Blogging, as you might have heard, is changing the face of the media. It may also be changing the face of mathematical research. For the first time ever, a substantial mathematical problem has been solved via an accumulation of blog comments, all building on each other. Could this be the future of mathematical research?
Before I explain the problem, let’s talk a little about tic-tac-toe. As you probably figured out long ago, intelligent players of ordinary tic-tac-toe (on a 3 by 3 board) will invariably battle to a draw. But, as you probably also figured out, not every game ends in a draw, because not every player is intelligent.
On the other hand, if we blacken out the three squares on the main diagonal and don’t allow anyone to play there (so the game ends when the remaining six squares are filled, then every game is sure to end in a draw. There’s simply no way to get three in a row when you’re not allowed to play on the diagonal:
Okay. Now let’s play tic-tac-toe in three dimensions, with three ordinary boards stacked on top of each other (giving you a total of 27 places to place your X.) How many of those 27 squares would I have to blacken to insure that winning is flat-out impossible? The answer, it turns out, is 9—as long as you choose the right 9. And if we go to 4 dimensions? Now there are 81 squares, and if you want to prevent any possibility of winning, you’re going to have to blacken at least 29 of them. In 5 dimensions, you’ve got to blacken at least 93.
(See yesterday’s post if you’re puzzled about how to play tic-tac-toe in four dimensions).
The density Hales Jewett Theorem says that as you go to higher and higher dimensions, the number of squares you must black out to prevent a win gets arbitrarily close to 100% of the squares available. In some high enough dimension, you’ll have to black out at least 90% of the squares; in some higher dimension, you’ll have to black out 95%, and then 99% and 99.9999%. If you’re not sure why anybody would care about such a thing, take my word for it—there are many applications to other areas of mathematics.
(My statement of the theorem glosses over some minor technicalities; the actual theorem is slightly stronger than what I’ve quoted here.)
Now until a few months ago, the only known proof of the density Hales Jewett theorem was extremely difficult. But Tim Gowers, a Fields-Medal winning Cambridge mathematician, thought there ought to be an easier proof. So he did what everyone with an opinion about anything does nowadays; he posted his opinion on his blog. He also did what no mathematician had ever done before, and invited the entire world to collaborate with him in proving his opinion correct. Following an initial post asking “Is Massively Collaborative Mathematics Possible?”, he posted a description of the problem and invited his readers to have at it in comments.
Commenters leapt in. In response to a couple of dozen blog posts by Gowers and others, roughly spanning the calendar year 2009, commenters continued to build on each others’ ideas until they produced the (relatively) simple proof Gowers had been hoping for. Along the way, they accomplished a lot more—for example, we now know that in 5 and 6 dimensions, you’ve got to black out exactly 93 and 279 squares (again, glossing over some minor technicalities); these numbers were not known before the blogging project. For any single mathematician—or team of mathematicians—this would have been a singular accomplishment. It’s not clear it would ever have happened in a world without blogs.
Gowers believes this could be the beginning of a whole new way of doing mathematics, allowing hundreds or thousands of mathematicians to contribute to the solution of a single problem. Of course this raises all sorts of questions about rewards and incentives, many of which are addressed (but not, or course, settled) in Gowers’s “Massive Collaboration” post. Still, I have an inkling that this is a big freaking deal.
Isn’t this just a speeded up version of what normally happens? Problems used to disseminate through letters from Mersenne, and then through printed journals, and now http.
Its new connectivity in the human supermind. Whether it will be productive, or create more noise is to be seen. With the old printed journals, there was a distilling and selecting process that is not always present in the blogosphere.
It has always seemed strange to me that mathematical theorems are named after the thinkers who “discovered” them when (1) they are often the product of several generations worth of collaboration, and (2) they transcend human thought — indeed, the universe, as Landsburg postulates — altogether. Associating a name with a concept seems like a trivial exercise.
Why not assign these theorems random numbers, if not something more descriptive? You might argue it would take the incentive out of innovation, stripping geniuses of their name recognition and page immortality. But having virtually no market power, mathematicians have no better alternative than to spill the beans for whatever compensation they can get. It might dissuade some purely profit-driven folks, though. Where would we be without Stern Stewart’s Economic Value Added®?
Nick – if mathematician’s currently get some benefit from having theorems named after them, it seems like your proposal is just pure loss – mathematician’s don’t get the pleasure of eponymous theorems and, as far as I can tell, no-one at all is any better off.
Presumably you would in fact have to compensate mathematicians by raising their salaries if you wanted them to continue to do research, so it might not be the mathematicians who are worse off, but someone must have to pay the consequences, right?
John Faben: Good to see a reader who thinks like an economist :)
Whatever I may think of his proposal, I’ll give Nick consistency points for his choice of a nick.
The informational efficiency of language can hardly be discounted as a potential benefit. Sparing millions of children from having to learn and pronounce idiosyncratic names like Pythagoras, l’Hôpital, and Chebyshev (among the mildest transgressors), as unconscionable as this may seem, could remove barriers to entry and stimulate future interest in the field.
It is hypothesized, for instance, that idiosyncratic words such as “eleven” and “twelve” as opposed to decade consistent pronunciations “ten-one” and “ten-two” (as is the case in several East Asian languages) may contribute to the lackluster mathematical performance of the average American student.
Likewise, eponymous theorems serve to obfuscate rather than to reveal, synthesize, or catalogue. Naturally, such an overhaul would not be costless. Random character strings are another matter altogether — one I’ll save for Randall Munroe.
Congratulations on the beginning of Science 2.0, and not a minute too soon. Science 1.0 was a big success with the great help of the printing press which stole the Church’s gatekeeper function away from large scale communications. Science 2.0 will be an even bigger success thanks to a new printing press on steriods, squared: the Internet. Church 2.0, The Secular Church of the Public Sector, is goin’ down.
