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The Infinite In Mathematics

The Infinite in Mathematics PDF

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Author: Felix Kaufmann
Publisher: Springer Science & Business Media
ISBN: 9400997957
Size: 32.76 MB
Format: PDF, ePub, Mobi
Category : Science
Languages : en
Pages : 237
View: 4396

Book Description: The main item in the present volume was published in 1930 under the title Das Unendliche in der Mathematik und seine Ausschaltung. It was at that time the fullest systematic account from the standpoint of Husserl's phenomenology of what is known as 'finitism' (also as 'intuitionism' and 'constructivism') in mathematics. Since then, important changes have been required in philosophies of mathematics, in part because of Kurt Godel's epoch-making paper of 1931 which established the essential in completeness of arithmetic. In the light of that finding, a number of the claims made in the book (and in the accompanying articles) are demon strably mistaken. Nevertheless, as a whole it retains much of its original interest and value. It presents the issues in the foundations of mathematics that were under debate when it was written (and in some cases still are); , and it offers one alternative to the currently dominant set-theoretical definitions of the cardinal numbers and other arithmetical concepts. While still a student at the University of Vienna, Felix Kaufmann was greatly impressed by the early philosophical writings (especially by the Logische Untersuchungen) of Edmund Husser!' He was never an uncritical disciple of Husserl, and he integrated into his mature philosophy ideas from a wide assortment of intellectual sources. But he thought of himself as a phenomenologist, and made frequent use in all his major publications of many of Husserl's logical and epistemological theses.


The Art Of The Infinite

The Art of the Infinite PDF

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Author: Robert Kaplan
Publisher: Bloomsbury Publishing USA
ISBN: 160819888X
Size: 34.79 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 416
View: 5854

Book Description: A witty, conversational, and accessible tour of math's profoundest mysteries. Mathematical symbols, for mathematicians, store worlds of meaning, leap continents and centuries. But we need not master symbols to grasp the magnificent abstractions they represent, and to which all art aspires. Through language, anyone can come to delight in the works of mathematical art, which are among our kind's greatest glories. Taking the concept of infinity, in its countless guises, as a starting point and a helpful touchstone, the founders of Harvard's pioneering Math Circle program Robert and Ellen Kaplan guide us through the “Republic of Numbers,” where we meet both its upstanding citizens and its more shadowy dwellers, explore realms where only the imagination can go, and grapple with math's most profound uncertainties, including the question of truth itself-do we discover mathematical principles, or invent them?


To Infinity And Beyond

To Infinity and Beyond PDF

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Author: Eli Maor
Publisher: Princeton University Press
ISBN: 9780691025117
Size: 48.54 MB
Format: PDF, Kindle
Category : Mathematics
Languages : en
Pages : 284
View: 538

Book Description: Eli Maor examines the role of infinity in mathematics and geometry and its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the horror infiniti of the Greeks to the works of M. C. Escher; from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes the mathematician's fascination with infinity--a fascination mingled with puzzlement. Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified by the work of the Dutch artist M. C. Escher, six of whose works are shown here in beautiful color plates.--Los Angeles Times [Eli Maor's] enthusiasm for the topic carries the reader through a rich panorama.--Choice Fascinating and enjoyable.... places the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics.--Science -- "Science"


Georg Cantor

Georg Cantor PDF

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Author: Joseph Warren Dauben
Publisher: Princeton University Press
ISBN: 0691214204
Size: 14.99 MB
Format: PDF, ePub
Category : Science
Languages : en
Pages : 424
View: 6035

Book Description: One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula. Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.


The Mathematics Of Infinity

The Mathematics of Infinity PDF

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Author: Theodore G. Faticoni
Publisher: John Wiley & Sons
ISBN: 0470049138
Size: 42.40 MB
Format: PDF, Mobi
Category : Mathematics
Languages : en
Pages : 320
View: 5444

