The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics

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Publisher : Springer Nature
ISBN 13 : 3030187071
Total Pages : 313 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics by : John L. Bell

Download or read book The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics written by John L. Bell and published by Springer Nature. This book was released on 2019-09-09 with total page 313 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.

The Continuous and the Infinitesimal in Mathematics and Philosophy

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Author :
Publisher : Polimetrica s.a.s.
ISBN 13 : 8876990151
Total Pages : 354 pages
Book Rating : 4.8/5 (769 download)

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Book Synopsis The Continuous and the Infinitesimal in Mathematics and Philosophy by : John Lane Bell

Download or read book The Continuous and the Infinitesimal in Mathematics and Philosophy written by John Lane Bell and published by Polimetrica s.a.s.. This book was released on 2005 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt:

THE NATURE of INFINITESIMALS

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Publisher : Xlibris Corporation
ISBN 13 : 147970184X
Total Pages : 268 pages
Book Rating : 4.4/5 (797 download)

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Book Synopsis THE NATURE of INFINITESIMALS by : Peter F. Erickson

Download or read book THE NATURE of INFINITESIMALS written by Peter F. Erickson and published by Xlibris Corporation. This book was released on 2006-05-05 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Erickson explores and explains the infinite and the infinitesimal with application to absolute space, time and motion, as well as absolute zero temperature in this thoughtful treatise. Mathematicians, scientists and philosophers have explored the realms of the continuous and discrete for centuries. Erickson delves into the history of these concepts and how people learn and understand them. He regards the infinitesimal as the key to understanding much of the scientific basis of the universe, and intertwines mathematical examples and historical context from Aristotle, Kant, Euler, Newton and more with his deductions-resulting in a readable treatment of complex topics. The reader will gain an understanding of potential versus actual infinity, irrational and imaginary numbers, the infinitesimal, and the tangent, among other concepts. At the heart of Erickson’s work is the veritable number system, in which positive and negative numbers are incompatible for the basic mathematical operations of addition, subtraction, multiplication, division, roots and ratios. This number system, he demonstrates, can provide a new interpretation of imaginary numbers, as a combination of the real and the veritable. Erickson further explores limits, derivatives and integrals before turning his attention to non-Euclidean geometry. In each topic, he applies his new understanding of the infinitesimal to the ideas of mathematics and draws conclusions. In the case of non-Euclidean geometry, the author determines that it’s inconsistent with the infinitesimal. Erickson supplies illustrative examples both in words and images-he clearly defines new notation as needed for concepts such as eternity, the infinitesimal, the instant and an unlimited quantity. In the final chapters, the author addresses absolute space, time and motion through the lens of the infinitesimal. While explaining his deductions and thoughts on these complex topics, he raises new questions for his readers to contemplate, such as the origin of memory. A weighty tome for devotees of mathematics and physics that raises interesting questions.

The History of Continua

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Publisher : Oxford University Press
ISBN 13 : 0192537490
Total Pages : 320 pages
Book Rating : 4.1/5 (925 download)

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Book Synopsis The History of Continua by : Stewart Shapiro

Download or read book The History of Continua written by Stewart Shapiro and published by Oxford University Press. This book was released on 2020-12-01 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical and philosophical thought about continuity has changed considerably over the ages. Aristotle insisted that continuous substances are not composed of points, and that they can only be divided into parts potentially. There is something viscous about the continuous. It is a unified whole. This is in stark contrast with the prevailing contemporary account, which takes a continuum to be composed of an uncountably infinite set of points. This vlume presents a collective study of key ideas and debates within this history. The opening chapters focus on the ancient world, covering the pre-Socratics, Plato, Aristotle, and Alexander. The treatment of the medieval period focuses on a (relatively) recently discovered manuscript, by Bradwardine, and its relation to medieval views before, during, and after Bradwardine's time. In the so-called early modern period, mathematicians developed the calculus and, with that, the rise of infinitesimal techniques, thus transforming the notion of continuity. The main figures treated here include Galileo, Cavalieri, Leibniz, and Kant. In the early party of the nineteenth century, Bolzano was one of the first important mathematicians and philosophers to insist that continua are composed of points, and he made a heroic attempt to come to grips with the underlying issues concerning the infinite. The two figures most responsible for the contemporary orthodoxy regarding continuity are Cantor and Dedekind. Each is treated in an article, investigating their precursors and influences in both mathematics and philosophy. A new chapter then provides a lucid analysis of the work of the mathematician Paul Du Bois-Reymond, to argue for a constructive account of continuity, in opposition to the dominant Dedekind-Cantor account. This leads to consideration of the contributions of Weyl, Brouwer, and Peirce, who once dubbed the notion of continuity "the master-key which . . . unlocks the arcana of philosophy". And we see that later in the twentieth century Whitehead presented a point-free, or gunky, account of continuity, showing how to recover points as a kind of "extensive abstraction". The final four chapters each focus on a more or less contemporary take on continuity that is outside the Dedekind-Cantor hegemony: a predicative approach, accounts that do not take continua to be composed of points, constructive approaches, and non-Archimedean accounts that make essential use of infinitesimals.

