Reviewing the Reviewer's of Keynes's a Treatise on Probability: Ignorance Is Bliss

By Michael Brady

The standard view of the economics profession is that Keynes was a brilliant, intuitive, nonrigorous innovator. These essays show that Keynes backed up his intuitions with a rigorous mathematical and logical supporting analysis, which has been overlooked.

• Language: English
• Publisher: Xlibris US
• Release date: Sep 24, 2016
• ISBN: 9781524544898

Michael Brady

About the Author Michael Emmett Brady received his PhD degree in economics from the University of California. He received his BA and MA degrees from California State University as well as completing all requirements for a BA in mathematics. He has taught mathematics courses and graduate level courses in business statistics, operation management, production management, and mathematical economics.

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Reviewing the Reviewer’s of Keynes’s A Treatise on Probability

Ignorance is Bliss

MICHAEL BRADY

Copyright © 2016 by Michael Brady.

Library of Congress Control Number:  2016915699

ISBN:   Hardcover   978-1-5245-4491-1

             Softcover     978-1-5245-4490-4

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Rev. date: 09/24/2016

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Contents

Introduction

Essay 1 – Bertrand Russell on Keynes’s A Treatise on Probability: What one would expect from the Twentieth Century’s Greatest Philosopher

Essay 2 – F. Y.Edgeworth on J M Keynes’s A Treatise on Probability: The Mind and Journal

of the Royal Statistical Society Reviews

Essay 3 – C D Broad’s Review of J M Keynes’s A Treatise on Probability and the Role of William Ernest Johnson in that book

Essay 4 – E. B. Wilson on Keynes on Probability

Essay 5 – On Fisher’s Review of J M Keynes’s A Treatise on Probability‐A Fiasco

Essay 6 – A Study of Ramsey’s Extremely Poor Reading of Chapter III of J. M. Keynes’s A Treatise on Probability and the Refutations Made by J M Keynes and Bertrand Russell

Essay 7 – J M Keynes, like Benoit Mandelbrot, was right. They (Econometricians, Statisticians,) do not Know What They are Doing

Essay 8 – Raymond Pearl’s Review of J M Keynes’s A treatise on Probability: Another Fiasco

Essay 9 – Harold Jeffreys on J M Keynes’s A Treatise on Probability: Overlooking George Boole is a fatal error

Essay 10 – F. P. Ramsey’s 1922 Cambridge Magazine review of J M Keynes’s A Treatise on Probability (1921): An Intellectual Quagmire

Essay 11 – Arne Fisher’s Review of J M Keynes’s A Treatise on Probability: Fiasco Number Three

Essay 12 – Richard E. Braithwaite on J M Keynes’s A Treatise on Probability and Logical Theory of Probability: Ignorance is Bliss1

Bibliography

Introduction

The Edgeworth –Wilson Correspondence on the A Treatise on Probability (1921): What it tells us about current, academic assessments of that Work

Michael Emmett Brady,

Lecturer,

College of Business Administration and Public Policy,

Department of Operations Management,

California State University, Dominguez Hills,

Carson, California,

USA

Abstract-The Edgeworth – Wilson exchanges in 1923 reveal that neither Francis Ysidro Edgeworth, one of the top ten economists of all time in world history, nor Edwin Wilson, one of the very top, American, applied mathematicians in the USA in the first half of the twentieth century, had any idea about how to deal with Part II of J M Keynes’s A Treatise on Probability (1921). The same conclusion holds for Ronald Fisher, who Stephen Stigler claims was the greatest statistician of the twentieth century. Richard Dawkins claims he was also the greatest biologist since Darwin. In fact, it is Benoit Mandelbrot, by a longshot, who was the greatest statistician of the 20 th cen tury.

Part II of the A Treatise on Probability includes Keynes’s formal analysis of his Boolean, non additive, nonlinear, indeterminate, interval valued approach to logical probability, based on the use of upper and lower limits or bounds.

