scholarly journals BOUT GEORGE BOOLE’S CALCULUS OF LOGIC

2021 ◽  
pp. 267-275
Author(s):  
Анатолий Пушкарский
Keyword(s):  
Theory Of Mind ◽  
Mathematical Analysis ◽  
Philosophy Of Mathematics ◽  
Psychological Theory ◽  
Logical System ◽  
Mathematical Basis ◽  
Philosophical Foundations ◽  
Linear Differential ◽  
And Mathematics ◽  
Philosophical Doctrine

Проводится разбор и анализ небольшой статьи Джорджа Буля, опубликованной на следующий год после выхода его знаменитой работы «Математический анализ логики». Исследуется вопрос о том, насколько ее можно рассматривать как адекватный и репрезентативный обзор его логической концепции. Также исследуются математические, экстралогические и философские основания логики Буля. Указывается, что математическим основанием его логической системы послужили его исследования в области линейных дифференциальных уравнений. Кроме того, в основании его пионерских открытий лежали произвольные философско-методологическим допущения, которые напрямую не относились к области логики и математики. Такими допущениями следует признать некоторую оригинальную психологическую теорию сознания, его философию математики, предполагающую существование универсального математического исчисления, и некоторые положения логико-философской концепции И. Канта In the present paper, the small article by George Boole, published the next year after the publication of his famous work The Mathematical Analysis of Logic, is analyzed. We discuss to what extent it can be considered as an adequate and representative overview of his logical views. The mathematical, extralogical and philosophical foundations of Boole’s logic are explored. It is pointed out that the mathematical basis of his logical system was his research in the field of linear differential equations. Moreover, his pioneering discoveries were based on arbitrary philosophical and methodological assumptions that were not directly related to the field of logic and mathematics. Some of the original psychological theory of mind, its philosophy of mathematics, which presupposes the existence of a universal mathematical calculus, and some provisions of Immanuel Kant’s logico-philosophical doctrine should be recognized as such assumptions.

Control Reversal Effects on Swept-Back Wings

1949 ◽  
Vol 1 (1) ◽  
pp. 3-34
Author(s):  
Haydn Templeton
Keyword(s):  
Mathematical Analysis ◽  
Quantitative Estimation ◽  
Dynamic Condition ◽  
Loss Of Control ◽  
General Introduction ◽  
Mathematical Basis ◽  
Control Power ◽  
Physical Picture ◽  
The Subject ◽  
Definition Of

SummaryAileron reversal effects on swept-back wings in general and elevon reversal effects on tailless swept-back wings in particular are discussed on a non-mathematical basis, attention being confined to the orthodox flap type of control. The main purpose of the paper is to convey in the simplest terms possible a clear physical picture of the conditions producing loss of control power, emphasis being naturally laid upon the part played by structural wing distortion. Certain qualitative features relating to the two phenomena are also discussed. As a general introduction to the discussion on aileron reversal effects, the definition of “aileron power” in relation to the actual dynamic condition of rolling is described at some length. For elevon reversal effects on tailless aircraft the effect of wing flexibility on both “elevon power” and on trim in steady symmetric flight is considered. With the descriptive treatment adopted the analysis is of necessity broad and general but is designed to appeal to those not too familiar with the subject. The results of certain calculations on a hypothetical wing, which may be of interest, are included. A mathematical analysis for the quantitative estimation of both aileron and elevon reversal effects is given in the Appendix.

scholarly journals A Bird's-Eye View: Core Knowledge in Wild North Island Robins of New Zealand

2021 ◽  
Author(s):  
◽  
Alexis Garland
Keyword(s):  
Cognitive Abilities ◽  
Ecological Validity ◽  
Psychological Theory ◽  
Building Blocks ◽  
Gaze Direction ◽  
Core Knowledge ◽  
Human Infants ◽  
Evolutionary Context ◽  
In The Wild ◽  
And Mathematics

