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New Infinitary Mathematics

Autor: Petr Vopěnka

Número de Páginas: 352

The dominant current of twentieth-century mathematics, which simultaneously explores and applies infinity (albeit in bizarre ideal worlds), relies on Cantor's classical theory of infinite sets. Cantor’s theory in turn relies on the problematic assumption of the existence of the set of all natural numbers, the only justification for which – a theological justification - is usually concealed and pushed into the collective unconscious. This book begins by surveying the theological background, emergence, and development of classical set theory. The author warns us about the dangers implicit in the construction of set theory, traceable in his own and other eminent mathematicians' seminal works on the subject. He then goes on to present an argument about the absurdity of the assumption of the existence of the set of all natural numbers. However, the author’s contribution is not just a negation of current views and assumptions. On the contrary, the new infinitary mathematics that he proceeds to propose and develop is driven by a cautious effort to transcend the horizon bounding the ancient geometric world and pre-set-theoretical mathematics, whilst allowing mathematics to...

Paradoxes of Measures and Dimensions Originating in Felix Hausdorff's Ideas

Autor: Janusz Czy?

Número de Páginas: 768

In this book, many ideas by Felix Hausdorff are described and contemporary mathematical theories stemming from them are sketched.

Fantastic Numbers and Where to Find Them

Autor: Tony Padilla

Número de Páginas: 240

What are the mysterious numbers that unlock the secrets of the universe? In Fantastic Numbers and Where to Find Them, leading theoretical physicist and YouTube star Antonio Padilla takes us on an irreverent cosmic tour of nine of the most extraordinary numbers in physics. These include Graham's number, which is so large that if you thought about it in the wrong way, your head would collapse into a singularity; TREE(3), whose finite value could never be reached before the universe reset itself; and 10^{-120}, which measures the desperately unlikely balance of energy the universe needs to exist. . . Leading us down the rabbit hole to the inner workings of reality, Padilla demonstrates how these unusual numbers are the key to unlocking such mind-bending phenomena as black holes, entropy and the problem of the cosmological constant, which shows that our two best ways of understanding the universe contradict one another. Combining cutting-edge science with an entertaining cosmic quest, Fantastic Numbers and Where to Find Them is an electrifying, head-twisting guide to the most fundamental truths of the universe.

Philosophy of Mathematics

Autor: Thomas Bedürftig , Roman Murawski

Número de Páginas: 476

The present book is an introduction to the philosophy of mathematics. It asks philosophical questions concerning fundamental concepts, constructions and methods - this is done from the standpoint of mathematical research and teaching. It looks for answers both in mathematics and in the philosophy of mathematics from their beginnings till today. The reference point of the considerations is the introducing of the reals in the 19th century that marked an epochal turn in the foundations of mathematics. In the book problems connected with the concept of a number, with the infinity, the continuum and the infinitely small, with the applicability of mathematics as well as with sets, logic, provability and truth and with the axiomatic approach to mathematics are considered. In Chapter 6 the meaning of infinitesimals to mathematics and to the elements of analysis is presented. The authors of the present book are mathematicians. Their aim is to introduce mathematicians and teachers of mathematics as well as students into the philosophy of mathematics. The book is suitable also for professional philosophers as well as for students of philosophy, just because it approaches philosophy from the...

The Corpses of Times Generations

Autor: Richard J. Kosciejew

Número de Páginas: 589

Anyone who has ever tried to present a rather abstract scientific subject in a popular manner knows the great difficulties of such an attempt. Either he succeeds in being intelligible by concealing the core of the problem and by offering to the reader only superficial aspects or vague allusions, thus of deluding the reader by arousing in him the deceptive illusion of comprehension; Or else he gives an expert account of the problem, but as the untrained reader is unable to follow the exposition and becomes discouraged from reading any further. If these two categories are omitted from todays popular scientific literature, surprisingly little remains. But the little left is very valuable indeed. It is very important that the public is given an opportunity to experience-consciously and intelligently-the efforts and results of scientific research. It is not sufficient that each successive progression is taken up, elaborated, and applied by a few specialists in the field. Restricting the body of knowledge to a small group deadens the philosophical spirit of these people and leads to spiritual poverty. THE CORPSES OF TIMES GENERATIONS represents a valuable contribution to popular...

New Perspectives on Mathematical Practices

Autor: Bart Van Kerkhove

Número de Páginas: 248

This volume focuses on the importance of historical enquiry for the appreciation of philosophical problems concerning mathematics. It contains a well-balanced mixture of contributions by internationally established experts, such as Jeremy Gray and Jens Hoyrup; upcoming scholars, such as Erich Reck and Dirk Schlimm; and young, promising researchers at the beginning of their careers. The book is situated within a relatively new and broadly naturalistic tradition in the philosophy of mathematics. In this alternative philosophical current, which has been dramatically growing in importance in the last few decades, unlike in the traditional schools, proper attention is paid to scientific practices as informing for philosophical accounts.

Dictionary of Paradox

Autor: Glenn W. Erickson , John A. Fossa

Número de Páginas: 232

Dictionary of Paradox is a fascinating reference work for scholars, students, and the general public. It describes those paradoxes that are either especially interesting today or that have a continuing interest from the historical point of view. Each main entry consists of four parts: a statement of the paradox, an explanation of its paradoxicality, a discussion of attempted or accepted resolutions, and a list of readings. In addition, there are other entries explaining terms relating to paradoxes, listing alternative names for the main entries, or discussing variants of the entries. All information is extensively cross-referenced.

Bridges to Infinity

Autor: Michael Guillen

Número de Páginas: 222

This book is an endlessly fascinating journey through a mathematician's looking glass.

The Infinite

Autor: A. W. Moore

Número de Páginas: 296

This historical study of the infinite covers all its aspects from the mathematical to the mystical. Anyone who has ever pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of the subject. Beginning with an entertaining account of the main paradoxes of the infinite, including those of Zeno, A.W. Moore traces the history of the topic from Aristotle to Kant, Hegel, Cantor, and Wittgenstein.

A Brief History of Infinity

Autor: Brian Clegg

Número de Páginas: 276

Infinity is a concept that fascinates everyone from a seven-year-old child to a maths professor. So remarkable and strange is it that contemplating it has driven at least two great mathematicians over the edge into insanity. Where did the concept of infinity come from? Who were the people who originally defined and later refined this paradoxical quantity? Why is infinity, a concept we can never experience or truly grasp, at the heart of science? How can some infinities be bigger than others? An exploration of the most mind-boggling feature of maths and physics, this work examines amazing paradoxes, for example Hilbert's Hotel. This imaginary resort has an infinite number of rooms, which all happen to be occupied. Unfortunately an endless coach turns up carrying an infinite number of new guests. It's not a problem though - it's easy to prove they can all be accommodated. The book also looks at: the people who devised and refined the concept, the many mind-bending paradoxes of infinity, infinity's place at the heart of mathematics and science in processes such as calculus, how dividing by zero brings infinity into view and infinity and cosmos.

The Encyclopedia of Philosophy

Autor: Paul Edwards

Número de Páginas: 1112

Treats eastern & western philosophy; it deals with ancient, medieval and modern philosophy; and it discusses the theories of mathematicians, physicist, biologist, sociologists, psychologists, moral reformers and religious thinkers where these have had an impact on philosophy.

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