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Understanding the Infinite

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Review

Understanding the Infinite is a remarkable blend of mathematics, modern history, philosophy, and logic, laced with refreshing doses of common sense. It is a potted history of, and a philosophical commentary on, the modern notion of infinity as formalized in axiomatic set theory...An amazingly readable [book] given the difficult subject matter. Most of all, it is an eminently sensible book. Anyone who wants to explore the deep issues surrounding the concept of infinity...will get a great deal of pleasure from it. (Ian Stewart New Scientist)How, in a finite world, does one obtain any knowledge about the infinite? Lavine argues that intuitions about the infinite derive from facts about the finite mathematics of indefinitely large size...The issues are delicate, but the writing is crisp and exciting, the arguments original. This book should interest readers whether philosophically, historically, or mathematically inclined, and large parts are within the grasp of the general reader. Highly recommended. (D. V. Feldman Choice)

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From the Back Cover

An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions that have dogged this theory since its introduction in the late nineteenth century.

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Product details

Paperback: 376 pages

Publisher: Harvard University Press (January 13, 1998)

Language: English

ISBN-10: 0674921178

ISBN-13: 978-0674921177

Product Dimensions:

6.2 x 1 x 9.2 inches

Shipping Weight: 1.1 pounds (View shipping rates and policies)

Average Customer Review:

4.7 out of 5 stars

5 customer reviews

Amazon Best Sellers Rank:

#1,609,470 in Books (See Top 100 in Books)

This book I bought a few years ago but only started reading 4 months ago and just finished. I must say that it was not a light read and it requires certain mathematical maturity beyond undergraduate courses. The first part deals with Cantor and Zermelo set theories and axioms. It is very dry sometimes and chapters are long which was not good for me because I was only reading 10 - 12 pages per week while commuting. In many places the author assumes that a reader already knows a lot about logic and set theory, for example, at the end, he devotes a page or two about Putman modal logic and uses freely its quantifiers without explaining them. Some glossary at the end would have greatly benefited this book. What I found clarifying is the fact that there are two foundations of set theory: the notions of logical and combinatorial collections. For the latter the Axiom of Choice is self-evident and is no longer controversial. The second part starting from chapter VI is more philosophical and concerns with epistemology and ontology of the infinite. At least at the beginning it clarifies the difference between potential and actual infinity. In the middle we see the use of schemas to avoid quantifiers. At the end of the book the author discusses the theory of indefinite large and small, its extrapolations to infinite and provides examples from mathematical analysis. The main theme of the book, as I understand it, is that our intuition about infinity arises from intuitive understanding of indefinitely large sets, their hierarchies and extrapolations.Thanks,Dmitry VostokovFounder of Literate Scientist Blog

I have taken a lot of math, up to the advanced undergrad level, but not much on this development of set theory, though I have come across it quite a bit in my reading. I found the book quite approachable, though I don't claim to be a great judge of the author's approach to a difficult subject and how it compares to other material on the subject. I was just happy to (sort of) get it.I confess that the whole notion of the infinite fascinates me.

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To help understand how QM (quantum mechanics) and GR (general relativity) collide you need to understnad planck constants and what lay beyond (the very small and large). This book helps to lay that foundation. A must read for anyone trying to bridge QM/GR. HS/College math required to grasp hard concepts but a good read for lay people.

The 20th century saw more advances in knowledge than could filter down to general society. Relativity and Quantum Theory are part of the vernacular, even if the popular conceptions are not necessarily good generalizations of their counterparts in science. The corresponding advances in philosophy, however, have stayed more in the province of academia, largely because philosophy itself has become highly technical; but the physics of beyond-everyday-experience have demanded these advances, primarily in epistemology, because the fundamental questions of science today are of meaning and understanding.Understanding the Infinite is a work of epistemology. Its contribution to the foundations of general knowledge demand that it disseminate beyond academia, although the ground Lavine breaks requires the extensive citations and technical style he employs. The author poses and addresses the following question. If set theory is so intuitively self-evident and seemingly such a fundamental underpinning of all mathematics, why is it so hard to express technically and why has the axiomatization of set theory been so controversial? Set theory was the big idea which the mid-20th century educational establishment thought important enough to indoctrinate schoolchildren with in the guise of new math. Yet set theory never took root in popular consciousness, certainly not the notion of transfiniteness.Lavine starts out by dispelling the anecdotal account of the development of set theory, which has misled even professional mathematicians and philosophers to conclude "The fundamental axioms of mathematics...are to a large extent arbitrary and historically determined." He constructs what he claims is the correct historical development of set theory (I'll let historians of mathematics decide this) including sidetracks into Russell's failed program to equate mathematics and logic (and in the process dispels the significance of Russell's paradox), and von Neumann's axiomatization of set theory emphasizing functions. The outcome of his exposition is the Zermelo-Fraenkel axiomatization with the Axiom of Choice (ZFC), today's common form of set theory. These chapters by themselves could serve as an introduction to set theory, except that the Continuum Hypothesis is barely mentioned, since it plays no role in Lavine's program. Admittedly, he has nothing new to add.The main event is Lavine's epistemological tour-de-force. Building upon work of Jan Mycielski he introduces the reader to the concept of finitary mathematics and constructs a finitary ZFC, showing that this theory justifies the adoption of what he calls the "Axiom of Zillions" (indefinitely large sets) in which we have access to very large sets' ordinal, but not necessarily its predecessors. The final step is to show this all "intuitively" extrapolates to ZFC.QEF, QED.I introduced physics in the opening paragraph of this review because I see Lavine's rigorous treatise in the epistemology of mathematics as a contribution to the grand unification of physics, mathematics, and epistemology. Lavine treads lightly in the physical realm. He writes "...modern physics makes it seem likely that the physical universe is of finite extent..." All of the dominant cosmologies put forth in the 20th century incorporated this misdirection set off by general relativity. On a large scale the universe must be curved. Ironically Lavine published in 1994, just as new astronomical observations began whispering "in three dimensions the universe is Euclidean". If that whisper becomes a shout in the 21st century, as appears likely from the mounting evidence, physics will have to address the transfinite.The Calculus had to be put on a firm theoretical foundation so that it could be used as a tool to advance knowledge without justifying its use. We may see that Lavine's epistemology will do the same for set theory and transfinite numbers.

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