The Continuous The Discrete And The Infinitesimal In Philosophy And Mathematics

The Continuous  the Discrete and the Infinitesimal in Philosophy and Mathematics PDF Book Detail:
Author: John L. Bell
Publisher: Springer Nature
ISBN: 3030187071
Size: 56.54 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 313
View: 1899

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Book Description: 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 Nature Of Infinitesimals

Author: Peter F. Erickson
Publisher: Xlibris Corporation
ISBN: 147970184X
Size: 39.75 MB
Format: PDF, Kindle
Category : Mathematics
Languages : en
Pages : 268
View: 1010

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Book Description: 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.

Big Data Code And The Discrete City

Big Data  Code and the Discrete City PDF Book Detail:
Author: Silvio Carta
Publisher: Routledge
ISBN: 1351007394
Size: 44.46 MB
Format: PDF, ePub, Mobi
Category : Computers
Languages : en
Pages : 212
View: 5431

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Book Description: Big Data, Code and the Discrete City explores how digital technologies are gradually changing the way in which the public space is designed by architects, managed by policymakers and experienced by individuals. Smart city technologies are superseding the traditional human experience that has characterised the making of the public space until today. This book examines how computers see the public space and the effect of algorithms, artificial intelligences and automated processes on the human experience in public spaces. Divided into three parts, the first part of this book examines the notion of discreteness in its origins and applications to computer sciences. The second section presents a dual perspective: it explores the ways in which public spaces are constructed by the computer-driven logic and then translated into control mechanisms, design strategies and software-aided design. This perspective also describes the way in which individuals perceive this new public space, through its digital logic, and discrete mechanisms (from Wi-Fi coverage to self-tracking). Finally, in the third part, this book scrutinises the discrete logic with which computers operate, and how this is permeating into aspects of city life. This book is valuable for anyone interested in urban studies and digital technologies, and more specifically in big data, urban informatics and public space.

The Bulletin Of Symbolic Logic

The Bulletin of Symbolic Logic PDF Book Detail:
Size: 24.53 MB
Format: PDF, Mobi
Category : Logic, Symbolic and mathematical
Languages : en
Pages :
View: 159

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Book Description:

Oppositions And Paradoxes

Oppositions and Paradoxes PDF Book Detail:
Author: John L. Bell
Publisher: Broadview Press
ISBN: 1770486038
Size: 21.47 MB
Format: PDF
Category : Philosophy
Languages : en
Pages : 216
View: 3080

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Book Description: Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In Oppositions and Paradoxes, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, Bell not only motivates abstract conceptual thinking about the paradoxes at issue, but he also offers a compelling introduction to central ideas in such otherwise-difficult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the famous Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life — he would, if he had infinite time, presumably never complete the work, although no individual part of it would remain unwritten. Or think of an office mailbox labelled “mail for those with no mailbox”—if this is a person’s mailbox, how can they possibly have “no mailbox”? These and many other paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty in many of our most basic concepts.

The History Of Continua

The History of Continua PDF Book Detail:
Author: Stewart Shapiro
Publisher: Oxford University Press
ISBN: 0192537490
Size: 20.65 MB
Format: PDF, Kindle
Category : Philosophy
Languages : en
Pages : 320
View: 3184

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Book Description: 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 Concise History Of Mathematics For Philosophers

A Concise History of Mathematics for Philosophers   PDF Book Detail:
Author: John Stillwell
Publisher: Cambridge University Press
ISBN: 1108456235
Size: 11.35 MB
Format: PDF, Docs
Category : Science
Languages : en
Pages : 75
View: 7552

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Book Description: This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.

Herman Melville And The Politics Of The Inhuman

Herman Melville and the Politics of the Inhuman PDF Book Detail:
Author: Michael Jonik
Publisher: Cambridge University Press
ISBN: 1108420923
Size: 45.77 MB
Format: PDF
Category : Literary Criticism
Languages : en
Pages :
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Book Description: An ambitious, revisionary study of not only Herman Melville's political philosophy, but also of our own deeply inhuman condition.

The Blackwell Companion To Philosophy

The Blackwell Companion to Philosophy PDF Book Detail:
Author: Nicholas Bunnin
Publisher: John Wiley & Sons
ISBN: 9780631219071
Size: 63.81 MB
Format: PDF, ePub, Mobi
Category : Philosophy
Languages : en
Pages : 951
View: 463

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Book Description: An introduction to philosophy looks at key issues, figures, and movements in the field.