Infinity: A Very Short Introduction

Description

Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large (infinite) is intimately related to the infinitely small (infinitesimal). Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas of mathematics rest upon some version of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals. In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. Cover INFINITY: A Very Short Introduction Copyright Dedication Contents List of illustrations Introduction Outline of the book Chapter 1: Puzzles, proofs, and paradoxes Nine appeals to infinity Largest number Diagonal of a square Area of a circle Light switch Balls in the bag One third in decimals Squares and numbers Hilbert’s hotel Grandi’s proof of the Creation Solutions and comments Largest number Diagonal of a square Area of a circle Light switch Balls in the bag One third in decimals Squares and numbers Hilbert’s hotel Grandi’s proof of the Creation Chapter 2: Encounters with the infinite Finite and infinite Not just a big number Infinity in arithmetic Irrational numbers Numbers through the microscope Discrete and continuous Chapter 3: Historical views of infinity Warning: may contain infinity Preconceptions of the human mind Zeno’s paradoxes Achilles and the tortoise The dichotomy The arrow The stadium Discussion of Zeno’s paradoxes Achilles and the tortoise The dichotomy The arrow Philosophers and the infinite Infinity in Christian belief Modern era Could there be a largest number? Chapter 4: The flipside of infinity Proof of Archimedes’s theorem Calculus and its precursors Limits Infinite series Infinitesimals revenant Non-standard analysis Chapter 5: Geometric infinity Linear perspective Beyond the blue horizon The line at infinity Parallels Directions Perspective drawing Chapter 6: Physical infinity Infinity in optics Infinity in Newtonian gravity Infinity in relativity Infinite universe? Curvature Chapter 7: Counting infinity Counting and matching Cantor’s transcendence proof Fourier analysis Set theory Transfinite cardinals Precursors to Cantor Transfinite ordinals Cantor and Wittgenstein Mathematics, philosophy, and religion Processes and things Mathematical existence References Further reading Chapter 1: Puzzles, proofs, and paradoxes Chapter 2: Encounters with the infinite Chapter 3: Historical views of infinity Chapter 4: The flipside of infinity Chapter 5: Geometric infinity Chapter 6: Physical infinity Chapter 7: Counting infinity Publisher’s acknowledgements Index SOCIAL MEDIA: Very Short Introduction