The big question: how long is a piece of string theory? (can we ever understand the universe?)

Sydney Morning Herald ^ | February 25, 2004

[Bikers4Bush]

All you need is a white towel.

[PatrickHenry]

PING. [This ping list is for the evolution side of evolution threads, and sometimes for other science topics. FReepmail me to be added or dropped.]

[LiteKeeper]

INTREP - PHILOSOPHY OF SCIENCE - COSMOLOGY - COSMOGONY

[old-ager]

Polite question: I genuinely don't understand how Godel's discovery could support evolution, yet you are pinging your evolution list with this article. How do you see Godel's proof as supporting evolution? Thank you in advance.

[RightWingAtheist]

We aren't seeing the end of physics, (contra James Horgan), but rather, simply the end of logical positivism.

[LibWhacker]

Any given mathematical statement (eg, 11 is a prime number) must either be true or false, right? Wrong! Godel showed that however elaborate mathematics becomes, there will always exist some statements (not the above ones though) that can never be proved true or false.

Then why did he use it? What is it with the crappy math articles lately? Now we're going to have ignoramuses running around saying mathematicians can't prove 11 is prime and smirking at everything else they say as well.

[Honcho]

My head hurts.

[<1/1,000,000th%]

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

Whatever they are.

[stainlessbanner]

we will never be able to grasp in totality how the universe is put together.

bump

[longshadow]

Any given mathematical statement (eg, 11 is a prime number) must either be true or false, right?

Wrong! Godel showed that however elaborate mathematics becomes, there will always exist some statements (not the above ones though) that can never be proved true or false.

The "primeness" of "11" is no more in doubt than any other proven theorem in Mathematics. The author is very misleading in his presentation of the example, though he then correctly states the reality in the subsequent paragraph, though he leaves out an important caveat: Gödel showed that there will exist some statements derivable in a mathematical system which can't be decided based on the axioms of that system. There is nothing, however, to prevent you from proving the statement true by appealing to axioms outside of the system you are working in.

But the point remains: any statement which is proveable by standard Mathematical proof techniques is NOT a "Gödel statement" and hence is NOT undecideable. That some statements are undecideable cast no doubt on the ones we can, and do eventually prove. There is no evidence that reality will turn out to be based on Gödelian statements, so it may not even be an issue for physicists.

[longshadow]

And once you bring in other axioms, there will exist some other derivable statements which can't be decided based on the expanded set of axioms.

That's true, but the original statement in question will none-the-less have been proven, and thus is no longer a Gödel statement, though there will now be new ones as you correctly point out.

[tcuoohjohn]

I confess...I am a mathematical dunderhead. Can anyone tell me in layman's terms what a "manifold" is?.. Given my limitations I understand what an automotive manifold is but haven't a clue what a mathematical manifold is. I have tried websites but the discussions of manifolds are very much " inside baseball" stuff.

[LibWhacker]

But there are plenty of relatively simple unproven conjectures, such as Goldbach's conjecture (that every even integer greater than two is the sum of two primes), that people have speculated may in fact be true but not provable in Godel's sense. Instead he throws in the ridiculous and misleading statment that the primality of 11 may not be decidable; i.e., that perhaps mathematicians may have been wrong in thinking it was, lol! Davies is a bit of a joke sometimes.

[LibWhacker]

Never had topology myself, so I can't help you. But I did find this explanation that seems fairly approachable.

[longshadow]

Can anyone tell me in layman's terms what a "manifold" is?..

"A manifold is a topological space which is locally Euclidean ..." source: http://mathworld.wolfram.com/Manifold.html

"locally Euclidean" simply means that on short distance scales ("local") it exhibits the topological characteristics of flat Euclidean Geometry.

A topological space is the most primitive (least complex) mathematical structure:

In the chapter "Point Sets in General Spaces" Hausdorff (1914) defined his concept of a topological space based on the four Hausdorff Axioms.

1. To each point x there corresponds at least one neighborhood U(x), and U(x) contains x.
2. If U(x) and V(x) are neighborhoods of the same point x, then there exists a neighborhood W(x) of x such that W(x) is a subset of the intersection of U(x) and V(x).
3. If y is a point in U(x), then there exists a neighborhood U(y) of y such that U(y) is a subset of U(x).
4. For distinct points x and y, there exist two disjoint neighborhoods U(x) and U(y).

source: http://mathworld.wolfram.com/TopologicalSpace.html

I'm afraid there's no good way to state this in laymans' terms without losing the precision of the axioms.

[Physicist]

Instead he throws in the ridiculous and misleading statment that the primality of 11 may not be decidable

Well, no. He didn't say that. He gave the primality of 11 as an example of a mathematical truth, and then went on to say that the truth of some mathematical statements is undecidable.

I do disagree with Davies, however, that Gödel's theorem has anything to do with physics. Gödel's theorem only applies to certain types of formal systems. It is by no means clear that there exists no formal system appropriate for describing physics that is free from undecidable propositions. Furthermore, if even if all appropriate systems suffer this blind spot, it isn't clear that it would conceal anything important about the universe.