Intuition And The Axiomatic Method

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Intuition and the Axiomatic Method

Author : Emily Carson
Publisher : Springer Science & Business Media
ISBN 13 : 1402040407
Total Pages : 328 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Intuition and the Axiomatic Method by : Emily Carson

Download or read book Intuition and the Axiomatic Method written by Emily Carson and published by Springer Science & Business Media. This book was released on 2006-07-02 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kant’s theory of intuition to be obsolete. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, Gödel, Poincaré, Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kant’s own approach. By way of these investigations, we hope to understand better the rationale behind Kant’s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.

Intuition and the Axiomatic Method

Author : Emily Carson
Publisher : Springer Science & Business Media
ISBN 13 : 9781402040399
Total Pages : 356 pages
Book Rating : 4.0/5 (43 download)

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Book Synopsis Intuition and the Axiomatic Method by : Emily Carson

Download or read book Intuition and the Axiomatic Method written by Emily Carson and published by Springer Science & Business Media. This book was released on 2006-01-24 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kant’s theory of intuition to be obsolete. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, Gödel, Poincaré, Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kant’s own approach. By way of these investigations, we hope to understand better the rationale behind Kant’s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.

Intuition in Science and Mathematics

Author : H. Fischbein
Publisher : Springer Science & Business Media
ISBN 13 : 0306472376
Total Pages : 298 pages
Book Rating : 4.3/5 (64 download)

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Book Synopsis Intuition in Science and Mathematics by : H. Fischbein

Download or read book Intuition in Science and Mathematics written by H. Fischbein and published by Springer Science & Business Media. This book was released on 2005-12-19 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: In writing the present book I have had in mind the following objectives: - To propose a theoretical, comprehensive view of the domain of intuition. - To identify and organize the experimental findings related to intuition scattered in a wide variety of research contexts. - To reveal the educational implications of the idea, developed for science and mathematics education. Most of the existing monographs in the field of intuition are mainly concerned with theoretical debates - definitions, philosophical attitudes, historical considerations. (See, especially the works of Wild (1938), of Bunge (1 962) and of Noddings and Shore (1 984).) A notable exception is the book by Westcott (1968), which combines theoretical analyses with the author’s own experimental studies. But, so far, no attempt has been made to identify systematically those findings, spread throughout the research literature, which could contribute to the deciphering of the mechanisms of intuition. Very often the relevant studies do not refer explicitly to intuition. Even when this term is used it occurs, usually, as a self-evident, common sense term.

Axiomatic Method and Category Theory

Author : Andrei Rodin
Publisher : Springer Science & Business Media
ISBN 13 : 3319004042
Total Pages : 285 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Axiomatic Method and Category Theory by : Andrei Rodin

Download or read book Axiomatic Method and Category Theory written by Andrei Rodin and published by Springer Science & Business Media. This book was released on 2013-10-14 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.

Rethinking Knowledge

Author : Carlo Cellucci
Publisher : Springer
ISBN 13 : 3319532375
Total Pages : 427 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Rethinking Knowledge by : Carlo Cellucci

Download or read book Rethinking Knowledge written by Carlo Cellucci and published by Springer. This book was released on 2017-03-29 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality.

Naturalizing Logico-Mathematical Knowledge

Author : Sorin Bangu
Publisher : Routledge
ISBN 13 : 1351998447
Total Pages : 319 pages
Book Rating : 4.3/5 (519 download)

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Book Synopsis Naturalizing Logico-Mathematical Knowledge by : Sorin Bangu

Download or read book Naturalizing Logico-Mathematical Knowledge written by Sorin Bangu and published by Routledge. This book was released on 2018-02-01 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato’s dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this important philosophical query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. Naturalizing Logico-Mathematical Knowledge contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers’ willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies.

Great Currents of Mathematical Thought

Author : François Le Lionnais
Publisher : Courier Corporation
ISBN 13 : 9780486495798
Total Pages : 280 pages
Book Rating : 4.4/5 (957 download)

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Book Synopsis Great Currents of Mathematical Thought by : François Le Lionnais

Download or read book Great Currents of Mathematical Thought written by François Le Lionnais and published by Courier Corporation. This book was released on 2004-01-01 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: 50 essays by eminent scholars include meditations on "Structures," Disciplines," "Space," "Function," "Group," "Probability," and "The Mathematical Epic" (Volume I) and on "Mathematics and the Human Intellect," "Mathematics and Technology," and "Mathematics and Civilization" (Volume II). 1962 edition.

