The Poincaré Conjecture has Been Proved

Martin Dunwoody, a famous mathematician who works in the field of topology has a preprint that provides a proof of the Poincaré conjecture. This was one of the seven Clay Mathematics Institute millenium prize problems (reported on Slashdot here). The solution to each of the problems carries a monetary reward of 1 million dollars. However there are a number of conditions that still need to be met for the prize to be awarded in the case of the Poincaré conjecture.

Comments

Wierd Problem

[KagatoLNX]

If you follow the link to the description of the problem, it gets really wierd. Apparently this is one of those problems where you have to prove it for 1=7} but no one ever managed n=3 (which was the original, non-generalized conjecture anyways). Funny that this guy just had to fill in the last blank.

Re:Wierd Problem

[squidinkcalligraphy]

seems rather an inelegant way to make a general proof; general proof for n>=5, the seperate proofs for n=1, n=2, n=3 and n=4. Does this new proof just do n=3, or is it a _nice_ general proof?

Re:Wierd Problem

[leviramsey]

seems rather an inelegant way to make a general proof; general proof for n>=5, the seperate proofs for n=1, n=2, n=3 and n=4. Does this new proof just do n=3, or is it a _nice_ general proof?

I see no inelegance to this method. One of the steps in the general proof may only work if n>=5. This does not mean that the general proof is invalid.

Essentially, the same method underlies inductive proof (e.g. a general proof that holds for n>s, and a demonstration that n=s combine to n>=s).

Re:Wierd Problem

[Gary Yngve]

Without reading the preprint, I cannot say (not that I could understand it anyway :) ). But it wouldn't surprise me if the proof was just for 3.

R^3 is kind of a magical place. R^2 might not have enough wiggling room, but R^4 might have too much. There exists a cross product in only R^3.

Re:Wierd Problem

[splorf]

Here's some further info on the Poincare conjecture.

This proof does just d=3 and it's interesting that it's essentially combinatorial. Smale's proof for d>=5 was based on differential topology, a grand and beautiful branch of pure higher math. Freedman's proof for d=4 used Yang-Mills theory developed in particle physics. d=3 looks like essentially a computer scientist's proof.

Disclaimer: I don't understand this stuff in any detail--these remarks are based on looking at the preprint and remembering stuff that I heard in math class long ago. Also, I think I'll wait to hear what the math community says, before believing the problem is really finally solved.

Re:Wierd Problem

[richard-parker]

seems rather an inelegant way to make a general proof; general proof for n>=5, the seperate proofs for n=1, n=2, n=3 and n=4. Does this new proof just do n=3, or is it a _nice_ general proof?

M.J.Dunwoody's proof is restricted to the n = 3 case.

Re:Wierd Problem

[NMSpaz]

Given that one of the links states that all cases other than 3 had already been proven, it's quite likely that the proof indeed for that case. </sarcasm>

Re:Wierd Problem

[avsed]

No, one can have a cross (outer) product in any number of dimensions, just as one can have an inner ("dot") product in any number of dimensions. Tensor calculus is the generalisation of ordinary geometric calculus that describes this.

Dan

Re:Wierd Problem

[Anonymous Coward]

Actually, if you read up on the problem, you'll see that although the proof is for n=3, that actually means he's working in R^4.

Re:Wierd Problem

[crawling_chaos]

This is an inductive proof. Anyone who finds the technique inelegant better worry, since pretty much the entire field of Computer Science is held up by similiar "inelegant" logic.

Come to think of it, most of Abstract Algebra is as well. Someone want to help me out here? I only audited AA at eight in the morning and dated a math major. I wasn't paying much attention to math on either occasion.

Re:Wierd Problem

[squidinkcalligraphy]

I'm not suggesting the general proof is not valid, nor am I questioning induction (and, indeed, I find induction quite elegant), but I can't consider a general proof which consists of 5 completely different proofs for different ranges of numbers to be elegant. its the same kinda thing as writing a whole bunch of if-then-elseif-elseif... statements as opposed to a one-line-of-code general statement.

Re:Wierd Problem

[Shaheen]

Lots of problems are like this. Namely, every NP-Complete problem is like this.

You can prove that a problem is NP-Complete by restating the problem as a polynomial-time manipulation of another problem. Sort of like "if the red marbles in this problem are considered the nodes of the tree in this other problem, then I can say this because I know something about red marbles."

In this case, n > 3 is simply a problem that can be stated as a manipulation of n = 3.

Re:Wierd Problem

[JanneM]

On the contrary, inductive proofs are often among the most elegant ones I've seen. Just establish a base case, then show that if the previous case is true, then so must the next one.

/Janne

Re:Wierd Problem

[caffeined]

Yes, the proof is only for n=3. Poincare's original conjecture was only for n=3, but was later extended to be for all n. The other cases have been proven already, so this proof takes care of the remaining (original) case.

If you're interested in reading up on it a bit, the link in the original post to "http://mathworld.wolfram.com/PoincareConjecture.h tml" is excellent. (And it's where I learned the above stuff about the various cases.

Re:Wierd Problem

[Anonymous Coward]

I've discovered a truly remarkable proof myself. I just can't fit it into this HTML text box.

Re:Wierd Problem

[BlowCat]

That site doesn't define 3-manifold and 3-sphere. Is it a 3D sphere in 4D or a 2D sphepe in 3D? It it's the former, then the result hardly has any practical use, since we live in 3D and mostly deal with 2D manifolds :-)

Re:Wierd Problem

[Anonymous Coward]

Have you ever considered that the universe isn't one general if statement? Seriously, get a grip. What you think of such proofs, and thusly mathematics is clearly unimportant, since you obviously have no idea what the fuck you're going on about.

Re:Wierd Problem

[Anonymous Coward]

Yet another slashbot displays his awe inspriring ignorance.

Re:Wierd Problem

[SIGFPE]

If by cross product you mean a bilinear map (a,b)-> a x b on some vector space where the product is orthogonal to the arguments and |a x b|^2+(a.b)^2=|a|^2|b|^2 then you may be surprised to hear that there are many cross products in dimension 7. They come from the octonions, the non-associative 8D generalisation of the quaternions. See here.

Re:Wierd Problem

[Anonymous Coward]

"You can prove that a problem is NP-Complete by restating the problem as a polynomial-time manipulation of another problem."

With rigor, damnit! "of another NP-Complete problem."

Without the correction, you just told everyone that any NP-Complete problem is solvable in polynomial time, which I'm pretty sure isn't what you meant.

"Without mathematical rigor, you are nothing but a numerologist!"

Re:Wierd Problem

[Anonymous Coward]

The 3-sphere lives in R^4, not in R^3. The Poincare conjecture was solved in R^n for all n > 4 by Smale in the sixties leaving open only the original problem for n = 4. If Dunwoody is correct, he will be immensely famous.

R^4 is however incredibly weird. There are nontrivial statments about R^n true for all n except n = 4.

It is possible to generalise cross products to higher dimensions using tensor products.

Re:Wierd Problem

[colmore]

3 dimensional spaces are exceptionally strange. there are a large number of theorems that have trivial proofs for 1 or 2 dimensional spaces, and work out very easily for 4 or more, but for some reason, 3-space is highly irregular.

My knowledge of topology isn't terribly deep, so you'll have to ask others for more info.

Re:Wierd Problem

[mreece]

Uh... sort of. First of all, the cross product isn't just any outer product, it's an antisymmetrised one, more of a wedge product than a tensor product. Anyhow, you can certainly do similar things in other dimensions, but the objects you get won't all be "of the same type", in a loose way of speaking. Somewhat more precisely, only in 3 dimensions do we have that the spaces of 1-forms and 2-forms have the same dimension, so that if a and b are one forms, (a /\ b) can be somehow identified with another one form. Even in 3 dimensions there really isn't a canonical way to do this, which is why cross products end up needing an orientation (the whole right-hand rule thing, which is rather arbitrary).

Re:Wierd Problem

[Anonymous Coward]

Ooh, look how fucking huge your mathcock is---you can tell someone they're an idiot without even pretending that you know the right answer.

Fucktard.

Re:Wierd Problem

[ajmarks]

One can easily define a cross product in R^4. It would however operate on three vectors, not 2. In fact, it is quite simple to define a function CrossP(v1,v2,...,vn) that returns a vector normal to v1,...,vn in R^(n+1) (what length you give the vector is a slightly different matter).

Re:Wierd Problem

[Anonymous Coward]

That's just plain funny.

Re:Wierd Problem

[Anonymous Coward]

Can you say "nonassociative"??

Re:Wierd Problem

[Gary Yngve]

Sure, and by the Celebrated Theorem of Froebenius, there is no such thing in a higher dimension. :)

Re:Wierd Problem

[SIGFPE]

Why not? Here's a book on the subject.

Re:Wierd Problem

[Gary Yngve]

Whew, it's been a long time since I've taken algebra... Froebenius's result just applies to associative division algebras. Octonions are the only other finite-dimensional real division algebra.

http://mathworld.wolfram.com/DivisionAlgebra.htm l

now I've seen it all

[squidinkcalligraphy]

The Poincaré Conjecture proved, and microsoft ads on slashdot

Re:now I've seen it all

[Anonymous Coward]

ditto - time to go somwhere else.
Can the world have ONE geek place that is pure,
i.e. the only mention of MS is when we complain
about the virus-of-the-day. And I don't want any of the editors to be also writing one-line-plot pet-books influenced by Oprah and available for the M$ bookreader.

Re:now I've seen it all

[Anonymous Coward]

I have definitely seen one MS ad here on slashdot. This cannot be happening.

Well..

[autopr0n]

Thanks for explaning what it is... or at least what it applies to/why it's important.

Re:Well..

[Anonymous Coward]

Yeah, fucking hell. I read that Mathworld stuff three times and I'm still barely any the wiser.

Anyone want to take a shot at explaining this without using "homeomorphic" and "simply-connected."

Re:Well..

[nucal]

All right, I'll bite - and please correct me if I'm wrong. Basically I think that the gist of the Poincaré's Conjecture is that in a given dimension all surfaces of a certain class are equivalent. In other words, you could (virtually) remold a solid cube into a solid sphere - but if you have a cube with a hole running through it, you can't remold the cube into a solid sphere without closing the hole, but you could remold it into a donut. So my simple-minded way of looking at this is that all non-holed objects are equivalent, all one-holed objects are equivalent, all two holed objects are equivalent, etc., and represent distinct classes of objects.

How'd I do math-jocks?

Re:Well..

[danielrose]

I don't get it, what is so difficult? Here is one more time for you slow guys:

The n = 1 case of the generalized conjecture is trivial, the n = 2 case is classical, n = 3 remains open, n = 4 was proved by Freedman (1982) (for which he was awarded the 1986 Fields medal), n = 5 by Zeeman (1961), n = 6 by Stallings (1962), and n >= 7 by Smale in 1961. Smale subsequently extended his proof to include n >= 5.

Now what part doesn't make sense? *efg*

Re:Well..

[Anonymous Coward]

Because not everyone is busy running porn sites like you. There are others here who'd contribute a lot more constructively to society, like doing maths & physics, and leading a life that's intellectually satisfying.

:-/
(disgruntled nerd)

Re:Well..

[Anonymous Coward]

For the record, from appearances, he doesn't run a pr0n site for a living. Plus, it's free-as-in-free-beer pr0n (+1 nerd), and he's looking at women, which means he's either male, horny, and without outlet for sexual tension (in varying degrees, +1 to +5 nerd), or a horny bi/lesbian fem-nerd (+500 nerd, in the hope one will sleep with me in return for my free nerd-points).

Re:Well..

[ethereal]

I tried that, but I really couldn't accomplish anything by providing free nerd-points. I wish you luck with it, though - and if you find that island or planet where the women do appreciate nerd-points, let us all know, OK?

Re:Well..

[Anonymous Coward]

if you cut'n'paste please give correct credits (in this case MathWorld).

Re:Well..

[Anonymous Coward]

The Poincare conjecture is more restrictive than that. It just says that in 3 dimensions, all finite, "zero-holed" (a rough description of "simply connected"), unbounded surfaces are topologically equivalent.

Re:Well..

[danielrose]

I didn't cut and paste, I copied and typed! Does this make me exempt?

Re:Well..

