Development Of Mathematical Logic Research Articles

Forming the competence of physics and mathematics students using information technology

Relevance. The relevance of the study arises as computers in educational institutions have ceased to be a mere subject of study and have become a tool for learning. The main purpose of using computer-based learning technologies is to increase the efficiency of the learning process. This task is also solved by increasing the motivation of students to study technical graphics. Graduates should be able to use graphics systems to create both blueprint and design documentation and to solve three-dimensional graphic modelling tasks. Purpose. The purpose of the study was to identify the features of the formation of professional competence of non-humanitarian specialities through information technology. Methodology. The methodological basis of the study was general theoretical methods, through which it was possible to determine the fact that the use of information and communication technologies in the classroom allows raising the level of information culture, realising the potential of each student, and forming the professional skills of future specialists. Results. This article covered the main methods of using information technology in mathematics and physics classes, the features of the competency-based approach, the advantages of using innovative computer technology, and the types of technologies used. It has been found that the use of information technology in mathematics and physics classes affects: the activation of students' cognitive activity, development of mathematical logic, variability of reasoning, and concentration of students' intellectual activity on research and investigation. Conclusions. The practical significance of the study lies in the fact that its findings and conclusions improve the training in the development of competence of students of physics and mathematics specialities using information technology in educational activities.

IVAN SLESHYNSKY AS A POPULARIZER OF THE IDEAS OF MATHEMATICAL LOGIC IN UKRAINE

The first half of the twentieth century was marked by the simultaneous development of logic and mathematics. Logic offered the necessary means to justify the foundations of mathematics and to solve the crisis that arose in mathematics in the early twentieth century. In European science in the late nineteenth century, the ideas of symbolic logic, based on the works of J. Bull, S. Jevons and continued by C. Pierce in the United States and E. Schroeder in Germany were getting popular. The works by G. Frege and B. Russell should be considered more progressive towards the development of mathematical logic. The perspective of mathematical logic in solving the crisis of mathematics in Ukraine was noticed by Professor of Mathematics of Novorossiysk (Odesa) University Ivan Vladislavovich Sleshynsky. Sleshynsky (1854 –1931) is a Doctor of Mathematical Sciences (1893), Professor (1898) of Novorossiysk (Odesa) University. After studying at the University for two years he was a Fellow at the Department of Mathematics of Novorossiysk University, defended his master’s thesis and was sent to a scientific internship in Berlin (1881–1882), where he listened to the lectures by K. Weierstrass, L. Kronecker, E. Kummer, G. Bruns. Under the direction of K. Weierstrass he prepared a doctoral dissertation for defense. He returned to his native university in 1882, and at the same time he was a teacher of mathematics in the seminary (1882–1886), Odesa high schools (1882–1892), and taught mathematics at the Odesa Higher Women’s Courses. Having considerable achievements in the field of mathematics, in particular, Pringsheim’s Theorem (1889) proved by Sleshinsky on the conditions of convergence of continuous fractions, I. Sleshynsky drew attention to a new direction of logical science. The most significant work for the development of national mathematical logic is the translation by I. Sleshynsky from the French language “Algebra of Logic” by L. Couturat (1909). Among the most famous students of I. Sleshynsky, who studied and worked at Novorossiysk University and influenced the development of mathematical logic, one should mention E. Bunitsky and S. Shatunovsky. The second period of scientific work of I. Sleshynsky is connected with Poland. In 1911 he was invited to teach mathematical disciplines at Jagiellonian University and focused on mathematical logic. I. Sleshynsky’s report “On Traditional Logic”, delivered at the meeting of the Philosophical Society in Krakow. He developed the common belief among mathematicians that logic was not necessary for mathematics. His own experience of teaching one of the most difficult topics in higher mathematics – differential calculus, pushed him to explore logic, since the requirement of perfect mathematical proof required this. In one of his further works of this period, he noted the promising development of mathematical logic and its importance for mathematics. He claimed that for the mathematics of future he needed a new logic, which he saw in the “Principles of Mathematics” by A. Whitehead and B. Russell. Works on mathematical logic by I. Sleszynski prompted many of his students in Poland to undertake in-depth studies in this field, including T. Kotarbiński, S. Jaśkowski, V. Boreyko, and S. Zaremba. Thanks to S. Zaremba, I. Sleshynsky managed to complete the long-planned concept, a two-volume work “Theory of Proof” (1925–1929), the basis of which were lectures of Professor. The crisis period in mathematics of the early twentieth century, marked by the search for greater clarity in the very foundations of mathematical reasoning, led to the transition from the study of mathematical objects to the study of structures. The most successful means of doing this were proposed by mathematical logic. Thanks to Professor I. Sleshynsky, who succeeded in making Novorossiysk (Odesa) University a center of popularization of mathematical logic in the beginning of the twentieth century the ideas of mathematical logic in scientific environment became more popular. However, historical events prevented the ideas of mathematical logic in the domestic scientific space from the further development.

