A Brief History of Infinity

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Zellini Thoughtfully Distills the Wisdom of Philosophers, Theologians, and Mathematicians.......2006-01-29

A Brief History of Infinity is a serious, in depth study of man's evolving concept of infinity. Paolo Zellini's thoughtfully examines and distills the ideas of philosophers, theologians, writers, and mathematicians.

Zellini's terse style requires close attention and I found it necessary to reread many sections. Zellini begins with Aristotle's negative notion of infinity (apeiron) as an incomplete and unrealized potentiality. This character of non-existence implicit in apeiron largely explains the inability (or refusal) of Greek mathematicians to introduce a concept of an actual, or real infinity.

The inexhaustibility of the unlimited and the impossibility of finding an absolute minimum or maximum became focal points of discussion in Oxford and Paris in the fourteenth century. Discourse on these topics remained important in the Renaissance, continued with Leibniz and Newton, culminating in the nineteenth century with Cauchy's and Weierstrass' definitive formulation of infinitesimal calculus.

Having less familiarity with philosophy, I found it profitable to skip (only for awhile) to later chapters that more directly addressed mathematical infinity, a topic of especial interest to me. These chapter included The Principle of Indiscernibles - Classes; The Actual Infinite - Indefinite and Transfinite; and The Antinomies, or Paradoxes of Set Theory.

Paolo Zellini's sources are wide ranging, almost intimidatingly so. We readers encounter the philosophical thoughts of the Platonists, Aristotle, the Pythagoreans, Anaximander, the Chaldeans, Duns Scotus, St. Thomas Aquinas, Giordano Bruno, Nicholas of Cusa, Raymond Lull, Descartes, Leibniz, Goethe, Kant, Hegel, Russell, Simone Weil, Quine, Popper, Wittgenstein, and many others. Similarly, on the literary front we meet Cervantes, Kafka, Borges, Musil, and others.

Mathematicians are prominent also. Zellini discusses the provocative ideas of Descartes, Newton, Leibniz, Dedekind, Poincare, Cauchy, Weierstrass, Bolzano, Frege, Du Bois-Raymond, Cantor, Russell, Whitehead, Godel, Von Neumann, Zermelo, Skolem, Brouwer, and many others.

In the end Paolo Zellini's analysis leans away from Cantor's actual mathematical infinity and toward a potential infinity, somewhat in accordance with Brouwer's finite constructive methods (intuitionism).

Key Idea: I was intrigued with Hermann Weyl's conciliatory observation: the infinite is intuitively accessible as an indefinitely open field of possibility, and in this respect would seem analogous to a series of numbers that can be extended unlimitedly. Yet completeness, the so-called actual infinite, lies beyond our reach. Nonetheless, the demands for a totality impel the mind to imagine the infinite, using some symbolic construction, as a closed entity. Hence, the primary philosophical interest of mathematics should consist in attaining a fundamental understanding of these symbolic constructions.

More a "tour de force" than a book to learn from.......2005-10-03

This book tells the history of infinity and the ways it was dealt with by thinkers of different times and cultures.
The subjects touched upon by this (relatively short) book extends overs an astonishingly wide range: philosophy, medieval theology, mathematics, logic, litterature
But although it is well written and in an engaging style, it is certainly no light reading
My main critic is that it is too elliptic: many items are introduced matter-of-factly without an attempt to an expository introduction. To a person not acquainted with the domain these remain just words without meaning.
For instance, what is the "axiom of choice" or Cantor's "transfinite" or a "monadology"? These words just appear "out of the blue". This book expects its reader to be familiar with philosophy (from Aristotle to Leibniz), medieaval theology ( A third of the book), modern mathematics and logic and what else; do you know anybody who is, except the author?
To sum it up, I didn't understand 60% of this book although I must say it awoke my curiosity about all those subjects (esp a writer named Musil who is often mentionned)

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Enjoyable Excursion into the Realm of Infinity
To infinity and beyond .................

A Brief History of Infinity
Brian Clegg
Manufacturer: Constable and Robinson
ProductGroup: Book
Binding: Paperback

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ASIN: 1841196509

Customer Reviews:

Enjoyable Excursion into the Realm of Infinity .......2006-08-06

A Brief History of Infinity - The Quest to Think the Unthinkable by Brian Clegg is a good introduction for the layman to the historical development of the concept of infinity. Clegg's book is a good book, but not an exceptional one. I found the first half to be less engaging, as perhaps the sections on Greek mathematics and the historical development of calculus were too familiar. The second half is much better, especially the discussion of Cantor's transfinite numbers. On balance, A Brief History of Infinity rates four stars and should appeal to a wide audience.

