Why is the Poincaré conjecture important?

August 24, 2006 10:17 AM

What benefits will solving the Poincaré conjecture bring to mankind?

There's been a few news items recently about this Russian chap who has apparantly solved this riddle. I don't even understand what the problem is that has apparantly been solved, but I would be interested in knowing what benefits its resolution will bring to the world. Can anyone explain in layman's terms?
Many thanks in advance.

posted by chill to science & nature (17 comments total)

In terms of material benefits, none whatsoever.

It's pure knowledge. Not all knowledge brings direct benefits to mankind. But to mathematicians, it's a great achievement that closes a major hole in their understanding.
posted by cgs06 at 10:39 AM on August 24, 2006

Indirect benefits. The greater the understanding that mathematicians have of their science, the more they can figure out about physics and the universe, which eventually trickles down to use pesants after it goes through practical applications like the US Space Program or the US Military.
posted by SpecialK at 10:44 AM on August 24, 2006

The interesting thing with conjectures is that you can just assume they're right and do all the work that would have followed it anyway. You don't need to wait until it's actually proved to build upon it.

Of course your new work isn't proven until the conjecture is, but that doesn't really hold anybody up.
posted by smackfu at 10:58 AM on August 24, 2006 [1 favorite]

this is a pretty good layman's-terms description of it. as for benefits to the world - probably nothing directly, as the difference in topology between a sphere and a donut is obvious (if not really expressible) to most people.
posted by sergeant sandwich at 11:09 AM on August 24, 2006

Knowing the answer to Poincare's conjecture won't build you a better toaster oven tommorow. There's no way that not knowing the answer to a "yes/no" question (which is what Poincare is, really; "Is everything that with the same algebraic data as a sphere necessarily a sphere?" "Yes," as it turns out.) can really impede technology. If some technology really depended on knowing the answer to Poincare you'd just go ahead and build it, assuming it was true, and if your gadget failed unexpectedly, well, then you'd have learned something.

So the primary benefit of Perelman's work is not the answer to the a big question, but the techniques he developed in the process of tackling it. The machinery of Ricci flows almost certainly will benefit applied mathematicians in a lot of unanticipated ways, and mathematicians and engineers who could not possibly have refined the machinery the way Perelman did can still use it now that he has.

That's the practical benefit of big, "pure" problems like Poincare and most of the Clay Institute problems. The answer to the question itself isn't as important as the machinery that has to be built to get there. Once an engine is built, industrious people will always find a way to drive it interesting places, even if the guy who invented it regarded it as an end in its own right and doesn't care where it goes.

By way of comparison, look at how much effect the answer to Fermat's Last Theorem has had on the world (I would say essentially none), versus how much effect the machinery of elliptic curves, which was refined enormously because of pure mathematicians' interest in the Fermat problem, has had.

Sorry if that's a bit more vague and philosophical than what you wanted; hope it still says something useful.
posted by Wolfdog at 11:09 AM on August 24, 2006 [5 favorites]

Not to mention the 13-year-old kid who's inspired to go into math or engineering by this.
posted by callmejay at 11:47 AM on August 24, 2006

I don't know that, in math, they ever have the applications ready before the math work is done, i.e. no one's sitting around saying, "We can build this air-car as soon as the math guys get done formulating Foo-Bar Theory."

It's more like after Foo-Bar theory is invented, someone realizes it can be used to build an air-car.
posted by TheOnlyCoolTim at 12:04 PM on August 24, 2006

Thanks all, that pretty much answers my question.
posted by chill at 12:23 PM on August 24, 2006

Your question is a variant of a question that mathematicians have to deal with all of their lives. Someone asks you to explain what it is you're working on and you do a better or worse job of explaining what it is, either way. Then, usually, comes the question "but why?"; "what is it good for?".

It isn't always good for something, and some of the time it will never be good for anything, at least not in the sense that the askers of such questions usually mean. There is a close analogy with certain kinds of physics; if you're trying to figure out what's going on inside of black holes, or whether or not there really is dark matter, or whether the universe is expanding or contracting, you're usually not doing it with an application in mind. You're studying it for its own sake, and the reward at the end is a greater understanding of the universe. And people don't ask physicists "but what is it good for", or at least I don't have the impression that it's as common. (Any theoretical physicists out there want to correct me on this one?)

