One of the earlier and also most persistent problems in philosophy is that of the One and the Many. In natural philosophy that problem emerges especially as the question: does a compound or aggregate have characteristics which do not simply follow from or supervene on the properties of its parts? Is an aggregate a 'mere' aggregate? Are there principles which are trivial or vacuous as far as non-compound systems are concerned (if there be any!) but take on force when compounds are considered?
We have already encountered a holism in quantum theory at a number of junctures. Specifically, the state of X + Y determines but is not determined by the reduced states for its parts X and Y. In Chapter 3 we saw that this sort of holism characterizes even classical probability functions; it becomes non-trivial in quantum theory because there probability is 'irreducible'. But even when all that is understood, there is something more.
The something more is the principle of Permutation Invariance for compounds with distinct but 'identical' parts. The sense of 'identical' needs to be made precise; two conjectures suggest themselves and must be investigated. The first is that Permutation Invariance accounts for the well-known but puzzling departures from classical statistics of aggregates. The second is that, when 'identical' is properly understood, it entails Permutation Invariance tautologically. In this chapter and the next, I shall argue that both conjectures are false, though each has a core of truth. 1
An atom . . . possesses two kinds of symmetry properties: (1) the laws governing it are spherically symmetric, i.e.