Clegg touches on a wide range of topics, often exploring unexpected connections and tangential matters. Sometimes his side trips are diverting and enjoyable, but at other times I found these digressions to be overly distracting. (The discussions on quantum physics seemed unnecessary; there so many good books for the layman available.) But to be fair, infinity is a broad topic and other readers may well find Clegg's far ranging approach to be stimulating and enjoyable.

Clegg addresses in a credible fashion Galileo's investigations of infinity, the battle between Newton and Leibniz (and Bishop Berkeley too) over infinitesimals, Cantor's transfinite numbers, and Robinson's non-standard analysis. These non-technical discussions are neither too superficial, nor too vague.

For readers willing to delve deeper into the mathematics of infinity, I highly recommend The Philosophy of Set Theory - An Historical Introduction to Cantor's Paradise by Mary Tiles. This is not an easy book as it primarily targets advanced students in philosophy and mathematics. Nonetheless, it is within reach of a persistent reader. Some sections can be read stand-alone; three mid-chapters (Numbering the Continuum, Cantor's Transfinite Paradise, and Axiomatic Set Theory) are outstanding and in themselves are worth the price of this book.

Going a step farther, a math major might wish to tackle Georg Cantor's Contributions to the Founding of the Theory of Transfinite Numbers. I particularly recommend the Dover reprint of the same title as it contains a lengthy introduction by the mathematician Philip E. B. Jourdain.

There is yet a third book to consider, one that has the same title as Brian Clegg's book. A Brief History of Infinity by Paolo Zellini (translated by David Marsh) "explores every aspect of infinity, distilling the wisdom of philosophers, artists, mathematicians, and theologians over the millennia". Less mathematical than my other two recommendations, Zellini's work is nonetheless quite challenging. It provides a detailed examination of philosophical issues underlying the concept of infinity.

To infinity and beyond ........................2006-06-15

There is almost a pun in the title: how can you have a brief history of infinity? However, this book by Brian Clegg is part of a series, so it is stretching bounds of probability to suggest that the whole series is named just to get a pun in a title! I have not read any of the others in the series, but from their titles, they seem to be of a historical nature, rather than concepts and ideas. The sub-title of this volume is also interesting: "the quest to think the unthinkable". That statement gives a key to the book. What does the notion of infinity give us?

For me the first 100 pages of Clegg's book do not really work. It is both slow, and nebulous. However, after that, it opens up, and gives a good narrative of the progress towards the unreachable. Strangely, that is not just the infinitely large, but also the infinity small. Somehow "proper" numbers are sandwiched between the immeasurable at either ends of the spectrum. Clegg falls on the side of the usefulness of ideas, rather than the sheer understandability. Calculus, in both the Newtonian and Leibnizian forms, is introduced as "something that works". It was only later that the need for infinitely small is eliminated from calculus - by then this mathematical tool had been very useful in solving many real problems.

After the slow labouring start, the volume CAN be a beginner's guide to the subject, but is probably not be for the faint-hearted. It will help your understanding if you are familiar with ideas from modern mathematical thinking. If not, the notion of there being different `levels' of infinity can come as rather a jolt. How can two numbers, both of which are not countable, be different? How can there be the same number of odd numbers as there are of odd-and-even numbers? It is not surprising that the man behind set theory, Georg Cantor, lost his sanity when investigating the infinitely large.

In essence, the book covers so much more than just infinity. It really does go "to infinity and beyond". Infinity is a strange idea, but one that underlies many of the advances that have taken place since the start of the 20th century. Advances in number theory, cosmology, physics and atomic physics can be better understood if a basic understanding of infinity is grasped. Clegg covers these parts well.

The title was not a pun. However, Clegg does not disappoint in his last sentence. "When it comes to infinity, the possibilities are, perhaps inevitably, endless". I suppose he had to say that, didn't he?

Peter Morgan, Bath, UK [...]

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Brief History of Infinity
Brian Clegg
Manufacturer: Constable & Robinson
ProductGroup: Book
Binding: Hardcover
ASIN: B000K85VG0

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A Brief History of Infinity
Paolo Zellini
Manufacturer: Penguin Global
ProductGroup: Book
Binding: Paperback
ASIN: B000OJHPD2

Book Description

Deregulation in the European Union examines the environmental implications of economic deregulation through case studies of energy, transport and water sectors, and discusses options for deregulation, such as self-regulation, negotiated agreements and environmental management systems.

This book represents the most up-to-date research from Europe on environmental deregulation. It also covers more general issues, including links between regulation, environmental protection and competitiveness, and the importance of better availability of environmental information.

