Nope. But it looks interesting =] Thanks.
I have some problems with this. If you speak of the halting problem, that is about self-reference as far as I can see. How does randomness enters the picture and becomes the main culprit?
Ooh! Ooh! Perpetual motion really is possible but there’s a government conspiracy lead by the reptilian Atlanteans to cover it up!
Clayton -
I took a quick look at that page and saw no mention of perpetual motion. If they believe in perpetual motion they are of course ‘misguided’, just like people who waste their time with word games…
How is the halting problem any more self-referential than the gravitational problem, except that we are physical beings asking a question about the behavior of a (imaginary) physical object? The proof that the halting problem is unsolvable relies on giving HALT(x) itself as input, i.e. HALT(HALT) but even this is not self-reference per se, it’s just providing a bit pattern to a function that happens to be the same as the bit pattern comprising the function itself. Its purpose is not self-reference but diagonalization.
As I said, that is a separate thread unto itself. Basically, randomness enters the picture because most mathematical facts are true for no reason. This statement (Chaitin’s) is not an attempt to be avant-garde or postmodern, it’s a statement about the statistical behavior of an exhaustive enumeration of mathematical theorems (or programs, as the case may be). Now, this statement is (unintentionally) confusing because when Chaitin says “most mathematical theorems” he does not mean most mathematical theorems which we find interesting, but most mathematical theorems in an objective, exhaustive enumeration of all possible mathematical theorems. We humans just happen to find the theorems which are true for a reason (compressible) interesting exactly because they’re true for a reason (our brains look for things which are “patternful” and ignore things which are apparently patternless or try to impose patterns on them).
Read about Chaitin’s constant for a concrete example of a mathematical object which is defined “almost constructively” (to use Chaitin’s words), yet whose value is as random as the flipping of a metaphysical coin. Chaitin has even used his constant to demonstrate that number theory contains pure randomness. This is a surprising result because we would expect that structure should be all-pervasive in mathematical systems which contain no apparently chaotic or disordered elements. What Chaitin has found is that order is infinitesimally rare in the universe of all mathematical truths and this is why there are always facts which are true, but not provable (or “true for no reason” where “for no reason” means “without possibility of being proven true”).
Clayton -
I was not giving the link for serious considerations, but to show that math cranks are actually out there.
Well maybe Goedel is one of them ?
edit :
The thing is, there’s no 100% correct set theory out there. And set theory deals all the time with ‘infinity’ in a way that frankly doesn’t make sense as far as I’m concerned. Recursion works great in real computers - that doesn’t mean that these metaphysical theories about infinity, the foundations of mathematics, etc, are right.
I think Goedel is pretty smart, and I think many mathmeticians would agree. I would assume the vast majority of those who discredit him are unfamiliar with the details of his theorems and mathematics in general.
When you are dealing with a discrete, deterministic system, theorems inside this system can be proven true or false. Goedel, to my knowledge, has proven his theorem inside such an environment. There is no necessity for measurement or other such flexible observations. You “test” the theorem by determining if it is consistent with the system’s axioms (required basis for any such system). There are no real world discrepancies with empirical data or the truthfulness of theories upon which a new theory is based.
Attempting to apply the system to the real world would be required to introduce the problems I believe you are describing.
Ah yes. If the majority of the establishment claims ‘X’ is true, it logically follows ‘X’ is true…
Yeah, that’s why I came here - because the establishment told me that this was the place for sound economic theory. Give me a break.
Do you have an argument against Goedel’s theorem, or do you just enjoy being the last skeptic in the room? Do you have an argument or evidence to reinforce your viewpoint? Do you believe that theory inside discrete, deterministic systems based upon axioms have the same obstacles as theory based upon the behavior of the real world, such as physics or economics?
Your point ? You’re claiming Goedel is clever because the establishment says so - that’s a ridiculous non-sequitur.
But this is not mathematics - that’s the problem.
http://www.cs.umaine.edu/~chaitin/summer.html
That’s not mathematics - that’s metaphysics. I can safely laugh at claims such as " And I learned that physics says that the ultimate nature of reality is mathematical"…
Of course, ‘physics’ says no such thing.
So, again we’re not discussing maths, but something else.