[ajmarks]

R^3 (the minimal space in which the n=3 case exists) is rather odd. In R^0, R^1, and R^2, most things are pretty simple. In R^4+, there's enough room to avoid problems (in R^4 or higher, for example, you can neglect any countable set of 3 dimensional spaces while integrating, etc.). R^3 is probably the hardest space of finite integral dimension in which to work, which is unfortunate, because that's where almost all of engineering takes place.

Re:Well..

[Anonymous Coward]

Maybe, the Original Beings existed in higher-dimensional spaces, but we evil humans were banshed to three-dimentional space and trapped here, as a punishment, until we wisen up. (Think of that Superman movie, in which the three evil super-beings from Krypton were trapped in that crystal, imprisoned for their crimes, before being accidentally freed by a nuclear explosion.)

teacup == donut

[Anonymous Coward]

so we finally have mathematical proof that a teacup is a donut for every teacup in the known (Euclidean) universe

Re:teacup == donut

[Quirk]

Charles Dodgson, somewhere thru the looking glass, is at tea with the Mad Hatter discussing this very matter.

:)

Re:teacup == donut

[JabberWokky]

No, we're doing soirees and diagramming whether or not the bat can be black at dawn.

--
The JabberWokky (yes, I know. It's intentional to create a unique string.)

Re:teacup == donut

[Jouster]

Actually, a teacup does not equal (!=, or <> for you freaky folks) a donut. Properly wrapped, a "rubber band" could become smaller and smaller while maintaining definition, since the bottom of the teacup is a definite surface. Now, were said rubber band wrapped around the rim, you'd be correct.

What does this amount to? Look at the difference between a donut and the "donut holes" sold by some bakeries. A donut is a surface with a hole in the center; a donut hole is a hole with a surface in the center. From this, is becomes clear that ANYTHING can be said to be encased within a larger "hole".

Feel free to do your own extrapolation from there--personally, I stop just short of a unified field theory before logic stemming from that conjecture breaks down.

Jouster

Re:teacup == donut

[djmurdoch]

Teacups have handles.

Re:teacup == donut

[Anonymous Coward]

Teacups have handles you know, that's where the hole is.

Re:teacup == donut

[Jouster]

I'm aware of the existence of handles on teacups. However, by wrapping such that the rubber band extends over the top of the cup, across the bottom of the cup, and along a side that does not additionally protrude to form a handle, one has a definate surface (the bottom of the cup) with which to work.

Jouster

Re:teacup == donut

[Anonymous Coward]

A teacup is topologically equivalent to a donut. Yes, you can find contractible loops on a teacup that enclose a "definable surface", such as the one you describe. But you can also find them on a donut.

I think you're confused about what the words "topologically equivalent" (homeomorphic) mean.

Re:teacup == donut

[Jouster]

Please, by all means, describe a contractible loop on a donut that produces a "definable surface" and doesn't cheat in its construction (e.g., assume the donut cannot be cut, and neither can the rubber band). It has been my belief such a loop does not exist, but I am, as always in such matters, eager to have someone prove me wrong.

Jouster

Re:teacup == donut

[Anonymous Coward]

Pick any point on the donut and draw a little circle around it. Unless you're using some weird definition of "definable surface", this circle will bound a simple closed disc on the "inside", and the rest of the donut on the "outside". This is opposed to, say, a noncontractible circle that wraps the donut hole and splits it into a cylinder-with-boundary.

Re:teacup == donut

[Jouster]

Granted, this works, but it's breaking the rules, so to speak. We are using physical analogies, and one cannot--without breaking the continuity of one or the other--wrap a rubber band around the outside ring of a donut. (Ignore momentarily that one could conceivably do this prior to cooking. :>)

Jouster

Re:teacup == donut

[Anonymous Coward]

What "rules"? The ones you made up? It is a mathematical fact that a teacup and a donut are homeomorphic. If you want to alter the mathematical definitions involved, and say that according to your definitions, a teacup and a donut are not homeomorphic, that's your business -- but it has nothing to do with what everyone else in the world means by "homeomorphic".

Re:teacup == donut

[Quirk]

...soirees and diagramming whether or not the bat can be black at dawn

Too brillig at dawn to empirically test the theory? Surely not for the JabberWokky, but, if so, then merely fall back on 'What I tell you three times is true'.

Proof

[droyad]

Anything can be proved with enough flawed mathematics. Think how many times things have been proven, only to be found flawed later on? That is the foundation of the scientific method.

Re:Proof

[Quirk]

Popper fan I presume.

:)

Re:Proof

[phooka.de]

Anything can be proved with enough flawed mathematics. Think how many times things have been proven, only to be found flawed later on? That is the foundation of the scientific method.

Nope, this is where you're wrong. Math is different from any other science when in comes to "proving" things.Compared to a mathematical proof, any other scentific "proof" is just a currently accepted working theory.

That's the strangth and beauty of mathematics: once it's proven, we know that it's true until the end of days, not even God could change it if he exists. not even if the laws of physuics suddenly changed and altered all we know about the universe would our mathematical proof become untrue.

Compared to mathematics, even physics is - as a science - not much more proovable than sociology.

Re:Proof

[Anonymous Coward]

Of course it depends what axioms you take to be true to start with.

Re:Proof

[phooka.de]

For which reason all mathematic proofs should start with something like "given that the following assumptions are true" or "based on the following axioms...".

I guess calling it a mathematical proof includes this, though.

Re:Proof

[bentini]

Wow. I wish I could highlight a section of your post to point out as being wrong as you did the the grandparent. Unforutnately, I can't. You're wrong all throughout.

First, how do you show something is proven? Well, you give a proof. How do I know the proof is correct? I work through all the steps... But what if I mess up and sneeze and my thinking gets confused and I accept something that isn't true? It could happen. Well, I'll just push it through a formal logic computer program that checks it.
But what if the computer has a glitch and a 0 or a 1 gets accepted. Or worse, I made the error while programming the formal logic system. Or more subtly, the compiler or hardware.

Basically, it's like this, proofs are as much a social event as a mathematical cedrtainty. Proofs are presented, and believed, and then refuted. Mathematical proof is a social process carried on by mathamaticians, and you can't forget that. I'm sure that I've proved things incorrectly before, and believed them. Just because nobody's found an error in a published and accepted proof doesn't mean one doesn't exist. If you think that humans can do ANYTHING with probability 1, you're sorely mistaken and are seeing the world in too convenient terms.

Sorry to burst your bubble, but there's a lot of thinking in this. Peer review does not imply flawlessness.

Re:Proof

[cheezehead]

Anything can be proved with enough flawed mathematics.

No, it's the other way around. Kurt Gödel proved that there will always be mathematical truths that cannot be proven with a mathematical system, no matter how the mathematical theorems and rules are extended. This caused quite a shock in the mathematical circles early last century. It took Douglas Hofstadter (See: Gödel, Escher, Bach : An Eternal Golden Braid, ISBN 0465026567) an entire book to explain this. No way I could explain it here, even if I could remember :-).

Re:Proof

[mizhi]

Isn't this just Godel's theorem? That in any axiomatic schema, there will be certain propositions that can not be proven within it.

Re:Proof

[ariels]

Of course this is untrue. Using flawed Mathematics, one can do anything. Specifically, Gödel showed that a first-order logic axiomatization of any "sufficiently complex" system (arithmetic is more than complex enough) is either inconsistent or incomplete. What you describe is incompleteness. But of course any system which can prove P and also ~P (i.e. is inconsistent) can prove anything (and its negation); you don't need Gödel for that!

As I always say, "if 2+2=5, then the Poincar Conjecture is true".

Re:Proof

[khuber]

An error in John Nash's 1956 "The Imbedding Problem for Riemannian Manifolds" wasn't found until Solovay reported it in 1998.

http://www.math.princeton.edu/jfnj/texts_and_graph ics/erratum.txt

-Kevin

Re:Proof

[squidinkcalligraphy]

In mathematics, mistakes are usually picked up quickly. Once something is proven, it usually stays proven (in maths). The only thing it relies on are the underlying axioms, which rarely change in mathematics.

However, it is provable that any formal system (maths, logic, etc) cannot be both coherent and consistent (see Godel or Turing). So u get a necessary paradox underlying any formal system, including mathematics, so its all bullshit anyway.

Which is why I'm a fan of absurdist logic. All successful arguments must contain a contradiction/paradox somewhere. Politicians are followers of this; they have acquired the fine skill of disguising this contradiction (much like the proof of 5=4 elsewhere in this discussion).

Re:Proof

[perky]

Godel's incompleteness theorem in two sentences: Within any self-consistent system of formal logic there will be theorems that are true, but which cannot be proven within that system. This is such a theorem.

Re:Proof

[da cog]

Wow... so much for my blind faith in logic...

I mean, who really takes the idea seriously that just because

*) ALL DOGS ARE MAMMALS
*) FLUFFY IS A DOG

that it MUST follow that Fluffy is a mammal? After all, there is absolutely NO WAY that we can know this for sure. For all we know, Fluffy could really be a bird, or even a space alien trying to take over the world. In fact, we don't even know that Fluffy is a dog. After all, the person asserting that he was COULD have been wrong, couldn't he have? Despite those arrogent logicians and their assertion of provable certainty, we still know ABSOLUTELY NOTHING about Fluffy, nor will we ever.

All of these years, I had believed that, if nothing else, 1+1 must equal 2... but recently I have started to doubt this. Big Brother's assertion that 2+2 actually equals 5 just makes more and more sense every day that I think about it... perhaps more of the things he says are really true.

Nothing... NOTHING, can ever be known with 100% uncertainty--and since that includes this sentence, we don't even know for a fact that nothing can truly be known! Feeling depressed at this? Hopeless? Well, look at the bright side: you might not be feeling anything at all, since there exists the very distinct possibility that you don't actually exist. After all, nothing can be known for certain, so your own belief that you exist may turn out to be fundamentally flawed.

Now, if you'll excuse me, I am going to go back to throwing myself at the ground in the hope that I'll miss it and be able to fly... people keep telling me that this is impossible, but when I succeed, ho ho!, then all of you scientists, mathemeticians, and logicians will be revealed for the flawed fools you really are...

*climbs up steps*
*jumps off building*
"I'm flying!!! I'm..."
*THUD*
"Dammit... well, as they always say, 3,333,333rd time's a charm!"
*climbs up steps...*

Re:Proof

[caffeined]

Your comments are good as far as they go, *but* they can be taken too far.

Yes, it is true that a proof might be mistaken and that the mistake might not be caught. This is much like the scientific process, though, in that later work which builds on it can lead to a result which is inconsistent with other accepted proofs, leading to the original proof being re-questioned. Just as in science, the bedrock proofs, from which other proofs build, are constantly being implicitly re-tested.

I agree that you can never be 100% certain of anything (other than the base axioms which are simply defined as being true), but the probability asymptotically approaches 100% the longer that the proof stands without producing a contradiction of some sort.

To me, it's like what Popper said about the scientific process - things can be disproved by coming up with a counter-example, but you can never definitively prove something because that would imply testing/checking all possible situations - an impossibility.

But, to say that this means that "truth" is "socially constructed" takes this too far. It appears to imply that *any* result could be arrived at and be allowed to stand. Since math is a competitive process (like science) in which you can make your reputation by showing that an accepted "fact" is not really true, any statement which doesn't have some intrinsic merit will eventually be shown to be bogus.

Many of the thinkers who have come up with these theses of "socially-constructed truth" tend to come from the "soft"-er disciplines, such as lit crit and philosophy. I think that many of them suffer from a sort of "credibility envy" in which they are uncomfortable with the fact that the results of their studies are not accorded the same degree of respect as those of say, physics, or math. Therefore, in order to elevate their disciplines to the same level of respect as the "hard"-er disciplines, they need to show one of two things - either that their disciplines are just as rigorous as the "hard"-er ones, or that the so-called "hard" discplines aren't really all that "hard" and are in fact just "soft" disciplines in disguise. They have opted for the second line of attack.

Re:Proof

[Florian Weimer]

Gödel's Incompleteness Theorem does not apply to any set of axioms; the set of axioms has to be consistent and you must be able to express certain properties of the integers.

Re:Proof

[Anonymous Coward]

You should check out
Dan Willard's work he is exploring the boundaries of Gödel's incompleteness theorem and finding some (restricted but still interesting) systems with stronger self consistency than predicted by the incompleteness theorem.

