Thursday, April 17, 2014

Gauss, the fox who decided to be a hedgehog

As mentioned before, I'm not entirely comfortable with the fox/hedgehog spectrum -- this isn't a concept that reduces readily to a scalar -- but as long as we're here...

One of the minor revelations of the recent discussion of the new 538 was that Andrew Gelman had posted on the subject of foxes and hedgehogs way back in 2005:
This got me thinking about statisicians. I think we’re almost all foxes! The leading stasticians over the years all seem to have worked on lots of problems. Even when they have, hedghehog-like, developed systematic ideas over the years, these have been developed in a series of applications. It seems to be part of the modern ethos of statistics, that the expected path to discovery is through the dirt of applications.

I wonder if the profusion of foxes is related to statistics’s position, compared to, say, physics, as a less “mature” science. In physics and mathematics, important problems can be easy to formulate but (a) extremely difficult to solve and (b) difficult to understand the current research on the problem. It takes a hedgehog-like focus just to get close enough to the research frontier that you can consider trying to solve open problems. In contrast, in statistics, very little background is needed, not just to formulate open problems but also to acquire many of the tools needed to study them. I’m thinking here of problems such as how to include large numbers of interactions in a model. Much of the progress made by statisticians and computer scientists on this problem has been made in the context of particular applications.

Going through some great names of the past:

Laplace: possibly hedgehog-like in developing probability theory but I think of him as foxlike in working on various social-statistics applications such as surveys, that gave him the motivation needed to develop practical Bayesian methods.

Gauss: least-squares is a great achievement, but developed as a particular mathematical tool to solve some measurement error problems. In the context of his career, his statistical work is foxlike.

Galton: could be called a “hedgehog” for his obsession with regression, but I think of him as a fox with all his little examples.

Fisher: fox. Developed methods as needed. Developed theory as appropriate (or often inappropriate).

Pearson: the family of distributions smells like a hedgehog, but what’s left of it, includng chi-squared tests, looks like fox tracks.

Neyman: perhaps wanted to be a hedgehog but ultimately a fox, in that he made contributions to different problems of estimation and testing. I’d say the same of Wald and the other mid-century theorists: they might have wanted to be hedgehogs but there was no “theory of relativity” out there for them to discover, so they seem like foxes to me.
I can't really argue with as framed here. Gelman seems to be using a slightly different definition than the standard know-about-many-things/know-about-few-things, but the idea of foxes coming up with advances to respond to different applications in different fields. Still, there is a certain irony in describing Gauss as one who was "interested in everything, and move[d] easily from one problem to another." In terms of ability, this is undeniably true, but in another sense you could argue that few have ever sacrificed more to specialize.

Perhaps the ultimate fox was Leibniz who, in addition to that whole calculus thing...
made major contributions to physics and technology, and anticipated notions that surfaced much later in philosophy, probability theory, biology, medicine, geology, psychology, linguistics, and computer science. He wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts. He wrote in several languages, but primarily in Latin, French, and German. There is no complete gathering of the writings of Leibniz.
Gauss very probably could have given him a run for his money because (and this is the amazing part) he was roughly as gifted with language as he was with math. He appears to have been conversant in well over a dozen. Gauss was almost twenty (and, among other things, already had least squares under his belt) before he finally decided he should leave philology as a hobby and focus on the sciences. I can't find a reference for this but I seem to recall that he explicitly said he didn't want to repeat Leibniz's mistake and allow himself to be distracted by tackling too many subjects.

The standard response here is to chuckle and say it looks like he made the right choice, but I'm not so sure. People who actually know what they're talking about might disagree, but I wonder how long it would have taken to fill the hole that would have been left if Gauss had diverted a few years of serious thinking from mathematics to linguistics. Not to say that any of his mathematical work was unimportant, but, with the notable exception of number theory where he still casts a long and very distinct shadow, wouldn't most of the things we call Gaussian have still arrived, albeit a few years later (and at the same time in those cases where researchers came up with the same ideas later but published them first)?

By comparison, think about what someone like Gauss might have done with five or ten years of serious linguistic research. The results could have disappointed but just think of the potential.

I'm way out of my depth here, so I'm just hoping to raise some points for discussion (and not say anything stupid in the process). There is, however, one point I would like to make. Hedgehog/fox conversations get complicated quickly and can shift radically when you change the way you frame a question. As a mathematician and a physicist, Gauss was certainly a fox, but when you look at the constraints he put on himself -- choosing not to do any work in any area where he had such incredible aptitude -- he certainly did his best to be a hedgehog.

1 comment:

  1. I think Gauss made the right choice. I don't think he would have gotten very far advancing linguistics. There was still a lot of field work to be done. Whole continents full of languages had been barely studied, let alone categorized. Further, he was at least a century early for any serious semantic modeling and probabilistic analysis that have since been enabled by computer and functional neural analysis.

    Mathematics, in contrast, was rapidly coalescing with algebra, geometry, analysis and other fields rapidly unifying, and here Gauss had a real talent. He was a veritable steamroller, if only because others had dumped out enough hot asphalt for him to chip seal his way across the field.

    I suspect that if he had spent more time on linguistics he would have developed something all too focused on the limited language data then available or, worse, come up with yet another of the seriously flawed semantic schemes developed in that era e.g. tagmemes. There were buttes and mesas back then, but the chasms were too great for the rope bridges of the era.