The Kalam Cosmological Argument for God 9781591024736, 2006027553

Citation preview

The

K al a m

Cos m ol og i c al A rg u m e nt f o r G od

M

STUDIES IN ANALYTIC PHILOSOPHY Quentin Smith, Series Editor The Double Content of Art John Dilworth God and Metaphysics Richard M. Gale The Ontology of Time L. Nathan Oaklander The Kalam Cosmological Argument for God Mark R. Nowacki Philosophy of Religion, Physics, and Psychology: Essays in Honor of Adolf Grunbaum Aleksandar Jokic Reference and Essence, 2nd edition Nahan U. Salmon

STUDIES IN ANALYTIC PHILOSOPHY Quentin Smith, Series Editor

The

K al a m

Cos m ol og i c al A rg u m e nt f o r G od

Mark R.

Nowacki

Prometheus Books 59 John Glenn Drive Amherst, New York 14228-2197

Published 2007 by Prometheus Books The Kalam Cosmological Argument for God. Copyright © 2007 by Mark R. Nowacki. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, digital, electronic, mechanical, photocopying, recording, or otherwise, or conveyed via the Internet or a Web site without prior written permission of the publisher, except in the case of brief quotations embodied in critical articles and reviews. Inquiries should be addressed to Prometheus Books 59 John Glenn Drive Amherst, New York 14228–2197 VOICE: 716–691–0133, ext. 207 FAX: 716–564–2711 WWW.PROMETHEUSBOOKS.COM 11 10 09 08 07

5 4 3 2 1

Library of Congress Cataloging-in-Publication Data Nowacki, Mark R. Kalam cosmological argument for God / Mark R. Nowacki. p. cm. Includes bibliographical references and indexes. ISBN 978–1–59102–473–6 (hardcover : alk. paper) 1. God—Proof, Cosmological. 2. Cosmology. 3. Creation. 4. Religion and science. I. Title. BT103.N69 2006 212'.1—dc22 2006027553 Printed in the United States of America on acid-free paper

To Chuen

CONTENTS

Acknowledgments

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Introduction 1. What Is the KCA? 2. Why This Book Was Written 3. Order of Treatment and How to Use This Book

13 13 15 16

PART 1. THE KALAM COSMOLOGICAL ARGUMENT CONTEMPORARY ANALYTIC PHILOSOPHY: THE STATE OF THE QUESTION IN

Chapter 1. The Kalam Argument Described 4. Craig’s Version of the KCA as Presented in TKCA 4.1 Craig on Premise 1: Whatever Comes to Be Has a Cause of Its Coming to Be 4.2 Craig on Premise 2: The Universe Came to Be 4.2.1 Preliminary Remarks and Terminology

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27 27 27 30 31

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CONTENTS

4.2.2 Selective Exposition of Cantorian Transfinite Number Theory 4.2.2.1 Brief History of Mathematical Speculation on the Infinite 4.2.2.2 Cantor and the Development of Set Theory 4.2.3 Reflections on and Reactions to Cantor 4.2.3.1 Taxonomy of Positions within the Philosophy of Mathematics 4.2.3.2 Paradoxes in Paradise 4.2.3.3 Reactions to the Paradoxes 4.2.4 Craig’s Argumentative Strategy for Premise 2 4.2.5 Argument (A) 4.2.5.1 Premise (ii) of Argument (A) 4.2.5.2 Premise (i) of argument (A) 4.2.6 Argument (B) 4.2.7 Argument (C) 4.3 Craig on the Conclusion: The Universe Has a Cause of Its Coming to Be Chapter 2. A Taxonomy of Objections and Replies Argument Guide 5. Purpose, Method, and Notation 6. Division I—Objections Propaedeutic to the KCA 7. Division II—Objections to Premise (1) of the KCA 8. Division III—Objections to Premise (2) of the KCA 8.1 Division III.a—Objections to Argument (A) of the KCA 8.1.1 Objections to Premise (i) of Argument (A) 8.1.2 Objections to Premise (ii) of Argument (A) 8.2 Division III.b—Objections to Argument (B) of the KCA 8.2.1 Objections to Premise (a) of Argument (B) 8.2.2 Objections to Premise (b) of Argument (B) 8.3 Division III.c—Objections to Argument (C) of the KCA

33 33 35 42 44 46 48 52 54 57 62 65 69 71 103 103 104 106 111 114 115 115 116 123 124 127 134

CONTENTS

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8.3.1 Objections to Premise (a) of Argument (C) 135 8.3.2 Objections to Premise (b) of Argument (C) 136 9. Division IV—Objections to Conclusion (3) of the KCA 137

PART 2: FACTUAL MODALITY, SUBSTANCE METAPHYSICS, AND THOUGHT EXPERIMENTS IN THE KALAM COSMOLOGICAL ARGUMENT Chapter 3. Understanding the Modal Requirements of the KCA 10. Appeals to Logical Possibility in Objections to the KCA 11. Outline of the Argument 12. Establishing That the KCA Requires More Than Logical Possibility 13. Modal Distinctions according to Braine 14. Thought Experiments and the KCA Chapter 4. Substances and Substantial Possibility 15. Elements of a Theory of Substantial Possibility 16. A Metaphysics of Substances 16.1 Which Theory of Substance? 16.2 Why an Ontology of Substances? 16.3 Reply to Common Objections 17. Substantial Nature and the Manifestation of Causal Power 17.1 Connecting Substance and Active Power 17.2 Active Power and Natural Necessity 18. Substantial Possibility Chapter 5. Substantial Modality in KCA Thought Experiments 19. Applications of Substantial Possibility and a Substance-Based Metaphysics 20. Why Evaluation of the KCA Requires Substantial Possibility 21. Events and Temporal Marks

165 165 168 170 174 178 198 198 200 200 203 208 211 211 213 218

232 232 233 235

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CONTENTS

21.1 Clarification of the Notion of “Event” 21.2 Clarification and Defense of “Temporal Mark” 22. A New KCA Thought Experiment 22.1 The Hyperlump Thought Experiment 22.2 Consistent Mathematical Description Insufficient for Factual Possibility 22.3 No Determinate Shape Consistent with the Hyperlump 22.4 Hyperlumps and Time Past

235 237 242 243 244 249 252

Chapter 6. Conclusion 23. Summary of Results 24. Prospects and Diagnosis

261 261 264

Appendix 25. Cantor’s Theory of the Actual Infinite 26. Operations with Transfinite Numbers

267 276

Bibliography

283

Index of Names

303

Index of Subjects

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ACKNOWLEDGMENTS

I

have been turning over the ideas in this book for some time, and I have had the benefit of discussing them with a number of inordinately patient and helpful individuals at every step of the way. Since the present text is a revised version of a dissertation written for the Catholic University of America, my greatest proximate debt is to Riccardo Pozzo, whose great kindness and generous guidance as my director can never be fully repaid. I would also like to thank Michael Gorman and Timothy Noone, readers of the dissertation, who did much to help me clarify key points in the argument. Jean DeGroot and Thérèse-Anne Druart provided insight and guidance during the early stages of the work. My debt to Kurt Pritzl, OP, runs deep: With supererogatory patience he supported my work both within and without the School of Philosophy, helping me to secure scholarship funding and then teaching opportunities that kept food on the table and a roof over my family’s head. I have also enjoyed the collegial support of the philosophy departments at both Howard University and George Washington University, and I would like to mention my special appreciation for the help and advice I have received from Charles

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ACKNOWLEDGMENTS

Verharen, Paul Churchill, Ilya Farber, and Tatiana Romanovskya. At Georgetown University I had the good fortune to work with Wilfried Ver Eecke, who has been my much-appreciated mentor and guide ever since. I would also like to thank John Williams and Tan Yoo Guan, formerly my teachers at the National University of Singapore and now my colleagues at the Singapore Management University: their support and encouragement have been unwavering from the very first. On a more personal level, I would like to thank Gretchen Gusich and Ingrid Genzel for their constant friendship and unfailing willingness to help in times of need. My thanks also to Erik Tozzi, Loy Hui Chieh, and Mitch Jones for their philosophical contributions to this work. My numerous other intellectual debts should be apparent from even a casual perusal of the book, but I would like to mention here that I have been much inspired by the high philosophical standards set by Bill Craig, Quentin Smith, Adolf Grünbaum, and Graham Oppy. I am grateful to be able to follow in their footsteps. Riccardo Pelizzo, Renaissance man and colleague, provided key advice during the final stages of manuscript revision. My thanks also to Jared Poon, who in preparing the indices and in helping with the final editing has achieved the highly dubious honor of being almost as familiar with this book’s contents as I am. Some last-minute—and hence all the more appreciated—editing help was supplied by Steven Burik and Nagarajan s/o Selvanathan. Lastly, but most importantly, I would like to thank my family: my dear wife, Chuen-Yee, to whom this book is dedicated; and our children, Diogenes, Xanthippi, and Calliope, who together make life such a joy. The final preparation of the book manuscript was significantly aided by a research grant (04-C208-SMU-010) from the Office of Research, Singapore Management University.

INTRODUCTION

1. WHAT IS THE KCA?

A

pproximately fifteen hundred years ago John Philoponus, a Christian Neoplatonist, proposed a simple argument for the existence of God. His argument runs thus: 1. Whatever comes to be has a cause of its coming to be. 2. The universe came to be. 3. Therefore, the universe has a cause of its coming to be.

As a result of the influence of William Lane Craig, a contemporary analytic philosopher who defends a version of Philoponus’s argument, this argument and the family of subarguments that support it have come to be known as the kalam cosmological argument (KCA).1 The KCA has profound implications for philosophy of nature and for philosophy of religion. First, the argument shows that the universe did not exist forever but instead came to be. When properly understood, this coming to be of the universe is recognized as a coming to be sim-

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INTRODUCTION

pliciter, that is, a coming into being ex nihilo. Second, the argument entails that the coming to be of the universe was caused by something that transcends the universe itself. As this transcendent cause brought the universe into existence ex nihilo, it is appropriate to describe this transcendent cause as a creator. Third, as there are good reasons for holding that only a being who possesses all of the pure perfections can have the power to create, and further, that only a deity possesses all of the pure perfections, it follows that the transcendent cause and creator of the universe is none other than God.2 Although the KCA is of ancient provenance and the cluster of issues raised by the argument have been discussed at a high level of sophistication from the argument’s very inception, the argument fell out of favor during the nineteenth century; serious interest in the KCA such as is found in the contemporary literature is of comparatively recent origin. Three factors may be singled out as especially important in explaining the remarkable resurgence of interest in the KCA: publication of extensive historical studies have revealed the rich and varied history of the argument;3 advances in mathematical approaches to the infinite in the late nineteenth and twentieth centuries have allowed clearer presentation of the more abstract philosophical forms of the argument;4 and the timely development of big bang cosmology has promised a surprising empirical confirmation of the central contention of the KCA, namely, that the past existence of the universe is finite. All three approaches, the historical, the mathematical, and the scientific, are woven into Craig’s 1979 work The Kalåm Cosmological Argument (TKCA). It is difficult to overstate the importance Craig’s work has had in contemporary discussions of the KCA.5 While Craig would be the first to acknowledge his philosophical debts to previous thinkers, the shape and trajectory of the last twenty years of analytic speculation on the KCA bear the unmistakable stamp of Craig’s influence. The formal shape of the argument, the variety of issues considered to be significant in discussions of the KCA, and the very name of the argument under discussion all find their origin in Craig. In addition to the early presentation of the argument found in TKCA, Craig has authored an impressive number of articles and book contributions that elaborate upon and defend his position. He has also written (with Quentin

INTRODUCTION

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Smith) a second book dedicated to the KCA, entitled Theism, Atheism, and Big Bang Cosmology.6 Besides Quentin Smith, who is the foremost critic of the KCA, significant contributions to the contemporary understanding of the KCA have been made by (among others) Adolf Grünbaum, Norman Kretzmann, J. P. Moreland, Wesley Morriston, David Oderberg, Graham Oppy, and Robin Small. Their investigations, which are subsequent to Georg Cantor’s mathematical researches into the nature of the infinite, have advanced the KCA far beyond its previous formulations: Issues surrounding the argument that were once major sources of philosophical puzzlement—such as how an actually infinite set can be given a consistent description and how the actual infinite and the potential infinite are mutually related—can now be handled with clarity and precision. What is more, the current situation is unusually promising in that there is a general agreement about what tools need to be brought to bear on the problem (e.g., transfinite mathematics, thought experiments), even if there is still little consensus on how those tools should be applied. Though it would be imprudent to expect a definitive resolution concerning how the KCA should be assessed any time soon, the current state of research is ripe for a consolidation and clarification of the key points at issue.

2. WHY THIS BOOK WAS WRITTEN Much good work has been done on the KCA, and several key themes in the argument have been identified and pursued in depth. However, it has been more than twenty-five years since any systematic presentation of the argument as a whole has been attempted.7 The body of literature relating to the KCA has grown to a significant bulk, and there is now a real danger of fragmentation, duplication, and misassessment because of the plurality of interpretations that the argument has received. One major aim of this book is to perform the scholarly service of laying bare the logical structure of the KCA that has emerged from and been continuously refined in philosophical discussion. There is thus a doxographical dimension to this book, which, it is hoped, will make the work useful to future researchers. For it must be admitted

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INTRODUCTION

that it is quite time-consuming to track down all of the relevant articles needed to build up a complete picture of the KCA. The difficulty arises from two sources. First, many important articles are located in lesser-known and hard-to-obtain journals. Second, the journals and monographs wherein topics touching on the KCA are discussed cut across several disciplines: philosophy of science, philosophy of religion, philosophy of logic, metaphysics, philosophy of mathematics, history of philosophy, philosophy of time, and various departments of theology—a list that is by no means exhaustive, for the KCA also appears in recent works of Christian apologetics, popularizations of contemporary science, and specialized scientific journals. Thus, another service performed by this book is to provide a report on the “state of the question” in contemporary analytic philosophy and thereby furnish a convenient guide to the significant literature on the KCA. Even with these fairly clear scholarly goals, I have yet been obliged to focus my attention on but one important strand of the KCA literature—a concession to practical limits of space and time whose implications I elaborate more precisely in section 3. The key to future progress, I think, lies in explicitly situating the KCA within its appropriate metaphysical context. By placing the KCA within a substance-based metaphysics, it is possible to come to a more precise understanding of how and why the argument reaches the theistic conclusion that its defender claims. Providing an adequate description of the KCA’s proper metaphysical home in turn requires the articulation of an analytic account of substance as well as the construction of a modal theory sensitive to the possibilities and necessities that obtain for substances. An overview of how I go about fulfilling these tasks is given in the next section.

3. ORDER OF TREATMENT AND HOW TO USE THIS BOOK Contemporary support for the KCA originates from two sources: philosophical argument, which draws upon the traditional disciplines of metaphysics, philosophy of nature, and philosophy of mathematics; and empirical confirmation, which draws upon contemporary advances in astrophysics and scientific speculation concerning big bang cos-

INTRODUCTION

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mology. This book is intended as a critical investigation and development of the contemporary philosophical strand of the KCA. While occasional mention will be made of the history of the argument, this will always be done within the context of a contemporary assessment of the KCA. This is appropriate because one frequently encounters historically derived responses to the KCA being advanced as guides to the proper evaluation of the KCA in its contemporary form.8 In all such cases I will simply accept the historical interpretation given and assess the resultant argument on its own merits. For while it is true that getting the historical story straight is both desirable and interesting in its own right, the state of the question concerning the contemporary KCA is largely unaffected by the correctness of the various historical interpretations encountered.9 My treatment of explicitly scientific versions of the KCA will be conducted along the same lines: whenever possible I will avoid entering the current debate over how particular results of current scientific cosmology are to be interpreted. This approach is motivated first, by the tendency of the latest scientific scholarship to change only slightly less rapidly than the styles of Parisian fashion designers, and second, by the consideration that the arguments found within the philosophical strand of the KCA are largely detachable from the specific commitments of contemporary science.10 It may also be noted that the philosophical strand of the KCA itself affords considerable room for clarification and development. This book is divided into two main parts. Part 1, comprising chapters 1 and 2 sets forth the logical structure of the KCA as it is found in the literature and then presents an organized taxonomy of the major objections to the KCA that have been advanced in the literature. Part 2, comprising chapters 3 through 6, focuses upon one of the most important species of objection to the KCA and then attempts to reformulate the KCA in a way that allows the objections to be met. In this second part of the book I argue that the introduction of a metaphysics of substances allows the KCA defender to effectively answer certain challenges concerning the logical possibility of instantiating an actual infinite in nature. In chapter 1 the KCA is presented in a form that closely tracks the original 1979 version of the argument Craig gives in TKCA. Later developments in Craig’s approach are taken up in chapter 2, as are cer-

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INTRODUCTION

tain detailed responses he develops in the appendices of TKCA.11 It is to be noted that Craig’s initial version of the KCA is developed without any explicit allegiance to a metaphysics of substances.12 Chapter 2 lays out a taxonomy of the various objections that have been made to the KCA, distinguishing (for example) among objections that call premise (1) into doubt, objections that call premise (2) into doubt, and objections that argue that what conclusion (3) arrives at is not really God.13 Various subspecies of each of these kinds of objections are also distinguished. Chapter 3 focuses on one particular kind of objection to the KCA, clear examples of which may be found in the work of Graham Oppy. Objections of this type involve the rejection of premise (2), and they rely upon certain thought experiments designed to show that the existence of an actual infinity is possible. Now, I willingly concede that these antiKCA thought experiments show that an actual infinity is possible in the sense of being consistently conceivable. But, as I will argue, there is yet another kind of possibility, one that is stronger than mere consistent conceivability, and, moreover, that this is the kind of possibility that is relevant for the KCA. It follows that the defender of the KCA can refute Oppy-style objections by showing that they make use of the wrong kind of possibility. Adopting a suggestion made by David Braine, I name the stronger sort of possibility required for the KCA “factual possibility.” Chapter 4 lays the metaphysical groundwork needed to justify and clarify this stronger kind of possibility. The idea advanced is not only that possibility should be conceived of in terms of factual possibility, but also that the specific subdomain of factual possibility that applies to the KCA is a type of possibility that is grounded in the causal powers of substances. This subdomain of factual possibility I name substantial possibility. The notion of “substance” required for a proper understanding of substantial possibility is worked out by drawing upon resources available from within analytic philosophy. The specific analytic theory of substance I defend is developed on the basis of the work of G. E. M. Anscombe, David Braine, and Sarah Broadie. Anscombe reintroduces analytic philosophy to the notion of substance as it is found in Aristotle’s Categories; namely, substance is that which functions as a subject of predication but is not predicated of other things. Braine elaborates Anscombe’s position, noting connections this idea of sub-

INTRODUCTION

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stance has with linguistic philosophy and bringing into view the idea of substance as the locus of active causal powers. Broadie furnishes additional arguments for the metaphysical primacy of substance and also helps fill in some useful detail regarding how the active powers of substances are expressed in causation. The connection between substance and causality is further developed on the basis of the work of Rom Harré and E. H. Madden on causal powers. My position is that the notions of substance and causality already available within contemporary analytic philosophy are sufficient for the purposes of the KCA. Chapter 5 applies the results of chapters 3 and 4 to the KCA. Three points will be made. First, it is explained why factual possibility, not mere logical possibility, is the right notion of possibility to use in discussions of the KCA. Second, the notion of substantial possibility developed in chapter 4 will be used to show that the anti-KCA thought experiments discussed in chapter 3 do not yield the correct conclusion, that is, they do not demonstrate that an actual infinite is substantially possible. Third, I will deploy the notions of factual and substantial possibility to show that the pro-KCA thought experiments discussed in chapters 1 and 2 do use the correct notion of possibility. Chapter 6 contains a summary of the results obtained. Briefly put, investigations such as the present one test the adequacy of the resources of analytic metaphysics in one branch of natural theology. Within the present work I hope to make two broad contributions to the field. First, Craig’s version of the KCA is reworked and clarified, and second, currently outstanding objections to the KCA that have not yet been adequately met within the literature are resolved. Finally, the book’s appendix contains additional material on transfinite number theory that should be of use to those who are either unfamiliar with Cantorian transfinite mathematics or those who would desire a brief overview of those aspects of transfinite arithmetic that are immediately relevant to an assessment of the KCA. The arrangement of material just outlined has implications for how this book can be used most profitably. Few readers, for instance, would derive great pleasure from reading chapter 2 in a straightforward, linear fashion: Taxonomies are marvelous reference tools, but providing a gripping narrative is not exactly their chief virtue. Therefore, the following approach is suggested.

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INTRODUCTION

To begin with, readers who are not familiar with Craig’s TKCA should work through chapter 1, perhaps occasionally referring to the appendix to fill in details concerning how Cantor’s theory of the infinite plays into an assessment of the KCA. Readers who are familiar with TKCA are advised to at least skim chapter 1 to observe how I have reorganized and, in places, updated Craig’s original presentation. Chapter 2 should be approached rather differently. As one moves through chapter 1, a number of questions or objections will naturally arise. At this point one should then note the particular premise to which the question or objection applies and then, using the guide at the beginning of chapter 2, locate the section wherein objections to that premise are treated. Part 1 of the book, in sum, is designed to serve as a reference tool for the speedy retrieval of information. Part 2 of the book is more conventional in its presentation and may profitably be read straight through. Chapters 3 through 5 in particular form a thematic unit, where the basic thrust of the argument is to develop a conceptually clarified and strengthened version of the KCA by situating the argument within its proper metaphysical and modal context. A new family of thought experiments appears in chapter 5, along with discussions of key notions such as “events” and “temporal marks.” Chapter 6 summarizes the results of this study, and suggests some lines for future research.

NOTES 1. In this introduction I will use KCA as shorthand for the entire family of arguments that attempt to prove the existence of God from the finitude of the past. In the remainder of the book I will focus exclusively upon contemporary variants of the argument that either arise from or make reference to Craig’s work. Unless specifically noted, the terms KCA and kalam should not be read as historical allusions to an important branch of medieval Islamic religious and philosophical speculation but instead should be understood to refer to the argument in its specifically contemporary setting. 2. While I believe that the God whose existence is proven through the KCA can ultimately be identified with the God who is worshiped in the three great monotheistic faiths, I also accept a division of labor between philosophy and theology similar to that espoused by Aquinas. The philosopher is not in a

INTRODUCTION

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position to demonstrate certain revealed truths, e.g., the doctrine of the Trinity, because such revealed truths exceed the capacity of natural reason to demonstrate. Truths of this sort are not properly said to be irrational but rather suprarational. See Summa contra gentiles I.3. 3. Pride of place must be given to the extensive translation efforts of the Aristotelian commentators of late antiquity initiated under the leadership of Richard Sorabji. See, for instance, John Philoponus, Against Aristotle, on the Eternity of the World, trans. Christian Wildberg (London: Gerald Duckworth, 1987) and John Philoponus and Simplicius, Place, Void, and Eternity—Philoponus: Corollaries on Place and Void; Simplicius: Against Philoponus on the Eternity of the World, trans. David Furley and Christian Wildberg (Ithaca, NY: Cornell University Press, 1991). For the medieval Islamic, Jewish, and Christian periods, key studies include Herbert A. Davidson, Proofs for Eternity, Creation and the Existence of God in Medieval Islamic and Jewish Philosophy (New York: Oxford University Press, 1987); R. C. Dales, Medieval Discussions of the Eternity of the World (Leiden: Brill, 1990); and Ernst Behler, Die Ewigkeit der Welt (München: F. Schöningh, 1965). Underappreciated discussions of the KCA appear in early modern philosophy, for example in the writings of the Cambridge Platonist Ralph Cudworth. A noteworthy, though somewhat truncated, version of the argument appears as the thesis of Immanuel Kant’s First Antinomy: see his Critique of Pure Reason A426/B454. After Kant’s critique discussion of the argument abated, and the distinctive KCA approach to proving the existence of God fell into neglect for approximately 150 years. A handful of philosophical discussions of the KCA appeared sporadically in the early and mid-twentieth century, any list of which should include the treatments of the argument by the Thomist Fernand Van Steenberghen and also the discussions of G. J. Whitrow and Pamela Huby, who were much influenced by Kant’s version of the argument. It was not until the publication of Craig’s work, however, that interest in the KCA blossomed. Consideration of various issues surrounding the KCA now accounts for a steady stream of articles published in analytic philosophy of religion and philosophy of science. 4. Georg Cantor’s development of transfinite mathematics is crucial here. More will be said about Cantor and the implications of his work in the presentation of Craig’s 1979 version of the argument in chapter 1. 5. It is to be noted that interest in Craig’s contemporary restatement of the KCA is almost exclusively an Anglophone phenomenon. Judging from a survey of the literature, it can be said that Continental Europe has essentially ignored Craig’s work for the past twenty-five years. 6. William Lane Craig and Quentin Smith, Theism, Atheism, and Big Bang Cosmology (New York: Oxford University Press, 1993); henceforth TA&BBC. 7. The brief summary of the KCA given in chapter 1 of TA&BBC is simply an excerpt of TKCA. This is appropriate given the aims and audience of

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INTRODUCTION

TA&BBC, but much important augmentative detail is omitted. The omissions of TA&BBC are unfortunate in that some participants in the debate appear to have taken TA&BBC for their starting point without going back to TKCA for important clarifications. This situation may be rectified somewhat in the near future with the recent (2000) paperback reprint of TKCA. 8. A good example of this may be found in J. J. MacIntosh, “St. Thomas and the Traversal of the Infinite,” American Catholic Philosophical Quarterly 68, no. 2 (1994): 157–77. 9. It is preferable to approach Craig’s own work this way: scholarly reception of the historical part of TKCA has been less enthusiastic than is the case for the rest of the book. 10. Which is not to suggest that it is possible to detach the philosophical strand of the KCA from all forms of empirical confirmation altogether. The KCA defender relies upon certain basic facts about the world (and even certain interpretations of those facts) that are commonly appealed to or simply assumed within contemporary science. For instance, any plausible scientific physics will have to give some account of change, and the philosophical strand of the KCA appeals to facts about change. Nor, in setting aside consideration of the scientific strand of the KCA, do I mean to suggest that there are no interesting or important arguments to be found in that strand of the KCA. Craig has developed a number of intriguing arguments in support of the KCA by drawing upon the resources of contemporary scientific cosmology. The issue, rather, is one of emphasis. Instead of working out a defense of the KCA from within the philosophy of science, I propose to investigate that strand of the KCA situated within a broader philosophy of nature. 11. In the first appendix of TKCA Craig relates the KCA to Zeno’s paradoxes and the analytic literature on “supertasks.” In the second appendix Craig discusses the thesis of Kant’s First Antinomy. The germs of Craig’s more detailed responses to critics is often to be found in these appendices and so reserving detailed consideration of their contents until chapter 2 will help avoid needless repetition. 12. Based solely on texts available in TKCA it may reasonably be suspected that Craig rejects a metaphysics of substances conceived along Aristotelian lines. For instance, Craig rejects Aquinas’s position that God enjoys an eternal mode of existence after creation because Aquinas’s arguments for God not being really related to creatures turn upon Aristotelian conceptions of substance, relation, and accident: Although Aquinas argues that God remains timeless after creation because He sustains no real relation to the world, Aquinas’s solution is singularly unconvincing. For his solution is system-dependent

INTRODUCTION

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upon a peculiar understanding of relation as an accident inhering in a substance. Abandon these Aristotelian categories and it seems foolish to say God is not really related to the world as Creator to creature. If God is related to the world, then it seems most reasonable to maintain that God is timeless prior to creation and in time subsequent to creation. (TKCA, p. 152; I have omitted Craig’s footnote reference) 13. For these premises see the argument given in section 1 as well as the “Argument Guide” presented at the beginning of chapter 2.