In case you are wondering who Willard is, he is the guy who determined that sequential sorting could be done faster than O(N log N) (asymptotically, not practically unfortunately as the constants are large) without resorting to radix sorting.

Re:Proof

[Anonymous Coward]

If you think that humans can do ANYTHING with probability 1, you're sorely mistaken and are seeing the world in too convenient terms.

with probability 1, a human takes a pair
of integers m and n and then a mystery happens, namely the following holds: (m+n)^2=m^2+2mn+n^2

Actually, (m+n)^2=m^2+2mn+n^2 is an example of
a theorem that is certainly true.
Certainly, few thousands years ago this theorem
was quite a mind-boggling statement.

Poincare theorem(or conjecture) will be in time
almost certain kind of statement (or it will be
disproved). Or it will be forgotten.

Please don't mix up the real world and the
mathematical abstractions...

Re:Proof

[23]

Sorry to burst *your* bubble, but you're confusing 2 things: the _nature_ of mathematical proofs and the 'complexity' of recent deep/famous ones.

A mathematical proof is a logical consequence, given some axioms, including: 'Given, that we follow a certain scheme of logic'. That a^2 + b^2 = c^2 in a triangle with a 90 angle is a consequence of certain axioms and some logic. And, as the above poster pointed out, that will be so to eternity, precisely for the reason, that axioms do not have anything to do with the world per se and thus are not required to be checked against reality and/or changed.
That was all the above poster said and being a physicist, I admit that that's true with all ad- and disadvantages.

Now on the other hand, if a complicated proof can only be be peer-reviewed by so many people there is of course a danger, that it might be no proof, because they all make the same error. But that's a fact of life, and the probability of a proof being a proof goes to one, as the number of reviewers goes to infinity. Same thing applies btw. to your formal logic checking soft/hardware.

Your rant on non-perfectness of peer review has nothing to do with what the above poster said, i.e. that there is an inherent difference between math and any science, that tries to explain/explore the real world. One is a formal construction, the other tries to explain what is plainly there, with or without human brains trying to interpret it. You can not 'prove' anything in science, since the world doesn't give a fuck about the way you think about it and you can never be sure, that your axioms are correct. Conversely pure math doesn't give a fuck about being applicable to the real world.

Amen

Re:Proof

[Anonymous Coward]

We could also put it this way - mathematical proofs are by far the easiest to verify compared to any other proofs, as they rely only on abstract concepts and can be verified objectively, not observations that may be incorrectly interpreted, coincidental or experienced by someone else who you would simply have to trust.

Re:Proof

[SIGFPE]

The likelihood that the Fundamental Theorem of Algebra (FTA) is false is so small, notwithstanding what you say about drunkenness, that it's more likely that you don't understand English. In fact it's more likely that your understanding of English is so poor that you've said the opposite of what you mean by accident rather than that FTA is false. Given these odds it's probably better to keep quiet than spout stuff about how FTA could be false don't you think?

Re:Proof

[Darby]

Think how many times things have been proven, only to be found flawed later on?

That's the strangth and beauty of mathematics: once it's proven, we know that it's true until the end of days

Well, in a sense, you're both right.
There have been many cases where a proof has been given for a problem which is accepted at first and only later found to be flawed.
Now, if the proof is correct, then it is *true*.
If it is flawed, but not known to be flawed then it might not be true, but it is thought to be.

Re:Proof

[colmore]

yes, but most math is based on Peano and Euclid (or variations)

those axioms are pretty hard to deny.

Re:Proof

[norton_I]

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.
-- Albert Einstein

Really, we do have proofs in physics(for example) that are just as provable as those in mathematics. You just have to understand that proofs of any kind are made based on certain assumtions (axioms + rules of logic).

For instance, the quantum no-cloning theorom states that you cannot exactly duplicate an unknown quantum mechanical state. This is an absolutely proven theorom -- one of the axioms of which is the Schrodinger equation. If we ever find that quantum mechanics is not the correct description for our universe, the no-cloning theorom will still be entirely valid within the constructs of QM, as well as the regime of the universe under which QM is applicable.

Likewise, Euclid said the sum of the angles of a triangle is Pi, but this is only true for trinagles in spaces that have a certain structure, which is why we call it Euclidian. It turns out that in general, space is non-Euclidian, though unless you are near a black hole or a neutron star, the difference is hardly noticable.

Computer scientists have "proven" using very general methods, that there are no algorithms for computing certain things that are faster than a given bound -- There is no way to search an unordered list in faster than O(N) time, no way to sort arbitrary numbers in less than O(N*Log(N)) time, etc. However, this is based on a Turing machine model of computation, and the laws of quantum mechanics as we understand them allow computers intrinsically more powerful than a turing machine. We still don't understand much about what these quantum computers can and can't do better than a classical computer, but we do know that they can search unordered lists faster than any classical computer, though I think it has been shown that they cannot sort lists faster than a classical computer.

Re:Proof

[Anonymous Coward]

If you think that humans can do ANYTHING with probability 1, you're sorely mistaken and are seeing the world in too convenient terms.

Ah, but you've forgotten the simple beauty of a tautology! Humans can be human with probability 1.

Re:Proof

[Anonymous Coward]

With probability 1, a human picks a number, and it is not the largest currently known merseinne prime plus 5.

Re:Proof

[nihilogos]

Basically, it's like this, proofs are as much a social event as a mathematical cedrtainty. Proofs are presented, and believed, and then refuted.

Most proofs are presented, not believed, and refuted. Good proofs are presented, not believed, subjected to scrutiny, bolstered by alternative proofs and finally accepted. The peer review process is remarkably successful because most mathematicians and scientists actively pratice skepticism. Name one proof that has been accepted for 50 years and then shown to be incorrect.

I'm sure that I've proved things incorrectly before, and believed them

That's not much of a peer review process is it?

Peer review does not imply flawlessness.

The original poster didn't actually mention peer review - you did. He/she was referring to the absolute certainty that seems to be inherent in some mathematical theorems. I put it to you that Euclid's proof that there are infinitely many prime numbers is flawless.

Re:Proof

[nihilogos]

50 years?

Re:Proof

[ajmarks]

Mathematicians don't make names for themselves by disproving things. They earn fame for proving new things (or outstanding conjectures). If Theorem A has a valid proof, than there cannot be a counterexample to it, or the system is flawed. Since it is most likely that most of mathematics is not done in a flawed system, there is no way to disprove something once once it has been proven mathematically.

Re:Proof

[ajmarks]

Integers can be sorted in O(N) time (actually, so can floats, but unless you're sorting trillions of items, go with the O(n*log(n)) algorithms.

The algorithm for integers is simple (no value is passed for x on execution):
1. If the list contains 0 or 1 element, return.
2. Set a counter to 0, unless a value is given for x, set x=2^(n-1), where your ints have n bits.
3. Iterate through the list:
3a. If !(a[i] & x), then
3aa. swap a[i] and a[j]
3ab. increment j.
4. Apply algorithm to the sub-array a[0..j-1] and a[j..n].

This impliments nicely in C where pointers take care of the dirty work involved in passing parts of arrays (pass a pointer to the middle of the array).

Re:Proof

[markmoss]

proofs are as much a social event as a mathematical cedrtainty.

You are going a bit too far. Proofs do require a social activity before they are accepted, namely rigorous checking by a number of other mathematicians. But most proofs have survived such checking as to be mathematical certainties -- within a given system of axioms.

For instance, the Pythagorean theorem has a simple one page proof that has been reviewed by every mathematics student for over 2,000 years, and no flaw has been found. It's certain -- within Euclidean geometry. It also is known to have limited real-world applicability: it won't be exact for right triangles drawn on the curved surface of the earth, nor (according to General Relativity theory) will it be exact for large triangles in space. But it's close enough for most surveying work, and more than close enough for machine shop work.

On the other extreme, there is the computer-generated proof of the 4-color theorem. IIRC, one mathematician could not read the whole proof in a lifetime. Merely understanding how they formulated the problem into a computer program will take up more of your life than most people want to spend on a single abstract problem. Certain computer bugs can be ruled out by re-compiling the program for different computers and comparing results, but the real question is whether the program is correct -- and apparently mathematicians who have reviewed it think it is correct, with a lot higher probability than is needed to execute a man in Texas, but not everyone agrees it is _proven_.

And what's the real-world applicability? In theory a mapmaker could get along with just 4 colors, but it's easier and clearer to use more...

Re:Proof

[benhaha]

So you think people can never be inhuma?

Re:Proof

[benhaha]

A human picks a number, and with probability 1 it is not the largest currently known merseinne prime plus 5.

Actually I don't think that would hit probability 1. It would have a lot of nines to be sure, maybe over 3,500,000, but it would not be 1. There are two obvious ways this can happen:

1. Once you have a human who just might reel off a very long random string of digits for fun, and keep going at one digit per second for eight hours per day, for 121 days, there is a non-zero probability of that sequence being your number.
2. The person could use another means of specifying the number, for example by saying "The largest currently known merseinne prime plus five".

Now that you have set the challenge, I expect the probability of option 2 to go right up.

Re:Proof

[streetlawyer]

Nope, that is where you're wrong, at least when you start talking about "the end of days". Anything you say is entirely dependent on what you mean by "proof", and that is a concept which has been protean in its meaning since the start of the human activity known as Mathematics. It appears to be true for Church's concept of a proof, but since 1933 we've known that this proof-concept is not entirely satisfactory, because of Goedel's result. It is entirely possible that some future genius of mathematical logic will come up with a new concept of what it is to prove a mathematical statement which solves the questions of the Continuum Hypothesis and the Axiom of Choice, and that this unimaginable future proof-concept will not have the properties you claim for mathematical proof.

Interesting...

[Thaidog]

...I've been told I'm homeomorphic myself...

Re:Interesting...

[Krapangor]

Everybody is homeomorphic to himself because the identity mapping is continuous for E->E.

Re:What's the problem?

[jojor]

uhm....yeah but who streches rubberbands around donuts?

Let's wait on calling it "proved"

[Anonymous Coward]

Nothing is proven until it is peer reviewed and published in a prestigious journal, and then it must to be out there for some time before it is truly accepted. Also, there may be a mistake that throws the proof off for a few years.

Re:Let's wait on calling it "proved"

[signifying nothing]

Nothing is proven until it is peer reviewed and published in a prestigious journal

Does that mean I still have the chance to be first with a proof of Pythagoras's theorem?

GOOD LORD!

[Anonymous Coward]

My eyes are bleeding from clicking on that link! Is the PBS webmaster blind?

Re:GOOD LORD!

[DGolden]

Or just Xft on X. No need to subject yourself to the nightmare of running a Mickeyshaft "O"S

Re:What's the problem?

[psavo]

Interesting stuff.
isn't it 'better' to not think about rubberband at outer surface bat at 'outer rim'. At about below surface of apple/doughnut?
Then one will see that in apple rubberband (even in 3D) is convexish (I mean infinitely thin rubberband), but in doughnut, there is no way to see some part of rubberband unless it's quantized.
Same applies fo 'standard' universe and with one which has a 'pen'-hole which goes straight through rubberband (some odds for that..).

In related news.... 4 = 5

[dimator]

Here's the proof:

assume a, b, c such that: a + b = c

then 5a + 5b = 5c
and 4c = 4a + 4b

adding the two: 5a + 5b + 4c = 4a + 4b + 5c

shifting some terms around: 5a + 5b - 5c = 4a + 4b - 4c

simplifying: 5 (a + b - c) = 4 (a + b - c)

dividing by the common factor (a + b - c): 5 = 4

:)

Re:In related news.... 4 = 5

[phooka.de]

Always nice to see how division by zero can be masked ;-)

Re:In related news.... 4 = 5

[Anonymous Coward]

Except a+b-c = 0. Therefore you are dividing by zero, yielding infinity.

Infinity * 4 = Infinity * 5

is by definition

Infinity = Infinity

Your proof is flawed QED.

Re:In related news.... 4 = 5

[nucal]

If a + b = c

then (a + b - c) = 0

so 5*0 = 4*0

someone had to do it ....

Re:In related news.... 4 = 5

[nucal]

someone had to do it ....

make that everyone had to do it

Re:In related news.... 4 = 5

[Mindfool]

The problem with this proof is that, since a+b=c, a+b-c must be 0 (zero), and you can't divide by zero.