Foundational Theories of Classical and Constructive Mathematics

Author : Giovanni Sommaruga
Publisher : Springer Science & Business Media
ISBN 13 : 9400704313
Total Pages : 312 pages
Book Rating : 4.4/5 (7 download)

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Book Synopsis Foundational Theories of Classical and Constructive Mathematics by : Giovanni Sommaruga

Download or read book Foundational Theories of Classical and Constructive Mathematics written by Giovanni Sommaruga and published by Springer Science & Business Media. This book was released on 2011-03-24 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book "Foundational Theories of Classical and Constructive Mathematics" is a book on the classical topic of foundations of mathematics. Its originality resides mainly in its treating at the same time foundations of classical and foundations of constructive mathematics. This confrontation of two kinds of foundations contributes to answering questions such as: Are foundations/foundational theories of classical mathematics of a different nature compared to those of constructive mathematics? Do they play the same role for the resp. mathematics? Are there connections between the two kinds of foundational theories? etc. The confrontation and comparison is often implicit and sometimes explicit. Its great advantage is to extend the traditional discussion of the foundations of mathematics and to render it at the same time more subtle and more differentiated. Another important aspect of the book is that some of its contributions are of a more philosophical, others of a more technical nature. This double face is emphasized, since foundations of mathematics is an eminent topic in the philosophy of mathematics: hence both sides of this discipline ought to be and are being paid due to.

Transition to Advanced Mathematics

Author : Danilo R. Diedrichs
Publisher : CRC Press
ISBN 13 : 1000581667
Total Pages : 552 pages
Book Rating : 4.0/5 (5 download)

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Book Synopsis Transition to Advanced Mathematics by : Danilo R. Diedrichs

Download or read book Transition to Advanced Mathematics written by Danilo R. Diedrichs and published by CRC Press. This book was released on 2022-05-22 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis, a standard fare for a transition course, but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics. The authors implement the practice recommended by the Committee on the Undergraduate Program in Mathematics (CUPM) curriculum guide, that a modern mathematics program should include cognitive goals and offer a broad perspective of the discipline. Part I offers: An introduction to logic and set theory. Proof methods as a vehicle leading to topics useful for analysis, topology, algebra, and probability. Many illustrated examples, often drawing on what students already know, that minimize conversation about "doing proofs." An appendix that provides an annotated rubric with feedback codes for assessing proof writing. Part II presents the context and culture aspects of the transition experience, including: 21st century mathematics, including the current mathematical culture, vocations, and careers. History and philosophical issues in mathematics. Approaching, reading, and learning from journal articles and other primary sources. Mathematical writing and typesetting in LaTeX. Together, these Parts provide a complete introduction to modern mathematics, both in content and practice. Table of Contents Part I - Introduction to Proofs Logic and Sets Arguments and Proofs Functions Properties of the Integers Counting and Combinatorial Arguments Relations Part II - Culture, History, Reading, and Writing Mathematical Culture, Vocation, and Careers History and Philosophy of Mathematics Reading and Researching Mathematics Writing and Presenting Mathematics Appendix A. Rubric for Assessing Proofs Appendix B. Index of Theorems and Definitions from Calculus and Linear Algebra Bibliography Index Biographies Danilo R. Diedrichs is an Associate Professor of Mathematics at Wheaton College in Illinois. Raised and educated in Switzerland, he holds a PhD in applied mathematical and computational sciences from the University of Iowa, as well as a master’s degree in civil engineering from the Ecole Polytechnique Fédérale in Lausanne, Switzerland. His research interests are in dynamical systems modeling applied to biology, ecology, and epidemiology. Stephen Lovett is a Professor of Mathematics at Wheaton College in Illinois. He holds a PhD in representation theory from Northeastern University. His other books include Abstract Algebra: Structures and Applications (2015), Differential Geometry of Curves and Surfaces, with Tom Banchoff (2016), and Differential Geometry of Manifolds (2019).

