The Continuous The Discrete And The Infinitesimal In Philosophy And Mathematics

Author : John L. Bell
Publisher : Springer Nature
Page : 313 pages
File Size : 54,9 Mb
Release : 2019-09-09
Category : Mathematics
ISBN : 9783030187071

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The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics by John L. Bell Pdf

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.

Author : John Lane Bell
Publisher : Polimetrica s.a.s.
Page : 354 pages
File Size : 42,7 Mb
Release : 2005
Category : Mathematics
ISBN : 9788876990151

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The Continuous and the Infinitesimal in Mathematics and Philosophy by John Lane Bell Pdf

Author : John L. Bell
Publisher : Cambridge University Press
Page : 7 pages
File Size : 49,5 Mb
Release : 2008-04-07
Category : Mathematics
ISBN : 9780521887182

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A Primer of Infinitesimal Analysis by John L. Bell Pdf

A rigorous, axiomatically formulated presentation of the 'zero-square', or 'nilpotent' infinitesimal.

Author : Stewart Shapiro,Geoffrey Hellman
Publisher : Oxford University Press
Page : 320 pages
File Size : 54,6 Mb
Release : 2020-12-01
Category : Philosophy
ISBN : 9780192537492

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The History of Continua by Stewart Shapiro,Geoffrey Hellman Pdf

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.

Author : Peter F. Erickson
Publisher : Xlibris Corporation
Page : 260 pages
File Size : 52,7 Mb
Release : 2006-05-05
Category : Mathematics
ISBN : 9781479701841

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The Nature of Infinitesimals by Peter F. Erickson Pdf

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 Ericksons 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 its 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.

Author : Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga
Publisher : Springer Nature
Page : 293 pages
File Size : 49,5 Mb
Release : 2022-09-17
Category : Mathematics
ISBN : 9783030777999

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Axiomatic Thinking II by Fernando Ferreira,Reinhard Kahle,Giovanni Sommaruga Pdf

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.

Author : Mark Burgin
Publisher : World Scientific
Page : 370 pages
File Size : 55,6 Mb
Release : 2022-04-22
Category : Mathematics
ISBN : 9789811236853

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

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.

Author : Mohua Banerjee,A. V. Sreejith
Publisher : Springer Nature
Page : 232 pages
File Size : 55,8 Mb
Release : 2023-02-22
Category : Mathematics
ISBN : 9783031266898

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Logic and Its Applications by Mohua Banerjee,A. V. Sreejith Pdf

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.

Author : Paul Redding
Publisher : University of Chicago Press
Page : 303 pages
File Size : 43,5 Mb
Release : 2023-06-12
Category : Philosophy
ISBN : 9780226826066

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Conceptual Harmonies by Paul Redding Pdf

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.

Author : Stewart Shapiro,Geoffrey Hellman
Publisher : Oxford University Press, USA
Page : 593 pages
File Size : 49,9 Mb
Release : 2021
Category : Mathematics
ISBN : 9780198809647

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The History of Continua by Stewart Shapiro,Geoffrey Hellman Pdf

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.

Author : Gregory S. Moss
Publisher : Springer Nature
Page : 499 pages
File Size : 40,8 Mb
Release : 2022-11-13
Category : Philosophy
ISBN : 9783031138621

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The Being of Negation in Post-Kantian Philosophy by Gregory S. Moss Pdf

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.

Author : Amir Alexander
Publisher : Simon and Schuster
Page : 368 pages
File Size : 48,6 Mb
Release : 2014-07-03
Category : Science
ISBN : 9781780745336

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Infinitesimal by Amir Alexander Pdf

On August 10, 1632, five leading Jesuits convened in a sombre Roman palazzo to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that fateful day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world. Amir Alexander takes us from the bloody religious strife of the sixteenth century to the battlefields of the English civil war and the fierce confrontations between leading thinkers like Galileo and Hobbes. The legitimacy of popes and kings, as well as our modern beliefs in human liberty and progressive science, hung in the balance; the answer hinged on the infinitesimal. Pulsing with drama and excitement, Infinitesimal will forever change the way you look at a simple line.

Author : Michael McGillen
Publisher : Cornell University Press
Page : 190 pages
File Size : 41,6 Mb
Release : 2023-12-15
Category : Literary Criticism
ISBN : 9781501772832

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Shapes of Time by Michael McGillen Pdf

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.

Author : Bertrand Russell
Publisher : Courier Corporation
Page : 230 pages
File Size : 42,5 Mb
Release : 1993-01-01
Category : Mathematics
ISBN : 0486277240

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Introduction to Mathematical Philosophy by Bertrand Russell Pdf

In the words of Bertrand Russell, "Because language is misleading, as well as because it is diffuse and inexact when applied to logic (for which it was never intended), logical symbolism is absolutely necessary to any exact or thorough treatment of mathematical philosophy." That assertion underlies this book, a seminal work in the field for more than 70 years. In it, Russell offers a nontechnical, undogmatic account of his philosophical criticism as it relates to arithmetic and logic. Rather than an exhaustive treatment, however, the influential philosopher and mathematician focuses on certain issues of mathematical logic that, to his mind, invalidated much traditional and contemporary philosophy. In dealing with such topics as number, order, relations, limits and continuity, propositional functions, descriptions, and classes, Russell writes in a clear, accessible manner, requiring neither a knowledge of mathematics nor an aptitude for mathematical symbolism. The result is a thought-provoking excursion into the fascinating realm where mathematics and philosophy meet — a philosophical classic that will be welcomed by any thinking person interested in this crucial area of modern thought.

Author : Benjamin Halligan,Alexei Penzin,Stefano Pippa,Rebecca Carson
Publisher : Bloomsbury Publishing
Page : 370 pages
File Size : 45,8 Mb
Release : 2021-09-09
Category : Philosophy
ISBN : 9781350105669

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Politics of the Many by Benjamin Halligan,Alexei Penzin,Stefano Pippa,Rebecca Carson Pdf

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.