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."

Comments

Math cannot exist before wind.

[blackraven14250]

How can math problems exist before people start using mathematics? Last I checked, math was nothing more than a representation of the hypothetical that as closely models our universe as possible.

Re:Math cannot exist before wind.

[John Hasler]

Some say math is discovered. Others say it is invented. You are one of the latter.

Re:Math cannot exist before wind.

[invisiblerhino]

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.

Re:Math cannot exist before wind.

[jd]

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.)

Re:Math cannot exist before wind.

[Obfuscant]

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.

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.

Re:Math cannot exist before wind.

[Evil Pete]

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.

Re:Math cannot exist before wind.

[Evil Pete]

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.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? — Albert Einstein

There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology. — Alexandre Borovik

Quotes shamelessly stolen from here.

Re:Math cannot exist before wind.

[Anonymous Coward]

Axioms are invented. Theorems are discovered.

Re:Math cannot exist before wind.

[Boronx]

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.

Re:Math cannot exist before wind.

[anarchyboy]

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.

Check out the Collatz Conjecture...

[Oxford_Comma_Lover]

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.

Re:Check out the Collatz Conjecture...

[ACS Solver]

I have fond memories of that one. On the subject of teaching and education...

One of my math teachers once showed me the problem. The teacher knew I'm decent at math and would occasionally show me interesting or unusual problems. The interesting part is, the teacher told me to have a try at proving the proposition of this problem, without telling me that it's an unsolved problem. So I had a good amount of fun trying to prove this. Of course, it's not like I could make a proof with my high school knowledge, but it challenged my mind and was a fun thing to do. And had the teacher told me right away that it's an unsolved problem, I wouldn't have had the motivation to think about it, knowing beforehand that I wouldn't be able to find a proof.

That was one of my educational highlights, though. Way to provide a mental challenge!

I'm still amazed by how part of the problem's beauty is that it's easy to understand the actual proposition. That isn't true for most unsolved problems, after all. Take the recently proven Poincaré conjecture, just understanding what it states takes some math knowledge, though it has a nice approximation in layman's terms. As for the example of the Hodge conjecture, I probably don't know half the mathematical concepts required to understand the problem.

Re:Check out the Collatz Conjecture...

[Shikaku]

http://pastebin.com/m67281bd6 :D I have nothing better to do. Optimized to save known values and not recalculate them.

Strange point

[2.7182]

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.

Re:Strange point

[Anonymous Coward]

Nobody bothers to check it because it's published on Arxiv. In the math community, Arxiv is basically a very detailed blog. Say anything you want, and some people will read it and maybe be interested. But nobody will really take you seriously. After all, there's tons of flat-out wrong papers on Arxiv and no form of quality control whatsoever.

There are many, many peer-reviewed journals. If this paper is good, it should be published in one of those. The fact that it's not raises doubts about its quality.

Massively collaborative "Polymath" efforts

[Anonymous Coward]

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.

Encyclopedia of Integer Sequences

[Rockoon]

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.

Re:I have this proof.

[Anonymous Coward]

I proved that there are infinite primes in a twitter comment for a friend doing her homework:

Assume some set of primes is all of them. Multiply them all. Add 1. None of primes go into new #; its factors are additional primes

Hilbert problems

[aws4y]

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

Disambiguation reveals the simple answer

[Brain-Fu]

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.

Here's mine...

[rayharris]

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...