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Understanding the Infinite Paperback – February 12, 1998

ISBN-13: 978-0674921177 ISBN-10: 0674921178

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Product Details

  • Paperback: 376 pages
  • Publisher: Harvard University Press (February 12, 1998)
  • Language: English
  • ISBN-10: 0674921178
  • ISBN-13: 978-0674921177
  • Product Dimensions: 9.2 x 6.4 x 1 inches
  • Shipping Weight: 1.1 pounds (View shipping rates and policies)
  • Average Customer Review: 4.8 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Best Sellers Rank: #466,287 in Books (See Top 100 in Books)

Editorial Reviews

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)

About the Author

Shaughan Lavine is Associate Professor of Philosophy at the University of Arizona.

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25 of 25 people found the following review helpful By Jack Fox on March 7, 2004
Format: Paperback
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.
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5 of 5 people found the following review helpful By Dmitry Vostokov on July 25, 2009
Format: Paperback Verified Purchase
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 Vostokov
Founder of Literate Scientist Blog
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Format: Kindle Edition Verified Purchase
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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2 of 4 people found the following review helpful By Dennis K. Morgan on October 7, 2009
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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.
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