What does Polymath tell us about problem solving?

Gowers and Nielsen have written a nice opinion piece (Nature 461, 879-881, 15 October 2009) on The Polymath Project, an open-source and collaborative attempt at solving an unsolved math problem – to find a new proof of a result in ergodic theory called the density Hales-Jewett theorem using only ‘elementary’ building blocks. The protocol for the collaboration was that each participant (who could be anyone – from beginning student to Fields medalist – with an interest in the topic, from anywhere in the world) could post one nugget of an idea at a time, to the weblog. There was no requirement regarding the individual contributions. Indeed, it seems like some of the posts were just comments clarifying small points. In this first trial, it took a group of 27 people 37 days and approx. 800 serious comments to get the desired result.

The authors ask, who would have guessed that the working record of a mathematical project would read like a thriller? But, of course, both of them  know very well (as does every serious scientist) that this is exactly the nature of research – this is why we do what we do.

To me, there were two important messages to take away:

(a) The protocol begins to demystify the process of creativity. I have always been turned off by the macho posturing by researchers who deny that big ideas are, in the end, just clever compositions of carefully chosen smaller ones. Instead, this project strongly suggests that the collaborative effect of multiple incomplete but properly diverse viewpoints is what it takes, much like Minsky’s Society of Mind.

(b) As the article notes,

“Although DHJ Polymath was large compared with most mathematical collaborations, it fell short of being the mass collaboration initially envisaged. Those involved agreed that scaling up much further would require changes to the process. A significant barrier to entry was the linear narrative style of the blog. This made it difficult for late entrants to identify problems to which their talents could be applied.”

With my AI researcher hat on (i.e., as someone who looks at this project as inspiration for the design of corresponding ‘intelligent’ computational systems), I find that this is the exactly the challenge that an autonomous agent must come to terms with when trying to learn useful skills in a lifelong sense. It is not that hard to devise learning procedures that can do the equivalent of making nuggets of suggestions. The harder problem is to learn context and measures of appropriateness for the individual components. However, if we begin to understand how people really do this it shouldn’t be impossible to get machines to follow (although, as someone used to ask me in response to such bold statements, ‘…famous last words?!’).

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2 thoughts on “What does Polymath tell us about problem solving?

  1. More interestingly: how might a machine determine what is interesting by itself? In the above example, a known end to end was carried out and show to be possible.

    Maybe the kind of “mathematics” that would arise if devised by machines does not have much to do with the way we see math anyway… In some sense it goes to the heart of what is the meaning of math and physics? Say physics is only caring about compact descriptions of natural phenomena. But today, compact descriptions can be purely data based, since most of want we want to do is to predict or control.

    But of course, some of physical laws gives nature to insights: for ex, the huge repulsion among close molecules can be used for self assembly… Judging and representing this has been a “human” task. And, at times we are great at it, at times, we put our heads in the sand and rest on the safety of belonging to some pet problem group. Might this change? Or put in another way, would machines really care to fly?

    • My professional curiosity is precisely regarding your last paragraph. What is the human doing when (s)he gets that ‘insight’ connecting molecular forces and self-assembly, and is there a nice computational account of that? It is fine if the resulting program is only occasionally successful – understanding that process still seems like a scientific question that is worthy in and of itself! I think that if people can get a desire to fly then there ‘must be’ some version of an intelligent machine that also (just like real people, perhaps only sometimes and depending on a variety of external influences) gets a similar desire.

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