A Mathematician in the Land of Physicists

First note: I don't know what to call myself. I don't think it's fair to call myself a mathematician. But I have had little formal training in statistics, either. I suppose the closest thing I could think of would be 'computational scientist.' But that's next to meaningless.

I've noticed this during my time at Santa Fe. Many people introduce themselves as a physicist, or a microbiologist, or an economist. I would never call myself a mathematician. I would say that I study mathematics. I don't know when that will change. It's not as if once I get my PhD a switch will flip and I will become a different person, a 'mathematician.' Perhaps it's all in the head.

But that's not what I want to talk about. It was a roundabout way of getting to my real topic: what it feels like to be a mathematician / statistician in a school predominantly run by physicists1.

First, I should clarify what I mean by 'physicist,' since that's such a broad term as to be nearly useless. I'm referring to a mentality that in my own personal experience I've found to be more common among physicists2. The mentality involves a cavalier use of mathematics, without a meticulous attention to details. Physicists go in, guns blazing, with simple theories that they claim can explain complicated phenomenon. This xkcd comic sums up the stereotype.

Compare this to the mathematician, who isn't satisfied until some conjecture has been proved. (Why else are people still working on the twin prime conjecture?) Mathematicians are very careful to state all of the hypotheses of their theorem, and careful to make sure these hypotheses are met before applying the theorem to something else. Physicists don't care about theorems. Empirical evidence is enough. Take the odd primes joke as an example.

So, what particular instances do I have in mind? One of the lecturers today invoked the law of large numbers to claim that distributions get more and more spread out over time. That couldn't be further from the statement of that theorem. He also tried to argue (partially in jest [I hope]) that all distributions should be normal because of the Central Limit Theorem. Again, a complete misunderstanding of the theorem. The lecturer was a mathematician, and I'm sure he knows this material, but when he talked about it to a general audience, he let the facts get away from him. So, despite the fact that he's a mathematician, he seems to very much have what I would call the physicist's mind set.

Another example. One of the graduate student lecturers commented that Nonlinear Time Series Analysis3 by Kantz and Schreiber is the 'bible of time series analysis.' I think that's a gross misrepresentation of the field of time series analysis. Kantz and Schreiber is the bible for people in the dynamical systems / nonlinear dynamics community, but that hardly makes up the majority of people interested in time series analysis. (See statisticians.) This statements shows a very skewed perception of the tools of time series analysis4.

The graduate student did have a candid moment when he was talking about methods for choosing the embedding dimension and time delay in a delay-coordinate embedding. He commented that he came from a purely theoretical background, and felt bad any time he used a heuristic. This, again, I would say separates the mathematician mindset from the physicist mindset. The mathematician may, every now and again, break the rules and use a result without proof. But s/he feels bad about it.

I realize I'm playing a game of 'us v. them' where I'm framing the mathematicians as the good guys and the physicists as the bad guys. There are certainly advantages to the physicists' approach. It may produce much more junk research, but in the process it probably produces more good research, and more quickly, than the methodical approach of the mathematician.

I say all of this to make clear the strange discomfort I've felt over the past few days. I consider myself part of the 'complex systems' community. But every time a speaker invokes a 'power law' when speaking of a function of the form \(x^{\alpha}\) (we just call those polynomials in math...), or begins to fit a line through any arbitrary plot, I can't help but feel out of place. I'm beginning to realize that even within so specialized a community, vastly different cultures can quickly form.

Physicists have had a long run of discovering simple, universal truths about the universe. But I really doubt such simple, universal truths will be so easily discovered in biological and social systems. The real world is just too messy. It's not a spherical cow in a vacuum. And we shouldn't try to force it into that mold just because that has worked in the past.

  1. I should preface this with the fact that I'm really enjoying myself in Santa Fe. And heck, my advisor is in a physics department. I just want to record some cultural differences that I've noticed.

  2. Just as in a band, the people who played trumpet had a 'type.'

  3. A whole new cohort of students have learned how to embed their scalar time series and compute fractal dimensions. Without a single mention of surrogate data, to make sure you're not bullshitting yourself. Or even an explicit definition of mutual information. Oy.

  4. Not to mention that a lot of the ideas about embedding really amount to inferring an autoregression function. Which falls well within the wheelhouse of things statisticians have done a lot of great work on.