CC BY
45
Why did Kolmogorov use a dependent requirement to probabilities?
ate conditions for personal development and overcoming the In whole as a philosophical movement formed in the
conflict in society. It is conceived as a result of the spread of connection with Christianity personalism has become close
personalistic doctrine among the people. enough to a large number of Christians, especially in the West.
References:
1. Дворецкая Е. В. Персонализм и христианство. (Религия и нравственность в секулярном мире. Материалы научной конференции. 28-30 ноября 2001 года. Санкт-Петербург. СПб. Санкт-Петербургское философское общество. 2001. С. 163 http://anthropology.ru/ru/texts/dvoretsk/secular_38.html
2. ПСРЛ. - Т. 1. Лаврентьевская летопись. - Л., 1926. 426c.
3. Mounier E. Revolution personnaliste et communautaire: http://www.uqac.uquebec.ca/zone30/Classiques_des_sciences_ sociales/index.html
4. Лега В. П. Современная западная философия. Персонализм [Электронный ресурс] http://www.sedmitza.
ru/lib/text/431862/
5. Паскаль Блез. Мысли. С-Пб. 1888. 358 с.
6. http://azbyka.ru/dictionary/15/personalizm-all.shtml
7. Достоевский Ф. М. Братья Карамазовы. - М.: Художественная литература, 1973. 816 с.
8. Hocking W. E. The Meaning of God in Human Experience https://archive.org/stream/meaningofgodinhu027626mbp /meaningofgodinhu027626mbp_djvu.txt
Reznikov Vladimir Moiseevich, Institute of Philosophy and Law of the SBRAS, Senior Researcher, Novosibirsk State University, Department of Philosophy, Associate Professor,
E-mail: [email protected]
Why did Kolmogorov use a dependent requirement to probabilities?
Abstract: An explanation due to Shafer and Vovk is studied for Kolmogorov’s using the requirement ofproximity of theoretical probability to frequencies, which can be derived on the basis of Bernoulli’s theorem and Cournot’s principle. To some extent, this explanation is subjective. Some new philosophical and formal arguments are proposed in favor of rationality of this explanation.
Keywords: probability, independence, Cournot’s principle, Bernoulli’s theorem, frequency interpretation.
In 1933 A. Kolmogorov published a book, where an axiomatic theory of probability was proposed, which was eventually accepted by the mathematical community [1]. In 1936 the book was published in Russian [2]. This small-volume monograph was not purely mathematical; it also examined some issues related to the methodology of application of probability theory. In particular, Kolmogorov formulated two statements, describing the properties of probabilities, for the events being studied in the applications of probability theory. These requirements by Kolmogorov are the following:
«A. One can be pretty much sure that if a set of conditions S is repeated a large number of times n, and m denotes the number of cases in which the event A occurred, then the ratio m/n will differ little from P(A).
B. If P(A) is very small, then one can be pretty much sure that for a single realization of the conditions S the event A will not take place» [2, 13].
In the literature, the condition B is known as the principle of A. Cournot [3]. In the case of several requirements there naturally arises the question of their compatibility. According to G. Shafer and V. Vovk, it was noted already by Kolmogorov’s contemporaries, in particular, by E. Borel, P. Levy and others
that condition A is derivable from condition B and Bernoulli’s theorem. Even though Kolmogorov formulated the requirements to probabilities informally, it is naturally assumed in the statement about derivability of one condition from the other that these conditions are given in a formal way. It should be noted that condition A is the conclusion of Bernoulli’s theorem. The reason for using condition B is that the conclusion of Bernoulli’s theorem is true on any typical sample. The fact that the conditions A and B turn out to be dependent gives raise to several questions. Why did Kolmogorov finally decide to use dependent requirements for probabilities? Why didn’t this fact appear in the center of attention of mathematicians long ago and was not thoroughly studied?
In two interesting papers Shafer and Vovk proposed several explanations of the linkedness of Kolmogorov’s requirements [4-5]. The strongest explanation is based on a special significance for Kolmogorov of requirement A, since it has a frequency nature, and Kolmogorov noted that in the application of probability theory he followed, in general, R. von Mises, the founder of the frequency interpretation. This explanation is plausible, but it is based on Kolmogorov’s preferences and, therefore, has a somewhat subjective character.
83
Section 11. Philosophy
The purpose of the work is to substantiate the assumption that the appeal to the frequency interpretation to demonstrate a special role of assertion A can be justified and has a completely objective character. The proposed demonstration of the significance of statement A in the frequency interpretation is based on the arguments of both methodological and computational nature.