PART 1

THE KALAM COSMOLOGICAL ARGUMENT IN CONTEMPORARY ANALYTIC PHILOSOPHY THE STATE OF THE QUESTION

1

4. CRAIG’S VERSION OF THE KCA AS PRESENTED IN TKCA

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his chapter presents a logically streamlined version of the KCA as originally introduced and defended by William Lane Craig. An Argument Guide summarizing the key arguments of the KCA is given at the beginning of chapter 2. Henceforth, all numbered premises refer to the numbering given in the Argument Guide. For reasons discussed in the introduction, I diverge from Craig’s actual order of presentation in TKCA and treat each step of the argument in the order in which it appears in the Argument Guide.

4.1 Craig on Premise 1: Whatever Comes to Be Has a Cause of Its Coming to Be Craig’s defense of premise 1 in TKCA is extremely sketchy. He begins with the following comments: We may now return to a consideration of our first premiss, that everything that begins to exist has a cause of its existence. The

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CHAPTER 1 phrase “cause of its existence” needs clarification. Here I do not mean sustaining or conserving cause, but creating cause. . . . Applied to the universe, we are asking, was the beginning of the universe caused or uncaused? . . . I do not propose to construct or elaborate defence of this first premiss. Not only do considerations of time and space . . . preclude such, but I think it to be somewhat unnecessary as well. For the first premiss is so intuitively obvious, especially when applied to the universe, that probably no one in his right mind really believes it to be false. Even Hume himself confessed that his academic denial of the principle’s demonstrability could not eradicate his belief that it was nonetheless true. Indeed the idea that anything, especially the whole universe, could pop into existence uncaused is so repugnant that most thinkers intuitively recognize that the universe’s beginning to exist entirely uncaused out of nothing is incapable of sincere affirmation.1

As might be expected, this approach provoked strong and immediate reaction. In a series of journal exchanges, most notably with Quentin Smith, Craig has been pressed to define and defend his views on causation.2 (Craig’s exchanges with Smith and other philosophers on the subject of causation are surveyed in chapter 2.) For my own part, Craig’s curious lack of suasive power on this subject was one of the factors that motivated development of the theory of substantial possibility presented in chapters 3 and 4 of this book. Craig’s underdetermined and somewhat quirky views on causation make it difficult for him to respond to critics who base their objections to the KCA on the mere logical possibility of instantiating an actual infinite. Although Craig himself is aware of the differences between logical possibility and stronger notions of possibility—he firmly asserts that the KCA must be situated within a modal context richer than that of logical possibility—just what Craig means by his frequent invocations to stronger notions of possibility is unacceptably vague.3 Continuing our investigation of premise 1, we find that Craig begins his defense with an appeal to authority. He canvasses a diverse wrangle of philosophers, including Anthony Kenny, C. D. Broad, and P. J. Zwart, who have all commented on the intuitive plausibility of the principle ex nihilo nihil fit.4 The purpose of the assemblage is to create a presumption against David Hume’s famous challenge to that prin-

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ciple. In his response to Hume, Craig draws upon the work of G. E. M. Anscombe, who, in a well-known article, argues that Hume’s criticism of the principle “whatever comes to be has a cause of its coming to be” involves an illegitimate inference from the imaginability of some state of affairs to its factual possibility.5 As Craig remarks, All Hume has really shown is that the principle ‘everything that begins to exist has a cause of its existence’ is not analytic and that its denial, therefore, does not involve a contradiction or a logical absurdity. But just because we can imagine something’s beginning to exist without a cause it does not mean that this could ever occur in reality. There are other absurdities than logical ones. And for the universe to spring into being uncaused out of nothing seems intuitively to be really, if not logically, absurd.6

Having concluded his response to Hume, Craig proceeds to sketch two lines of argument in favor of the causal principle enshrined in premise 1.7 The first line of argument he calls “the argument from empirical facts,” the second “the argument from the a priori category of causality.” According to the first line of argument, denying the reality of causality is arbitrary because there is no better-confirmed empirical proposition. “The empirical evidence in support of the proposition is absolutely overwhelming, so much so that Humean empiricists could demand no stronger evidence in support of any synthetic statement.”8 The second line of argument Craig develops from the work of Stuart Hackett.9 Briefly, Hackett defends a neo-Kantian epistemology within which “the categories have application beyond the realm of sense data . . . [and] furnish knowledge of things in themselves.”10 The specific position from Hackett that Craig endorses is that “the categories are both forms of thought and forms of things—thought and reality are structured homogeneously.”11 Craig then argues that since causality is a validly derived category and further that since validly derived categories reveal the real structure of both thought and world, it follows that the causal principle “whatever comes to be has a cause of its coming to be” must be a synthetic a priori proposition. As this principle is both universal and a necessary condition of thought it is a priori, and as this principle is not analytic (as Hume teaches us), it

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must be synthetic. With the remark that Hackett has provided an attractive defense of the reality of causation that merits further research, Craig brings his discussion of premise 1 to a close.12

4.2 Craig on Premise 2: The Universe Came to Be Developing a case for premise 2 requires the introduction of some technical apparatus. Most of the KCA defender’s work occurs in this preliminary stage, for the actual arguments for premise 2 are relatively straightforward once the proper interpretive framework is in place. In the subsequent sections of this chapter, I address the following topics. First, I introduce various distinctions that must be made in discussion of the infinite. Second, I discuss those portions of both set theory and Cantorian transfinite number theory required for evaluating the KCA.13 Third, I present a slightly emended version of Craig’s canvassing of the various camps of philosophical thought regarding the ontological status of mathematics. Fourth, I present Craig’s justification of premise 2. To keep the argumentative moves Craig makes in their proper perspective, it is useful to bear in mind the overall strategy of the KCA. The key point to keep in view is that Craig will actually offer two strands of argument in support of premise 2.14 The first such strand of reasoning consists in arguments showing that it is impossible for an actual infinite to obtain in nature.15 The second strand of argumentation aims at demonstrating the impossibility of forming an actual infinite through a process of successive addition. In both cases it is crucial that the mathematical notion of the actual infinite be precisely understood. To clarify this crucial concept Craig turns to the work of Georg Cantor and the development of axiomatic set theory that followed Cantor’s foundational efforts. Once the Cantorian notion of the actual infinite is properly understood, Craig argues that while it may well be the case that the notion of the actual infinite is conceptually consistent and mathematically fruitful, it is intuitively obvious that the actual infinite cannot have extramental existence. The inapplicability of Cantor’s actual infinite to the actual world is brought out by presenting several thought experiments wherein a literal application of Cantor’s work to the realm of nature yields absurd results. Since Cantor’s math-

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ematical description of the actual infinite is the clearest and most plausible understanding of the actual infinite we currently have or indeed are likely ever to possess,16 the absurd results generated by Craig’s thought experiments together yield a conclusive reductio ad absurdum in favor of premise 2.17

4.2.1 Preliminary Remarks and Terminology Craig identifies the main argumentative burden of the KCA to fall on premise 2 and it is to the defense of this premise that he dedicates the bulk of his book. It should be noted that the account of Craig’s position I give below departs from the order of presentation adopted in TKCA. This is done with a view toward clarifying and improving the taxonomic efforts of chapter 2. Since a number of authors have commented on the KCA and not all of these commentators have restricted themselves to Craig’s idiom, a few preliminary remarks on terminology are apposite. Unless otherwise noted, whenever the term infinite appears this should be understood as a reference to the mathematical infinite and not to the metaphysical infinite.18 By metaphysical infinite I denote an ontologically positive qualitative lack of limit. Metaphysical infinity thus marks off a family of concepts that find their proper home within philosophical theology: omnipotence, omniscience, and omnibenevolence are examples. Within philosophical theology, talk of the metaphysical infinite is associated with notions of perfection, unity, wholeness, and completeness. Another term for the metaphysical infinite (commonly found in discussions either of or by Cantor) is absolute infinite.19 By way of contrast, the term mathematical infinite denotes a purely quantitative lack of limit.20 This is the kind of infinity associated with the process of counting that is commonly appealed to in explanations of the continuum and of open geometric curves, and that appears in discussions of limits in algebra and the calculus. The mathematical infinite is further distinguished into the potential infinite and the actual infinite. The potential infinite denotes a limitless quantitative process. Potential infinity is therefore a dynamic concept: endless addition, endless division, endless succession: when one element is given another always follows. The potential infinite, of its very nature, is

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never complete. The potential infinite is also known as the improper infinite and, more suggestively, as the variable finite.21 The actual infinite, on the other hand, denotes an unlimited, simultaneously presented, quantitative whole. The concept of an actual infinite is therefore static: The whole in question is taken as a complete quantitative unity given all at once. An example of the actual infinite, derived from the science of continuous quantity (i.e., geometry), would be a Euclidean line that lacks endpoints. From the science of discrete quantity (i.e., arithmetic) we have the example of Cantor, who stood outside of (so to speak) the set of natural numbers and grasped them as a completed totality. The actual infinite is also known as the proper infinite.22 To add further precision to the notion of the actual infinite, Cantor introduces the idea of quantities that are greater than any finite quantity. Such quantities he calls transfinite, and the numbers that stand for such quantities he calls transfinite numbers. (Cantor’s theory of transfinite numbers is treated in section 4.2.2 below and in the appendix.) The actual and potential infinite can also be mapped onto what amounts to a grammatical distinction. Consider the different structural parsings of “this body can move infinitely fast.” Understood in its categorematic sense, this means that the body in question is capable of achieving a speed that surpasses any finite measure; that is, the body can move at an actually infinite speed. On the other hand, if it is understood in its syncategorematic sense, what this means is that there is no limit to the finite speeds that this body is capable of obtaining; that is, the speed of the body is potentially infinite. Finally, in deference to an important body of related literature, I will also employ the term supertask.23 Supertask denotes a completed infinite task. Since the infinite task realized through the performance of a supertask is conceptualized as a completed whole, the term infinite in completed infinite task must be construed in a categorematic sense. As will become apparent later on, it is important to note that the realization of a categorematically infinite supertask is supposed to be brought about through a dynamic, syncategorematically infinite process.24 In the literature on supertasks it is common to find an additional element in the definition of supertask, namely, that the infinite task is to be completed in a finite amount of time. I will not include the note of temporal finitude as a necessary condition in my use of the term

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supertask but will instead use the term to cover cases of infinite tasks that are supposed to be completed in either finite or infinite amounts of time. Typical examples of supertasks include Zeno’s paradoxes of motion, Thomson’s lamp,25 and Bertrand Russell’s fable about a man who counts through all of the natural numbers in two minutes flat.

4.2.2 Selective Exposition of Cantorian Transfinite Number Theory The purpose of this section is to give a summary of Cantorian transfinite number theory as it applies to the KCA.26 Given this pragmatic focus, I will be glossing over a number of mathematically significant details. For instance, I will not be concerned here with the continuum problem,27 and I will give a brief treatment of only one of the betterknown paradoxes of the infinite, namely, Russell’s paradox.28 Although Craig gives an accessible and remarkably clear exposition of transfinite number theory in TKCA, he omits or states too briefly some technical points that were to become relevant to the KCA debate in its subsequent development. Instead of reserving discussion of such technical issues to chapter 2, I will include them here for ease of reference.

4.2.2.1 Brief History of Mathematical Speculation on the Infinite The majority view concerning the nature of the infinite that prevailed from the time of Aristotle until the middle of the nineteenth century was that of the potential infinite.29 It was widely agreed that the actual infinite was both impossible to realize in nature and dispensable in mathematical practice.30 According to Aristotle, the infinite exhibits itself in different ways—in time, in the generation of man, and in the division of magnitudes. For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different.31

So, for more than two thousand years the standard model for mathematical infinity—a model well established by the time Aristotle mentions it in the quotation above—was that of a magnitude possessing a potency for being indefinitely divided or extended. A magnitude suc-

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cessively operated upon (i.e., repeatedly augmented or divided) will at each stage be finite and will at no stage achieve actual infinity.32 Although later mathematicians would eschew Aristotle’s physical examples in favor of more abstract illustrations (e.g., psychological examples were especially prevalent in the nineteenth century, the mathematical infinity of the natural numbers being exemplified by an enumeration of successive thoughts), a conceptual connection with some sort of successive process remained. Now, in its connection with the notion of process the infinite was hardly unique as a mathematical concept: a vivid, dynamic strain of mathematical reasoning is evident throughout the history of mathematics, present everywhere from Newton’s development of the calculus as a theory of “fluxions” to contemporary elementary geometry texts that explain the congruency of figures as the ability to lift one figure out of the plane and superimpose it upon another figure without remainder.33 What is peculiar about pre-Cantorian mathematical speculation about the infinite is the repeated insistence that only a process-based account is appropriate. For instance, in a well-known letter the great mathematician Carl Friedrich Gauss addresses Heinrich Christian Schumacher thus: I must protest most vehemently against your use of the infinite as something consummated; this use is never permitted in mathematics. The infinite is but a façon de parler indicating a limit to which certain ratios may approach as closely as desired when others are permitted to increase indefinitely.34

Interestingly, contemporary mathematicians have defended Gauss’s view—at least in part. The situation regarding the interpretation of infinity that currently prevails is nicely captured by Abraham Fraenkel: In almost all branches of mathematics, especially in analysis (for instance, in the theory of series and in calculus, also called “infinitesimal calculus”), the term “infinite” occurs frequently. However, mostly this infinite is but a façon de parler . . . the statement lim 1⁄n = 0 n®➝ `

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asserts nothing about infinity (as the ominous sign ` seems to suggest) but is just an abbreviation for the sentence: 1⁄n can be made to approach zero as closely as desired by sufficiently increasing the positive integer n. In contrast herewith the set of all integers is infinite (infinitely comprehensive) in a sense which is “actual” (proper) and not only “potential.”35

In short, standard mathematical practice dictates that one should dispense with appeals to the actual infinite whenever possible. This is an understandable strategy, for the legitimacy of mathematical explanations based upon the potential infinite is largely unquestioned.36 The situation is decidedly different in the case of the actual infinite: it took the heroic intellectual efforts of Cantor to secure the intellectual legitimacy of the notion of the actual infinite.37

4.2.2.2 Cantor and the Development of Set Theory It is to Georg Cantor, more than anyone else, that contemporary mathematics owes its understanding of the infinite. In opposition to longstanding mathematical tradition, Cantor defended and gave precise formulation to a mathematical notion of actual infinity. He distinguished the actual infinite from the absolute infinite and explained how it is possible to talk meaningfully about a number that is greater than any finite number. Such a number, transcending the realm of all finite quantity, he termed transfinite. Cantor then proceeded to show that it is possible to speak in a precise way about different sizes of infinity. He further demonstrated that there is an inexhaustible supply of actually infinite numbers, thus revealing the shocking richness of the transfinite realm: each transfinite finds its unique place within a hierarchy of magnitudes wherein each transfinite number (with the exception of the first transfinite number) is greater than the last. Finally, Cantor worked out a calculus for the actual infinite, demonstrating with full logical rigor how operations analogous to addition, multiplication, and exponentiation are possible in the transfinite realm. The mathematical world Cantor opened up for exploration presents a dizzying view, and it is fortunate that not all of it needs to be presented for an evaluation of the KCA. For the presentation of set theory and the theory of transfinite numbers that follows I borrow

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heavily from Craig with supplemental material drawn from A. W. Moore and other sources.38 Cantor’s work on the actual infinite depends on two key notions: (a) the notion of a set, and (b) the principle of correspondence as the sufficient condition of quantitative equivalence. Upon these deceptively simple and intuitive notions Cantor grounds the entirety of set theory and the theory of transfinite numbers. I will treat each in turn. Cantor defines a set thus:39 By a “set” we are to understand any collection into a whole M of definite and distinct objects m of our intuition or our thought. These objects are called the “elements” of M.40

Although difficulties with this so-called naive notion of a set were soon recognized, Cantor’s definition has great intuitive plausibility. At heart, a set is any collection of objects treated as a unity. The letters of the alphabet, the planets in the solar system, the days of the week, the books in a library, or the natural numbers could each be taken as a set. Further, a set is determined by its members. This determination applies in three distinct senses: first, the elements of a set are in some way prior to their set; second, two sets with precisely the same elements are identical; third, two sets with the same quantity of elements are equivalent.41 There are two basic ways of constructing a set.42 The first and most straightforward way of specifying set membership is to just list all of the members of the set.43 For instance, one could construct a set with the following elements: red, orange, yellow, green, blue, indigo, violet. This would then be written as: (red, orange, yellow, green, blue, indigo, violet)

The second means of specifying set membership is to identify some trait that all and only the members of the set have in common.44 For instance, the set given above could be characterized as the set of colors belonging to the visible rainbow. In either case, whether the set be specified in the first way, that is, by simple enumeration of elements, or in the second way, that is, by picking out a common feature, the identity of the set remains the same, for the identity of the set is determined by its members and not by the means of its construction.

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mined by its members and not by the means of its construction. Two features of Cantor’s definition deserve particular attention: the elements of a set must be definite and they must be distinct. As Fraenkel explains: The meaning of the latter [term, i.e., distinct] is simple. It states that any two objects which appear as members of the same set are different; in other words, that an object may belong, or not belong, to a set but cannot ‘more than belong,’ for instance belong repeatedly, as may be the case in a sequence such as the sequence (1⁄2 , 1, 2⁄3, 1, 3⁄4, 1, . . .) which contains the member 1 infinitely many times. . . . The meaning of ‘definite’ is more involved. It expresses that, given a set s, it should be intrinsically settled for any possible object x whether x is a member of s or not. Here the addition ‘intrinsically’ stresses that the intention is not actual decidability with the present (or with the future) resources of experience or science; a definition which intrinsically settles the matter, such as the definition of ‘transcendental’ in the case of the set of all transcendental numbers, is sufficient.45

Commenting on this passage, Craig remarks that although contemporary mathematicians have abandoned Cantor’s definition of a set in favor of an axiomatic approach it is still the case that “the characteristics of definiteness and distinctness are . . . considered to hold of the members of any set.”46 Craig draws an immediate corollary: The potential infinite has no place in set theory. Since the identity of a set is determined by its definite and distinct elements, any change in those elements results in a new set. Hence, no set could capture the essentially dynamic character of the potential infinite.47 While it is possible (leaving intuitionist worries about constructability aside) to have a set with any finite quantity of elements that one wishes, the fundamental notion of a set that Cantor and his followers work with is a static one, and hence the concept of process essential to the potential infinite is positively excluded. The actual infinite, in sharp contrast with the potential infinite, is given all at once and may be treated as a completed whole. The very feature of set identity that allows Cantor to compare sizes of sets, namely, the determination of a set by its elements, precludes discussion of the potential infinite in set theory. Being able to say that set theory does not deal with the potential

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some objectors, notably Bertrand Russell, have alleged that the consistency of Cantor’s mathematical treatment of the actual infinite is sufficient to secure the factual possibility of an actually infinite past.48 This is a remarkably unsophisticated objection on several counts, but perhaps the best way of pointing out the inadequacy of such a rejoinder is to stress that it is a non sequitur.49 Some of the more important arguments against the possibility of a sempiternal past turn upon the ontological interpretation of the potential infinite. Since the potential infinite is not dealt with in set theory, appeals to Cantor in hopes of solving the problem are unavailing. The situation is no longer as clear-cut today as it was in 1979. Craig’s argument does hold good for the classical set theories he mentions, and in particular the argument applies to Zermelo-Fraenkel set theory (ZF), which is perhaps the most popular variety of set theory investigated by contemporary mathematicians.50 However, in 1965 Lotfi Zadeh introduced the world to fuzzy set theory (FST), and in FST the classical criterion of definiteness does not apply in as obvious a way.51 Craig can hardly be blamed for failing to anticipate the explosive development of FST during the 1980s and early 1990s. Nevertheless, if the condition of definiteness does not apply in the usual sense, then Craig’s argument from the identity of a set being determined by its members to the impossibility of finding the potential infinite within set theory needs to be rethought. I believe that it is possible to show that FST does not constitute an exception and that Craig’s basic insight still applies. I will argue this point after a few brief remarks on the theory of fuzzy sets. FST is a generalization of classical set theory, it being possible to generate all of the results of classical axiomatic set theories like ZF within FST.52 From one convenient perspective, FST may be understood as the theoretical expression of what happens when the classical notion of definiteness is loosened. Under classical set theory, membership in a set is not a matter of degree: as Fraenkel puts it, it is intrinsically settled for each object whether it is an element of a particular set or not. In FST, however, set membership does come in degrees. Specification of a fuzzy set always involves specifying for each of its elements a particular degree of membership.53 To see how this works, consider the following case.

