FROM THE LOGICAL TO THE STATISTIC PROBABILITY

IN THE SCIENTIFIC RESEARCHES

Nikolay Ivanov Petrov

Technical UniversitySofia

Abstract: The article features the ideas of an aviation engineer, specialist in reliability and management systems. These ideas have been formed on the basis of quotations of world known scientists in the fields of philosophy, mathematics and the theory of management.

The following thesis is supported: “The measurement of the probability degree for making a hypothetical conclusion is defined by the information possibility for generalization found in its contents.”

Key words: logical probability; statistic probability; information possi-bility for generalization

Introduction: The idea of probability appeared in the remote past. Then it was the characteristic of human knowledge. Besides, the existence of probable knowledge was recognized in contrast with the true (real) knowledge and the false (delusion) knowledge. As Bertrand Russel points out two scientists – skeptics, Charneades and Chlitomachus, “revolted against the belief in gods, magic and astrology, which had been gaining popularity. They developed a con-structive doctrine which defined the probability degree as a sense of  trust.”1, 2

The role of the probability theory in the structure of contemporary mathematics is essential. “Every specialist in the probability theory knows well that mathematics is actually part of the probability theory.” -   these words were the beginning of the lecture delivered by the world famous American scientist George Doob in front of the Moscow Mathematics Society in 1960.3 Of course, his words were accepted as a joke but everybody knows the old saying that “in every joke there is some truth”. It is a fact that in modern mathematics we can bear witness to an intensive development of probability theory.

Isaac Yaglom thinks that “the relative weight of the probability theory has dramatically increased in comparison with the other mathematic sciences.”3

The famous Bulgarian scientists Academic Ivan Popchev, Prof. Yordan Zapryanov and Prof. Stoyan Markov say that: “The formal theory can, not only, provide for the building of large system models, but also, because of its specific structure, strict logic of conclusion and simplicity of terms is a tool of knowledge, allowing the achievement of new results which cannot be possible within the content description.  Mathematic models as a class of sign models are actually an illustrative, concrete and simplified image of the original. If there is compliance between the model and the original,   the model can be used for experimental research, its properties can be studied and transferred to the original, and thus a theory can be build. A model is a mathematical description of the information and logical side of the formal system structure.4

ExposeThe use of probability and its distributions is extremely wide – from the every day language to all scientific researches. Together with that, the interpretation of the probability is well-founded and it shows the large scale of interaction with the fundamental models of being and knowledge.

The development of the probability idea had led to the fact that in the philosophy’s methodological literature of the 30s of 20th century were differentiated two types of probability – logical (inductive) and statistic (frequency) probability. The scientist Rudolph Carnap points out: “There exist two fundamentally different types of probability and I call the first one “statistic probability” and the other one - “logical probability”. Unfortunately, the word “probability” is used with two completely different meanings. This interpretation of the different meanings of the term “probability” is the reason for dramatic contradictions in the books of philosophic sciences. It is also the reason for philosophic discussions due to the different speculations of the scientists.5

As he mentioned his interest in probability interpretation, Karl Popper says that together with Rudolph Carnap they decided, in the 30s, “to differ sharply probability as a term used in the physics hypotheses (especially in the quantum theory) on the one hand, and on the other hand – the so called probability hypothesis or a degree of its proof (a degree of confirmation).6 The idea of inductive (logical) probability is older than the idea of statistic (frequency) probability. In the remote past the term “degree of probability” was used as a compliant characteristic of some or other hypothetical statements. A similar approach to probability is quite popular nowadays, too, especially in every day life.

In the spoken language the phrase “quite probably”, “probably”, “unlikely, improbably” and “impossible” are frequently used. They are used in describing the perspective of the development of some events – from the simple life is truth to the scientific and political prognoses. For example, it is said: “It is very probable NASA to build a space base on the Moon in 2020.”, “It is quite possible a man to land on Mars in the middle of 21st century.”, “It is absolutely improbable that the idea of trajectory has played an important role in the development of the elementary part theory”, etc. In the above mentioned cases probability is shown as a measurement for subjective confidence, focusing on different outcomes of the events. It makes it possible to express an attitude towards the reliability of particular single statements.

Such an attitude towards probability, as a characteristic of logical relations, is practically independent from the direct experience appeal. This problem is typical for the period of recognition of the probability theory and its classical stage of development. The language of such theory is formed on the basis of analogy with the separate single statements and the analyzing of the different degree of their confirmation. For the scientists of the classical period”, Rudolph Carnap says, “ probability is a degree of reliability of future events or the trust in them. That is a logical probability, not a probability in the statistic sense.5 Correspondently, an event with a degree of probability 1(one) is considered reliable, while an event with zero probability – improbable and an event with a probability degree different from zero – as a probable one with a certain degree of reliability. It is interesting that the term possibility is divided into two types. Max Bunge states: “There exist two radically different types of possibilities- conceptual and real.7  The real possibilities refer to the formal interpretation of probability, while conceptual possibilities – to the logical probability. By considering the two approaches to probability interpretation, Yan Haking discusses its dual nature.8

Inductive probability expresses the logical relation between two opinions (statements). The mutual subordination between the opinions (statements) has different character and they are mostly divided into deductive and inductive.

