we want to apply this answer technologically, do we not have to approximate this answer? When bankers or physicist calculate with the number e, they have to use an approximation at some point. When one wants to apply a number, it has to be put in a closed form. But transcendental numbers aren't closed, really. We are never finished calculating them, even if we have all the precision we need. A transcendental number is something like an algorithm. An implied set of instructions for calculating digits.

Take e for example, defined as the limit n --> infinity (1+1/n)^n. Infinity is the instruction in this algorithm (metaphor, of course) to increase n to the precision desire, with no upper bound.

We fly high into abstract realms, which are great, but must always return to a finite number of precise digits to interact with the real world.

---------- Post added 05-29-2010 at 04:23 PM ----------

ughaibu;170347 wrote:
What's your definition of finiteness?No. Here's a fun read: http://philsci-archive.pitt.edu/archive/00001164/00/formfiz_preprint.pdf


How surprising! I enjoyed that paper. Yes, I'm in the intuitionist camp somewhere. I do see the attractions of formalism, but I also see objections.

To look at a page of symbols and see particular symbols is already, in my opinion, the work of that founding intuition I have argued for. I argue that human experience is automatically discrete. We break the world into pieces, into objects. We break a page of math into individual symbols visually, and then conceptually "decode" the relationships represented by/in these symbols.

In that paper, chess is talked of as if it is not considered a mathematical system. I personally consider it to be one. Because it's a precise abstract system.

Anyway, thanks for the links. I would like to hear your opinions in more depth sometime, if you ever feel like sharing them...Smile
 
 

 
  1. Philosophy Forum
  2. » Logic
  3. » Incompleteness, Omega, Chaitin, AIT
Copyright © 2021 MadLab, LLC :: Terms of Service :: Privacy Policy :: Page generated in 0.02 seconds on 09/23/2021 at 12:52:53