The Making of Mathematics

Author : Carlo Cellucci
Publisher : Springer Nature
ISBN 13 : 3030897311
Total Pages : 457 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis The Making of Mathematics by : Carlo Cellucci

Download or read book The Making of Mathematics written by Carlo Cellucci and published by Springer Nature. This book was released on 2022-03-07 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an alternative to current philosophy of mathematics: heuristic philosophy of mathematics. In accordance with the heuristic approach, the philosophy of mathematics must concern itself with the making of mathematics and in particular with mathematical discovery. In the past century, mainstream philosophy of mathematics has claimed that the philosophy of mathematics cannot concern itself with the making of mathematics but only with finished mathematics, namely mathematics as presented in published works. On this basis, mainstream philosophy of mathematics has maintained that mathematics is theorem proving by the axiomatic method. This view has turned out to be untenable because of Gödel’s incompleteness theorems, which have shown that the view that mathematics is theorem proving by the axiomatic method does not account for a large number of basic features of mathematics. By using the heuristic approach, this book argues that mathematics is not theorem proving by the axiomatic method, but is rather problem solving by the analytic method. The author argues that this view can account for the main items of the mathematical process, those being: mathematical objects, demonstrations, definitions, diagrams, notations, explanations, applicability, beauty, and the role of mathematical knowledge.

The Philosophers and Mathematics

Author : Hassan Tahiri
Publisher : Springer
ISBN 13 : 3319937332
Total Pages : 320 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis The Philosophers and Mathematics by : Hassan Tahiri

Download or read book The Philosophers and Mathematics written by Hassan Tahiri and published by Springer. This book was released on 2018-08-14 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the unique relationship between two different approaches to understand the nature of knowledge, reality, and existence. It collects essays that examine the distinctive historical relationship between mathematics and philosophy. Readers learn what key philosophers throughout the ages thought about mathematics. This includes both thinkers who recognized the relevance of mathematics to their own work as well as those who chose to completely ignore its many achievements. The essays offer insight into the role that mathematics played in the formation of each included philosopher’s doctrine as well as the impact its remarkable expansion had on the philosophical systems each erected. Conversely, the authors also highlight the ways that philosophy contributed to the growth and transformation of mathematics. Throughout, significant historical examples help to illustrate these points in a vivid way. Mathematics has often been a favored interlocutor of philosophers and a major source of inspiration. This book is the outcome of an international conference held in honor of Roshdi Rashed, a renowned historian of mathematics. It provides researchers, students, and interested readers with remarkable insights into the history of an important relationship throughout the ages.

Identity? Metaphysical Approach

Author : Jure Zovko
Publisher : LIT Verlag Münster
ISBN 13 : 3643912714
Total Pages : 160 pages
Book Rating : 4.6/5 (439 download)

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Book Synopsis Identity? Metaphysical Approach by : Jure Zovko

Download or read book Identity? Metaphysical Approach written by Jure Zovko and published by LIT Verlag Münster. This book was released on 2021-01-25 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: Identity is considered prima facie the most important philosophical principle in traditional logic and metaphysics. The reflexive ability of the subject implies that he or she can relate to himself or herself and to others, thus establishing the basis for the formation of individual and social identity. The present essays, which contains papers held at the conference of the "Institut International de Philosophie" at the University of Zadar, in 2007, present a range of positions and arguments regarding the possibilities of philosophical interpretation of identity.

János Bolyai Appendix

Author : F. Kárteszi
Publisher : Elsevier
ISBN 13 : 9780080872490
Total Pages : 236 pages
Book Rating : 4.8/5 (724 download)

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Book Synopsis János Bolyai Appendix by : F. Kárteszi

Download or read book János Bolyai Appendix written by F. Kárteszi and published by Elsevier. This book was released on 1987-01-01 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: The epoch-making work of János Bolyai is presented here, together with a supplement outlining Hungarian political and science history to help the reader to get acquainted with the miserable fate of János Bolyai and with his intellectual world. A facsimile of a copy of Bolyai's original 1831 Scientia Spatii (also known as the Appendix) is included, together with a translation. Comments and notes, and a survey of the effects of his work, complete the volume.

Neuroeconomics

Author : Paul W. Glimcher
Publisher : Academic Press
ISBN 13 : 008092106X
Total Pages : 557 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Neuroeconomics by : Paul W. Glimcher

Download or read book Neuroeconomics written by Paul W. Glimcher and published by Academic Press. This book was released on 2008-10-10 with total page 557 pages. Available in PDF, EPUB and Kindle. Book excerpt: Neuroeconomics is a new highly promising approach to understanding the neurobiology of decision making and how it affects cognitive social interactions between humans and societies/economies. This book is the first edited reference to examine the science behind neuroeconomics, including how it influences human behavior and societal decision making from a behavioral economics point of view. Presenting a truly interdisciplinary approach, Neuroeconomics presents research from neuroscience, psychology, and behavioral economics, and includes chapters by all the major figures in the field, including two Economics Nobel laureates. * An authoritative reference written and edited by acknowledged experts and founders of the field * Presents an interdisciplinary view of the approaches, concepts, and results of the emerging field of neuroeconomics relevant for anyone interested in this area of research* Full-color presentation throughout with carefully selected illustrations to highlight key concepts

