"Iz no eazy vay in Matematics, iz only der rrite vay." -- Prof. Izaak Wirszup, U of Chicago, 1975.

One thing left out of the discussion of undecideablilty is that undecideable statements in one system may be decideable with a different set of axioms. (Sometimes you can use the undecidable statement as an axiom, giving different systems-- the parallel postulate, for example, gives rise to three different systems of geometry.) What Godel showed is that every system, no matter how constructed, will have some statement, somewhere, which is beyond it. A "Unified Theory of Math" is what's impossible. What I've never seen is a proof that you can't multiple math systems, each which has undecidable statement, all of which can intersect or be disjoint, but every statement is provable in at least one system. (In other words, there exist statements that are never provable in any system.) Then again, maybe I just dropped out before they got to that part.

Another fact is that some questions can be show to be false, but never shown to be true. For a long time, the four-color map theorem fell into this category, as you could decide against it by simply drawing a five color map, even though a proof that five colors are necessary may have been impossible to create. (That can be done because such a map would have a finite number of regions, and so in the worst case, you could five colors necessary by just enumerating all the possibilities. (And any real mathematicians out there will know that I've just engaged in a huge amount of handwaving.)) This is why falsifiability is important in science, because it's usually a lot easier and possible.

*shakes head*, the writer isn't exactly correct about what Godel proved in his theorem. Simple mathematics is NOT shot through with logical contradictions. Rather, Godel was able to use simple mathematics to encode mathematical axioms and, more importantly, their proofs. The Computer Science spin on the proof is that, at the time, the theory was that Mathematics was mechanical: It would be possible to feed axioms into some sort of algorithm, turn the crank, and out would pop All True statements. Godel showed that, for any algorithm for cranking out true statements, there was one statement that was demonstrably true, but was also demonstrably unprovable. I.e., no matter how long the crank was turned, that demonstrably true statement would never appear. It would have to be an AXIOM, a statement that had to be true, AND included in the initial state of the algorithm. Godel then showed that, for the NEW and IMPROVED algorithm, there existed yet another statement that was demonstrably true, but was also demonstrably unprovable. He showed that you could keep "improving" the algorithm, but never arrive at the "final" algorithm that would prove all true statements.

(Keep in mind that the crucial distinction is between "true" (I.e. something that is not contradictory or demonstrably false), and "provable" (I.e. being able to go through a reasoning process from a set of axioms to produce that "true" statement.)

The word for this situation is not "undecidable", but "incomplete": A system is incomplete if it cannot prove all truths given the set of axioms and a process of derivation that does not produce contradictions.

"Undecidability" came from a different attack on the problem: Why not build on Godel's system for mapping numbers to symbols that represented axioms and proof processes and develop a meta-process that would generate all possible sets of axioms and proof processes, toss out the "invalidly formed" ones, and analyze the remainder? Alan Turing, IIRC, put the kabosh on that by showing, using the famous Turing Machine, that any metaprocess would be fooled by certain sets. The only way to REALLY determine if the member was valid was to "execute" the proof algorithm on the set (I.e. run the program). He was able to show that some members of the set would cause any proof algorithm to enter an infinite loop: I.e. you could never tell if the program was bad (infintely looping), or was just taking a long time to prove a particularly difficult truth. He boiled this last obstacle down to the question of whether there was a metaprocess for determining if a program halted or not. He was able to prove that any process of proof that was able to determine whether a program could halt did not exist, because you could be able to construct a set of axioms and a proof process that would be false, but which the program would say was true. I.e. it was contradictory, and thus false. Thus, the halting problem is known to be not solvable because any system that claims to do so can be made to produce false statements. It is that finding that causes the whole enterprise of mathematical proof to be "undecidable", since you will never know if you've hit a truth statement that is tough to prove, or which causes your proof system to run endlessly, because it is one of those statements that either makes it cycle in an infinite loop, or happens to be the truth that the system is missing to make it complete.

Whew! Well, THAT'S the computer science version. I'm sure a pure mathematician would cringe at that explanation, because computer science has a very wierd way of proving things, which I never got the hang of...

Oh, the tie in to Hawkins: The holy grail of Mathematics is to develop a branch of mathematics that is totally useless as a means to explain nature or a natural law. They've never found it: Hilbert once developed a branch of mathematics that everyone thought met the criterion, but upon discovering that the math described the behavior of csubatomic particles in a certain state, reportedly muttered, "DAMN!".

It appears that Hawkins is thinking that the reverse is true: If every truth in mathematics has an analog of some sort in physical nature, then there will always be something new to explain in nature, since Godel's theorem says there will always be new truths that cannot be proved.

"But dig deeper, and the richness and variety of nature are found to stem from just a handful of underlying mathematical principles."

Typical confused Platonic Idealism. Nature may be described by a handful of mathematical principles, but they do not "underlie" it, nor does Nature stem from them.

"It turns out that mathematical systems rich enough to contain arithmetic are shot through with logical contradictions."

No, no, no - this is not even remotely what G�del's Theorem states; it is not a question of contradiction (if it were, arithmetic would be worthless), only completeness. G�del's (First) Theorem states that any sufficiently rich formal system, if consistent, is incomplete - i.e., there are true statements which can not be derived from the axioms. His second Theorem states that this deficiency can not be solved by the addition of axioms - if you add an axiom making a formally undecidable statement true, then there always exist other formally undecidable statements, ad infinitum.

Alternatively (my view, by the way), G�del's Theorems can be interpreted as a formal criticism of Tertium non Datur. If you admit undecidable statements in the formal logic, then consistency and completeness both become decidable.

Finally, this application of G�del's Theorems to physics is bogus. The real problem in physics is the inability to unify Relativity and Quantum Mechanics - a problem which I believe is intractable with the current formulation of Quantum Mechanics.

I believe Ptah is incorrect in stating that all mathematical systems have natural analogues.

I'm not a professional, but my physics adviser says there are many that do not. A friend of his wrote a book, "Mathematics: Queen of the Sciences," to make the point that because physical systems can be described mathematically, and often by apparently abstract math that was developed before the physical system was discovered, people get confused and think that the universe is mathematical.

The universe is the universe. It makes the rules, it does not follow some rules external to it. I read Godel's Theorem last year and could not understand what all the fuss is about. His idea applies to systems created by humans. It says nothing that I can see about natural systems.

Natural systems are describable mathematically, but nature does not need to "decide" the truth of those statements.

Anyhow, the meaning of final theory is beyond the understanding of journalists. The final theory, if found, will be final because the speed of light is constant and therefore no information can be recovered from a space much smaller than the Planck radius.

Therefore, science stops at the Planck mass and Planck radius.

It does not follow, though, that having the final theory on paper allows us to derive from it the character of the Universe. The calculations are too complicated, even if we could get all the preliminary data, which we cannot.

As Steven Weisberg puts it, knowing the genome of a fruit fly does not tell you whether it is alive or not.

Raoul, Ptah, JD, Harry:

Wow--thanks for the outstanding discussion. Speaking as someone who has a (11 yr old) MS in Computer Science, you all refreshed my memory, and taught me a few things.

Much thanks.

What Jeff said. I'm blown away.

Raoul:

I don't think having multiple systems helps; the aggregate of those systems is itself a system, and you won't be able to work out a consistant way of resolving all the contradictions between them.

I believe Ptah is incorrect in stating that all mathematical systems have natural analogues.

I did not say that. I said that It looked to me as if HAWKINGS was saying that.

Sorry.

Since posting (and misspelling Weinberg's name), I also found this statement of his: mathematics does not explain anything.

But mathematics does provide astonisingly beautiful descriptions.