Not much time to post today, so I thought I’d just put up a couple of nice little quotes about the Central Limit Theorem. In case you don’t know it, this theorem explains why so many phenomena result in measurable things whose frequencies of occurrence can be described by the Normal (Gaussian) distribution, with its characteristic Bell-shaped curve. I’ve already mentioned the role that various astronomers played in the development of this bit of mathematics, so I won’t repeat the story in this post.

In fact I was asked to prove the theorem during my PhD viva, and struggled to remember how to do it, but it’s such an important thing that it was quite reasonable for my examiners to ask the question and quite reasonable for them to have expected me to answer it! If you want to know how to do it, then I’ll give you a hint: it involves a Fourier transform!

Any of you who took a peep at Joan Magueijo’s lecture that I posted about yesterday will know that the title of his talk was *Anarchy and Physical Laws**. *The main issue he addressed was whether the existence of laws of physics requires that the Universe must have been designed or whether mathematical regularities could somehow emerge from a state of lawlessness. Why the Universe is lawful is of course one of the greatest mysteries of all, and one that, for some at least, transcends science and crosses over into the realm of theology.

In my little address at the end of Joao’s talk I drew an analogy with the Central Limit Theorem which is an example of an emergent mathematical law that describes situations which are apparently extremely chaotic. I just wanted to make the point that there are well-known examples of such things, even if the audience were sceptical about applying such notions to the entire Universe.

The quotation I picked was this one from Sir Francis Galton:

I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the “Law of Frequency of Error”. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along

However, it is worth remembering also that not everything has a normal distribution: the central limit theorem requires linear, additive behaviour of the variables involved. I posted about an example where this is not the case here. Theorists love to make the Gaussian assumption when dealing with phenomena that they want to model with stochastic processes because these make many calculations tractable that otherwise would be too difficult. In cosmology, for example, we usually assume that the primordial density perturbations that seeded the formation of large-scale structure obeyed Gaussian statistics. Observers and experimentalists frequently assume Gaussian measurement errors in order to apply off-the-shelf statistical methods to their results. Often nature is kind to us but every now and again we find anomalies that are inconsistent with the normal distribution. Those exceptions usually lead to clues that something interesting is going on that violates the terms of the Central Limit Theorem. There are inklings that this may be the case in cosmology.

So to balance Galton’s remarks, I add this quote by Gabriel Lippmann which I’ve taken the liberty of translating from the original French.

Everyone believes in the [normal] law of errors: the mathematicians, because they think it is an experimental fact; and the experimenters, because they suppose it is a theorem of mathematics…

There are more things in heaven and earth than are described by the Gaussian distribution!