Archive for March, 2011

Pi is still wrong

In anti-honor of “Pi Day,” I’d like to direct your attention to Michael Hartl’s The Tau Manifesto. The Manifesto is inspired by Bob Palais’ article \pi is wrong! and presents a list of simple, but compelling, reasons that 2 \pi, not \pi, is the more fundamental constant.

These ideas have been discussed on the blathosphere before, e.g. on Bill Gasarch and Lance Fortnow’s blog Computational Complexity. There Terence Tao makes the following remark:

It may be that 2 \pi i is an even more fundamental constant than 2 \pi or \pi. It is, after all, the generator of \log(1). The fact that so many formulae involving \pi^n depend on the parity of n is another clue in this regard.

The basic argument for this point of view can be summarized as follows: e^z : \mathbb{C} \to \mathbb{C} is a special function because it is the unique eigenvector of eigenvalue 1 of the derivative operator acting on, say, complex-analytic functions on \mathbb{C}, and this function has period 2 \pi i. So we see that this constant pops directly out of a definition of \mathbb{C} and a definition of the derivative of a complex-analytic function: no arbitrary choices were necessary. (The closest thing to an arbitrary choice here is the decision to identify the tangent space of a point in a vector space with the vector space itself, but this is completely invariant.)

The 2 \pi here is precisely the circumference of a unit circle, which is distinguished among all circles because in \mathbb{C} it is the only circle of positive radius closed under multiplication. This is a fundamental number because of the relationship between the unit circle and Pontrjagin duality (which has the Fourier transform and Fourier series as special cases), and is responsible for all appearances of 2 \pi in mathematics that I know of.

For example, the reason there is a factor of \sqrt{2\pi} in the definition of the Gaussian distribution (which is where the factor of \sqrt{2\pi} comes from in Stirling’s formula) is that the Gaussian distribution is its own Fourier transform. This factor is commonly cited as an application of \pi that has nothing to do with circles, but of course the Fourier transform has everything to do with circles.

Edit, 3/15/11: Vi Hart also explains the wrongness of \pi in video form. I have to admit I think I read the title of her post and then promptly forgot I had done so when writing this post.

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There’s a lot of food for thought in John Baez’s latest post on Azimuth, an interview with AI researcher Eliezer Yudkowsky.

Eliezer Yudkowsky happens to be one of the most interesting people I know of. In addition to his work on friendly AI, he helped found the community blog Less Wrong. The material in his Sequences there describes what Yudkowsky calls “the art of rationality,” but if you’re not up to reading several long sequences of blog posts, you might be interested in his enormously popular Harry Potter fanfic, Harry Potter and the Methods of Rationality, which explores many of the same ideas in a more fun and accessible setting. I think this was an enormously clever ploy on his part.

Since this is a math blog, I’d also like to highlight the following part of the interview above:

In Silicon Valley a failed entrepreneur still gets plenty of respect, which Paul Graham thinks is one of the primary reasons why Silicon Valley produces a lot of entrepreneurs and other places don’t. Robin Hanson is a truly excellent cynical economist and one of his more cynical suggestions is that the function of academia is best regarded as the production of prestige, with the production of knowledge being something of a byproduct. I can’t do justice to his development of that thesis in a few words (keywords: hansom academia prestige) but the key point I want to take away is that if you work on a famous problem that lots of other people are working on, your marginal contribution to human knowledge may be small, but you’ll get to affiliate with all the other prestigious people working on it.

Words to ponder in light of the refusal of famous mathematicians like Grothendieck and Perelman to associate with academia.

Edit, 3/14/11: Part two of the interview is up.

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