## The Poincaré Conjecture has Been Proved 307

Posted
by
chrisd

from the np-is-the-problem-for-me dept.

from the np-is-the-problem-for-me dept.

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.

## Wierd Problem (Score:3, Interesting)

## Re:Wierd Problem (Score:1)

## Re:Wierd Problem (Score:2, Informative)

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 (Score:2)

isan 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 (Score:3, Funny)

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 (Score:2, Informative)

Dan

## Re:Wierd Problem (Score:2)

## Re:Wierd Problem (Score:2, Informative)

If you're interested in reading up on it a bit, the link in the original post to "http://mathworld.wolfram.com/PoincareConjecture.

## Re:Wierd Problem (Score:2)

## Re:Wierd Problem (Score:2)

## Re:Wierd Problem (Score:2)

## Re:Wierd Problem (Score:2)

http://mathworld.wolfram.com/DivisionAlgebra.ht

## Re:Wierd Problem (Score:2, Informative)

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 (Score:2)

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

## For 1=7 (Score:2)

## Re:For 1=7 (Score:2)

## Re:Wierd Problem (Score:2)

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.

## now I've seen it all (Score:3, Funny)

## teacup == donut (Score:1, Funny)

## Re:teacup == donut (Score:2, Funny)

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

:)## Re:teacup == donut (Score:2)

--

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

## Re:teacup == donut (Score:2)

## Let's wait on calling it "proved" (Score:2, Informative)

## In related news.... 4 = 5 (Score:1, Funny)

assume a, b, c such that:

a + b = cthen

5a + 5b = 5cand

4c = 4a + 4badding the two:

5a + 5b + 4c = 4a + 4b + 5cshifting some terms around:

5a + 5b - 5c = 4a + 4b - 4csimplifying:

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

5 = 4:)

## Re:In related news.... 4 = 5 (Score:1)

## Re:In related news.... 4 = 5 (Score:2, Interesting)

## Re:In related news.... 4 = 5 (Score:2)

Zero: The Biography of a Dangerous Idea [amazon.com]

## Re:In related news.... 4 = 5 (Score:1, Redundant)

a + b = cthen

(a + b - c) = 0so

5*0 = 4*0someone had to do it ....

## Re:In related news.... 4 = 5 (Score:1)

someone had to do it ....make that everyone had to do it

## And all horses are the same color. (Score:2)

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.## Cows have an infinte number of legs (Score:2, Funny)

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.

## Re:In related news.... 4 = 5 (Score:2, Funny)

## Re:In related news.... 4 = 5 (Score:2, Insightful)

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 (Score:3, Informative)

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

because, whenyapproaches zero,x/ywill obviously become larger. But that is not the same as0*infinity is undefined, however, continuing the example above, I could write:

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->0which means thatyis a sequence of numbers approaching zero, and noty = 0. A suitable sequence might e.g. bey[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 (Score:2)

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 (Score:2)

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 [slashdot.org] 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 (Score:2)

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,...

## Pah, anybody with a time machine (Score:3, Funny)

## proof has been announced (Score:5, Insightful)

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 (Score:2, Insightful)

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 (Score:2)

## Statement of conjecture on wolfram incorrect? (Score:5, Informative)

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? (Score:2)

## Re:Nah (Score:5, Funny)

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

isobvious."## Re:Nah (Score:3, Insightful)

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:Statement of conjecture on wolfram incorrect? (Score:2, Funny)

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

## books on this stuff (Score:2)

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 (Score:3, Informative)

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 (Score:2)

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.

## I second that (Score:2)

## English please! (Score:2, Interesting)

## Re:English please! (Score:2, Informative)

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! (Score:2)

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! (Score:4, Insightful)

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! (Score:2)

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! (Score:2, Informative)

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! (Score:2)

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

## Re:English please! (Score:2)

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

## the eric conspiracy (Score:4, Funny)

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

## Poincare conjecture cases (Score:3, Informative)

## ok. I have no idea what this is about (Score:2)

## Here's a couple books (Score:2, Informative)

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 [amazon.com] is fairly accessible.

From Here to Infinity [amazon.com] 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 [amazon.com]

## Help, I don't get it (Score:2)

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 (Score:2)

Isn't a sphere with a bubble in it (say, A = {x in R^n: 1/2Sure, but I think the whole point is to prove that acompact3d 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 (Score:2, Informative)

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

## Thanks, I think I got it now (Score:2)

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.

## Are there any physical science advances (Score:2)

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

Winton

## Re:Proof (Score:1)

Popper fan I presume.

:)## Re:Proof (Score:1)

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

knowthat 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 (Score:4, Insightful)

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 (Score:2)

## Nope (Score:2, Insightful)

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 :-)

## Re:Proof (Score:2)

## Re:Proof (Score:3, Informative)

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

-Kevin

## Re:Proof (Score:2)

## Re:Proof (Score:2, Insightful)

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 (Score:2)

## Re:Proof (Score:2)

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 themThat'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 (Score:2)

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...

## Blind faith in Mathematics (Score:2, Insightful)

And faith in Mathematical proof is counterproductive at a level beyond that

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 [amazon.com] 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.

## Um... (Score:2)

halfan education in mathematics...## Re:Proof (Score:3, Interesting)

-- 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 (Score:2)

you'rewrong, 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.## Re:Proof (Score:1)

"given that the following assumptions are true"or"based on the following axioms...".I guess calling it a

mathematicalproof includes this, though.## Re:Proof (Score:2)

those axioms are pretty hard to deny.

## Troll, or uninformed post? [*free clue enclosed*] (Score:2)

"Anything can be proved with enough flawed mathematics." How does one prove something with flawed mathematics? Certainly, one can

attemptto 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.

## Re:Proof (Score:2)

## Re:What's the problem? (Score:2, Insightful)

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..).

## Re:What's the problem? (Score:5, Informative)

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

## Re:What's the problem? (Score:2)

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

## Re:What's the problem? (Score:3, Informative)

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.

## Re:The problem is... (Score:2, Funny)

## Re:Well.. (Score:2, Informative)

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*

## Yeh, okay (Score:2)

## Re:Old news... (Score:2, Insightful)

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... (Score:2)

## Re:I gave up being a math major (Score:2, Insightful)

## Re:I gave up being a math major (Score:2)

when writing for a knowledgeable audience, it makes much more sense to refer to things in technical terms than to describe everything in detail.

*especially* in math, where things generally have very precise definitions.

## Re:...has been "PROVEN", ...has been "PROVEN" (Score:3, Interesting)

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.

## Re:...has been "PROVEN", ...has been "PROVEN" (Score:2)

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.