<p>A prominent psychological theory on early cognitive development is Spelke’s Core Knowledge (CK) hypothesis (Spelke, Breinlinger, Macomber, & Jacobson, 1992), which posits that human infants, and possibly other species, are guided by innate understandings of how object movements, classification and quantification are governed by physical laws and, further, how agents are capable of perceptions and purposive action. CK is a set of cognitive building blocks, which serve as the foundation for more complex cognition such as acquisition and use of symbol systems pertaining to language and mathematics (Spelke, 2000). Evidence points to four core systems of knowledge: representation of number, object, space (or geometry) and agency. Investigation of spontaneous CK in nonhuman species in the wild is fundamental to understanding the ecological validity and evolutionary context for a set of systems that is proposed to be universally embedded. The bold, inquisitive manner, naïve fearlessness and unique insect caching behaviour of wild North Island robins (Petroica longipes) presents a unique opportunity to identify and characterise CK in a new model system. Six studies were conducted with the aim of investigating core developmental cognition in robins. The first three studies focused on perception of numerical quantity. Study 1 investigated the ability to discriminate between both large and small quantities, finding that robins successfully discriminate between unusually large quantities independent of ratio. Study 2 explored quantity discrimination in which summation of items is spatially distributed across an array, and found that while robins perform successfully with small numbers, the task presented substantially more cognitive demand. Study 3 measured robins’ reactions to computation by presenting simple addition and subtraction problems in a Violation of Expectancy (VoE) paradigm, finding that robins search longer when presented with a mathematically incongruent scenario. The last three studies focused on perception of agency. Study 4 investigated robins’ response to gaze direction in humans in a competitive paradigm, and found that they were sensitive to human gaze direction in all conditions but one. Study 5 explored perception of physical capability in humans, and results indicated that limb visibility significantly influences pilfering choice. Study 6 examined robins’ perception of animacy in prey, finding that in a VoE paradigm, robins’ expectation of hidden prey continuity varies depending on mobility and animacy. Taken together, the results of these six studies suggest that while supportive of fundamental characteristics defining basic Core Knowledge in many ways, some unique results in the cognitive abilities of this biologically naïve species shed new light on our growing understanding of the shared basis of cognition. A deeper look at avian performance in core developmental tasks, especially in a naïve wild population, can offer new insights into sweeping evolutionary theories that underpin basic cognitive mechanisms.</p>

scholarly journals PERKEMBANGAN MATEMATIKA DALAM FILSAFAT DAN ALIRAN FORMALISME YANG TERKANDUNG DALAM FILSAFAT MATEMATIKA

Sepren ◽  
2021 ◽  
Vol 2 (2) ◽  
pp. 17-22
Author(s):  
Robin Tarigan
Keyword(s):  
Learning Outcomes ◽  
Historical Perspective ◽  
Philosophy Of Mathematics ◽  
Long Time ◽  
And Mathematics ◽  
The Relationship

Philosophy of mathematics does not add a number of new mathematical theorems or theories, so a philosophy of mathematics is not mathematics. The philosophy of mathematics is an area of ​​reflection about mathematics. After studying for a long time, one needs to reflect on learning outcomes by reflecting on the philosophy of mathematics. Mathematics and philosophy are closely related, compared to other sciences. The reason is that philosophy is the base for studying science and mathematics is the mother of all sciences. There are also those who think that philosophy and mathematics are the mother of all existing knowledge. From a historical perspective, the relationship between philosophy and mathematics underwent a very striking development. This article discusses the development of mathematics in philosophy and the flow of formalism contained in the philosophy of mathematics in particular

Mathematics Based on Fuzzy Logic

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Mathematical Analysis ◽  
Classical Logic ◽  
Mathematical Reasoning ◽  
Narrow Sense ◽  
And Mathematics ◽  
Truth Degrees

Mathematical reasoning is governed by the laws of classical logic, based on the principle of bivalence. With the acceptance of intermediate truth degrees, the situation changed substantially. This chapter begins with a characterization of mathematics based on fuzzy logic, an identification of principal issues of its development, and an outline of this development. It then examines the role of fuzzy logic in the narrow sense for developing mathematics based on fuzzy logic and the main approaches developed toward its foundations. Next, some selected areas of mathematics based on fuzzy logic are presented, such as the theory of sets and relations, algebra, topology, quantities and mathematical analysis, probability, and geometry. The chapter concludes by examining various semantic questions regarding fuzzy logic and mathematics based on it.

