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16 August 2006
While I'm sure that many readers of this blog saw this article in the August 15 NYT science section, it's worth noting anyway as an insight into the sociology of mathematics and a look at some interesting pure mathematics as well (plus the graphics therein were really cool).
It seems that Perelman solved a 102 year old mathematical question of huge importance but wants to have nothing to do with the affect on the field and the resulting acclaim (including an almost certain Fields Medal) since he's disappeared into the Russian forest. Nonetheless, other mathematicians have taken on the task of writing up his results producing proofs in three books that are now available online. These are fascinating to read, even though much of the discussion is at the highest mathematical level, since some of the principles are familiar to us (Cauchy-Schwartz, gradients, Hessians, etc.) from routine work, but obviously appearing in wildly different contexts.
So here's the related question. Suppose, like mathematics, we could list the "big" unsolved problems in political science. What would this list look like? Personally, I'd love to see such a thing. Of course it is unclear whether we have a David Hilbert-like figure to say "As long as a branch of science offers an abundance of problems, so is it alive" and then to go on and identify the 23 most important unsolved problems (the 1900 "Hilbert Challenge":). In this vein, my list of unsolved problems would start with why does the discipline cling to the bankrupt NHST and continue to worship "stars"?
Posted by Jeff Gill at August 16, 2006 10:34 PM
According to widely circulated rumors, the Fields Medal this year will be awarded to Terence Tao (australian mathematician of chinese descent, who started hight school at the age of 8, got his Ph.D. from Princeton when he was 21, and became a full professor at UCLA at the age of 24. He is famous for proving that prime numbers can form arbitrarily long arithmetic progressions) and Grisha Perelman (a weird russian guy with long fingernails who proved the notorious Poincaré conjecture. It is rumored that he will not make the appearance at the Congress and will decline the prize, although I doubt it). In any case, we just have to wait till Tuesday and see for ourselves.
Posted by: Philip Sidney at August 17, 2006 7:17 PM
jeff, this is a great post. what are the big polisci questions? many would be causal questions. perhaps the causal questions of greatest interest probably would relate to the causes of effects (as opposed to estimating the effects of causes) and we don't know how to credibly answer those questions. actually, we don't even know how to coherently ask questions about causes. i'm hoping your post prompts people to propose questions of critical importance that can't be appropriately addressed with available statistical methods. Here's one I'm curious about: What are the effects of democracy + capitalism on all the outcomes we typically care about (employment, GDP, life expectancy, education, crime, investment, etc.)?
The thought experiment is to imagine that the regime in North Korea, or Burma, or Belarus, were to crumble and a democratic/capitalistic transformation were to occur. what would we see? nobody really believes that cross-country OLS or tall-tale IV estimates can produce credible answers to these questions. we need something new.
by the way, i was looking at prior blog postings about gary's redistricting work, and it prompted me to think that this is the *type* of question that would make it onto the Hilbert-polisci question list. the "what is a fair standard for evaluating partisan gerrymandering and how should it be operationalized?" question is particularly cool because: (1) it has important normative aspects and real-world relevance across countries and across time; (2) it's NOT posed as a causal question; (3) and yet we see the merits of a problem-solving framework that is both formal and quantitative.
Posted by: alexis Diamond at August 18, 2006 12:14 PM
That's what we would do in sociology, though, if people listed problems. They would be problems about the infrastructure of how sociology thought about or studied problems, and not about problems in the social world itself. Imagine if the Hilbert Challenge had been a list of twenty-three problems with how mathematicians went about proving things.
Posted by: Jeremy Freese at August 19, 2006 2:23 PM
Well, as a person possessing BA in political science, I regard your idea as extremely interesting, but...well, you know that social sciences are quite different. At this point I'm not sure it is all possible to make such lists. Polisci deals with very dynamic patterns and problems, though they do have some perpetual roots, so to say. If you think of any problem that can be put in the list of such kind, do let me know!maybe I'll even join in composing such a list.
Posted by: essay lover at August 22, 2006 7:38 AM