Goedels Incompleteness Theorem

[humanino]

We certainly have to understand non-perturbative features of QFT. I know Marlon will one day solve the confinement problem. This should indeed come from topological properties. At least everybody hopes so by studying toy models for QCD.

But this is far from Godel ...

 

[master_coda]

we cannot tell nature how to work or what to do

Of course you can. I do it all the time. Nature doesn't seem to listen too well, though.

 

[humanino]

Let's get back to Godel !
Could someone explain how diophantine equations raised interest in Godel's theory ?

 

[humanino]

Of course you can. I do it all the time. Nature doesn't seem to listen too well, though.

Master ! I trusted you ! you cannot fail ! :surprise:

 

[master_coda]

Let's get back to Godel !
Could someone explain how diophantine equations raised interest in Godel's theory ?

One of Hilbert's problems was to produce a general method for determining if a given diophantine equation was solvable or not. It was not actually proven that no such method exists until the 70s.

Clearly if someone wants to prove that a particular problem in arithmetic is unsolvable, it's probably worthwhile to study the results of people like Godel. The incompleteness theorem was directed at arithmetic, after all.

 

[Eye_in_the_Sky]

1)YES
2)NO
3)YES

...

 

[matt grime]

This is precisely Godel's theorem. I say "any demonstration is obvious once you know how to demonstrate it". I claim that mathematics are obvious. The reason for that, the reason why mathematics are so easy compared to physics, is that in mathematics, one knows what one is talking about. There are definitions. In physics, we can only applies clever ideas to model reality.

I actually went through Godel's demonstration. Several times. It's technical, but one can understand every single step.

Now one can understand Godel's argument without understanding the technical demonstration. Actually, I would probably never have undestood the technical part of it, without the first section of the article, where he explains the "lier paradox" thing. Even though, of course, I had already heard it somewhere else, would it just be in the beginning of my book where Godel's article is reprinted

I see no argument there supporting your assertion it is obvious. If mathematics is easy can you post a proof of the Riemann Hypothesis then? Or do you mean the mathematics of the proof of Goedel's theorem is obvious? Or that each step is simple? "obvious" is a subjective statement, and tends to indicate when used in mathematical proofs that the result can be deduced with less than an hour of thinking about it.

 

[arildno]

This reminds me of how a certain Dr. John Watson thought once the steps of a deduction had been explained to him ("Gosh, was it that easy?" (or elementary..))

 

[humanino]

Well, I am quoting Michel Berger. Besides, I remember my math exams a few years ago. One can search for an answer really hard for several days without finding it. And then, when it comes to working the answers with the teacher, and once you get the answer, it is pretty obvious.

I don't mean to be provocative here.

 

[arildno]

No, you're not provocative.
You just sound like Dr. Watson..

 

[matt grime]

There is a difference between thinking, "gosh, why didn't I think of that" and the problem/theorem/solution actually being obvious in a sense that a professional mathematician would use, since there's not guarantee that you would *ever* have actually thought of it without hindsight. It can only be obvious if you've thought of the step without being told. So, I will accept that you consider Goedel's theorem to be obvious if you can with hand on heart say that you would, without prompting, have firstly thought it up as a conjecture, and then proved it. Thinking that the steps in someone else's proof are obvious is not a particularly note worthy thing since, if it is a basic explanation (ie omits no steps), written by a good mathematician, then it *will* seem obvious, but that is a function of the writer not the reader. If you're using it in that sense then you don't mean it's "easy" in the sense you could have proved it, you mean you've read a good explanation if it.

 

[humanino]

If mathematics is easy can you post a proof of the Riemann Hypothesis then?

Stay tuned !

Seriously, I feel sad everybody is taking what I said the wrong way

Or that each step is simple?

Yes, it would be closer to that. But once you know each single step, you get a general overview of a proof. I guess I just have difficulties to express myself in a foreign language. Obvious might ot be the word I need.

 

[humanino]

So, I will accept that you consider Goedel's theorem to be obvious if you can with hand on heart say that you would, without prompting, have firstly thought it up as a conjecture, and then proved it.

So I must admit I was wrong I guess.

I was basically trying to single out a specific feature of mathematics as compared to physics for instance. But then, mathematicians most often tend to think physics is just applied math.

 

[humanino]

When confronted to a new result in math, one can work out the proof and understand it because everything is clearly defined.

When confronted to a new result in physics, one can undestand the mathematics without getting the feel for the physical process involved.

Well, maybe I should just forget it...

