Thursday, May 10, 2012

Frozen numbers

Yesterday I came across an interesting article about prime numbers (via a Naked Capitalism links post -- pretty much the only part of NC I still find worth reading, so long as you can skip over the inevitable kvetching about how little sleep she's gotten, how much traveling she's done, and how bad her internet connection is ... which and but does this belong even in every goddamn links collection too?).

But I digress.  

The basic idea is here:

The same freezing which is responsible for transforming liquids into glasses can help to predict some patterns observed in prime numbers, according to a team of scientists from Queen Mary, University of London and Bristol University.
At a low enough temperature, water freezes into ice by arranging its molecules into a very regular pattern called crystal. However many other liquids freeze not into crystals, but in much less regular structures called glasses—window glass being the most familiar example. Physicists have developed theories explaining the freezing phenomena, and built models for understanding the properties of glasses.
Now, a researcher from Queen Mary's School of Mathematical Sciences, together with his colleagues from Bristol have found that frozen glasses may have something common with prime numbers and the patterns behind them.
Dr Fyodorov explained: "The prime numbers are the elements, or building blocks, of arithmetic. Our work provides evidence for a surprising connection between the primes and freezing in certain complex materials in physics."

This is interesting because it tells you that some other physical system is running the generate-prime-numbers algorithm.  Up till now, pretty much the only physical systems we knew of that were running this algorithm directly were: 1) the human brain, and 2) my computer.  I think the fact that prime numbers are still used in cryptography indicates that neither is running an especially computationally efficient implementation of the algorithm.  Maybe the frozen stuff has found a shortcut.

I also found it interesting because the last comment in this quote is tellingly odd. It brings up one of my favorite questions: what is a number?  I mean, if you have this simple machine that spits out a remarkably complicated and interesting pattern (prime numbers) I can see how you might be tempted to conclude that you had found "the elements, or building blocks" of the universe.  But don't you have to re-evaluate how special and fundamental you think the building blocks are when you find them keeping your Coca-Cola in the bottle?  This seems like the reaction of someone who just found the world's best seashell -- it's pretty alright, but I'm not sure you can consider it fundamental.

Rather than seeing prime numbers as the building blocks, doesn't this discovery reinforce the thought that they are the outcome of an algorithmic process?  In some sense this is obvious -- they taught you the algorithm for generating these in grade school -- but it makes you start to wonder whether even the other things we take to be simple building blocks and obvious starting points are actually the endpoints of giant complicated systems.  

Along these lines, I started wondering a while back about the complexity of some of the basic shapes like circles, squares, etc ...  Naturally, we consider something like a grid to be a very simple form.  And yet, it doesn't seem to arrive in nature all that frequently; if you look around, all the grids that meet your naked eye will have been built by humans.  The same would be true of prefect circles or spheres.  If these shapes are so simple, why aren't there more of them?  I don't see quite to the end of this argument yet, but I have discovered the first objection only reinforces my sneaking suspicion that the circle is actually really complicated.  One of the places you can find prefect geometric forms that have nothing to do with humans are in the orbits of planets or the regularities of crystals.  Those aren't simple machines at all though!  If you need half the bloody cosmos to draw a circle, you can't very well call it simple.  Similarly, you need a lot of atoms to get a crystal formation.  The grid may be an apparently simple pattern that actually requires an enormous machine to produce it.  Humans may finally have been around long enough to truly become the salt of the earth.

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