Re:In related news.... 4 = 5

[Xerithane]

The horrible thing about this, I saw that proof and looked at it. The first thought that popped into my head was, "No.. that can only be true if it's an illegal divison by zero and then it would crash."

Life as a programmer is fun, especially when your mind fails to seperate out normal things that don't crash. Like paper for instance.

Re:In related news.... 4 = 5

[Rhinobird]

oh sure you can, but then winston churchill becomes a carrot...and other such nonsense.

Re:In related news.... 4 = 5

[frinsore]

Actually dividing by zero doesn't give you infinity, it yeilds an undefined. If 4 / 0 was infinity then 0 * infinity would be 4, which it's not.

Also Infinity doesn't always equal Infinity. There are many different types of infinity that may or may not equal. Consider all the counting numbers, thats an Infinity. Now consider all the real numbers, that's a different Infinity. The second Infinity is greater then the first (counting numbers are a subset of all real numbers), hence Infinity doesn't equal Infinity

Re:In related news.... 4 = 5

[Anonymous Coward]

Actually 4/0 is infinity, but 0*infinity is undefined, because it could be 4, or it could be 5, or it could be a lot of things.

Re:In related news.... 4 = 5

[Anonymous Coward]

x/0 is undefined!!!! for all x real

Re:In related news.... 4 = 5

[ethereal]

0*anything is 0. Why should infinity get to be any different? :)

Re:In related news.... 4 = 5

[ethereal]

I think "cardinality" is the word you're looking for. Various infinities are distinguished by their cardinality, aren't they? That's where transcendental numbers come from IIRC, although I could be wrong about that.

Re:In related news.... 4 = 5

[j7953]

Actually 4/0 is infinity, but 0*infinity is undefined

No, x/0 is undefined. However, you can do things like

lim y->0 of x/y = infinity (for x > 0)

because, when y approaches zero, x/y will obviously become larger. But that is not the same as

x/0 = infinity (this is wrong!)

0*infinity is undefined, however, continuing the example above, I could write:

lim y->0 of 0 * x/y = lim y->0 of 0/y = 0

i.e. in that example, "0*infinity" would be zero.

The problem with infinity is that you can't use it like a number, because it isn't one. Infinity literally means that there is an infinite number of things, e.g. the set of integers is infinite, meaning you can never list all integers because there is always a successor. You'll never "arrive at infinity" when listing integers. This means you can calculate with infinity only with equations that involve sequences and their limits. (Like the above-mentioned lim y->0 which means that y is a sequence of numbers approaching zero, and not y = 0. A suitable sequence might e.g. be y[n] = 1/n with n = 1, 2, .... Obviously, this sequence is approaching zero, but will never be equal to zero.)

Re:In related news.... 4 = 5

[Anonymous Coward]

Yes, and taking 1/ these different inifinities gives different values of 0.

Re:In related news.... 4 = 5

[sydneyfong]

You're an idiot...
anybody who knows basic arithmetic knows you can't divide by zero....

I guess it's an attempt to be funny, but i just think it's sorta lame...

Re:In related news.... 4 = 5

[Anonymous Coward]

0*infinity is undefined, however, continuing the example above, I could write:

lim y->0 of 0 * x/y = lim y->0 of 0/y = 0

i.e. in that example, "0*infinity" would be zero.

Yes, but

lim x->0 of x * 4/x = 4
lim x->0 of x * 5/x = 5

so "0*infinity" can also be 4 or 5. But "4/0" can only be infinity. To just say that both are undefined is to ignore a fundamental difference between the two expressions.

Re:In related news.... 4 = 5

[Anonymous Coward]

Anything*infinity is infinity. Why should 0 get to be any different? :) c=m=8

Re:In related news.... 4 = 5

[InadequateCamel]

I recently read a book called "Zero", which was (predictably) about the history of the number zero. There are a few appendicies in the book, one of which is entitled "Build Your Own Wormhole Time Machine", but the fun one is "Animal, Vegetable, or Minister?" where the author divides by zero (a-b, where a=b=1) and goes on to prove that Winston Churchill is a carrot. A good read.

Re:In related news.... 4 = 5

[Anonymous Coward]

You meant "transfinite" numbers, I guess. Trascendental numbers have nothing to do with cardinals. A number is trascendental if it is not algebraic (i.e. if it is not a root of a polynomial with rational coefficients). Pi and e are examples, but in fact "most" numbers are trascendental.

Re:In related news.... 4 = 5

[Anonymous Coward]

0*(any number) = 0. Infinity is not a number.

Re:In related news.... 4 = 5

[josse]

The problem with infinity is that you can't use it like a number, because it isn't one. Infinity literally means that there is an infinite number of things, e.g. the set of integers is infinite, meaning you can never list all integers because there is always a successor.

At the same time there _are_ numbers larger than infinity. The set of intergers is countably infinite, while the set of reals is uncountably infinite. Meaning you can put as many reals as you want between two other reals. No matter how close the two reals you pick are, you still can't count the number of reals between them.

A countable set is defined as a set where there exist a 1-1 mapping into the set of integers.

The cardinality (number of members in a set) of N (the set of integers) is called aleph_0. One does not know which aleph that corresponds to the cardinality of reals. One does not think it is aleph_1

Read the excellent book "The Mystery of the Aleph" by Amir D. Aczel for more information about an exciting branch of mathematics.
--
josse

Re:In related news.... 4 = 5

[Anonymous Coward]

Wow, you caught it, you are SO SMART!!!

Re:In related news.... 4 = 5

[EvlG]

Would that happen to be this book, for the interested reader?

Zero: The Biography of a Dangerous Idea

Re:In related news.... 4 = 5

[Tony-A]

Cardinality for "how many elements are in the set".
Two sets have the same cardinality iff there exists a one-one function from one set onto the other set. Thus there are exactly as many primes as there are rationals. In all cases, the power set (set of all subsets) has strictly more elements than the original set. The power set of the null set has exactly one element, the null set. The null set has no elements.

You can also have infinite ordinals. Addition defined but not necessarily subtraction. 1,2,3,...,INFINITY,INFINITY+1,...,2INFINITY,...

Re:In related news.... 4 = 5

[slamb]

But "4/0" can only be infinity.

No, that's not true.

lim x -> 0 of 4/x = undefined
lim x -> 0+ of 4/x = +inf
lim x -> 0- of 4/x = -inf

One statement says "as x approaches 0 from the left (x is an extremely small positive number), 4/x approaches positive infinity." The other says the same thing, except from the left (negative). Only if they are the same is the general limit true.

lim x -> x_0 = f(x) <=> lim x -> x_0+ = lim x -> x_0- = f(x)

Re:In related news.... 4 = 5

[tlk nnr]

No, x/0 is undefined.

Compile and run this app:

void main()
{
printf("1/0 is %f.\n", 1.0/0.0);
printf("0/0 is %f.\n", 0.0/0.0);
}

The output is

1/0 is inf.
0/0 is nan.

nan stands for NotANumber. According to IEEE floating point arithmetics, 1/0 is infinite, and 0/0 is undefined.

But I wouldn't dare to say that aloud if matematicians are around.

Re:In related news.... 4 = 5

[Darby]

At the same time there _are_ numbers larger than infinity.

Technically, there are different "infinities"

One does not know which aleph that corresponds to the cardinality of reals.

Actually, One (I forget which One specifically) has shown that the continuum problem (is the cardinality of the reals equal to aleph1) is independent of the axioms meaning you can either accept it as true or accept it as false and either way if the other axioms are consistent then the other axioms plus the new one will still be consistent.

One will get different mathematics depending upon which one is chosen.

One does not think it is aleph_1

Actually Many (the majority of the mathematical community) have arbitrarily decided that it *is* Aleph1. The mathematics are "cooler" this way.

Re:In related news.... 4 = 5

[Repton]

1 = sqrt(1)
= sqrt( (-1) * (-1) )
= sqrt(-1) * sqrt(-1)
= i * i
= -1

Re:In related news.... 4 = 5

[Anonymous Coward]

The problem is that since a+b=c and you are dividing by a+b-c, what you are essentially doing is dividing by zero..... That is very, very, very bad in math. The world falls apart when things like that are allowed

Re:In related news.... 4 = 5

[j7953]

nan stands for NotANumber. According to IEEE floating point arithmetics, 1/0 is infinite, and 0/0 is undefined.

Well, I'm not familiar with the reasoning behind IEEE floating point arithmetics, but x/0 yields infinity in many programming languages. My guess is that it was defined this way because it makes more sense for most pratical applications, since otherwise a number x/y (with a small, variable y) might suddenly become undefined when y becomes so small that the computer's precision doesn't suffice and the computer thinks that y=0. In this case, x/y shouldn't be undefined, it should be "very large, beyond the computer's precision." This is better expressed by saying x/y=inf instead of x/y=nan.

Concerning mathematics, read slamb's reply to an AC who replied to my post, he explains why x/0 has to be undefined: depending on what sequence you use, it could be either positive or negative infinity.

Re:In related news.... 4 = 5

[ethereal]

Transfinite, exactly. Thanks for the correction. How come all the smart people in this thread are ACs? :)

Re:What's the problem?

[Anonymous Coward]

You show great skill at cut & pasting from http://www.claymath.org/prizeproblems/poincare.htm : )

Just kidding. Go ahead, enjoy the cut & paste karma.

Re:What's the problem?

[Gary Yngve]

Here's an algebraic topology version of the problem:

Given a simply connected tetrahedral mesh, show that the mesh can be collapsed by topologically invariant operations to a single tetrahedron.

busted

[Anonymous Coward]

good catch, i don't mind when people do that, but they should give props to the reference material and not just post it as their own.

Re:What's the problem?

[metlin]

Which is exactly what they've done in the paper. They've depicted on how the mesh could possibly collapse.

They have depicted an 8-gon curve which satisfies the intersection properties, extrapolate using a 2 vertex model and use that to show the possible collapse. They've not depicted the collapse per-se in action tho. :-(

For 1=7

[XNormal]

You can prove anything :-)

Re:For 1=7

[Frank.B.Parker]

Maybe he meant n >= 7, and the HTML just ate the >.

Re:For 1=7

[DNAGuy]

You can use facts to prove anything that's even remotely true!

-- Homer Simpson

Re:For 1=7

[Anonymous Coward]

Fellow named Godel would have argued with Homer on that one.

Re:For 1=7

[Anonymous Coward]

Incorrect. You cannot even "prove" your own existence. Think about it.

Re:For 1=7

[jnana]

I've thought about it, and I must exist, 'cause if I don't, then my girlfriend has been a very bad girl.

Re:For 1=7

[streetlawyer]

Strange how the proof of your existence causes me to immediately doubt the existence of "your girlfriend".

Re:For 1=7

[jnana]

Strange how your comment causes me to doubt you understood my comment. I do have a girlfriend, but that is irrelevant. Clearly you have yet to develop a sense of humor, otherwise you would have seen my comment as the humorous self-referentially contradictory statement that it was meant to be. Perhaps you needed a smiley :-), like so many other humorless souls.

Old news...

[Junior J. Junior III]

I already read about this, like, three zillion years from now. Can't you find anything to report about that HASN'T already happened?

Re:Old news...

[Beautyon]

Can't you find anything to report about that HASN'T already happened?

How can ANY editor report something that HASN'T YET HAPPENED??

Re:Old news...

[ethereal]

I think he was playing off of this comment :)

Re:Old news...

[Anonymous Coward]

How can ANY editor report something that HASN'T YET HAPPENED??

On stock market sites people demand this all the time.

Re:Old news...

[Junior J. Junior III]

You just get that time machine from the previous story, and...

Vindication

[PhysicsGenius]

I've been saying for years that our combinatorial place value system of numbers locks people into a limited mindset of numerical thinking. So I feel vindicated by the fact that this guy solved the Poincare Conjecture using Roman numerals. They are better all round and easily manipulable.

Blind faith in Mathematics

[ynotds]

I write this as a reformed Mathematician of sorts, which is analogous to being a reformed smoker ... the expectations that half an education in Math gives as to the existence of right and wrong answers sure looks ugly once you can escape its grip.

