The Elusive Law: Chaos and the Limits of Pluralism in Physical Description — Epoche C2
The question, and a repair to the essay's foundations A regularity stated at one level — the equation of state of a fluid, the motif on a dyed cloth, the reliability of a trained movement — counts as a law only if it survives variation in whatever realises it at the level below. This note asks when that survival is actually secured rather than assumed, and argues that it always requires a specific mechanism, one which is absent in exactly the systems that chaos theory describes. Before the argument, a repair. The version of this essay published earlier attributed the pluralist position it opposes to a 2018 article in Synthese by Peter Simons entitled 'The Multiple Realizability of Physical Laws'. I have not been able to verify that such an article exists. Peter Simons is a real philosopher whose principal work is in mereology and formal ontology, and Synthese is a real journal, but the paper as cited cannot be confirmed, and a second entry in the same list was a hybrid of two real things: the 2002 Oxford University Press book by Robert Batterman is The Devil in the Details , not 'The Measure of Minds'. An argument needs an opponent who actually holds the position attacked, so what follows is rebuilt on three works that certainly exist and that state the pluralist case in stronger forms than the attributed article was made to state it. The thesis defended is unchanged. What the multiple realisability argument actually claims The canonical statement is Fodor's (1974). Reduction of a special science to physics, on the model then standard, requires bridge laws pairing each predicate of the special science with a predicate of physics. Fodor's objection is not that the pairing is hard to find but that it cannot be a law. Take his own example, monetary exchange: the physical events that realise it include the passing of coins, of notes, of shells, of signed paper, of electronic register entries, and indefinitely many others. The physical predicate corresponding to 'monetary exchange' is therefore an open disjunction, and a disjunction of that kind is not a natural kind — it figures in no physical law, supports no counterfactuals as such, and does not project onto new instances. No bridge law, hence no reduction, hence the special science's generalisation stands on its own. The argument turns on a failure of kindhood at the lower level, not on complexity. Kim (1992) turns the same observation against Fodor's conclusion, and this is the dispute a pluralist has to win. If a higher-level property is realised by physically heterogeneous states, and if causal powers are conferred by physical realisers, then the higher-level property is itself causally heterogeneous — and a causally heterogeneous class is no better a kind at its own level than the disjunction was at the level below. Kim's remedy is local reduction: not one law about the higher-level property, but a family of laws, one per realiser type. Multiple realisability, on this reading, is an argument against the unity of the higher-level kind rather than for its autonomy. A third position, and the one this essay ends up closest to, is Cartwright's (1999). Her pluralism is about laws rather than about levels: a law holds where a nomological machine is in place — a stable arrangement of components with fixed capacities, shielded from interference — and outside such arrangements it makes no claim at all. The resulting picture is a dappled world of lawful patches with no universal cover. Cartwright's position matters here because it already contains the thought this essay will make precise: lawfulness is something a situation has to be arranged to possess. A worked case that contains both regimes: the batik cloth One correction of fact before the example is used. The published essay referred to 'batik weaving'. Batik is not a weaving technique but a wax-resist dyeing technique applied to cloth already woven: hot wax is laid down with a spouted pen ( canting ) or a copper stamp, the cloth is dyed, and the wax is removed, the waxed areas having been protected. Taken properly, the example is better than the use originally made of it, because a single cloth exhibits two opposite behaviours. The motif is robust: the same design emerges whatever the exact fibre-by-fibre distribution of the wax, because the process has a threshold. A region is either sealed against the dye or it is not, and once the wax is comfortably above the sealing thickness, further variation makes no difference to the outcome. That threshold is a contraction mechanism — a wide set of microstates is carried onto one macroscopic result. The crackle, by contrast, is the fine veining produced where the wax fractures before dyeing, and it differs on every cloth; it is generated by a process that magnifies rather than absorbs small differences, which is exactly why the crackle serves to identify an individual piece. Two regimes, one artefact, and the difference between them is not a difference of level but a difference of mechanism. Chaos, stated with the quantities it turns on Chaos theory bears on the first of these regimes by describing the second. Its central quantity is the rate at which nearby states separate. For a flow $\varphi_t$ on a state space and an infinitesimal displacement $\delta(t)$ carried along a trajectory, the largest Lyapunov exponent is $$\lambda = \lim_{t\to\infty}\frac{1}{t}\ln\frac{\|\delta(t)\|}{\|\delta(0)\|},$$ and the system is chaotic on the relevant set when $\lambda > 0$, so that $\|\delta(t)\| \approx \|\delta(0)\|e^{\lambda t}$. The system in which this was first exhibited numerically is Lorenz's (1963) three-variable truncation of thermal convection, $$\dot{x} = \sigma(y-x), \qquad \dot{y} = x(\rho-z)-y, \qquad \dot{z} = xy - \beta z,$$ at the parameter values he used, $\sigma = 10$, $\rho = 28$, $\beta = 8/3$. Here $\lambda \approx 0.906$ per unit of the model's dimensionless time, so an initial separation doubles in $\ln 2/\lambda \approx 0.77$ time