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3.4 Dispositions in Mathematical Physics

A mathematical physicist might argue that what is objective and essential about a physical theory is a certain differential equation and its family of solutions in phase space. I agree that it would be a significant scientific achievement to have devised such an equation that correctly predicts events and movements etc.   But if we are not satisfied with a purely instrumentalist account of scientific theories, we can still ask `what is it about the physical world that makes this theory a correct description?' Why does the differential equation correctly predict the world? Perhaps because it describes correctly the time evolution of dispositions such as forces, potentials, or quantum wave functions, in which case it can be construed as a good description of these real dispositions.

Perhaps, however, it cannot be simply interpreted in this way, yet still gives good predictions of what actually happens. I would then ask whether the equation can also describe what particular objects or particles might do if put in new situations? If it cannot, then it clearly falls short of the tasks physicists set themselves, and it would be useless for planning in engineering. If it does answer hypothetical questions, I ask: which properties of the objects involved determine the parameters of the equation? How, for example, is the equation set up with the correct strengths of couplings to the gravitational, electromagnetic and/or nuclear fields? I claim that it is precisely these dispositional properties of the objects that are needed to set up hypothetical test cases, with purely structural properties being inadequate. That is, it would appear that any mathematical description can answer all hypothetical questions correctly only if it has embedded within it (explicitly or implicitly) a description of causes and their operations. The equation, for example, would have different structures if vector influences combined in different ways (e.g. by vector sum, or by largest effect dominating, or by random selection of effects from different sources).

    One kind of physical theory that appears to avoid real dispositions is Einstein's general theory of relativity, and its modern variant, geometrodynamics.   Both of these theories are usually interpreted in terms of a `block universe' with time as if spatially extended, as they reduce all dynamics to the geometric properties of spacetime. As however both theories can and have been used to answer hypothetical questions (e.g. what would happen to the solar system should the sun suddenly become deformed), they can both be reformulated as sets of rules for deriving the state of the universe (along some spacelike hypersurface) at some time t', given its state at some earlier time t < t'. In a realistic interpretation of this time-dependent formulation, it turns out that dispositions (or something very similar) reappear. For now general relativity describes how the mass-energy tensor causes spacetime curvature, and the curvature itself in turn describes how objects would move in spacetime if they were present. That is, matter can be regarded as influencing the dispositions of objects to move in straight or curved paths.

  Geometrodynamics (assuming the theory can be worked out in adequate detail) has a similar `thick sandwich' time-dependent interpretation, but is different from general relativity in that now physical objects are not in spacetime (as we have always imagined), but simply are regions of spacetime with certain patterns of curvature. In the time evolution from t to t', these patterns interact in a non-linear fashion, and attract and repel each other as do physical objects. But this means that implicit in the non-linear field equations are rules for determining how a given pattern of curvature would interact in various circumstances, on the basis of its nature as a pattern. That is, these patterns do have dispositional properties. I agree that in this theory the nature of objects would be a set of structural properties of spacetime curvature, and not a set of dispositional properties. But since these structural patterns only have significance in conjunction with the field equations, and because this conjunction results in true dispositions, the theory is not incompatible with reality of dispositions. It is just that, if geometrodynamics were correct, the dispositions would be properties of spacetime itself, not of physical objects in spacetime (as there are no such things).

A significant part of ordinary physics is the explanation of macroscopic dispositional properties in terms of the dispositional properties of the components and the configuration of these components in the whole. Thus the elasticity of a solid, for example, is explained in terms of the attractions between the electrons and their neighbouring atoms. Note that it is not enough to say that the elasticity can be explained simply in terms of the `electronic structure', as purely structural properties cannot explain dispositional features without assuming some dispositions (such as charge, mass etc.) inherent in the electrons themselves.

There have been very few attempts in physics to deny that the constituent parts do have causal properties -- i.e. that electrons do not really have electric charge, mass, and spin.   Even the proposed quarks have these dispositional properties, along with `colour charge', `strangeness', `charm', etc. It is an empirical question which causal properties are basic, but physicists nearly always rely on some causal properties being fundamental. The form of the postulated basic causal powers (nuclear and electric fields, etc.) may be constrained by mathematical laws of symmetry, so that for example there are only discrete values for charge, spin etc., but the existence of these causal powers is something that must be assumed in order to provide a basis for physical accounts of the dispositions evident in nuclear, chemical, and biochemical systems.

The task of physics must therefore be to relate causal properties to what is known about the actual forms of the objects under investigation. In a way, physics need not be concerned with the ultimate nature of dispositions, but only with knowing that they do exist, and then with investigating their properties, locations, interactions, effects, changes with time, etc. in as much detail as possible. Mathematical physics therefore describes the numerical features of natural objects, and simply assumes that there are dispositional properties that exist according to the described forms. For example, the attribution of electric charge is purely formal until it is assumed either that there is a real dispositional property (e.g. a force) with corresponding features, or that there is a corresponding coupling to a potential-energy field.

    `Forces' and `potentials' are equivalent descriptions of the same disposition, as forces are spatial gradients of potential energy field. One could alternatively argue3.6 that it is the forces that are real, as it is the forces which actually operate between corpuscles, and that the `potentials' and `energy' are only mathematical constructs. Although the total energy is conserved, it is not really conserved as substance is conserved. However, a realistic ontology of forces or energy requires concepts beyond the original scope of the corpusclur theory. I argued in section 2.2, for example, that `potential energy', as the ability to do work, is a `second order disposition', in that it is a disposition to produce forces.

  In Newtonian physics the mechanical corpuscules had the dispositional properties of impenetrability, durability, and perfect elasticity according to their spatial shape. These causal properties cannot however be logically derived from the shape alone, as Descartes was forced to acknowledge. Newton also realised that other causal powers must be attributed to the atoms, in order to explain gravity and the tensile strength of materials. Gravitational attraction could be made according to mass and distance, but the short-range attractions could not be attributed according to any form known at that time. The fact that the attribution of gravitational powers was in strict accordance to some known form may have contributed to the impression that gravity was satisfactorily explained, but this attribution does not remove the need for a dispositional category. For Newton's law of gravitation does not say what always does happen, but only what would happen in suitable circumstances (e.g. no interference from outside influences: in Newton's case, no nuclear or electromagnetic forces).   This fact was one of Nancy Cartwright's [1983] objections to the reality of physical laws, but is understandable when laws, even Newton's law, are regarded as laws of causes, not simply as laws of effects.    

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Prof Ian Thompson


Author: I.J. Thompson (except as stated)