Knowing the future: the devil in the detail

Graphic - knowingthefuture
Henri Poincare

This is the right moment to stop and reflect on the sense in which a knowledge of physics allows us to understand the world.

At first sight it may seem that this understanding is retrospective: we are confronted with some phenomenon (the fall of an apple, the rainbow in the sky …) and we explain what we have observed. But our understanding is also predictive: we know what will happen (..to the next apple, when we next see rain-in-sunshine …). The capacity to foretell the future that will follow from given present conditions is the source of the practical power conferred by our understanding.

From the perspectives of classical mechanics in particular (we will not open the quantum box here) it would appear that we could predict our destiny entirely. Laplace thought that way:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.
[Laplace]

Of course most of us have to settle for intellects that are less-than-vast; and we recognise that Laplace’s picture is an unattainable (and not altogether attractive) extreme. Nevertheless we have tended to hold on to the idea that the future of (at least) simple mechanical systems is predictable.

Henri Poincare (shown above) was probably the first to recognise that this predictability is an illusion:

If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible …
[Poincare]

That was written at the beginning of the twentieth century and (it seems) largely forgotten until the arrival of computers-and-thence computer simulation- showed that Poincare had been right.

We now recognise that the overwhelming majority of physical systems we have to deal with (even seemingly simple mechanical ones) exhibit behaviour that is extremely sensitive to its initial (current) state Typically the differences between initially similar systems grow exponentially fast as their futures unfold. The precision which which initial conditions are known therefore sets a horizon of predictability; to extend the horizon even a very little (ie to see a just a little further into the future) requires a huge improvement in our knowledge of what is going on ’now’. One can still say that Laplace was right; he just didn’t realise how ’vast’ an intellect we would require (how many significant figures we would need to specify the initial conditions) to predict even the first 10 cycles of our driven pendulum.