Contemporary Mathematical Thinking

Author : Frédéric Patras
Publisher : Springer Nature
ISBN 13 : 3031275489
Total Pages : 133 pages
Book Rating : 4.0/5 (312 download)

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Book Synopsis Contemporary Mathematical Thinking by : Frédéric Patras

Download or read book Contemporary Mathematical Thinking written by Frédéric Patras and published by Springer Nature. This book was released on 2023-04-13 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the evolution of mathematical thought during the 20th century. Representing a unique point of view combining mathematics, philosophy and history on this issue, it presents an original analysis of key authors, for example Bourbaki, Grothendieck and Husserl. As a product of 19th and early 20th century science, a canon of knowledge or a scientific ideology, mathematical structuralism had to give way. The succession is difficult, still in progress, and uncertain. To understand contemporary mathematics, its progressive liberation from the slogans of "modern mathematics" and the paths that remain open today, it is first necessary to deconstruct the history of this long dominant current. Another conception of mathematical thought emerged in the work of mathematicians such as Hilbert or Weyl, which went beyond the narrow epistemological paths of science in the making. In this tradition, mathematical thought was accompanied by a philosophical requirement. Modernity teaches us to revive it. The book is intended for a varied public: mathematicians concerned with understanding their discipline, philosophers of science, and the erudite public curious about the progress of mathematics.

The Logical Foundations of Scientific Theories

Author : Decio Krause
Publisher : Routledge
ISBN 13 : 1315535203
Total Pages : 162 pages
Book Rating : 4.3/5 (155 download)

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Book Synopsis The Logical Foundations of Scientific Theories by : Decio Krause

Download or read book The Logical Foundations of Scientific Theories written by Decio Krause and published by Routledge. This book was released on 2016-07-15 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses the logical aspects of the foundations of scientific theories. Even though the relevance of formal methods in the study of scientific theories is now widely recognized and regaining prominence, the issues covered here are still not generally discussed in philosophy of science. The authors focus mainly on the role played by the underlying formal apparatuses employed in the construction of the models of scientific theories, relating the discussion with the so-called semantic approach to scientific theories. The book describes the role played by this metamathematical framework in three main aspects: considerations of formal languages employed to axiomatize scientific theories, the role of the axiomatic method itself, and the way set-theoretical structures, which play the role of the models of theories, are developed. The authors also discuss the differences and philosophical relevance of the two basic ways of aximoatizing a scientific theory, namely Patrick Suppes’ set theoretical predicates and the "da Costa and Chuaqui" approach. This book engages with important discussions of the nature of scientific theories and will be a useful resource for researchers and upper-level students working in philosophy of science.

Discipline Filosofiche (2015-1)

Author : Mario Alai
Publisher : Quodlibet
ISBN 13 : 8874627998
Total Pages : 193 pages
Book Rating : 4.8/5 (746 download)

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Book Synopsis Discipline Filosofiche (2015-1) by : Mario Alai

Download or read book Discipline Filosofiche (2015-1) written by Mario Alai and published by Quodlibet. This book was released on 2015-01-15 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contents: Mario Alai, Andrea Sereni and Giorgio Volpe, Guest Editors’ Preface • Ernest Sosa, Philosophical Intuitions and Metaphysical Analysis • Jonathan M. Weinberg, The Methodological Necessity of Experimental Philosophy • Steven Bland, Conceptual Analysis, Analytic Philosophy, and the Psychologistic Turn • Bryce Huebner, The Construction of Philosophical Intuitions • Alfredo Tomasetta, Physicalist Naturalism in the Philosophy of Mind (far less Warranted than Usually Thought) • Markus Pantsar, Assessing the “Empirical Philosophy of Mathematics” • Huginn Freyr Thorsteinsson, Experimental Philosophy and the Importance of Intuitions in the Philosophy of Language • Francesca Ervas, Elisabetta Gola, Antonio Ledda and Giuseppe Sergioli, Lexical Ambiguity in Elementary Inferences: an Experimental Study • Richard Davies, How to Point a Philosophical Armchair