To say “Kurt Godel was smart” is an understatement of cosmic proportions. Godel possibly thought more deeply about mathematics than any man ever has.
It’s also not Godel’s 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems. Please read the paper - the introduction is not inaccessible even to a layperson - and any one of its many online (or hardcopy) expositions and tell me what specific error Godel makes in his reasoning. 
Most of Chaitin’s online articles are metaphysical discussions - he makes no claims to the contrary. But he is discussing what he sees as the philosophical implications of his mathematical work (most of which was done during the late 60’s and early 70’s). I fail to see the problem.
By the way, with Chaitin, I hold that the physical universe is fundamentally mathematical. It’s a kind of Pythagoreanism called “digital physics.” Look it up.
In the meantime, I’d like to see a specific objection to Godel’s reasoning.
Clayton -
I would agree that popularity is not necessarily correlated to correctness, although in academic communities this is more often more likley the case. (Note - there is a huge difference between completely independent cranks and conflicting schools.) What I meant about Goedel and the establishment is that I have not been made aware of any dissenting opinions that have been able to convince me Goedel is wrong. In 70+ years since this widely known theorem has been around, you’d think you’d at least come into contact with movements that present a strong case. Maybe I’m wrong. Do you know of any strong rebuttals?
I read the paper. I’m not arguing about mathematics, but axiomatic systems. That is, the resulting system when certain things are accepted as rather than proven to be true. I don’t believe anyone claimed Goedel’s theory definitely applied to general reality, only systems that were based upon axioms derived from real life observations.
Chaitin says that the theorem can be applied to things that are undefined. I don’t think this is exactly true. Without any axioms, nothing really means anything. And he seems to define randomness. His examples still seem to rely on axioms. As far as randomness goes, I find the work of Wolfram interesting on this one (note: I know it is not completely his work, and that he is a huge prick about that).
Chaitin is saying that he believes reality is an axiomatic system. He’s not going to be able to prove this ever, I don’t think. Goedel’s incompleteness theory, nonetheless, does not simply invalidate all theorems inside such a system, only saying that it is impossible to know if all theorems inside this system are true or false, or whether the axioms themselves can be proven consistent. For most people’s purposes, Godel’s work changes nothing.
If reality is an axiomatic system and Goedel’s incompleteness theory does apply, this probably won’t have any fundamental effect on science in general anyway. I believe most people already assume that we can’t know if all theories can be verifiably proven true or false.
It is important to realize that Godel’s incompleteness theorem may or may not apply to nature, because there are no means to determine if it is an axiomatic system. Currently, M-theory’s failure to incorporate the 4 forces into a set of compatible and consistent equations shows that if nature is axiomatic, these axioms are unknown. Further, things like fluctuating cosmic constants and such would lead us to believe that if there are axioms, we are a long way away from knowing them.
By the way, as Chaitin discusses in the video lecture I linked, Godel’s work on incompleteness has been largely ignored by the mathematical establishment. Chaitin’s work, which is even worse for the establishment, is - as he says in the video - considered “pornographic” in the mathematical community… it’s something most people know about but nobody discusses.
Nevertheless, the work of both scholars has not been refuted (there is one guy who rejects some of Chaitin’s arguments, but I think his reasoning is incorrect). In that sense, they are “mainstream” that is, they are not cranks like the “Guage Society” which cineram linked to. That in itself doesn’t make it true and you’re welcome to be the first to find an error in their reasoning. Call me credulous, but I’m persuaded.
Clayton -
Most of the Incompleteness theorem deals with the problem of how certain sorts of closed logical arguments are flawed in as much as they require external referents to make them ‘normal’ (or complete). There are exceptions, which I believe Godel recognized too, such as the Pythagorean Theorem of Right Triangles, and I think one other mathematical relationship (I can’t remember the name of it right now…Sorry). Anyways, if you try to apply the Incompleteness Theorem to a given logical relationship, you’re trying to show that its attempt to be ‘closed’ it simply setting itself up to be unproven or more exacting to state unprovable (as it will just go around and around in deductive breakdowns where it will reference itself forever). Oddly, if you look at the work of Topos Theory, it hints at the same situation, but in a more obvious manner than the Incompleteness Theorem itself (that some sets of logic have to remain ‘open’ to be consistent).