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Let us start with the set S: (Abel, Bob, Charles, Dan, Ed, Fred, George, Ham); this set is generated from a list of all men within a particular café. Suppose we then wish to describe how tall each of the members of S happens to be. To do this, we construct a fuzzy set T. The members of S, with their respective heights, are as follows: Abel Bob Charles Dan Ed Fred George Ham

3 feet 11 inches 5 feet 1 inch 5 feet 7 inches 5 feet 11 inches 6 feet 1 inch 6 feet 3 inches 6 feet 9 inches 7 feet 10 inches

In this list there are some men who would clearly count as tall, some who are clearly not tall, and some who exhibit the trait of being tall to some degree. Let us represent the degree to which an individual is tall by a real function that maps onto the interval [0,1], where 0 represents definite nonmembership and 1 represents definite membership. The fuzzy set T can now be constructed by mapping each element of S onto T with an ordered pair. The first member of each pair will be an element of S and the second member of each pair will be a value within the interval [0,1] that has been determined by a real function denoting the degree of membership associated with the first member of the pair. The fuzzy set T might then be thought (for ease of exposition) to look something like this: Tmembers : ( Abel, Bob, Charles, Dan, Ed, Fred, George, Ham ) Tdegrees : (

0,

.12,

.35,

.56, .75,

.8,

.97,

1

)

It is now possible to explain how the problem for Craig’s argument arises. Suppose we have another fuzzy set, T1, that comprises the same elements as T above except that Dan belongs to T1 to the degree .65 instead of to the degree .56. One might then say that sets T and T1 are coextensive, that is, they contain exactly the same individuals that were elements of set S. It then appears possible to have two sets with the

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same elements, and hence it appears possible to have modifications of the elements of a set without creating a new set. Craig’s argument can be rescued by introducing two restrictive clauses. Instead of trying to argue with what is, in effect, a stipulative definition of sameness on the part of his opponent, Craig may admit that not just any old change to the elements of a set results in the creation of a new set. Instead, since the identity of a set is at least partially determined by its distinct elements, any change in the quantity of those elements results in a new set. With these minor additions, Craig’s original argument against finding the potential infinite in set theory (classical or otherwise) becomes sound. Craig could assay a stronger response. Even though it might be the case that S is equally a subset of T and T1, it yet remains that T and T1 are themselves nonidentical sets. On the metalevel of FST, sets T and T1 have distinct and fully definite elements, and it is in virtue of these characteristics that the two sets are distinguished from one another.54 The situation, I would argue, is akin to that faced when discussing polyvalent logics. While it is possible to construct a consistent logical system based upon any number of values (e.g., T, F, and U in a trivalent logic), in the metadescription of any such logic one will ultimately appeal to the principle of excluded middle and a bivalent logical scheme. Definiteness and distinctness reappear in FST; it is just that FST allows us to relativize the membership conditions of subordinate systems like ZF. While more could be said about the notion of a set, it is necessary to introduce the second pillar upon which Cantor erected his theory of the transfinite numbers, namely, the principle of correspondence. Cantor formally introduces the principle of correspondence thus: We say that two aggregates M and N are “equivalent” . . . if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. To every part M1 of M there corresponds, then, a definite equivalent part N1 of N, and inversely.55

The basic idea Cantor appeals to is quite familiar from everyday experience. Craig calls the underlying notion Cantor relies on for his definition of equivalence the principle of correspondence. The principle of

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correspondence states that two sets are equivalent if all of the members of the first set can be put in a one-to-one correspondence with the members of the second set such that there are no members left over on either side. Suppose two children have been out trick-or-treating on Halloween. When they get home, they decide to compare their loot and see who got the most candy. Both children spill their candy out on the kitchen table and arrange it so that each piece from one child matches up with one piece of candy from the other child. There is no counting involved here: if all of the candy pieces pair up and none are left over on either side then the two children have exactly the same amount of candy; if the first child has candy left over when the second child’s supply has been exhausted then the first child has more candy. A point that bears keeping in mind is that equivalence is not the same as identity. Two children may have equivalent quantities of candy without having the same kinds of candy. Similarly, to say that two sets are equivalent does not imply that the elements of each set are the same; rather, all that a claim of set equivalence tells us is that under some arrangement it is possible to set up a one-to-one correspondence between the elements of the sets. Often, such a correspondence can only be arrived at by rearranging the elements of one of the sets: a fact that can be significant when one attempts to use set theory to model certain physical processes or states of affairs.56 In itself the principle of correspondence seems unobjectionable. However, Cantor’s application of the principle yields startling results.57 Following Richard Dedekind’s lead, we can say that Cantor uses the principle of correspondence to define an actually infinite set as “a set having a proper subset that is equivalent to the original set.”58 By a proper subset is meant a subset that does not exhaust all the members of the original set: At least one member of the original set is not a member of the subset. A sufficiently rigorous presentation of the Cantorian distinction between finite and infinite sets is captured by the following definitions drawn from Fraenkel: DEFINITION VI. A set I is called finite and more strictly, inductive if there exists a positive integer n such that I contains just n members. The null-set O is also called finite.—A set which is not inductive is called infinite (non-inductive).59

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CHAPTER 1 DEFINITION VII. A set R is called infinite and more strictly, reflexive if R has a proper subset that is equivalent to R. A set which is not reflexive is called finite (non-reflexive); in other words, a nonempty set is non-reflexive if every mapping into itself is a mapping onto itself.60

What is noteworthy about Fraenkel’s definitions—definitions that are, incidentally, very much alive in contemporary mathematical practice—is that they explicitly preserve the notion of equivalence arising from application of the principle of correspondence while at the same time jettisoning the traditional idea that “the whole is greater than its part.”61 In the case of actually infinite sets Euclid’s fifth axiom goes by the boards. In any actual infinite there will be a part that is not less than but rather equal to the whole to which it belongs.62 From a careful consideration of the implications of the principle of correspondence as it applies to infinite sets, Cantor was able to work out an arithmetic of the infinite. Once granted the principle of correspondence, Cantor is able to show that actually infinite sets come in different sizes, so that some infinities are larger than other infinities. For instance, the sort of infinity found in the natural numbers is smaller than the sort of infinity found in the real numbers. (Translated into Cantorian notation, the cardinality of the set of natural numbers, ℵ is less than the cardinality of the set of real numbers, ℵ1.) Cantor is also able to show that, in the case of the actual infinite, there is an interesting divergence in the way cardinal and ordinal numbers behave. For instance, there can be different orderings, and hence different transfinite ordinal numbers associated with the same transfinite cardinal. For further details, please consult the appendix.63

4.2.3 Reflections on and Reactions to Cantor After having presented an overview of Cantor’s work on the actual infinite, Craig succinctly rehearses the main positions in the philosophy of mathematics. The reason for Craig’s doing so is clear in the light of the argumentative strategy of the KCA: Wishing to prove that the past existence of the universe is finite, Craig’s argumentative burden would be considerably lightened if his opponent were willing to admit that the actual infinite is nowhere to be met with in nature.64

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Now, as it turns out, a significant proportion of the major competing philosophical accounts of mathematics assert that the actual infinite has no place outside the realm of mathematical speculation.65 Such positions in the philosophy of mathematics either reject the existence of the actual infinite or so qualify their understanding of the actual infinite that their acceptance of the actual infinite becomes metaphysically benign from the standpoint of the KCA. Therefore, by providing a taxonomy of the major families of opinion in the philosophy of mathematics Craig is able to focus his presentation of the KCA without having to treat in detail the positions of thinkers who are, in effect, fellow travelers. Having sifted out his most prominent opponents, whom he collectively labels mathematical realists, or “realists” for short, Craig argues that the various paradoxes besetting naive set theory pose insurmountable difficulties for the realist position.66 The seriousness of this challenge is generally conceded by mathematical realists, and over the years a variety of realist responses to the paradoxes have been attempted. The next step in Craig’s analysis consists in a survey of realist reactions to the paradoxes of set theory. His survey is then followed by arguments for the inadequacy of these realist responses. Once the difficulties facing the realist approach have been brought to light, Craig goes on to argue that accepting a nonrealist interpretation of mathematics—or, at least, of the actual infinite—is philosophically preferable. However, in adopting a nonrealist view of mathematics and of the actual infinite, one implicitly accepts a fundamental thesis of the KCA, namely, that the actual infinite does not obtain in the realm of nature. For, if the universe were supposed to exist eternally a parte ante, then the events that make up its past would constitute an actually infinite totality.67 Denying the possibility of finding the actual infinite in nature rules out such a totality. Consequently, the universe’s past existence must be finite, which is precisely the result the KCA defender hopes to establish. Having explained Craig’s argumentative strategy, I will now work through his position in greater detail. The remainder of this section divides into three subsidiary sections. In the first section I present a slightly expanded version of Craig’s taxonomy of positions in the philosophy of mathematics. In the second section I provide a brief dis-

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cussion of what is arguably the most important paradox of set theory, namely, Russell’s paradox. In the same section, I present one strand of realist response to Russell’s paradox. In the third section I present Craig’s analysis of this realist response to Russell’s paradox and summarize Craig’s position concerning what is the correct ontological interpretation of the actual infinite.

4.2.3.1 Taxonomy of Positions within the Philosophy of Mathematics Given the prominence of set theory in contemporary mathematics, Craig allows the following question to guide the construction of his proposed taxonomy of positions in the philosophy of mathematics: What is the ontological status of sets?68 Since sets are abstract entities, the possible responses to this question parallel the answers given to the traditional problem of universals.69 Without trying to do full justice to the lush variety of positions available, Craig divides the various philosophies of mathematics into four schools: platonism, nominalism, conceptualism, and formalism. According to Craig, “Platonism, or realism, maintains that corresponding to every well-defined condition there exists a set, or class, comprised of those entities that fulfil this condition and which is an entity in its own right, having an ontological status similar to that of its members.”70 Kurt Gödel is an oft-cited example of a mathematical platonist because of his doctrine of “mathematical intuition,” a faculty whereby the mathematician is held to be in intuitive contact with abstract mathematical entities.71 The subgroup of platonists Craig singles out for attention are the logicists, theoreticians who “attempted to reduce the laws of the mathematics of number to logic alone.”72 The preeminent logicists are Gottlob Frege and Bertrand Russell; more will be said about their brand of platonism in the next two sections. “Nominalism” Craig defines as the position that “there are no abstract entities such as numbers or sets, but that only individuals exist.”73 Craig’s remarks on the nominalist position are exceedingly brief, but the main conclusion he wishes to draw is that nominalist theories of mathematics are “only too glad to jettison the whole [Cantorian] system as a mathematical fiction.”74 Since Cantorian transfinite

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arithmetic cannot, under a nominalist interpretation, come out to be literally true, it is difficult to see how a nominalist would be attracted by or could appeal to the actual infinite.75 The third position Craig discusses is that of conceptualism, which he defines as follows: “Conceptualism contends that abstract entities such as numbers and sets are created by and exist in the mind only, and have no independent status in the real world. A well-defined condition produces a corresponding set, but this set has mental existence only—the mathematician creates his mathematical entities; he does not discover them.”76 The particular conceptualists Craig discusses are Leopold Kronecker and L. E. J. Brouwer, who both belong to the school of intuitionism.77 Intuitionists hold that constructibility in terms of the intuitive human activity of counting is a necessary condition for admitting the existence of any mathematical entity.78 Proofs that would require an infinite number of steps, like Cantor’s diagonal proof, would not be admitted by an intuitionist since the human mind cannot actually execute such proofs. According to Craig, intuitionists would be inclined to dismiss the actual infinite on the ground that it is nonconstructible, and conceptualists in general would accord only conceptual existence to the actual infinite.79 The fourth and last philosophy of mathematics Craig discusses is formalism. Formalists “eschew all ontological questions concerning mathematical entities and maintain that mathematical systems are nothing but formalised systems having no counterparts in reality.”80 Mathematics is just a symbolic system of marks on paper devoid of metaphysical import, and consistency is the necessary and sufficient condition for mathematical existence.81 David Hilbert is the typical representative of this school of thought. Since formalists are willing to admit that formal mathematical systems have no metaphysical import, their position on the actual infinite is not hostile to the central KCA contention that it is impossible to instantiate an actual infinite in the realm of nature.82 In line with the development of the agnostic position of the formalists, it is useful to remind oneself that working mathematicians never need enter the fracas over how their discipline should be interpreted. So on one level, the formalist is correct: How one answers the question of the existence of sets is of no moment, since it is possible to churn out lovely mathematics without ever considering concrete application.83 With particular regard to the use of the word existence as it appears in

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expositions of set theory, it is entirely possible (and, it seems to me, preferable) for practicing mathematicians to work with an extremely formal notion of existence. In point of fact, several mathematicians are content to get by with an extremely thin notion of existence.84 The upshot is that accepting a negative judgment on the instantiability of the actual infinite need not have a detrimental impact on current mathematical practice. The noninstantiability of Cantor’s work in our universe no more delegitimizes research into transfinite numbers than our universe’s possessing a particular physical geometry delegitimizes mathematical research into alternative geometries. However, the temptation to philosophize is great, and it is not unusual for philosophical considerations to intrude even in the most technical mathematical contexts: The work of Gödel, for instance, is rife with speculation concerning mathematicians as “discoverers” as opposed to “inventors,” and the mathematical output of Brouwer is famously guided by his intuitionist scruples.85 Moreover, the numerous connections between mathematics and the physical world are impossible to ignore. There is a legitimate distinction to be drawn between pure and applied mathematics, and the connections between theory and application are often as surprising as they are profound. Cantor himself believed that his theory had important extramathematical applications.86 And what should philosophers say about cases like that mentioned by Friedrich Waismann, where the behavior of a real function is best explained by observing how that function behaves when i is substituted as its value?87 How can it be that the complex numbers, which are also aptly named the imaginary numbers, have explanatory value in concrete applications?88 These are, it seems to me, real puzzles calling for further philosophical reflection.

4.2.3.2 Paradoxes in Paradise Craig’s next move in the argument makes the reason for his detour through the philosophy of mathematics even clearer. It is often alleged, contra the position of the KCA, that Cantor’s work legitimized the notion of the actual infinite. Thus, from the consistency of Cantor’s mathematical work the factual possibility of instantiating the actual infinite is supposed to follow.89 Now, as the taxonomy given

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above amply illustrates, such a judgment is unwarranted. Even among mathematicians the assumption that one can make a smooth and easy transition from the realm of mathematics to the extramental world has not gone unchallenged: only one of the four major schools of thought, that of the platonists, affirms the extramental existence of the actual infinite. (And, it may be noted, even among mathematical realists there is some hesitancy regarding whether the actual infinite ever assumes a nonabstract form.)90 With his taxonomy Craig has opened up the conceptual space required to argue that Cantor’s development of a consistent account of the actual infinite has not eo ipso rendered the instantiation of the actual infinite in nature factually possible.91 The temptation to assert the existence of an actual infinite in nature is greatest for those who already accept the existence of an actual infinite in another ontological domain, which is the case for mathematical realists.92 To further undermine this natural temptation, Craig attacks the realist position itself. He focuses his criticism on what is perhaps the greatest difficulty for the realist position that arises strictly from within the domain of mathematics: the antinomies or paradoxes of set theory discovered at the beginning of the twentieth century.93 Should Craig’s criticism of mathematical realism be successful, then he is entitled to a much stronger version of his thesis than is strictly required for the KCA: Instead of the relatively weak claim that the actual infinite has no concrete instantiation, Craig can make the stronger claim that the actual infinite lacks any extramental existence whatsoever. In his critique of mathematical realism Craig restricts himself to a consideration of the logicist position of Russell. He begins his critique with a review of some of the better-known paradoxes besetting naive set theory. While Craig mentions three antinomies, the BuraliForti paradox, Cantor’s paradox, and Russell’s paradox, it is upon the last of these that he lavishes most of his attention.94 This paradox, first discovered by Russell in 1901, may be expressed as follows: Some sets have themselves for members and others do not. For instance, the set of all the sets discussed in this chapter will have itself for a member whereas the set of all blue objects will not be a member of itself since sets are not blue. Now, let R be the set of all sets that are members of themselves. Does R belong to itself or not? If R does not include itself as a member then it fulfills the condition for membership in R and

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hence should belong to itself. On the other hand, if R does have itself for a member then it does not fulfill the condition of membership in R and hence should not have itself for a member. So, R both must and must not be a member of itself, which is a contradiction.95 Russell’s paradox threatens the very core of a platonist interpretation of set theory where numbers and/or sets are held to be independently existing entities. This is because, as Craig notes, “if numbers and sets do exist extra-mentally, then such sets as are encountered in the antinomies seem inevitable.”96 In response to the antinomies of set theory Russell proposed his famous theory of types. According to this theory, self-referring expressions, and in particular self-referring expressions that employ the logical operator all, are meaningless. Stated baldly, Russell’s guiding principle is that “‘[w]hatever involves all of a collection must not be one of the collection’; or, conversely: ‘If, provided a certain collection had a total, it would have members only definable in terms of that total, then the said collection has no total.’”97 The basic strategy of the theory of types is to assign each entity within set theory a particular logical level, that is, a type, within a hierarchy of types. At the lowest level (call it level 0) are the individuals that are not sets; the next level up (level 1) is composed of sets containing the entities of level 0 as their members; the next level (level 2) consists of sets containing the sets of level 1 as their members, and so on. A set can contain members only from the type level immediately below itself. Under such a scheme the Russellian paradox of the “set of all sets that do not contain themselves” cannot arise because the statement describing the problematic set is strictly nonsense: no set could be a member of itself because its description clearly involves trespassing the rule that sets can only contain entities from the immediately lower level in the hierarchy of types.98

4.2.3.3 Reactions to the Paradoxes Craig’s reaction to Russell’s defense of the logicist program is well within the mainstream of contemporary analytic philosophy: Craig criticizes the theory of types for being a blatantly ad hoc patch-up lacking any principled justification.99 Considered on its own terms, a realist philosophy has no principled reason for disallowing impredicative defini-

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tions. True, many sets are not members of themselves; but why is it nonsense to talk about, say, the set of all sets referred to in this chapter? As Stephen Barker puts it: “If a set has independent reality, then why may not members of the set be defined by reference to the set itself ?”100 Again, it is sometimes the case that self-reference is not vicious but rather virtuous: The statement “All true statements are true” is, to all appearances, well formed even though it makes reference to itself. (It would, in fact, be rather disturbing if this statement did not include itself in its scope.)101 As Barker puts it: “Russell surely was exaggerating when he claimed that the theory of types was inherently reasonable. On the contrary, its character is that of an arbitrary makeshift device for stopping the paradoxes.”102 Craig concludes his argument with the observation that both the axiomatic (formalist) and intuitionist approaches escape the paradoxes of set theory in a principled way and so are preferable to Russell’s jury-rigged mathematical realism.103 With mathematical realism eliminated, none of the remaining philosophies of mathematics depend on or provide a priori support for the extramental existence of the actual infinite. Since the philosophy of mathematics no longer presents an obstacle, discussants of the KCA are in a position to advance independent arguments for or against the extramental existence of the various types of mathematical infinity. At this point Craig moves beyond his critique of logicism to suggest an explanation for why the success of transfinite number theory is not a sufficient guarantor of the existence of the actual infinite. He writes: It seems to me that the surd problem in instantiating an actual infinite in the real world lies in Cantor’s principle of correspondence. . . . This principle is simply adopted in set theory as a convention; for how could it be proved? One may cite empirical examples of the successful use of the principle for comparing finite real collections . . . but it would be impossible to conduct such an empirical proof for infinite collections. Therefore, in the mathematical realm equivalent sets are simply defined as sets having a one-to-one correspondence. The principle is simply a convention adopted for use in the mathematical system created by the mathematician.104

Craig has no quarrel with the mathematician’s use of the principle of correspondence taken as a convention. The difficulty comes, according

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to Craig, when this mathematical convention is projected onto the real world we inhabit. Granted, in our real world we do have experiences that justify the legitimacy of the principle of correspondence; however, all of these empirical confirmations are experiences of finite collections of objects and not actually infinite collections. Craig continues: Should someone naively object to Cantor’s system on the basis that in it the whole is not greater than a part, mathematicians will remind him that Euclid’s maxim holds only for finite magnitudes, not infinite ones. But surely the question that then needs to be asked is, how does one know that the principle of correspondence does not also hold only for finite collections, but not for infinite ones? Here the mathematician can only say that it is simply defined as doing so. For all the finite examples in the world cannot justify the extrapolation of this principle to the infinite; its proveability [sic] is precisely the same as Euclid’s maxim.105

Application of the principle of correspondence to actually infinite collections is thus revealed to be a mathematical convention. The principle of correspondence is decidedly not an unchallengeable, universal metaphysical principle that, having been imported into mathematics, can without further ado be exported elsewhere. The principle of correspondence is, in fact, only one of the conventions that could be adopted in a mathematical account of actually infinite sets. Euclid’s axiom that the whole is greater than the part is, as Craig notes, an equally likely candidate, for the Euclidean principle (as I will call it) is equally well confirmed within experience. (Other mathematicians have attempted to construct systems that preserve Euclid’s principle. Most notably, Bernard Bolzano sketched the rudiments of a theory of the infinite that allowed for different sizes of infinity based on the part/whole relationship instead of the principle of correspondence.106 Cantor’s influence has been so decisive that Bolzano’s suggestion is hardly discussed in contemporary mathematics.) It can be shown that it is inconsistent to apply both Euclid’s principle and the principle of correspondence to infinite sets at the same time: The former principle holds that the whole is greater than the part, whereas the latter principle, when applied to infinite sets, implies that the whole is not greater than the part. Mathematicians are therefore

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correct insofar as they recognize that only one of the two principles can be adopted in set theory. But which principle? And what ontological weight should be given to the mathematician’s choice? Craig argues: Both [the principle of correspondence and Euclid’s principle] seem to be intuitively obvious principles in themselves, and both result in counter-intuitive situations when either is applied to the actual infinite. The most reasonable approach to the matter seems to be to regard both principles as valid in reality and the existence of an actual infinite is impossible.107

One way of capturing the spirit of Craig’s remarks is to compare his evaluation of Cantorian set theory with the approach that many analytic philosophers take with respect to the different systems of geometry that have been developed. There are several distinct and mutually exclusive accounts of geometry, with those of Euclid, Bernhard Riemann, and Nikolai Lobachevski being the best known. Only one (if any) of these geometries accurately reflects the true physical geometry of our universe. Nonetheless, all of these geometries, Euclidean and non-Euclidean, are internally consistent and mathematically fruitful topics of research, and the mathematician is free to explore any or all of them as he or she is so inclined. There should not, however, be any prima facie expectation on the part of the mathematician that the particular geometry being explored yields an accurate description of our actual world; considerations beyond that of mere mathematical consistency must come into play before one may claim that a particular geometry is a suitable candidate for the task of physical description. It is also possible that some of the geometries investigated by the mathematician could not, even in principle, apply to our physical world because of the conventions adopted within that geometry: for instance, the world we inhabit could never be adequately described by a two-dimensional geometry.108 The situation regarding Cantorian transfinite mathematics parallels the geometric case just described. Something more than formal consistency is required for models of the actual world. As Craig forcefully argues, it is absurd to suppose that the actual infinite can be met within the real world. Pace Cantor and his followers, both Euclid’s principle that the whole is greater than the part and the principle of corre-

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spondence apply to real-world collections.109 Cantor’s set theory is properly likened to a consistent geometrical system that has no realworld application: it is interesting, but devoid of any necessary metaphysical import. Craig’s position as presented thus far may be summarized in the following way.110 First, mathematical existence, that is, the sort of existence required by working mathematicians, is not sufficient for actual existence. An immediate corollary is that the jointly sufficient conditions of mathematical existence are not jointly sufficient conditions for actual existence. Second, in the actual world both the principle of correspondence and Euclid’s principle obtain. Third, in the mathematical realm only one of these two principles may obtain at any one time for actually infinite sets. Which principle applies in cases of the actual infinite is a matter of mathematical convention. Fourth, as a consequence of the foregoing points, Cantor’s account of the actual infinite applies exclusively to the mathematical realm and not to the actual world.

4.2.4 Craig’s Argumentative Strategy for Premise 2 Craig develops three major strands of argument for premise 2 (viz., that “[t]he universe came to be”). I have labeled these arguments, imaginatively enough, arguments (A), (B), and (C).111 In arguments (A) and (B) Craig advances reasons for maintaining that the quantity of past events in the history of the universe is finite. Neither (A) nor (B) alone, nor (A) and (B) taken in conjunction, are sufficient to secure premise 2. Craig therefore introduces argument (C) to bridge the gap. In argument (C) he argues that the universe’s having a past composed of a finite quantity of events implies that the universe came to be simpliciter (that is, ex nihilo). The argumentative strategies Craig adopts in (A) and (B) are similar. In each case Craig introduces thought experiments that are designed to show the absurdity of supposing that the universe has always existed. Since, as the thought experiments illustrate, it is absurd to suppose that the universe has an infinite past existence, it follows that the past existence of the universe must be finite. The specific thought experiments offered in support of argument (A)—discussed in section 4.2.5—are intended to demonstrate the impossibility of sup-

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posing an actual infinite to exist in the actual world. The picture that argument (A) presupposes is a static one: all attention is focused on tickling out the absurdities inherent in applying the term infinite categorematically to our world. The thought experiments of argument (B)—discussed in section 4.2.6—are intended to show that it is impossible to form an actual infinite by means of successive addition. The picture presented in argument (B) is thus a dynamic one. Argument (B) makes no claims regarding whether the actual infinite can be found anywhere outside of the conceptual realm. Instead, the cogency of argument (B) turns upon the absurdity of saying that it is possible to complete a supertask.112 One way of understanding the interplay of the potential or syncategorematic infinite and the actual or categorematic infinite in relation to the KCA is this: if the world has existed eternally it must be possible to translate the syncategorematic infinity of past events into a fully realized categorematically infinite collection of events that finds its completion in the present moment. So in order for the world to be eternal a parte ante, it would in principle have to be possible to reach the actual infinite through a process of successive addition. But, as Craig goes on to argue, it is in principle impossible for a syncategorematic infinite to be transmuted into a categorematic infinite. Hence, the world cannot be eternal a parte ante. Argument (C), which is detailed in section 4.2.7, is quite different from arguments (A) and (B). Argument (C) is designed to show that the coming to be of the universe is actually a creation ex nihilo. Change in the universe did not arise from a prior quiescent state, nor was the universe generated from preexisting matter. Without argument (C) an opponent of the KCA could maintain the possibility that the existence of the universe is detachable from the collection of events that constitutes its temporal history. To see how this could be, suppose for the moment a relational A-theory of time like that espoused by Gottfried Leibniz.113 According to such a theory it is true that, on some level, time simply is change: no change, no time. An opponent of the KCA might then assert that arguments (A) and (B) only establish that there must have been a first event in the universe’s past, that is, the successive elapsing of events must have a beginning. It is still possible that prior to the universe’s first event (in some special, nontemporal sense of prior), the universe existed in a perfectly changeless quiescent state.