A Primer of Infinitesimal Analysis

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Publisher : Cambridge University Press
ISBN 13 : 0521887186
Total Pages : 7 pages
Book Rating : 4.5/5 (218 download)

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Book Synopsis A Primer of Infinitesimal Analysis by : John L. Bell

Download or read book A Primer of Infinitesimal Analysis written by John L. Bell and published by Cambridge University Press. This book was released on 2008-04-07 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: A rigorous, axiomatically formulated presentation of the 'zero-square', or 'nilpotent' infinitesimal.

Axiomatic Thinking II

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Publisher : Springer Nature
ISBN 13 : 3030777995
Total Pages : 293 pages
Book Rating : 4.0/5 (37 download)

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Book Synopsis Axiomatic Thinking II by : Fernando Ferreira

Download or read book Axiomatic Thinking II written by Fernando Ferreira and published by Springer Nature. This book was released on 2022-09-17 with total page 293 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Göttingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.

Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective

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Publisher : World Scientific
ISBN 13 : 9811236852
Total Pages : 370 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective by : Mark Burgin

Download or read book Trilogy Of Numbers And Arithmetic - Book 1: History Of Numbers And Arithmetic: An Information Perspective written by Mark Burgin and published by World Scientific. This book was released on 2022-04-22 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is the first in the trilogy which will bring you to the fascinating world of numbers and operations with them. Numbers provide information about myriads of things. Together with operations, numbers constitute arithmetic forming in basic intellectual instruments of theoretical and practical activity of people and offering powerful tools for representation, acquisition, transmission, processing, storage, and management of information about the world.The history of numbers and arithmetic is the topic of a variety of books and at the same time, it is extensively presented in many books on the history of mathematics. However, all of them, at best, bring the reader to the end of the 19th century without including the developments in these areas in the 20th century and later. Besides, such books consider and describe only the most popular classes of numbers, such as whole numbers or real numbers. At the same time, a diversity of new classes of numbers and arithmetic were introduced in the 20th century.This book looks into the chronicle of numbers and arithmetic from ancient times all the way to 21st century. It also includes the developments in these areas in the 20th century and later. A unique aspect of this book is its information orientation of the exposition of the history of numbers and arithmetic.

Introduction to Mathematical Philosophy

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Publisher :
ISBN 13 :
Total Pages : 224 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Introduction to Mathematical Philosophy by : Bertrand Russell

Download or read book Introduction to Mathematical Philosophy written by Bertrand Russell and published by . This book was released on 1920 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The History of Continua

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Publisher : Oxford University Press, USA
ISBN 13 : 0198809646
Total Pages : 593 pages
Book Rating : 4.1/5 (988 download)

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Book Synopsis The History of Continua by : Stewart Shapiro

Download or read book The History of Continua written by Stewart Shapiro and published by Oxford University Press, USA. This book was released on 2021 with total page 593 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.