You’ll need to explain this statement - Godel’s theorems apply to formal systems, there are no “undefineds” in formal systems (their intent is to avoid ambiguity).
None of Chaitin’s or Godel’s work has anything to do with having no axioms.
Kind of… randomness in Chaitin’s view is the complete exhaustion of order and structure and, by virtue of this, is undefinable. His definition is not a “definition” in the sense of making randomness well-defined, which would be an obvious inconsistency.
Of course.
Note that Wolfram is an intellectual debtor to Chaitin. His ANKOS cites Chaitin many times.
More specifically, he believes that the physical world is a computer. This is not so extreme, a mainstream theoretical physicist (Seth Lloyd) has written a small book on the subject, Programming the Universe.
It’s not about proof. It’s about constructing a new paragdigm from which to do mathematics (and physics) to, hopefully, gain new insights into the structure of the physical world.
The axioms can only be proven consistent if the system itself is inconsistent (2nd incompleteness theorem).
Watch the video - Chaitin work refutes this widely held view. Incompleteness is pervasive in mathematics. The vast, vast majority of mathematical facts are true “for no reason.”
Chaitin sees the primary impact of his work to be on mathematics itself. He advocates doing math in a “physics-like” manner, i.e. creating hypotheses, performing experiments and then drawing (tentative) conclusions on the basis of the results.
Clayton -
Clayton,
I was referring to his section on “some things are true by accident”. Obviously, there are still axioms, even if they are not expressed as mathematical equations. I believe we are in complete agreement, although you seem to have a much more fundamental grip on why you believe you are correct, whereas I am just blabbering.
Yes, random is definable as being undefinable, or something like that. It’s a tough idea to take on face value. You kind of figure that randomaeity might be a human thought phenomenon rather than a mathematical concept. One dimensional cellular automata based upon 8 simple rules produces “randomness,” but this obviously isn’t random. Is random really unpredictable, or just to human perception?
Wolfram - Chaitin connection! I now know why Chaitin’s name sounds so familliar. [:D]
Very interesting on the rest of your post. I will look more into this. Wolfram seems to have argued in the opposite direction for experimentation → he sought to bring his principles based in the math world into the real world, while Chaitin wants to bring the scientific method into mathematics.
What I meant by Godel’s work changing nothing, I was referring to applying the scientific method to nature. Godel’s work didn’t suggest that scientific reasoning upon nature was a waste of time or should be conducted differently. Yes, it SHOULD have quite an impact on the math world. Correct me if I’m wrong here.
Randomness is what is left when there is no more order or structure to be found. Program-length gives us an objective, but uncomputable way to “define” randomness. It’s the fact that the definition is uncomputable (no program or algorithm can be written to recognize random vs. non-random numbers) that saves it from inconsistency. [:)]
Well, what mathematics is not a human thought phenomenon. Think about it. [;)]
Well, you have to distinguish between randomness and non-determinism. It’s deterministic, but that doesn’t mean that it’s non-random. That’s the key result of Godel’s, Turing’s and Chaitin’s work… deterministic systems generate true randomness.
Predictability and randomness are related but not synonymous. There’s another name which Chaitin mentions in the video, that of Ray Solomonoff. Solomonoff applied the ideas of computability theory to induction and developed a theory of algorithmic prediction. Essentially, he formalized what we mean when we talk about induction. In my opinion, Solomonoff has resolved the problem of induction which is the source of much hand-wringing in philosophical circles. But that’s just my opinion.
You could say that. [:)]
Here’s an interesting paper by Cristian Calude, From Heisenberg to Godel via Chaitin. He argues that the Heisenberg Uncertainty Principle and logical incompleteness are directly related (you’ll find the story in the opening paragraph interesting…) Mind-blowing stuff. [;)]
Clayton -
For anyone intersted, Peter Smith, logician and senior lecturer at Cambridge, posted in a blog the other day his notes (PDF) for his four lectures on Gödel’s theorems this term. He also says to watch this space for supplemental things such as exercises, handouts and what not. If you haven’t visted his blog before, there is a wealth of resources there.
Just scanned Godel Without (Too Many) Tears… this looks like a very solid technical (but not too technical) introduction.
Clayton -