And faith in Mathematical proof is counterproductive at a level beyond that ... it hides the beautiful truth that Math is something that can be joyously explored in its multitudinous riches without any need for the reality checking of the (would be) sciences.

Personally I have come to see both Math and Science (or more strictly the scientific method) as but potent toolsets, and to confine my own quest for more profound truths to those "interdisciplinary" comparisons that have been called anything from "complex systems" to "general evolution".

This step is a bit like the step from geometry to topology which has clearly escaped the wit of the moderator who took offense at a not quite successful attempt to make something funny out of teacups and donuts.

Re:Blind faith in Mathematics

[Anonymous Coward]

Well, mathematics differs from physics in one sense; it is defined based on abstract concepts that are true by definition.

I can always be certain that ab = ba, whether or not quarks exist or they are manifestations of something entirely different. I can be certain that ab = ba even if I don't exist and am merely someone else's dream.

Re:Blind faith in Mathematics

[Darby]

I can always be certain that ab = ba,

No, no, no.

You can only be sure of this if a and b are points in a space in which the operation is commutative.
If a and b are integers and you're using normal multiplication, then yes you can be sure the statement holds.
If a and b are matrices and you're using matrix multiplication then you can be somewhat sure it's not true for any given a and b and absolutely sure that it isn't true in general.

The problem is...

[Anonymous Coward]

They're offering $1m for Clay mation. Hell I can wad up a ball of playdoh in my basement and get prettier pictures.

It just goes to show if it isn't one thing, its another. If it isn't a ball of clay, its, its...

Oh, Clay Mathematics. That's different.

Never mind.

Re:The problem is...

[nucal]

Still, by the Poincaré Conjecture - Gumby is equivalent to Pokey.

proof has been announced

[call -151]

Normally it take a while for a proof to be verified- a better title would be `A Proof has been announced for the Poincare Conjecture.' The Poincare conjecture has attracted a great deal of attention and lots of remarkable, deep work, but it has also had its fair share of proofs which fell apart under serious scrutiny. Most notably, Colin Rourke and a co-author I can't remember had claimed a proof of the Poincare conjecture in 1987 which took something like a year-plus before the mistakes were found, and took a great deal of energy by a number of mathematicians to find the errors.

That being said, Martin Dunwoody is a remarkable researcher and this work relies on important, ground-breaking work of Abby Thompson and Hyam Rubenstein, and this preprint sounds very promising!

Re:proof has been announced

[sydneyfong]

That would be a proof being proved....

Re:proof has been announced

[Anonymous Coward]

At the very least, the past participle form should be used, making it "The Poincaré Conjecture has Been Proven"

Re:proof has been announced

[jacobb]

Well, actually, it's a beautifully simple, short-and-sweet, easy-to-follow 6 page proof. Most students of topology can easily follow it (well, pretty easily anyway).
I highly doubt that any errors will pop up at all simply because the proof is elementary. (note to non-mathematicians... elementary and simple or easy are two very different things in math).
And it's only 6 pages!

Re:proof has been announced

[Anonymous Coward]

Rubbish. Dunwoody has posted *six* versions of this supposed proof already, because people have found errors in the previous versions. It is quite a leap of faith to suppose that no errors remain.

Re:proof has been announced

[jacobb]

Ahh... but were the errors errors in the proof or just syntactical? I don't think that the proof has been seriously revised or changed. That having been said, please note I haven't read a previous version.
After all, this is a "preprint" just for that reason - so this type of thing can be ironed out before publishing.

It's extremely rare that a preprint gets published verbatim - they usually go through a couple versions.
Errors may remain, but i'd say it's more of a leap of faith to say that this version is mathematically erroneous.

Re:proof has been announced

[colmore]

as Thelonious Monk once said "Simple ain't easy"

Re:proof has been announced

[Anonymous Coward]

I was a math major in college, but forgot most
of the stuff. Could you suggest a book to refresh my memory before reading this proof? Thanks.

Troll, or uninformed post? [*free clue enclosed*]

[rakslice]

Don't confuse mathematics with science. The scientific method likes induction from a limited set of cases. Mathematical methods of proof won't touch that kind of reasoning with an 10-foot pole.

"Anything can be proved with enough flawed mathematics." How does one prove something with flawed mathematics? Certainly, one can attempt to prove something with flawed mathematics, but if the mathematics are flawed, what does it prove?

"Think how many times things have been proven, only to be found flawed later on?" Okay. Zero. See above.

Mmmmm... Math...

[Anonymous Coward]

Doh!

Re:What's the problem?

[Anonymous Coward]

>uhm....yeah but who streches rubberbands around donuts?

Man that has to suck, losing karma by getting modded offtopic for telling someone else they're offtopic. Next time instead of trying to scrounge a 'funny' point in something you have no idea about, be a proper karma whore and do a google search. I mean if you're going to make the effort to post why not go the extra distance?

Personally I like to mod people down as overrated when they try to post something funny, although I'm guilty of whoring through humour myself. However if I had it I'd give you that extra point you need just for your anti-american comments, but chances are you'd lose it again next week trying to whore another funny point. Oh well.

Nope

[dark-nl]

This is very different. Bentini's theorem is simply "Mathematicians can be wrong" :-)

I agree with that one. Some proofs are large and complicated, and they might have bugs in them that haven't been noticed yet. I even think it's possible that human minds have bugs which makes them incapable of noticing certain kinds of errors.

More straightforwardly, some proofs have computer-generated parts and their verification is computer-assisted (the four-colour problem, IIRC), and we all know that computer programs have bugs :-)

Statement of conjecture on wolfram incorrect?

[Anonymous Coward]

Surely it should read:

The conjecture that every *compact* simply connected 3-manifold is homeomorphic to the 3-sphere,

Normal euclidean space R^3 is simply connected,
and definitely NOT homeomorphic to to the
3-sphere !!

(That they are not homeomorphic can be proved by
comparing their homotopy or homology groups).

Liam.

Re:Statement of conjecture on wolfram incorrect?

[p3d0]

Mods, I think the parent may be a gem lost in the abyss of anonymity. Do what you think is right.

Re:Statement of conjecture on wolfram incorrect?

[damas]

hell, boy, add infinity to R^3 and you got your compact, don't ye?

but the question is correct and the answer is:
compact simply connected 3-manifold without _a boundary_ (any closed simply connected 3-manifold) and stuff :)

Re:Statement of conjecture on wolfram incorrect?

[Anonymous Coward]

Well, DUH.

Thank you, Captain Obvious.

Re:Statement of conjecture on wolfram incorrect?

[lyosha]

Here's the better proof that R^3 and S^3 are not homeomorphic. Here it goes:

S^3 is compact and R^3 is not.

Re:Statement of conjecture on wolfram incorrect?

[drini]

If I had a theorem for each mistake on MW...

ehhehe
well guys, give the guy a break.
Send a correction, besides talking here
so his site gets improved. It's an
awesome site

------------
drini@math.com
http://planetmath. org

books on this stuff

[danny]

Maybe it's time for me to make a fourth attempt at reading Harper and Greenberg's Algebraic Topology: A First Course. I think I got three quarters of the way through last time...

Can anyone recommend any other books on algebraic topology?

Danny.

Re:books on this stuff

[Anonymous Coward]

Try M.A.Armstrong's Basic Topology published by Springer-Verlag New York Inc., ISBN 0387908390.

It's a fairly solid book on the subject. I haven't seen Harper and Greenberg though so I can't say how it compares.

Re:books on this stuff

[Pepeee]

Look for W. Massey's books: "Algebraic Topology, A First Course" and "Singular Homology Theory" in Springer's Graduate Texts in Mathematics. That's where I learned all my Algebraic Topology.

Re:books on this stuff

[jasoegaard]

Bredon's "Topology and Gemetry" is a modern classic.

The old one was Spanier's "Algebraic Topology".
(which is also quite nice).

Re:books on this stuff

[Pepeee]

How can you call Spanier nice?? I don't know of anybody who's been able to read more than two pages! It's dense and very confusing in its "great generality"! Completely unreadable!

Re:books on this stuff

[jasoegaard]

> How can you call Spanier nice??
> I don't know of anybody who's been able
> to read more than two pages! It's dense and
> very confusing in its "great generality"!
> Completely unreadable!

I didn't say it was easy. From the preface:

| The reader is not assumed to have prior
| knowledge of algebraic topology, but he is
| assumed to be mathematically spohisicated.

The topics come in a orderly fashion. And there no gaps in the proofs. The original book is from 69 and there is a reason it is still in the shops.

Re:books on this stuff

[Tityrus]

Alan Hatcher's "Algebraic Topology" is, besides being freely available on his homepage, one of the best & most elegant textbooks I've ever come across.

He also has some other books on more advanced topics in algebraic topology, in various stages of completion, but I haven't read those yet.

Re:books on this stuff

[spectatorion]

my university uses munkres's topology for its intorductory (senior undergraduate-level) course in topology. the book's first section is mostly point-set and the second seems to be algebraic. unfortunately, i have not gotten much beyond chapter two, but an older undergrad whom i have spoken to says it is "very nice." munkres also has a book on algebraic topology alone called elements of algebraic topology.
best of luck.

Re:books on this stuff

[mreece]

I agree that Hatcher's book is good, although I'm just beginning to learn this stuff. We're using that as the text in Math 263 at Chicago - taught by J.P. May. May has a book as well, "A Concise Course in Algebraic Topology," which is highly categorical in its perspective. I'm struggling to figure out limits and colimits, so I'm afraid I haven't made it past the second chapter yet, but there seems to be a lot of good stuff in that book.

At a more basic level, Munkres "Topology" is good for point-set stuff, but also has some algebraic topology.

It isn't about algebraic topology, but I very highly recommend Milnor's "Topology from the Differentiable Viewpoint" and his book on Morse Theory. Guillemin & Pollack also give a very good treatment of differential topology. And Thurston's book on three-dimensional geometry and topology is awesome, but I think I would have had a very hard time getting through much of it without the class I took on it in the fall.

Re:books on this stuff

[Darby]

Can anyone recommend any other books on algebraic topology?

For my Topology course at U.C. Santa Barbara we used Czes Kosniowski's A First Course in Algebraic Topology, Cambridge University Press.

I thought it was very good.

Re:books on this stuff

[mpsmps]

Most of the recommendations here have been similar in flavor to Harper and Greenberg. If you've had trouble with that, you're probably better off with a wholly different approach. Try Bott and Tu, Differential Forms in Algebraic Topology.

This book covers a lot of ground, but avoids a lot of messy algebra by restricting (until late in the book) to the intuitive differential forms case. For example, most algebraic topology books spend a chapter constructing products using messy (and unmotivated to the beginner) homological algebra. By sticking with differential forms, Presto! cup products are just the exterior product of differential forms. Want a two line proof of Poincare duality? Cap product is integration along the fibre: Verify it on an open disk and apply Mayer-Vietoris. You get all this in the first couple of chapters. Of course, if you are interested in torsion, you have to work harder, but that can wait.

Re:books on this stuff

[MrFredBloggs]

"Matt Reece If you can't figure out how to e-mail me, don't."

If they cant figure it out, how are they going to email you in the first place? Or are you talking to Matt Reece? Isnt that you? You`re telling yourself not to email yourself if you can`t figure it out? Eh? EH?!

Um...

[rakslice]

Are the expectations you speak of about mathematical truth? Or about truth in general? If you have expectations about truths about the properties of the universe, I don't see what that has to do with math; perhaps these expectations are less the result of an education in mathematics, and more the result of half an education in mathematics...

English please!

[prestwich]

Can someone explain what the hell this problem is about in English please? (Preferably avoiding the word manifold).

Re:English please!

[kordless]

I'll try, but I get WAY out of my league when you talk about anything bigger than n=3.

n represents dimensions. i.e. n=1 is one dimension, n=2 is two dimensions, n=3 is three dimensions, etc.

"simply connected" just means that the boundary surrounding something is connected. For example, in n=2 space (a piece of paper for example), if you drew a line around a bunch of ants, and connected the ends, it would be simply connected. If your line was actually two lines, and weren't connected (you had two groups of ants) then you have a multiply connected boundary.

A manifold (sorry, had to use it) is just an object without a boundary. The earth is a manifold, as is any other n=3 space (3d) object that is connected to itself. In n=3 space, the only way you can have a boundary is to have two different objects, in two different locations.

