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

Sydney Morning Herald ^ | February 25, 2004

[RightWingAtheist]

I'm enjoying being a spectator to this debate so much, I think I'll join in. My own view is that the ability to do science is hardwired in our brains, and as such, may possibly be the result of evolutionary pressures. If are to assume that the neo-Darwinist approach to cognitive psychology is true (as I do) then we do NOT possess a general all-purpose brain which is capable of deducing or understanding everything. Instead, we have evolved very specific mental structures (or mental algorithims, or what ever term you prefer) which have proved useful to our survival and sustenance, and which through some way, some combination of adaptative and exaptationist processes, have been useful beyond their original purpose. And yet, the limitations are still there; there are certain phenomena which are minds are unable to ever completely discern or explain.

If this seems like an unduly pessimistic expression of biological determinism and fatalism, then allow me to conclude on an upbeat note. The biological apparatus allowing us to science encompasses our perceptual faculties as well as our cognitive ones. We have been able to move beyond the innate limitations of our five senses through the instruments we have devised-everything from telescopes to microscopes, thermometers to chromatographs, radio telescopes to bubble chambers. We may be able to similarly go beyond the biological limitations of our masses of gray mush by developing computers and other artifical intelligence systems capable of doing science, rather than simply serving as assistants. How this may be done-well, that's an answer which is beyond me.

[LibWhacker]

That's an interesting website, thanks! I'll print out some of his papers and read them. I have heard the quote before that anything that is sufficiently complex can pass for random. Didn't realize he said it.

But it's still not clear to me that there are necessarily proofs out there that exceed the capacity of man's brain to understand, but I suppose that'll come out as I read some of his articles. It seems to me that he'll never be able to prove that a given theorem is beyond mankind's ability to prove. Of course, there are lots of unproven conjectures bouncing around out there, but when it comes to many of them, we just don't know if it's because we're not and never will be smart enough to prove or disprove them (I sure hope that's not the case, though it does appear to be what Chaitin is saying), or if it's because math isn't sufficiently developed to allow us to do much with them yet (another way of saying this is that the right genius hasn't come along yet and showed us the correct way to approach the problem), or if it's because it's one of Godel's undecidables.

[Virginia-American]

Is it known that there are provable theorems out there that are hopelessly beyond the human intellect to prove?

Consider the four-color theorem. The original proof requires the use of a computer to check billions of cases; here's a link to another proof, where the author says

"The first proof needs a computer. The second can be checked by hand in a few months, or, using a computer, it can be verified in about 20 minutes."

As a general rule, given any formal system, and given any positive N, there will be theorems in it whose proof requires at least N steps. That's any N - eg 10 to the 10 to the 10 ... to the 10.

[Physicist]

By composition, the number system is a natural product of physical law. Ergo, physical law contains Godel statements.

But by the same argument, human reasoning would contain Gödel statements, which is patently absurd. Gödel statements are not a general feature of mathematical systems. They only occur in certain classes of formal axiom sets, such as number theory.

Human reasoning has bounded algorithmic complexity.

Show me. Remember that we are talking about human reasoning in principle, so don't bother with in-practice arguments like the limited human lifespan.

The nature of Godel statements is that they are neither true nor false

No, the nature of Gödel statements is that their truth or falsity is not provable by the set of axioms of the formal system in which they appear.

[Physicist]

Consider the four-color theorem. The original proof requires the use of a computer to check billions of cases

I maintain that computer programs are implementations of human reasoning, thus all of their results fall under that heading.

[Doctor Stochastic]

I maintain that computer programs are implementations of human reasoning, thus all of their results fall under that heading.

Church's Thesis (if true) would certainly support this claim. So far, all models of computation (for example: Turing machines, recursive function theory, Post normal forms, quantum computers, probabilistic turing machines, stack machines, etc.) all compute exactly the same set of functions. No model of computation has been found to extend this set.

[RadioAstronomer]

bttt for later read this evening. :-)

[LTCJ]

Is it true that because balancing your checkbook is cast in the language of mathematics, it is subject to the limitations of Gödel's theorem?

Gödel's theorem? Of course not.
The balancing of my checkbook is, however, an excellent example of Heisenberg's Uncertainty Principle.

[tcuoohjohn]

So a manifold is a Euclidean description of a non-Euclidean space?

