Principle of Uniformity: Keynes’ Treatise
In his Treatise on Probability Keynes is critical of the principal of uniformity, suggesting the importance complexity.
Ch. XIX The Nature of Argument by Analogy
“The law of the Uniformity of Nature appears to me to amount to an assertion … of a generalised judgment of irrelevance, namely, of the irrelevance of mere position in time and space to generalisations which have no reference to particular positions in time and space. It is in respect of such position in time or space that ‘nature’ is supposed ‘uniform.’ …”
Ch. XXI The Nature of Inductive Argument Continued
This implicitly refers to Whitehead’s Holism:
“The kind of fundamental assumption about the character of material laws, on which scientists appear commonly to act, seems to me to be much less simple than the bare principle of Uniformity. They appear to assume something much more like what mathematicians call the principle of the superposition of small effects, or, as I prefer to call it, in this connection, the atomic character of natural law. The system of the material universe must consist, if this kind of assumption 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. We do not have an invariable relation between particular bodies, but nevertheless each has on the others its own separate and invariable effect, which does not change with changing circumstances, although, of course, the total effect may be changed to almost any extent if all the other accompanying causes are different. 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.
Perhaps it has not always been realised that this atomic uniformity is in no way implied by the principle of the Uniformity of Nature. Yet there might well be quite different laws for wholes of different degrees of complexity, and laws of connection between complexes which could not be stated in terms of laws connecting individual parts. In this case natural law would be organic and not, as it is generally supposed, atomic. 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. Yet nature might still be uniform, causation sovereign, and laws timeless and absolute.
The scientist wishes, in fact, to assume that the occurrence of a phenomenon which has appeared as part of a more complex phenomenon, may be some reason for expecting it to be associated on another occasion with part of the same complex. Yet if different wholes were subject to different laws quâ 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.
We do habitually assume, I think, that the size of the atomic unit is for mental events an individual consciousness, and for material events an object small in relation to our perceptions. These considerations do not show us a way by which we can justify Induction. But they help to elucidate the kind of assumptions which we do actually make, and may serve as an introduction to what follows.”
Ch. XXII The Justification of These Methods
But in the end Keynes puts forward a kind of ‘small world’ argument:
The ultimate constituents together with the laws of necessary connection make up what I shall term the independent variety of the system.
[I]f the premisses of our argument permit us to assume that the facts or propositions, with which the argument is concerned, belong to a finite system, then probable knowledge can be validly obtained by means of an inductive argument.
The Principle of the Uniformity of Nature, as I interpret it, supplies the answer, if it is correct, to the criticism that the instances, on which generalisations are based, are all alike in being past, and that any generalisation, which is applicable to the future, must be based, for this reason, upon imperfect analogy.
We judge directly that the resemblance between instances, which consists in their being past, is in itself irrelevant, and does not supply a valid ground for impugning a generalisation.
That is, we judge that the future will be just an extension of the past.
As a logical foundation for Analogy, therefore, we seem to need some such assumption as that the amount of variety in the universe is limited in such a way that there is no one object so complex that its qualities fall into an infinite number of independent groups (i.e. groups which might exist independently as well as in conjunction) ; or rather that none of the objects about which we generalise are as complex as this ; or at least that, though some objects may be infinitely complex, we sometimes have a finite probability that an object about which we seek to generalise is not infinitely complex.
Thus, if we are seeking to reason probabilistically about a group of people, we have to suppose that some, at least, behave like automata.
Perhaps our generalisations should always run: ‘It is probable that any given φ is f.’ rather than, ‘It is probable that all φ are f.’
That is, we are more justified in saying that most φ are f than suggesting that all are, and more justified in assigning a probability to what might happen in the short run than in assigning a probability to a claim about what will always happen. Thus instead of saying that Boyle’s law is almost certainly true one might say that the predictions of Boyle’s law will almost certainly hold for the foreseeable future.
Our assumption [is] that we have a finite a priori probability in favour of …there being some limitation of independent variety … in the objects of our generalisation. Our experience might have been such as to diminish this probability a posteriori. It has, in fact, been such as to increase it. It is because there has been so much repetition and uniformity in our experience that we place great confidence in it.
This passage was presumably written before Keynes’ involvement in the Great War. It seems to me that those who experience has been quite contrary to ‘repetition and uniformity’ and who have found the hypothesis that others have some ingenuity and perhaps even craftiness might be excused for being less than confident in the hypothesis. See also Ramsey on the limitations of probabilistic reasoning.
The assumption … that the system of Nature is finite is in accordance with the analysis of the underlying assumption of scientists … .
Keynes may mean natural scientists, as against human scientists or economists, say.
Now an assumption, that all systems of fact are finite … cannot … be regarded as having absolute, universal validity … . The most which can be maintained is that this assumption is true of some systems of fact, and, further, that there are some objects about which, as soon as we understand their nature, the mind is able to apprehend directly that the assumption in question is true.
[It is] difficult … to remove from the organon of thought the inductive method which can only be based on it or on something like it.
Thus Keynes is not arguing that the hypothesis is necessarily true in all cases, but that we need something like it to support inductive or probabilistic reasoning.
We may see induction (and hence probabilistic influence) as a kind of extrapolation based on the status quo, which will be good up to the point where there is a disruptive innovation.
If we are only really justified in making probabilistic claims about ‘the foreseeable future’, then what is beyond is ‘uncertain’. There may be situations in which we cannot foresee very far, so that such uncertainty is much more significant than simple (numeric) probability.
Sometime people start from an assumption that reality is probabilistic and deduce, for example, that free will does not exist. Keynes is taking the converse approach: he has to suppose that ‘the world of interest’ is effectively ‘small’, with no creativity and hence no scope for the exercise of free will, in order to be able to perform induction.
Much of Keynes’ account makes more sense if we just think in terms of physics and chemistry, which seem to be adequately described by universal laws which scientists gradually come to approximate. In the human sciences there has been some utility in applying probabilistic reasoning, but that does not mean that the implicit assumptions are well-founded. Instead such sciences tend to use probabilistic methods to make limited extrapolations, but then to have to modify their theories substantially in the light of events. The significance of the use of probability in the two cases is thus quite different.
Some people regard inductive and probabilistic reasoning as the hall-marks of ‘being rational’, and hence start by assuming a ‘small world’. But in the light of Keynes’ subsequent work one can take a different view: that one should always be on the look out for innovation beyond one’s imagination, recognise when ‘the foreseeable future’ might be not very far away, and use less mechanistic approaches to grappling with uncertainty. Alternatively, if you don’t want a probabilistic continuation of the status quo, one should look out for the opportunities for disruptive innovation, undermining rationalities based on induction and probabilistic reasoning.