Slashdot Mirror
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.
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.
I think Mauve has the most RAM. --PHB (Dilbert Comic)
The Poincaré Conjecture proved, and microsoft ads on slashdot
"I think it would be a good idea" Gandhi, on Western Civilisation
Charles Dodgson, somewhere thru the looking glass, is at tea with the Mad Hatter discussing this very matter.
:)"Academicians are more likely to share each other's toothbrush than each other's nomenclature."
Cohen
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.
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..).
fucktard is a tenderhearted description
You show great skill at cut & pasting from http://www.claymath.org/prizeproblems/poincare.htm : )
Just kidding. Go ahead, enjoy the cut & paste karma.
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.
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.
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.
You can prove anything :-)
Stop worrying about the risks of nuclear power and start worrying about the risks of not using nuclear power.
Isn't this just Godel's theorem? That in any axiomatic schema, there will be certain propositions that can not be proven within it.
Humorless sig goes here.
oh sure you can, but then winston churchill becomes a carrot...and other such nonsense.
If Mr. Edison had thought smarter he wouldn't sweat as much. --Nikola Tesla
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*
i hate pansy republicans
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.
... 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.
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 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.
-- Our systemic servants do not good masters make.
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!
It's psychosomatic. You need a lobotomy. I'll get a saw.
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.
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
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 :-)
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.
Can anyone recommend any other books on algebraic topology?
Danny.
I have written over 900 book reviews
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...
Can someone explain what the hell this problem is about in English please? (Preferably avoiding the word manifold).
Can't you find anything to report about that HASN'T already happened?
How can ANY editor report something that HASN'T YET HAPPENED??
ATH0 Bitcoin: 1DnwFLXczVZV8kLJbMYoheUrpqHesjxrSi
--
The JabberWokky (yes, I know. It's intentional to create a unique string.)
"$30 for the One True Ring. $10 each additional ring!" -- JRR "Bob" Tolkien
Still, by the Poincaré Conjecture - Gumby is equivalent to Pokey.
http://www.math.princeton.edu/jfnj/texts_and_graph ics/erratum.txt
-Kevin
Maybe we should give these problems to the people at the next ACM International Programming Contest.
No, x/0 is undefined. However, you can do things like
because, when y approaches zero, x/y will 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->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.)
Sig (appended to the end of comments I post, 54 chars)
Teacups have handles.
"The new wave is not value-added; it's garbage-subtracted" - Esther Dyson, Dec 1994
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.
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.
Sigh. My id isn't prime. 2 2 2 2 2 3 5 313
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.
Toronto-area transit rider? Rate your ride.
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.
You just get that time machine from the previous story, and...
You see? You see? Your stupid minds! Stupid! Stupid!
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 choose to remain celibate, like my father and his father before him.
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.
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.
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.
-
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?
that can be made because of this potential breakthrough?
Just curious, or whether it is just an annoying abstract problem that was solved?
Winton
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."
Would that happen to be this book, for the interested reader?
Zero: The Biography of a Dangerous Idea
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
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,...
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?
-- SIGFPE
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)
yes, but most math is based on Peano and Euclid (or variations)
those axioms are pretty hard to deny.
In Capitalist America, bank robs you!
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.
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.
Matt Reece
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.
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.
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.
:wq
50 years?
:wq
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.
autopr0n is like, down and stuff.
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.
Sig (appended to the end of comments I post, 54 chars)
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...
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.
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.
-- the most controversial site on the Web