Paul Benacerraf and Hilary Putnam. Introduction. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, 1964, pp. 1–27. - Rudolf Carnap. The logicist foundations of mathematics. English translation of 3528 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 31–41. - Arend Heyting. The intuitionist foundations of mathematics. English translation of 3856 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 42–49. - Johann von Neumann. The formalist foundations of mathematics. English translation of 2998 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 50–54. - Arend Heyting. Disputation. A reprint of pages 1-12 (the first chapter) and parts of the bibliography of XXI 367. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 55–65. - L. E. J. Brouwer. Intuitionism and formalism. A reprint of 1557. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 66–77. - L. E. J. Brouwer. Consciousness, philosophy, and mathematics. A reprint of pages 1243-1249 of XIV 132. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 78–84. - Gottlob Frege. The concept of number. English translation of pages 67-104, 115-119, of 495 (1884 edn.) by Michael S. Mahoney. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 85–112. - Bertrand Russell. Selections from Introduction to mathematical philosophy. A reprint of pages 1-19, 194-206, of 11126 (1st edn., 1919). Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 113–133. - David Hilbert. On the infinite. English translation of 10813 by Erna Putnam and Gerald E. Massey. Philosophy of mathematics, Selected readings, edited by Paul Benacerraf and Hilary Putnam, Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, pp. 134–151.

1969 ◽  
Vol 34 (1) ◽  
pp. 107-110
Author(s):  
Alec Fisher
Keyword(s):  
New Jersey ◽  
English Translation ◽  
Philosophy Of Mathematics ◽  
Bertrand Russell ◽  
Hilary Putnam ◽  
Foundations Of Mathematics ◽  
Rudolf Carnap ◽  
Von Neumann ◽  
David Hilbert ◽  
And Mathematics

Zen, Mathematics, and Rāmānujan: Uncommon Links

2016 ◽  
Vol 4 (1) ◽  
pp. 25-47
Author(s):  
Masato Mitsuda
Keyword(s):  
Scientific Revolution ◽  
Technological Progress ◽  
18Th Century ◽  
Philosophy Of Mathematics ◽  
Zen Buddhism ◽  
Fundamental Difference ◽  
The Third ◽  
Late 18Th Century ◽  
And Mathematics ◽  
Celestial Sphere

For centuries, religion has been the main impulse for moral and humanistic advancement, and ever since the rise of the Scientific Revolution (from 1543, the year Copernicus published De revolutioni bus orbium coelestium [On the revolution of the celestial sphere] – to the late 18th century), mathematics has been the cardinal element for scientific and technological progress. Mathematics requires a logical mind, but religion demands a receptive and compassionate mind. Even though there is a fundamental difference between the two subjects, the aim of this essay is to explore the relationships between Zen, mathematics, and Rāmānujan. The first section expounds on Bodhidharma’s and Hui neng’s notions of “no mind” and the “essence of mind,” as they are deemed an important bridge between Zen and mathematics. The second section presents how mathematics and Zen Buddhism relate to each other. Accordingly, the views on intuition, imagination, freedom, and language based on Einstein, Cantor, Brouwer, Poincare, et al. are discussed. The third section discusses the work of the most renowned mathematician in modern India in relation to Zen Buddhism. Rāmānujan’s unparalleled accomplishment in the field of number theory is well known among mathematicians. However, he is not well presented in the philosophy of mathematics, because of his unusual approach to mathematics.

scholarly journals Theological Underpinnings of the Modern Philosophy of Mathematics.

2016 ◽  
Vol 44 (1) ◽  
pp. 147-168
Author(s):  
Vladislav Shaposhnikov
Keyword(s):  
Twentieth Century ◽  
Modern Philosophy ◽  
Philosophy Of Mathematics ◽  
Main Concern ◽  
Early Twentieth Century ◽  
Pure Mathematics ◽  
And Mathematics

Abstract The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

scholarly journals Taking Plurals at Face Value

2021 ◽  
pp. 8-30
Author(s):  
Salvatore Florio ◽  
Øystein Linnebo
Keyword(s):  
Philosophy Of Mathematics ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Logical System ◽  
Plural Logic ◽  
First Order ◽  
Philosophical Significance ◽  
Pure Logic

Plural logic is a logical system in which plural terms and predicates figure as primitive expressions alongside the singular resources of ordinary first-order logic. The philosophical significance of this system depends on two of its alleged features: being pure logic and providing more expressive power than first-order logic. This chapter first introduces the language and axioms of plural logic and then analyzes this logic’s main philosophical applications in metaphysics, philosophy of mathematics, and semantics.

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