This inability to deal with Part II of the A Treatise on Probability would explain why the Keynesian Fundamentalists (the Robinsons, R Skidelsky, P Davidson, GLS Shackle, J Runde, S Dow, R O’Donnell, A Carabelli, T Lawson, and hundreds of other Post Keynesian and Institutionalist economists) have completely failed to grasp the interval valued approach of Keynes. They simply lacked the necessary training in mathematical logic, probability, and statistics needed to comprehend what Keynes was doing. The result is a gigantic, intellectual mess based on (a) their misbelief that Keynes’s theory is an ordinal theory that can only be applied some of the time because of their confusion regarding unknown and indeterminate probabilities and (b) their inability to find the very large number of errors made in F P Ramsey’s two very poor reviews of Keynes’s A Treatise on Probability. Again, Ramsey was simply overwhelmed by Part II of the A Treatise on Probability and could not follow Keynes’s analysis of interval valued probability. Ramsey’s claims about Keynes’s mysterious non numerical probabilities not obeying the laws of the calculus of probability and the Stohs-Garner discussions about what is a non numerical probability? are the foundation for the Fundamentalist position of Robert Skidelsky that Keynes’s approach was an ordinal approach that could only be applied some of the time. Nothing is further from the truth.

Section 1. Introduction

Section 2 will cover the Edgeworth –Wilson correspondence. This correspondence demonstrates that neither Edgeworth or Wilson had mastered Part II of the TP. Edgeworth, but not Wilson, had figured out from chapter 3 that Keynes’s non numerical probabilities were intervals. I will show that Wilson did not figure this out until 1934,when he wrote a disguised, second review of the TP while pretending to be dealing with a problem from Boole. Neither Edgeworth or Wilson understood the many intellectual gaffes that were contained in the reviews of Ronald Fisher, Raymond Pearl, or Arne Fisher. Section 3 will demonstrate how the ignorance about Part II of Keynes’s A Treatise on probability (TP,1921) spread over time and led to the fiasco in the time period 1980-2016 about the confusion concerning Keynes’s unknown probabilities and his indeterminate probabilities made by all Post Keynesian and Institutionalist economists. Section 4 will conclude the Introduction to the first volume of my Reviewing the Reviewers by briefly summarizing each essay in the book and rating the author with a grad e from A to F.

Section 2. The Importance of the Edgeworth-Wilson correspondence

The great importance of the Edgeworth –Wilson exchange over Keynes’s TP in 1923 stems from the candid admission on the part of both writers that they could not follow /understand what Keynes was doing in Part II of the TP. This inability to grasp Keynes’s fundamental logical analysis is exactly the same problem underlying all academic work, with the exception of myself and my co-authors, that has appeared in the economics and philosophy literature since the late 1970’s.

Edgeworth starts the exchange with the following statement:

Let me suggest your giving an opinion of the value of Keynes’ new symbolism. I do not feel that I had mastered it sufficiently to criticize it… (Mirowski (ed.),1994,p.433)

Of course, Edgeworth wrote to the wrong mathematician. Wilson was an applied mathematician and not a mathematical logician. The correct persons to have written to for an explanation were named Bertrand Russell and Alfred North Whitehead.

Wilson’s response was the following:

"As a matter of fact I am not much of a philosopher. That is one reason I should not have undertaken to review Keynes’s book for the Bulletin of the American Mathematical Society. I have no adequate background for judging the validity of the author’s philosophical discussion. On the other hand probably nobody in the world is better qualified to discuss Keynes’ book in its entirety than you are …" (Mirowski (ed.).1994,p.433)

Later, Wilson states that

It seemed to me that Keynes’ symbolism was a natural extension and application to the case in hand of the sort of symbolism used by Boole, C.S. Peirce, Schroeder, Busseli and Whitehead, without becoming as involved as the symbolism of the last two and of Peano… (Mirowski (ed.) .1994,p.435).

Wilson is correct here also, although it is Bertrand Russell and Boole who Keynes is adapting in order to provide a interval values probabilistic foundation for inductive logic.

Continuing, Wilson states that

To date Keynes’s treatment of probability seems to be the clearest cut attempt to put the whole matter on a strictly logical foundation but I am not at all sure that this part of his work is really well done. I am not myself a specialist in postulate theory… (Mirowski (ed.) .1994,p.435).

This misses the mark. Wilson needed merely to have written to Bertrand Russell, the world’s leading mathematical logician at that time, to find out how well done this part of the book was.

