Induction and the 'atomic' universe

Starting from evidence shown in the similarity (or positive analogy) and dissimilarity (or negative analogy) observed between particular instances, one may arrive at or infer a probability of a general conclusive proposition - what is known as a method of 'inductive generalization'. It is interesting in this connection to note that Keynes explicitly states: 'An inductive argument affirms, not that a certain matter of fact is so, but that relative to certain evidence there is a probability in its favour' (TP, p. 245).

There are two kinds of hypothesis to rationalize the validity of inductive methods: the hypothesis of 'atomic uniformity' and the hypothesis of 'the limitation of independent variety'.5 The hypothesis of atomic uniformity is, it would seem, reflected in a scientific view of the universe and nature. In this regard, Keynes states

The system of the material universe must consist, if this kind of assumption [about the atomic character of natural law] is warranted, of bodies which we may term (without any implication as to their size being conveyed thereby) legal atoms, such that each of them exercises its own separate, independent, and invariable effect, a change of the total state being compounded of a number of separate changes each of which is solely due to a separate portion of the preceding state. [...] Each atom can, according to this theory, be treated as a separate cause and does not enter into different organic combinations in each of which it is regulated by different laws.

(TP, pp. 276-7)

However, what if a natural law holds an organic rather than an atomic character? Keynes would obviously respond to this question as follows:

If every configuration of the universe were subject to a separate and independent law, or if very small differences between bodies - in their shape or size, for instance - led to their obeying quite different laws, prediction would be impossible and the inductive method useless ... if different wholes were subject to different laws qua wholes and not simply on account of and in proportion to the differences of their parts, knowledge of a part could not lead, it would seem, even to presumptive or probable knowledge as to its association with other parts. Given, on the other hand, a number of legally atomic units and the laws connecting them, it would be possible to deduce their effects pro tanto without an exhaustive knowledge of all the coexisting circumstances.

(TP, pp. 277-8, emphasis added)

In short, if the parts-whole relation concerning the universe has an organic, not an atomic, character and, along with this, natural laws have organic, not atomic, features, it would be impossible to deduce probable knowledge about the system as a whole from knowledge of the parts constituting that system. In a nutshell, inductive inference would be useless. To confirm the validity of inductive methods, the universe must be composed of 'legally' atomic units and the laws connecting them must be atomic, certainly not organic. Keynes's answer is that the method of inductive generalization does not subsist without the assumption of atomic uniformity: 'The system of nature is finite' (TP, p. 290). A finite system in this sense renders inductive generalization effective. In Keynes' view, the independent variety of the universe as well should not be infinite for inductive generalization to be possible.

It should be noted that Keynes, on this point, asserts that

[T]he almost innumerable apparent properties of any given object all arise out of a finite number of generator properties, which we may call фг ф2 ф3 ... Some arise out of фг alone, some out of ф: in conjunction with ф2 , and so on. The properties which arise out of ф: alone form one group [fj; those which arise out of ф1 ф2 in conjunction form another group [f2 ] and so on. Since the number of generator properties is finite, the number of groups also is finite.

(TP, p. 282; emphasis and bracketed notation added)

In this context, a categorical statement that 'the number of generator properties is finite' plays, as it were, a definitive role. However, this is, it would appear, an a priori intuition, or rather a kind of axiom; moreover, it is not based on empirical understanding. In any event, if the number of such generator properties ф1, ф2, ... ф... is n, the number of generated groups [f1, f2, f3...] must be 2n. Thus, whenever the number of ф is finite, the number of f is necessarily finite; and therefore one never encounters infinite quantities in this field. It is perhaps in view of this effect that Ramsey in his review article of TP (Ramsey, 1999 [1922], p. 255), explicitly states that 'the Hypothesis of Limited Variety is simply equivalent to the contradictory of the Axiom of Infinity'. In such a manner, 'the system of nature is finite', that is to say, 'the hypothesis of limited independent variety' generating a finite system of nature is, in fact, the assumption sine qua non for the validity of inductive methods.6

Although, as shown above, the method of inductive generalization is based on 'the hypothesis of atomic uniformity' and 'the hypothesis of limited independent variety', these hypotheses, according to Keynes, result basically in the same conception of the system of nature. It is interesting to single out his assertion in this regard:

The hypothesis of atomic uniformity, as I have called it, while not formally equivalent to the hypothesis of the limitation of independent variety, amounts to very much the same thing. If the fundamental laws of connection changed altogether with variations, for instance, in the shape or size of bodies, or if the laws governing the behaviour of a complex had no relation whatever to the laws governing the behaviour of its parts when belonging to other complexes, there could hardly be a limitation of independent variety in the sense in which this has been defined. And, on the other hand, a limitation of independent variety seems necessarily to carry with it some degree of atomic uniformity. The underlying conception as to the character of the system of nature is in each case the same.

(TP, p. 290, emphasis added)

In sum, these two inductive hypotheses are based purely upon an atomistic conception of the system of nature and of its natural laws. Part III in TP ('Induction and Analogy') forms, it would seem, the hard core - the main theme of which is the fundamental relationship between the validity of inductive methods and Keynesian probability. There Keynes undoubtedly adopts his atomistic views about 'universe', 'nature' and 'natural laws'.

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