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Collaborative science and the central limit theorem

For an outsider, say a politician, science may seem like a pretty random activity. Not only the positive results are rare and come at a random rate, but the choice of topics appears to be random as well. Although this may look almost like a crime to those who consider themselves to be science creditors (i.e. to tax payers), there is a deep reason behind this behavior. Science is supposed to produce regularity and determinism from the chaos of facts and observations. Yet, to do so it has to tread in uncharted territories. And there is very little you can or should do about it, except sticking to a few well-known guiding principles and trying to be as honest as possible. Regularity, working-hours regime, orientation on practical problems have very little positive influence on the outcome.

The amount of uncertainty and randomness in the daily research activities while proving a statement, deriving a formula, or tuning an experimental setup is surprisingly huge. Hence, in principle, the desire for more order in the chaos of science on the side of science managers or those who observe it from a close distance is easy to understand. Yet, even the most innocent-looking and logically plausible changes to the natural scientific process may lead to strong undesired outcomes. Take, for instance, the idea of cooperative research, strongly promoted here in the Netherlands on the state level. Over the past ten years all research has been actively clustered in large centers, competing groups in and across universities were either eliminated or joined, and research of individuals outside the agreed mainstream directions has all but vanished. Officially, there are several lofty fellowships available to individual researchers with one of the conditions being the scientific originality of the proposal. Practically, however, only mainstream topics are eligible, due to the lack of qualified and generous referees in the case of unusual projects. This is a self-enforcing system and it must be the same everywhere. Truly original research has never been well-financed. This is not my point, however. What I am trying to examine here is the idea of a joint research effort as such. Not the naturally occurring collaborations of like-minded individuals, but the cooperation as a scientific method.

As I said above, randomness seems to be inherent to scientific research at the most elementary level, i.e. at the level of an individual researcher. The range of topics, knowledge and creativity available to a single scientist are all, unfortunately, finite. The output of a cooperative research effort may be viewed as a sum of random variables. If not the final result, then at least the necessary consensus reached at some stage. For example, at the stage of the problem statement. The behavior of such a “collective” is governed by the law of large numbers and the variants of the central limit theorem. Moreover, as the size of the collective grows, and everybody tries to maintain the independence of their scientific judgment, we should observe the phenomenon of the concentration of measure – with overwhelming probability the result of the collective effort will be concentrated around the mean value. That is despite the fact that the total sum of knowledge and the total range of ideas of all the participants are huge. While this is good news for managers who strive for stability, order, and predictability, it is certainly bad news for those who expect a technological breakthrough or a revolutionary theory.

Recently I have witnessed this effect in all of its glory. The cooperation in question was not even enforced by the authorities, but was a completely voluntary effort of mathematicians called the Polymath project. The initiative came from Timothy Gowers and has already resulted in a “probably solved” Hales-Jewett theorem. I did not see how that one worked, but was pretty curious when Gowers was ready to start a new polymath project. This time he suggested several possible topics. This choice reflected the range of his personal interests, which appears to be pretty broad. I think this is true for many scientists. The topics were:

  1. Littlewood’s conjecture and related problems
  2. Another project related to Hales-Jewett theorem
  3. Four Erdos-style combinatorial problems
  4. Non-boring chess-playing program
  5. The origin of life

I put it here exactly as he posted it. Let me guess which one has caught your eye… Is it number 3? Just kidding. The obvious outlier, statistically speaking, is the last one, where Gowers suggested to look for a mathematical model or a computer algorithm which with random initial conditions would produce an evolving system of growing complexity with overwhelming probability. That would be something! Yet, statistics works against Gowers. With overwhelming probability the Polymath community converged to … yes, topic number 3.

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