scholarly journals Evolution of teaching the probability theory based on textbook by V. P. Ermakov

2021 ◽  
Vol 11 (2) ◽  
pp. 300-314
Author(s):  
Tetiana Malovichko
Keyword(s):  
Probability Theory ◽  
19Th Century ◽  
Mathematical Expectation ◽  
Random Variable ◽  
Total Probability ◽  
Creative Development ◽  
Educational Literature ◽  
And Mathematics ◽  
Kyiv Polytechnic Institute ◽  
The 19Th Century

The paper is devoted to the study of what changes the course of the probability theory has undergone from the end of the 19th century to our time based on the analysis of The Theory of Probabilities textbook by Vasyl P. Ermakov published in 1878. In order to show the competence of the author of this textbook, his biography and creative development of V. P. Ermakov, a famous mathematician, Corresponding Member of the St. Petersburg Academy of Sciences, have been briefly reviewed. He worked at the Department of Pure Mathematics at Kyiv University, where he received the title of Honored Professor, headed the Department of Higher Mathematics at the Kyiv Polytechnic Institute, published the Journal of Elementary Mathematics, and he was one of the founders of the Kyiv Physics and Mathematics Society. The paper contains a comparative analysis of The Probability Theory textbook and modern educational literature. V. P. Ermakov's textbook uses only the classical definition of probability. It does not contain such concepts as a random variable, distribution function, however, it uses mathematical expectation. V. P. Ermakov insists on excluding the concept of moral expectation accepted in the science of that time from the probability theory. The textbook consists of a preface, five chapters, a synopsis containing the statements of the main results, and a collection of tasks with solutions and instructions. The first chapter deals with combinatorics, the presentation of which does not differ much from its modern one. The second chapter introduces the concepts of event and probability. Although operations on events have been not considered at all; the probabilities of intersecting and combining events have been discussed. However, the above rule for calculating the probability of combining events is generally incorrect for compatible events. The third chapter is devoted to events during repeated tests, mathematical expectation and contains Bernoulli's theorem, from which the law of large numbers follows. The next chapter discusses conditional probabilities, the simplest version of the conditional mathematical expectation, the total probability formula and the Bayesian formula (in modern terminology). The last chapter is devoted to the Jordan method and its applications. This method is not found in modern educational literature. From the above, we can conclude that the probability theory has made significant progress since the end of the 19th century. Basic concepts are formulated more rigorously; research methods have developed significantly; new sections have appeared.

Types in Logic and Mathematics Before 1940

2002 ◽  
Vol 8 (2) ◽  
pp. 185-245 ◽  
Author(s):  
Fairouz Kamareddine ◽  
Twan Laan ◽  
Rob Nederpelt
Keyword(s):  
19Th Century ◽  
Type Theory ◽  
Simple Theory ◽  
Formal Systems ◽  
Modern Notation ◽  
And Mathematics ◽  
Logical Paradoxes ◽  
The 19Th Century

AbstractIn this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910–1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's Grundgesetze der Arithmetik for which Russell applied his famous paradox and this led him to introduce the first theory of types, the Ramified Type Theory (RTT). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Church's own simply typed λ-calculus (λ→C) and we finish by comparing RTT, STT and λ→C.

Presocratic Philosophy

Classics ◽  
2013 ◽  
Author(s):  
Richard D. McKirahan
Keyword(s):  
19Th Century ◽  
Social Practices ◽  
Greek Philosophy ◽  
Beliefs And Practices ◽  
Presocratic Philosophy ◽  
History Of ◽  
And Mathematics ◽  
The 19Th Century ◽  
First Time ◽  
Source Materials

