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Jiro comments on No Universally Compelling Arguments in Math or Science - Less Wrong
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It requires a population that's capable cumulatively, it doesn't require that each member of the population be capable.
It's like arguing a command economy versus a free economy and saying that if the dictator in the command economy doesn't know how to run an economy, how can each consumer in a free economy know how to run the economy? They don't, individually, but as a group, the economy they produce is better than the one with the dictatorship.
Democracy has nothing to do with capable populations. It definitely has nothing to do with the median voter being smarter than the average politician. It's just about giving the population some degree of threat to hold over politicians.
"Smarter" and "capable" aren't the same thing. Especially if "more capable" is interpreted to be about practicalities: what we mean by "more capable" of doing X is that the population, given a chance is more likely to do X than politicians are. There are several cases where the population is more capable in this sense. For instance, the population is more capable of coming up with decisions that don't preferentially benefit politicians.
Furthermore, the median voter being smarter and the voters being cumulatively smarter aren't the same thing either. It may be that an average individual voter is stupider than an average individual politician, but when accumulating votes the errors cancel out in such a manner that the voters cumulatively come up with decisions that are as good as the decisions that a smarter person would make.
I'm increasingly of the opinion that the "real" point of democracy is something entirely aside from the rhetoric used to support it ... but you of all people should know that averaging the estimates of how many beans are in the jar does better than any individual guess.
Systems with humans as components can, under the right conditions, do better than those humans could do alone; several insultingly trivial examples spring to mind as soon as it's phrased that way.
Is democracy such a system? Eh.