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Tracking the World's Great Unsolved Math Mysteries
coondoggie writes "Some math problems are as old as the wind, experts say, and many remain truly unsolved. But a new open source-based site from the American Institute of Mathematics looks to help track work done and solve long-standing and difficult math problems. The Institute, along with the National Science Foundation, has opened the AIM Problem Lists site to offer an organized and annotated collection of unsolved problems, and previously unsolved problems, in a specialized area of mathematics research. The problem list provides a snapshot of the current state of research in a particular research area, letting experts track new developments, and newcomers gain a perspective on the subject."
http://en.wikipedia.org/wiki/Collatz_conjecture Speaking of unsolved math mysteries, the 3n+1 problem is a fabulous way to spend days and days of your life. It's particularly fun if you think about it in binary. Whatever the answer is, it's either simple and elegant or complex beyond imagination.
-- IANAL, this isn't legal advice, and definitely isn't legal advice for you. Also, Squee!
I have this wonderful proof for this conjecture, but unfortunately the 80 char limit for sig in slashdot is too small for it.
sed -e 's/Chuck Norris/Rajnikant/g' joke > fact
Some say math is discovered. Others say it is invented. You are one of the latter.
Warning: this article may contain humor, sarcasm, parody, and perhaps even irony. Read at your own risk.
Julius Shaneson and Sylvain Cappell claimed to have solve a famous problem about counting the lattice points in a circle. It's been out for years, even earlier than this arxiv paper:
http://arxiv.org/abs/math/0702613
Thing is, even though it is a famous problem, no one cares enough to check. So this notion of "famous" is shaky.
The requirement to model our universe as closely as possible is a requirement of physics, not mathematics. The fact that mathematics, even very abstract mathematics, accurately models the natural world is a deep mystery.
xterm -n 8
their servers will explode when they take a stab at Navier-Stokes. I asked Wolfram-Alpha, but it simply returned the exact solution of a degenerate case, the solution being 'Fuck you.'
'We are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress.' RPF
they're a dime a dozen, too.
rewriting history since 2109
See http://rjlipton.wordpress.com/2009/11/12/more-on-mathematical-diseases/ for unsolved problems which are all really simple and also really addicting to think about. For many of these, the best way to stop thinking about one of them is to start thinking about another.
I'm actually a bit puzzled as to why this is a Slashdot article. If I wanted to point to something new in the way people are doing math I'd point to Math Overflow http://mathoverflow.net/ where many professional mathematicians, grad students and others are active. It is essentially a centralized system for people to post math questions and get math answers from people who know. It is very cool. It also is highly addictive to read.
or small values of 3
rewriting history since 2109
As of about a year ago, a new kind of collaborative math project known as "polymath" is emerging. These research projects are completely open for any interested scholar to drop in and make contributions to the problem at hand. The technical infrastructure is based on well-known tools such as wikis and forum discussions
The very first such project successfully explored a new approach to the density Hales-Jewett thorem--a significant problem in combinatorics--in about six weeks of effort, with a fully preserved record of about a thousand contributions from dozens of participants.
See Polymath Wiki for the details. This new contribution from the AIM will provide a focus point for such efforts and encourage similar massively collaborative projects.
And of course, the emerging field of computer-verified mathematics is also dependent on massive collaboration, in order to translate existing results into a fully-formalized form that computes will understand and verify as correct. A wiki-based project could be a great help there as well.
If it is discovered, the solution already exists and the problem was solved before wind existed, because the problem never existed in a state where it didn't have a solution.
If it is invented, the problem didn't exist before the wind.
In either case, the problem isn't older than the wind.
I would claim that the ratio of a circle to its diameter is independent of being observed, or indeed there being an observer. I would also claim that the laws of geometry, probability and topology are universal and also do not depend on the existence of observers, let alone their ability to perform maths.
Radioactive decay follows an exponential decay curve. It will have done so long before anyone could add, let alone handle irrational numbers like e.
This puts me firmly in the category of maths being discovered, not invented. Mathematical tools, however, are invented and not discovered. I consider these to be quite different. If you were to imagine an alien lifeform on some distant world, they'll have an identical math but their experience of it, the way they treat it, the systems they use, those will all be unique to them because those are inventions and not anything fundamental to maths itself.
In a simpler example of the same concept, we can use ancient Greek maths today even though they didn't have a concept of zero and had (to modern eyes) very alien views on the way maths worked. We can use ancient Greek maths because the results don't depend on any of that.