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In this scenario the basic conditions of arguments (A) and (B) are fulfilled yet the universe did not come to be ex nihilo, which would imply that it is not necessary to posit a cause of the universe’s coming to be. Unless the possibility of a quiescent universe is eliminated the inference from the necessary finitude of the past to the coming-to-be of the universe is unsound, leaving premise 2 of the KCA insecure. The logical relationships among the arguments offered in support of premise 2 may be described as follows. The success of argument (C) is a necessary condition for proving premise 2; the success of at least one of arguments (A) and (B) is a necessary condition for proving the truth of premise 2; and the success of either argument (A) or argument (B), taken in conjunction with the success of argument (C), is sufficient to prove the truth of premise 2. I begin with a consideration of Craig’s presentation of argument (A).

4.2.5 Argument (A) In argument (A) Craig argues that the actual infinite cannot exist outside the mathematical realm. He summarizes his intentions in these words: I have no intention whatsoever of trying to drive mathematicians from their Cantorian paradise. While such a system may be perfectly consistent in the mathematical realm, given its axioms and conventions, I think that it is intuitively obvious that such a system could not possibly exist in reality. The best way to show this is by way of examples that illustrate the various absurdities that would result if an actual infinite were to be instantiated in the real world.114

Craig refrains from challenging Cantor the mathematician and instead concentrates his energies on Cantor the metaphysician. The absurdities of the Cantorian system do not appear in the abstract realm of mathematics; contrary to the position of the intuitionists, Craig admits the formal cogency of the proofs employed in transfinite number theory. Rather, absurdities arise only when Cantor’s abstractions are projected onto the concrete world we inhabit. Once the possibility of the concrete existence of the actual infinite has been eliminated, Craig can then make the characteristic KCA move of showing

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that an eternal past existence of the universe would involve just such an actual infinite. This result would then support the basic KCA position expressed in premise 2 (namely, that “the universe came to be”). The structure of the argument Craig advances to show this may be formalized thusly: Argument (A) (i) An infinite temporal regress of events would constitute an actual infinite. (ii) An actual infinite cannot exist. (iii) Therefore an infinite temporal regress of events cannot exist.115

Before proceeding, a few remarks are appropriate concerning how the terms within argument (A) are employed. Craig explains the meaning of the term exist in premise (ii) and conclusion (iii) thus: “by ‘exist’ we mean ‘exist in reality,’ ‘have extra-mental existence,’ ‘be instantiated in the real world.’ We are contending, then, that an actual infinite cannot exist in the real world.”116 The notion of existence Craig works with here is very broad, and it should be noted that the defender of the KCA may find it strategically useful to incorporate a sharper notion of existence into a future revised version of the KCA. To appreciate this last point, note that formulating a more precise notion of existence can help bring out the fact that the ultimate success of the KCA does not hinge upon Craig’s arguments against mathematical realism. For instance, argument (A) has as its goal the elimination of the possibility of an actual infinite of past events, and events (past or present) are concrete in a way that mathematical abstracta are not. So what a defender of the KCA might find useful is a notion of existence that distinguishes between the abstract mode of existence proper to mathematical entities (and other abstracta such as propositions) and the concrete (i.e., nonabstract) mode of existence enjoyed by nonabstract beings. Craig, however, does not accept a platonist account of mathematics, and so he phrases the argument in a more general way, leaving it to the reader to make any necessary precisions.117 Craig explains the meaning of the word event in premise (i) and conclusion (iii) as follows: “By ‘event’ we mean ‘that which happens.’ Thus . . . premiss [(i)] is concerned with change, and it asserts that if the

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series or sequence of changes in time is infinite, then these events considered collectively constitute an actual infinite.”118 Again Craig prefers to work with a broad, largely intuitive notion. This is understandable considering the great number of precisions that can be made regarding events, most of which do not effect an evaluation of the KCA. To circumvent trivial objections to the KCA, the following understanding of event, provided by Quentin Smith in TA&BBC, should suffice: Let an event be a complex that includes whatever happens at one time. . . . Let each time be an extended interval of some given length, say an interval of 1 second. The set of past events form a series (which has sequentially ordered members), since past events are sequentially ordered by the relation is later than. We may say that the set of past events is infinite if there are an infinite number of events that are past, with each such event being of 1 second’s duration and being later than some other event.119

Unless specifically noted, I will employ the word event along the lines Smith suggests.120 At this point someone might wish to object to the theory of time implicit in Craig’s reference to a “sequence of changes,” as any talk about the reality of change tends to place one within the A-theoretic camp of philosophers of time. Now, on the one hand, it is true that Craig is a noted philosopher of time (to date he has published at least four significant monographs in the philosophy of time) and it is also true that he subscribes to an A-theory of time. On the other hand, it is possible to detach the KCA from its traditional association with the Atheory, and reformulate the argument in such wise that it is neutral with respect to both A- and B-theories of time. To emphasize the logical independence of the KCA from any particular temporal theory one could, for instance, substitute the relatively neutral term eventstates for events in argument (A). However, in this book I shall not attempt to carry out the numerous subtle revisions that translating the argument into event-states would require but instead shall follow Craig’s preferred expression and retain the term event.121 The last bit of terminological clarification that should be introduced here, even though it does not appear explicitly in argument (A),

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is the word eternal. In discussions of argument (A) there will be frequent invocations of the idea that the existence of the universe lacks a beginning. Other formulations include: the universe has an infinite past; the universe has existed everlastingly; the universe possesses a sempiternal past; the universe has always existed. When I write of the universe as having an eternal past it is to these sorts of endless existence that I refer. The meaning of eternal that is expressly excluded is the notion of eternity found in theological discussions of the timeless mode of God’s existence. This second notion of eternal one could fairly term Boethian eternity.122 With these distinctions in hand I turn now to an examination of Craig’s defense of argument (A). For the moment I would like to pass over consideration of premise (i), that an infinite temporal regress of events would constitute an actual infinite, and begin instead with premise (ii), namely, the claim that the actual (categorematic) infinite cannot exist.

4.2.5.1 Premise (ii) of Argument (A) Premise (ii) asserts that an actual infinite cannot exist. Craig does not so much argue for this premise as paint evocative images of what it would be like for actual infinite to actually exist. He begins by asking us to suppose that an actually infinite library exists.123 This library is filled with a denumerably infinite number of books. (That is, the number of books in the library is isomorphic with the set of natural numbers and hence has ℵ1 as its cardinal.) The books come in two colors, black and red, and are arranged so that the two colors alternate, with every other book being a different color. Given this description of the books in the library, Craig points out that we would probably not hesitate to say that the number of red books is equal to the number of black books. But, continues Craig, would we not hesitate to say that there are as many red books as there are black and red books taken together? For, in the latter collection, the red books are clearly a subset of the total collection of books: After pairing off all of the red books with each other, would there not be an actually infinite quantity of black books left over? And what if another color, say, green, is added to the collection, such that every third book

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in the total collection is now green? A denumerably infinite quantity of books has just been added to the collection—and yet according to Cantor the number of books has not changed: The original collection of red books remains equal in quantity to the totality of red, black, and green books in the new collection. (ℵ0 + ℵ0 + ℵ0 = ℵ0) Again, what if there were a collection consisting of a denumerably infinite variety of colors of books: Would it not be reasonable to suppose that there is only one book of each color in the collection? But then suppose that for each of these different colors there were a denumerably infinite number of books of that color. Would we believe someone who told us that adding all of these infinites upon infinites of books increases the total number of books not one jot over the quantity of books we started with? Strange results indeed, but perfectly in line with Cantor’s account of the actual infinite. Consider again a denumerably infinite collection of books, and let us further stipulate that a unique number has been printed on the spine of each book. The numbers are assigned so that there is a one-to-one correspondence between the books and the set of natural numbers. Then, since “the collection is actually infinite, this means that every possible natural number is printed on some book.”124 The implication is that it is impossible to add another book to the library. Why? Because there is no unused number we could assign the new book: Every possible natural number has already been used up by the original number assignment; any natural number one might propose already has a corresponding book on the shelf. As Craig points out, this is absurd: Real things can always be numbered, yet here we are, book in hand, with no number to assign to it. In response to the above it might be suggested “that we number the new book ‘no 1’ and add one to the number of every book thereafter.”125 While this would be a perfectly acceptable maneuver in the mathematical world, it is impossible to suppose that such a strategy would work in the actual world. “For an actual infinity of objects already exists that completely exhausts the natural number system— every possible number has been instantiated in reality on the spine of a book.”126 Only in the case of a potential infinite, where there is a new number generated with each dynamic step taken, could the proposed recount be implemented. The openness required by the recount

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simply does not obtain for an actually infinite collection, for the definite and distinct elements of an actual infinite are all already given as a static whole. In short, the proposed recount requires the creation of a new number to put on the spine of the book to be added to the collection, but there is no such number forthcoming. And if there were such a number, Craig would simply point out that the collection either did not fulfill the original conditions stipulated or it was not actually infinite in the first place. What if, instead of adding a book at the beginning of the collection and renumbering from 1, we attempt to add the new book to the other end of the collection? For instance, it might be suggested that we number the new book ω + 1 or ℵ0 + 1. However, neither number will do. First, ω + 1 has the same cardinal number as ω, and what is needed is a new cardinal number, not a new ordinal number. Second, ℵ0 + 1 will not do: “[ℵ0 + 1] reduces to ℵ0, and yet we do have an extra, irreducibly real book on our hands.”127 Moreover, as Craig points out, even though the collection of books is supposed to be actually infinite, there is no book numbered ℵ0 in the collection. ℵ0 has no immediate predecessor (for reasons explained in the appendix), yet all of the books (possibly excepting the first book if there is one) and the spine numbers associated with them would have immediate predecessors. All that saying the collection has ℵ0 elements tells us is that the collection of books constitutes a determinate whole with a denumerably infinite number of elements. But we already knew that ex hypothesi. Therefore, Craig concludes, it is impossible to add another book to the library’s collection.128 He summarizes the situation as follows: These illustrations show that if an actual infinite could exist in reality, it would be impossible to add to it. But it obviously is possible to add to, say, a collection of books: just take one page from each of the first hundred books, add a title page, and put it on the shelf. Therefore, an actual infinite cannot exist in the real world.129

The discussion with which Craig follows this observation serves to reemphasize the points he has already made. Suppose, per impossibile, that we did augment the collection of books with a new book in the manner Craig describes. That new book would have the ordinal number ω + 1 (even though, of course, there is no prior book in the

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ωth position). Despite having a unique ordinal number, adding this new book to the shelf augments the total number of books not at all. “But how can this be? We put the book on the shelf: there is one more book in the collection; we take it off the shelf: there is one less book in the collection.”130 The new book can be added and removed from the shelf as often as we wish, yet the quantity of books does not change. If Cantor’s mathematics can indeed be instantiated in the real world then we need not restrict ourselves to adding and removing a single book. Let us therefore add a denumerable infinity of new books to the actually infinite collection of books resting on the library shelves. The ordinal number of the augmented collection would be ω + ω. Are there really no more books than before? How could a collection of real books numbered (1, 2, 3, . . . , 1, 2, 3, . . .) have the same cardinal number (namely, ℵ0) as a collection of books numbered (1, 2, 3, . . .)? For it seems that all of the natural numbers should have been exhausted in the first ω series: There simply should not be any more natural numbers to go around as each and every one of them has already been uniquely paired off with an element of the first series. And now suppose that an actual infinity of actual infinities of books is added to the original collection (yielding a collection of ordinal type ω + ω2). Would one still wish to maintain that the number of books has not been increased by a single book? Craig avers: “Clearly, something must be amiss here. What is it?—we are trying to take conceptual operations guaranteed by the convention of the principle of correspondence and apply them to the real world of things, and the results are just not believable.”131 It was said above that adding and removing a book from the library shelf does not change the total number of books. But is this true? Suppose once more that the shelves of our library are filled with an actual infinity of books which have been so numbered that each of the natural numbers is represented somewhere within the collection. Suppose book 1 is loaned out. Has the number of books decreased? Judging from what happens when books are added to actually infinite collections, it would seem not: The number of books in our library remains the same. What if instead books (1, 3, 5, . . .) are loaned out? The shelves are depleted by an actually infinite quantity of books, yet the number of books in the collection remains constant. Moreover, the “cumulative gap created by the missing books would be an infinite distance, yet if we push the

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books together to close the gaps, all of the infinite shelves will remain full.”132 Surely this is strange: looked at naively, how can removing half of the books in the library leave one with no extra shelf space? Suppose next that the person who borrowed books (1, 3, 5, . . .) returns them and another patron comes along who checks out the same number of books, except this time it is books (4, 5, 6, . . .) that are borrowed. Suddenly, from having a denumerably infinite number of books, our collection is wiped out and only three lonely books sit on the shelf, namely, books (1, 2, 3). This is absurd. How can removing the same quantity of books make no difference to the cardinality of the collection in the first instance yet make a profound difference to the cardinality of the collection in the second instance? One might object that in transfinite cardinal arithmetic inverse operations are prohibited. It is not permitted, for instance, to subtract quantities from ℵ0, for the perfectly legitimate reason that consistent results are not obtainable for the subtraction operator.133 This is, of course, a good point to make—if all one were worried about is developing a consistent mathematical account. The trouble is that Cantor’s transfinite number theory is here being tested for its applicability to the actual world. So, while “we may correct the mathematician who attempts inverse operations with transfinite numbers, we cannot in the real world prevent people from checking out what books they please from our library.”134 Craig concludes his case against the actual infinite with a consideration of nondenumerable infinite collections.135 Illustrating a nondenumerable infinite is challenging, for while it is generally admitted that things existing in the real world can be numbered, instantiation of a nondenumerable infinite precisely requires that we not be able to number (i.e., count) all of the elements in the collection. For example, supposing that a book has ℵ0 pages is absurd. First, because in the real world it is possible to count pages; and second, because concrete existents like pages in a book take up space, and a single book with ℵ0 pages of uniform thickness would be infinitely thick. Mathematical examples of the nondenumerable infinite, such as points and functions, do not enjoy actual existence according to Craig and hence do not constitute counterexamples.136 And with these last remarks Craig closes his defense of premise (ii) of argument (A).

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4.2.5.2 Premise (i) of Argument (A) The next task facing Craig is the defense of premise (i), which asserts that an infinite temporal regress of events would constitute an actual infinite. Translating this into a slightly less cumbersome formula, premise (i) may be expressed thus: “An infinite past is an actually infinite past.” By premise (i) Craig intends to capture the idea that argument (A) is concerned with change, and it [i.e., premise (i)] asserts that if the series or sequence of changes in time is infinite, then these events considered collectively constitute an actual infinite. The point seems obvious enough, for if there has been a sequence composed of an infinite number of events stretching back into the past, then the set of all events would be an actually infinite set.137

Craig finds premise (i) obvious, and expresses surprise that some philosophical heavyweights have had trouble with it.138 Aristotle and Aquinas, two prominent philosophers whom Craig claims fall under this description, characterize the succession of past events as a potential infinite. Two suggestions may be made in favor of this view. First, it may be suggested that the proper way to talk about an infinite past time is to say that the past is beginningless.139 If the past is beginningless, this means that there is no first event in the history of the universe. Lack of a first event does not of itself immediately entail that the universe has existed eternally, so arguments against the eternal past existence of the universe are misconceived.140 Second, and this is a position Craig claims was explicitly defended by both philosophers, an eternal past does not result in an actual infinite because past days have only successive existence and hence do not exist all at once. Since the existence of one day successively lapses into the existence of another, past days constitute a potential and not an actual infinite.141 Let us examine these two suggestions in order. The first suggestion, namely, that the infinite past of the universe is best described as beginningless, sounds quite plausible at first. If successful, this maneuver would sidestep the family of problems associated with the actual infinite and would effectively place the entire argumentative burden of the KCA on the acceptability of argument (B). However, as

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Craig indicates, asserting that time is beginningless is not a sufficient answer to the problem of an eternal world. This, he writes, is because it is in analysing what a beginningless series of events involves that the absurdities [of supposing the past to be eternal] become manifest. As G. E. Moore indicates, if we grant that events really occur in time, then only two alternatives are possible: either there was a first event or there has been an actually infinite series of events prior to the present one. For if there was no first event, then there must have been an event prior to any given event; since this one also could not be first, there must be an event prior to it, and so on ad infinitum.142

So it appears that a beginningless series of events would prima facie involve the existence of an actual infinite. It is the very question of how time can be beginningless that motivates the KCA in the first place. With respect to the second suggestion, Craig maintains that the fact that past events do not exist simultaneously is irrelevant to the issue at hand. Past events are determinate parts of reality and as such may in principle be collected into a totality.143 To clarify and support this response to Aquinas, Craig cites the authority of the prominent Thomist Fernand Van Steenberghen. As Van Steenberghen notes, [A] universe eternal in the past implies an infinite series in act, since the past is acquired, is realized; that this realization has been successive does not suppress the fact that the infinite series is accomplished and constitutes quite definitely an infinite series in act.144

Since the past events of the universe would constitute an actual infinite, Van Steenberghen draws the conclusion that Aquinas contradicts his own teaching in S.T. I.7.4, where Saint Thomas argues against the possibility of instantiating an actual infinite in nature. Aquinas’s own example of a blacksmith who works through all time past and who breaks an endless series of hammers furnishes a counterexample to Aquinas’s position: The collection of hammers would be an actual infinite.145 After defending the legitimacy of collecting past events into a totality that would be actually infinite if the world were eternal, Craig introduces an important consideration regarding future events.146 Whereas the past represents a completed totality, the future is open-

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ended and is correctly described as potentially infinite. The reason Craig offers for this is that past events have actually existed and hence have some purchase on the real world. Future events, on the other hand, do not and never have existed; future events have yet to occur, and as such have no claim to real existence but only to possible existence. “In no sense does the future actually exist—we must not be fooled by Minkowski diagrams . . . into thinking that future events somehow subsist further down the line, waiting for us to arrive at them. . . . Only the sequence of past events can count as an actual infinity.”147 The last argument Craig offers in favor of premise (i) is a variant of the Tristram Shandy paradox.148 Tristram Shandy is the famous literary character created by Laurence Sterne who writes his autobiography at the rate of one year of writing for each day recorded. Thus, if Shandy started writing when he was twenty years old, he would be twenty-seven when he finished recording the first week of his life.149 Although Craig does not draw out the implications of Tristram’s plight in this degree of detail, the Tristram Shandy paradox can be analyzed in two stages. The first stage involves displaying how the Tristram Shandy paradox explicitly supports premise (i). The second stage involves manifesting how the Tristram Shandy paradox supports the general KCA position that an actually infinite past is impossible.150 My argument relating to the first stage of analysis is brief. An accurate mathematical description of Shandy’s task will involve positing a collection of past events of order type (*ω + *ω). This is clearly a case wherein appeal must be made to the actual infinite, which is what was to be proved. Craig understandably emphasizes the second stage of analysis, where the Tristram Shandy paradox is shown to count against the possibility of an eternal past. While the description of Tristram Shandy’s literary activity clearly is coherent in the finite case, how does Shandy fare if we suppose that he has been writing through an eternal past time? According to Craig, Shandy does not fare very well.151 As the collection of days in an eternal past would constitute an actual infinite, the principle of correspondence guarantees that a one-to-one correspondence can be established between past days and past years, implying that on Cantor’s theory Shandy should have completed his autobiography. But this is manifestly absurd: Shandy cannot have

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written about today’s events because each day he lives through creates another year of work. On the contrary, it seems that if the past is actually infinite, Shandy must have fallen infinitely far behind in his narrative and today he must be writing about some past day that is at an actually infinite remove from today.152 Craig mentions another difficulty with the Tristram Shandy paradox.153 If the series of past events is actually infinite, why should one suppose that Shandy is only finishing his autobiography today? Certainly the world was just as eternal yesterday as it is today. But if that is the case, Shandy should have finished writing about his life yesterday, not today. In fact, no matter how far back we regress through the past days of Shandy’s life he should have been finished with his writing. Therefore, at no time in the eternal past can Shandy be said to be writing the last page of his book: Whatever past day one picks it is always correct to say that Shandy should have finished writing the day before. Thus, at no past day will one discover Shandy in the process of writing, which is absurd since Shandy is, ex hypothesi, writing through all of past eternity. What is more, at no past day will Tristram Shandy be in the process of finishing his writing, which is also absurd, since for the book to be completed he must have finished it on some day. As Craig summarizes: “What the Tristram Shandy story really tells us is that an actually infinite temporal regress is absurd. . . . [We] not only find that the absurdities pertaining to the existence of an actual infinite apply to it, but also that these absurdities are actually heightened because of the sequential character of the series.”154 Since the actual infinite cannot exist and the collection of past events would constitute an actual infinite, Craig concludes that argument (A) is sound: An infinite temporal regress of events cannot exist.

4.2.6 Argument (B) The second strand of argument Craig offers in support of premise (2) takes as its inspiration traditional arguments against the possibility of realizing or attaining an actual infinite by means of some dynamic, successive process.155 The following exposition will focus exclusively upon the argument Craig develops in the main text of TKCA.156 In addition to his main discussion Craig dedicates two lengthy appen-

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dices to related topics.157 I adopt this approach because the core of Craig’s position is to be found in the argument I report and because this core argument clearly guides Craig’s responses to the various positions he examines in the appendices.158 The structure of argument (B) may be presented thus: Argument (B) (a) The temporal series of events is a collection formed by successive addition. (b) No collection formed by successive addition can be an actual infinite. (c) Therefore the temporal series of events cannot be an actual infinite.

This kind of argument has received much attention down through the years: in addition to being defended by Philoponus, closely related variants appear in al-Ghazali, Bonaventure, and the Thesis of Kant’s First Antinomy. It is commonly known as the impossibility of traversing the infinite and also as the impossibility of counting to infinity. As mentioned above, the success of argument (B) turns upon the cogency of denying the possibility of supertasks. If the world were to have existed eternally it would have to be possible, in principle, to realize an actual infinite through some process of successive addition, that is, it would have to be possible to translate the successively given syncategorematic infinite of past events into a fully realized, categorematically infinite collection of past events that finds its completion in the present moment. It is useful to note that argument (B) takes no position on the question of whether an actual infinite can exist. On the contrary: “Even if an actual infinite can exist, the temporal series of events cannot be one, since an actual infinite cannot be formed by successive addition, as the temporal series of events is.”159 Craig’s terse comment may be explained thus. Let us assume, for the sake of developing argument (B) more fully, that an actual infinite can exist. Still, the possibility of an actual infinite existing as yet tells us nothing about how an actual infinite can be brought into existence. While it may be possible to bring an actual infinite into being all at once (for instance, God might be able to create an actually infinite library),160 no process of successive addition could transform a potentially infinite collection into an actually infinite col-

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lection (even God could not fill an empty library with an actually infinite collection of books by adding one book after another to the empty shelves).161 Now, it seems that the temporal series of past events is a collection formed by successive addition. No collection so formed could achieve actual infinity. It follows that the temporal series of past events is not actually infinite. Since supposition of an eternal past requires that the collection of past events be actually infinite, it likewise follows that the temporal series of past events cannot add up to an eternal past. But if the past is not eternal then the quantity of past events within the temporal series must be finite and premise (2) of the KCA is secure. I turn now to a consideration of premise (a) of argument (B). The two key terms in this premise are successive and addition. In claiming that the temporal series of past events is successive, Craig means to assert that not all of the events of the past coexist. Events are realized sequentially, with one event following upon another.162 Second, by saying that the collection of past events is formed through addition, Craig wishes to exclude certain misunderstandings of his position. The central error he warns against is that of mistaking the way we think about past events for the way in which those past events exist in themselves. For instance, it is possible to regress mentally through the past, moving backwards in thought from the current moment to past events ever more distant from today. This mental regress would, in effect, be a subtractive process: As we regress pastward in thought, we subtract one event after another from the collection of past events. However, the series of temporal events is not itself formed through a subtractive process. Rather, the temporal series itself consists in one event following another, each passing event adding to and building upon what went before.163 As Craig notes: Even the expression “temporal regress” can be misleading, for the events themselves are not regressing in time; our thoughts regress as we mentally survey past events. But the series of events is itself progressing in time, that is to say, the collection of all past events grows progressively larger with each passing day.164

The temporal series of events is, therefore, a collection formed by successive addition. With these remarks Craig concludes his discussion of premise (a).