We’re already seeing the initial application of Science 2.0 with regard to climate. I expect the internet will replace traditional journals more quickly than many imagine.
Consider the political implications of the successes of distributed computing and crowdsourcing. Somewhere, Friedrich Hayek is smiling.
I agree with Ken: the innertubes merely allow the process to be sped up. But I’m not sure I’m as hopeful as Neil… what’s the old saying? “Never underestimate the power of stupid people in large numbers?”
And who’s checking the arithmetic, anyway?
It’s an extension of brain storming. And you have to look at it as, “of course it will work.” If you have players who both contribute and play by the rules.
Cheap technology, the open source philosophy, and social networks (from irc to facebook) are bringing hard science research into the basements of amateur hobbyists. Math will succumb as well, with or without the “professionals”.
i.e. http://tidbit77.blogspot.com/2010/02/fusion-reactors-first-light.html
Bzzzt.
Actually, the etymology of “eleven” is that it comes from *ainlif,
“one left over (after ten)”. Similarly, twelve is from *twalif, “two left over (after ten)”.
So it’s not a failure of language, but rather a failure of English and math teachers to teach etymology and historical linguistics. Learning etymology would also vastly improve spelling and help language skills in general, but noooooo. We have to wait till college to learn the answers to basic questions which occur to every young speaker of English.
“You can achieve anything you want in the world as long as you don’t mind who takes the credit”
— Many attributions, including Harry Truman, Ronald Reagan
The experiment itself proves that mathematical minds (if not employed, tenured mathematicians; but some of the participants must have been such, they weren’t all geek geniuses in their garage apartments in their pajamas) can’t resist solving a problem for the intrinsic reward of doing so.
“Now let’s play tic-tac-toe in three dimensions, with three ordinary boards stacked on top of each other (giving you a total of 27 places to place your X.) How many of those 27 squares would I have to blacken to insure that winning is flat-out impossible? The answer, it turns out, is 9—as long as you choose the right 9.”
I give up – what are the right 9 squares to block any win?
Seems to me that you need 11. First, every column must have at least one blocked square somewhere in it, giving a minimum of 9 blocks. Next, each horizontal board must be blocked as with the 2D board – and a 2D board can only be fully blocked using 3 blocks if the center square is used. Since the center column will have 3 blocks, the 3D version would require 11 blocked squares – 3 in the center column, and 1 in each of the other columns (for instance, having opposite diagonal lines on levels 1 and 3, and a 5-block cross shape on level 2). With 2 players, one could win with less than 11 blocked squares if the other player just played on squares not in the winning path.
But please correct me if I’ve misinterpreted the problem.
WG: I was careless, and you are absolutely right.
The discrepancy is related to this parenthetical statement that I slipped into the middle of the post:
(My statement of the theorem glosses over some minor technicalities; the actual theorem is slightly stronger than what I’ve quoted here.)
The minor technicality is that for density-Hales-Jewett-theorem purposes, not quite every line counts as a line. To get a “Hales Jewett line” you must first pick one corner of the cube, and then only count lines that are translations of lines passing through that corner. Therefore on a 3 x 3 board, all the verticals and horizontals count (because they are carbon copies of the vertical and the horizontal that pass through the special corner) but only one of the two diagonals counts (because the other diagonal is not a mere translation of the main diagonal—it’s a translation together with a rotation). So in this slightly modified tic-tac-toe, getting all three squares along the off-diagonal does not count as a win.
Nick: But how else would we remember that Nicolai Ivanovich Lobachevsky is his name, Hi!
jms: Surely not the jms?
isn’t this just converging on a solution in possibility space via a type of genetic algorithm?
I haven’t read the blog detailing the process of solving this problem but it seems to me that each post could be considered an iteration i.e. applying a fitness function, formulation a new offspring via a cross-over/mutation of earlier posted solutions and then birthing an offspring via a new post?
no?
Indeed, the disutility caused by the risk of plagiarism, misattribution, and otherwise arbitrary naming conventions imposed on forgotten collaborators provides another case against attaching names to concepts. Who likes to see one’s rival take credit for a stolen idea, particularly if that idea was the product of one’s own life’s work? The only defense against this outcome is keeping research close to the chest, which is the antithesis of collaboration.
Opensource mathematics! What an idea.
Present the problem and have members collaboratively
work on the project.
Nick,
As someone with a PhD in mathematics, I find it much easier to remember the Pythagorean Theorm, l’Hôpital’s Rule, and Chebyshev polynominals than geometry theorem #12, calculus rule #42, and polynomial class #536. Additionally, eponymous naming provides hooks to students interested in learning more about how all of these concepts arose in history – and anything that discourages curious students is an unacceptable cost in my book. Also, I’m don’t think that the informational efficiency of language is improved by removing the hooks that help place these theorems into their proper context. I proved a (minor) theorem for my dissertation and if I ever get around to publishing it, it’s at least going to be called ‘Kesseler’s Lemma’!
How are the theorems named? Is it still Fermat’s last theorem, or is it Wiles’ theorem now?
Neil: Your example illustrates the impossibility of attributing credit accurately by appending names. If I were going to rename Fermat’s Last Theorem, I’d call it the Frey-Ribet Theorem, not WIles’s. “Wiles’s Theorem” should refer to (what was) an important special case of the Taniyama-Shimura Conjecture (or the Taniyama-Shimura-Weil Conjecture?). But of course this no longer needs a name, since it is subsumed by the Breuil-Conrad-Diamond-Taylor Theorem.
“If I were going to rename Fermat’s Last Theorem”
The Landsburg Renaming.
Which inevitably leads one to think of Robert Ludlum and eg “The Dunsinane Reforestation.”