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This book examines the environmental implications of economic deregulation through case studies of the energy, transport and water sectors, and particularly the challenges facing the countries of Central and Eastern Europe.

Environmental Costs and Liberalization in European Air Transport: A Welfare Economic Analysis (Transport Economics, Management, and Policy)

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Environmental Costs and Liberalization in European Air Transport: A Welfare Economic Analysis (Transport Economics, Management, and Policy)
Youdi Schipper
Manufacturer: Edward Elgar Pub
ProductGroup: Book
Binding: Hardcover

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Book Description

With air transport becoming an increasingly vital part of the economy, the regulatory reform of this market has been a major development in European political economy. This book focuses on two market failures within the airline industry - market power and environmental externalities - and analyses how they have been affected by deregulation. The author employs economic models complemented by extensive empirical research, to demonstrate how the introduction of competition, brought about by liberalization, has resulted in considerable consumer benefits.

The author argues that these benefits, such as increased choice through the expansion of operations, must be off set against increased environmental costs including greater noise pollution and emissions, not to mention the reduction of profits that often accompany market liberalization. In the process the book tackles a number of important issues including the background and history of airline regulation in the EU, the basic policy trade-off between monopoly power and external costs, monetary valuation of externalities, and the relationship between airline scheduling and external costs. Perhaps surprisingly, the author concludes that even in the presence of environmental costs, the introduction of competition in airline markets has resulted in net welfare improvements.

Book Description

The second in CEPR's annual Monitoring European Deregulation (MED) series, this report explores the economic and regulatory aspects of a single European market for electricity and provides a basis for policy choices both at national and EU levels. The report combines analyses of key issues in electricity market integration and liberalization with evaluations of practical experiences in selected European countries: France, Germany, Norway, Spain, Sweden, and the UK. Key issue include: to what extent competition on national electricity markets is a necessary requirement for the integration of these markets, and the design of national electricity markets in which competition in generation and supply is allowed.

Customer Reviews:

information and presentation.......2000-04-28

This book is strong with data and valuable information collected from various journals and reports. Taking this into consideration a reader can find easy to understand information on the political, social, economic, legal, and structural aspects of the European electricity market (in the first part of the book). The second part of the book presents studies of markets in countries including France, the UK, Nordic countries, and Hungary (I think the publisher forgot about hungary..but it's in there). The only weakness I have noticed in this book is the presentation. Having numerous authors may have made it difficult to avoid repetitive analysis and redundancies. Some sections treat similar issues in similar manners while some other sections treat them in very different ways making it hard to find continuity through the whole book (actually through any part at all). Although the reader will want to start with part-one before attacking part-two, he will be able to read chapters in each part in any order. Overall, the topic of this book is so broad and complex that a reader can only expect a good bigger-picture understanding of the subject. The appendix section containing definitions makes this book a challenging but feasible venture for a beginner in the field.

information vs. presentation.......2000-04-28

This book is strong with data and valuable information collected from various journals and reports. Taking this into consideration a reader can find easy to understand information on the political, social, economic, legal, and structural aspects of the European electricity market (in the first part of the book). The second part of the book presents studies of markets in countries including France, the UK, Nordic countries, and Hungary (I think the publisher forgot about hungary..but it's in there). The only weakness I have noticed in this book is the presentation. Having numerous authors may have made it difficult to avoid repetitive analysis and frequent redundancies. Some sections treat similar issues in similar manners while some other sections treat them in very different ways making it hard to find continuity through the whole book (actually through any single part at all). Although the reader will want to start with part-one before attacking part-two, he will be able to read chapters in each part in any order. Overall, the topic of this book is so broad and complex that a reader can only expect a good bigger-picture understanding of the subject. The appendix section containing definitions makes this book a challenging but feasible venture for a beginner in the field while the references and links provided in this book offer resources for further studying.

European Telecommunications Liberalisation (Routledge Studies in the European Economy)

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European Telecommunications Liberalisation (Routledge Studies in the European Economy)
Kjell Eliassen
Manufacturer: Routledge
ProductGroup: Book
Binding: Library Binding

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All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books
ASIN: 0415187818

Book Description

This book examines the process and consequences of telecommunications liberalization in the context of an ever closer European Union. The creation of a single market for telecommunications and of a wider European single market mirror one another. Telecommunications are also something of a test case for the privatization process, as this sector has traditionally been a state monopoly. The volume approaches the European experience from three angles; the politics of regulation and the process of liberalization in the EU (including case studies of the UK, France, and Germany); and increasing global economic interdependence makes international comparisons essential. It compares the EU experience with that of the Czech Republic, Israel and Thailand and examines the consequences of technology and continuous innovation.