Homeomorphic just means one object is like another.

They generalize the whole idea to one where all the objects are compact. That just means that the objects "surface" area is as small as it can get for a given internal volume. For example in n=3 space, you can minimize an area (like the material of a balloon) in relation to the volume inside (the helium). Circles are compact for n=2 space, and spheres are compact for n=3 space. BTW, even though I state this as if it were a fact, we don't know about all the compact spaces where n > 2. It would *seem* to make sense that a sphere is the only compact object to 3 space, but stating that as a truth, as of today, isn't possible. Maybe we can do that after they win their million bucks....

So, the whole thing boils down to showing that a compact 3d object is the same as a sphere.

Kord

Shameless plug, check out Grub!

Re:English please!

[jso888]

The way I'm thinking about the problem, is this. Given the condition that no point on the rubber band can ever break contact with the surface of the object it's wrapped around (sphere or doughnut -- I think that's a torus):

You could move the rubber band towards an arbitrary apex of a sphere until the rubber band condenses to a single point at the apex. This applies to other volumes such as cubes and cones or even a randomly squeezed bit of toothpaste.

On the other hand, this can't be done for a torus when you've stretched a rubber band around the wide way, because dealing with the hole in the doughnut would mean having to break contact with the surface of the dougnut.

They're asking for topological proof that this is the case. Don't ask me to describe simple connectedness in plain English; it's an intuitive thing for me -- someone whose last math course was calculus 101.

What I don't get is why you can't cheat when initially placing the rubber band on the doughnut, and stick it to one side of the hole so that the shrinking process never has to cross the chasm, as it were. Or is that besides the point?

Also, what are the real world applications of this proof?

Re:English please!

[jso888]

Er, that exceptional case about placing the rubber band on the doughnut off to one side of the hole is the point.

For a sphere, you could start wrapping the rubber band around any part of the sphere, and be able to converge it to a point.

This is always the case, and that's simple connectedness. They're asking for proof of this concept.

This is not always the case for a doughnut. There are exceptional cases where you can converge the rubber band to a point without breaking contact, like wrapping the rubber band around the doughnut off to the side of the hole. But there are also cases where this can't be done because of the doughnut hole. The doughnut shape isn't simply connected.

Re:English please!

[nels_tomlinson]

They generalize the whole idea to one where all the objects are compact. That just means that the objects "surface" area is as small as it can get for a given internal volume.

I suppose that this is a trivial quibble, but my understanding was that a compact set was (in N-space, N\lt\infty) a closed, convex, bounded set. Thus, an egg is a compact set in R^3, despite not having minimum surface area. For spaces which are not vector spaces, or don't have topologies, it's more complex, but now I'm telling you what I don't know, instead of what I do.

Re:English please!

[nivedita]

4 (Insightful)!? Almost every statement in this post is incorrect.

The description of simply connected is a description of connectedness. Simply connected means your space doesn't have holes in it, in addition to being connected. This is required, since there are obviously 2-D surfaces (think of donuts) that are connected, yet not homeomorphic to a 2-sphere.

A manifold is a space that is locally homeomorphic to Euclidean space. i.e. if you take a very small piece of the space near a point, it looks like a small piece of R^n. A figure 8 curve is an example of a 1-dimensional space that is not a manifold.

Homeomorphic means that there exists a bicontinuous (continuous in both directions) one-one correspondence between the spaces.

Compactness has precisely nothing to do with surface areas and volumes. If an objects surface area is as small as it can get wrt its volume, it's a sphere, and this has been known for a long time. Secondly, circles are 1-D, not 2-D.

Intuitively the notion of compactness corresponds to being `finite'. In R^n, a set is compact if it is closed (i.e. contains its boundary) and bounded (doesn't stretch off to infinity). The general definition of compactness is more hairy: one way of stating it is that every infinite sequence in the set has a convergent subsequence (note that the limit also has to be in the set).

What the Poincare conjecture states, roughly, is that any closed bounded d-dimensional object in R^n that doesn't have any holes in it (this makes it homotopy equivalent to a d-sphere) is actually homeomorphic to a d-sphere. (Note: it's non-trivial to prove that a compact d-dimensional manifold can actually be embedded in R^n for some n).

Re:English please!

[Anonymous Coward]

I'll try, but I get WAY out of my league when you talk about anything bigger than n=3.

**You're out of your league already

n represents dimensions [wolfram.com]. i.e. n=1 is one dimension, n=2 is two dimensions, n=3 is three dimensions, etc.

"simply connected" just means that the boundary surrounding something is connected. For example, in n=2 space (a piece of paper for example), if you drew a line around a bunch of ants, and connected the ends, it would be simply connected. If your line was actually two lines, and weren't connected (you had two groups of ants) then you have a multiply connected boundary.

**No, simply connected means that any (n-1)-loop embedded in the topological space is homotopic to a point

A manifold (sorry, had to use it) is just an object without a boundary. The earth is a manifold, as is any other n=3 space (3d) object that is connected to itself. In n=3 space, the only way you can have a boundary is to have two different objects, in two different locations.

**No, a n-manifold is an object such that every point is locally homeomorphic to R^n

Homeomorphic just means one object is like another.

**No. Homeomorphic means that there is a bijective continuous function between two spaces such that its inverse is also continuous.

They generalize the whole idea to one where all the objects are compact. That just means that the objects "surface" area is as small as it can get for a given internal volume. For example in n=3 space, you can minimize an area (like the material of a balloon) in relation to the volume inside (the helium). Circles are compact for n=2 space, and spheres are compact for n=3 space. BTW, even though I state this as if it were a fact, we don't know about all the compact spaces where n > 2. It would *seem* to make sense that a sphere is the only compact object to 3 space, but stating that as a truth, as of today, isn't possible. Maybe we can do that after they win their million bucks....

**No. Compactness means that any open cover of the space has a finite subcover.

Re:English please!

[Anonymous Coward]

"simply connected" just means that the boundary surrounding something is connected.

No, it means that every loop in the space can be smoothly contracted to a point. This can happen in a plane, for instance. But suppose you punctured the plane with a hole; a loop surrounding the puncture would get "caught" on it if you tried to shrink the loop down to a point. A punctured plane is not simply connected.

A manifold (sorry, had to use it) is just an object without a boundary.

An "object"? That's kind of vague. A manifold is, roughly, just an n-dimensional "surface" -- e.g., a 0D point, a 1D line or curve, a 2D plane or sphere, a 3D Euclidean volume, etc. Somewhat more precisely, it's something that "locally looks like" an n-dimensional Euclidean space. (e.g., if you look at a piece of a sphere, it looks like a piece of a 2D Euclidean plane, as far as topology is concerned; it's only when you look at the whole sphere globally at once that you see that it's not a plane.)

Sometimes the definition of "manifold" includes the possibility of a boundary; sometimes they are separated denoted "manifolds-with-boundary".

Homeomorphic just means one object is like another.

Pretty vague again. There are many ways an "object" can be "like" another object.

Two topological spaces are "homeomorphic" if one can be continuously deformed into the other (e.g., by stretching, squeezing, and bending, but not cutting, pasting, or tearing.)

(A manifold is a topological space that is locally homeomorphic to an n-dimensional Euclidean space.)

They generalize the whole idea to one where all the objects are compact. That just means that the objects "surface" area is as small as it can get for a given internal volume.

That's completely wrong. The notion of "compactness" exists even when no geometry (and hence no measure of area or volume) is defined. What you are describing is not a compact space but a "minimal surface".

Compactness roughly means "finite", as in the surface of a sphere, compared to an infinite plane. More precisely it means that if you can cover the space with a bunch of (maybe infinitely many) "patches" (open sets or neighborhoods), you can always make do with only finitely many of them in order to cover the space. ("Able to be partrolled by finitely many near-sighted policemen" is how one professor put it.)

Re:English please!

[Anonymous Coward]

Nice. Thanks.

Re:English please!

[Anonymous Coward]

Roughly, it says that every finite 3-dimensional surface without boundary, within which every loop can be smoothly contracted to a point without getting "caught" on any holes or anything, can be continuously deformed to have the shape of a 3-sphere (the surface of a 4-dimensional hyperball).

Re:English please!

[Anonymous Coward]

Being simply connected is not a property ofthe boundary. For example, a 3-sphere has no boundary, yet it is simply connected. Even if you think as the sphere as the boundary of the volume it encloses, things don't work. If you think of a torus as the bounday of the volume it encloses, it is connected, yet the torus is not simply connected.

It would be more accurate to think of "simply connected" as meaning "without holes". Or, if you put a rubber band around an object, you can just remove it by pulling it. That's clearly the case for a sphere. It is clearly NOT the case for a torus.

Compactness has nothing to do with "tightness", although the name may suggest otherwise. _Any_ surface is "as small as it can get" for the volume it encloses. A paraboloid, for example, satisfies this, yet it is not compact. Compactness in R^3 is equivalent with "bounded" (and closed, if you want to be precise). A paraboloid is clearly unbounded, but a sphere is not.

Homeomorphic means that one object can bed eformed continuously into another. Stretching, bending, scaling are all continuous transformations, but cutting a piece off an object is not. For example, a circle is homeomorphic to a square (if you imagine a circle made of rubber, just stretch it to form a square). But a circle is NOT homeomorphic to a line, because to deform the circle into the line you need to use scissors.

Without boundary means that it is impossible to get to the space "inside" the surface from outside without crossing the surface at some point. For example, a sphere, a torus or a paraboloid have no boundary, but a sphere with a small hole near the north pole has a boundary.

Poincare's conjecture can be phrased as saying that any bounded surface made of rubber without holes and without boundary can be deformed into a sphere. If you think of it this way, it's pretty clear that the conjecture must be true. On the other hand, "obvious" things are often very hard to prove.

Re:English please!

[$uperjay]

While a circle can be described with only one dimension, that doesn't make it one-dimensional. It has length and width, just like any other 2D shape. A true one-dimensional object is a line, which has only length. If a circle was 1D, a sphere would also be 1D by the same reasoning. Which it isn't.

Re:English please!

[John Miles]

A figure 8 curve is an example of a 1-dimensional space that is not a manifold... Secondly, circles are 1-D, not 2-D.

Could you clarify those points? I don't see how either a circle or a curve can be parameterized with a single variable in either an oriented or a non-oriented space, which is (or at least should be) the criterion for single-dimensionality.

Re:English please!

[Anonymous Coward]

Mathematically, a circle is 1D and a sphere is 2D, because it requires one number to specify one's location on a circle (e.g., an angle), and two numbers to specify one's location on a sphere (e.g., latitude and longitude).

It's true that a circle cannot be embedded in a 1D line, but can be embedded in a 2D plane, but that does not make a circle 2D.

Re:English please!

[Anonymous Coward]

Any point in the circle can be specified by a single arc parameter around the circle (a mapping from S^1 to [0,1)). You don't need to embed a circle in some higher-dimensional space in order to define it.

The criterion for "single-dimensionality" is: it is possible to find an open neighborhood of any point that is homeomorphic to R^1 (the real line). And that is true of a circle.

Re:English please!

[Anonymous Coward]

http://www.claymath.org/prizeproblems/poincare.htm

This is the official site for the contest.

"The Poincaré Conjecture:
If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since."

Re:English please!

[Anonymous Coward]

Excellent explanation!

I think even I, a biologist, got that. But, coming to think of it, how do I verify that?

Re:English please!

[John Miles]

Cool, that makes sense... I guess you're saying that a single parameter (the diameter) is sufficient to describe any circle, and that you only need two parameters if you're going to describe the circle's position in >1-space.

Re:English please!

[Arkaein]

More specifically, the circumference of a circle is 1D and the surface of a sphere is 2D. However in most cases in which a circle or sphere are considered in a more complete system a circle is described in 2D and a sphere in 3D.

After all, any closed parametric equation of the form x=x(t), y=y(t) can have its position described in terms of t. A circle fits this description. However the properties that define a circle must be defined in 2D (i.e. all circles can be represented by the above form, but not all equation of this form are circles, because f(t) must specifically describe 2 dimensions).

Re:English please!

[snarkh]

This has nothing to do with parametrizing the circle itself. Rather, intrinsically, to describe a point on a circle you just need one parameter
(angle). On the sphere you need two parameters
(two angles, for example).

Re:English please!