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The defense of premise (b) is likewise succinct. The first point to grasp is that the potential infinite and the actual infinite are conceptually distinct: they differ in kind, not in degree, and no viable offspring could issue from their union because they could never mate.165 Neither ℵ0 nor ω has an immediate predecessor. But without an immediate predecessor it is impossible to pass through the domain of the potential infinite to enter the realm of the actual infinite: actual infinity is unreachable from any finite vantage, no matter how exalted that finite vantage may be.166 Craig’s point may be paraphrased in this way. Any supertask that is supposed to be instantiated in the real world requires the completion of an ωth task. But ω has no immediate predecessor, and with no immediate predecessor it is impossible to reach the required last task through any amount of successively realized discrete acts. Moreover, if ω did have an immediate finite predecessor it would be inappropriate to talk about ω as being above and beyond all of the finite numbers. Since ω has no immediate predecessor yet one must complete an ωth task to complete any supertask it is obvious that performing supertasks is impossible. The second point Craig underscores is that the truth of premise (b) has nothing to do with any time factor.167 No matter how much time is involved,168 one cannot turn a syncategorematic infinite into a categorematic infinite.169 Regardless of how many elements one adds to a finite collection it is always possible to add one more; this expresses the very essence of the syncategorematic infinite. A syncategorematic, or potential, infinite is always finite at every stage; there is no point at which it ceases to be able to get larger. The actual infinite stands in stark contrast to the potential infinite on this score. Recall Craig’s example of an infinite library: Unlike a potentially infinite collection of books, which gets larger with every book added, an actually infinite collection of books does not increase in number with the addition of new volumes. It is always possible to add another element to a potentially infinite collection, and it is this openness to further addition that precludes completion of any potentially infinite process. Without being completed a potentially infinite process never attains to actual infinity, for the potential infinite is, at every stage of its unfolding successive existence, actually finite. Stopping a potentially infinite process of suc-

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cessive addition short of its completion would merely leave one with a vast yet finite collection of elements. Craig illustrates this with the following example: Imagine a man running through space. With each step that the man takes a stone slab appears beneath his foot, and as soon as he removes his foot from the slab the slab disappears. It is clear, Craig says, that even if the man runs for all eternity he will never run out of new stone slabs to run on. The man will never run through all the stone slabs; another step is always possible, and with that step comes a new slab. It is therefore impossible to traverse all of the possible elements of a potentially infinite collection.170 Finally, Craig argues that the only way a collection to which new members are successively added could be actually infinite is if that collection possessed an actual infinity of elements to begin with.171 But then, Craig observes, it would not be the case that we are dealing with a collection formed by successive addition. Nor would a collection built upon an actually infinite base serve as an appropriate model of past events, for the temporal series itself is successively formed throughout. Craig’s defense of premise (b) is now finished, and with it the central philosophical pillars of the KCA have been presented.

4.2.7 Argument (C) With arguments (A) and (B) the case for the (necessarily) finite quantity of the universe’s past events is complete. From both arguments (A) and (B) one learns that the collection of past events cannot be actually infinite and, in the course of developing argument (B), one further learns that there are insuperable difficulties in supposing the past to be potentially infinite. Since the quantity of past events can be neither actually nor potentially infinite, it follows that the quantity of past events is finite. As Craig notes: “This conclusion alone will be sufficient to convince most people that the universe had a beginning, since the universe is not separate from the temporal series of events.”172 However, despite their practical persuasive power, arguments (A) and (B) do not by themselves conclusively establish that the universe came to be ex nihilo. Craig offers, for the sake of completeness, an alterative scenario. The basic idea behind the new scenario is that it is conceptually possible to separate events (and changes in time) from

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the notion of duration or continuity of existence. Thus, it may be possible to conceive of a perfectly quiescent, changeless universe that suddenly begins to change, and with that first sudden change the finite series of past events is initiated.173 Formal presentation of the quiescent universe scenario is given in what I have named argument (C). The purpose of argument (C) is to logically secure the desired theistic interpretation of premise (2), that is, that the universe came to be simplicter. But if the universe came to be simpliciter, then it must have come to be from no pre-existing matter. If so, then the coming-to-be of the universe is not a making (i.e., a change involving the rearrangement of preexisting stuff) but rather constitutes a creation ex nihilo.174 Once it is proven that there exists a cause capable of creating ex nihilo, it is possible to show that this creating cause must be God. Argument (C) can be expressed as a disjunctive syllogism: Argument (C) (a) Either the universe came to be simpliciter or the finite temporal regress of events was preceded by an eternal quiescent universe. (b) It is not the case that the finite temporal regress of events was preceded by an eternal quiescent universe. (g) Therefore the universe came to be simpliciter.175

The import of premise (a) is straightforward. Assuming that either argument (A) or argument (B) is successful, premise (a) summarizes the two disjunctive possibilities: either the universe always existed (in the fashion indicated) or the universe came to be simpliciter. Craig develops his argument for premise (b) as a disjunction. If the universe existed in an eternal quiescent state, then the first event that arises in the quiescent universe must either be caused or uncaused. Suppose that the first event is caused. Then all of the necessary and sufficient conditions for the first event arising must either have been eternally present in the quiescent universe or they were not eternally present. If they were eternally present, then their effect would have been eternally present as well, which would render the existence of a first event impossible. But from arguments (A) and (B) we know that there must be a first event in the history of the universe, so it cannot be the

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case that the necessary and sufficient conditions for the first event to arise were eternally present. On the other hand, if the necessary and sufficient conditions for the first event were not eternally present, then these conditions themselves must have come to be. The result is that the historical series of past events has been pushed back one step. However, this argumentative dodge cannot be repeated indefinitely: As arguments (A) and (B) show, the series of past events must be finite and hence must come to a stop somewhere with a first event. Therefore, it is impossible to account for the arising of the first event in the history of the universe on the supposition that it was caused. Consider next the remaining disjunct, namely, that the arising of the first event within the history of the universe is uncaused. As Craig writes: While this is logically possible, it does not seem very reasonable. For this means that this event occurred entirely without determinate conditions for its happening. The universe existed in a static, absolutely immobile state from eternity and then inexplicably, without any conditions whatsoever, a first event occurred . . . such a picture of the universe is singularly unconvincing.176

Craig, in short, appeals to some form (perhaps a highly attenuated form) of the principle of sufficient reason (PSR). Given some version of the PSR, it follows that a quiescent universe could never stir from its frozen state. Since the universe is now active, it is apparent that the universe never existed in a perfectly quiescent state.177 But if the universe never existed in a quiescent state, premise (b) of argument (C) is secure. Therefore, the conclusion stated in (g) obtains: the universe came to be simpliciter.178 Taken together with arguments (A) and (B), Craig argues that premise (2) of the main KCA is demonstrated to be true: the universe came to be.

4.3 Craig on the Conclusion: The Universe Has a Cause of Its Coming to Be Both premise (1) and premise (2) of the KCA have been defended. All that remains is an evaluation of the conclusion Craig draws from these two premises. What then is implied by the result that there must be a cause of the universe’s coming to be?179

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The most important thing to grasp about Craig’s evaluation of the conclusion of the KCA is that he does not believe the KCA provides adequate philosophical warrant for the conclusion that the creator of the universe (whose existence the KCA proves) possesses the full panoply of divine attributes (e.g., omnipotence, omniscience) traditionally discussed within theistic apologetics. Rather, Craig is content with the claim that the KCA proves that a personal Creator of the universe exists.180 Craig’s contention that the creator of the universe is a personal being is supported by two arguments.181 The first argument appeals to the reasoning presented in argument (C), which suffices to prove that the creator of the universe cannot be a “mechanically operating” or non-voluntary cause.182 Non-voluntary causes are always active when situated within a context that permits their operation. Were a non-voluntary cause to bring about the existence of the universe ex nihilo it would have done so from all eternity, for all of the necessary and sufficient conditions for its operation would have been present.183 Now, if the cause of the universe’s coming-to-be were eternally active the universe itself would have been brought into existence an eternity ago. But this is impossible, as the KCA demonstrates. Hence, the cause of the creation of the universe could not be a non-voluntary cause but rather must be a voluntary cause. Since any being capable of voluntary activity possesses a will (for the will is the principle of voluntary activity) and also an intellect (for intellect is a faculty required for the operation of will), it follows that the voluntary creator of the universe must be a personal being.184 Craig offers a second argument for the personal nature of the creator of the universe. This argument takes its starting point from the observation that the coming to be of the universe was caused. Adopting an argument from al-Ghazali, Craig argues that only an appeal to (what he calls) the Islamic principle of determination will afford an answer to the question: Why did the universe begin to exist when it did instead of existing from eternity?185 The principle of determination states that when two outcomes are equally possible and one results this occurrence must be due to the action of a personal agent who freely chooses one of the two possible outcomes. In the present case, it may be thought that the existence and the nonexistence of the universe

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were equally likely prior to the universe’s being caused to exist. Since the two equally likely possibilities were resolved into a single result, it follows that the creative cause of the universe must be a freely acting personal agent.186 After establishing that the creator of the universe must be a personal agent, Craig raises the question of whether this personal creator is really related to that which has been created. Thomas Aquinas, for reasons bound up with his particular theory of relations, holds that God is not really related to the world. Abandoning the opinion of Aquinas, Craig asserts that it is reasonable to say that the personal creator of the universe (who may plausibly be identified with God) is related to the world as Creator to creature.187 This relation does not imply the existence of any real change in God.188 However, in being related to the world God can be said to have voluntarily entered the stream of time: It is more accurate, according to Craig, to characterize the divine mode of existence after creation as an endless temporal existence.189 Craig concludes his work with the claim that he has philosophically demonstrated the existence of “a personal Creator of the universe who exists changelessly and independently prior to creation and in time subsequent to creation.”190 Having discovered this, “it is incumbent upon us to inquire whether He has specially revealed Himself to man in some way that we might know Him more fully or whether, like Aristotle’s unmoved mover, He remains aloof and detached from the world that He has made.”191

NOTES 1. William Lane Craig, The Kalåm Cosmological Argument (London: Macmillan, 1979), p.141; hereafter TKCA. I have omitted Craig’s footnote citing the relevant passages in Hume. 2. The fullest treatment Craig presents in favor of premise 1 can be found in essays 5 and 10 of William Lane Craig and Quentin Smith, Theism, Atheism, and Big Bang Cosmology (New York: Oxford University Press, 1993); hereafter TA&BBC. 3. Craig uses several different terms for the restricted domain of possibility he has in mind. In TKCA he typically employs variants of “real possibility.” In articles, he sometimes speaks of “metaphysical possibility” and at

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other times he speaks of “broadly logical possibility.” Detailed exposition of what Craig means and how these differently named sorts of possibility might (or might not) differ from one another are not forthcoming in his work. Would Craig endorse the accounts of factual and substantial possibility I present in chapters 3 and 4? Perhaps not—but where he would distance himself from my position is an open question. To simplify matters, I will consistently refer to “factual possibility” in my presentation of the KCA. This is not what Craig literally says. It is, however, what I think he should say. (My apologies to Bill Craig for adopting this device.) An example of Craig’s use of the distinction between logical and (what I term) factual possibility is apposite here. In a brief survey of analytic authors who have dismissed the KCA, Craig offers the following critique of W. I. Matson: Matson similarly asserts that since there is no logical inconsistency in an infinite series of numbers, there is no logical inconsistency in an infinite series of events, and therefore the first cause argument is incurably fallacious. . . . Matson fails to understand that the kalåm argument holds that the existence of an actual infinite is really, not logically, impossible. That there is a difference can be seen in the fact that God’s non-existence, if He exists, is logically, but not really, possible; if He does not exist, His existence is then logically, but not really, possible. Analogously, the existence of an actual infinite is really impossible, even if it may not involve logical contradiction. (TKCA, p. 155 n. 17) 4. TKCA, pp. 141–44. 5. See G. E. M. Anscombe, “Whatever Has a Beginning of Existence Must Have a Cause: Hume’s Argument Exposed, ” in The Collected Philosophical Papers of G. E. M. Anscombe: Volume One: From Parmenides to Wittgenstein (Minneapolis: University of Minnesota Press, 1981), pp. 93–99. 6. TKCA, p. 145. 7. It should be noted that Craig’s argument is overly compressed here, for he shifts without warning from discussing the factual impossibility of Hume’s imagined cases to a defense of the reality of causation broadly construed. The jump is not much softened by the fact that Craig explicitly rejects certain causal theories, specifically the occasionalism of al-Ghazali and Nicolas Malebranche, later in the book. See TKCA, p. 181. 8. TKCA, p. 145. 9. Craig’s references are to Stuart Hackett, The Resurrection of Theism (Chicago: Moody, 1957). 10. TKCA, p. 146.

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11. Ibid., p. 147. 12. Ibid., pp. 147–48. 13. Additional details concerning Cantor’s system are presented in the appendix. 14. See, e.g., TKCA, pp. 65, 69, and pp. 102–103. 15. The requisite definition of actual infinite is given in the next section. 16. A good case can also be made in favor of the hyperreal numbers introduced by Abraham Robinson. Unlike Cantor’s transfinite numbers, ordinary arithmetic operations work perfectly well for hyperreals. However, the Cantorian distinction between the variable finite (traditionally represented by `) and the actually infinite stands. Hyperreal numbers are defined in terms of sequences of real numbers, and a hyperreal number is said to be infinite when it is either greater than all real numbers (for positive hyperreals) or less than all real numbers (for negative hyperreals). Given that the distinction between potential infinity and actual infinity remains intelligible, the same absurdities one encounters in attempting to instantiate an actual infinite (as in Argument A) or transmute a potential infinite into an actual infinite (as in Argument B) arise. For a convenient introduction to hyperreal numbers, see James M. Henle and Eugene M. Kleinberg, Infinitesimal Calculus (Mineola, NY: Dover, 2003), chaps. 3 and 4. Less mathematically and ontologically benign are the surreal numbers introduced by David Conway. An objection to the KCA, based on an oddity of the surreal number system, is developed in chapter 2 below. 17. Craig details this strategy in TKCA, pp. 69–72. Of course, more work needs to be done by the defender of the KCA than merely showing the inherent absurdity of instantiating an actual infinite in rerum natura. I would argue, however, that in securing this key point one has progressed significantly along the path to proving the existence of God. The position of the KCA defender is analogous to that of Thomas Aquinas after the presentation of the Five Ways: If the arguments of Summa Theologiae I.2.3 are sound, then extracting all of the pure perfections from what has gone before is a relatively straightforward task. 18. I borrow this terminology from A. W. Moore. I do not, however, accept Moore’s definitions of these terms. 19. In this book the term metaphysical infinite will be preferred. This is for two reasons. First, it is easy to confuse the terms absolute infinite and actual infinite, and the latter plays a key role in Craig’s version of the KCA. Second, although “absolute infinite” is the preferred language of Cantor, Cantor himself often fails to distinguish between metaphysical and mathematical senses of infinity. The confusion in Cantor’s thought seems to arise through his neglect of the analogies embedded in the particular formula that guides his thinking on the absolute infinite, viz., that the absolute infinite is “that than which

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nothing greater can be thought.” Thus, Cantor moves easily from discussions of God’s absolute infinity to discussions of a (supposed) quantity greater than any other possible quantity. This latter notion, as Cantor himself notes, is absurd: There can be no greatest transfinite quantity. For more comments on this subject see Robin Small, “Cantor and the Scholastics,” American Catholic Philosophical Quarterly 66, no. 4 (1992): 407–28. 20. Failure to distinguish the metaphysical from the mathematical infinite can lead the unwary into serious confusion. A notable example of how not to talk about the infinite may be found in Morris Lazerowitz, “On a Property of a Perfect Being,” Mind 92 (1983): 257–63. 21. Although in some contexts Craig employs the Cartesian term indefinite as a synonym for the potential infinite (see, e.g., TKCA, p. 69), I will avoid using “indefinite” in this way. 22. Similar distinctions are to be found in Cantor. Philip Jourdain summarizes Cantor’s position as follows: The Grundlagen begins by drawing a distinction between two meanings which the word “infinity” may have in mathematics. The mathematical infinite, says Cantor, appears in two forms: Firstly, as an improper infinite (Uneigentlich-Unendliches), a magnitude which either increases above all limits or decreases to an arbitrary smallness, but always remains finite; so that it may be called a variable finite. Secondly, as a definite, a proper infinite (Eigentlich-Unendliches), represented by certain conceptions in geometry, and, in the theory of functions, by the point infinity of the complex plane. In the last case we have a single, definite point, and the behaviour of (analytic) functions about this point is examined in exactly the same way as it is about any other point. Cantor’s infinite real integers are also properly infinite, and, to emphasize this, the old symbol “`,” which was and is used also for the improper infinite, was here replaced by “ω.” (Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans. Philip E. B. Jourdain [New York: Dover, 1955], pp. 55–56) For references to Cantor, see Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. Ernst Zermelo (Hildesheim: Georg Olms, 1962). The distinction between the improper and the proper infinite may be found in Gesammelte Abhandlungen, pp. 165–66. Cantor’s novel use of the ` symbol is discussed in ibid., p. 195. 23. Two excellent anthologies of the “supertask” literature are available: A. W. Moore, ed., Infinity (Brookfield, VT: Dartmouth Publishing, 1993); and Wesley C. Salmon, ed., Zeno’s Paradoxes (Indianapolis: Hackett, 2001).

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24. This immediately raises the problem of how a supertask is supposed to be completed. Craig addresses this problem several times; see section 4.2.6 for further details. 25. J. F. Thomson, “Tasks and Super-Tasks,” Analysis 15 (1954): 1–13. This paper is reprinted in the anthologies by Moore and Salmon mentioned in n. 23 above. 26. It might be mentioned here that in drawing upon the work of Cantor for his proof of the finitude of the past Craig is on good historical ground: Cantor himself “suggested that his theory of transfinite numbers could also be used to demonstrate the absolute impossibility of the eternity of time, space, and matter” (Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite [Cambridge, MA: Harvard University Press, 1979]). 27. That is, that 2ℵ0 = ℵ1. Succinctly stated, Cantor’s continuum hypothesis is that the cardinality associated with the real numbers (i.e., the set of numbers denoting a continuum) constitutes the next largest infinite cardinal after ℵ0. Assuming only the standard nine axioms of Zermelo-Fraenkel set theory it is impossible to prove either that the continuum hypothesis is true or that it is false. Introduction of a tenth axiom would allow derivation of 2ℵ0 = ℵ1, but none of the tenth axioms proposed thus far have the same intuitive appeal as the first nine. Without such intuitive appeal the situation appears to be much like that encountered by Russell, who found himself obliged to introduce an axiom of infinity and thereby vitiated his project of reducing all of mathematics to self-evident rules of logic. A convenient exposition of the continuum problem may be found in A. W. Moore, The Infinite (London: Routledge, 1995), pp. 154–55. For more details see Raymond M. Smullyan, “The Continuum Problem,” in Encyclopedia of Philosophy, ed. Paul Edwards (New York: Macmillan, 1967), 2:207–12. There is also a generalized form of the continuum hypothesis which asserts that for every ordinal n it is the case that ℵn + 1 = 2ℵn. For a compact discussion of some technical aspects of the continuum hypothesis, see Alexander Abian, The Theory of Sets and Transfinite Arithmetic (Philadelphia: W. B. Saunders, 1965), pp. 392–94. 28. Craig himself focuses on Russell’s paradox although he also mentions the antinomies of Burali-Forti and Cantor. See TKCA, p. 90. 29. There were, of course, some notable exceptions. The actual infinite was championed by such thinkers as Giordano Bruno, Baruch Spinoza, and Gottfried Leibniz. Aristotle is careful to note that the use of potential in “potential infinite” does not perfectly parallel the usage of potential in other contexts. He writes: “But we must not construe potential existence in the way we do when we say that it is possible for this to be a statue—this will be a statue, but something infinite will not be in actuality. Being is spoken of in many ways, and we say that the infinite is in the sense in which we say it is day or it is the

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games, because one thing after another is always coming into existence” (Physics III.6, 206a19–23). Unless otherwise noted, all citations of Aristotle refer to The Complete Works of Aristotle: The Revised Oxford Translation, 2 vols., ed. Jonathan Barnes (Princeton, NJ: Princeton University Press, 1984). 30. As Aristotle summarizes: Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable. In point of fact they do not need the infinite and do not use it. They postulate only that a finite straight line may be produced as far as they wish. It is possible to have divided into the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them whether the infinite is found among existent magnitudes. (Physics III.7, 207b27–34) 31. Physics III.6, 206a25–29. A slightly more compressed version may be found in Physics III.6, 207a7–8: “[S]omething is infinite if, taking it quantity by quantity, we can always take something outside.” 32. As Aristotle writes: “By addition then, also, there is potentially an infinite, namely, what we have described as being in a sense the same as the infinite in respect of division. For it will always be possible to take something ab extra. Yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every determinate magnitude is surpassed and there will always be a smaller part” (Physics III.6, 206b16–20). 33. Neither of these examples would meet the standards of rigor demanded of contemporary mathematical practice. Nevertheless, both approaches have heuristic value. I have encountered texts written in the late twentieth century that still explain the differential calculus as “slope finding” (a notion much akin to Newton’s fluxions). The example of congruent figures is adapted from a popular geometry text that explains reflection through a line in this way: triangle ABC is a proper reflection of triangle DEF if and only if it is possible to lift ABC out of the plane and flip it over onto DEF. From a contemporary perspective this is a shockingly loose explanation, but the student gets the point readily enough. Moreover, it is quite plausible to argue that a dynamic understanding of mathematics is a valuable aid to mathematical discovery. 34. C. F. Gauss, Briefwechsel, 6 vols., ed. C. A. F. Peters (Altona: Gustav Esch, 1860–1865), 2:269. The letter is dated July 12, 1831. To make his own view more palatable Cantor is willing to employ the traditional language of “approaching a limit” to explain the theory of transfinite numbers: “It is even permissible to think of the newly and [sic] created number ω as the limit to

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which the numbers v strive [where v denotes a finite positive integer], if by that nothing else is understood than that ω is to the first integer which follows all the numbers v, that is to say, is to be called greater than every v” (Cantor, Contributions, pp. 56–57, n. †). The German reads: “Es ist sogar erlaubt, sich die neugeschaffene Zahl ω als Grenze zu denken, welcher die Zahlen v zustreben, wenn darunter nichts anderes verstanden wird, als daß ω die erste ganze Zahl sein soll, welche auf alle Zahlen v folgt, d. h. größer zu nennen ist als jede der Zahlen v” (Cantor, Gesammelte Abhandlungen, p. 195). Certain important qualifications must be made here; I will have more to say about this topic in the next section and in the appendix. 35. Abraham Adolf Fraenkel, Abstract Set Theory, 2nd ed. (Amsterdam: North-Holland, 1961), pp. 5–6. The resolution of the foundational problems in mathematical analysis referred to was due to Karl Weierstrass, who clarified the key mathematical notions of minimum, function, and differential quotient. 36. Possible exceptions might arise from the intuitionist and conventionalist camps. An extreme intuitionist might claim that the reliability of the successor function (i.e., the “+ 1” function) is not amenable to a satisfying finite constructive proof and hence the potential infinity of the natural numbers cannot be established. It should be remembered, however, that even L. E. J. Brouwer accepted the potential infinite: See L. E. J. Brouwer, “Intuitionism and Formalism,” in Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), pp. 77–89, 80–81, and passim. On the other hand, an extreme conventionalist (Ludwig Wittgenstein has been portrayed as such) might claim that the successor function depends on empirical verification and hence, since no such verification is forthcoming, the potential infinite must be rejected. However, philosophers and mathematicians who have conventionalist leanings tend to take a fairly benign view of the potential infinite. One may cite here the case of Friedrich Waismann, who articulates a hybrid position in the philosophy of mathematics somewhere between conventionalism and intuitionism. Waismann is willing to accept the potential infinite even though he is prepared to jettison Russell’s axiom of infinity on empirical grounds. See Friedrich Waismann, Lectures on the Philosophy of Mathematics, ed. Wolfgang Grassl (Amsterdam: Rodopi B.V., 1982), pp. 33, 118–23. 37. For a succinct overview and appreciation of Cantor’s contribution to mathematics, see David Hilbert, “On the Infinite,” in Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), pp. 183–201. 38. The literature on the infinite, both popular and technical, is (fittingly enough) vast. The most obvious place to start for our present purposes is Craig’s excellent survey in TKCA, pp. 65–82. Also very readable, with an interesting Wittgensteinian slant, is A. W. Moore’s detailed historical exposition in

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The Infinite. Eli Maor’s elegant To Infinity and Beyond: A Cultural History of the Infinite (Princeton, NJ: Princeton University Press, 1991) can also be recommended. José Benardete provides a wealth of arresting thought experiments as well as a solid rejoinder to the mid-twentieth-century trend toward verificationism in his Infinity: An Essay in Metaphysics (Oxford: Clarendon, 1964). Rudy Rucker’s Infinity and the Mind: The Science and Philosophy of the Infinite (Princeton, NJ: Princeton University Press, 1995), while brisk and entertaining, is unfortunately too philosophically unsophisticated to serve as a reliable guide. Important selections from Cantor are available in a convenient English translation by Philip Jourdain in Contributions to the Founding of the Theory of Transfinite Numbers. Several important mathematical results have come to light since Cantor, many of which are gathered in Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge, MA: Harvard University Press, 1967). For clear and rigorous introductions to set theory in general and to transfinite arithmetic in particular, one may consult the excellent works of Alexander Abian and Abraham Fraenkel. In Abstract Set Theory, Fraenkel gives the classic formulation of what is perhaps the most influential variant of set theory in current use. Though somewhat dated, the bibliography in Fraenkel is of considerable value as a guide to the literature on set theory. Abian’s The Theory of Sets and Transfinite Arithmetic (Philadelphia: W. B. Saunders, 1965) gives an accessible and focused account of set theory that conveniently provides most of the material needed for work on the KCA. 39. In my exposition of Cantor’s set theory I will work exclusively with this definition of set. Cantor did propose other informal ways of defining a set. Variations within Cantor’s writings arise more from a difference of emphasis than from a change in his basic position. Thus, Cantor writes in a note to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre: Mannichfaltigkeitslehre. Mit diesem Worte bezeichne ich einen sehr viel umfassenden Lehrbegriff, den ich bisher nur in der speziellen Gestaltung einer arithmetischen oder geometrischen Mengenlehre auszubilden versucht habe. Unter einer „Mannigfaltigkeit“ oder „Menge“ verstehe ich nämlich allgemein jedes Viele, welches sich als Eines denken läßt, d. h. jeden Inbegriff bestimmter Elemente, welcher durch ein Gesetz zu einem Ganzen verbunden werden kann . . .” (Gesammelte Abhandlungen, p. 204. Jourdain’s translation and discussion of this passage may be found in Contributions, p. 54) I will not be using the above definition of set in my exposition of set theory since it omits a characteristic, namely, that of distinctness, important to the KCA. However, it is worth noting that the notion of binding elements into a