We can use Roman results, too, despite the fact that their numbering system doesn't really follow a number base in any way we'd understand. It doesn't matter, though, because the important stuff all takes place below such superficial details. Even more remarkable, we can read many of the numbers written in Linear A, even though we can't read the language itself and know very little about the culture or people.
None of this would be possible if what lay under maths was invented. It's very hard to rediscover lost inventions, as there's many ways of producing similar results. But when you can rediscover lost number systems with comparative ease - well, doesn't that tell you there has to be something a bit more universal to it?
(I won't get into parrots being able to discover the notion of zero, but it's again pertinent as it's an example of a universality that transcends the invented language it's described in.)
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
why not hide them in video games so we can get more people to look at them.
Only (very) loosely related but deserving mention is the Encyclopedia of Integer Sequences.
This encyclopedia has proven very useful for me in that I have avoided 'solving' many problems with it.
"His name was James Damore."
Since the existance of a perfect circle depends on the thoughts of an observer, the ratio of the diameter to the circumference of such an object must depend on there being an observer. Nature can produce approximate circles, but not perfect ones.
Radioactive decay follows an exponential decay curve. It will have done so long before anyone could add, let alone handle irrational numbers like e.
"Exponential decay curve" and "irrational numbers" are two different concepts. (1/2)^N is an exponential decay curve -- which defines the half-life of a radioactive substance. For no integer value of N is the result "irrational".
This puts me firmly in the category of maths being discovered, not invented.
Right destination, wrong reason.
Some say math is discovered. Others say it is invented.
And still others (especially those in grade school and high school) say that math should neither have been invented nor discovered.
Mathematically modelling the brain would seem to be a very trivial problem. The problem is that there's a lot of brain to model. I've posted (admittedly non-rigorous) mathematical models of the brain on Slashdot before, but narry a grant check from it. Bah.
Computational fluid dynamics for foams, liquid crystals, et al, isn't any harder than for anything else. The equations are chaotic by nature, but chaotic systems can be well-behaved on aggregate under certain conditions. CFD as generally done relies on some specifically hand-picked special case or cases being universally true. They never are, which is why most CFD differs from how systems actually behave in practice.
If you were to treat CFD as a problem in chaos theory, rather than as isolated collections of imperfect examples of special cases, there would be no problem. It is always when engineers try to take shortcuts and oversimplify the maths to make it easy on themselves that they run into problems. They should be locked up for their own safety. If you want to really annoy them, lock them up with some airgel foam.
The problem with chaotic systems is that the system is sensitive to initial conditions, which means the only way to get "correct" results is to use infinite precision and zero step sizes. This isn't useful, but is a good way to annoy experts in CFD.
This leaves two options - use very very big, very very fast computers (the option used by F1 teams), or find an equivalent problem you CAN solve (the idea behind transforms).
Ok, does chaos look like a good place to use transforms? If you could identify and classify the Strange Attractors in the system, can you do anything useful? Probably not, at least not in the "solving the problem" sense. Chaos is fully deterministic, but it is utterly unpredictable. The only solution is the whole solution.
What knowing the Strange Attractors might tell you is how to vary the precision and step-size to get the best results for a given level of compute power. But it's going to be all raw horsepower from thereon out.
The best way to invest money on such work is to design a co-processor that performs a handful of fairly high-level maths functions directly (optimized purely for speed, not physical or logical space) so that you can do Navier-Stokes almost at the level of raw hardware rather than through clunky software. Then cluster the living daylights out of the co-processor.
It's necessary to optimize commodity hardware for space, because chip real-estate is expensive. However, if you're building what is basically a SOP (single-operation processor) for a dedicated market that can afford things like Earth Simulator, the only time you care about space is when it impacts speed.
Ideally, if the speed of light wasn't an issue, you'd want each bit in the output to be produced by wholly independent logic, duplicating the input bits as necessary to accomplish this. In practice, you'd probably want to start with that conceptually but in reality have something that was somewhere between that and a highly compressed form. Too parallel and the delays in communication exceed the benefits from the parallelization.
But this is all obvious. Anyone here who has done multi-threading or any other form of parallelization knows about synchronization issues and communication overheads. It's even one of the biggest chunks of any course on the subject of parallel design. There's nothing new there, certainly nothing "unsolved".
But, yeah, a well-designed Navier-Stokes co-processor would likely give you greater accuracy and greater performance than the modern pure software solutions. Especially those using ugly protocols to do the communications.