Linguistic units and their systems

AbstractA theoretical discussion of units in linguistic theory would be, in a sense, incomplete without a discussion of the systems, whether overt or implied, that the units are associated with. This paper traces conceptualizations of units and their accompanying systems in several disciplines. We identify some important problems with rule-based accounts (Parsons 1937) of social action and discuss the transition to non-rule-based theory afforded by ethnomethodology (e.g.Garfinkel 1963,1967;Heritage 1984,2011). We draw direct parallels between these issues and analogous developments in mathematical logic (Gödel 1992) and philosophy of mind (Fodor 1968,1983;Lucas 1961;Putnam 1960,1967etc.), and argue that these stem directly from fundamental properties of a class of all formal systems which permit self-reference. We argue that, since these issues are architectural in nature, linguistic theory which postulates that linguistic units are the outputs of a consistent, self-referential, rule-based formal systems (e.g.Hauser, Chomsky & Fitch 2002) will inevitably run into similar problems. This is further supported by examples from actual language use which, as a class, will elude any theoretical explanation grounded in such a system.

Logic in Poland in the Twentieth Century: An Introduction

A. Fraenkel and Y. Bar-Hillel in their classic Foundations of Set Theory (published in 1958) wrote ‘There is probably no country which has contributed, relatively to the size of its population, so much to mathematical logic and set theory and Poland. Leaving the explanation of this curious fact to sociology…’.1 However, the development of logic (and any other scientific field as well) in a given country, is not a linear function of the size of its population. In fact, there are several factors that conditioned the enormous development of mathematical logic and the foundations of mathematics in Poland, particularly in the years 1918–1939. The following circumstances can be taken into account as elements for a sociological explanation of this fact (curious or not, but that is another matter): the philosophical context (the rise of the Lvov-Warsaw School), the mathematical context (Polish Mathematical School and the Janiszewski program, the co-operation of philosophers and mathematicians in teaching and doing logic, the amount of logical teaching in secondary schools and universities, and several organizational enterprises (the place of logic at universities, journals, scientific societies).

VỀ SỰ HÌNH THÀNH, PHÁT TRIỂN CỦA LOGIC HÌNH THỨC VÀ VAI TRÒ CỦA NÓ TRONG NHẬN THỨC KHOA HỌC VÀ THỰC TIỄN ĐỜI SỐNG

VỀ SỰ HÌNH THÀNH, PHÁT TRIỂN CỦA LOGIC HÌNH THỨC VÀ VAI TRÒ CỦA NÓ TRONG NHẬN THỨC KHOA HỌC VÀ THỰC TIỄN ĐỜI SỐNG

Meta- and Object-language in Information Retrieval Research

The distinction between a meta- and an object-language has become increasingly familiar in information science, through the diffusion of the concept of metadata. A significant antecedent can be found in the development of formal logic. This paper proposes an analogous distinction for information retrieval research, between discourse about and transformations within systems. The aim is to obtain analytical clarity and economy in research effort.

On formalism freeness: Implementing Gödel's 1946 Princeton bicentennial lecture

In this paper we isolate a notion that we call “formalism freeness” from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the 20th century.

On Formalism Freeness: Implementing Gödel's 1946 Princeton Bicentennial Lecture

AbstractIn this paper we isolate a notion that we call “formalism freeness” from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability. We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic in the 20th century.