Book Description: A balanced and clearly explained treatment of infinity in mathematics. The concept of infinity has fascinated and confused mankind for centuries with concepts and ideas that cause even seasoned mathematicians to wonder. For instance, the idea that a set is infinite if it is not a finite set is an elementary concept that jolts our common sense and imagination. the Mathematics of Infinity: A guide to Great Ideas uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing. Writing with clear knowledge and affection for the subject, the author introduces and explores infinite sets, infinite cardinals, and ordinals, thus challenging the readers' intuitive beliefs about infinity. Requiring little mathematical training and a healthy curiosity, the book presents a user-friendly approach to ideas involving the infinite. readers will discover the main ideas of infinite cardinals and ordinal numbers without experiencing in-depth mathematical rigor. Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun your intuitive view of the world. With a thoughtful and balanced treatment of both concepts and theory, The Mathematics of Infinity focuses on the following topics: * Sets and Functions * Images and Preimages of Functions * Hilbert's Infinite Hotel * Cardinals and Ordinals * The Arithmetic of Cardinals and Ordinals * the Continuum Hypothesis * Elementary Number Theory * The Riemann Hypothesis * The Logic of Paradoxes Recommended as recreational reading for the mathematically inquisitive or as supplemental reading for curious college students, the Mathematics of Infinity: A Guide to Great Ideas gently leads readers into the world of counterintuitive mathematics.


Cantor S Lovely Game

 Cantor s Lovely Game  PDF

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Author: Catherine MacHale
Publisher:
ISBN:
Size: 48.40 MB
Format: PDF, Kindle
Category :
Languages : en
Pages : 266
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Book Description: Though I have studies mathematics, I had never encountered the name Georg Cantor until I began to study literature. Then, within the space of a few weeks, I found one reference to Cantor in a short story of Jorge Luis Borges and another in an essay by Julia Kristeva. I was curious. What was it about this relatively obscure nineteenth century German mathematician that drew the attention of an Argentinean writer of fantastical fiction and a Bulgarian feminist literary theorist? "Infinity" would be a short answer to the question but hopefully this thesis will provide a longer answer. Cantor developed the branch of mathematics known as set theory and an essential aspect of his set theory was the recognition of infinity as a legitimate concept within mathematics. Borges and Kristeva share a fascination with infinity, which would seem to explain their interest in Cantor, but I believe that it is not just the fact that Cantor deals with infinity in his work but the extraordinary way in which he does so that forms the basis for the attraction. Many people have studied the infinite and many more have refused to do so on the grounds that they consider it a topic incompatible with logical thought, though the latter group seem to have discussed the infinite quite as much as the former in an effort to demonstrate that the infinite should not be discussed. The first chapter of this thesis addresses the fears that generally characterize any interaction with the infinite and the second chapter examines how Cantor contended with these fears. What seems most remarkable to me about his approach to the infinite is the courage and generosity he displays when faced with an ostensible nightmare for the mathematical mind. Rather than viewing the paradoxes and inconsistencies generated by infinity as problems to be overcome, he accepts them as attributes that determine what it means for something to be infinite. Both Borges and Kristeva have a taste for the revolutionary and unorthodox, whether it be in politics or philosophy, and there is a sense in which their admiration for Cantor is simply the result of an appreciation of his bravery in contending with a subject commonly condemned in scholarly circles, as well as an interest in the singular way in which he does so. However, as I examined Cantor's set theory in relation to the writings of Kristeva and Borges, it became apparent that there was something in Cantor's system and in the concept of infinity itself that was related to the most fundamental interests of both writers. The infinite has been called unmentionable, even unthinkable, yet Cantor spoke of it. To speak of infinity is to say the allegedly unsayable. For Kristeva and for Borges, the infinite is inseparable from the concerns of every writer because it is intimately connected with language.


How To Measure The Infinite Mathematics With Infinite And Infinitesimal Numbers

How To Measure The Infinite  Mathematics With Infinite And Infinitesimal Numbers PDF

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Author: Mauro Di Nasso
Publisher: World Scientific
ISBN: 9813276606
Size: 77.95 MB
Format: PDF, Docs
Category : Mathematics
Languages : en
Pages : 348
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Book Description: 'This text shows that the study of the almost-forgotten, non-Archimedean mathematics deserves to be utilized more intently in a variety of fields within the larger domain of applied mathematics.'CHOICEThis book contains an original introduction to the use of infinitesimal and infinite numbers, namely, the Alpha-Theory, which can be considered as an alternative approach to nonstandard analysis.The basic principles are presented in an elementary way by using the ordinary language of mathematics; this is to be contrasted with other presentations of nonstandard analysis where technical notions from logic are required since the beginning. Some applications are included and aimed at showing the power of the theory.The book also provides a comprehensive exposition of the Theory of Numerosity, a new way of counting (countable) infinite sets that maintains the ancient Euclid's Principle: 'The whole is larger than its parts'. The book is organized into five parts: Alpha-Calculus, Alpha-Theory, Applications, Foundations, and Numerosity Theory.


