[8] Keynes's conception of this generalised notion of probability is that it is a strictly logical relation between evidence and hypothesis, a degree of partial implication.
[9][notes 4] In a 1922 review, Bertrand Russell, the co-author of Principia Mathematica, called it "undoubtedly the most important work on probability that has appeared for a very long time," and said that the "book as a whole is one which it is impossible to praise too highly.
"[17] [notes 5] With recent developments in machine learning to enable 'artificial intelligence' and behavioural economics the need for a logical approach that neither assumes some unattainable 'objectivity' nor relies on the subjective views of its designers or policy-makers has become more appreciated, and there has been a renewed interest in Keynes's work.
[41] According to Whitehead Chapter 12 'The Definition and Axioms of Inference and Probability' 'has the great merit that accompanies good symbolism, that essential points which without it are subtle and easily lost sight of, with it become simple and obvious.
The very certainty and ease by which he is enabled to solve difficult questions and to detect ambiguities and errors inn the work of his predecessors exemplifies and at the same time almost conceals that advance which he has made.
[42]Chapter 14 'The Fundamental Theorems of Probable Inference' gives the main results on the addition, multiplication independence and relevance of conditional probabilities, leading up to an exposition of the 'Inverse principle' (now known as Bayes Rule) incorporating some previously unpublished work from W. E. Johnson correcting some common text-book errors in formulation and fallacies in interpretation, including 'the fallacy of the middle term'.
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.
This notes that Francis Bacon and John Stuart Mill had implicitly made assumptions similar to those Keynes criticised above, but that nevertheless their arguments provide useful insights.
In Chapter 28 'The Law of Great Numbers' Keynes attributes to Poisson the view that 'in the long ... each class of events does eventually occur in a definite proportion of cases.
He notes a significant limitation of conventional statistical methods, as then used: Where there is no stability at all and the frequencies are chaotic, the resulting series can be described as 'non-statistical.'
The latter belongs to Inductive Correlation or Statistical Induction, an attempt at the logical analysis of which must be my final task.His final paragraph reveals Keynes views on the significance of his findings, based on the then conventional view of classical science as traditionally understood at Cambridge: In laying the foundations of the subject of Probability, I have departed a good deal from the conception of it which governed the minds of Laplace and Quetelet and has dominated through their influence the thought of the past century, though I believe that Leibniz and Hume might have read what I have written with sympathy.
But in taking leave of Probability, I should like to say that, in my judgment; the practical usefulness of those modes of inference, here termed Universal and Statistical Induction, on the validity of which the boasted knowledge of modern science depends, can only exist and I do not now pause to inquire again whether such an argument must be circular if the universe of phenomena does in fact present those peculiar characteristics of atomism and limited variety which appear more and more clearly as the ultimate result to which material science is tending ….
If the contemporary doctrines of Biology and Physics remain tenable, we may have a remarkable, if undeserved, justification of some of the methods of the traditional Calculus of Probabilities.
[notes 20]The above assumptions of non-organic ‘characteristics of atomism and limited variety’ and hence the applicability of the then conventional statistical methods was not long to remain credible, even for the natural sciences,[58][59][60] and some economists, notably in the US, applied some of his ideas in the interwar years,[61][62] although some philosophers continued to find it 'very puzzling indeed'.
[67][68] Keynes developed this point in his more well-known General Theory of Employment, Interest and Money and subsequently, specifically in his thinking on the nature and role of long-term expectation in economics,[69] notably on Animal spirits.
[74][notes 24] The significance of 'true' uncertainty beyond mere precise probabilities had already been highlighted by Frank Knight[76] and the additional insights of Keynes tended to be overlooked.
[79] But subsequently there was a partial 'return of the master'[3] leading to calls for a 'paradigm shift' building further on Keynes' insights into 'the nature of behaviour under conditions of uncertainty'.
The fundamental uncertainty proposed in both works has then deeply influenced the development of economic and probability theory in the past century and it still resonates with our lives today, considering the ups and downs that the world economy is experiencing.However it has often been regarded as more philosophical in nature despite extensive mathematical formulations and its implications for practice.