Meta‐ and object‐language for information retrieval research: proposal for a distinction

The distinction between a meta‐ and an object‐language has become increasingly familiar in information science, through the diffusion of the concept of metadata. A significant antecedent to this distinction can be found in the development of formal logic. This paper proposes an analogous distinction for information retrieval research, between the metalanguage of discourse about information retrieval systems and the object‐language of transformations within systems. In formal logic, acceptance of a meta:object distinction has had a clarifying and simplifying effect. An understanding of potential object‐language transformations as the writing, erasure, and substitution of symbols has also been developed. The existing metalanguage of information retrieval research has displayed a founding assumption (the value of delivering all, and possibly only all, the records relevant to a given query), some central concepts, entities for evaluative purposes, and derived measures. An alternative founding principle of enhanced informed choice is endorsed. The emerging view of operations within information retrieval systems such as transformation, sorting, and partitioning is strongly analogous to the more fully established account of possible object‐language transformations in formal logic. Analytical clarity has been obtained and economy in research effort is made possible.

Bertrand Russell and the introduction of mathematical logic in China

Bertrand Russell was one of the most prominent figures in the formation and development of mathematical logic. It is widely acknowledged that his work in this field exerted tremendous influence in the West, especially in the first three decades of the twentieth century. The important role he played in inspiring Chinese interest in this subject, however, is virtually unknown. This paper describes Russell's contributions to the introduction of mathematical logic in China through a discussion of his lectures in Beijing in March of 1921, and the subsequent Chinese translations of his Introduction to Mathematical Philosophy and Principia Mathematica.

The Empty Set, The Singleton, and the Ordered Pair

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.

Is a final theory conceivable?

Some physicists believe that a final theory, which will unify separate branches of theoretical physics, including quantum field theory and general relativity, is imminent. The physicists who expect such a theory typically assume that ultimate natural laws will be expressed by the same mathematical formalism which is associated with present-day physics. This view is questionable, because mathematics itself evolves and is not currently in a finished form. Historically, a succession of discoveries in physics has unveiled new laws of nature, each stage being made possible by the development of new mathematics. Evolutionary biology and interbehavioral psychology, ordinarily overlooked by the philosophy of science, reveal further inconsistencies inherent in the concept of a terminal stage of scientific discovery. Another troublesome implication of this notion has come to light recently: Finality of a conceptual scheme comprised of a finite number of natural laws is incompatible with recent developments in mathematical logic and the theory of computational irreducibility.

A survey of mathematical logic, part II: post-1931

In part I of this article [1], we described developments in mathematical logic before 1931. Gödel’s famous paper On formally undecidable propositions of Principia Mathematica and related systems, was published in January of that year. In it Gödel established two theorems:(1)Incompleteness: For any consistent theory at least as strong as arithmetic there are sentences which can neither be proved nor disproved in the theory.(2)Unprovability of consistency: The consistency of such a theory cannot be proved within the theory.We are going to outline, with minor adjustments to notation, Gödel’s actual proof of (1) given, in translation, in [2, pp. 5–38].

What is the role of model theory in the study of meaning?

One of the most important events in the development of modern linguistics was the break-through of model-theoretic semantics in the 1970s which is so intimately connected with the works of Richard Montague (1974). But ever since its first applications to linguistics, the adequacy of model theory for the study of natural language semantics has been doubted (cf. Jardine 1975; Potts 1975 and with special reference to Montague’s version of that approach Potts 1976), and although model theory is a well-established subdiscipline of mathematical logic, its proper role for the foundations of logic is as much debated as its adequacy in the study of natural language meaning. Constructivists (Kamlah and Lorenzen 1967: 1933195; Martin-Liif 1984: 3) and finitists (Hilbert and Bernays 1934: 125-131), as well as logicians who, like Hinst (1982) conceive of logical rules as based on pragmatic principles, are more often sceptical about the explanatory value of model theory as an account of meaning. Admittedly, also these critics may use model-theoretic means as technical tools in their formal work, and there is something like a constructive model theory. But what is at issue here is not the technical apparatus of model theory but its status as an explanatory account to meaning. When the eminent logician A. Mostowski (1965), in his lectures on the development of mathematical logic from 1930 to 1964, opens his lecture on the theory of models1 with the statement, ‘the modern form of semantics is the theory of models’, this seems to represent a problem as settled which is still being discussed. This is even more true for linguistics than for logic and metamathematics. So, Bickhard and Campbell’s (B&C’s) foundational article on the application

The Development of Logic in the Twentieth Century

Logic is an old science. It had developed rather slowly over a very long period; in the last one hundred years, however, it has made swift progress. That part of logic which developed rapidly in the last one hundred years is usually called modern logic. The principal part of modern logic is mathematical logic. Mathematical logic, according to its properties, is both logic and mathematics. The establishment of mathematical logic is a leap in the history of logic. Its development is essential to the study of modern mathematics. It is one of the foundation theories of computer science. The development of mathematical logic has progressed in three different directions and may be classified into three aspects.

