Talk:Gregory Chaitin
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[edit] Dates needed
It might help to add a date, which i assume is late twentieth century for most of his work so far. Also the external web page cited is in New Zealand. Is he really American? Maybe an American currently resident in New Zealand? Or just an American but his fan club is based in New Zealand?
- He resides in the US. I added birthdate information. --AxelBoldt
[edit] Chaitin and randomness
I removed the sentence Chaitin's work has profound consequences for our ideas of randomness. The main contribution to mathematical understanding of randomness are Kolmogorov's axiom of a probability space (and, arguably, the possibility of measuring randomness through the notion of Kolmogorov complexity). --Aleph4 13:36, 21 May 2004 (UTC)
[edit] Pronunciation
Is his name pronounced CHAY-tin? It would be nice if someone could verify this and add it in the entry.
[edit] Importance
I changed "important contributions" to "contributions". It seems that his most important work are attempts at popularizations of well known results from Gödel, Kolmogorov, Matijasevitch etc. If anybody knows about important results, please undo the change but also list at least one important contribution (and please do not take Chaitin's Omega-number as example, which is just a completely trivial formulation of old knowledge.) --131.130.190.55 21:29, 24 January 2006 (UTC)
- There was a long discussion of Chaitin on the FOM mailing list in 2001. The consensus seems to be that his Omega stuff is correct and nontrivial, but that his interpretation of it is wrong. I've added a statement to that effect, and references to two papers criticizing his interpretation.--Bcrowell 19:41, 24 February 2006 (UTC)
[edit] I disagree
I think that Chaitin's Omega number is an important contribution to mathematics and is not a trivial formulation of old knowledge. For any math problem, the bits of his Omega number completely determine whether the problem is solvable or not. Chaitin proved that this number is random and that only a finite number of bits of Omega are even knowable. This is a huge result, as shows that mathematics is random and most of it is unknowable. In my opinion, this the greatest theorem in all of mathematics, and no, I'm not Gregory Chaitin.
- Could you please give me a proof of "...mathematics is random and most of it unkowable.'?
You have to read Chaitin's work to get the full proof of this. In my opinion, most of Chaitin's critics don't understand Chaitin's work and its significance.
- No, I'm asking for a discrete, definite proof. I have read Chaitin's work and I don't think he has a proof for that kind of assertions. Maybe Chaitin's 'proof' is unprovable too..
OK, I'll give it a shot. The number Omega describes the nature of mathematics because for any mathematics problem, the bits of Omega completely determine whether that problem is solvable or not. So mathematics is random because Omega is random. And most of mathematics is unknowable because only a finite number of bits of Omega are knowable.
[edit] Middle name
Someone added John as Gregory J. Chaitin's middle name, but I haven't been ablo to find any source to support it. Anyone has one? Mariano(t/c) 10:37, 12 April 2006 (UTC)
- Since noone provided any source (or comment at all) on the addition of John as his middle name, I reverted that edit. Mariano(t/c) 08:50, 20 April 2006 (UTC)
- I re-changed the name to John, here is my source http://www.fcen.uba.ar/consdire/sesion02/04-03-02/0403-9.htm GalGross 03:51, 26 March 2007 (UTC)
- Great! 'chas gracias.--Mariano(t/c) 19:41, 26 March 2007 (UTC)
[edit] Panu Raatikainen gets it all wrong
Here's a quote:
But why does Chaitin think so? It is because he interprets his own variants of incompleteness theorems as follows: “The general flavor of my work is like this. You compare the complexity of the axioms with the complexity of the result you’re trying to derive, and if the result is more complex than the axioms, then you can’t get it from those axioms” (The Unknowable, p. 24). Or, in other words: “my approach makes incompleteness more natural, because you see how what you can do depends on the axioms. The more complex the axioms, the better you can do” (The Unknowable, p. 26).
But appearances notwithstanding, this is simply wrong. In fact, there is no direct dependence between the complexity of an axiom system and its power to prove theorems. On the one hand, there are extremely complex systems of axioms that are very weak and enable one to prove only trivial theorems. Consider, for example, an enormously complex finite collection of axioms with the form n < n+ 1; even the simple theory consisting of the single generalization “for all x, x < x+ 1” can prove more. On the other hand, there exist very simple and compact axiom systems that are sufficient for the development of all known mathematics (e.g., the axioms of set theory) and that can in particular decide many more cases of program-size complexity than some extremely complex but weak axiom systems (such as the one above).[1]
But if anyone is "simply wrong" here, it's Raatikainen: these "extremely complex systems" he gives as an example are in fact extremely simple. One can say:
1 < 2 2 < 3 3 < 4 ... 1000000 < 1000001
but this is - precisely in terms of AIT! - hardly more complex than saying just:
for all x, x < x+ 1
This only shows that Raatikainen doesn't understand elementary concepts of AIT, yet he ventures to criticize Chaitin's work. Personally, I find it a bit disturbing... GregorB 16:38, 23 April 2006 (UTC)
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- The point is that there are finitely axiomatized theories that prove much more than other theories that have many complicated (but proof theoretically weak) infinite axiom schemes. In any natural programming language, the finitely axiomatized theory will have a much smaller index than the theory with the schemes, but it will still be much stronger in terms of what it can prove. Thus there is not a direct relation between the smallest index for a theory and the number of statements K(n) > k that the theory can prove; this interpretation of Chaitin's result is not valid. CMummert 04:54, 16 June 2006 (UTC)
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- Well... Unless we pick a million random numbers, and use them to form a million axioms. Then Chaitin is in trouble, since it is clear that these axioms are not too powerful, but still are quite complex... GregorB 16:37, 24 April 2006 (UTC)
[edit] Argumentum ad personam
The article says: Chaitin is frequently criticized as having an inflated ego and being a relentless self-publicist.[2][3][4]
Neither are the sources (comments at Amazon) any reliable nor is there a point of including that into encyclopedia, unless the overwhelming majority of mathematicians would think the same. The article already lists critics of his work. -- ZZ 09:55, 12 June 2006 (UTC)
[edit] Misleading sentences
I refer to the following two sentences.
Chaitin has defined Chaitin's constant, a real number whose digits are equidistributed and which expresses the probability that a random program will halt. has numerous remarkable mathematical properties, including the fact that it is definable but not computable.
The first sentence is not right; there is no such thing as a random program, because there is no nontrivial probability measure on the set of programs. The word remarkable in the second sentence seems to imply that few such numbers are known, which is not true. There are many definable uncomputable numbers. Some of them, such as 0ˈˈˈ, have definitions which might be considered much simpler than the definition of one of Chaitin's numbers. CMummert 04:54, 16 June 2006 (UTC)
