simplicity, unity, explanation, beauty *
4.4 Further issues
in not containing any implicit reference to specific times or places (or hypersurfaces of S that distinguish one region of S from another).
Physical properties must be invariant in a sense that corresponds to the invariance of allowed functions in physics. The more general notion of property, which includes Goodmanesque properties, is excluded, just as the more general notion of function, which includes (2) above as an
“unchanging” function, is excluded.
4.4 Further issues
What of the other aspects of the problem of non- empirical requirements in science mentioned in the introduction? Here are a few brief remarks concerning some of these further issues.
Most of the other terms used to refer to non- empirical requirements can be straightforwardly related to unity. We have seen that this is true of symmetry and invariance. Non- ad hoc, organic, inwardly perfect and non- aberrant can be interpreted as appealing to unity, and harmonious, beautiful and conceptually coherent can be interpreted as presupposing unity. A dynamical physical theory can be held to be explanatory in char-acter to the extent that it is (1) empirically contentful, and (2) unified.
Simplicity, however, is quite different. The simplicity of a theory can be interpreted as having to do, not with whether the same laws apply throughout the space S, but rather with the nature of the laws, granted that they are the same. Some laws are simpler than others. In order to overcome the objection that simplicity is formulation- dependent it is essential, as in the case of unity, to interpret “simplicity” as applying to the content of theories, and not to their formulation, their axiomatic structure, etc. Theories can only, at best, be partially ordered with respect to degrees of simplicity. Even when two theories are amenable to being assessed with respect to relative simplicity, there is always the problem that a change of variables may reverse the assessment. Let the two the-ories be (1) y = x, and (2) y = x2. We judge (1) to be simpler than (2).
Let x2 = z. We now have (1) y = z, and (2) y = z. Now (2) is simpler than (1). Assessment of relative simplicity of two theories may only be unambiguous when restrictions are placed on the form that physical vari-ables can take, so that only linear transformations of the type z = Ax + B (where A and B are constants) are permitted, for example. It is a further great success of the theory presented here that it succeeds in distinguish-ing sharply between these two aspects of physical theory, the unity and simplicity aspects, and succeeds in explicating both.10
We can use these two notions to solve the problem of ambiguity of judgement concerning the relative non- empirical merits of Newton’s and Einstein’s theories of gravitation. Newton’s theory is simpler, but Einstein’s is more unified in that it eliminates gravitation as a force, and reformulates Newton’s first law so that it becomes the assertion that bod-ies move along geodesics in curved space- time, curvature being caused by mass, or by stress- energy- density more generally. As theoretical physics draws closer to capturing the true theory of everything, it is reasonable to expect that the totality of fundamental physical theory will become increasingly unified and complex.
So far I have stressed that terminological unity and simplicity are irrelevant when it comes to assessing unity and simplicity in a physically significant sense. In scientific practice, however, terminology is chosen so as to reflect physically significant unity and simplicity (Maxwell, 1998, 110– 3). Thus if the content of a theory exhibits certain symmetries, ter-minology is chosen so that it too exhibits these symmetries, so that if the theory is invariant with respect to position or orientation in space, ter-minology is chosen which reflects this fact. Once a theory is formulated in such “physically appropriate” terminology (as it may be called), two versions of symmetry operations arise as a result: “active” (which make changes to physical systems) and “passive” (which make corresponding changes to the description of unchanged physical systems). Granted that we formulate physical theories exclusively in such “physically appro-priate” terminology, then terminological unity and simplicity comes to reflect physical unity and simplicity, and is thus, to that extent, physically significant.
What of the simplicity and unity of theories in sciences other than fundamental physics? Much needs to be said on this topic; the following brief remarks can serve only as pointers to a more adequate treatment.
