Free The Science of Discworld II by Terry Pratchett

different ways depending on what it sees.
    Complexity guru Stuart Kauffman has taken this difficulty a stage further. He points out that while in physics we can expect to pre-state the phase space of a system, the same is never true in biology. Biological systems are more creative than physical ones: the organisation of matter within living creatures is of a different qualitative nature from the organisation we find in inorganic matter. In particular organisms can evolve, and when they do that they often become more complicated. The fish-like ancestor of humans was less complicated than we are today, for example. (We’ve not specified a measure of complexity here, but that statement will be reasonable for most sensible measures of complexity, so let’s not worry about definitions.) Evolution does not necessarily increase complexity, but it’s at its most puzzling when it does.
    Kauffman contrasts two systems. One is the traditional thermodynamic model in physics, of N gas molecules (modelled as hardspheres) bouncing around inside their 6 N -dimensional phase space. Here we know the phase space in advance, we can specify the dynamic precisely, and we can deduce general laws. Among them is the Second Law of Thermodynamics, which states that with overwhelming probability the system will become more disordered as time passes, and the molecules will distribute themselves uniformly throughout their container.
    The second system is the ‘biosphere’, an evolving ecology. Here, it is not at all clear which phase space to use. Potential choices are either much too big, or much too limited. Suppose for a moment that the old biologists’ dream of a DNA language for organisms was true. Then we might hope to employ DNA-space as our phase space.
    However, as we’ve just seen, only a tiny, intricate subset of that space would really be of interest – but we can’t work out which subset. When you add to that the probable non-existence of any such language, the whole approach falls apart. On the other hand, if the phase space is too small, entirely reasonable changes might take the organisms outside it altogether. For example, tiger-space might be defined in terms of the number of stripes on the big cat’s body. But if one day a big cat evolves that has spots instead of stripes, there’s no place for it in the tiger phase space. Sure, it’s not a tiger … but its mother was. We can’t sensibly exclude this kind of innovation if we want to understand real biology.
    As organisms evolve, they change. Sometimes evolution can be seen as the opening-up of a region of phase space that was sitting there waiting, but was not occupied by organisms. If the colours and patterns on an insect change a bit, all that we’re seeing is the exploration of new regions of a fairly well-defined ‘insect-space’. But when an entirely new trick, wings, appears, even the phase space seems to have changed.
    It is very difficult to capture the phenomenon of innovation in a mathematical model. Mathematicians like to pre-state the space of possibilities, but the whole point about innovation is that it opens up new possibilities that were previously not envisaged. So Kauffman suggests that a key feature of the biosphere is the inability to pre-state a phase space for it.
    At risk of muddying the waters, it is worth observing that even in physics, pre-stating the phase space is not as straightforward as it might appear. What happens to the phase space of the solar system if we allow bodies to break up, or merge? Supposedly 5 the Moon was splashed off the Earth when it collided with a body about the size of Mars. Before that event, there was no Moon-coordinate in the phase space of the solar system; afterwards, there was. So the phase space expanded when the Moon came into being. The phase spaces of physics always assume a fixed context. In physics, you can usually get away with that assumption. In biology,