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whole by means of a law is commonly appealed to in contemporary philosophical expositions of Cantor. 40. Cantor, Contributions, p. 85. I have modified Jourdain’s translation slightly: Where I have set Jourdain has aggregate, and where I have distinct he has separate. The German reads: “Unter einer „Menge“ verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die „Elemente“ von M gennant werden) zu einem Ganzen” (Cantor, Gesammelte Abhandlungen, p. 282). Jourdain’s translation, while quite reliable, has become somewhat dated since the standard English translation of Mengenlehre is now set theory. If the current mathematical idiom were to be consistently adopted, then the German plural die Mengen would, in many contexts, be best translated by the singular English set. However, despite this shift in terminology, except in cases where confusion might arise, I have left Jourdain’s translation untouched. (I will note all such departures.) I do so because Jourdain captures the literal sense of Cantor quite nicely and, moreover, it is useful to preserve the obsolete terminology as a constant reminder that in reading Cantor one is, despite the polish of Cantor’s style, still at the rough beginnings of a new science. 41. I will not enter here into the interesting question of how the priority of elements to their set is to be construed. More recent writers on set theory are apt to substitute equipollent or equipotent where Cantor and Craig use equivalent. 42. A classic discussion of the two methods of forming a set can be found in Bertrand Russell’s Introduction to Mathematical Philosophy (London: George Allen & Unwin, 1956), pp. 12–13. The first means of specifying set membership Russell calls extensional, the second he calls intensional. Russell argues that the intensional approach is more fundamental. (I agree with Russell on this point.) 43. Such an approach to sets is familiar from computer programming. Suppose I were to write a data compression algorithm that takes as its input any alphanumeric data file. The data file would constitute in concrete form a finite set of enumerated elements. There need be no logical relation obtaining between the elements of an enumerated set: The algorithm might apply equally well to a file of randomly generated ASCII characters and compression to a database of carefully entered information. 44. This second way of specifying a set is the method usually presupposed in contemporary philosophical discussions of Cantor’s set theory. In such discussions it is usually combined with an appeal to the second definition of set that was mentioned in n. 40 above. To see how this works, first suppose that the law of excluded middle holds. Then, for any meaningful predicate, it will be the case either that the predicate applies to a given thing or it does not apply. The collection of all and only those things to which the predicate applies will then constitute a set. This seems to be a very natural way of going about building a

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set, and it came as something of a shock to discover that there are contradictions lurking within this method when it is considered in its full universality. 45. Fraenkel, Abstract Set Theory, p. 10. Fraenkel gives a similar exposition in his article on “Set Theory” in The Encyclopedia of Philosophy, vol. 7, ed. Paul Edwards (New York: Macmillan, 1972). 46. TKCA, p. 68. 47. One might argue, however, that the idea of the potential infinite is important to or even presupposed by classical set theory. For instance, if one adopts what has come to be called “the iterative conception of set,” then it appears that sets are constructed in a stepwise fashion, each set building upon the sets that go before. Such a stepwise constructive method resembles the dynamic succession characteristic of the potential infinite. For an excellent discussion see the articles by George Boolos, Charles Parsons, and Hao Wang in Paul Benacerraf and Hilary Putnam, eds., Philosophy of Mathematics (Cambridge: Cambridge University Press, 1983). 48. Craig makes this point in TKCA, pp. 71–72, and cites Pamela Huby as an authority. For an analysis of Russell, see appendix 2 of TKCA. 49. I will have more to say about this sort of objection in chapter 2. Briefly stated, the main difficulty with such an objection is that being able to give a consistent mathematical description of the infinite does not suffice to establish the factual possibility of its obtaining in this universe. (A consistent mathematical description may, of course, be a necessary condition for something to exist.) Something more is needed, as is apparent in the case of geometry, where it is acknowledged that care must be exercised to determine which, if any, of the competing geometries obtains as the actual physical geometry of our universe. 50. Forced to abandon naive set theory because of the various paradoxes discovered, mathematicians who wanted to preserve Cantor’s treatment of the infinite took to axiomization. The best-known system is ZF (with or without the axiom of choice); however, there are other widely used axiomatic systems besides ZF, e.g., NBG (von Neumann-Bernays-Gödel) and MK (MorseKelley) set theory. Both NBG and MK are extensions of ZF in the sense that the same theorems provable about sets in ZF are provable in NBG and MK. There are also set theories that are incompatible with ZF, such as Quine’s NF (“New Foundations”: see Willard Van Orman Quine, From a Logical Point of View, 2nd ed. [Cambridge, MA: Harvard University Press, 1980], pp. 80–101). The paradoxes that arise in Cantor’s system do not occur in NF. This is hardly surprising, since NF was specifically constructed so as to preclude formal derivation of the paradoxes. For a discussion of the philosophical limitations of NF, see George Boolos’s judicious comments in “The Iterative Conception of a Set,” in Philosophy of Logic, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), p. 490.

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Note that in its customary axiomatic formulation ZF assumes an ontology of sets; an applied version of set theory would have to allow for the existence of individuals that are not sets. To simplify matters I will assume that the necessary modifications to the axioms of ZF (or whatever set theory one prefers) have been made to allow for such individuals. Vide the pertinent remarks in Charles Parsons, “What Is the Iterative Conception of Set?” in Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), pp. 503–506; and Hao Wang, “The Concept of Set,” in ibid., pp. 530–70. 51. See Lotfi Zadeh’s classic paper “Fuzzy Sets,” Information and Control 8 (1965): 338–53. A convenient and reasonably detailed explanation of fuzzy set theory is given in George Bojadziev and Maria Bojadziev, Fuzzy Sets, Fuzzy Logic, Applications (Singapore: World Scientific, 1995). 52. Since most of mathematics can be expressed in set theory, it proves possible to “fuzzify” other branches of mathematics. One thus finds fuzzy calculus, fuzzy topology, and so on. There is also fuzzy logic, a discipline within which there is much ongoing debate concerning how to interpret Zadeh’s fuzzy notion of truth. A trenchant, and I think convincing, criticism of fuzzy logic is developed in Susan Haack, Deviant Logic, Fuzzy Logic: Beyond the Formalism (Chicago: University of Chicago Press, 1996), pp. 229–58. As Haack puts it, “I remain convinced, first . . . that truth does not come in degrees, and, second, that fuzzy logic is not a viable competitor of classical logic” (p. 230). See also her brief remarks in Philosophy of Logics (Cambridge: Cambridge University Press, 1978), pp. 162–69. 53. One interesting implication of this is that fuzzy sets do not have a single cardinality. Rather, fuzzy sets may be said to have a cardinality of n, where n is the result of a function expressing the various degrees of membership of the set’s elements. See Peter van Inwagen, Material Beings (Ithaca, NY: Cornell University Press, 1990), pp. 221–27. 54. We may leave aside iterated versions of this puzzle where the degree of membership is determined by multiple factors (e.g., membership might be made dependent upon a prior membership) instead of a single-dimensioned function. 55. Cantor, Contributions, pp. 86–87. The German reads: Zwei Mengen M und N nennen wir „äquivalent“ . . . wenn es möglich ist, dieselben gesetzmäßig in eine derartige Beziehung zueinander zu setzen, daß jedem Element der einen von ihnen ein und nur ein Element der andern entspricht. Jedem Teil M1 von M entspricht alsdann ein bestimmter äquivalenter Teil N1 von N und umgekehrt. (Gesammelte Abhandlungen, p. 283, original emphasis omitted)

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This definition of equivalence, along with the principle of correspondence that undergirds it, is taken up in ZF as well as other varieties of set theory; e.g.: DEFINITION V. If the members of a set T can be related to the members of a set S in a one-to-one (biunique) correspondence, i.e., so that a single member of T corresponds to each member of S and vice versa, we speak of a (one-to-one) mapping of S onto T. In this case the set T is called equivalent to the set S, written T∼S. (Fraenkel, Abstract Set Theory, p. 24; I have omitted Fraenkel’s footnote references) 56. See, for example, objection III.b.(b).8 in chapter 2. 57. Since Cantor’s formulation is uncharacteristically verbose (see Contributions, pp. 103–104, 108), I will rely upon the logically equivalent but more succinct expressions of other authors. 58. “A system S is said to be infinite when it is similar [i.e., equivalent in the sense expressed by the principle of correspondence] to a proper part of itself; in the contrary case S is said to be a finite system.” (Richard Dedekind, Essays on the Theory of Numbers, trans. Wooster Woodruff Beman [New York: Dover, 1963], p. 63. The gloss on “similar” may be justified by consulting ibid., pp. 53–55.) The German reads: “Ein System S heißt unendlich, wenn es einem echten Teile seinter selbst ähnlich ist; im entgegengesetzten Falle heißt S ein endliches System” (Richard Dedekind, Was sind und was sollen die Zahlen [Braunschweig: Friedr. Vieweg & Sohn, 1960]. For Dedekind’s technical definition of “ähnlich” see ibid., pp. 7–8). Interestingly, Dedekind provides an argument for the existence of actual infinities to which this definition applies. Reduced to its essential features, his argument is strongly reminiscent of one Saint Augustine used to show, contra the Academic skeptics, that we know infinitely many things. For instance, I know that I exist. But if I know that I exist, then I also know that I know that I exist. But then I also know that I know that I know that I exist, and so on. Dedekind’s own argument is somewhat more complex in its expression: Satz. Es gibt unendliche Systeme. Beweis. Meine Gedankenwelt, d. h. die Gesamtheit S aller Dinge, welche Gegenstand meines Denkens sein können, ist unendlich. Denn wenn s ein Element von S bedeutet, so ist der Gedanke s’, daß s Gegenstand meines Denkens sein kann, selbst ein Element von S. Sieht man dasselbe als Bild φ(s) des Elementes s an, so hat daher die hierdurch bestimmte Abbildung φ von S die Eigenschaft, daß das Bild S’ Teil von S ist; und zwar ist S’ echter Teil von S, weil es in S

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Elemente gibt (z. B. mein eigenes Ich), welche von jedem solchen Gedanken s’ verschieden und deshalb nicht in S’ enthalten sind. Endlich leuchtet ein, daß, wenn a, b verschiedene Elemente von S sind, auch ihre Bilder a’, b’ verschieden sind, daß also die Abbildung φ eine deutliche (ähnliche) ist. Mithin ist S unendlich, w. z. b. w. (Was sind und was sollen die Zahlen, 14, Dedekind’s references omitted. For translation see Dedekind, Essays on the Theory of Numbers, p. 64) 59. Fraenkel, Abstract Set Theory, p. 28. 60. Ibid., p. 29. 61. Craig places great emphasis on this point. See, for example, TKCA, pp. 94–95. For a sensitive and illuminating discussion of the basic issues involved, see Stephen F. Barker, Philosophy of Mathematics (Englewood Cliffs, NJ: Prentice-Hall, 1964), pp. 65ff. 62. For a proof of this, see Cantor, Contributions, p. 108. 63. Cantor also claimed to see profound connections between transfinite number theory and other parts of mathematics. For instance, Cantor claimed that the irrational numbers are conceptually dependent upon the transfinite numbers. Quoting from a letter Cantor wrote in 1884, Jourdain writes: At the end of this letter, Cantor remarked that . . . “ω is the least transfinite ordinal number which is greater than all finite numbers; exactly in the same way that 2 is the limit of certain variable, increasing, rational numbers, with this difference: the difference between 2 and these approximating fractions becomes as small as we wish, whereas ω – v is always equal to ω. But this difference in no way alters the fact that ω is to be regarded as as definite and completed as 2 , and in no way alters the fact that ω has no more trace of the numbers v which tend to it than 2 has of the approximating fractions. The transfinite numbers are in a sense new irrationalities, and indeed in my eyes the best method of defining finite irrational numbers is the same in principle as my method of introducing transfinite numbers. We can say that the transfinite numbers stand or fall with finite irrational numbers, in their inmost being they are alike, for both are definitely marked off modifications of the actually infinite.” (Cantor, Contributions, p. 77) Unfortunately, the original letter from which Jourdain quotes from does not appear to have survived. As Ivor Grattan-Guinness remarks in his introduction to Jourdain’s collected works:

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64. One can understand Craig’s delight when he encounters passages like the following in the writings of a mathematician of David Hilbert’s stature: We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can thought about things be so much different from things? Can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality? (“On the Infinite,” p. 191) 65. This way of phrasing the matter papers over a multitude of significant differences, for the variety of positions adopted by thinkers who fall under the description is quite wide. Some mathematicians, like Hilbert, grant that Cantor’s theory is mathematically valid but claim that the concept of the actual infinite has no ontological application; other mathematicians, like Brouwer, reject Cantor’s work even when restricted to the conceptual domain of mathematics. A thinker like Brouwer who rejects the actual infinite (or, more precisely, the nondenumerable actual infinite) in mathematics will a fortiori reject the possibility that the actual infinite (or at least some varieties of the actual infinite) is to be met with in nature. 66. Those who fall under the “mathematical realist” rubric are also dubbed “platonists.” It is hopeless to try to justify the label on scholarly grounds, but the appellation has become traditional within the philosophy of mathematics. Craig himself uses the label, as the next section shows. 67. A natural objection to this claim is that past events cannot constitute an actually infinite totality because past events no longer exist. Craig addresses this concern in the course of his presentation of the KCA; for more on how Craig’s thesis might be defended, please refer to chapter 5, section 21 below. 68. TKCA, p. 87. The actual taxonomy proposed by Craig is heavily dependent upon the philosophical accounts developed by Stephen Barker in Philosophy of Mathematics and by Abraham Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy

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in Foundations of Set Theory (Amsterdam: North-Holland, 1958); see TKCA, p. 156 nn. 29–32 for Craig’s references. In addition to these excellent sources one might also call attention to the introduction in Benacerraf and Putnam, Philosophy of Mathematics, to Stephan Körner, The Philosophy of Mathematics: An Introductory Essay (New York: Dover, 1968), and to Stewart Shapiro, Thinking about Mathematics: The Philosophy of Mathematics (New York: Oxford University Press, 2000). As Shapiro ably illustrates, the philosophy of mathematics has undergone considerable development since the late 1980s. There are, for instance, new realist positions that do not develop their ante rem realism in terms of sets. And new life has been breathed into the realist project by Penelope Maddy: See especially her Realism in Mathematics (Oxford: Oxford University Press, 1990). (Maddy has since modified her position in important details: See Maddy, Naturalism in Mathematics [Oxford: Oxford University Press, 1997].) 69. TKCA, p. 87. 70. TKCA, p. 88. Given Craig’s definition of platonism and also given that it is unlikely a platonist will believe the natural numbers are not well defined, we are unlikely to meet a platonist who does not accept the existence of an actual infinite. It is possible, however, to conceive of a weaker form of mathematical realism that does not commit itself to the actual infinite. In that case, the mathematical realist would count as an ally of the KCA and no further argument against mathematical realism is necessary. 71. Gödel’s appropriation of Cantor’s transfinite numbers is complex. On the one hand, he is happy to accept that Cantor’s work represents a discovery and not an invention: For someone who considers mathematical objects to exist independently of our constructions and of our having an intuition of them individually, and who requires only that the general mathematical concepts must be sufficiently clear for us to be able to recognize their soundness and the truth of the axioms concerning them, there exists, I believe, a satisfactory foundation of Cantor’s set theory in its whole original extent and meaning, namely axiomatics of set theory interpreted in the way sketched below. [Gödel then details his responses to the paradoxes of set theory.]” (“What Is Cantor’s Continuum Problem?” in Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam [Cambridge: Cambridge University Press, 1983], p. 474) On the other hand, Gödel is very hesitant about admitting that Cantor’s system has a nonabstract realization:

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CHAPTER 1 the objects of transfinite set theory . . . clearly do not belong to the physical world and even their indirect connection with physical experience is very loose (owing primarily to the fact that set-theoretical concepts play only a minor role in the physical theories of today). (Ibid., p. 483)

72. TKCA, p. 88. 73. Ibid. 74. Ibid. Craig gives very short shrift to the mathematical nominalists and gives no example of someone who has maintained a nominalist account of mathematics. What seems to underlie Craig’s understanding of nominalism is the discussion found in Barker, Philosophy of Mathematics, pp. 69–72. In concluding his discussion of nominalism Barker writes: “It seems impossible to avoid the conclusion that number theory cannot be given any thoroughly nominalistic interpretation under which it will come out literally true. The convinced nominalist will have to view the system of number theory as incapable of having any true interpretation. . . . Some non-nominalists would regard this conclusion as a reductio ad absurdum of nominalism” (ibid., p. 72). As the work of Hartry Field has shown, a nominalist approach to the philosophy of mathematics is capable of more promising and sophisticated development than either Barker or Craig anticipate. See Field, Science without Numbers (Princeton, NJ: Princeton University Press, 1980). 75. A mathematical nominalist might try to rescue Cantorian transfinite arithmetic by claiming that it is simply a brute fact that there is an actually infinite number of things in the universe. (This would be a nominalist take on Russell’s axiom of infinity.) Once granted this contingent fact, it may also be a brute fact that the parts of the universe can be paired off with each other in the way Cantor describes. There seem to be serious difficulties with this sort of approach. How can a Cantorian nominalist make sense of what are necessarily nonempirical applications of the principle of correspondence? The question does not appear to be intractable, but a fair amount of explaining on the part of the nominalist seems to be in order. Hartry Field offers a compelling view of how mathematical nominalism can be made respectable. Field takes the substantivalist line that space-time points and space-time regions are not abstract but concrete. This yields an ontology of physical objects the size of the powerset of the continuum (ℵ2). For concise summary of Field’s philosophy of mathematics and some pointed criticism, see Shapiro, Thinking about Mathematics, pp. 226–37. One obvious problem with Field’s ontology is that it is very difficult to see why we should treat extensionless space-time points as physical objects. 76. TKCA, p. 88. 77. Another important figure Craig might have mentioned is Arend

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Heyting. Both Heyting and Brouwer have representative articles in Benacerraf and Putnam, Philosophy of Mathematics. Although Craig does not explicitly say so, those who accept an explanation of the foundation of mathematics that draws upon the Aristotelian and Thomistic traditions would presumably be classified as conceptualists. For discussions of Aristotle see Thomas Heath, Mathematics in Aristotle (Bristol, UK: Thoemmes, 1998), and Jonathan Lear, Aristotle: The Desire to Understand (New York: Cambridge University Press, 1992); for Thomas Aquinas see Armand Maurer, “Thomists and Thomas Aquinas on the Foundation of Mathematics,” Review of Metaphysics 47, no. 1 (1993): 43–62. A Thomistic theory in particular has much to recommend it, especially in regard to its ability to resolve questions concerning the ability of mathematics to accurately model the physical world. I am, however, unaware of any published work that explicitly defends the Aristotelian or Thomistic accounts from the pointed criticism that Frege (in The Foundations of Arithmetic [New York: Harper & Brothers, 1960]) and Peter Geach (in Mental Acts: Their Content and Their Objects [London: Routledge & Kegan Paul, 1957]) offer against the mental abstraction on which these theories appear to rely. (The case of Geach is further complicated by his rather idiosyncratic position that Aquinas is not an abstractionist in the relevant sense. More work needs to be done before it can be said with confidence that Aquinas’s account of mathematics escapes Geach’s critique of abstractionism.) It also seems possible to develop a basically Aristotelian position along structuralist lines. (See n. 83 below.) 78. For a clearer understanding of the intuitionist position it is useful to turn to the remarkably lucid presentation by Heyting. Speaking through the voice of a character named “Int.” Heyting writes: If “to exist” does not mean “to be constructed,” it must have some metaphysical meaning. It cannot be the task of mathematics to investigate this meaning or to decide whether it is tenable or not. We [intuitionists] have no objection against a mathematician privately admitting any metaphysical theory he likes, but Brouwer’s program entails that we study mathematics as something simpler, more immediate than metaphysics. In the study of mental mathematical constructions “to exist” must be synonymous with “to be constructed.” (“Disputation,” in Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam [Cambridge: Cambridge University Press, 1983], pp. 66–67) 79. TKCA, pp. 88–89. Brouwer himself admits the acceptability of denumerably infinite sets, such sets being actually infinite. See L. E. J. Brouwer, “Intuitionism and Formalism,” in Philosophy of Mathematics, ed. Paul Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), p. 81.

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80. TKCA, p. 89. 81. Ibid. 82. Since the publication of TKCA, a fifth important approach to the philosophy of mathematics, called structuralism, has come into its own. The inspiration behind structuralist approaches is that mathematics is fundamentally about formal structures and the various mappings among those structures. There is little agreement among structuralists as to which ontological account is supposed to undergird mathematical structures: some structuralist philosophies are readily classified as platonist; others, such as Geoffrey Hellman’s modal structuralism, are developed along nominalist lines. For assessments of the competing brands of mathematical structuralism, see Hellman, “Three Varieties of Mathematical Structuralism,” Philosophia Mathematica 3, no. 9 (2001): 184–211, and Shapiro, Thinking about Mathematics, chap. 10. 83. Although Craig does not emphasize the point, formalism’s disconnect from the world is also its fatal flaw as a philosophy of mathematics. We are left without any account of the applicability of mathematics. This is a general difficulty besetting formalism and is not limited to the formalist account of transfinite number theory. 84. A good example is Abian, who writes: Thus, for instance, whenever in the Theory of Sets we are confronted with a statement such as there exists a set x whose elements are sets b and c, and there exists a set u whose elements are sets x, b and m, then we may take this statement as implying that a table such as the following appears as a part of the illusory table which describes the Theory of Sets (i.e., when the latter is treated as though it were an illusory model): a b∈a c∈a

e a ∈e b ∈e m ∈e

...