In this paper a formal argument is based on the frequency interpretation of probability theory, since it is adequate for the purposes of the paper for several reasons. Firstly, Kolmogorov pointed out to the adequacy of the frequency interpretation of probability theory in the context of applications. Secondly, the mathematicians, actively engaged in the applications, highly appreciate the role of von Mises’ concept in the practice of scientific research. For example, V. Tutubalin writes: “It is now believed that von Mises’ approach describes the properties of real phenomena, to which the mathematical theory of probability is applicable” [6, 14]. And then he continues: “at the
same time, Kolmogorov’s approach creates a very convenient scheme, on the basis of which there is developed a mathematical theory, which is formally independent from any application” [ibid].
The best known adaptation of von Mises’ theory for applied theories was carried out by Yu. Alimov [7]. Alimov’s approach is not about a new mathematics, but about a specific relationship between the world of experience with mathematics. A fundamental problem of applied statistics is to obtain the stable frequency estimates of a studied random variable. The role of mathematics reduces to determining stable averaged characteristics of another random variable, if the operator connecting these variables is known. In this article, following the works ofvon Mises, Alimov and Tutubalin, we assume that all theoretical probabilities are unknown and determined on the basis of the frequency estimates. In the beginning we give a methodological analysis of Kolmogorov’s requirements and Bernoulli’s theorem.
Methodological analysis of statement A
From the methodological positions, in the framework of frequency interpretation, statement A by itself, regardless of Bernoulli’s theorem, has a well-founded, natural-science character. There exist two approaches to the correct verification of statement A. In the first approach, the empirical evaluation of the theoretical probability based on stable frequency
Methodological analysis of Cournot’s principle
characteristics and the qualitative assessment of verification of requirement A are assumed. The second approach involves the empirical determination of the theoretical probability and the use of frequency interpretation of Bernoulli’s theorem to verify the statement.
A critical analysis of Cournot’s principle is offered. Firstly, a characteristic feature of low-probability events is that they occur rarely, but they can occur as a result of any test, including the very first experiment. In a sense, this is not consistent with the principle of Cournot, prohibiting realization of a low-probability event in a single test. Secondly, realization of low-probability events is a manifestation of chance. Cournot’s principle prohibits realization of low-probability events, and thereby limits the manifestation of chance, though in science the realizations of extremely improbable events are known. Thirdly, Cournot’s principle is in the foundations of statistical hypothesis testing. According to Cournot’s principle, if an
unlikely event occurs under the assumption of the hypothesis being correct, then the hypothesis is rejected. It is believed that the correct hypotheses are rarely rejected on the basis of this principle; however, there are data structures where the application of Cournot’s principle in the case of unlikely event realization leads to systematic errors. Here is a famous example demonstrating the shortcomings of the methodology of hypotheses testing, which uses the principle of Cournot. If John is an American, then it is unlikely that he is a member of the US Congress. However, John is a member of the US Congress, so the correct hypothesis of his being an American is not confirmed [8, 32-33].
Methodological analysis ofJ. Bernoulli’s theorem
Bernoulli's theorem. There were conducted n independent trials of event A, and this event occurred in m experiments. It is known that the theoretical probability of event A occurring in each experiment is equal to p (A), m/n is the frequency of the event A, e is the calculation accuracy. Then, for the infinite number of experiments, the following equality holds:
lim P(
m
---p( A)
n
<s) = 1
(1)
cance is overestimated. Firstly, in the context of empirical interpretation, theoretical quantities are never known, while in Bernoulli’s theorem a theoretical quantity, the probability of success, is known. Secondly, in the framework of empirical interpretation, one can talk about the external probability P in the theorem only in the case of existence of stable estimates for the theoretical probability p (A). Furthermore, as will be shown later, verification of requirement A on the basis of definition of the external probability P requires much more calculations than on the basis of stable estimates for the probability p (a).
Bernoulli's theorem has played an outstanding role in the development of probability theory as a theoretical science and its development. However, its epistemological signifi-
Frequency interpretation of Bernoulli's theorem
The frequency interpretation of the theorem is modeled on results. One series of experiments consists of n tosses, k is the
the basis of the following experiment. The experiment consists planned number of series of experiments; thus one needs to
of tossing a regular coin and is considered successful if the head carry out kxn experiments altogether. Note that in each series of
n——k>
84
Why did Kolmogorov use a dependent requirement to probabilities?
tests one frequency characteristic of the event A is determined. Obviously, k frequencies will be obtained by carrying out k series of experiments. In the proposed experiment, the theoretical probability p (A) is determined by the frequency characteristics of the event under study; namely, if the observed frequencies are really close to each other, i. e. fall within an interval of the length which is less than or equal to the observation accuracy s, then we will take any of the frequencies as the probability. If the frequency characteristics are highly variable, then no good estimation of the frequency exists.