If Intel can conceptulalize 80 Pentium 4 cores on a wafer, it would seem reasonable enough to imagine modern fabrication methods being able to put at least a couple of hundred dedicated Navier-Stokes processors into the same space. Since the input for an iteration would be based on output from that and other processors, there's no
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
YES! This has long been acknowledged by people who we usually assume know a little bit about the physical world. It seems reasonable to me, but demonstrating why it is reasonable is another thing.
Bitter and proud of it.
Ideal circles do not exist. Before human beings they didn't exist, and they still do not exist. We define them. Can you think of a perfect circle? If you can you must have perfect visual processing in your brain. This is a hard problem I admit, and I'm not going to pretend my answer is absolutely correct. However, mathematics proceeds from axioms, which are fundamental assumptions ... sometimes based on physical intuitions, but sometimes not.
I think mathematics is so effective because in the realm of physics our discoveries have few degrees of freedom and can therefore be represented by simple rules. Since the rules must be consistent we have the basis for physics and a tie in to mathematics.
Quotes shamelessly stolen from here.
Bitter and proud of it.
I am pretty sure that some of the problems at least will be Hilbert Problems that do not currently have a solution. http://en.wikipedia.org/wiki/Hilbert_problems
Did Glenn Beck rape and kill a girl in 1990? gb1990.com
This is a great idea. It whould promote more interest in the specific problems and unsolved math problems in general. Besides, more collaboration should result in better research.
Loban Amaan Rahman ==> Anagram of ==> Aha! An Abnormal Man!
I would argue the opposite: a problem is something which has a solution, something without a solution is not a problem but a circumstance.
Those things which we use mathematics to describe (relationships of every variety) are discovered (by observation and experience)
The language with which we describe them (symbols, axioms, and rules of transformation) is invented (and refined over time, as a quick review of the history of mathematics will promptly reveal)
Additional products of this language (logical consequences of the axioms we have invented) are subsequently discovered.
We equivocate the term "Mathematics" to mean all three of these things (that described, the language of description, and logical consequences of the axioms of that language). When the word means all three of these things at once, it seems that we have both discovered and invented it, and lively (though misguided) debate ensues.
When we establish clarity about our topic of discussion (through disambiguation of our terms), then whether it was invented or discovered becomes clear, as I have just demonstrated.
No, the bicycle is equivalent to a number base or a mathematical system. It is an implementation OF an underlying system (in this case, Newton's Laws), but Newton's Laws would still remain exactly the same whether Newton - or indeed bicycles - had ever existed.
The definition is also immaterial, as that too is an implementation detail. The underlying principle would remain unaltered whether the definitions of circumference, diameter or pi had ever been developed.
You are confusing the overlaid system with what it overlays. I'm saying you don't need to. Your argument is that the overlaid system is artificial, an invented product. I'm saying you're entirely correct on that. But what I am also saying is that what the product overlays, what is beneath the terms, the dynamics and the fancy Greek lettering is not artificial but exists whether it is known to exist or not.
The problem with assuming the two layers are the same is that you run into the Anthropomorphic Principle - the universe is the way it is because it produced people capable of seeing it. Let us, for a moment, assume the Many Worlds theory of Quantum Mechanics is correct. Then there are universes OTHER than the one we see and the theory falls down. The same would be true if the model of a multiverse as a foam (where each universe is a bubble in that foam) is correct.
But if you're on this site, you should be familiar with layering anyway. Maths - the fundamental, overarching thing that is shown in all mathematical systems that exist, will exist or ever have existed - is a Layer 1 concept in the OSI model. Concepts like numbers and other fundamental but artificial building blocks are Layer 2, which makes Group Theory a layer 2 switch. Anything and everything that MUST be true because of something in layer 2 is arguably also layer 2, which would include Goedel's Theorum. Anything that is true only in a specific implementation of mathematics is layer 3 or above.
Does using an OSI representation make it easier to see how not all maths is the same?
It's a small world and it smells funny; I'd buy another if it wasn't for the money; Take back what I paid (SoM)
First of all "as old as the wind" is just an expression means "really fricken old". It's obviously not meant to be taken literally, so get off your high horse.
Secondly,
If it is discovered, the solution already exists and the problem was solved before wind existed
Just because a solution exists, does not mean you have solved the problem. Think of it this way. You are looking through your telescope at night up at the stars and you notice a new star you have never seen before. You look at all the star-charts you can find and realize that no one has ever documented this star. You've just discovered it.