. . . The above considerations show how we may interpret more concretely the notion of existence in the Theory of Sets. In short, if an axiom or a theorem of the Theory of Sets asserts that if certain sets (with such and such sets as their elements) exist, then a certain set (with such and such sets as its elements) also exists, we shall interpret this as: if certain sets (with the elements of each set written under the corresponding set) are listed in the above illusory table, then a certain set (with its elements written under it) must also be listed in the same illusory table. (Abian, The

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Theory of Sets and Transfinite Arithmetic, pp. 67–68, emphasis in the original. Craig remarks on this passage in TKCA, pp. 70–71) 85. For examples see the collections of articles by Gödel and Brouwer in Benacerraf and Putnam, Philosophy of Mathematics. 86. It seems therefore that Craig misspeaks when he claims: “Cantor’s definition of a set made it clear that he was theorising about the abstract realm and not the real world for, it will be remembered, he held that the members of a set were objects of our intuition or of our thought” (TKCA, p. 70). For a good discussion of Cantor’s attitude toward his own work, see Small, “Cantor and the Scholastics.” See also Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge, MA: Harvard University Press, 1979). 87. Waismann argues that the series 1/(1 + x2) diverges for x = 1 because it diverges for x = i, since when x = i the denominator goes to 0 and the function goes to `. From this it follows that the function will also diverge for all values where |x| > 1. Thus, the fact that there is a singularity for the function in the complex domain can be used to explain the behavior of the function in the real domain. See Waismann, Lectures on the Philosophy of Mathematics, pp. 29–30. 88. One could also mention the various uses electrical engineers have found for the complex numbers. 89. Among the various examples one might educe to show this, Craig cites the case of R. L. Sturch, who dismisses the KCA in a single sentence: “the result of applying Cantorian theory to [these] paradoxes is to resolve them” (quoted in TKCA, p. 155 n. 17. See also the example of W. I. Matson given in n. 3 above). 90. Gödel, as mentioned in n. 72 above, is an example of such a thinker. Even Bertrand Russell was agnostic about demonstrating the existence of the actual infinite: It cannot be said to be certain that there are in fact any infinite collections in the world. The assumption that there are is what we call the “axiom of infinity.” Although various ways suggest themselves by which we might hope to prove this axiom, there is reason to fear that they are all fallacious, and that there is no conclusive logical reason for believing it to be true. At the same time, there is certainly no logical reason against infinite collections, and we are therefore justified, in logic, in investigating the hypothesis that there are such collections. (Bertrand Russell, Introduction to Mathematical Philosophy [London: George Allen and Unwin, 1956], p. 77)

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See also ibid., pp. 137–41, where Russell distances himself from positive arguments for the existence of the actual infinite. A better example of the ready assimilation of the mathematically consistent to the factually possible is to be found in the publications of Graham Oppy, discussed in chapter 3. 91. Craig marshals an impressive list of prominent mathematicians who are willing to distance themselves from a commitment to the extramental existence of the actual infinite: see TKCA, pp. 70–71. 92. It is possible for members of at least two of the three nonrealist positions Craig discusses (viz., the nominalists and formalists) to hold that as a matter of contingent fact there does exist an actual infinite in the realm of nature. Craig’s response would be that the possibility of an actual infinite existing in nature is not a thesis that gains a priori support from any of the three nonrealist philosophies of mathematics. Moreover, while asserting the extramental existence of the actual infinite might be logically consistent with nominalism or formalism, such an assertion would be empirically false, as can be shown by the various thought experiments of the KCA. 93. There are other well-known objections to the platonist approach to abstracta. Positing the extramental existence of abstracta has been objected to on the grounds that such abstracta are causally inert, epistemologically inaccessible, and explanatorily otiose. It is not the task of the present book to develop these traditional lines of criticism. 94. For Craig’s comments on the first two paradoxes, see TKCA, pp. 89–90. 95. Russell describes the paradox that has come to be named after him in his Introduction to Mathematical Philosophy, p. 136. In that work Russell poses the problem in terms of classes instead of sets. 96. TKCA, p. 91. 97. Bertrand Russell, “Mathematical Logic as Based on the Theory of Types,” in Logic and Knowledge, ed. Robert Charles Marsh (London: Unwin Hyman, 1956), p. 63. He glosses the second formulation (stated in terms of a “total”) with the following note: “When I say that a collection has no total, I mean that statements about all its members are nonsense” (p. 63 n.*). For present purposes it is not necessary to distinguish between the simple and ramified versions of the theory of types. 98. The claims Russell makes for his theory of types are quite sweeping. For instance, the theory of types is claimed to resolve not only the paradoxes of set theory but is also alleged to take the sting out of the Liar paradox. Russell summarizes his discussion of the theory of types thus: After stating some of the paradoxes of logic, we found that all of them arise from the fact that an expression referring to all of some

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collection may itself appear to denote one of the collection; as, for example, ‘all propositions are either true or false’ appears to be itself a proposition. We decided that, where this appears to occur, we are dealing with a false totality, and that in fact nothing whatever can significantly be said about all of the supposed collection. In order to give effect to this decision, we explained a doctrine of types of variables, proceeding upon the principle that any expression which refers to all of some type must, if it denotes anything, denote something of a higher type than that to all of which it refers. (“Mathematical Logic as Based on The Theory of Types,” p. 101) 199. Craig’s criticism of Russell, presented at TKCA, pp. 91–92, follows the critique offered by Barker, whom I quote below. 100. Barker, Philosophy of Mathematics, p. 87. 101. This example of a self-referential statement does not appear in Craig but is, I think, a congenial development of the line of criticism he offers. Another virtuous example: Without impredicative definition it would be awkward to logically paraphrase the statement “Michael Jordan has the highest free-throw percentage on the basketball team.” One way of capturing this claim would be: M = Michael Jordan T = the team (∀x) (x ∈ T . (x ≠ M . FT%M > FT%x)) The difficulty, of course, lies with the T. It is perfectly natural to think of Michael Jordan as a member of his basketball team. Why should we accept that it is necessary in principle to find some way of paraphrasing away the team? 102. Barker, Philosophy of Mathematics, pp. 87–88. 103. TKCA, p. 92. 104. TKCA, p. 94. 105. TKCA, pp. 94–95. 106. As Moore notes: “[Bolzano] seemed to take it for granted that there were ‘true’ criteria for comparing sets in size, and that these were the ‘subset’ criteria. On Bolzano’s view there just were fewer even natural numbers than natural numbers altogether, irrespective of the fact that they could be paired off ” (Moore, The Infinite, p. 113). 107. TKCA, p. 95. 108. Our world is not and could not be the world of the Square described in Edwin A. Abbott’s Flatland. 109. In the remainder of this chapter and where appropriate in the

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remainder of this book I will use the words collection and group where I have heretofore been employing the word set. This is to help mark the difference between the mathematical account of the actual infinite, which I have been discussing hitherto, and its supposed existence in the realm of nature, which is the subject of discussion henceforward. When talking of a collection or group some ontological import is at issue; use of the term set will not carry any such ontological import. 110. It is to be noted that some of these points are only fully established after considering the material in sections 4.2.5 and 4.2.6 below. 111. In addition to being presented below, each of these arguments is also given in the argument guide at the beginning of chapter 2. 112. I depart slightly from the conventional usage of this term in that I do not presuppose that a supertask must be completed within a finite span of time. As Craig argues, it is not the length of time that renders a supertask impossible since a supertask is incompletable even given an actually infinite amount of time. The difficulty lies in the basic description of a supertask. It is impossible to complete any supertask because it is impossible to transmute a potential infinite into an actual infinite via some sequential process. 113. Craig explains A and B theories of time in this manner: Following McTaggart, contemporary philosophers of space and time distinguish between an A-theory of time, according to which events are temporally ordered by tensed determinations of past, present, and future, and temporal becoming is an objective feature of physical reality, and a B-theory of time, according to which events are ordered by the tenseless relations of earlier than, simultaneous with, and later than, and temporal becoming is subjective and minddependent. (TA&BBC, p. 94) 114. TKCA, p. 82. 115. This is a slightly reworded version of the argument Craig gives at TKCA, p. 69. 116. TKCA, p. 69. 117. Although Craig does not accept a realist account of mathematics, he is willing, on occasion, to entertain the realist hypothesis to make his point. See, for instance, his ingenious argument in TA&BBC, p. 97, where he supposes a platonist account of mathematics to distinguish between numbers and numerals. 118. TKCA, p. 95. 119. TA&BBC, p. 77. 120. I offer further precisions concerning what is meant by an event in

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chapter 5. Evidence that Craig would find the above definition acceptable may be gathered from some of his remarks on the use of the word event in his presentation of (what I have called) argument (B): But the series of events is itself progressing in time, that is to say, the collection of all past events grows progressively larger with each passing day. Nor is the series of events a continuum from which events are formed by division. We are speaking in this argument [viz., argument (B)] primarily of events, definite and distinct happenings, that occur in time; we are not speaking of time itself, which is at least mentally infinitely divisible. (TKCA, p. 103) 121. I do not mean to suggest that the choice of a particular temporal theory is of no consequence. On the contrary: one interesting result obtained by situating the KCA within a B-theoretic context is that, in addition to the past existence of the universe being finite, the future existence of the universe is demonstrated to be finite as well. (Some may take this as a reductio ad absurdum of the B-theory.) 122. For Boethius’s original formulation, see The Consolation of Philosophy V.5. An important discussion of Boethius is to be found in Thomas Aquinas, Summa Theologiae I.10.1 and 4. 123. All of the library and book examples mentioned in this and the succeeding few paragraphs are taken from TKCA, pp. 82–84. The objections and replies that follow are also due to Craig. Although Craig does not cite his source for the infinite library thought experiment, the idea of an infinite library was famously developed by Jorge Luis Borges: See “The Library of Babel,” in Labyrinths (New York: New Directions, 1962), pp. 51–58. Quine has some interesting remarks related to Borges’s infinite library: See Willard Van Orman Quine, Quiddities (Cambridge, MA: Harvard University Press, 1987), s.v. “Universal Library.” 124. TKCA, p. 83. Another way to think of this is from the standpoint of a publisher who has to assign ISBNs to her books. Publish one more new book and there is no ISBN available to assign to it. Or, to anticipate a possible objection based on Robinson’s hyperreal numbers, there is no standard integer to print on the spine of the book. This qualification is required because, according to one of the stranger results of nonstandard analysis, there are nonstandard infinite integers that are not real numbers. Such hyperreal infinite integers would not be accessible from the real integers, as infinite hyperreals lie beyond all real numbers; there is no immediate real predecessor to an infinite hyperreal integer, and thus Craig’s assessment of the thought experiment is correct. 125. TKCA, p. 83.

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126. TKCA, pp. 83–84. 127. TKCA, p. 84. 128. At this point in the argument Craig briefly abandons his collection of books in order to discuss the famous thought experiment known as Hilbert’s Hotel. (See TKCA, pp. 84–85.) As the example of Hilbert’s Hotel adds nothing new to the points made earlier with respect to the infinite library, I will not reproduce Craig’s discussion here. For an explanation of the Hilbert’s Hotel thought experiment, see George Gamow, One, Two, Three . . . Infinity (Mineola, NY: Dover, 1988), p. 17. 129. TKCA, p. 85. 130. Ibid. 131. TKCA, pp. 85–86. 132. TKCA, p. 86. 133. See appendix for details. 134. TKCA, p. 86. 135. See TKCA, pp. 86–87. 136. TKCA, p. 87. See also sections 4.2.3.1–4.2.3.3 above. Craig is on good Aristotelian ground here: see, e.g., his analysis of potentially infinite divisibility in TA&BBC, pp. 93–94, where he defends the Aristotelian position that a point is not a point until you pick it. (In support of this interpretation of Aristotle, see De Anima III.6, 430b11–14.) 137. TKCA, p. 95. 138. Craig writes: “But manifest as this may be to us, it was not always considered so. The point somehow eluded Aristotle himself, as well as his scholastic progeny, who regarded the past sequence of events as a potential infinite” (TKCA, p. 95). 139. This way of conceptualizing the problem may be implicit in S.T. I.46.2 ad 6, where Aquinas rejects Bonaventure’s argument for the finitude of the past because, given an infinite past, it would be impossible to fix a first term from whence to begin calculating the traversal of time to the present day. The inability to fix the required first term at an infinite remove from today could be said to be due to the fact that the universe’s history is beginningless. The other considerations Aquinas’s argument raises for the notion of a traversal are more fully dealt with in the context of argument (B) and in chapter 2. Sobel gives a nice presentation of Aquinas’s position: See Jordan Howard Sobel, Logic and Theism: Arguments For and Against Beliefs in God (Cambridge: Cambridge University Press, 2004), p. 182. 140. For more on the “beginningless” objection to the KCA, see objection III.a.(i).1 in chapter 2 below. 141. In support of his interpretation, Craig cites Phys. III.6, 206a25–206b1, and S.T. I.7.4.

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142. TKCA, p. 157 n. 34. See G. E. Moore, Some Main Problems of Philosophy (New York: Macmillan, 1953), pp. 174–75. 143. For Craig’s comments, see TKCA, p. 96. To expand on what he writes there: Everything that exists in rerum natura is denumerable. This is, perhaps, the conceptual point behind the scholastic dictum that unum is a transcendental predicate. Unity applies to every category of being, and past events, as determinate parts of reality, can be numbered and hence legitimately collected into a denumerable totality. It may be admitted, however, that for someone who is a noted philosopher of time, Craig is oddly terse regarding what he means by the reality of past events. Unfortunately, getting sidetracked into a discussion of Craig’s voluminous writings on the philosophy of time would defeat the purpose of this chapter. My discussion of temporal marks in chapter 5, section 21.2, is intended to help alleviate the problem. 144. Quoted in TKCA, p. 96. In addition to this passage one might also cite Fernand Van Steenberghen, Thomas Aquinas and Radical Aristotelianism (Washington, DC: Catholic University of America Press, 1980), pp. 16–17: “[T]he infinite series of past events implied by the hypothesis of an eternal world is clearly an infinite in act, not in potency. Indeed, it is a realized and achieved infinite, already produced in reality.” 145. TKCA, p. 96. Craig also argues that the successive nature of time presupposed by Aristotle and Aquinas in fact exacerbates their position. The actual infinite cannot be added to, yet the past is constantly being augmented by the addition of new days. Again, if the past is actually infinite, the quantity of past events never changes: the quantity of past events is the same no matter how far one regresses into the past. (For these issues, see TKCA, p. 97. Hyperreals may be accounted for by rephrasing the argument in terms of the impossibility of modifying the cardinality of an actually infinite collection of days.) 146. TKCA, pp. 96–97. 147. TKCA, p. 97. Note that Craig’s argument clearly presupposes an Atheory of time. For the B-theorist this explanation would not avail since future events would have an existence on par with past events, and the same KCA reasonings that establish the finitude of the past relative to today would apply to future events. When combined with a B-theoretic ontology the KCA establishes that the entire existence of the universe, past and future, is of finite duration. 148. The Tristram Shandy paradox is originally due to Bertrand Russell. For Russell’s description and analysis of the paradox, see The Principles of Mathematics, 2nd ed. (London: George Allen & Unwin, 1937), pp. 358–59. 149. Russell’s own take on the paradox, is not terribly interesting since it is manifestly fallacious. According to Russell, if Tristram lived for only a finite amount of time, he would never finish recording all the days of his life because he would keep falling further and further behind in his writing. (Thus far Russell’s

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characterization of the Shandy case is not in dispute. Nor is there any dispute over the inherent coherence of the finite Shandy case. The existence of slow writers is more than a mere logical possibility: As the scholastic dictum has it, ab esse ad posse valet consequentia.) However, Russell continues, were we to suppose that Tristram is immortal then he could finish his task: As an immortal being, Tristram would live through an actually infinite number of days and an actually infinite number of years, and as both years and days are denumerably infinite it is possible to set up a correspondence between them such that every day is written about in some year. Among other difficulties one might mention with Russell’s view, it is clear that he fails to notice that the future is only potentially infinite, and so the future years that are supposed to correspond to the present days Shandy is living through do not exist. If the future years do not exist, then they cannot legitimately be gathered into a collection called “the days of Tristram’s life” and placed in correspondence with past or present days. See TKCA, pp. 97–98, for Craig’s analysis; see also Craig’s references to and comments on G. J. Whitrow and David Conway in TKCA, p. 157 nn. 41 and 42. It should be observed that this reply to Russell presupposes an A-theory of time. On a B-theory of time one might instead note various causal impossibilities implied by Russell’s proposal. 150. It may well be the case that the Tristram Shandy paradox can be adapted so as to support both arguments (A) and (B). See n. 153 below. 151. For the argument that follows, see TKCA, p. 98. 152. This interpretation of the Tristram Shandy paradox could be used to support the contention of argument (B) that it is impossible to traverse the actual infinite by means of some successive process. As the Shandy paradox indicates, the structure of time is sufficiently rich to permit mapping collections of order type *ω + *ω onto it. Should one suppose the world to be eternal, then there would be past days at an actually infinite remove from today that must have been traversed on the way from the infinite past to the present moment. 153. TKCA, pp. 98–99. 154. TKCA, p. 99. 155. Or, should one prefer Kantian language, argument (B) affirms the soundness of the Thesis of Kant’s First Antinomy. As Kant would have it: “But now the infinity of a series consists precisely in the fact that it can never be completed through a successive synthesis” (Immanuel Kant, Critique of Pure Reason, trans. Paul Guyer and Allen W. Wood [Cambridge: Cambridge University Press, 1998], A426/B454). The German has: “Nun besteht aber eben darin die Unendlichkeit einer Reihe, daß sie durch sukzessive Synthesis niemals vollendet sein kann” (Kritik der reinen Vernunft, 2 vols., ed. Wilhelm Weischedel [Frankfurt am Main: Suhrkamp, 1995]). 156. The argument is found in TKCA, pp. 102–110.

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157. The appendixes treat (1) the KCA in relation to Zeno’s paradoxes (with an excursus through the analytic literature on supertasks) and (2) contemporary treatments of Kant’s First Antinomy. 158. Additional material drawn from the appendices of TKCA may be found in chapter 2. 159. TKCA, p. 103. In characterizing past events in this fashion Craig is clearly taking an A-theory of time as his paradigm. Since B-theorists must explain the phenomenon of the arrow of time (e.g., D. H. Mellor, a noted Btheorist, takes pains to develop a causal account of time that rules out the possibility of causal loops: Effects and causes are so related that causes always precede effects and never vice versa), it seems that there are resources available for translating argument (B) into a form that is acceptable to a B-theorist. I will not undertake the task of providing such a translation, not least because I myself subscribe to a relational A-theory of time. 160. This is, of course, only an illustration. As will become apparent in part 2 of this book, I reject the possibility that God could create an actually infinite library. 161. See TKCA, pp. 104–105. Craig writes that “even God could not instantiate the infinite library volume by volume, one at a time” (TKCA, p. 105). 162. TKCA, p. 103. 163. A clarification of what counts as an “event” for the KCA and some brief remarks concerning the metaphysical status of the temporal marks left by past events can be found below in chapter 5, section 21. 164. TKCA, p. 103. 165. TKCA, p. 104. 166. “Therefore one can never reach ℵ0 by successive addition or counting, since this would involve passing through an immediate predecessor to ℵ0” (TKCA, p. 104). Cantor himself would be suspicious of transforming a variable finite, that is the potential infinite, into an actual infinite by means of some successive process. He writes in a letter of 1886: Accordingly I call ω the limit of the increasing, finite, whole numbers v, because ω is the least of all numbers which are greater than all the finite numbers. But ω – v is always equal to ω, and therefore we cannot say that the increasing finite numbers v come as near as we wish to ω; indeed any number v however great is quite as far off from ω as the least finite number. Here we see especially clearly the very important fact that my least transfinite ordinal number ω, and consequently all greater ordinal numbers, lie quite outside the endless series 1, 2, 3, and so on. Thus ω is not a maximum of the finite numbers, for there is no such thing. (Quoted by Jourdain in Contributions, p. 78)

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167. TKCA, pp. 104–105. 168. Craig cites Russell and Matson as examples of philosophers who have erred in thinking that the reason why a potential infinite cannot become an actual infinite is due to an insufficiency of time (TKCA, p. 159 n. 46). It might also be suggested that the entire strategy of appealing to infinite past time is wrongheaded. Whether an infinite amount of past time can elapse is precisely the point at issue. Superimposing a problematic collection of, say, seconds on top of a collection of discrete past events does not settle the matter: “[F]or if one [such collection] is possible, both are possible, and if one is absurd, both are absurd. Since any collection formed by successive addition cannot be infinite, an infinite number of seconds cannot have elapsed” (TKCA, p. 159 n. 47). 169. This fact is recognized by set theory: a denumerable number of finite sets forms a set that is itself denumerable, and the addition of two finite sets always results in a set that is itself finite. 170. TKCA, p. 104. 171. TKCA, p. 105. 172. TKCA, p. 99. 173. Craig attributes the original version of this argument to al-‘Allåf (TKCA, p. 99). One might plausibly push the attribution back to Plato in the Timaeus. 174. I am presupposing here that premise (1) has already been secured, viz., that whatever comes to be has a cause of its coming to be. 175. For Craig’s version of argument (C) and his discussion of the argument, see TKCA, pp. 99–102. 176. TKCA, pp. 100–101. 177. Craig develops a second strand of argument in favor of premise (b). Boiled down to its essentials, his argument is that supposing the universe existed in a quiescent state prior to the first event would contradict the best empirical evidence available concerning the state of the early universe. The only way the universe could exist in a perfectly quiescent state is if all of the matter of the universe existed at 0K, i.e., at absolute zero. However, the matter existing under the conditions posited by big bang cosmology was very hot indeed, and so the hypothesis that there was a prior existence at 0K is unwarranted. See TKCA, pp. 101–102. 178. In an interesting section (TKCA, pp. 106–109) Craig takes up the question of whether the soundness of the KCA implies that there must be a beginning of time as well as a beginning of existence for the universe. His answer can be summarized in two points: First, the notion of a first instant of time can be made coherent (consider, for example, a relational A-theory of time: where there is no change there is no time; yet, the idea of duration, understood as uninter-

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rupted existence, could still apply); second, it is possible to conceptually disconnect the existence of time from the existence of the universe (as, for example, Richard Swinburne has done by introducing the idea of a Newtonian absolute time that precedes the proper time of our changing universe). 179. Craig’s comments on the conclusion of the KCA are more suggestive than rigorously argued. This is understandable in the light of Craig’s (I think reasonable) opinion that the majority of his argumentative work is complete and that it would be a desperate nontheist indeed who would accept the KCA yet deny that it is the existence of God that has been proven. (Cf. Craig’s remarks on the problem of evil at pp. 173–74 n. 172.) One can gain much insight into Craig’s theology from reading pages 149–53 and their associated footnotes. (For instance, Craig remarks: “I think that it is within the context of Trinitarian theology that the personhood and timelessness of God may be most satisfactorily understood” [TKCA, p. 171 n. 166].) However, given the overtly philosophical focus of the present book, I will pass over most of this interesting material in silence. 180. Craig makes the following remarks in TKCA, p. 173 n. 171: I find the modesty of the kalam argument’s conclusion one of its appealing features; the argument does not try to prove too much. I am very suspicious of marvelous arguments of natural theology which purport to prove from reason alone a whole catalogue of divine attributes. If theologians dogmatically insist, however, that only a being who is proved to be omnipotent, omniscient, and so forth, deserves to be called “God,” then it is of course true that the cosmological argument as I have presented it does not prove the existence of “God.” I would be perfectly happy to say only that the argument shows that a personal Creator of the universe exists. We must then seek to discover by means of reason or revelation whether the Creator is good, omnipotent, and so forth. 181. In his presentation Craig runs these two distinct arguments together. See TKCA, pp. 149–51. 182. Craig develops this argument at TKCA, pp. 150–51. 183. Supposing a nonvoluntary cause to be inactive would imply that prior to the creation of the universe some change must have occurred either within the nonvoluntary cause or within its environment. But this prior change would itself count as an event and the KCA proves that the total quantity of past events must be finite. So, to circumvent this objection, all the defender of the KCA has to do is push the argument back to a consideration of the cause of the first event simpliciter and enquire into the necessary and sufficient con-

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ditions that obtain for the first operation of the cause of the first event simpliciter. 184. While the creator of the universe may not have been demonstrated to fulfill all of the criteria for personhood delineated by Boethius (that is, while it has been shown that the creator has a rational nature it has yet to be proven that the creator is an individual substance), in contemporary analytic discussions of personhood possession of both an intellect and will are taken to be jointly sufficient conditions for establishing personhood. 185. TKCA, p. 150. 186. For Craig’s characterization of the Islamic principle of determination and his statement of the argument, see TKCA, pp. 150–51. 187. TKCA, 152. Craig cites the position of Swinburne with approval: See TKCA, p. 152, 172 n. 169. 188. TKCA, p. 152. 189. TKCA, pp. 152–53. While Craig’s remarks here raise some interesting issues, insofar as the agreement between Craig and Aquinas goes beyond the purely verbal level I think that Thomas has the better of the argument. 190. TKCA, p. 152. 191. TKCA, p. 153.

2

ARGUMENT GUIDE Objections Propaedeutic to Main KCA

Division I

Main KCA (1) Whatever comes to be has a cause of its coming to be. (2) The universe came to be. (3) Therefore the universe has a cause of its coming to be.

Division II Division III Division IV

Argument A (i) An infinite temporal regress of events would constitute an actual infinite. (ii) An actual infinite cannot exist. (iii) Therefore an infinite temporal regress of events cannot exist. Argument B (a) The temporal series of events is a collection formed by successive addition. (b) No collection formed by successive addition can be an actual infinite. (c) Therefore the temporal series of events cannot be an actual infinite. Argument C (a) Either the universe came to be simpliciter or the finite temporal regress of events was preceded by an eternal quiescent universe. (b) It is not the case that the finite temporal regress of events was preceded by an eternal quiescent universe. (g) Therefore the universe came to be simpliciter.