[mreece]

You read a math text and find out that almost everything that post said is wrong. The definitions of simply connected and compact are, as others pointed out, not at all what the previous poster claimed they were.

mathworld.wolfram.com is generally a good reference for looking up definitions...

Re:English please!

[Anonymous Coward]

That wasn't quite what I was saying. Given a circle, you need only one parameter (the angle) to say where on the circle you are. Therefore a circle is one-dimensional.

(The diameter of the circle is a red herring; you might give it in order to specify which circle you're talking about, but once you've fixed that, you only need one parameter to say where on the circle you are. Anyway, the notion of distance is only introduced in geometry; in topology, all circles are equivalent to each other, and have no defined "diameter".)

Re:English please!

[The_Laughing_God]

A circle is 1-D because topology makes a distinction between a 'circle' (a line figure) and the region bounded by it. The region inside the circle isn't part of the circle, any more than the region outside the circle is. Casual English uses the word 'circle' for the line figure and the region it encloses.

Similarly, if we call a basketball a 'sphere', we are discussing its 2-D surface. If we want to talk about a basketball as a 3-D object, including its internal volume, we must call it a 'ball'. There is such a thing as a 3-D sphere (3-sphere), but it is the surface of a four dimensional 'ball', which, I assure you, is like nothing you've ever seen. Many posters seem to have forgotten this today, and are speaking of 3D spheres, when they mean 2-D spheres enclosing 3-D balls.

2-D spheres have more interesting properties than 3-D balls - which you might not suspect if you think of them (as many do) as mere surfaces (i.e a part or property) of 3-D balls.

Re:English please!

[Alexander Poquet]

No, actually, that's not really what he's saying. It's important to remember that when we talk about a circle, we are talking about the space commonly refered to as S^1 -- that is, it's the boundary of the 2-ball. Just the line part that you draw when you draw a circle, if you will, not the area it encloses.

S^1 is a space in and of itself -- it does not need to be embedded in another space for it to make sense. As such, talking about "where" the circle is not very useful in the context, not any more than you need to talk about where R^2 is (the plane).

Intuitively, S^1 (which is basically like a line, looped back on itself) only has one dimension because you can map R (which is one dimensional) onto it in a continuous way.

Continuity is important because, for example, the cardinality of R^1 and the cardinality of R^2 can be shown to be the same -- which means there exists a bijection between them. Thus, there is a function which maps all of R onto all of R^2 -- and so if you didn't require continuity you would wind up having to say that R and R^2 are the same dimension, which clearly isn't true.

Dimension in topological spaces is necessarily more complex than dimension in vector spaces, since there are many many more topological spaces than vector spaces. I'm only trying to give you an intuitive grasp of what is meant; don't take this as definition.

Re:English please!

[Anonymous Coward]

I wouldn't be too high on your horse. Your two statements

1. Simply connected means your space doesn't have holes in it and
2. doesn't have any holes in it (this makes it homotopy equivalent to a d-sphere)

sure make it sound like you believe that "simply connected" means "homotopy equivalent to a d-sphere," which is False.

Homotopy equivalent to a sphere is a much stronger condition that simply connected. For instance, homotopy equivalent to a sphere implies that higher homology spaces (H^k for k smaller than d=dim of space) vanish, which isn't true for all simply connected spaces (eg, K3 surfaces).

Re:English please!

[Anonymous Coward]

Topologically, the "region inside the circle" isn't even defined (unless you happen to be embedding the circle in some other space), so there's no point in even talking about it.

Re:English please!

[EvilGwyn]

• No, simply connected means that any (n-1)-loop embedded in the topological space is homotopic to a point
• No, a n-manifold is an object such that every point is locally homeomorphic to R^n
• No. Homeomorphic means that there is a bijective continuous function between two spaces such that its inverse is also continuous.
• No. Compactness means that any open cover of the space has a finite subcover.

Thanks for the plain english explanation! Without this I would never have understood it.

Re:English please!

[Anonymous Coward]

Simply connected means that any loop is homotopic to a point... what is your definition of "(n-1)-loop"?

Re:English please!

[Anonymous Coward]

You're entirely correct. I was thinking of higher homotopy groups when I wrote that and I slipped.

Re:English please!

[Anonymous Coward]

if you live in the line that is the outside edge of a circle, you are living in a one dimensional space. you can move to the left or to the right.

locally, it looks like you are in R1 space

Re:English please!

[soybean]

Are you trying to tell me that donuts don't have holes in them? Geez, no wonder you're bad at math.

Summary of incompleteness in 1 sentence

[Kimble]

"This statement cannot be proved."

Of course, Gödel had a lot of i-dotting and t-crossing (heck, even o-dotting!) to do to turn that into the Incompleteness Theorem, but that's what it boils down to. Another good lay-person explanation (along with about 200 logic puzzles to boot) is in Raymond Smullyan's What is the Name of This Book?, ISBN 0139550623.

Encoding...

[JacobO]

I was wondering what kind of strange language the title of this conjecture was written in - instead of the e with the acute, I was seeing a rather less roman and definitely more asian pictograph. As it turns out, my browser thought this page was encoded in UTF-8. Switching back to the regular ISO-8859-1 encoding everything seemed to make more sense. Did any one else notice that or was it just my browser?

I realize that this is not entirely on-topic, just curious tho!

Re:Encoding...

[J'raxis]

Slashdot does not specifically set the charset (neither in a META tag, nor in the HTTP response headers), which is a Bad Thing when using eight-bit characters, considering any given octet (such as 0xE9, é in ISO-8859-1) is something different in nearly every character encoding. The author should have just used the ASCII-encoded HTML entity é (see ISO-8879) and it would work everywhere.

You should keep your browsers default character set as either ISO-8859-1 or Windows-1252 (these are nearly the same, except ISO undefines the characters 128159). UTF-8 is still such a marginal encoding that any page that uses it will say it does so in the headers, so you shouldnt use this as the default fall-back.

BTW: did your browser just display the single 0xE9 octet as a UTF-8 character? Mine (MSIE/5.0 for Macintosh) ate up the next two octets also. I think that 0xE9 is an octet in UTF-8 that indicates the character is 3 octets long (read the standard, it gets messy how this works), which explains why MSIE chomped the space and C following it. However, in UTF-8, all octets that are pieces of multibyte characters must have their highest bit on, so space and C are not legally allowed to follow an 0xE9.

Nah

[splorf]

It's not a gem, just sort of a mathematical troll. If Mahtworld forgot to say the manifold had to be compact, they made a minor misstatement that they probably ought to corrected in the interest of painstaking precision, but it's no big deal. It's obvious what they meant.

Re:Nah

[Cowculator]

Be careful how you phrase that last sentence - your carefree use of the word "obvious" in reference to math calls to mind an old joke:

Two mathematicians were talking one day about some recent work they'd done. One described a proof to the other but quickly glossed over a complicated step. The second one said, "Wait a minute - you didn't prove your last assertion." The reply: "It's obvious."

So the second mathematician wordlessly took a piece of chalk, went to the nearby blackboard, and began to fill it with long statements full of obscure symbols. Nearly half an hour later, he stopped writing, turned around, and said, "You're right. It is obvious."

Re:Nah

[mreece]

This reminds me of another anecdote - which I believe is true. I don't recall who it is about, though. The story is that at a seminar, a respected mathematician was giving a proof when someone questioned one step. The speaker said, "it is clear," and moved on. A bit later, he turned back to the questioner and said "it can be shown," then continued once more with the talk. A few minutes later, he paused, thought for a few seconds, turned to the questioner, and said "It is well-known." Moving on with the argument, a few minutes later he paused again, turned once more to the questioner, and said: "It is wrong."

It's always easy to take things for granted that look obvious; to some extent one always has to do this. The trick is knowing when you can do it and be right.

Re:Nah

[mamba-mamba]

Richard Feynman related a similar story in one of his autobiographical books ( _Surely you're joking, Mr Feynman_ and _What do you Care what Other People Think_ ).

It was something like this. He said that whenever he or one of his physicist friends questioned a math student about any assertion, the math student always said "it's trivial." So they always teased the math students by saying that mathematicians could prove only trivial theorems. It was all good natured, though. Feynman was not anti-mathematician or anti mathematics.

MM
--

Re:What's the problem?

[col_the_limey]

Mmmmm...... doughnuuut....

the eric conspiracy

[the eric conspiracy]

Maybe we should give these problems to the people at the next ACM International Programming Contest.

Poincare conjecture cases

[Anonymous Coward]

I'm somewhat familiar with this proofs used in different dimension ranges. It's absolutely necessary to separate out the proof into separate cases because the topology changes wildly with dimension. Roughly speaking in dimensions 4 there is so much room that certain powerful general techniques become possible (essentially, half the dimension of the manifold is more than 2 dimensions away from the full dimension --- so submanifolds of half the dimension cannot be KNOTTED). In dimension 3 and 4 special techniques must be used (and they are different in each case). In dimension 4, a submanifold of half the dimension (i.e, 2) can be knotted in the full manifold, but one can analyze the types of knotting that occurs. Manifolds of dimension 3 need techniques UNIQUE to this dimension (incompressible surfaces, etc.). The case of dimension 3 has been the hardest.

Re:Poincare conjecture cases

[esonik]

Hmm, very insightful comment. One question: what is so special about submanifolds of half the dimension of the manifold ?

And all horses are the same color.

[s20451]

Theorem: All horses are the same color.

Proof: By induction. First consider the case of one horse. Clearly, one horse is the same color as itself. Now suppose any set of k horses is the same color. If we take a set of k+1 horses, there are k ways to create sets of k horses, all of which must be the same color under the inductive hypothesis, and all of which contain common horses. Therefore any set of k+1 horses are the same color. Therefore all horses are the same color, by induction.

Re:And all horses are the same color.

[Anonymous Coward]

...and all of which contain common horses

Wrong when k=1, yawn...

Re:And all horses are the same color.

[CmdrSam]

Clever, but your proof falls down going from k=1 to k=2, since your two sets of 1 horse don't overlap in this case.

--Sam L-L

Re:And all horses are the same color.

[Tom7]

That's a good one. I'll have to give that to my students some time. =)

Re:And all horses are the same color.

[$uperjay]

Therefore all horses are the same colour as themselves, you mean?

Been Proved?

[Anonymous Coward]

Don't you mean, "has been proven"? Bad editor! No cookie for you.
This sentence will be no sense made.

Cows have an infinte number of legs

[Jhan]

1) Cows have an even number of legs.
2) Cows have forelegs and two back legs, equalling six legs.
3) Six is an odd amount of legs for a cow.
4) By 1 and 3 cows have both an even number of legs and an odd number of legs.
5) The only number that is both odd and even is infinity.

Cows have an infinite number of legs. QED.

I second that

[roystgnr]

Rice U. breaks Munkres' first book up into two classes, calling the second "Geometric Topology". It's a very clear discussion of the subject. I found "Elements of Algebraic Topology" much harder, but that may just be because we only had one semester to deal with that one.

ok. I have no idea what this is about

[RestiffBard]

I can appreciate that it is very interesting to mathematics folks. thats easy. no one knows what I'm talking about when I mention quantumn physics (I'm not a physicist but I can wrap my head around what I read). Mathematics however just befuddles me to no end. Could several of you math junkies point me in the direction of a good starter text on Mathematics? Something I can pick up at Barnes and Noble. Not the Knuth of Mathematics either. Knuth's titles are enough to make my toes curl and my brain fry. Just a layman's intro to Math will do. I'll ask again when I've figured out the first one.

Help, I don't get it

[roystgnr]

It's only been a couple years since I took geometric topology; I shouldn't have forgotten this much, this fast.

Isn't a sphere with a bubble in it (say, A = {x in R^n: 1/2 < d(x,0) < 3/2}) a 3-manifold? It's an open subset of 3-space.

Isn't that set A simply connected? You can deformation retract it down to S^2, which is simply connected.

And yet, even if the fundamental group pi_1(A) = 0, the higher homotopy groups aren't trivial: pi_2(A) isn't zero, so A can't be homeomorphic to a 3-sphere.

So why isn't this a counterexample to the Poincare conjecture?

Re:Help, I don't get it

[Anonymous Coward]

Open subsets aren't compact. I believe the Poincare conjecture is restricted to compact manifolds.