Shapes of Time

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Publisher : Cornell University Press
ISBN 13 : 150177283X
Total Pages : 190 pages
Book Rating : 4.5/5 (17 download)

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Book Synopsis Shapes of Time by : Michael McGillen

Download or read book Shapes of Time written by Michael McGillen and published by Cornell University Press. This book was released on 2023-12-15 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Shapes of Time explores how concepts of time and history were spatialized in early twentieth-century German thought. Michael McGillen locates efforts in German modernism to conceive of alternative shapes of time—beyond those of historicism and nineteenth-century philosophies of history—at the boundary between secular and theological discourses. By analyzing canonical works of German modernism—those of Karl Barth, Franz Rosenzweig, Siegfried Kracauer, and Robert Musil—he identifies the ways in which spatial imagery and metaphors were employed to both separate the end of history from a narrative framework and to map the liminal relation between history and eschatology. Drawing on theories and practices as disparate as constructivism, non-Euclidean geometry, photography, and urban architecture, Shapes of Time presents original connections between modernism, theology, and mathematics as played out within the canon of twentieth-century German letters. Concepts of temporal and spatial form, McGillen contends, contribute to the understanding not only of modernist literature but also of larger theoretical concerns within modern cultural and intellectual history.

Logic and Its Applications

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Publisher : Springer Nature
ISBN 13 : 3031266897
Total Pages : 232 pages
Book Rating : 4.0/5 (312 download)

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Book Synopsis Logic and Its Applications by : Mohua Banerjee

Download or read book Logic and Its Applications written by Mohua Banerjee and published by Springer Nature. This book was released on 2023-02-22 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: Edited in collaboration with FoLLI, this book constitutes the refereed proceedings of the 10th Indian Conference on Logic and Its Applications, ICLA 2023, which was held in Indore, India, in March 2023. Besides 6 invited papers presented in this volume, there are 9 contributed full papers which were carefully reviewed and selected from 18 submissions. The volume covers a wide range of topics. These topics are related to modal and temporal logics, intuitionistic connexive and imperative logics, systems for reasoning with vagueness and rough concepts, topological quasi-Boolean logic and quasi-Boolean based rough set models, and first-order definability of path functions of graphs.

Conceptual Harmonies

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Publisher : University of Chicago Press
ISBN 13 : 0226826066
Total Pages : 303 pages
Book Rating : 4.2/5 (268 download)

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Book Synopsis Conceptual Harmonies by : Paul Redding

Download or read book Conceptual Harmonies written by Paul Redding and published by University of Chicago Press. This book was released on 2023-06-12 with total page 303 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new reading of Hegel’s Science of Logic through the history of European mathematics. Conceptual Harmonies develops an original account of G. W. F. Hegel’s perplexing Science of Logic from a simple insight: philosophical and mathematical thought have shaped each other since classical times. Situating Science of Logic within the rise of modern mathematics, Redding stresses Hegel’s attention to Pythagorean ratios, Platonic reason, and Aristotle’s geometrically inspired logic. He then explores how later traditions shaped Hegel’s world, through both Leibniz and new forms of algebraic geometry. This enlightening reading recovers an overlooked stream in Hegel’s philosophy that remains, Redding argues, important for contemporary conceptions of logic.

The Being of Negation in Post-Kantian Philosophy

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Publisher : Springer Nature
ISBN 13 : 3031138627
Total Pages : 499 pages
Book Rating : 4.0/5 (311 download)

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Book Synopsis The Being of Negation in Post-Kantian Philosophy by : Gregory S. Moss

Download or read book The Being of Negation in Post-Kantian Philosophy written by Gregory S. Moss and published by Springer Nature. This book was released on 2022-11-13 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: By drawing on the insights of diverse scholars from around the globe, this volume systematically investigates the meaning and reality of the concept of negation in Post-Kantian Philosophy—German Idealism, Early German Romanticism, and Neo-Kantianism. The reader benefits from the historical, critical, and systematic investigations contained which trace not only the significance of negation in these traditions, but also the role it has played in shaping the philosophical landscape of Post-Kantian philosophy. By drawing attention to historically neglected thinkers and traditions, and positioning the dialogue within a global and comparative context, this volume demonstrates the enduring relevance of Post-Kantian philosophy for philosophers thinking in today’s global context. This text should appeal to graduate students and professors of German Idealism, Post-Kantian philosophy, comparative philosophy, German studies, and intellectual history.

