Massive Open Collaboration In Math Declared a Success

nanopolitan writes "In late January, Tim Gowers, a Fields Medal winner at Cambridge University, used his blog for an experiment in massive online collaboration for solving a significant problem in math — combinatorial proof of the density Hales-Jewett theorem. Some six weeks (and nearly 1000 comments) later, Gowers has declared the project a success, and some of the ideas have already been written up as a preprint."

Comments

Too bad he used a blog

[bendytendril]

I read TFA (I didn't understand it, but at least I read it). The entire discussion was a typical linear blog where they had keep numbering items and referring back to them (i.e. "regarding item #32 I think....")

What he really needed is a threaded message board ala Slashdot.

Re:Too bad he used a blog

[morgan_greywolf]

How about Subversion or GIT?

Massive open collaboration in math

[morgan_greywolf]

I wonder if you could do massive open collaboration for software? You could probably write an OS kernel, maybe even an entire operating system!

List of authors

[pimpimpim]

I am all in favor of this, as it allows people outside scientific communities to join in with a low barrier (that is, if you happen to be a math wizard). But is, and if so, how is he going to ensure that the right people will be mentioned as co-authors in the paper?

Re:List of authors

[Sheafification]

Gowers mentioned this in one of his early posts about the project. Publishing will probably be under some sort of pseudonym (think Bourbaki), with a link given to the blog entries. If you're curious about how much someone contributed, you can go check it out exactly.

Prior Art

[D Ninja]

I totally tried "Massive Open Collaboration" on my homework and tests in high school. I most definitely came up with this idea first.

And, no, I still don't understand basic algebra? Why do you ask?

Surprise result

[Hwatzu]

It turns out that 2 + 2 actually = 5.
I know; I'm surprised, too. Well, I'm off to patch my calculator.

Why now?

[Anonymous Coward]

It would have been nice, had this been posted before they declared success.

Now all we have is a blog post with a gazillion comments, and all the interesting work has already been done.

Maybe next time we can all join the fun?

Re:Why now?

[Anonymous Coward]

Well, given that as far as I can tell everyone involved was a research mathematician, that is, a professor of mathematics, and that it included a few Fields medalists, that is the equivalent of Nobel Prize winners in maths, I would say that for this problem slashdot noise would have been highly counterproductive.

This is something for the cathedral, not the bazaar. That said he did mention intending to try to make the next problem more accessible.

It's about n-dimensional tic-tac-toe.

[bcrowell]

What you won't be able to figure out from the slashdot summary, or from either of the links (unless you're a specialist) is that this is a theorem about n-dimensional tic-tac-toe. The idea is that you make an n×n×n×...×n in some number of dimensions, and then you fill in some fraction of the boxes with x's and o's (or possibly some set of more than 2 symbols). The theorem says that if the dimension is high enough, and the fraction of the boxes that get filled in is high enough, you're guaranteed to have a line of symbols (possibly diagonal) that wins the tic-tac-toe game. In other words, a tic-tac-toe game in a high-dimensional space can't end in a draw, and can't even go on for very long before someone wins. The definition of "high enough" is what they're trying to pin down. They apparently proved it (or just made some progress toward proving it?) in a particular special case.

Re:It's about n-dimensional tic-tac-toe.

[kramulous]

If that is the interpretation, then where is the crossover point between lower dimensional and higher dimensional space where the draws stop?

Re:It's about n-dimensional tic-tac-toe.

[psnyder]

Now they can move onto more important things, like n-dimensional chess.

I postulate that with enough dimensions, my opponent's king will be in checkmate before I even make a move. If said number of dimensions are found to be within the confines of string theory, I would not owe my friend 20 bucks nor the sexual favors agreed upon in the rematch. Finally! A useful implication of string theory.

Re:It's about n-dimensional tic-tac-toe.

[snorb]

That's the Hales-Jewett theorem. The density Hales-Jewett(3) is a little different. Suppose you're putting X's on a 3x3x...x3 (n-dimensional) grid. Such a grid has 3^n points. What the theorem says is that if you've put X's on at least, say, 1% of the 3^n points, then you must have made a line somewhere if n is large enough. You can replace the 1% with whatever fraction you please, but that will change how large n has to be. No, the theorem doesn't state exactly how large n has to be, but already it's a challenging problem.

Re:halp!

[Anonymous Coward]

A good way to think about it is a very abstract and sophisticated version of the "Pigeonhole principle" :

Lets say you have k + 1 balls and only k bins. If you want to put all the bills into those bins then you are going to have to put at least 2 balls into one of the bin.

Ramsey theory deals with general problems of this type where if you have too much of one thing (balls) then something else is bound to happen (two balls forced to share a bin).

e.g if you have at least six friends at a party then they are bound to be a subset of 3 friends who either all know each other or all don't know each other.

The idea is that once you get a certain density or to a certain quantity of something, some other structure is bound to appear.

This Density Hales-Jewett theorem specifically deals with playing tic-tac-toe in high dimensional cubes

i.e if you have a D dimensional hyper cube or whatever and the dimension D is sufficiently large then there is guaranteed to be a win for one player (unlike the regular version which can end in a draw).

Disclaimer: IANAM

Massive?

[Anonymous Coward]

How was it massive? In the author's own words:

"There seemed to be such a lot of interest in the whole idea that I thought that there would be dozens of contributors, but instead the number settled down to a handful, all of whom I knew personally."

So a guy had a problem to solve, and batted some ideas back-and-forth amongst a few of his mates. Why is this newsworthy?

Re:halp!

[snorb]

Terry tao's blog has an explanation that's about as simple as you're going to find. (At least one that actually explains the math without handwaving).

http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/

Re:halp!

[Anonymous Coward]

Your first paragraph is wrong (of course they're trying to prove a theorem, one that asserts that parameters exist for which the result they want is true), the method isn't "simple enough" unless you're a Ph.D.-level expert in combinatorics, and the rest of your post is absolute nonsense. Linux doesn't have the same rigid structure as the combinatorial objects being studied, and even if it did the constants involved in this theorem would very quickly get much bigger than the number of modules or even lines of code in the Linux kernel.

Exactly what is an "arc through a module" supposed to mean, anyway, and how does knowing the existence of a "buggy arc" give you even the slightest idea of where it is?