Quick notes and reflections on Eugen Slutsky’s papers

These are, mainly, disjoint notes and reflections inspired by my reading of some of E. Slutsky’s papers: E.E. Slutsky (2010), Collected Statistical Papers, selected and translated by Oscar Sheynin, Berlin. (All pages referred below are from this collection. Sheynin’s notes are in brackets. Mine too, but with my initials.)

Although these notes are mostly for self clarification, I’m making them public so readers may ask for clarification, or just post their comments. My main goal here is to explore the connection between, on the one hand, mathematical statistics and, on the other hand, the materialist conception of history and the radical critique of classical political economy and modern economics. But I make lots of digressions that will test the patience of casual readers. Beware.

E.E. Slutsky (1916), “Statistics and mathematics. Review of Kaufman” (“Statistika i matematika”), in Statistichesky Vestnik, No. 3–4, 1915–1916, pp. 104–120.

  1. Slutsky rejects Kaufman’s discouraging remarks:

“It is hardly possible to master consciously the principles of the statistical
theory […] without [its] connection with the main principles of the theory of probability.” (p. 22)

This is particularly ominous:

“Thoroughly perceive the boundaries of your competence. […] In
particular, certainly abstain from mechanically applying final formulas
provided by mathematical statistics without being quite clearly aware of
their intrinsic meaning and sense, otherwise misunderstanding can often
result.” (Ibidem)

This is the “Little knowledge is worse than no knowledge” doctrine. Well, historically, the little knowledge of our ancestors led to the greater knowledge we now have, with all its constructive and destructive potential. So, little knowledge may well be, often enough, better than no knowledge at all. More importantly, this doctrine is not likely to deter any human from trying to go from ignorance to little knowledge to greater knowledge.

2. Slutsky seems to suggest here (p. 31) that, to a better way to introduce statistics is to leave aside the philosophical discussion on the definition of probability:

“I imagine that it would have been more advantageous to expound the principles of the theory of probability by examples with balls etc. making use of the most elementary concept of probability as the ratio of the favorable cases to all of them, and only to deepen this idea afterwards by indicating other possibilities.” [I fixed the punctuation. JH]

Discussing the “logical foundations” of probability is fine, but its ontological content is too controversial and subtle a discussion to be placed up and front in an introductory book.

Following Pareto, in his paper on consumer choice, he will make a similar argument regarding the concept of utility. Better to avoid the Austrian discussion on the psychological content of utility and focus on its formal properties as a relation or function between consumption and some abstract index of well-being. That is certainly the tried and true method of science: break down a complex phenomenon into its parts, and go after the “lowest hanging fruit” first. Again, little knowledge improves (marginally) on no knowledge. And soon, by Hegel’s law of transition from quantity to quality, you make leaps in understanding.

3. Slutsky’s alludes to the “methodological essence of statistics” (p. 24):

“As a method, statistics is certainly not a science, but a technique, that is, a system not of reasoning, but of tricks, rules and patterns of practical cognizing work, whether applied systematically or not, conscientiously or unconscientiously, for scientific or practical goals. Just the same as addition and subtraction remain arithmetical operations independently from who is applying them and what for.” (pp. 24–25)

This stands in contrast with what are now called the Laplacian and Bayesian doctrines of probability, say as codified by Jaynes’ (2003), Probability Theory: The Logic of Science, where probability widens its scope to become the universal method of cognition, as a opposed to a particular method.

4. Also of note is the reference to A. A. Chuprov’s view of statistics as an “ideographic” (as opposed to “nomographic”) science. This is how Oscar Sheynin, the compiler and translator of Slutsky’s papers, puts it:

“According to Chuprov (1909), who followed the German philosophers Windelband and Rickert, various sciences are either ideographic or nomographic (rather than nomothetic, as those philosophers called it). The former described reality (history), the latter studied regularity.” (p. 32)

So, the German historical school (predecessor of today’s “institutional economics” school), which emphasizes the fragmentary, supposedly irreducible diversity of social phenomena in space and time, is “ideographic,” while the Hegelian and Marxian approach, pointing out the unity that underlies the diversity of social phenomena, is “nomographic.”

5. Most fascinating to me is the choice of terms by these early theoreticians. Clearly, there’s a history that led to the adoption of today’s clunky terminology in probability theory and statistics. It is what it is. But I particularly appreciate Slutsky’s remark on statistics as a field concerned with “the general properties of totalities” (p. 25).

