What makes us take note? And elephants.

Among all the stuff that passes by our senses, some penetrates. Which and why? And what about the elephants in the room that we don't notice.

Once again this is a germ of a thought, that leads all sorts of places and which other people must know far more about. But, well anyway, here's a few strands.

- Sometimes there are things that I know, but they don't seem important until something or someone moves it to the foreground

- There are things which are big, really big, yet most of the time, in most contexts, they are ignored. The elephants in the room.

I think these are connected. Probably the best person for pointing out elephants is George Monbiot. You can look at any of his blogposts or Guardian articles, but I'm prompted by today's, about the number of people we lock up in jail, and even more dramatically, the number that the USA locks up. There's no point me reproducing what he's said, look for yourself, but this is huge, surely? The USA, land of the free, locks up more people (a higher proportion of its population) that any other country in the world?

What you can do with Information Theory

You can get some bounds on the number of prime numbers, for one thing.

In the latest newsletter of the IEEE Information Theory Society there are some really neat calculations of lower bounds on the number of prime numbers under N [1]. I've been thinking about what significance there is to the fact that it is information theory that is being used to do this. Is it just a set of maths tools, like any others, or is it using the fact that it is 'information'.

FWIW, this is where I've got to.

A difficulty I've always had with any mathematical proof is what goes on between the lines. I write:

a < b + 3
therefore
b > a - 3

The 'therefore' seems pretty obvious to me, and I know that mathematicians will be able to come up with rigorous arguments as to why and under what conditions that 'therefore' is valid or not, as the case may be, but it seems to me that there will always be steps that take place inside your head. The reasoning will always require some sort of mental assumptions.

The proof you use in any specific case is therefore sitting on top of some internal reasoning. So, if you use information theory in the proof, there are some assumptions about information used in the argument. And in answer to my question; yes, it is a mathematical proof just like any other, but also, yes it does make use of 'information'.

Like everything I put in this blog, this needs working through more rigorously, in that mythical future when I've got time to do it.

[1] Ioannis Kontoyiannis. Counting Primes Using Entropy. IEEE Information Theory Society Newsletter Vol. 58, No. 2, June 2008. It will eventually appear here.