Politics of the Many

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Publisher : Bloomsbury Publishing
ISBN 13 : 135010566X
Total Pages : 370 pages
Book Rating : 4.3/5 (51 download)

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Book Synopsis Politics of the Many by : Benjamin Halligan

Download or read book Politics of the Many written by Benjamin Halligan and published by Bloomsbury Publishing. This book was released on 2021-09-09 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Politics of the Many draws inspiration from Percy Bysshe Shelley's celebrated call to arms: 'Ye are many – they are few!' This idea of the Many, as a general form of emancipatory subjectivity that cannot be erased for the sake of the One, is the philosophical and political assumption shared by contributors to this book. They raise questions of collective agency, and its crisis in contemporary capitalism, via new engagements with Marxist philosophy, psychoanalysis, theories of social reproduction and value-form, and post-colonial critiques, and drawing on activist thought and strategies. This book interrogates both established and emergent formations of the Many (the people, classes, publics, crowds, masses, multitudes), tracing their genealogies, their recent failures and victories, and their potentials to change the world. The book proposes and explores an intense and provoking series of new or reinvented concepts, figures, and theoretical constellations, including dividuality, the centaur, unintentional vanguard, insomnia at work, always-on capitalism, multitude (from its 'voiding' to a '(non)emergence'), crowds, necropolitics, and the link between political subjectivity and value-form. The contributors to Politics of the Many are both acclaimed and emergent thinkers including Carina Brand, Rebecca Carson, Luhuna Carvalho, Lorenzo Chiesa, Jodi Dean, Dario Gentili, Benjamin Halligan, Marc James Léger, Paul Mazzocchi, Alexei Penzin, Stefano Pippa, Gerald Raunig, and Stevphen Shukaitis.

Consciousness, Time, and Scepticism in Hume’s Thought

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Publisher : Taylor & Francis
ISBN 13 : 1040015646
Total Pages : 329 pages
Book Rating : 4.0/5 (4 download)

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Book Synopsis Consciousness, Time, and Scepticism in Hume’s Thought by : Lorne Falkenstein

Download or read book Consciousness, Time, and Scepticism in Hume’s Thought written by Lorne Falkenstein and published by Taylor & Francis. This book was released on 2024-04-02 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: David Hume’s philosophical work presents the reader with a perplexing mix of constructive accounts of empirically guided belief and destructive sceptical arguments against all belief. This book reconciles this conflict by showing that Hume intended his scepticism to be remedial. It immunizes us against the influence of “unphilosophical” causes of belief, determining us to proportion our beliefs to the evidence. In making this case, this book develops Humean positions on topics Hume did not discuss in detail but that are of interest to contemporary philosophers: consciousness and the unity of consciousness, temporal experience, visual spatial perception, the experience of colour and other qualia, objective experience, and spatially extended minds. It also challenges currently accepted interpretations of Hume’s views on the finite divisibility of space and time, vacuum, the duration of unchanging objects, and identity over time. It deals with criticisms of Hume that were raised by his contemporaries, notably by Thomas Reid, draws attention to earlier seventeenth‐ and eighteenth‐century work that has bearing on the interpretation of Hume’s thought, and compares Hume’s achievements with those of later nineteenth‐century psychologists and philosophers. Consciousness, Time, and Scepticism in Hume’s Thought will appeal to scholars and advanced students interested in Hume, history of philosophy, and early modern theories of perception, time, and consciousness.

Deleuze and the History of Mathematics

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Publisher : A&C Black
ISBN 13 : 1441113894
Total Pages : 225 pages
Book Rating : 4.4/5 (411 download)

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Book Synopsis Deleuze and the History of Mathematics by : Simon Duffy

Download or read book Deleuze and the History of Mathematics written by Simon Duffy and published by A&C Black. This book was released on 2013-05-09 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon B.Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seemingly incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it.

Real Analysis Through Modern Infinitesimals

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Publisher : Cambridge University Press
ISBN 13 : 1107002028
Total Pages : 587 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Real Analysis Through Modern Infinitesimals by : Nader Vakil

Download or read book Real Analysis Through Modern Infinitesimals written by Nader Vakil and published by Cambridge University Press. This book was released on 2011-02-17 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: A coherent, self-contained treatment of the central topics of real analysis employing modern infinitesimals.