CONCLUSION: THE METAPHYSICS OF CLASSICAL PHYSICS
Let us briefly return to the argument with which we began this chapter: classical particles should be regarded as individuals because of the way permutations are counted in classical statistical mechanics. As we indicated, this 22422p158w argument has been resisted on the grounds that, for example, the Gibbs Paradox indicates that this counting is deeply problematic. These grounds can be spelled out as follows: we begin with the requirement that the entropy of a system be 'extensive'. This means that if the volume of some system (a gas, say) is divided into two halves, then the entropy of the whole system must equal the sum of the entropies of the respective halves. Unfortunately, the expression for the entropy obtained from Maxwell-Boltzmann statistics does not in fact meet this requirement. Thus when it is actually applied to the situation where a sample of gas is mixed with a sample of the same kind of gas, it yields an entropy increase. This is unsatisfactory on two grounds. First, and foremost, it just does not agree with the results of classical thermodynamics. Secondly, and more dubiously, it might be argued that in so far as classical thermodynamics
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is a 'phenomenological' theory in which no reference to particles need be made (and of course such an argument might well draw on the history of the subject at this point), nothing 'physical' can be said to happen when two samples of the same gas are mixed. Thermodynamically speaking, 'mixing' is a concept which is only applicable when different kinds of gases are brought together and so from this perspective, the mixing of a gas with itself is not a physical event; hence the entropy change should be zero.
Standardly, the 'paradox' is resolved by excluding the relevant permutations from the calculation of the statistical entropy. This can be achieved by simply dividing the classical expression for the number of possible complexions by N yielding an expression for the entropy-known as the Sackur-Tetrode equation-which is extensive as required. 197 The justification that is usually given is that this procedure eliminates 'redundant' complexions obtained through a permutation of the particles. The issue now is, how are we to interpret this procedure? One proposal is that it shows that we should not take classical particles to be individuals, at least not on the basis of arguments involving their permutation. 198 As we shall see, the usual contrast with quantum physics is then lost. However, this proposal radically misconstrues the historical situation and, by blurring the ontological character of classical particles, blurs the distinction between classical and quantum statistical mechanics in general. Historically, the failure of extensivity and the Gibbs Paradox in general were seen as revealing a fundamental flaw in the Maxwell-Boltzmann definition of entropy and one which is corrected by shifting to quantum statistical mechanics at the theoretical level and an understanding of the particles as quantum in nature at the ontological level. It is in this sense that quantum mechanics, in general, has been generally taken to 'resolve' the problem. 199 In other words, the force of the proposal can be turned around: what it suggests is that the world is actually quantum in nature, as one would expect. What the exclusion of the permutations (by dividing by N!) is a manifestation of is precisely that the particles are not just indistinguishable in the classical sense. And if this aspect is incorporated into the analysis from the word go, then Gibbs' 'Paradox' simply does not arise. Thus, we take the import of the paradox to be not that we must give up an understanding of classical particles as individuals, but that we must shift to a quantum understanding of particles in general.
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Given, then, the permutation argument, how should we understand the individuality of the particles? We have argued that although some form of non-spatio-temporal TI may be sufficient for classical statistical mechanics, it is not necessary. The weighting assignments required to get the right (Maxwell-Boltzmann) statistics can always be obtained either by transforming to an appropriate phase space where the relevant trajectories are made apparent or by insisting that at most one particle can occupy a given space-time point. In either case, the individuality of the particles is conferred via some form of the Space-Time view. 200 Further articulation of this latter account has taken us into another whole debate about the nature of space-time itself and there we need to call a halt. Our principal conclusion in this chapter is that what we have in the classical case is a kind of underdetermination of the metaphysics of individuality by the physics: the physics by itself cannot tell us which package-TI or STI-we should prefer. As we shall now see, a much more radical form of underdetermination exists in the quantum domain, where the physics cannot tell us whether the particles are even individuals or not.
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