 

[matt grime]

"obvious" is perhaps not the right word for *mathematical* reasons (but as I hope I said, this is very subjective, and it is only MY belief that that is what obvious *should* be reserved for) and is I suggest not reflective of the fact that just because there appears post facto no other way of thinking that that was how you'd have thought about it. Here's an example of someone missing the point, it's hearsay, allegedly said by a fellow at Trinity Hall at high table one evening. After watching Panorama program about Wiles's proof of FLT, the fellow, not a mathematician, said to a mathematician, "yes, I think I understand the proof".

An obvious proposition is, say, if n is even, then n^2 is even.
Goedel is not obvious in the sense I understand the word. If it were obvious how come there were no well examples of statements that were unprovable before the conjecture was proven, that it caused some mild outrage (mainly for philosphical reasons, apparently), and that for a while no reasonable mathematical statements were known to unprovable? Ok, so you've read a good explanation of the proof, that doesn't make it an obvious result which was how it sounded to me.

 

[matt grime]

When confronted to a new result in math, one can work out the proof and understand it because everything is clearly defined.

When confronted to a new result in physics, one can undestand the mathematics without getting the feel for the physical process involved.

Well, maybe I should just forget it...

It is only a result AFTER it has been proven. Mathematics behaves like physics in this regard. You need to look at the evidence and formulate a reason as to why it's true, then prove that the conjectured reason is true. If you think you can quickly work out the proof, then I suggest you've not done any difficult mathematics.

Let me give you an example of what part of my PhD is motivated by.
Several people in the 80s (and before) noticed that there were many similarities between representation theoretic facts of a group and normalizers of some p-subgroups. The so-called local representation theory. However there were no known links between the actual theories that worked in all situations. For instance a "Morita equivalence" would explain it, but it was true for things that were known to be not morita equivalent. Eventually M. Broue conjectured that a "derived equivalence" was the reason for it. He made this in the early 90s. Now, more than ten years on no one has proved the conjecture in full generality but they have for almost every reasonable group (reasonable in terms of size or important characteristics, for instance it has been shown for all permutation groups). If you still think maths is simple then heaven help anyone who does a difficult subject like physics (hint, tongue in cheek parting shot). But perhaps it might convince you research mathematics is more like a science than you realize.

 

[mathwonk]

While it is fun philosophically to play with the scary implications of Goedel's theorem, i.e. that there may exist lots of true but unprovable theorems in a given system,

in real life, it seems harder to find interesting problems or conjectures that some smart mathematician cannot eventually (or rather quickly) prove or disprove.

I.e. it often seems the problem we have is the exact opposite of the one we are worried about here. It is hard to think up good problems that actually last awhile in the face of the onslaught of research progress.

Of course there are a few famous problems that have lasted quite a while, like Goldbach's conjecture, and the Hodge conjecture, but it may be that Goldbach (although not Hodge) has lasted so long because not too many people are
interested in it.

FLT (Fermat) of course lasted a while, but finally bit the dust, so these time frames may actually be measured in hundreds of years, making it hard to know.

But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel.

I think Hilbert showed his greatness with his famous list of problems in 1900. They really inspired people. The centennial celebration of that event in 2000 at UCLA failed pretty badly in my opinion to replicate his feat with a new list of interesting problems, although there were some nice talks.

What do you guys think?

 

[matt grime]

"But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel."

However, they aren't the only ones. Continuum hypothesis, Collatz type conjecture, etc. The type of set theory you use can be important and not just to set theorists (Shelah showed that Ext_{\mathbb{Z}}^1(A,\mathbb{Z})=0 implies A is free, depends on the set theory you adopt.)

 

[marlon]

"But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel."

However, they aren't the only ones. Continuum hypothesis, Collatz type conjecture, etc. The type of set theory you use can be important and not just to set theorists (Shelah showed that Ext_{\mathbb{Z}}^1(A,\mathbb{Z})=0 implies A is free, depends on the set theory you adopt.)

Matt Grime...snizzlefizzle, that the answer!

regards,
MaRLoN

 

[humanino]

"But as long as the only unprovable statements are barn burners like "I am unprovable", many people have somewhat lost interest in Goedel."

However, they aren't the only ones. Continuum hypothesis, Collatz type conjecture, etc. The type of set theory you use can be important and not just to set theorists (Shelah showed that Ext_{\mathbb{Z}}^1(A,\mathbb{Z})=0 implies A is free, depends on the set theory you adopt.)

Yes yes yes. Those are the important ones.

At some points, mathematicians get into a position like "we have that very important assumption : let us look closer if it is independent of our already classified axioms". And : either the new assumption is demonstrable, or it is not in which case we are glad to say "We found a new axiom to classify"