Individuality in Classical Physics
I see no reason why energy shouldn't also be regarded as divided atomically.
Boltzmann,
Let us begin by recalling the distinction between distinguishability, understood as involving more than one object and individuality, understood as pertaining to that object taken by itself. Epistemologically, it is through distinguishability that we become aware of something as an individual. Ontologically we may then go on to analyse that individuality either in the same terms or via some other conceptual principle, such as Suarez's 'individual unity' or some form of Transcendental Individuality, for example. Now this is all very well for 'macroscopic' objects, to which we have relatively straightforward epistemic access, but what about their microscopic counterparts, such as atoms and electrons? These are not observable, or at least, not in the way that tables, rocks and other people are, and if we cannot get a similar epistemic grip to begin with, how are we to proceed to consider their individuality?
The answer, as Reichenbach correctly realized, is to look at how such objects behave collectively and, from that epistemic standpoint, come to some conclusion regarding their individuality. In other words, we replace the examination of the object itself with inferences based on the statistical behaviour of an assemblage of such objects. 1 Thus, consider the distribution of two particles of the same kind-two electrons for example-over two 'boxes' or energy states, where the particles are indistinguishable in the sense we noted in the previous chapter; that is, in the sense of possessing the same intrinsic properties such as (rest) mass, charge, etc. We can generate the following arrangements.
In classical physics, (3) (see Fig. 2.1) is given a weight of twice that of (1) or (2), corresponding to the two ways the former can be achieved by permuting
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Figure 2.1. Two particles over two energy states.
the particles. This gives us four combinations or complexions in total and hence we can conclude that the probability of finding one particle in each state, for example, is 1/2. This probability assignment effectively characterizes Maxwell-Boltzmann statistics, the historical origins of which we will briefly consider below. As we shall see in the next chapter, we obtain quite different probability assignments when it comes to quantum statistics. Note that it is assumed that none of the four combinations is regarded as privileged in any way, so each is just as likely to occur. 2
The question now arises: on what grounds do we assign (3) a weight twice that of (1)? Such grounds cannot involve the distinguishability of the particles, since they are indistinguishable, in the above sense. Hence, it has been concluded, the increased weight given to (3)-that is, the counting of particle permutations-is accounted for by the claim that the particles are individuals. In other words, from this aspect of their collective behaviour, it can be inferred that the particles must be individuals, since their permutation makes a difference to the counting. Thus, it is typically concluded, classical particles must be regarded as individuals, in some sense.
Now, this conclusion can be resisted. One might insist that the 'extra' weight given to (3) is just some kind of 'brute fact', for which no account should be given. However, that response effectively blocks any further metaphysical exploration of the foundations of classical statistical mechanics. A more effective response would be to point out that there may be grounds, other than the statistics of the particles, for not counting their permutations, thus removing any need to account for the weight given to (3). The infamous Gibbs Paradox, for example, suggests that on thermodynamical grounds such permutations need to be 'factored out' of the relevant mathematical expressions. 3 We shall defer a full consideration of this response for two reasons: first
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of all, we need to delineate the foundations of classical statistical mechanics a little further and set down these mathematical expressions before we tackle the paradox itself. Secondly, the effectiveness of this response in general hinges on one's understanding of the import of the paradox: does it reveal something profound about the foundations of classical statistical mechanics or was it a harbinger of the shift to an alternative form of statistics, underpinned by quantum mechanics? Those who are keen to resist the claim that classical particles are individuals can be understood as answering 'yes' to the first question, while we prefer to take the second option. Hence we will leave this response for now and return to it at the end of the chapter, before moving on to quantum physics.
If the argument from permutations itself is accepted, there is the further question: how should this individuality be understood? As we shall see, different answers to this question can be given in terms of the views given in Chapter 1. However, we shall argue, the physics itself does not unequivocally support one of these answers over the others, contrary to what is often suggested in the literature.
This form of underdetermination of the metaphysics of individuality was reflected in the historical development of classical statistical mechanics, so we shall digress to consider the context in which the balls and boxes argument is put forward. This will not only allow us to present the relevant physical framework in terms of which recent discussions of these issues have been presented, but it will also reveal how the metaphysics featured in the construction of the theory itself. What Reichenbach's manoeuvre effectively does is to give us epistemic access to the particles by taking us down the famous reduction of the macroscopic properties of thermodynamics to the microscopic ones covered by statistical mechanics. Let us now consider this route in more detail.
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