Infinity And The Mind

Infinity and the Mind PDF

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Author: Rudy von Bitter Rucker
Publisher:
ISBN: 9780691001722
Size: 33.34 MB
Format: PDF, ePub
Category : Nature
Languages : en
Pages : 342
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Book Description: In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the Mindscape, where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.


In Search Of Infinity

In Search of Infinity PDF

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Author: N.Ya. Vilenkin
Publisher: Springer Science & Business Media
ISBN: 1461208378
Size: 59.34 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 146
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Book Description: The concept of infinity is one of the most important, and at the same time, one of the most mysterious concepts of science. Already in antiquity many philosophers and mathematicians pondered over its contradictory nature. In mathematics, the contradictions connected with infinity intensified after the creation, at the end of the 19th century, of the theory of infinite sets and the subsequent discovery, soon after, of paradoxes in this theory. At the time, many scientists ignored the paradoxes and used set theory extensively in their work, while others subjected set-theoretic methods in mathematics to harsh criticism. The debate intensified when a group of French mathematicians, who wrote under the pseudonym of Nicolas Bourbaki, tried to erect the whole edifice of mathematics on the single notion of a set. Some mathematicians greeted this attempt enthusiastically while others regarded it as an unnecessary formalization, an attempt to tear mathematics away from life-giving practical applications that sustain it. These differences notwithstanding, Bourbaki has had a significant influence on the evolution of mathematics in the twentieth century. In this book we try to tell the reader how the idea of the infinite arose and developed in physics and in mathematics, how the theory of infinite sets was constructed, what paradoxes it has led to, what significant efforts have been made to eliminate the resulting contradictions, and what routes scientists are trying to find that would provide a way out of the many difficulties.


Eight Lessons On Infinity

Eight Lessons on Infinity PDF

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Author: Haim Shapira
Publisher: Watkins
ISBN: 1786782340
Size: 64.24 MB
Format: PDF, Kindle
Category : Mathematics
Languages : en
Pages : 208
View: 6727

Book Description: A fun, non-technical and wonderfully engaging guide to that most powerful and mysterious of mathematical concepts: infinity.in this book, best-selling author and mathematician Haim Shapira presents an introduction to mathematical theories which deal with the most beautiful concept ever invented by humankind: infinity. In this book, best-selling author and mathematician Haim Shapira presents an introduction to mathematical theories which deal with the most beautiful concept ever invented by humankind: infinity. Written in clear, simple language and aimed at a lay audience, this book also offers some strategies that will allow readers to try their ability at solving truly fascinating mathematical problems. Infinity is a deeply counter-intuitive concept that has inspired many great thinkers. In this book we will meet many sages, both familiar and unfamiliar: Zeno and Pythagoras, Georg Cantor and Bertrand Russell, Sofia Kovalevskaya and Emmy Noether, al-Khwarizmi and Euclid, Sophie Germain and Srinivasa Ramanujan.The world of infinity is inhabited by many paradoxes, and so is this book: Zeno paradoxes, Hilbert's "Infinity Hotel", Achilles and the gods paradox, the paradox of heaven and hell, the Ross-Littlewood paradox involving tennis balls, the Galileo paradox and many more. Aimed at the curious but non-technical reader, this book refrains from using any fearsome mathematical symbols. It uses only the most basic operations of mathematics: adding, subtracting, multiplication, division, powers and roots – that is all. But that doesn’t mean that a bit of deep thinking won’t be necessary and rewarding. Writing with humour and lightness of touch, Haim Shapira banishes the chalky pallor of the schoolroom and offers instead a truly thrilling intellectual journey. Fasten your seatbelt – we are going to Infinity, and beyond!


Infinity A Very Short Introduction

Infinity  a Very Short Introduction PDF

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Author: Ian Stewart
Publisher: Oxford University Press
ISBN: 0198755236
Size: 70.21 MB
Format: PDF, ePub, Docs
Category : Mathematics
Languages : en
Pages : 144
View: 4165

Book Description: Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large (infinite) isintimately related to the infinitely small (infinitesimal). Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas ofmathematics rest upon some version of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals.In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectualexercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, andenthusiasm to make interesting and challenging topics highly readable.