Solutions to the equations of fundamental physical theory, specifying precisely how increasingly complex physical systems evolve in space and time, rapidly become horrendously complex in character. In carrying out derivations, physicists invariably “simplify” results obtained by discard-ing variable quantities or higher order terms judged to be insignificant in the physical situations under considerations. Just this is done when NT is
“derived” from Einstein’s theory, or Kepler’s and Galileo’s laws of motion are “derived” from NT.11 The outcome is a range of more or less termino-logically simple phenomenological laws of only approximate validity. But the simplicity is not, here, merely pragmatic, since such a law has been
“approximately derived” from some fundamental physical theory for-mulated in a “physically appropriate” way, the “approximate derivation”
showing what the range of applicability of the law is with what degree of accuracy. Even though such laws are incompatible with the fundamen-tal physical theory from which they have been “approximately derived”, nevertheless what the “derivations” reveal is that pragmatic simplicity has been obtained by sacrificing strict derivability and precise empirical accuracy, there being nothing here to counter the underlying unity in nature postulated by fundamental physical theory (in so far as it does postulate this). Laws such as these are prevalent throughout phenom-enological physics, astrophysics and parts of physical chemistry. Even where such “approximate derivations” cannot be carried through, for large parts of chemistry, and for biology, nevertheless, as I have already remarked, laws and theories of these sciences are constrained by funda-mental physics, and must endeavour to be compatible with fundafunda-mental physics, at least in the qualified way just indicated in connection with phenomenological physics. Thus, much of the great explanatory power of Darwinian theory stems from the fact that it postulates mechanisms for evolution – random inheritable variation and natural selection – which are capable of designing living things able to pursue the goals of survival and reproductive success in their given environments, these mechanisms nevertheless being compatible with the purposeless cosmos depicted by physics. Biology must accord with physics in much more specific ways as well, in that the mechanisms of inheritance and development must accord with physics, and so too the multitude of processes that take place in living things.
What implications does the account of non- empirical requirements for theories, given here, have for science? How can justice be done to evolving non- empirical requirements? How is persistent preference for unified theories, even against the evidence, to be justified? I take these three problems together.
At the beginning of this chapter I demonstrated that, in physics, theories that are unified, in senses (1) and (2) at least, are persistently chosen in preference to available, empirically more successful, but disunified theories. To proceed in this way is to make the permanent assumption that the phenomena under consideration are such that all theories of these phenomena that are disunified in senses (1) and (2) are false. If physicists persistently accepted theories that postulate atoms in preference to available, empirically more successful field theo-ries, it would be clear that physicists are thereby assuming that all field theories are false. Just the same holds for the persistent rejection of empirically more successful disunified theories.
But rigour demands that assumptions that are substantial, influen-tial, problematic and implicit need to be made explicit, so that they can be critically assessed, so that alternatives can be developed and consid-ered, the hope being that in this way such assumptions can be improved.
Thus rigour demands that science makes explicit, and so criticizable and improvable, the substantial, problematic, influential and implicit assumption that the universe, or the phenomena, are such that all dis-unified theories are false. This assumption, M, can easily be shown to be metaphysical, as follows. Persistent acceptance of theories unified in ways (1) and (2) involves rejecting infinitely many empirically more successful disunified rivals, T1, T2, … T∞, because they clash with M. In effect, M = notT1 and notT2 … and notT∞. In order to verify M we would need to falsify all of T1, T2, … and T∞, but as there are infinitely many theories, this cannot be done. In order to falsify M we need to verify just one of T1, T2, … or T∞, but physical theories cannot be verified. Hence M, being neither verifiable nor falsifiable, is metaphysical. It is a perma-nent metaphysical assumption of science – permaperma-nent, at least, as long as all theories disunified in senses (1) and (2) are rejected whatever their empirical success might be.
At once the question arises: how is this assumption M to be criti-cally assessed and, perhaps, improved? In Chapter 2 I argued that once the metaphysical assumption implicit in persistent preference in sci-ence for unified theories is acknowledged, it becomes apparent that we need to adopt a new conception of science, which construes science as making a hierarchy of such assumptions, these assumptions asserting less and less as one goes up the hierarchy, and thus becoming more and more likely to be true.12 These are assumptions about the knowability and comprehensibility of the universe. As we descend the hierarchy, assumptions become more substantial and specific, and much more likely to be false, and in need of revision. Revision is, however, kept as low down in the hierarchy as possible. Those physical theories are accepted which best accord with the evidence and the best available metaphysical assumption, B, say, lowest down in the hierarchy. But B may itself be revised if a rival assumption, B*, is developed which (a) is compatible with the assumption above it in the hierarchy, and (b) supports an empirical research programme that is more successful than the one supported by B. Relatively problematic assumptions high up in the hierarchy thus form a fixed framework within which much more specific, problematic assumptions can be revised in the light of empirical success and failure. As knowledge improves, assumptions
and associated methods improve as well; there is something like positive feedback between improving knowledge and improving knowledge- about- how- to- improve- knowledge, the methodological key to the success of modern science. Non- empirical requirements for theory acceptance, corresponding to metaphysical assumptions, improve with improving knowledge. Newton’s requirements of sim-plicity evolve into the symmetry principles of modern physics. For my view as to how acceptance of the hierarchy of metaphysical assump-tions is to be justified, see Chapter 7 (see also Maxwell, 1998, ch. 5;
2007, ch. 14; 2017a).