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.

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.

On the theory of types

1938 ◽  
Vol 3 (4) ◽  
pp. 125-139 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Infinite Series ◽  
Type Theory ◽  
Simple Theory ◽  
Minimal Set ◽  
Logical Constants ◽  
Formal Restriction ◽  
Russell's Paradox ◽  
Logical Paradoxes ◽  
Finite Arithmetic ◽  
Logical Types

In this paper the theory of logical types will be examined, and certain departures from it will be suggested. Though the purpose of the paper is not primarily expository, an approach has been possible which presupposes no familiarity with special literature. Matters at variance with such an approach have been confined to appendices and footnotes.In the early pages the logical paradoxes will be considered—an infinite series of them, of which Russell's paradox is the first. Then Russell's simple theory of types will be formulated, in adaptation to a minimal set of logical primitives: inclusion and abstraction. Two aspects of the theory will be distinguished: an ontological doctrine and a formal restriction. It will be found that by repudiating the former we can avoid certain unnatural effects of the type theory—notably the reduplication of logical constants from type to type, and the apparent dependence of finite arithmetic upon an axiom of infinity. But the formal restriction itself has unnatural effects, which survive, even in an aggravated form, after the type ontology has been dropped. A liberalization of the formal restriction will be proposed which removes the more irksome of these anomalies.

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.

A formulation of the simple theory of types

1940 ◽  
Vol 5 (2) ◽  
pp. 56-68 ◽  
Author(s):  
Alonzo Church
Keyword(s):  
Type Theory ◽  
Point Of View ◽  
Simple Theory ◽  
And Mathematics

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.

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?

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.

Immigrant Arab Women in Chile during the Renaissance (Nahda), 1910s-1920s

1970 ◽  
pp. 47-55
Author(s):  
Sarah Limorté
Keyword(s):  
20Th Century ◽  
19Th Century ◽  
Thematic Analysis ◽  
New World ◽  
Arab Women ◽  
Arab Woman ◽  
The Subject ◽  
The 19Th Century

Levantine immigration to Chile started during the last quarter of the 19th century. This immigration, almost exclusively male at the outset, changed at the beginning of the 20th century when women started following their fathers, brothers, and husbands to the New World. Defining the role and status of the Arab woman within her community in Chile has never before been tackled in a detailed study. This article attempts to broach the subject by looking at Arabic newspapers published in Chile between 1912 and the end of the 1920s. A thematic analysis of articles dealing with the question of women or written by women, appearing in publications such as Al-Murshid, Asch-Schabibat, Al-Watan, and Oriente, will be discussed.

THE IDEA OF «COMMUNITY» IN THE GOVERNMENT CONCEPTS OF GALICIA OF THE XIX – BEGINNING OF THE XX CENTURY

2019 ◽  
pp. 15-21
Author(s):  
Liubomyr Ilyn
Keyword(s):  
Nineteenth Century ◽  
19Th Century ◽  
Practical Significance ◽  
Distinctive Features ◽  
Legal Studies ◽  
Legal Principle ◽  
General Scientific ◽  
The Government ◽  
The Right ◽  
The 19Th Century

Purpose. The purpose of the article is to analyze and systematize the views of social and political thinkers of Galicia in the 19th - beginning of the 20th centuries. on the right and manner of organizing a nation-state as a cathedral. Method. The methodology includes a set of general scientific, special legal, special historical and philosophical methods of scientific knowledge, as well as the principles of objectivity, historicism, systematic and comprehensive. The problem-chronological approach made it possible to identify the main stages of the evolution of the content of the idea of catholicity in Galicia's legal thought of the 19th century. Results. It is established that the idea of catholicity, which was borrowed from church terminology, during the nineteenth century. acquired clear legal and philosophical features that turned it into an effective principle of achieving state unity and integrity. For the Ukrainian statesmen of the 19th century. the idea of catholicity became fundamental in view of the separation of Ukrainians between the Russian and Austro-Hungarian empires. The idea of unity of Ukrainians of Galicia and the Dnieper region, formulated for the first time by the members of the Russian Trinity, underwent a long evolution and received theoretical reflection in the work of Bachynsky's «Ukraine irredenta». It is established that catholicity should be understood as a legal principle, according to which decisions are made in dialogue, by consensus, and thus able to satisfy the absolute majority of citizens of the state. For Galician Ukrainians, the principle of unity in the nineteenth century. implemented through the prism of «state» and «international» approaches. Scientific novelty. The main stages of formation and development of the idea of catholicity in the views of social and political figures of Halychyna of the XIX – beginning of the XX centuries are highlighted in the work. and highlighting the distinctive features of «national statehood» that they promoted and understood as possible in the process of unification of Ukrainian lands into one state. Practical significance. The results of the study can be used in further historical and legal studies, preparation of special courses.

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