Deduction is a term which means that data, taken out of a batch of conclusions, are as true as the batch itself.

Induction is different from deduction as with it the state of the problem is completely different. The reliability of the inductive conclusion is always undefined, i.e. it is not a result reached by logical conclusion based on the batch, which makes it unreliable.

Here comes the question: Can the degree of reliability of the inductive hypothetical conclusion be evaluated in quantity? In the course of probability theory development such questions were frequently discussed each time the possibility to talk about a probability degree of hypothetical conclusion appeared in the study of nature objects.   In all such cases, the value of probability cannot be strictly measured or given quantity expression. Here we have to consider that the logical, i.e. inductive probability does not allow the development of   the mathematic apparatus of the probability theory.

The mathematical development of the probability theory is connected with the other type of probability – the statistic (frequency) probability in the study of phenomena. The achievements of the probability science are strongly connected with the fact that they allow the introduction of mathematical image (notion) of thinking in the study of the processes. That image is based on the development of the mathematical apparatus during the centuries. The development of the probability notion expresses mainly the strength and power of mathematics and its modern apparatus. Namely, the mathematical nature of probability defines its success. Without the probability idea of the mathematical basis of the model in use, the development of science is impossible.

The frequency approach to the interpretation of the term “probability” is defined in the process of its use and mathematical modeling.   Its defining was done in the 20s of 20th century and it is connected with the names of Robert Myses and Georg Reichenbach and nowadays with the scientistst Andrey Kolmogorov, Norbert Winer and Andrey Hinchin.

The establishment of the frequency approach corresponds in time with the transformation of the probability theory to the part of classic mathematics.  In its early years the probability theory was studied not as strict mathematics science but as a separate nature science. It was often compared to astrology and alchemy.

For the proving that the probability theory is a strict mathematics science, the scientific works of Andrey Kolmogorov had an important contribution. He stated the following: “If in 1920 we could say the probability theory was not part of mathematics, in 1936 it is not possible to say that any more.”9

The success of the mathematical formal expressions of knowledge is due to the fact that mathematics is closely connected with the nature of theoretical knowledge. It is now accepted that the probability theory has an independent value of its own. Its meaning is global and is defined by the analysis of its predicting function.  The nature of general knowledge is also important as well as the essence of its relations. The “independence” of the probability theory is expressed by the fact that mathematics exists and continues its development. Besides, it shows itself as a tool of knowledge. The development of the researches provokes the formation of new ideas, based on mathematics and its terms and notions.

As Freeman Dyson points out:Mathematics is the main source of notions and principles in science. It serves as means for the development of new nature theories.10

Mathematics is a science with an abstract structure. The laws of its functioning and development are actually a science of operations on a great number of terminable and interminable nature objects. In the study of the mathematical development of nature theories, the fact that mathematics is the main form for expressing the corresponding regularities becomes obvious.  With its help, the basic equations forming the core of the scientific theories are defined.

The main advantage of mathematics is that its abstract objects and relations express the inner organization of our knowledge and the corresponding nature processes. The historical character of the mathematic science is defined by its interaction with the other fields of knowledge.  That means that in the process of science development, a change of the mathematics disciplines which interact most closely with natural science occurs.

It is characteristic for classical mechanics to use the common classical analysis (differential and integral calculations); for classical electrodynamics of Maxwell – the vector analysis; for the theory of relativity – the tensor analysis; for quantum mechanics – the theory of Hilbert spaces; for the theory of elementary parts – the theory of the groups and the generalized functions.  In correspondence with that, the theory of probability is used in natural sciences and that theory is in constant progress, especially with the development of information and technology.

In order to understand the peculiarities of the use of mathematics in real knowledge, the problem of the sources of its development should be studied. As an example of the mathematization of physics is the fact that the development of mathematics is faster than that of physics. The mathematics’ disciplines, characteristic for most of the modern physics parts, have been “basically out-lined” independently from the corresponding physic theories.

Nikola Burbaki wrote the following: “Since the revolutionary development of modern physics started, a lot of efforts have been put in mathematics to give birth the convincing experimental truths (proofs).”11 He claims that in the development of quantum physics, the macroscopic intuition of reality was demonstrated, hidden by “microscopic” phenomena of a completely different nature. There is a need of such divisions of mathematics for the study of those phenomena which probably have not been developed to be used in the experimental study.  On the other hand, the axiom method shows that “the truths” which must be the bases of the conclusions in mathematics are in fact a particular aspect of the general concept of science development.

