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

Sydney Morning Herald ^ | February 25, 2004

[Unknowing]

It seems that no system can be fully understood from within the system. Russell's solution to the set pardox concluded that the set of all sets can't be a set; Emerson said "the field cannot be seen from within the field." Godel, Escher, Bach challenge our notions of completeness.

The fact that there always seems to exist extra-systemic data makes us aware that that we are "inside" the cosmos, and can't get out even in thought. Objectifying it to the point where we can imagine standing outside the universe observing it is a senseless but common mistake. Physics cannot issue a formula describing all phenomena at once because of these limitations.

Theoretical physics has been trending toward a kind of philosophical speculation, and BS is BS whether it's jibberish, gobbledegook, calculus or English.

[LibWhacker]

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

I think you're misreading it . . .

Okay, that's a good point. Let's take for example the statement

"One plus one is two."

That's a given mathematical statement. But according to Davies, if I say,

"Class, that statement must either be true or false,"

then I am mistaken because, according to him, you can't make that claim about ANY given mathematical statement, including one so trivial as this. At least that's how I read it, and I think that's how a lot of people will read it.

Do you think I'm making a mountain out of a molehill? Maybe so. It just struck me as sooooo wrong and misleading. Is Davies a Brit? Gotta make exceptions for those guys, lol.

[AFreeBird]

I'm sure math had something to do with it, but I'm with you, my idea of a manifold is:

[longshadow]

but let's be explicit: what is wrong about that statement, in your reading?

I found the statement to be misleading in the sense that to a casual reader, they might think that he's saying that "11 is a prime number" and similar simple straighforward true statements) are somehow in doubt.

You know that's no true, and I know it's not true, and we both know he couldn't have meant it that way, but for lay people not familiar with Gödel, it is way too easy to misinterpret what the sentence was intended to convey.

[Physicist]

according to him, you can't make that claim about ANY given mathematical statement

I read it differently. Davies has gainsaid the proposition that any given mathematical statement MUST necessarily be true or false. He presumably accepts that any given statement MAY be true or false. As it turns out, most mathematical statements are in fact either true or false. Davies is simply pointing out that there are exceptions.

Davies is using sloppy language, though. Gödel's theorem doesn't say that some mathematical statements are neither true nor false. What it says is that, in some formal systems, there exist some statements whose truth or falsity cannot be proven, one way or the other, within the system itself. But even these statements may themselves actually be true or false. In Davies's defense, however, I'll say that this distinction is lost on most people.

[Alamo-Girl]

Thanks for the ping!

[Consort]

A point is the beginning of everything; the departure of the unmanifested and the beginning of manifestation. Matter is formed point by point, by line, by angle, by surface, and by completing curve. When massed together, they appear as physical objects and are perceived as surfaces. And then, maybe not.

[edsheppa]

You're using terminology in a confusing way. 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).

[Rocky]

"The Greek philosophers Leucippus and Democritus suggested that however complicated the world might seem to human eyes, it was fundamentally simple. If only we could look on a small enough scale of size, we would see that everything is made up of just a handful of basic building blocks, which the Greeks called atoms. It was then a matter of identifying these elementary particles, and classifying them, for all to be explained.


"Today we know atoms are not the elementary particles the Greek philosophers supposed, but composite bodies with bits inside. "

I would rephrase it: Today we know that what "modern" science chose to call atoms are in fact not the fundamental building blocks that the Greeks envisioned. The real "atoms" are much smaller.

I would not suggest that the Greeks could have had any concept of nuclear physics, but let's not blame them for coopting their word to name something which turned out not to fit the original meaning.

[edsheppa]

Number systems, formal as they are, were arrived at via human reasoning, which is not a formal system.

Which has no bearing. The number system is formal and contains Godel statements. It is (assumption) the natural product of human reasoning which is (another assumption) a natural product of physical law. By composition, the number system is a natural product of physical law. Ergo, physical law contains Godel statements.

Human reasoning is, I maintain, complete, in that all possible truths are in principle available to it ...

Not so. Human reasoning has bounded algorithmic complexity. Any truth exceeding that complexity is not open to it. So far as I know, there's no way out.

... including the truth or falisity of formally undecidable Gödel statements

Again not so. The nature of Godel statements is that they are neither true nor false but may be assumed to be either true or false.

I don't think I need to make a case that human reasoning is inconsistent.

Actually I think "unreliable" is what you're after. Inconsistency would be believing X and not-X at the same time and not thinking that's a problem.

[BiffWondercat]

Thanks for the ping. What about Tesla? Not the band...

[Doctor Stochastic]

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

Not true. What is true is that such systems cannot within themselves be proven to be contratiction-free. Perhaps Paul Davies doesn't fully understand current mathematics. Of course, not all of mathematics is subject to these limitations. Pressberger arithmetnc (no multiplication, just addition, but you can multiply by any given number through iterated addition; you just can talk about multipllication as such) has no such problems. Euclidean geometry (and thus the non-Euclidean geometries too) are provably consistent.

[BiffWondercat]

But that damn cat works into all of this somehow, some way, doesn't it?

[Doctor Stochastic]

Actually, CH isn't a Goedel statement in that either CH or ~CH can be added to ordinary set theory (Zermel-Franco axiomatization to be more precise) and either both systems are consistent or neither is.

It's analogous to the Axiom of Parallels in Euclidean geometry; one can postulate 0, 1, or many parallel lines can be drawn through point not on a given line; there are thus three (at least) geometries.

[Doctor Stochastic]

With a manifold, one can join the Local Chapter of the Flat-Earth Society without being commited to the Global Organization.

[Doctor Stochastic]

It seems that no system can be fully understood from within the system.

Not necessarily true; Euclidean geometry (with the Hilbert axioms) is complete and consistent.

[LibWhacker]

Human reasoning has bounded algorithmic complexity. Any truth exceeding that complexity is not open to it. So far as I know, there's no way out.

Is it known that there are provable theorems out there that are hopelessly beyond the human intellect to prove? I've never heard that claim made before in quite the way you just made it. Of course, there are a whole host of things that are undecidable in Godel's sense of it, but that's a fundamental constraint having nothing to do with our intelligence or lack of it. I've often thought that perhaps there is some "critical mass" ito intelligence, and that once a species exceeds that critical mass, it can in principle prove anything that can be proven or, alternatively, understand any valid proof. As Albert Einstein was supposed to have said, "The most incomprehensible thing about the universe is that it is comprehensible." Hope that's true. It's gonna be depressing if we're just another species of dumb apes.

[BiffWondercat]

Can I use the word domain, instead of mainfold?

[edsheppa]

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

Have you looked at some of Chaitin's work? The digits of his Omega number are an interesting case. His take on it is interesting: anything sufficiently complex is essentially random.

But on a more mundane level, any finite theorem proving machine (and I'm assuming people are that) will be helpless in the face of a theorem whose smallest proof exceeds the capacity of the machine.

[MHGinTN]

Didn't Michio Kaku say that no manifold beyond 2D is local?