I think the reason that many people feel it's a justifiable question to ask mathematicians (though they would never ask the same question of a physicist) is because it's hard for most people to imagine that understanding more mathematics leads to a deeper understanding of the universe in any meaningful way.

It's true that "pure" math, at its heart, is devoid of any reference to the physical world; its laws are its own, and its truths are its own, and nothing that could change about the world can change that. But I believe that at its heart, when we study math we are studying something, not just playing with some arbitrary system of rules we set up (like some elaborately complicated game designed to trick funding agencies into giving us money). That something may be very hard for most people to grasp (for whatever reason -- I personally think it has a lot more to do with cultural taboos around the hardness of mathematics and with the way that mathematics is taught at a very early age, as a matter of rule learning and not as a creative endeavor), and many people, perhaps, don't even believe that it is something, but view it more as a game, like i suggested above. (For those interested in the philosophy of math, this was a major difference between the views of Godel and of Wittgenstein on what people are doing when they are doing math: Godel thought that they were finding truth, and Wittgenstein thought they were playing games.)

But I belive that there is something there (hard as it may be to define what it is -- but have you tried to define physical reality lately), and that the study of math is at its heart about the pursuit of knowledge and truth, and not simply a tool that is waiting to be applied to the "real" world.
posted by louigi at 12:50 PM on August 24, 2006 [3 favorites]

Sorry for posting after you said the question had already been answered; it's an issue I care about, and I'd already spent a bunch of time thinking about my response.
posted by louigi at 12:51 PM on August 24, 2006

Don't apologise! The more erudite responses the better!
posted by chill at 1:25 PM on August 24, 2006

Chill...

Advance apologies for attempting to answer and already answered question, too, but this brought to mind the same question, asked years ago, about a different topic.

I once had 60 acres in the mountains. Every year, I would hire a wildlife biologist to take friends and visitors on a wildflower hike in the spring. Invariably, once something was found, a guest would ask him, 'What is it good for?'. With a pained look, he would explain anthropocentric utility of the item, sometimes working really hard to do so.

In private, he once confided in me that all he ever wanted to do was ask people, 'What are you good for?'.

I know it sounds smart assed, and may be, but the good/bad of something isn't always measured by its human utility. ( I always thought it was funny, though!). That's my first observation.

Observation 2 is more to the point. It has the same utility as running a record marathon. The limits of reason had been run up against until our Russki pal did his thing or Wiles resolved Fermat's last. Now they've moved a little. And as we continue the process, we expand the universe of what reason can do. Practical? As practical as a marathon and to the normal human, pretty useless.

These things remind me of the last few words in a HUGE crossword puzzle (i.e., math). We invented it, we know how a lot (but not all) of what we invented works and (a lot but not all of ) what it implies, and we fill in the missing items slowly as new eyes, new intuition, and greater visualizing power comes on the scene. It is a fine time to be alive to see Poincare and Fermat resolved in the space of a few years.
posted by FauxScot at 1:43 PM on August 24, 2006 [1 favorite]

Listen people, that Transporter aint makin' itself.
posted by Kensational at 2:41 PM on August 24, 2006

FauxScott, you said what I was going to say much more politely, although I would have been more succint.
posted by Mr. Gunn at 5:01 PM on August 24, 2006

It's more like after Foo-Bar theory is invented, someone realizes it can be used to build an air-car.

I agree. Case in point -- mobile phones, people are always telling me, were developed with the help of quantum mechanics.

Once the solution to a mathematical problem is out there, the next time someone needs to solve that problem, whether they're inventing a mobile phone, or curing cancer, they can use it. On the shoulders of giants etc.
posted by AmbroseChapel at 5:28 PM on August 24, 2006

Specifically, the question "can a toroid be distorted into a sphere without tearing" was answered by Stephen Colbert this week, when he took a donut and crumpled it into a ball.

"I did not realize match could be so sticky", he concluded.
posted by Dunwitty at 2:10 AM on August 25, 2006

Actually, he said "I did not realize math could be so sticky", but he's a trained comedian, and I am not.
posted by Dunwitty at 5:17 PM on August 26, 2006