Springer Nature. - Approaching Infinity

Approaching Infinity

Also by Michael Huemer

ETHICAL INTUITIONISM

THE PROBLEM OF POLITICAL AUTHORITY

SKEPTICISM AND THE VEIL OF PERCEPTION

Approaching Infinity

Michael Huemer

Professor of Philosophy, University of Colorado at Boulder, USA

Michael Huemer 2016

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First published 2016 by
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ISBN: 9781137560858 hardback ISBN: 9781137560865 paperback

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Library of Congress Cataloging-in-Publication Data

Names: Huemer, Michael, 1969 author.

Title: Approaching infinity / Michael Huemer.

Description: New York : Palgrave Macmillan, 2016. | Includes index.

Identifiers: LCCN 2015040546| ISBN 9781137560858 (hardback) | ISBN 9781137560865 (pbk.)

Subjects: LCSH: Infinite.

Classification: LCC BD411 .H84 2016 | DDC 111/.6dc23

LC record available at http://lccn.loc.gov/2015040546

For Iskra

Contents

List of Figures

Preface

For many years, I have made a point of confounding my philosophy students with a variety of paradoxes, including several paradoxes of the infinite. And for many years, I tried myself to think through these paradoxes, without success. I list seventeen of these paradoxes in there were three philosophical questions about the infinite that exercised me.

Here is the first issue. It seems that there are some infinite series that can be completed. For instance, for an object to move from point A to point B, it must first travel half the distance, then half the remaining distance, then half the remaining distance, and so on. According to one famous argument, because this is an infinite series, the object can never reach point B. The proper response seems to be to insist that one can in fact complete an infinite series.

On the other hand, it seems that there are other infinite series that cannot be completed. For instance, imagine a lamp that starts out on, then is switched off after half a minute, then back on after another quarter minute, then off after another eighth of a minute, and so on. At the end of one minute, is it on or off? The proper response seems to be that one could not complete such an infinite series of switchings. We might think this is because an infinite series, by definition, is endless, and one cannot come to the end of something that is endless.

In other words, the solution to the first puzzle (one can complete an infinite series) seems to be the opposite of the solution to the second puzzle (one cannot complete an infinite series). Thats puzzling. What we would like to say is that some infinite series are possible, and others are impossible. But why? What is the difference between the infinite series of halfway-motions, and the infinite series of lamp-switchings? Thats the first philosophical question I wanted to answer.

This puzzle is tied up with a popular genre of arguments in philosophy: it is common to argue that some philosophical theory must be rejected because it leads to an infinite regress. For instance, if every event has a cause, and the cause of an event is also an event, then there must be an infinite regress of causes. To avoid this, some say, we should reject the idea that everything has a cause; instead, we should posit a first cause, something that caused everything else and was itself uncaused.

But almost as common as infinite regress arguments is a certain type of response, which claims that there is nothing wrong with the infinite regress. For instance, some say that we should simply accept that there is an infinite series of causes stretching into the past forever. Now, there seems to be wide agreement among philosophers that some but not all infinite regresses are bad; but there has been no consensus on which regresses are bad (vicious) and which benign. Hence, the second philosophical question I wanted to answer: What makes an infinite regress vicious or benign?

Finally, I started reflecting on infinities in science, where there seems to be an analogous issue. Some infinities are considered bad for instance, the infinite energy density and infinite spacetime curvature of a black hole are considered problems in astrophysics. They are thought to indicate a breakdown of accepted theories (notably, general relativity), and astrophysicists have been trying to devise new theories that eliminate these infinities. But there are other infinities that no one seems worried about: no one seems to consider the notion of an infinitely large universe, or the notion of an infinite future, to be problematic. Why is this? What makes some infinite quantities in a theory problematic while others are perfectly acceptable?

One can see that these three questions Why are some infinite series completeable and others not?, Why are some infinite regresses vicious and others not?, and Why are some infinite quantities problematic and others not? are quite similar, and so perhaps they have a common answer. After several years of puzzlement concerning the first two questions, I finally tried to connect them with the third question. It was then that I thought of a theory that seems to me to account for which sorts of infinities are possible and which impossible. After publishing one paper on the subject (Virtue and Vice among the Infinite), I decided to expand my ideas into this book.

I would like to thank Stuart Rachels for the conversations about the infinite that led to this expansion. In addition, I am grateful for the comments of Adrian Moore and Matt Skene on earlier drafts of the manuscript, and the questions and comments of Peter Klein, Ted Poston, Jeanne Peijnenberg, David Atkinson, and the other participants of the Infinite Regress Workshop organized by Scott Aiken at Vanderbilt University in October, 2013, where I presented my earlier paper on the subject. If there are any errors in this book, as surely there must be, they are the fault of one of these other philosophers most likely Peter Klein, because he supervised my PhD and should have trained me better.