# On J M Keynes's Rejection of the Moscow School of Probability's Limiting Frequency Approach to Probability and Kolmogorov's Axiom of Additivity (Countable Additivity): Non-Additivity was the Fundamental, Basic Axiom upon Which All of the Economics of Keynes was Built.

Scholedge International Journal of Management & Development. Vol. 02, Issue 11 (2015), pp.7-20.

28 Pages Posted: 29 Oct 2015 Last revised: 20 Jan 2016

Date Written: October 28, 2015

### Abstract

J M Keynes was an acknowledged, world renown, and internationally recognized expert in probability and statistics in the 1930’s based on his A Treatise on Probability (1921). Keynes had been selected by statistics journals to serve as a referee during the 1930’s. It is, therefore, no surprise that he was selected as the referee by the League of Nations to review Jan Tinbergen’s work on business cycles that used an econometrics approach based on The Law of Large Numbers, the Central Limit Theorem, and the Gaussian(Normal) Distribution. The fundamental axiom used by Tinbergen was additivity.

Kolmogorov and the Moscow School of Probability’s main innovation was to go from the axiom of additivity to the axiom of countable additivity. However, Keynes rejected additivity except in the special case that the weight of the evidence, w, which measured the relative completeness of the evidence, defined on the closed unit interval [0,1], equaled 1, approached 1, or approximated 1. Keynes also accepted goodness of fit tests, such as the Lexis-Q test, and exploratory data analysis as evidence that could be used to support using a particular probability distribution.

Keynes also rejected countable additivity for the same reasons he rejected additivity, in general. Keynes’s theory of probability and decision making is based on the interval valued approach of G. Boole, who also rejected additivity except as a limiting case. Thus, non-additivity and non-linearity, or imperfect and incomplete information, used as a shorthand description for economists, are the fundamental axioms on which the economics of Keynes, and Adam Smith before him, are based.

Ergodicity is based on a measure function, which is based on the Kolmogorov and the Moscow School of Probability’s Axiom of Countable Additivity. Therefore, it is obvious that Keynes would reject ergodicity (and non ergodicity) as general results unless the researcher provided extensive exploratory data analysis and goodness of fit tests supporting their claims. Paul Davidson and the Post Keynesian School have never supplied a shred of evidence to support their claims. The only group of researchers to do this are the econophysics followers of Benoit Mandelbrot and Nicholas Nassim Taleb, who were the first to supply empirical evidence to support their analysis.

Therefore, it is easy to completely reject the claims of Paul Davidson, made numerous times since 1983, that Keynes was unaware of the Kolmogorov-Moscow School of probability’s main result, which was to move from the axiom of additivity to the axiom of countable additivity, and then on to the concept of ergodicity, based on the definition of a measurable function.

Keynes’s main axiom was non additivity, which leads directly to nonlinearity. Keynes’s conclusions regarding liquidity preference, his rejection of the conclusion of gross substitutability, and his rejection of the meaningfulness of the ergodic-non ergodic distinction in stochastic process theory, are all based on non additivity and non linearity. All classical and neoclassical results are special cases based on additivity and/or countable additivity that lead to the linearity of the normal distribution. There are no missing axioms from Keynes’s work.

Paul Davidson has thus erred continuously over the last 33 years in confusing axioms with conclusions and claiming, without a shred of evidence, historical or otherwise, that Keynes was completely ignorant of the Kolmogorov-Moscow School of probability in the 1930’s. Keynes would never have been selected as the reviewer of Tinbergen’s work by the League of Nations if that was thought to be the case.

The fundamental axioms of neoclassical economics are additivity and linearity. From these axioms, and no other axioms, one can derive the conclusions of the neutrality of money, gross substitutability, and ergodicity. One could use the assumption of perfect and complete information for all decision makers as a shorthand description of the Neoclassical position. All neoclassical schools base their analysis on additivity and linearity. Therefore, Neoclassical economics is a special, limiting case of Keynes’s more general theory based on the axioms of non-additivity and non-linearity.

**Keywords:** additivity, nonadditivity, axioms, linearity, non linearity, Kolmogorov, ergodic, nonergodic, Keynes, Smith

**JEL Classification:** B10, B12, B14, B20, B22

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