Understanding the Infinite

by Shaughan Lavine

Overview

How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge.

Product Details

ISBN-13:	9780674921177
Publisher:	Harvard University Press
Publication date:	01/13/1998
Edition description:	REPRINT
Pages:	376
Product dimensions:	6.38(w) x 9.25(h) x 0.80(d)

About the Author

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

Table of Contents

Introduction

Infinity, Mathematics' Persistent Suitor

Incommensurable Lengths, Irrational Numbers

Newton and Leibniz

Go Forward, and Faith Will Come to You

Vibrating Strings

Infinity Spurned

Infinity Embraced

Sets of Points

Infinite Sizes

Infinite Orders

Integration

Absolute vs. Transfinite

Paradoxes

What Are Sets?

Russell

Cantor

Appendix: Letter from Cantor to Jourdain, 9 July 1904

Appendix: On an Elementary Question of Set Theory

The Axiomatization of Set Theory

The Axiom of Choice

The Axiom of Replacement

Definiteness and Skolem's Paradox

Zermelo

Go Forward, and Faith Will Come to You

Knowing the Infinite

What Do We Know?

What Can We Know?

Getting from Here to There

Appendix

Leaps of Faith

Intuition

Physics

Modality

Second-Order Logic

From Here to Infinity

Who Needs Self-Evidence?

Picturing the Infinite

The Finite Mathematics of Indefinitely Large Size

The Theory of Zillions

Extrapolations

Natural Models

Many Models

One Model or Many? Sets and Classes

Natural Axioms

Second Thoughts

Schematic and Generalizable Variables

Bibliography

Index

Editorial Reviews

This book is a defense of set theory that is more persuasive and compelling than many of the main-line defenses. In form it is an unusual combination of history, philosophy, mathematics, logic, and exposition, which is woven together in a very skillful and seamless way… If Lavine is right, received views of what various important figures in this history actually thought will have to be modified.

Solomon Feferman

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.

New Scientist - Ian Stewart

Highly original… [This book] deserves to become a focus of philosophical discussion.

Hartry Field

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.

Choice - D. V. Feldman

What one mostly finds in the recent philosophy of mathematics are variations on a few basic themes. Often the variations are pursued with originality, ingenuity, and verve, but, still, the number of fundamental ideas and positions is rather small. In Lavine’s book, we find a new idea. It is an important idea which is likely to significantly alter the way we think about the infinite, and it is developed with enormous clarity, both in thinking and writing. The result is a book which is sure to be widely read and discussed, and which is sure to leave a lasting impression on the philosophy of mathematics.

Vann McGee

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

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