Here, Slutsky seems to view statistics as broader than a body of methods, rather as a scientific discipline with a substantive content, calling it:

“the doctrine of the main formal properties of totalities; then, of their quantitative and structural forms (which now constitutes an essential part of the so-called Kollektivasslehre, that is, of the doctrine of frequencies and surfaces of distribution, of means etc).” (p. 26)

The doctrine of the formal properties of totalities. The term “totality” is heavily used by Hegel and Marx. In them, it has a strong connotation of concreteness. An abstraction is a one sided view of a complex concrete phenomenon. Thus, it is almost entirely devoid of specificity and concreteness. It is spectral. Totality is all the sides together. Here, Slutsky’s views on statistics as the doctrine of the chief formal properties of totalities connects with Hegel and Marx. How do you get to grasp a totality? You investigate it. And so on…

He continues:

“Finally, here also belongs the doctrine of the machinery of causes determining the frequency of phenomena rather than of separate events. All this is not a logical doctrine of the world of judgement and concepts, but statistical doctrines of the world of phenomena in [the entirety of] their forms and mutual conditionality.” (pp. 26–27)

Note that, in Marx’s doctrines, what makes phenomena appear fragmentary in our understanding is the limit of the productive force of our own labor, which bounds our knowledge. Our knowledge is always embodied in our physical products. Thus, “uncertainty,” our relative ignorance, the negative of our stock of knowledge, is simply one side, one aspect, of our wealth, i.e. of the productive force of our labor.

Regarding “separate events” as instances of one and the same entity or process presupposes a categorization, whereby the individual is tied to its universal. The self-same qualitative identity of the phenomenon in spite of its fragmentary appearance, of its quantitative variation, is what makes it possible to frame it as a population or stochastic process.

“Whether to separate them as a special subject for elucidation and
teaching, certainly depends on our arbitrary opinion, but a special science
emerges not by arbitrariness but [because of the existence of] intrinsic ties
[of the appropriate components], cognized as something objectively
compelling, as establishing a systematic likeness and unity of the
corresponding relations as well as of the considerations expressing our
knowledge of their properties and ties [between them].” (p. 27)

6. Here’s Slutsky on probability theory:

“Calculus of probability is a purely mathematical science. How something is occurring is of no consequence to it; it deals not with factual but possible frequencies, not with their real causes but their possible probabilities. And the concept of probability itself is there quite different, is generalized and abstract. As soon as some number is arbitrarily assigned as the weight of each possible event and a number of definitions is made use of, the basis is prepared for building in a purely abstract way infinitely many purely abstract castles of combinations in the air, and of going over from those weights to the weights of various derivative possibilities (for example, of some groups of repeated occurrences of events).

“For the calculus of probability, any enrichment of the concept of probability as compared with the above is useless, it would have nothing to do with it. Throwing a bridge from that ethereal atmosphere of mathematical speculations to the region of real events is only possible by abandoning the ground of the calculus of probability and entering the route of studying the real world with its machinery of cause and effect. Only thus we obtain knowledge about the ties between frequency and probability, justify [the assumptions of] the law of large numbers and find the basis for applying the calculus of probability to studies of reality.” (p. 27)

Again, emphasis on letting abstractions fly without loading it with empirical content. There’s nothing more concrete than an apt abstraction. There’s nothing more practical than a good theory.

7. Although these notes and reflections are mainly about Slutsky, I cannot resist quoting John G. Hibben’s (1902) Hegel’s Logic: An Essay in Interpretation (pp. 11–14), in extenso on the “dialectical method”:

“In order to understand the dialectic method, the following observations
mast be carefully considered:

“The first stage, that of the so-called thesis, is designated by Hegel as the stage of the abstract understanding; the second, the antithesis, which is a representation of the incompleteness of the first by showing its obverse side, is known as that of the negative reason; the third, the synthesis, is known as the speculative stage, or that of positive reason. The terms which are here employed — the abstract understanding, the negative reason, and the positive reason — are used in a sense peculiar to Hegel. There is a fundamental distinction drawn between abstract and concrete, a distinction which runs through the entire philosophical system of Hegel. Abstract is used always in the lease of a one-sided or partial view of things. Concrete, on the other hand, is used to indicate a comprehensive view of things which includes all possible considerations as to the nature of the thing itself, its origin, and the relations which it sustains; it is the thing plus its setting.

“The first of the three Stages is referred to also as the product of the understanding (der Verstand), the second and third, as that of the negative and positive reason (die Verunft) respectively. There is evidently a distinction drawn between the understanding and the reason, Hegel does not intend to leave the impression, however, that there is a certain definite faculty of the mind which we call the understanding, and still another quite distinct which we call the reason. Such a view fails wholly to grasp his meaning. Hegel maintains that the mind works as it were upon two levels, a lower and a higher, and yet one and the same mind withal.

