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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.
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..).
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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.
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.
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 :-)
Can't you find anything to report about that HASN'T already happened?
How can ANY editor report something that HASN'T YET HAPPENED??
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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
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).
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.
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