Consider the passage of red blood cells through arteries and arterioles. There are optical techniques for observing erythrocytes hurtling pell mell away from the thorax, propelled by the systole of the heart. We assume the cells progress passively, and might suppose that their behaviour could in theory, and very possibly in practice, at least approximately, be modelled mathematically. If this were to prove possible, it would be so because of the mechanical behaviour of the the cells. Now consider the movement of cars on an arterial road or motorway. There are differences, for instance cars move on the surface of the road, while blood cells move in three dimensions unless squeezing through the smallest capillaries. Also cars remain separate from each other, cells touch their fellows frequently. But the most important difference is that each car is self-regulating, observing or failing to observe speed limits, braking more or less quickly when a car in front slows, changing lanes to accelerate, slow down or simply overtake, stopping on the hard shoulder, or leaving the road altogether. It seems to me that the mathematics of describing the behaviour of the individual cars could not be generalised in the same way that the mathematics of the behaviour of the individual blood cells could be. And if this is true, then not because of the different complexities of cells and cars, but because of the difference between the necessitation of a physical law and the convention of following a rule. A rule clearly does not necessitate conforming behaviour. Wittgenstein reflected a great deal on rule following. In the case of rules which are important for the smooth running of society, they are usually only broken by crimainals, the incompetent, the intoxicated and the insane. ( Of course David Hume famously reached the sceptical conclusion that a particular cause did not necessitate a particular effect, and that it was repeated experience that resulted in inductive generalisation. The sun rises every day in the East, for example. Kant greatly admired Hume, and credited him with arousing him from his dogmatic slumbers, but after many years he did produce a response to Hume’s challenge in the Critique of Pure Reason. It is fun to suppose that Kant had some Scottish ancestry, and was Hume’s actual, as well as spiritual cousin. This of course is utterly fanciful.)
I recognise that the determinist/compatibilist can provide a response to this thought experiment in terms of individual pre-determining factors in the case of each driver. For example younger drivers may tend to drive faster and brake later. After a stop for a meal, all drivers may tend to drive more slowly. And regulations may be treated as causes rather than reasons. A sign warning of delays ahead, leading to large numbers of cars leaving the motorway at the next junction, can be treated as determinative of behaviour, even though some drivers soldier on. What of breakdown trucks, police activities, pile ups and how they are managed? And here I am approaching Wittgenstein’s language games, and those manifold forms of life which he felt could not be reduced to mathematics.
My main hope is this: that the vastness of the gulf between the mathematical description of cars on motorways and blood cells in arteries, invites the thought that self-determination is the most economical way in which to provide a mathematical description of the former. I draw a perhaps over optimistic analogy to the Copernican revolution, when a heliocentric model proved compelling because its mathematics was intuitively graspable, unlike the model where the movements of the sun and nearest planets were described from a terrestrial point of view, which though assuredly possible, was unwieldy and almost impossible to picture.