“Upon the lower certain considerations are overlooked which are the characteristic and essential features of the higher. Upon the lower level, that of the understanding, the mind employs one of its functions to the exclusion of the rest; namely, that of discrimination, the seeing of things in their differences, and therefore as distinct separate, and isolated, — out of relation to other things and to the unitary system which embraces them all. While, therefore, the function of the understanding may be regarded as a process of differentiation, that of the reason is essentially a process of integration. Reason is the synthetical power of thought. It is the putting of things together in their natural relations. The reason takes note, it is true, of the differences which are in the world of experience, and yet nevertheless is capable of apprehending the unity which underlies these differences. It sees things not as apart and separate, but as cohering in systems, and the distinct systems themselves as forming one all-comprehending system, the universe itself.” (pp. 11–14)

E.E. Slutsky (1939), “Autobiography” (“Zizneopisanie”), in Ekonomich Skola, vol. 5, No. 5, 1999, pp. 18–21.

1. As a young man from Yaroslavl (and Kiev and Munich and Kiev again and Moscow) was exposed to the revolutionary ideological and political ferment of the times, took part in student protests, was suspended, drafted by the army, etc. The 1905 revolution had a large impact on his formation. Here’s how he explains to Soviet bureaucrats his intellectual inclinations, perhaps in a manner that allowed him to rationalize how he came to work on economic planning and statistics:

“I very badly memorized people by sight and mistook one person for another one even if having met them several times so that I was unable to be a political figure either. A further analysis of my abilities confirmed this conclusion. I studied mathematics very well and everything came to me without great efforts. I was able to rely on the results of my work but I was slow to obtain them. A politician, a public speaker, however, needs not only the power of thought but quick and sharp reasoning as well. I diagnosed my successes and failures and thus basically determined the course of my life which I decided to devote exclusively to scientific work.

“”I became already interested in economics during my first student years in
Kiev. In Munich, it deepened and consolidated. I seriously studied Ricardo, then Marx and Lenin’s Development of Capitalism in Russia [1899], and other authors. [An editor mentions in a footnote that he also read Tugan-Baranovsky.] Upon entering the Law Faculty, I already had plans for working on the application of mathematics to economics. I only graduated from the University in 1911, at the age of 31. The year 1905–1906 [the revolutionary period] was lost since we, the students, barely studied and boycotted the examinations, and one more year was lost as well: I was expelled for that time period because of a boyish escapade. At graduation, I earned a gold medal for a composition on the subject Theory of Marginal Utility. However, having a reputation as a Red Student, I was not left at the University and [only] in 1916/1917 successfully held my examinations for becoming Master of Political Economy & Statistics at Moscow University.

“In 1911 occurred an event that determined my scientific fate. When beginning to prepare myself for the Master examinations, I had been diligently studying the theory of probability. Then, having met Professor (now, academician) A. V. Leontovich and obtaining from him his just appeared book on the Pearsonian methods, I became very much interested in them. Since his book did not contain any proofs and only explained the use of the formulas, I turned to the original memoirs and was carried away by this work. In a year, — that is, in 1912, — my book (Theory of Correlation) had appeared. It was the first Russian aid to studying the theories of the British statistical school and it received really positive appraisal.

“Owing to this book, the Kiev Commercial Institute invited me to join their staff. I worked there from January 1913 and until moving to Moscow in the beginning of 1926 as an instructor, then Docent, and, from 1920, as an Ordinary Professor. At first I took courses in mathematical statistics. Then I abandoned them and turned to economics which I considered my main speciality, and in which I had been diligently working for many years preparing contributions that remained unfinished. Because, when the capitalist economics [in the Soviet Union] had been falling to the ground, and the outlines of a planned socialist economic regime began to take shape, the foundation for those problems that interested me as an economist and mathematician disappeared. The study of the economic processes under socialism, and especially of those taking place during the transitional period, demanded knowledge of another kind and other habits of reasoning, other methods as compared with those with which I had armed myself. [What did he mean here? JH]

“As a result, the issues of mathematical statistics began to interest me, and it seemed to me that, once I return to this field and focus all my power there, I would to a larger extent benefit my mother country and the cause of the socialist transformation of social relations. After accomplishing a few works which resulted from my groping for my own sphere of research, I concentrated on generalizing the stochastic methods of the statistical treatment of observations not being mutually independent in the sense of the theory of probability.”

