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

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.

Hilbert's Programs: 1917–1922

Bulletin of Symbolic Logic ◽

10.2307/421139 ◽

1999 ◽

Vol 5 (1) ◽

pp. 1-44 ◽

Cited By ~ 50

Author(s):

Wilfried Sieg

Keyword(s):

Mathematical Logic ◽

19Th Century ◽

Proof Theory ◽

Foundations Of Mathematics ◽

Radical Constructivism ◽

Standard View ◽

Modern Logic ◽

Philosophical Reflection ◽

Lecture Courses ◽

The 19Th Century

AbstractHilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work.

Historical Face of Number Theory(ists) at the Turn of the 19th Century

Trends in Mathematics - Diophantine Analysis ◽

10.1007/978-3-319-48817-2_4 ◽

2016 ◽

pp. 175-229

Author(s):

Nicola Oswald

Keyword(s):

Number Theory ◽

19Th Century ◽

The 19Th Century

Introducing Turing’s mathematics

The Turing Guide ◽

10.1093/oso/9780198747826.003.0048 ◽

2017 ◽

Author(s):

Robin Whitty ◽

Robin Wilson

Keyword(s):

Probability Theory ◽

Mathematical Logic ◽

Number Theory ◽

Group Theory ◽

Statistical Analysis ◽

Numerical Analysis ◽

Riemann Hypothesis ◽

Large Scale ◽

Prime Numbers ◽

The Central Limit Theorem

Alan Turing’s mathematical interests were deep and wide-ranging. From the beginning of his career in Cambridge he was involved with probability theory, algebra (the theory of groups), mathematical logic, and number theory. Prime numbers and the celebrated Riemann hypothesis continued to preoccupy him until the end of his life. As a mathematician, and as a scientist generally, Turing was enthusiastically omnivorous. His collected mathematical works comprise thirteen papers, not all published during his lifetime, as well as the preface from his Cambridge Fellowship dissertation; these cover group theory, probability theory, number theory (analytic and elementary), and numerical analysis. This broad swathe of work is the focus of this chapter. But Turing did much else that was mathematical in nature, notably in the fields of logic, cryptanalysis, and biology, and that work is described in more detail elsewhere in this book. To be representative of Turing’s mathematical talents is a more realistic aim than to be encyclopaedic. Group theory and number theory were recurring preoccupations for Turing, even during wartime; they are represented in this chapter by his work on the word problem and the Riemann hypothesis, respectively. A third preoccupation was with methods of statistical analysis: Turing’s work in this area was integral to his wartime contribution to signals intelligence. I. J. Good, who worked with Turing at Bletchley Park, has provided an authoritative account of this work, updated in the Collected Works. By contrast, Turing’s proof of the central limit theorem from probability theory, which earned him his Cambridge Fellowship, is less well known: he quickly discovered that the theorem had already been demonstrated, the work was never published, and his interest in it was swiftly superseded by questions in mathematical logic. Nevertheless, this was Turing’s first substantial investigation, the first demonstration of his powers, and was certainly influential in his approach to codebreaking, so it makes a fitting first topic for this chapter. Turing’s single paper on numerical analysis, published in 1948, is not described in detail here. It concerned the potential for errors to propagate and accumulate during large-scale computations; as with everything that Turing wrote in relation to computation it was pioneering, forward-looking, and conceptually sound. There was also, incidentally, an appreciation in this paper of the need for statistical analysis, again harking back to Turing’s earliest work.

19th Century Logic Between Philosophy and Mathematics

Bulletin of Symbolic Logic ◽

10.2307/421117 ◽

1999 ◽

Vol 5 (4) ◽

pp. 433-450 ◽

Cited By ~ 14

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?

MATHEMATICAL PRACTICE AND MATHEMATICAL LOGIC: TO THE HISTORY OF RELATIONSHIP

10.47850/rl.2021.2.4.93-99 ◽

2021 ◽

pp. 93-99

Author(s):

Aleksandr Khlebalin

Keyword(s):

Mathematical Logic ◽

19Th Century ◽

Mathematical Practice ◽

Standards Of Proof ◽

History Of ◽

Epistemological Analysis ◽

The 19Th Century

The article annuls the role of practice in the development of mathematics in the 19th century in the formation of mathematical logic. It is shown that the revolutionary transformations of mathematics of the 19th century, which led to an increase in the abstractness of mathematical theories and concepts, was accompanied by an increase in uncertainty regarding the standards of proof, which led to the universal spread of anxiety (J. Gray) as an element of mathematical practice. It is argued that this element of practice was one of the sources of the emergence of mathematical logic, which claims to give rigor and accuracy to mathematics. The article argues that the socio- epistemological analysis of the practice of mathematics and the formation of mathematical logic will clarify the specifics of the development of relations between mathematics and mathematical logic.

Grammatical Descriptions in Manchu Grammar Books from the Qing Dynasty

Histoire Épistémologie Langage ◽

10.1051/hel/2019004 ◽

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.

Homeopathy on trial: prospective clinical studies in the 19th century

Allgemeine Homöopathische Zeitung ◽

10.1055/s-2005-868625 ◽

2005 ◽

Vol 250 (02) ◽

Author(s):

ME Dean

Keyword(s):

19Th Century ◽

Clinical Studies ◽

The 19Th Century