In the case when, under a large number of trials, the overwhelming majority of the frequency estimates belong to a distinguished interval of the length less than s, then, by virtue of this, verification is carried out of statement A without using the conclusion of Bernoulli's theorem. Then, the evaluation of the external probability P does not have any particular significance, since it is determined on the basis of the obtained estimate of the probability p (A) and requires additional calculations. However, in general, the estimation of the external probability matters. There are different approaches for determining the sought-for probability.
In the standard theory of probability the derivation of Bernoulli's theorem on the basis of the local theorem of A. Moivre -P. Laplace presupposes a large number of experiments. Even more experiments are necessary for obtaining this theorem on the basis of P. Chebyshev’s inequality. In order not to depend on these requirements, we will calculate the external probability in Bernoulli’s theorem on the basis of the frequency approach.
In the purely frequency-based experiment, the external probability P in Bernoulli’s theorem is defined by analogy with the probability p (A) and on its basis. Now let us carry out an analysis of the required number of tests to evaluate the external probability P. The external probability is calculated on the basis of the frequencies w, with which the following inequality holds:
m
---P( A)
n
< £
(2)
Note that, although finding the frequency characteristics w for the event, defined by expression (2), is more difficult
than determining the frequency of the event A, the determination of these frequencies w within Bernoulli's theorem presupposes conducting the same experiment, on the basis of which the frequencies of the event A are determined. Then, according to formula (2), the frequencies w will be calculated based on the frequencies of occurrence of the event A and the already known estimate of probability of the event A. Suppose that the i-th series of n observations has been held and a frequency mi/n of the event A has been found, then, knowing p (A) and s, verification is carried out of fulfilment of the inequality (2). Previously, carrying out each series of n experiments provided obtaining the frequency of the event A, now
the same series of experiments leads only to getting a singular estimate, defining a single satisfiability or non-satisfiability of the second inequality. If we would like to determine the frequency characteristics w with the same precision as the frequency of the event A was determined, i. e., on the basis of n evidences, then it is necessary to get each frequency w also on the basis n singular characteristics describing the satisfiability of inequality (2). Therefore, in order to obtain n evidences determining the satisfiability of(2), it is necessary to conduct n series of experiments, where each series consists of n tosses of a coin; thus, to obtain one frequency characteristic w, it is necessary to carry out n2 experiments. Since the probability p (A) was determined on the basis of k frequency characteristics mi/n, i=1, k, then, to determine the probability of inequality (2) being satisfied also on the basis of k frequency characteristics, kxn2 experiments are assumed to be conducted. And for each new estimate of probabilities based on the proposed experiment, n times more trials are required.
However, in reality, to obtain the frequency of a more
complex event defined by the expression (2), it is necessary that the corresponding series of experiments consisted of N observations, where N >> n, and also one will have to conduct a lot more series of observations K, where K >> k. Here, the expression X >> Y means that X is much more than Y. As a result, instead of nxk observations to determine the probability p (A), it will be necessary to carry out NxK observations in order to obtain the probability P, where NxK>>kxn2.
Thus, a rational explanation is proposed for Kolmogorov’s choice of the dependent condition A. Firstly, within the frequency interpretation, requirement A, as an independent condition, turns out to be correct in comparison with Cournot’s principle and Bernoulli's theorem, the requirement being
the conclusion of the latter. Secondly, within the framework of the proposed computational experiment, the theoretical probability of success, used in statement A, is determined on the basis of stable frequency estimates for this probability; whereas, verification of statement A is carried out qualitatively, using the length of the interval for the frequency estimates of theoretical probability. In the case of verification of statement A on the basis of the frequency interpretation of the external probability in Bernoulli’s theorem, there are needed much more trials in comparison with the qualitative approach to verification of statement A.
In our opinion, the lack of explanations of the linkedness of Kolmogorov’s requirements prior to the publications by Shafer and Vovk is due to two reasons. Firstly, this is connected to the unpopularity of Kolmogorovs conditions in modern literature, so we do not know any books or monographs, where both requirements of Kolmogorov are presented, with the exception of the well-known textbook of H. Cramer [9]. Secondly, it seems that these requirements are perceived informally.