One can only speculate about how Slutsky’s work on statistics, probability, and economics may have contributed to the development of socialist planning in the Soviet Union under a different set of circumstances more favorable to substantive grassroots democracy.

2. Here’s Slutsky’s famous theorem, a workhorse in the theory of stochastic processes and functions thereof, as represented today in textbooks. It is in LaTeX code, sorry. [To decipher it, highlight, copy (Control/Command+C), and paste (Control/Command+V) this paragraph on MathJax:]

If $g : R^K \rightarrow R^m$ is a continuous function, then (1) $x_T \overset{p}{\to} x \Rightarrow g(x_T ) \overset{p}{\to} g(x)$, i.e. $\textrm{plim} \ g(x_T) = g(\textrm{plim} \ x_T )$, (2) $x_T \overset{d}{\to} x \Rightarrow g(xT ) \overset{d}{\to} g(x)$, and (3) $x_T \overset{a.s.}{\to} x \Rightarrow g(xT ) \overset{a.s.}{\to} g(x)$.

Basically, it says that if you have a stochastic sequence that converges to some “center of gravity,” then any statistic (e.g. an average, variance, etc.) of the sequence will converge to the average of the “center of gravity.”

E. E. Slutsky (1942), “Autobiography” (“Zizneopisanie”), Ekonomich Skola, vol. 5, No. 5, 1999, pp. 21–24

Some repetition here, but it’s worth contrasting the emphases as his perspective changed a little in 3 years.

“I wrote my student composition for which I was awarded a gold medal [his essay on marginal utility, JH] from the viewpoint of a mathematician studying political economy and I continued working in this direction for many years. However, my intended [summary?] work remained unfinished since I lost interest in its essence (mathematical justification of economics) after the very subject of study (an economic system based on private property and competition) disappeared in our country with the revolution. [This makes it seem as if the analytics of consumer choice became irrelevant to planning, but they didn’t. They became even more relevant, though the dogmatism in the socialist ranks made it virtually impossible for Slutsky and others to do independent work. JH] My main findings were published in three contributions ([6; 21; 24] in the appended list [not available]). The first of these was only noticed 20 years later and it generated a series of Anglo-American works adjoining and furthering its results. [R. G. D. Allen’s (1936), Professor Slutsky’s Theory of Consumer Choice and Hicks’ (1939), Value and Capital. JH]

“I became interested in mathematical statistics, and, more precisely, in its then new direction headed by Karl Pearson, in 1911, at the same time as in economics. The result of my studies was my book Theory of Correlation, 1912, the first systematic explication of the new theories in our country. It was greatly honored: Chuprov [xviii, § 3] published a commendable review of it and academician Markov entered it in a very short bibliography to [one of the chapters of] his Calculus of Probability.

“The period during which I had been mostly engaged in political economy had lasted to ca. 1921–1922 and only after that I definitively passed on to mathematical statistics and theory of probability. The first work [8] of this new period in which I was able to say something new was devoted to stochastic limits and asymptotes (1925). Issuing from it, I arrived at the notion of a stochastic process which was later destined to play a large role. I obtained new results, which, as I thought, could have been applied for studying many phenomena in nature. [And social phenomena as well. JH] Other contributions [22; 31; 32; 37], apart from those published in the C. r. Acad. Sci. Paris (for example, on the law of the sine limit), covering the years 1926–1934 also belong to this cycle. One of these includes a certain concept of a physical process generating stochastic processes and recently served as a point of departure for the Scandinavian [Norwegian] mathematician Frisch and for Kolmogorov. Another one, in which I developed a vast mathematical apparatus for statistically studying empirical stochastic processes, is waiting to be continued.

“”Indeed, great mathematical difficulties are connected with such investigations. They demand calculations on a large scale which can only be accomplished by means of mechanical aids the time for whose creation is apparently not yet ripe. However, an attempt should have been made, and it had embraced the next period of my work approximately covering the years 1930–1935 and thus partly overlapping the previous period.” (pp. 228-230)

This last paragraph is interesting as he anticipates the development of modern computers. It should be noted here that in 1933, the Germans (under Nazi rule) had developed a telex messaging system, the predecessor of the modern Internet. Also in the early 1930s, H. G. Wells was trying to build a protocomputer. Bell Labs, started in 1925, by the late 1930s it was getting some results, and by the 1940s it was in full swing.

* * *

I shall add these notes with a link to a piece on Slutsky published by a newsletter of the Minneapolis Fed. Enjoy:

The Meaning of Slutsky

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