The word “Presocratic” was invented in the 19th century ce and does not represent a category recognized in antiquity. The expression “Presocratic philosophy” is misleading: first, because some “Presocratics” were Socrates’ contemporaries, some of them surviving him by decades, and second, because they did not call themselves philosophers and because the fields of inquiry they practiced extend far beyond what we think of as philosophy. Nevertheless, the label “Presocratic” is commonly applied to the intellectual figures of the 6th and 5th centuries bce (and a few that lived into the 4th) who dwelt in the Greek-speaking lands from what is now coastal Turkey to Sicily and who are included in this bibliography. Evidence of the influence of Presocratic thought on other areas of culture than philosophy is found in texts ranging from historical and rhetorical works to tragedy and comedy and beyond, to the Hippocratic medical writings and the Derveni Papyrus. Since no original texts of the Presocratics survive entirely, our knowledge of them is based on quotations (“fragments”) from their works and on reports (“testimonia”) about their views, lives, and writings in other authors whose works have been transmitted. Presocratic philosophy is the earliest phase of Greek philosophy; Plato and Aristotle were strongly influenced by the Presocratics and recognized them as their intellectual predecessors. The subsequent interest in the Presocratics in antiquity and in consequence our knowledge of them is largely due to Aristotle. In more recent times, systematic study of them began in the 19th century. Diels’s Doxographi Graeci (Diels 1879, cited under Source Criticism) for the first time permitted a rational reconstruction of much of the testimonial material, and Die Fragmente der Vorsokratiker (Diels and Kranz 1952, cited under Collections of Source Materials; first published in 1903) provided a collection of fragments and testimonia that brought the study of the Presocratics within the range of students and nonspecialist scholars of philosophy, classics, and the history of science. The study of “Presocratic philosophy” has traditionally extended to more subjects than we commonly consider philosophical. It includes topics not only in method, logic, metaphysics, epistemology, ethics, cognition, cosmology, and “psychology”—here meaning views about the nature of the psuchē (frequently translated “soul”)—but also examines connections with science and mathematics, and a variety of social practices. Recently this tendency has further expanded to include religious and mystical beliefs and practices, while by no means excluding the philosophical and scientific aspects of Presocratic thought, which remain the dominant topics of research.

Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory Probability Theory.

1994 ◽  
Vol 101 (4) ◽  
pp. 369
Author(s):  
Karen Hunger Parshall ◽  
A. N. Kolmogorov ◽  
A. P. Yushkevich
Keyword(s):  
Probability Theory ◽  
Mathematical Logic ◽  
Number Theory ◽  
19Th Century ◽  
The 19Th Century

19th Century Logic Between Philosophy and Mathematics

1999 ◽  
Vol 5 (4) ◽  
pp. 433-450 ◽  
Author(s):  
Volker Peckhaus
Keyword(s):  
Mathematical Logic ◽  
19Th Century ◽  
Early 20Th Century ◽  
Symbolic Logic ◽  
Modern Logic ◽  
History Of ◽  
And Mathematics ◽  
Correct Reasoning ◽  
The 19Th Century ◽  
The Relationship

AbstractThe history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, although logic is evidently one of the basic disciplines of philosophy. One needs only to recall some of the standard 19th century definitions of logic as, e.g., the art and science of reasoning (Whateley) or as giving the normative rules of correct reasoning (Herbart).In the paper the relationship between the philosophical and the mathematical development of logic will be discussed. Answers to the following questions will be provided:1. What were the reasons for the philosophers' lack of interest in formal logic?2. What were the reasons for the mathematicians' interest in logic?3. What did “logic reform” mean in the 19th century? Were the systems of mathematical logic initially regarded as contributions to a reform of logic?4. Was mathematical logic regarded as art, as science or as both?

On Choquet-Pettis Expectation of Banach-Valued Functions: A Counter Example

2018 ◽  
Vol 26 (02) ◽  
pp. 255-259 ◽  
Author(s):  
Hamzeh Agahi ◽  
Radko Mesiar
Keyword(s):  
Probability Theory ◽  
Mathematical Expectation ◽  
Choquet Integral ◽  
Random Variable ◽  
Pettis Integral ◽  
Counter Example ◽  
Correct Definition ◽  
Previous Definition ◽  
Choquet Expectation ◽  
Definition Of

In probability theory, mathematical expectation of a random variable is very important. Choquet expectation (integral), as a generalization of mathematical expectation, is a powerful tool in various areas, mainly in generalized probability theory and decision theory. In vector spaces, combining Choquet expectation and Pettis integral has led to a challenging and an interesting subject for researchers. In this paper, we indicate and discuss a failure in the previous definition of Choquet-Pettis integral of Banach space-valued functions. To obtain a correct definition of Choquet-Pettis integral, an open problem concerning the linearity of the Choquet integral is stated.

Grammatical Descriptions in Manchu Grammar Books from the Qing Dynasty

2019 ◽  
Vol 41 (1) ◽  
pp. 39-55
Author(s):  
Takashi Takekoshi
Keyword(s):  
19Th Century ◽  
Qing Dynasty ◽  
The Qing Dynasty ◽  
Chinese Scholars ◽  
High Degree ◽  
The 19Th Century

In this paper, we analyse features of the grammatical descriptions in Manchu grammar books from the Qing Dynasty. Manchu grammar books exemplify how Chinese scholars gave Chinese names to grammatical concepts in Manchu such as case, conjugation, and derivation which exist in agglutinating languages but not in isolating languages. A thorough examination reveals that Chinese scholarly understanding of Manchu grammar at the time had attained a high degree of sophistication. We conclude that the reason they did not apply modern grammatical concepts until the end of the 19th century was not a lack of ability but because the object of their grammatical descriptions was Chinese, a typical isolating language.

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