In order to develop new mathematical disciplines for working with creative imagination, particular mathematical models need to be attracted and applied. New mathematical models as well as entire new disciplines can appear in the process of analyzing factual material which has not been initially considered worthy. A typical example of that is the initial project of the theory of probability. It has to be pointed out that the basic language of studying this theory is the research of the ‘games of chance’. Of course, such a choice of elementary models does not contain any complicating factors and it serves as a basis for the development of new notions. It is typical not only for mathematics.

In that sense a famous example is the great “contribution” of genetics to the development of medical science and its further improvement – genetic engineering and the science of organ transplantation.

The general consideration of the nature of mathematics and its application has to be considered when the reasons for the success of the probability ideas are analyzed. The basic notions of the theory of probability  are similar to all basic mathematical notions. They carry an abstract character and, in their essence, they are away of the concrete nature of the real phenomena. The development and application of the probability theory is based on the research of more general forms and nature systems.

Including the ideas and methods of the the theory of probability in the development of the quantum theory was the reason for the establishment of the physics bases of the self-organization phenomena and then the synergetic science. That provokes the improvement of the respective concept by using probability. The abstract nature and the development of the particular methods are both reasons for the later success of the theory in question and knowledge of nature phenomena.12

The mathematical bases of the theory of subordination and the systems of high-level of organization are considered as another contribution to the theory of probability. That is taken as a base for the demonstration of the different degree of variability and versatility of the separate levels of the complex theoretical and technical systems. Having in mind that the stable levels are illustrated by generalized characteristics while the more changeable and versatile ones – by the language of the basic characteristics.

The possibility to synthesize logically different notions in a wholesome secure system is granted due to the fact that the corresponding rules are formed by existing probability distributions and their cognitive application.

Conclusions of the authors’ thoughts

The measure of the probability degree of a hypothetical conclusion is based on the information possibility found in its contents of notions and knowledge, world perspective and heuristic possibilities of mind and soul.

In the history of contemporary physics, mathematics and philosophy there has not been even a single concurrence between mathematics and the modern experimental physics.

Due to that particular “overtaking”, the strength of mathematical pro-bability theory is demonstrated and that defines its independent development.

Bibliography

1. Russel, B. History of Western Philosophy. M., 1959, p. 76, p. 257.

2. Russel, B. History of Western Philosophy. NovosibirskSiberian                   University Press, 2007, p. 301.

3. Doob,G, I.Yaglom. About some problems of modern mathematics and     Cybernetics. Moscow, 1965, p. 14-15.

4. Popchev,Iv.,Y.ZaprianovStMarkov. Hierarchical decentralized  mana-    gement systems. Technika Publishing House, S., 1985, p. 7.

5. Carnap, R. Philosophical Basis of Physics. Nauka Publishing House,     Moscow, 1971, pp. 63-81.

6. Popper,K. World’s predisposition. Two new points of view on     causality. Man and Philosophy. Part II, Nauka PH, Moscow, 1993, p.     122, p. 140, p. 142-143.

7. Bunge, M. Possibility and probability. Foundations of Probability The-              ory, Statistical Inference and Statistical Theories of Science. Vol. III,               Dordrecht-Holland, 1976, p. 31.

8. Hascing, I. The Emergence of Probability. Cambridge, 1978, p.11-17.

9. Kolmogorov,A.N. Main terms of the probability theory. 2 P.H.,     Moscow-Leningrad, 1936, p. 180.

10.Dyson,F. Mathematics and Physics. UFN Publishing House,      Moscow,1965, p.352

11.Burbaki, N. Articles on the history of Mathematics Nauka Publishing      House. Moscow, 1963, p.258.

12.Petrov,N. Probability, Independence, Information Society. Uchkov      Publishing House, Bulgaria, p. 37. Nikolay Ivanov Petrov received M. Sc. Degree from the National  University “V. Levski”, Aviation Faculty, specialty “Radio Equipping of Aircrafts”. He got Ph.D. with Doctorate Thesis “Optimizing and Control of the Technical Usage of Air Systems”. And he has defended Dissertation by Automated Systems for Information Technology and Management in the Institute for Perspective Defense Research, Military AcademySofia, Bulgaria. Since 2001 he has been working as an Assistant Professor and Associate Professor at the University “Professor D-r Assen Zlatarov”, Burgas (Bulgaria) and Thracian University, Stara Zagora, Yambol (Bulgaria). In 2004 he graduated UNWE-Sofia, subject “Economics of the Safety”. His research activities are centered on Automated Systems, Reliability and Risk of Technical Systems and Electronic Devices for Measuring.

In 2007 he awarded Professor by specialty, Atomized Systems for Treatment of Information and Control. He has more than 300 scientific works, publications and developments, 50 of which – abroad. He published 33 scientific books and textbooks, 10 of which – monographs.

# Contact:

Nikolay  Iv. Petrov,  Prof., Dr.Sc.

1784 Sofia, Poligona Str., № 8, Fl. 70

Phone: +359-2-974-46-91

E-mail: nikipetrov_1953@abv.bg