But you are saying that you did not discover the star, since the star already existed. Of course the solutions already exist for these math problems. However, discovery is the act of documenting an observation (ie, someone has to say "this is the answer"), so while they exist, no one has yet discovered them.
P = NP?
I so want it to be true. Quantum computing is our best hope right now of shedding light on this problem.
And it's not on their list...
I void warranties.
That wiki has a whiff of the tiger who said "It's well I'm named such, as I'm so fierce." Arithmetic, Geometry, Analysis, arose from careful observation of the universe. It's not really a mystery that they are well applied to the universe.
Play Command HQ online
Linkie. Socrates questions his victims, but only in a subjective way, that is, only about their opinions and perceptions. His view of the world below the veneer of Hellenic city-state culture was based on the cave of shadows.
Religion is what happens when nature strikes and groupthink goes wrong.
What is the difference between a mathematical concept representing or reflecting something physical and a mathematical concept that is used as a tool in a physical model?
Seems to me if you're using them in your physical theory then they have as much of a physical role (I'm not saying that they have to correspond to some observable quantity though) as say a symmetry group or any other mathematical object you use.
The only way it can be proven that mathematics is wholly artificial is to prove that the set of all mathematical "things" that are fundamental is equal to the empty set. ie: there is nothing - not a single property, not a single result - that is true everywhere, including Goedel's Theorum. If even something as simple as Goedel's Theorum is universal, then there exists at least one part of mathematics that is not invented but is wholly natural.
Since the only real constraint that Goedel's Theorem imposes is that there is not a finite set of axioms that can characterize all "sufficiently interesting" mathematics (i.e., that any finite axiomatization is necessarily incomplete) I don't see where you're going with that. It's a construct all the same and no amount of philosophical bullshit will change that.
Now, here we run into a problem. If Goedel's Theorum is not a universal result, but an artifice, then it is also false because it would have to be possible to create a counter-example and the theory states no counter-example of this kind can exist.
The problem is that you're into the space of self-referential mathematics when you're using Goedel's results (they're a necessary part of it, which is where the "sufficiently interesting" really comes from) so your argument just doesn't work too well. In particular, there most certainly is mathematics possible in systems that do not support the expression of Goedel's Theorem, but it's pretty dull stuff (no integers, for example). You can build on that base in various ways, but once you've got self-referentiality then you can get something equivalent to Goedel, and you're stuck. Or you can plunge on, by adding more rules and axioms which will either let you say more (but not everything) or render the whole edifice bollocks. But not one bit of this says anything about whether it is natural or not.
Surely that seals the argument right there and then. Those who argue mathematics is wholly artificial must be arguing Goedel's Theorum is false. All other cases do not prohibit the theorum from being true. Thus, if there is sound reason for believing the theorum true, there is sound reason for excluding the notion that mathematics is an artifice.
Your argument is full of bollocks. Goedel's Theorem is a consequence of a particular level of complexity of a formal system, and once it holds, you've got to a stage where you know there must be truths about the system that cannot be derived from a finite set of axioms about the system. But such a system can be pure artifice (e.g. by moving symbols around according to clearly stated rules, which is obviously non-real) and if one such system can be non-real, you've not proved that your favorite one ("mathematics") is real on that basis either.
Note that I'm not trying to demonstrate that mathematics is artificial. I'm just pointing out that you've not shown that it is not. (I actually suspect this is a matter for philosophers and not mathematicians since the observable effect is the same in either case.)
"Little does he know, but there is no 'I' in 'Idiot'!"
It seems to me that calling it a mystery implies you're not really sure why math can accurately model the natural world.
So, if you can explain it, it wouldn't be a mystery.
It seems that many scientists and mathematicians were confident that it could accurately model the natural world for one reason: it was "designed."
Basically, what I'm getting at is this: if your worldview includes a Creator/Designer/whatever of some sort (i.e., God) who created/designed and/or sustains the "natural" world, then it is actually not very surprising or mysterious that it is logical and consistent and able to be accurately modeled.
If your worldview tends toward thinking there was no design aside from what "design" evolution (by chance) "gives" it, then it would seem that accurately modeling the natural world would indeed by a mystery, as it ... does not entirely make sense. Laws do not really make "sense" unless there is an authoritative entity that put those laws in place... that entity could be God, a social construct, a human being, etc. But "laws of nature" don't really make sense if there is nothing actually consistent behind nature.
There's the long-winded version. :)