85
Section 11. Philosophy
References:
1. Kolmogoroff A. Grundbegrife der Wahrscheinlichkeitsrechnung. - Berlin: Springer, 1933.
2. Kolmogorov A. N. Basic Concepts of Probability Theory. - Moscow: Nauka, 1974. (In Russian).
3. Cournot A. Exposition de la theorie des chances et des probabilites. - Paris: Hachette, 1843.
4. Shafer G., Vovk V. The origins and legacy of Kolmogorov’s Grundbegriffe. URL: http://www.probabilityandfinance. com/articles/04.pdf (Access date: 22.05.2014)
5. Shafer G., Vovk V. The Sources of Kolmogorov’s Grundbegriffe//Statistical Science. 2006. Vol. 21, no. 1. P. 70-98.
6. Tutubalin V. N. Theory of Probability. - Moscow: Akademiya, 2008. (In Russian)
7. Alimov Yu. I. An alternative to the Method of Mathematical Statistics. - Moscow: Znanie, 1980. (In Russian).
8. Fidler F. From statistical significance to effect estimation: Statistical reform in psychology, medicine and ecology. Department of History and Philosophy of Science. The University of Melbourne, 2005. [Electronic resource]. URL: http://www. botany.unimelb.edu.au/envisci/docs/fidler/fidlerphd_aug06.pdf (Access date: 22.05.2014).
9. Cramer H. Mathematical Methods of Statistics. - Princeton: Princeton University Press, 1946.
Saliy Anatoliy Vladimirovich, Candidate of Philosophy, docent Higher state educational establishment of Ukraine "Ukrainian medical stomatological academy", Poltava, Ukraine
E-mail: [email protected]
Lev Shestov and Etienne Gilson: a dialogue against the background of the middle ages
Abstract: The article is devoted to analysis of the invisible dialogue between the two greatest thinkers of the twentieth century — Lev Shestov and Etienne Gilson. This is a conversation on the topic of the middle Ages: its problems and its features.
Keywords: Shestov, Gilson, neo-Thomism, faith, truth.
Lev Shestov’s interest in the medieval philosophy was determined by the fact that the major themes of his creativity were closely associated with problems of medieval philosophical thought. In this context, it is necessary to mention first of all the issues of cognition, the relationship between faith and reason, salvation, the problem of good and evil, etc. Lev Shestov elaborated his final understanding of the medieval thought after the acquaintance with the works of Adolf von Harnack and Etienne Gilson. And if one considers the issue of synthesis of Greek wisdom and Christianity, it does not matter for Shestov how it happens: whether there is the process of Hellinization of Christianity (Harnack), or Christianization of Hellenism (Gilson). One cannot distinguish any comprehensive and objective study of the medieval philosophy performed by Shestov. He never set up such problems. In the works of thinkers of one or another era, Shestov sought answers to his deepest questions, and saw the themes of consideration in which he was interested himself.
It is important that St. Anselm’s “credo ut intelligam” is not acceptable for Shestov. After all, this statement manifests that faith and reason cannot contradict each other; that reason leads to realization of the necessity of faith. By contrast, Shestov indicates that we certainly can realize something, but understanding will not add faith as such. Faith, for Shestov, is an insight which cannot be prepared or planned [4. p. 39-40].
And if the neo-Thomist position does not try to delimit the competence of faith and reason, seeking their complementarity, then Shestov sharply distinguishes them, and even opposes faith to reason. And if one considers the debate of Etienne Gilson and Lev Shestov in the context of problems of the Russian philosophy, Gilson’s views are quite similar to the positions of V. Solovyov and S. Trubetskoy [1. p.136].
The undisguised truth does not rely on anything, it does not prove anything, it is not justified to anyone; and yet in our mind it becomes a justified, proven and self-evident truth [5. p. 214].
Shestov interprets “The Spirit of Medieval Philosophy” and asserts that its central idea is as follows: the more true philosophy becomes a philosophy, the more it becomes Christian.
Shestov is impressed by Gilson’s objectivity in issues which are key for him, namely, in the question which the French thinker raises: is the Christian philosophy possible without the existence of Greek philosophy [5.p. 217]. Gilson supports Gilson in terms of the idea that not only the Middle Ages, but also modern people have inherited the basic philosophical problems, rational principles and all the techniques of our thinking from the Greeks [5. p. 220-221].
Shestov, nevertheless, is a thinker of one idea, and in Gilson’s works he focuses on those issues that have always
86