We can summarize at this point about the exchange between Wilson and Edgeworth. Wilson is certainly correct that he made a mistake in trying to evaluate Keynes’s book when he had no knowledge about the application by Keynes of Boolean algebra and logic to probability theory in Part II of the TP.

However, now Wilson makes the only dubious statement in his exchange with Edgeworth:

To me the most damaging aspect of the whole matter is found on page 427 of Keynes’ book in which he begs for generosity on the part of his readers… (Mirowski (ed.) .1994,p.435)

The statement above really doesn’t make any sense.

Another eleven years passed by before Wilson published his disguised and camouflaged, second review of the TP in 1934. Wilson had finally figured out what it was that Keynes was doing based on Boole’s analysis in chapters 16-21 of the Laws of Thought (Boole,1854). Wilson, after eleven years, had figured out that Keynes, like Boole, was applying an interval valued approach to probability using indeterminate probabilities. However, Wilson, in his article, then claimed that it was very easy to figure out what Boole and Keynes were doing. The only conclusion that can be reached is either that Wilson was lying to Edgeworth in 1923 or that he was lying in his 1934 article.

Another problem is that Wilson’s solution to one version of Boole’s Challenge problem in his 1934 article is presented while, at the same time, Wilson covers up the fact that Keynes’s chapter 15 solution of Boole’s Challenge problem on pp.162-163,which is superior to his own, is never mentioned. Wilson’s solution was based on the book by Yule, that used approximation techniques to solve Boole type problems, that Keynes recommended to the reader in a footnote on p.161 of the TP.

Finally, Wilson only grudgingly acknowledges Keynes’s chapter 17 demonstration, where Keynes used his own, original, conditional probability approach to solving a number of Boole’s problems. The only conclusion that can be drawn is that either Wilson had severe memory problems and completely forgot what he had written to Edgeworth eleven years earlier or that he was trying to cover up from the reader of his 1934 article the severe deficiencies he had made in 1923.

We should note, however, that by 1934 Wilson’s knowledge of Keynes’s approach was vastly superior to any Post Keynesian or Institutionalist economist, like Shackle, Robinson, Galbraith, Runde, Davidson, Skidelsky, S. Dow, O’Donnell, Lawson, Gillies or Carabelli, etc. There is no Post Keynesian or Institutionalist economist or philosopher alive today who had Wilson’s technical understanding of Keynes’s Boolean technique. Unfortunately, Wilson mars his article at the end of it by inquiring about how can uncertainty be a problem since statisticians always have all the relevant data and information available to solve their problems. This is, of course, an implicit claim that the weight of the evidence, w, equals 1,so that the problems become determinate and there is no uncertainty unless the problem has been mis-specified by statistical amateurs who have inadvertently forgot to obtain all of the necessary, statistical data to make the problem complete.

There are brief comments at the end of his exchange with Edgeworth indicating that Wilson substantially disagreed with the Arne Fisher and Raymond Pearl reviews that I have demonstrated in this book are completely worthless intellectually. Wilson is certainly correct here.

Wilson is also correct that the Edgeworth’s reviews of Keynes’s book were the most accurate overall. One can also mention Bertrand Russell’s review and Broad’s review. These reviews are the only A reviews. I give Wilson’s combined 1923 and 1934 reviews a C+ grade, a D in 1923 and an A- in 1934.

It is not possible to understand what Keynes is doing in Part III of the TP unless the reader has understood Part II of the TP. In Part I, Keynes supplies the reader with enough examples and illustrations so that the reader of chapter III could conclude that Keynes’s non numerical probabilities are interval valued probabilities.

Keynes also made it crystal clear that his principle of indifference is very different from Laplace’s approach based on unknown probabilities in chapters IV and VI.

Section 3) Why the Keynesian Fundamentalist view of Keynes’s TP approach being an ordinal theory makes no sense

The total and complete failure of the forty years of work done by Keynesian Fundamentalists from 1976-2016 on Keynes’s A Treatise on Probability revolves around the same problem that confronted R. Fisher, F. Y. Edgeworth and E. Wilson in 1923-what is it that Keynes was doing in Part II of the TP? The Keynesian fundamentalist response of Skidelsky, Runde, Carabelli, Dow, and O’Donnell has been to simply skip Part II and Part III, which is built on Part II. They then read their own subjective beliefs into chapter III of the TP. This is a recipe for intellectual chaos, since each Keynesian Fundamentalist brings his own beliefs about what he thinks Keynes must have meant to the table. This is the familiar What did Keynes really mean? approach.