[RightWingAtheist]

Do think this implies that there is a common universal logical structure underlying all forms of human reasoning, just as there exists a common grammatic structure underlying all language?

[longshadow]

So a manifold is a Euclidean description of a non-Euclidean space?

I'm not clear on what you mean by "Euclidean description"....

The simplest way I can think to say it is a manifold is a topological space which locally Euclidean. That is to say the space appears to be Euclidean if you don't stray too "far" away from the point you are examining.

I put "far" in parens because not all topological spaces have a notion of "distance" (as best I recall)....

[longshadow]

The status of the CH is definitely that it is undecidable. For example look here which I quote in relevant part.

Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the Axiom of choice).

I'm going to defer to "Doctor Stochastic" on this one, if he cares to address it. I think he's already stated the CH is NOT a Gödel statement, and I'm a bit perplexed by Wolfram's quote; I think he's saying the CH is undecideable in the same sense that the Parallel Postulate is in Geometry, which is definitely not the same as the Gödelian sense of the term.

[PatrickHenry]

... not all topological spaces have a notion of "distance" (as best I recall)....

I try to stay away from such places.

[tcuoohjohn]

I mean "is a manifold a Euclidean ( Planar) description of a Non-Euclidean ( curvilinear) space or form."?

Eg. Does Topology describe in a Euclidean way, by use of discreet cumulative adjacent planes a non-euclidean form?

Hope I'm making some sense here..as you can tell I am no mathematician so I have to use imprecise words and phrases to describe what I mean.

[longshadow]

The status of the CH is definitely that it is undecidable.

UPDATE:

The reference to Gödel, cited in your reference, is to his work on the consistency of CH with set theory, not to his Incompleteness Theorem.

Basically, Gödel showed that CH is consistent with formal set theory (Zermelo-Fraenkel) plus the Axiom of Choice (AC). Cohen later showed that ~CH is ALSO consistent with set theory + AC ("ZFC").

That means that CH is "undecideable" from within the framework of ZFC.

But that doesn't mean it is undecideable in an absolute sense, and work continues apace to find axioms that will lead to an answer to the continuum problem.

I hope that clears things up.....

[No Permission Needed]

Albert Einstein wrote an article for the 1926 edition of the Encyclopedia Brittanica explaining in simple terms space and time. It begans:

"All our thoughts and concepts are called up by sense-experiences and have a meaning only in reference to these sense-experiences. On the other hand, however, they are products of the spontaneous activity of our minds; they are thus in no wise logical consequences of the contents of these sense experiences. If, therefore, we wish to grasp the essence of a complex of abstract notions we must for the one part investigate the mutual relationships between the concepts and the assertions made about them; for the other, we must investigate how they are related to the experiences."

[DoctorMichael]

Bump.

[longshadow]

Eg. Does Topology describe in a Euclidean way, by use of discreet cumulative adjacent planes a non-euclidean form?

As best I can understand you question, no; I don't think that's it.

In topology, the word "manifold" refers to the subset of topological spaces that are locally Euclidean. IOW, it is a category of topological spaces that satisfy an additional constraint: they behave like Euclidean spaces if you don't stray too far -- IOW, Euclidean Geometry might work on short distance scales, but not between points in the space which are more widely separately.

The surface of the Earth is an example: locally, we can't tell that we aren't on a Euclidean plane, but if you zoom out far enough and watch ships sail over the horizon, you begin to realize it isn't really flat. Euclidean geometry doesn't work on large distance on the surface of the Earth. Shortest paths between two points are "Great Circle " routes instead of straight lines, but on small distance scales, the Great Circles become indistinguishable from straight lines connecting the two points. That's "locally Euclidean" but globally non-Euclidean.....

Does that help?

[GregoryFul]

Mathematically speaking, there is a discontinutity in logic, and anything can be emitted, or fall into a discontinutity. Anything goes at the discontinuity. That a discontinutity exists is suggestive that our logic is not correct. That is, something is or is not, is not necessarily true. Curiously, quantum physics postulates the same concept. A thing is not in a particular state until an observation is made.

[GregoryFul]

A simple example is:

Write on one side of a 3x5 card:
The statement on the other side of this card is a lie.
and on the other side:
The statement on the other side of this card is the truth.
Thats it.