In my mail: *The Best Writing On Mathematics 2011* (Mircea Pitici, ed.) from Princeton University Press. Just to get this out of the way: I’m in here! They reprinted my compressed sensing article from Wired.

You might now be wondering: are there really enough popular math articles published in a given calendar year to fill up an anthology? No. There are not. But this is part of the charm of what Pitici has done. His very broad definition of “writing on mathematics” allows him to include useful professional advice for young mathematicians from Andrew Schultz, reflections on a career in math education from John Mason, and academic-yet-readable philosophy (“What Makes Math Math?”) from Ian Hacking, whose *The Emergence of Probability* is my favorite book in history of mathematics.

I especially like Mel Nathanson’s pessimistic take on massive collaboration in mathematics — because it is a forcefully written, carefully argued case for a position with which I mostly disagree. “I would guess that even in the already interactive twentieth century,” he writes, “most of the new ideas in mathematics originated in papers written by a single author.” I would guess otherwise — at least if you restrict to the second half of the century, when joint papers started to become really common. Mel calls me out for writing about Tim Gowers’ Polymath Project in the New York Times with “journalistic hyperbole” — and here he is right! It is very hard, in the genre of 300-word this-year-in-science snippet, to keep the “gee whiz!” knob turned down and the “jury is still out” knob turned up.

Gowers claims the classification of finite simple groups as a pre-Internet example of massively collaborative mathematics. Nathanson agrees, but characterizes the classification as fundamentally uninteresting, “more engineering than art.” What would he say, I wonder, about recent progress towards modularity of Galois representations? It’s very hard to imagine him, or anyone, seeing everything that’s happened in the last 15 years as a mere footnote to Wiles. (But maybe some of the experts who read this blog would like to weigh in.)

Nathanson concludes:

Recalling Mark Kac’s famous division of mathematical geniuses into two classes, ordinary geniuses and magicians, one can imagine that massive collaboration will produce ordinary work and, possibly, in the future, even work of ordinary genius, but not magic. Work of ordinary genius is not a minor accomplishment, but magic is better.

Yes, but: magic can only happen in the already-enchanted environment created by the hard work of many minds, alone and in teams. Math is like earthball.

I wouldn’t go as far as to say “Yes but.” There is absolutely no evidence that the work of many minds can’t be magic, or even any a priori reason to think so. (I say this from the point of view that mathematical “magic” is in fact not magic when looked at in enough detail.)

To take the other extreme, if the work of many minds

canreliably produce magic, then we’d be all out of jobs when the singularity arrives.Evidence for whom? I guess Nathanson finds there is evidence, and some a priori reason, to think so.

Trying to find advantages of single-manned genius here is what comes first to my mind:

– Benedictine monks avoid idle talk, rescue efforts managers avoid giving optimistic estimates, surgeons give short-term goals (and lie) when performing painful operations. It is hard to resist temptation, we need very strong barriers. Another aspect of this is that stating one’s goals (of certain kinds -not all kinds) and partial results makes us less likely to achieve them.

Research depends heavily on social rewards/acknowledgement and it seems natural to expect collaboration to affect it.

(http://www.ted.com/talks/derek_sivers_keep_your_goals_to_yourself.html

http://danariely.com/)

– By definition, a crank thinks isolatedly from society’s masses. Magical genius may perhaps keep that kind of “trivially single person” property, we would not call it magical if people arrive at it progressively, in collaboration. Then the question is whether we may arrive collaboratively at the same discoveries, and with less effort. If so, we should review the positive connotations we associate with “magical”.

– There is a technical aspect of this question: information transmission and parallel computation. I don’t feel I have satisfactory insights about this, but I expect it would be relevant in justifying opinions more deeply.

– Returning to the question of motivation, it seems that much of (mathematical) research relies on exploring possibilities to find rewards consistently. There is an interplay between attempting far-reaching goals, pipe-dreaming, and making explorations or calculations that we know work. Collaboration may hinder the process of trying new things for extended periods of time, it seems to me much more probable to decrease the time (energy) an individual can keep trying than to increase it. For instance interacting with others requires a large amount of energy, attention, which prevents fuller investment into exploration (e.g. forgetting the norms); it forces us to use known rewards, like coming up with “well-known” facts quickly.

Thank you Jordan for the interesting post.

Oops, I should have said “single-(wo)manned”, or something else.

That was the “Yes but” of “Even were I to concede that point…,” not the “Yes but” of “You are correct, but should also consider that…”