Flying in the Face of Competition: The Policies and Diplomacy of Airline Regulatory Reform in Britain, the USA and the European Community 1968-94

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Flying in the Face of Competition: The Policies and Diplomacy of Airline Regulatory Reform in Britain, the USA and the European Community 1968-94
Alan P. Dobson
Manufacturer: Avebury Aviation
ProductGroup: Book
Binding: Hardcover

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ASIN: 0291398219

Flying into the Future: Air Transport Policy in the European Union

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Flying into the Future: Air Transport Policy in the European Union
Kenneth Button , Kingsley Haynes , and Roger Stough
Manufacturer: Edward Elgar Pub
ProductGroup: Book
Binding: Hardcover

General | Transportation | Nonfiction | Subjects | Books

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ASIN: 1858987997

Book Description

Flying into the Future explores the organization of air transport in the European Union. It analyses the nature of the industries supplying air transport services, the institutional structure of air transport services, and impediments to increased efficiency in the provision of air transport.

The reduction in institutional barriers and regulations has led to a more efficient provision of air transport services in the EU. This book assesses the improvements in efficiency of air transport services, and highlights institutional and physical problems impeding further efficiency gains. The authors examine airline operations, and the ability of two or more transport systems to operate effectively in tandem. They also consider how to make the boundaries between different transport networks invisible, as well as discussing issues of national organization and the juridical structures which impede operations. The analysis examines both the internal European Union market for air transport services and the links between it and the rest of the world. Other key issues discussed include:

- EU air transport developments in the context of global markets - Comparisons of recent developments in aviation policy between the EU and the United States - The problems of congestion in the air transport industry in Europe - The growth and significance of airline alliances

The authors not only consider the economics of European air transport but also legal, political, technical and geographical issues. They explore the problems of providing air transport in the context of inadequate information, institutional constraints, inherent market imperfections and imprecise objectives.

Book Description

How was the European airline industry transformed from national fragmentation in 1957 to a point in 2006 where the European Commission could negotiate with the US for an Open Aviation Area on behalf of all twenty five members of the European Union? What can explain the change in mindset that saw conservatism and the tight regulation of the airline industry replaced with increased competitiveness and the subsequent rise of the low cost no-frills airline? In his new book, Alan P. Dobson draws from a mass of European documentation, including interviews with officials and airline executives and a unique collection of personal papers, to answer these questions.

Dobson traces the liberalization of the airline industry from the 1970s right through to the present day, illustrating how integration came about and which forces were driving it. The approach is comprehensive, focusing on the work of the Commission, market forces, the voices of airline industry interests and on key individuals in the policy making process. The story concludes with an overview of the attempts by the EU and the US to create an Open Aviation Area that would embrace their respective airline industries allowing free operation of services throughout their territories and mutual ownership and control of each others airlines.

This book is a key resource for students and researchers interested in aviation and international policy, as well as academics engaged with European integration, globalism and economic history.

In Search of Open Skies:Law and Policy for a New Era in International Aviation

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In Search of Open Skies:Law and Policy for a New Era in International Aviation
Brian Havel
Manufacturer: Kluwer Law International
ProductGroup: Book
Binding: Hardcover

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ASIN: 9041103538

Book Description

European Transport Commissioner Neil Kinnock praises this book as `an important contribution' to the open skies debate in international aviation, noting that it `breaks new ground' in its comparative treatment of US and EU airline deregulation. The book exposes the anticompetitive bias of the archaic regime of bilateral air transport treaties launched over fifty years ago at Chicago. Repudiating bilateralism, it interweaves the successes and limitations of the US and EU air transport liberalization programs to define a 21st century multilateral open skies solution that, in Commissioner Kinnock's view, poses `intellectual challenges to all of us whose professional mission is the evolution of new law and policy for the international air transport industry.' This is the first full-length study of the world airline industry to evaluate: - the new American international aviation policy - the European Commission's controversial campaign for a mandate to enter multilateral air transport negotiations with the US - code-sharing and global airline alliances - the findings of important industry studies by the US presidential airline commission and the EU's Comite des Sages. A major feature of Professor Havel's analysis is his unprecedented use of the unpublished transcripts and archival materials of the blue-ribbon US presidential panel. The book is intended for an academic and professional audience, in both the United States and Europe, interested in the complex legal and policy issues that currently affect the world's most visible service industry.

Industrial Economics and Organization: A European Perspective

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Industrial Economics and Organization: A European Perspective
David Jacobson , and Bernadette Andreosso-O'Callaghan
Manufacturer: McGraw-Hill Companies
ProductGroup: Book
Binding: Paperback

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ASIN: 0077078896

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