The Keynesian Fundamentalists do not even have the knowledge of chapter 26 of the TP that Bertrand Russell had, which was that Keynes’s analysis in chapter 26 had to depend on some type of approximation ,interval valued probability technique. Wilson knew more about Keynes’s interval valued approach in1934 then any Post Keynesian in the year 2016,although he did not understand why uncertainty was a problem for a decision maker.

Consider the following confusion over unknown probabilities and indeterminate probabilities of the Keynesian Fundamentalists:

"In his Treatise on Probability (1921), Keynes set out an alternative: the landscape of chance. First, there is cardinal or measurable probability, e.g. ‘There is a one in six chance of your house catching fire in the next year’. This frequency view of probability partly derives from games of chance, partly from invariable connections found in the natural, and some parts of the human, world. ‘In actual reasoning… exact measures [of this kind] will occur comparatively seldom’, Keynes wrote². Second, is ordinal probability, in which we have some evidential basis for believing that something is more or less likely to occur without being able to attach numbers to ‘more’ or ‘less’. Most risk assessments used by non-financial firms are based on this informal procedure. However, there is a residual category of ‘unknown probabilities’, in which our evidence is too scanty even to say that something is ‘more likely than not, or less likely than not, or as likely as not’.³ For Keynes, probability is the hypothesis on which it is reasonable for us to act in conditions of limited knowledge.⁴ There is no presumption that our knowledge will be sufficient to give us calculable probabilities.

Here is Keynes’s canonical statement, from a 1937 essay:

By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of dice is not subject, in this sense to uncertainty; nor is the prospect of a Victory bond being drawn…." (Skidelsky, 2010,p.8).

The crucial error of Skidelsky, which he has repeated time and again since 1982, is his claim that

However, there is a residual category of ‘unknown probabilities’, in which our evidence is too scanty even to say that something is ‘more likely than not, or less likely than not, or as likely as not’.³ For Keynes, probability is the hypothesis on which it is reasonable for us to act in conditions of limited knowledge.⁴ There is no presumption that our knowledge will be sufficient to give us calculable probabilities.

The third category is indeterminate probabilities, which are interval estimates. If the intervals overlap, then the probabilities are non comparable.

Consider the same, exact, identical error made by O’Donnell since 1982:

….only a very restricted class of probabilities can be given numerical or cardinal magnitudes in Keynes’s theory. The vast majority of probabilities are non-numerical in nature, although subsets of these are ordinally comparable. Quantitatively, probabilities are classified into three subsets:

(i)  A very small set of numerical probabilities, only occurring in situations of mutually exclusive and exhaustive alternatives when Keynes’s principle of indifference holds…

(ii) A large set of non-numerical probabilities which can only be ordinally ranked.

(iii) A large set of non-numerical probabilities to which no quantitative ranking (ordinal or cardinal) can be applied." (O’Donnell,2014,p.128)

O’Donnell’s (iii) category is actually Keynes’s indeterminate, interval valued probabilities that are non comparable if there is any overlap in the intervals.

Consider the second assessment by O’Donnell:

"The measurement of probabilities is an intriguing aspect of Keynes’s theory. While other theories reduce probabilities to numerical (and hence universally comparable form), Keynes’s orderings of the probability space is far more complex. Three types of comparative relations are postulated:

Cardinal comparison, which generates the relatively minor class of numerical probabilities. These only exist under the restrictive condition of equi-probability, which is established by careful use of the ‘principle of indifference’.

Ordinal comparison, which generates the much bigger class of non-numerical probabilities. This class consists of many separate, incommensurable series whereby probabilities belonging to the same series are comparable in terms of greater or lesser, but probabilities belonging to different series are generally incapable of being compared in magnitude.

Non-comparability, which typically exists between numerical and non-numerical probabilities,