Re:Help, I don't get it

[Dyolf Knip]

Isn't a sphere with a bubble in it (say, A = {x in R^n: 1/2

Sure, but I think the whole point is to prove that a compact 3d shape, that is, the one with the greatest surface area in relation to its internal volume, is a sphere.

My question is, for the n>3 cases, were they basically doing geometry on hyperspheres? That's one thing I've never been able to wrap my head around.

Re:Help, I don't get it

[Anonymous Coward]

Your example is of a 3-manifold with boundary. The Poincare conjecture only considers 3-manifolds without boundary. (Obviously, no 3-manifold with boundary is homeomorphic to a 3-manifold without boundary, since "boundary" is a topologically defined concept.)

Re:Help, I don't get it

[snarkh]

It is not a compact manifold (as you said it is open), therefore Poincare conjecture does not apply.

And of course it is not homeomorphic to the 3-sphere, it is homotopic to the 2-sphere.

Yes and No

[Weezul]

Yes there are oftin errors in annonced proofs, but mathematicians rarely miss serious flaws in the proof. Once you understand what is going on (and have spend years working in the area) you can make little small mistakes and avoid making big mistakes.

Frequently, the guy to announce the proof has truely understood something deep about the problem, thus making asignificant contribution. Indeed, they frequently have understood so much that the rest of the mathematical community fills in even the serious gaps if their are any. The one notable execption to this is P=NP.. lots of people have announced results for that one.

not _quite_

[jacobb]

if it's simply connected: either it's not a 3-manifold (depending on whether the bubble is closed or not) or it's not a counterexample.

Are there any physical science advances

[wdavies]

that can be made because of this potential breakthrough?

Just curious, or whether it is just an annoying abstract problem that was solved?

Winton

Call -151

[Anonymous Coward]

Wow, somebody still remembers this. Course now we know how old you are. :)

It's a _preprint_

[Ulumuri]

Firstly, notice that this supposed "proof" is a preprint. That means it hasn't been peer-reviewed. As often happens with proofs of open problems, there appears to be a working proof and everyone gets excited, but on further reviewing, the proof is found not to work, at least without some modification. (This happened with Wiles' proof of Fermat's last theorem. I have a shirt of Wiles running after the equation x^n + y^n = z^n in a butterfly net which was made before the gap in his proof was fixed.)

6th revision

[Anonymous Coward]

If you look at the University of Southampton Mathematics Preprint page, you'll see that this is
the sixth revision of this preprint. Versions of this argument have previously been shot down by other experts.
There's no evidence this one has been accepted by any other expert.

Moderators!

[Anonymous Coward]

The parent post is complete nonsense...

Re:ok. I have no idea what this is about

[Anonymous Coward]

That's a pretty broad request. "A book on math." There are too many kinds of math; you won't find a book covering everything. What are you looking for? Arithmetic? Algebra? Geometry? Topology? Calculus?

Absurdist logic, or incosistent math?

[Anonymous Coward]

Which is why I'm a fan of absurdist logic.

How about inconsistent mathematics?

All successful arguments must contain a contradiction/paradox somewhere.

Hardly.

Here's a couple books

[Anonymous Coward]

Well sorry, but to truly understand this stuff you really do need to have studied a lot of mathematics. I'd say two years minimum of in depth, theory level college mathematics would allow you to read and at least get the gist of most mathematics texts/problems.
The poincare conjecture in the n=3 case is fairly simple to state, it's significance is what is more interesting, and that I cannot remember or find anything useful on at the moment.
Which is not to say you can't have a lot of fun trying to wrap your head around this stuff or other higher level mathematics anyway. Here's a couple general mathematics books with some fun problems in them.
Archimedes Revenge is fairly accessible.
From Here to Infinity By Ian Stewart, that is pretty in depth, but just trying to get the gist could be fun. It has a good chapter on Fermat's Last Theorem
And some of Ian's other books are probably good. Try here

more questions

[Anonymous Coward]

Then... is homotopic equivalence sufficient condition then for all d-dim objects in R^n (simply connected compact) to be homeomorphic to each other?

If this is true, then isn't the classificlaction problem solved for such objects?

(Disclaimer : am just a lousy physicist who dabbles into topology for fun.)

that's wrong..

[Anonymous Coward]

= sqrt( (-1) * (-1) ) = sqrt(-1) * sqrt(-1)

That is wrong. It's like saying: = sqrt(16) = sqrt(4 * 4) = sqrt(4) * sqrt(4) = 2 but square root of 16 is 4...

Re:that's wrong..

[Anonymous Coward]

= sqrt( (-1) * (-1) ) = sqrt(-1) * sqrt(-1)

That is wrong. It's like saying: = sqrt(16) = sqrt(4 * 4) = sqrt(4) * sqrt(4) = 2 but square root of 16 is 4...

Uh, it's right. It's like saying: = sqrt(16) = sqrt(4 * 4) = sqrt(4) * sqrt(4) = 2 * 2 = 4.

sqrt(1) = sqrt((-1) * (-1)) = sqrt(-1) * sqrt(-1) = (+/- i) * (+/- i) = +/- -1 = -/+ 1. But sqrt(1) = -/+ 1, so there is no problem.

Re:that's wrong..

[Anonymous Coward]

... no problem with the rest of the derviation, that is. The problem is in the first step, 1 = sqrt(1). It should be 1 = +/- sqrt(1), and then the answer is not inconsistent. You just end up with +/- 1 = +/- 1.

Thanks, I think I got it now

[roystgnr]

Basically my problems were:

The manifold needs to be compact for the conjecture to apply.

I was thinking of the "3-sphere" as B^3, not S^3.

Thanks, everyone.

Re:...has been "PROVEN", ...has been "PROVEN"

[pclminion]

From www.m-w.com:

Main Entry: prove
Pronunciation: 'prüv
Function: verb
Inflected Form(s): proved; proved or proven

You can say it either way. It's standard usage. Idiot.

some good books

[jacobb]

you're lucky... Algebraic Topology texts are much cheaper than the average math textbooks...
Here are 3 good books:

"Algebraic Topology," Hatcher, Allen ;
"A Concise Course in Algebraic Topology," May, Peter J. ;
"Algebraic Topology," Harper, J.R. & Greenberg, Marvin J.

Here are the links from amazon: 1 ; 2 ; 3.
Good Luck and Cheers!

Re:...has been "PROVEN", ...has been "PROVEN"

[Anonynnous Coward]

Fair enough. But the first one is more likely the correct one, while the second one was added as a concession to the fact that so few could get it right. It's like "data are" and "data is"--both are considered correct now, but literate people use the first.

Re:...has been "PROVEN", ...has been "PROVEN"

[Anonynnous Coward]

D'oh!

But the first one is more likely the correct one

Which, I see now, is the one that had been used. Sorry.

Re:ok. I have no idea what this is about

[Anonymous Coward]

There's a book translated from the German called Invitation to Mathematics by Konrad Jacobs. It's published by Princeton University Press and you can get it in softcover. Probably from bn or something.

The book was intended for high level students, but from some other discipline besides mathematics (philosphy maybe?) to get a brisk introduction to modern mathematics. There are chapters on topology, dynamical systems, game theory, and numbers.

You'll like it a lot.

Thank you for pointing this out

[Anonymous Coward]

If it gets published in a decent journal, then get excited.

Re:Thank you for pointing this out

[Anonymous Coward]

yeah, get excited if you want. But don't get a shirt with math 'jokes' on them and look like a dork.

And 1=2

[Anonymous Coward]

2^2 = 2 + 2
3^2 = 3 + 3 + 3
4^2 = 4 + 4 + 4 + 4

Therfore
x^2 = x + x + x +.. [x times]

so:
d/dx(x^2) = d/dx(x + x + x +... [x times])
2x = 1 + 1 + 1 [x times]
2x = x

Therefore 2 = 1
QED :)

Re:...has been "PROVEN", ...has been "PROVEN"

[Anonymous Coward]

*sigh* They're different tenses... "proved" is past, "proven" is past-participle

"I proved X", "X has been proven"
</grammar-nazi>

Yeh, okay

[autopr0n]

I have nothing against "intelectuals" but simply saying the "poincare conjecture" to me meas as little as "clitoris" probably means to you. It would help if they had had at least one or two sentances explaning what it was, or why it was important.

Goldbach Conjecture in our lifetimes??

[jnana]

I'm waiting for the Goldbach Conjecture to be proved, but not holding my breath.

From mathworld: "Schnirelman (1939) proved that every even number can be written as the sum of not more than 300,000 primes (Dunham 1990), which seems a rather far cry from a proof for two primes!" Still a ways to go, gents.

Re:Goldbach Conjecture in our lifetimes??

[Anonymous Coward]

But these days it has been got down to every even number is the sum of a prime, and an almost-prime (a number C that is the product of two primes).

Well, it IS a simple if statemement....

[da5idnetlimit.com]

You remember the beginning af fractal calculus ?

At first it was just a way to get infinite zoom.

Then a way to get pretty pictures.

Then somebody came up with an idea and found a general equation for simulating the groowing of tree and most plants, aand also an equation to calculate icing rate on a given surface...

And fractal calculus is just a SIMPLE thing.

Get some news of the guy who invented a simple yet oversophisticated mathematical programming langage (some time ago on /. don't have it just now)

What did he do ? he translates everything in his new language, which gives him a usable algo that is quite easy to programm.

=> Statement simplification using a full change of definition sets.

Just like going from the Greeks math and discovering relativity.

So, we go from singularity studies to broader and broader concepts, and with the time thoses get more application.

Just like the guy trying to build a time machine with Lasers.

the theory is old. But he had the idea of putting it all together.

Welcome into Evolution, Friend 8)

Re:ok. I have no idea what this is about

[paprika]

For many years, the standard book of this sort was "What is Mathematics?" by Richard Courant and Herbert Robbins. It's not really armchair reading, but if you're willing to pick up a pencil (and get stuck on tough points for a day or so), most people who passed calculus should be able to get through it.

Finally!

[Anonymous Coward]

Now I can sleep at night!

Answer this though

[Hoi Polloi]

What's in my pocket?

(Apologies to Bilbo Baggins)

Re:...has been "PROVEN", ...has been "PROVEN"

[Anonymous Coward]

Hope you enjoyed wasting your points modding me down, asshole. I'll just use some of my non-troll ID's to mod myself back up.

~~~

Re:...has been "PROVEN", ...has been "PROVEN"

[pclminion]

I agree, the use of mod points on this comment was inconceivable.

Sorry that I called you "idiot." I've been reading K5 recently and my policy (call it stupid) has been to reserve sarcasm for /. only.

Human language is a chaotic, natural system. To attempt to apply hard-and-fast rules to it seems silly.

Teachers should do this to their students

[A55M0NKEY]

High school algebra teachers should do problems like this ( and the 1=sqrt(1)*sqrt(1) thing ) on the board and get the class to try and figure out what happened. I remember beating my head over results like 4=5 and not even knowing what I did wrong. Then when I asked the teacher they would just do it the right way and not show me exactly where the division by zero was. They always magically avoided these kinds of things but never would tell us how to avoid them ourselves, I think they had learned 'habits' that they did not themselves understand to avoid this kind of thing. They would just say that it takes lots of 'practice' [to learn these habits]. Answers like 4=5 can destroy any confidence in adding linear equations that a novice might have, and make them think that they naturally suck at math!

In the 'proof' above there are two errors. One is a real mathematical error: the division by zero, but the other is a legal math move which is a strategic error ( adding the equations so that all three variables are on both sides gets you further from a solution ). This should be discussed explicitly in high school algebra class.

I don't even blame the teachers! I doubt I would do much better!

I now have a BA in math from a good school, and I was not a bad student either, but I still don't know any straight forward algorithm for doing basic high school algebra! I know that mathematica and other computer programs can solve equations, and simplify stuff. I've used them. They're awesome! But when I've browsed the web to find out how they work, I haven't found much.

Basically all the info I've found says that automatic algebra has to do with Groebner Bases, and a lot of abstract algebra. Sure I got an A- in Abstract Algebra, but this stuff is kinda thick for bathroom reading.. I need to read a 'Groebner Bases for Dummies' I guess... I think everyone should have an algorithm at their disposal that tey can be confident solves most commonly encountered algebra problems, and they ought to know why it works.

I wish I did