Friday, 10 May 2013

Models and Science

We love to make simplified models. We still use Newtonian models and there is reason. They work most of the time. Even these fall over and we cannot calculate a generalised three body problem of gravitational attraction as put forth by Newton now. If we tried this using Relativistic equations, well we do not have the computational power with all the computer systems on earth and a few lifespans to do that.

Back in 1887, mathematicians Ernst Bruns and Henri Poincaré demonstrated an elegant generalised system that offered proof showing that there is no general analytical solution for the three-body problem when defined using by algebraic expressions and integrals. This does not say that one could not exist, but that it cannot be completed using the mathematics we have at our disposal.

In this, they demonstrated that the motion of three bodies is generally non-repeating, except in special cases. Right now (and as last I know of) we have a total of 16 specific solutions to the three-body problem. The last 13 of these only in the last year (http://arxiv.org/abs/1303.0181).

These are great and have a wonderful purpose, but we need to remember the world is bigger and more complex than we can understand.

Models are just that. When we lose sight of this, we start to lose sight of what we can achieve.

Many models of reality are based on Euclidian space (geometry). The Friedmann–Lemaître–Robertson–Walker metric is an exact solution of Einstein's field equations of general relativity. From it and the general relativistic formula, we find that space is only approximately flat. A good approximation for most purposes, but flat it is not. To really model the world, we have to start with CAT(k) spaces, Hadamard spaces, and constructs such as Hilbert spaces in the Quantum mechanical world.

For the most part, the error rate is small and the calculation cost is such that we use a classical model. This does start to fail in modern applications. For example, the time system on the GPS we need to us a relativistic calculation as the time difference experienced is significantly affected by the differential velocity of the Earth to the satellite. The result would be a large error that continued to grow with the use of a classical model.

Science is all about models. We like to believe we can know it all, but this is most like something that will always lie outside our grasp.

For more on Hilbert Space see:

http://www.math.kun.nl/~landsman/HSQM2006.pdf

http://econ.la.psu.edu/~hbierens/HILBERT.PDF

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