Relativity in late Wilhelmian Germany: The appeal to a preestablished harmony between mathematics and physics

Most students of intellectual life in Wilhelmian Germany have at one time or other attempted to gauge the magnitude of their task by reading in Traugott Konstantin Oesterreich's massive bibliography and synopsis of philosophical currents. More than any other source, his text provides a measure of the bewildering profusion of competing philosophical systems that appeared around 1900.1 There one sees theories of myriad shades and complexions, bright patches of color that fade one into the other's periphery. Never before, the impression arises, had there been such a large and apparently anarchic outpouring of concern about philosophical truth and how to achieve it. In flogging his own work, every obscure professor or social malcontent seemed eager to gather in his train, Friedrich Paulsen noted in 1907, a gaggle of disciples.2 Fame and fortune rewarded those, like Wilhelm Ostwald, Ernst Haeckel, and Rudolf von Eucken, who gave the public what they wanted. From our privileged vantage point we see the superficiality of philosophical texts published at that time. Leaving aside major developments in mathematical logic, one is especially struck by the extent to which philosophers failed to arrive at formulations commensurate with achievements in other fields. Creative thinkers early in the twentieth century would have agreed: philosophy only lightly scarred the enduring accomplishments of the period. If one finds little echo of the special world systems of a Windelband, a Nietzsche, a Tonnies, or a Weber in literary fiction or scientific theories, there

The Principle of Sufficient Reason and the Development of Formal Logic

Whether or not principle of sufficient reason is a fundamental rule of formal logic is a question that merits serious discussion. In debates from as early as 1960s, when discussing subject and functions of formal logic, some comrades pointed out that formal logic cannot study just forms of thought alone. One of their basic arguments was that the principle of sufficient reason demands that content of a premise must be true. In discussions concerning truth and validity of formal logic, some comrades likewise suggested that validity of thought forms and truth of their content were identical, their reason being that one of demands of principle of sufficient reason is that a reason must be true. These discussions have already touched on question of nature of principle of sufficient reason. Thus, this discussion now under way is a continuation and deepening of past discussions on logic. Not only does it involve content of principle of sufficien...

The Principle of Sufficient Reason and the Development of Formal Logic

Whether or not the principle of sufficient reason is a fundamental rule of formal logic is a question that merits serious discussion. In debates from as early as the 1960s, when discussing the subject and functions of formal logic, some comrades pointed out that formal logic cannot study just the forms of thought alone. One of their basic arguments was that "the principle of sufficient reason demands that the content of a premise must be true." In discussions concerning the truth and validity of formal logic, some comrades likewise suggested that the validity of thought forms and the truth of their content were identical, their reason being that "one of the demands of the principle of sufficient reason is that a reason must be true." These discussions have already touched on the question of the nature of the principle of sufficient reason. Thus, this discussion now under way is a continuation and deepening of the past discussions on logic. Not only does it involve the content of the principle of sufficient reason and its role and function in formal logic, it also involves the object of study and the direction of the future development of this science, formal logic. The discussion of this problem is undoubtedly highly significant.

Development of Mathematical Logic.

Development of Mathematical Logic.

Tadeusz Kotarbiński. Introduction. Notes on the development of formal logic in Poland in the years 1900–39. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 1–14. - Jan Łukasiewicz. On the notion of possibility. English translation of the first half of 1864 by H. Hiż. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp.

Tadeusz Kotarbiński. Introduction. Notes on the development of formal logic in Poland in the years 1900–39. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp. 1–14. - Jan Łukasiewicz. On the notion of possibility. English translation of the first half of 1864 by H. Hiż. Polish logic 1920–1939, edited by Storrs McCall. The Clarendon Press, Oxford1967, pp.