WEYL AND THE ANALYSIS OF AGGREGATES

WEYL AND THE ANALYSIS OF AGGREGATES

The far-reaching nature of these issues was clearly perceived by Cassirer, and, even more acutely and perceptively perhaps, by Weyl. In remarking on the radical epistemological transformation effected by quantum mechanics, Cassirer notes that "[i]t seemed hitherto to be an axiom, not only of classical physics but of classical logic, that the state of a thing in a given moment is completely determined in every way and with respect to all possible predicates". ²¹¹ This 'axiom' must now be abandoned. Likewise, we must revise our understanding of the whole-part relationship. ²¹² In particular, Cassi 15215s185p rer notes, we must be careful to distinguish various possible meanings of 'inclusion' in this context. Thus we should distinguish 'containment' from 'being a part of'-a point is 'contained' in a line but should not be considered as forming a part of it. Since a whole may contain more than parts, the whole may be larger than the sum of its parts and this is exemplified by quantum mechanics, where a two-electron system determines the state of the electrons but not vice versa: "A knowledge

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of the states of the two parts does not determine the state of the joint system, and a derivation of the latter from the former is out of the question" ²¹³

The problem, then, is how such an aggregate may be "differentiated and 'individualized' ", and Cassirer notes, presciently perhaps, that

The ordinary method of counting, which presupposes that it is known from the beginning what is to constitute one thing and what two or more things, is here insufficient. Individual things are not separated from each other in as simple a manner as in the sensuous-spatial view; complicated theoretical considerations are thus always required in order to determine precisely what is to be treated as an individual, what is to be counted as a 'one'. ²¹⁴

The formal analysis of aggregates was taken further by Weyl in his classic 1949 work, The Philosophy of Mathematics and Natural Science. Weyl begins by emphasizing the structuralist understanding of objectivity:

Perhaps the philosophically most relevant feature of modern science is the emergence of abstract symbolic structures as the hard core of objectivity behind-as Eddington puts it-the colorful tale of the subjective storyteller mind. ²¹⁵

Among the 'simplest structures imaginable' are those which represent the combinatorics of aggregates and Weyl finds it gratifying that this 'primitive piece of symbolic mathematics' is so intimately related to philosophical issues having to do with individuality and probability, on the one hand, and fundamental physical phenomena in quantum mechanics and genetics, on the other.

Thus he begins by considering an aggregate of elements of the same kind and notes that this requires one to make the distinction we presented in Chapter 1, namely between indistinguishability, in the sense of possessing the same set of intrinsic properties, and identity per se. This raises the question of individuation, a question which, according to Weyl, Leibniz answered a priori with his Principle of Identity of Indiscernibles. An empirical answer, which is yet 'precise and compelling', is given by quantum statistics. ²¹⁶ The question of trans-temporal identity is closely related to such issues but for the case under consideration, the decision as to what is to be counted as equal and what as different influences the counting of arrangements on which the calculation of probabilities rests and hence "the problem of individuation touches the roots of the calculus of probability". ²¹⁷

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An aggregate can then be defined as a set S (whose cardinal is n, for example), on which is defined an equivalence relation such that elements which are in the same equivalence class are said to be of the same kind, or to be in the same state. Each element of S is labelled by an arbitrarily chosen distinct mark, p, and if each such element is capable of being in k classes, or states, then a definite 'individual state' of S is given if it is known, for each of the marks p, to which of the k states the element marked p belongs. ²¹⁸ This marking is subject to the 'principle of relativity' which states that the only statements and relations which have objective significance are those which are not dependent on any change in the choice of marks.

If it is known, for each of the n marks p, to which of the k classes the element marked p belongs, a definite individual state of S can be defined. However, if ". no artificial differences between elements are introduced by their labels p and merely the intrinsic differences of state are made use of", ²¹⁹ then S is characterized by its 'visible' or 'effective' state obtained by assigning to each class the number n _i of elements of the aggregate that belong to that class. Any two individual states are connected with the same effective state if and only if they are related by a permutation of the marks. In this way the above 'principle of relativity' comes to be manifested as a 'postulate of invariance' with respect to the group of permutations. Since it is the 'visible' state that is considered in quantum mechanics, it is obviously important to know how many different such states there are. This number equals that of the 'ordered' decompositions n = n ₁ + n ₂ + ·s + n _k of n into k summands n _i . And this has the value

• (3.7.1)
• 

that is, the number of ways of distributing k (P) energy elements over n (N) resonators, in Planck's expression, or of distributing n (bosonic) particles over k cells in Einstein's (from 1925).

Weyl then considers two processes the aggregate can effectively undergo. The first is a partition into two complementary sub-aggregates, where the number of distinct such partitions is 2 ⁿ . Since the elements are now discernible according to their 'kind', a sub-aggregate, S ₁ , say, can be characterized by assigning to each class C _i the number of elements, , with which that class is represented in S ₁ . has the n _i +1 values 0, 1, . n _i and hence the number of different possible effective partitions of S into such complementary sub-aggregates is (n ₁ +1) . (n _k +1).

This reaches a maximum of 2 ⁿ if all the n _i have the value 0 or 1. This in turn would correspond to a situation where no two elements of S are ever found in the same class, so one can effectively dispense with the labels. Weyl calls this a 'monomial' aggregate. ²²⁰ The second process is the inverse of partition, namely the union of two disjoint aggregates into a larger one.

So far Weyl's analysis has been conducted in entirely formal terms, but the application to quantum statistics is obvious. ²²¹ A gas of either photons or electrons can be described in terms of an aggregate as given above, since the particles have no identity ²²² and hence "[n]o specification beyond what was previously termed the effective state of an aggregate is therefore possible". ²²³ Thus the aggregates are characterized by giving, for each possible state, the number of elements (photons or electrons) in that state. The general case, as noted above, corresponds to Bose-Einstein statistics and the Fermi-Dirac form is obtained through the imposition of Pauli's Exclusion Principle, which, as Weyl notes, corresponds to the formation of a monomial aggregate (the two forms are related to the symmetry of the wave function for the aggregate in the following appendix). ²²⁴ He concludes by suggesting that "[I]n a profound and precise sense physics corroborates the Mutakallimūn (cf. footnote ²²¹); neither to the photon nor to the (positive and negative) electron can one ascribe individuality". ²²⁵

We suggest that what Weyl was trying to do in this formal analysis of the kinds of 'aggregates' suitable for quantum physics, was to arrive at a form of set

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theory appropriate for elements without identity. His 'principle of relativity' effectively allows him to 'forget' the 'nature' of the elements of S and by paying attention exclusively to the cardinality n ⁱ (i = 1,., k) of the equivalence class, he obtains the 'decomposition' n ₁ + · + n _k = n (which notably bears a certain resemblance to the occupation numbers) which, as Weyl emphasizes, is what is considered in the quantum context. What this procedure does is to effectively 'mask' the individuality of the elements of the set S. Weyl's 'aggregates' therefore constitute kinds of sets in which the elements are initially labelled, but then by focusing on what remains invariant under the action of the permutation group, the formal effect of these labels is 'wiped out'. We see this as an early attempt at a form of 'quasi-set' theory and in subsequent chapters we shall suggest a way in which Weyl's programme can be completed and the Received View underpinned with a secure formal basis. Before we do so we need to complete our version of the history of quantum statistics, since later developments in this history have been turned to in order to support an alternative to the Received View. These developments concern more general forms of quantum statistics, known as 'parastatistics', and not only did they play quite a significant role in elementary particle physics in the 1960s, but further generalized forms are still under active consideration today.