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III.a.(i) III.a.(ii) III.a.(iii)

III.b.(a) III.b.(b) III.b.(c)

III.c.(a)

III.c.(b) III.c.(g)

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5. PURPOSE, METHOD, AND NOTATION In this chapter I will catalog, in a systematic way, major objections that have been advanced against the KCA. Wherever appropriate I will also provide a sketch of the reply (often by Craig) that has been made to a particular objection.1 No attempt will be made to present every objection that has been made to the KCA, nor will an attempt be made to provide a final and definitive response to every objection recorded here.2 There are two main reasons for these limitations. First, not every rejoinder to the KCA has received a response. Second, the bulk of primary and secondary literature that has grown up around the KCA since Craig’s original publication of TKCA in 1979 precludes an exhaustive presentation of all the arguments that have made their way into print. Another necessary constraint is that, while I will occasionally mention objections that take the scientific (i.e., the big bang) version of the KCA as their target, no systematic presentation of objections to the scientific branch of the KCA will be attempted.3 This is in line with the stated focus of this book, which, as I indicated in the introduction, focuses on what I have called the philosophical branch of the KCA. The taxonomy of objections and replies presented therefore does not pretend to capture every nuance of the debate but rather is intended to reveal the broad contours of the discussion thus far. Capturing the essence of the KCA debate is complicated by the largely article-based format of the discussion: theistic rejoinders have (more often than not) generated atheistic surrejoinders, and these have their sur-surrejoinders, and so on. It is hoped that when a taxonomy of the major objections found in the literature (and, where appropriate, by mentioning important lacunae in the literature) is provided, the logical structure of the KCA will stand clearly exposed. It is further hoped that clarifying the logical structure of the argument will help bring some measure of order to the currently wide-ranging debate over the KCA and aid in assessing the argument. Some objections considered in this chapter do not apply uniquely to the KCA but apply equally well to other important families of argument within philosophical theology. For instance, arguments against the first premise of the KCA, namely, that whatever comes to be has a cause of its coming to be, are likely to apply to arguments for the existence of God that proceed from an analysis of motion or from a con-

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sideration of efficient causality.4 Or, since in part 2 of this book I develop a variant of the KCA that incorporates a metaphysics of substances, process-based metaphysicians might object that I have failed to locate the KCA within its proper metaphysical home.5 Other objections to the KCA that I will consider, while perhaps generalizable to other varieties of theistic argument, are especially pressing for the KCA. For instance, the defender of the KCA is committed to the proposition that the universe has not always existed. Now, according to certain temporal theories, time came into existence with the universe itself, which would entail that prior to the existence of the universe there was no time. On the basis of this view it has been objected that since voluntary choices require time for their realization, the cause of the universe’s coming into existence (if there be such) cannot have acted voluntarily. This conclusion, while formally consistent with the KCA, would not be consonant with the apologetic aims of the KCA and hence would seriously undermine the attractiveness of the argument for theists. Another example of an objection of this type is that of Adolf Grünbaum, who argues that the theist’s sudden appeal to a creative cause to explain the beginning of the universe is unwarranted.6 The sort of efficient causality that is well grounded in experience (and hence philosophically acceptable) is not this sort of causality, and so the theist, at the very least, cannot use the plausibility of conventional accounts of efficient causation to make plausible the claim that the coming to be of the universe ex nihilo was efficiently caused.7 Most of the arguments considered in this chapter are unique to the KCA. Clear examples of such would include arguments in favor of the possibility of an eternal past existence of the universe.8 Arguments of this type have been developed by a wide spectrum of philosophers, ranging from orthodox theists (e.g., Moses Maimonides and Thomas Aquinas) to committed atheists (e.g., Friedrich Nietzsche and Quentin Smith), and they cut to the heart of the KCA position. There are four major taxonomic divisions in this chapter. The first division considers objections that apply to the KCA as a whole and must be dealt with before the KCA’s specific premises and the arguments for those premises can be discussed. These objections may justly be called propaedeutic to the KCA. I will call this “Division I” (or “I” for short). The three remaining taxonomic divisions follow the three parts of the

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main KCA syllogism. Thus, the second taxonomic division treats those objections that call premise (1) into doubt; the third division addresses objections that call premise (2) into doubt; and the fourth taxonomic division treats objections to (or, perhaps, more properly, suggested limitations of) conclusion (3). These divisions will be named “Division II,” “Division III,” and “Division IV,” respectively. I will also distinguish subsidiary species of objections within these major taxonomic genera. For instance, objections falling under Division III may be further distinguished into objections to argument (A), objections to argument (B), and objections to argument (C). For these subdivisions I will employ the notations “III.a,” “III.b,” and “III.c,” respectively. The more important families of objection will be further distinguished and numbered, so that, for example, “III.b.(b).1” refers to the first objection to premise (b) of argument (B) for premise (2) of the KCA. It should be noted that while objections to conclusion (3) are fairly common in the KCA literature, I have not encountered any major objections that focus specifically upon the conclusions of arguments (A), (B), or (C). Hence, there are no subdivisions in this chapter corresponding to what the Argument Guide labels as III.a.(iii), III.b.(c), and III.c.(g). Future developments in the KCA literature may occasion changes in this regard.

6. DIVISION I—OBJECTIONS PROPAEDEUTIC TO THE KCA Since it is not practical to present a systematic taxonomy of all of the objections that could impact the KCA, in this section I will restrict myself to mentioning a small number of objections that have arisen directly within the context of evaluations of the KCA.

I.1 Objection: A theist cannot accept the soundness of the KCA.9 This is because, according to the theist, God has always existed. If God has always existed, then God must have existed throughout an infinite stretch of time prior to the arising of the first event. Therefore, theists cannot consistently assert both that God exists and that past time is finite.10

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Response: First, the KCA does not prove that the quantity of past time is finite; rather, the KCA proves that the quantity of past events is finite. The soundness of the KCA is, for instance, logically compatible with the position that God existed in an undifferentiated (and hence changeless and uneventful) Newtonian Absolute time prior to creation. The objection is therefore a red herring.11 Second, the objection presupposes that God exists in time, which is denied by many theists. Strictly speaking, it is not God’s existence that is measured by time but rather time that is measured by God’s existence. This is because time is created by God along with everything else.

I.2 Objection: The KCA proves too much, for if the argument is sound then the future must be finite as well.12 Response: Since past events and future events are on a par according to the B-theory of time, if the KCA can be soundly formulated within a B-theoretic ontology then the objection should be conceded.13 However, the result obtained is not a failure of the KCA but constitutes a reduction on the B-theory. If the KCA is considered within the context of an A-theory of time then the objection is denied. Unlike past events, which as determinate parts of reality may legitimately be collected into a whole, future events do not exist. (Even if past events cannot be said to exist now, they have determinately existed and have left determinate marks of their passing on what exists now.)14 The past, if eternal, would thus constitute an actual infinite, whereas the future, even if it were to continue sempiternally, would only constitute a potential infinite. Therefore arguments for the necessary finitude of the future that are based on the assumption that a sempiternal future implies the existence of an actual infinite are unsound.15

I.3 Objection: Drawing a theistic conclusion from the KCA is unwarranted because causal loops are possible and hence the same finite series of events could have been repeated eternally.16 Response: First, causal loops are impossible.17 Second, the KCA

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would be sound against an eternity of past causal loops because past events in prior cycles can be ontologically (if not empirically) distinguished from the events in other cycles. The total quantity of all such past events would be finite.

I.4 Objection: The probability that something will spontaneously come to be from nothing is not zero. Therefore there is no need to invoke an external cause of the universe’s coming to be.18 Response: First, it may be objected that it is not coherent to speak of probability prior to the existence of the universe. Second, prescinding from the positive arguments in favor of the weak version of the principle of sufficient reason (PSR) required by the KCA, the following response may be assayed: Probability requires a positive state description in order to be rendered meaningful. Relative to some positive state of affairs it may be possible to calculate the likelihood of some event occurring. On the hypothesis that neither the universe nor any cause of the universe’s coming to be exists there is, however, no positive state description available from which the various probabilities may be calculated. Therefore the suggestion that there is a greaterthan-zero probability that the universe will come to be spontaneously is unwarranted.19

I.5 Objection: The KCA assumes the ontological priority of “nothing” over “something.” This assumption of the normalcy of nothingness is unwarranted.20 Response: First, this position is not assumed by the KCA.21 Second, the truth of the normalcy of nothingness for finite beings is evident from the consideration that existence is not an essential property of any finite being and hence any appeal to the natures of finite beings cannot suffice to explain the continued existence of the universe from moment to moment.22

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I.6 Objection: The universe caused itself to exist.23 Response: In line with a limited application of the PSR, the defender of the KCA holds that the notion of causa sui (i.e, a thing bringing about its own beginning of existence) is metaphysically absurd.24

I.7 Objection: The demand for an external cause of the universe is a pseudoproblem; the natural realm sufficiently accounts for itself.25 Response: The question: What explains the coming to be of the universe? is both legitimate and unanswerable without reference to an external transcendent cause of the universe.

I.8 Objection: An atheistic interpretation of the coming to be of the universe is preferrable because it is simpler than a theistic explanation.26 Response: An explanation that claims that like arises from like is simpler than one that claims that unlike arises from unlike. The atheist claims that being arose from nonbeing, whereas the theist claims that being arose from being. Therefore the theist’s explanation is simpler and the argumentative burden lies on the side of the atheist.27

I.9 Objection: The KCA concludes that there must be a first event in the history of the universe. However, there can be no first event in the history of the universe. This is because events require time, and there is no space-time at the big bang singularity.28 Response: The KCA does not assert that the big bang is the first event; rather, the big bang is interpreted as the limit of past events.29 Nor is the possibility that past time is composed of an actual infinite of instants that asymptotically approach the big bang a difficulty for the KCA. This is because the events chosen as the basic metrical units

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for the KCA are temporally extended moments, that is, nonzero temporal durations.30 According to such a metric, if the universe came to be then there will be a first event and if the universe did not come to be then there will not be a first event. (If an ontological interpretation of past events is required, then greater clarity may be achieved by appealing to the past series of significant real changes in substances. See chapter 5, section 21 for details.)

I.10 Objection: It is possible both that there is a beginning to the universe (in the sense that there are no happenings prior to some limiting point) and that there is no beginning to the temporal series of prior, efficiently caused changes in the history of the universe. The universe might be structured such that caused changes occur with increasing rapidity and without finite limit the closer one gets to the big bang. Thus, the cause of an event occurring at time t was itself caused at time t/2, the cause of the event at t/2 was itself caused at time t/4, the cause of the event at t/4 was caused at t/8, and so on ad infinitum.31 Response: The envisaged scenario is logically but not factually possible. This may be demonstrated by means of a thought experiment. Imagine that Theseus has lived his life throughout the entire history of the universe up to time t. Theseus’s life is structured in the manner described by the objection, so that he moves with increasing rapidity the farther back one regresses in time. For each past division of his life, Theseus plays out ten yards of twine. At time t, Theseus cuts the twine, and then goes looking for the other end of the twine. Question: Does the twine have a second end? Neither an affirmative nor a negative answer is acceptable. If the twine does not have a second end then it cannot be a physical body, for all factually possible physical bodies have determinate surfaces. If the twine does have a second end, then have Theseus tie the ends together and unfurl the twine to make a great circle. Since the twine would be actually infinite in length, it should describe a circle having an actually infinite circumference. However, it is factually impossible for an actually infinite circle to exist.32 Hence, Theseus’s task is factually impossible. But if the proposed scenario is factually possible, then so is Theseus’s task. It follows

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that the beginningless series of causes mentioned in the objection is not factually possible.

7. DIVISION II—OBJECTIONS TO PREMISE (1) OF THE KCA Premise (1) of the KCA asserts that “[w]hatever comes to be has a cause of its coming to be.” In his explanation of premise (1) Craig expressly states that he is arguing for a first efficient cause of the entire universe.33 The defense of this premise raises a number of fundamental issues in metaphysics, the philosophy of nature, and the philosophy of science.34 Premise (1) expresses a weak form of the PSR, so discussions of the PSR in general will be relevant to the KCA. Discussion of the nature of causation is also relevant as certain causal theories are incompatible with the KCA.

II.1 Objection: The PSR is false; therefore premise (1) is false.35 Response: The weak formulation of the PSR required for the soundness of the KCA is true.36

II.2 Objection: In arguing that every event within the universe is caused, therefore the existence of the universe is caused, the theist commits a fallacy of composition. Response: The fallacy of composition is not a formal fallacy but a material fallacy. In some cases it is perfectly legitimate to infer a property of a whole from a common property of its parts. Suppose one has a puzzle every piece of which is triangular; does it follow that the assembled puzzle must itself be triangular? There is no necessity that it be so. However, if one has a puzzle each of whose pieces is red, then the whole puzzle will also be red. There is no known decision procedure that allows one to infer which predicates function like “red” and which like “triangular.”37 Therefore it has not been proven that the theist has committed a fallacy of composition.38

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II.3 Objection: Causes are essentially spatiotemporal and hence it is inappropriate to claim that the first event has a cause since no spatiotemporal cause can be assigned for the first event.39 Response: First, it may be denied that causes are necessarily spatiotemporal. Mental acts, which can function as causes at least with respect to other mental acts, are quite plausibly held to be nonspatial. Mental acts are also arguably nontemporal (in the sense of “occurring at a specific time”).40 Second, those who advance the objection may themselves accept as valid current scientific speculation concerning the state of the early universe. However, if causes are essentially spatiotemporal and it is also the case that the laws of physics “break down” when one approaches the initial big bang singularity, then it follows that the explanations physicists give when they talk about the early evolution of the universe cannot be causal. (At the very least, it will follow that scientific talk about the big bang cannot be harmonized with the nomic subsumption models favored by many philosophers of science.) But if the scientific explanations physicists give concerning the early history of the universe are neither convenient creation myths nor causal explanations, what sort of explanations are they? It is certainly odd to think that physicists, qua physicists, intend to give any other sort of explanation than causal ones. If the objector wishes to preserve the causal character of the cosmologist’s explanations, then the objector must furnish a principled way of distinguishing the causal appeals of the scientific cosmologist from those of the philosophical theist.

II.4 Objection: Premise (1) is false because of the counterexample of virtual particles that come to be without being caused.41 Response: The quantum vacuum within which virtual particles arise is not the empty void of Newton. Virtual particles come into being through borrowing energy from their immediate environment and the total mass/energy of the system within which they arise remains constant. So whatever the full causal story for virtual particles may be, it cannot be said that virtual particles come to be uncaused, for virtual

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particles do not come to be from nothing nor do they come to be without existing material preconditions.42 Assuming that virtual particles are substances, it can also be argued that at least a weak form of the PSR applies to their coming into being.43

II.5 Objection: Only events are causes; properly speaking, causation requires a correlation between two events, one serving as the cause, the other serving as the effect.44 Therefore there is no cause that may be correlated with the first event in the universe. Response: First, an event-based analysis of causation is incorrect. This is because “events and states presuppose the subjects with active power whose doings, states, and alterations they constitute, and it is . . . these subjects of active power that are, above all, causes.”45 Second, it has yet to be proven that a nuanced event-based account of causation is incompatible with the KCA. For instance, it may be possible to argue that God’s act of creation is an event but not an event in the universe.

II.6 Objection: Causes must temporally precede their effects.46 Since time begins with the first event, one cannot appeal to a temporally prior cause that brings about the first event. Response: First, the claim that the first event marks the beginning of time is not logically necessary, for it is logically possible that the cause of the first event existed in a prior undifferentiated Absolute time and so the cause of the first event can be said to temporally precede the first event. Second, simultaneous causation is possible.47 Third, the falsity of the position that “causes must temporally precede their effects” has been demonstrated.48

II.7 Objection: It is possible that the universe is infinitely old because the big bang may have been preceded by a prior big crunch; that big crunch was preceded by a big bang which itself was preceded by a big crunch; and so on.49

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Response: First, the oscillating universe model appealed to in the objection does not enjoy currency in contemporary scientific cosmology.50 It is necessary, on the oscillating universe model, to assume some sort of connection between our present universe and previous ones. (If there were no connection, and in particular no causal connection, that obtained between the repeated expansions and contractions, it is difficult to understand what would legitimize calling this the “oscillating universe” model instead of, say, the “successively existing independent universes” model.) But any such causal connection would allow for an assessment of the quantity of connected past events and then the KCA goes through in the usual way.

II.8 Objection: Backward causation is possible, therefore the universe could have brought about its own beginning in existence.51 Response: Backward or retro causation is impossible.52

8. DIVISION III—OBJECTIONS TO PREMISE (2) OF THE KCA Premise (2) of the KCA asserts that “[t]he universe came to be.” This premise is supported by three subsidiary arguments. The first argument, argument (A), supports premise (2) by arguing that an actual infinite cannot exist; however, an eternal past existence of the universe would entail the existence of an actual infinite; therefore, the universe cannot have an eternal past existence. Objections to argument (A) are discussed in division III.a, found below in section 8.1. The second argument, argument (B), supports premise (2) by arguing that it is impossible to successively traverse an actually infinite quantity; however, the universe’s having an eternal past existence would require such a traversal; hence, the universe does not have an eternal past existence. Objections to argument (B) are discussed in division III.b, found below in section 8.2. The third argument, argument (C), supports premise (2) by closing off a possible escape route left open by arguments (A) and (B):

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If it is possible to detach the existence of time from the existence of the universe, it may be possible for the universe to exist in a timeless, perfectly static state prior to the first event in the temporal history of the universe. Argument (C) contends that the universe could not exist in a perfectly static state prior to the first event. Moreover, the temporal existence of the universe is not detachable from the existence of the universe itself. Therefore, the coming to be of the first event in the history of the universe must be coincident with the coming to be ex nihilo of the universe itself. Objections to argument (C) are given in division III.c, found in section 8.3 below.

8.1 Division III.a—Objections to Argument (A) of the KCA The formal statement of argument (A) is as follows: (i) An infinite temporal regress of events would constitute an actual infinite. (ii) An actual infinite cannot exist. (iii) Therefore an infinite temporal regress of events cannot exist.

8.1.1 Objections to Premise (i) of Argument (A) Premise (i) asserts that “[a]n infinite temporal regress of events would constitute an actual infinite.” The following objections are made to this premise.

III.a.(i).1 Objection: An infinite past is not an actually infinite past. Precision: This assertion may be interpreted in various ways: 1. The past is not properly described as infinite (either actually or potentially) but is properly described as beginningless.53 2. The past cannot be said to be actually infinite because past events do not exist simultaneously.54 3. The past is potentially infinite.55

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General Response to (1–3): Craig’s responses are detailed in chapter 1 section 4.2.5.2. Response to (1): For additional responses to (1) see Moreland,56 Whitrow,57 and Craig.58 All charge that this objection is an attempt to dodge the problem instead of coming to grips with it. Response to (2): The response to (2) developed in TKCA follows the critique of Aquinas presented by Van Steenberghen.59 A succinct statement is found in Robert Prevost, who argues: [T]his fact about the past [viz., that past days no longer exist] would make no difference since it would be possible that each past moment would leave a permanent record which would exist on into the future. Hence, even though the past moments no longer exist, the permanent record of their existence would. Hence, if there were an infinite number of past moments, there would be an infinite number of permanent records.60

Response to (3): This issue is addressed several times by Craig61 and has been taken up again recently by Moreland.62 In general, it might be argued that any appeal to the potential infinite invokes a dynamic conception of an expanding finite collection. But how could this apply to past time? In order for the past itself to be an example of a potential infinite one would need to postulate the existence of new past days that lie beyond the furthermost past day from today. Even if such a picture can be made intelligible (e.g., tomorrow there would be two new past days: (a) today and (b) a new past day added at the far end of the temporal series), at the very least one would have to assume the possibility of dubious notions like backward causation. It would also be necessary to explain how this suggestion could be made to fit the empirical facts associated with the arrow of time, for it seems that our experience of time is such that time always proceeds toward the future and never toward the past. Opponents of the KCA who advance this objection have much work to do clarifying the suggestion before it can be counted as a serious ontological possibility.

8.1.2 Objections to Premise (ii) of Argument (A) Premise (ii) asserts that “[a]n actual infinite cannot exist.” The general strategy of those who have objected to this premise involves an appeal

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to the logical possibility of instantiating an actual infinite. Some philosophers have gone beyond asserting the mere logical possibility of instantiating the actual infinite and have suggested counterexamples.

III.a.(ii).1 Objection: The actual infinite is met with in nature. Precision: This thesis is held for a variety of reasons. 1. Because there are a priori arguments for actual infinities.63 2. Because some particular class or collection is thought to be infinite.64 3. Because of the transcendent determinacy of the real.65 Response to (1): The a priori arguments do not conclude with necessity to the existence of an extramental infinite multitude. Furthermore, mathematical realism is consistent with the soundness of the KCA.66 Response to (2): Contemporary physicists hold that the quantity of matter in the universe is finite and that the actual divisions of matter are likewise finite. Response to (3): The thesis of the transcendent determinacy of the real invites us to identify facts (i.e., what makes a true proposition true) as elements or real features of substantial reality.67 If one accepts a metaphysic of substances this is not necessary: The real features of the world are not dense (in the mathematical sense);68 rather, our way of breaking up the world and propositionally describing it is potentially infinite.69

III.a.(ii).2 Objection: It is logically possible for hypergunk to exist, where hypergunk is understood to be a gunk all of whose parts themselves have proper parts.70 Therefore it is possible for the actual infinite to exist. Response: This objection fails to consider the difference between logical and factual possibility. Even if one were to grant that hypergunk is logically possible, it is clearly not factually possible for hypergunk to exist. That it is factually impossible for hypergunk to exist fol-

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lows from a consideration of how hypergunk fails to conform to the basic requirements of a substance-based metaphysics.71

III.a.(ii).3 Objection: An actually infinite multitude of existing things is consistently conceivable. Therefore an actually infinite multitude of existing things is logically possible. Response: First, consistent conceivability is not a sufficient test for ontological possibility.72 Second, the KCA operates at the level of factual possibility, not logical possibility.73 In particular, the KCA typically proceeds by offering thought experiments involving physical entities that are then correlated with temporal events. Since the correlation of physical entities with past events is admitted to be possible, yet supposition of an actually existing infinite multitude of physical entities leads to absurdity, it follows that an actual infinity of past events is impossible. Such thought experiments indicate that an actually infinite multitude of past events and of physical things is not really consistently conceivable, and hence the objection is in error.

III.a.(ii).4 Objection: The KCA supposes that an actual infinite cannot exist. But there is no a priori argument against the existence of an actual infinite.74 Therefore the KCA is unsound. Response: This is a mischaracterization of the KCA, for the defender of the KCA argues not a priori but a posteriori. (The objection might be plausible if it were directed against certain historical forms of the argument.)75 In the KCA one does not argue that an actually infinite past is impossible in all logically possible worlds. Rather, what is argued is that this actual universe we inhabit could not have an actually infinite past.76

III.a.(ii).5 Objection: No one has ever demonstrated that an actually infinite multitude cannot exist.

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Response: This objection is met by considering both the previous responses and the KCA as a whole.

III.a.(ii).6 Objection: Theists cannot object to the existence of the actual infinite because of God’s omniscience. Precision: This objection has been developed in two directions: (1) God knows an actually infinite number of things. Since God’s mode of knowing is not successive, it follows that he knows the things that we would come to know in a potentially infinite way (such as the decimal expansion of p) in an actually infinite way (i.e., he knows the complete decimal expansion of p given all at once). God’s knowledge, therefore constitutes an example of an existing actual infinite.77 (2) God’s knowledge of the potentially infinite future guarantees the possibility of completing a potentially infinite series.78 The supporting argument runs thus: We may agree that the future is potentially infinite. Now, God is omniscient: He knows everything that occurs within the history of the universe, past, present, and future. Moreover, God is the creator of all things: Every being that exists in the universe does so as the terminus of God’s determinate creative intention.79 In God’s act of knowing, then, the potentially infinite multitude of future events finds its completion within a single act of knowing that treats that potentially infinite multitude as a completed totality. But if God can truly know the temporally spread potentially infinite multitude of future events as a completed totality, it must be logically possible for the potential infinite to be completed, for God’s knowledge of created things is both true and adequate. The completion of a potentially infinite series can result in nothing other than an actual infinite. Therefore, it must be possible for the actual infinite to be traversed by a process of successive addition. Response to (1) and (2): The infinity of God’s knowledge is best described as a metaphysical infinite rather than a quantitative infinite, so the conclusion does not follow.80 Moreover, appealing to God’s manner of knowing is a problematic guide for us to adopt in deciding how the underlying structure of the world must be. For example, God knows the truths expressed in propositions in a nonpropositional

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manner, God knows truths about temporal beings without himself being subject to time, and God knows potentially infinite things in a manner that is not itself potentially infinite—this for the reason that God’s cognitive activity is not successive. Arguing that God’s knowledge of future time somehow entails that he knows a potential infinite that has been converted into an actual infinite is a case of (illegitimately) imposing a human mode of knowing on God: In this case, God is supposed to think like Cantor. It is thus open to the theist to deny that how God knows things is an appropriate model for understanding the intrinsic logical connections among the distinctively human concepts of potential infinity and actual infinity. Various models of divine intellection have been proposed that get around the difficulties mentioned in the objections, the most promising theory being that of Brian Shanley, who suggests ways of understanding God’s knowledge of composite creation that avoids fracturing the divine ideas into a plurality in the manner both objections presuppose.81

III.a.(ii).7 Objection: The actual infinite must exist because the existence of the potential infinite presupposes the existence of the actual infinite.82 Response: The potential infinite does not presuppose the actual infinite either ontologically or conceptually.83

III.a.(ii).8 Objection: Many KCA thought experiments rely upon the correlation of past events with physical entities. However, there is no legitimate correlation of past events with physical entities that lends support to the KCA. This is because there have been infinitely many past events, but there are not now infinitely many physical entities.84 Response: The correlation of infinite past events with infinite physical entities is legitimate. If one is possible then the other is possible; if one is not possible then the other is not possible. The legitimacy of establishing the correlation between events and physical entities may be shown in three ways. First, it is generally conceded by opponents of the KCA that the existence of an actually infinite multitude of phys-

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ical entities is possible. The steady state cosmological model, which was the standard scientific account for much of the twentieth century, is today generally interpreted as a consistent description of a possible state of affairs that does not, for contingent reasons, happen to correspond to the state of our universe.85 If one accepts the steady state view as being possible, then there can be no principled objection to the use of this model in formulating thought experiments testing factual possibility. Second, the in-principle infinite divisibility of matter is generally conceded.86 The KCA defender is thus free to propose thought experiments that set up a correlation between past events and an actually infinite number of physical entities which, when collected together, occupy only a finite volume of space.87 Reasons for rejecting the possibility of a physical entity such as the thought experiments describe will then militate against the possibility of an actual infinite of past events. Third, the correlation of past events with physical substances is a metaphysical fact. Substances existing now bear the signs of their entire causal history in the form of temporal marks that correspond to real features now existing in the substance. These temporal marks explain both what a substance is and how the substance is. If the quantity of significant past events were actually infinite, there would be an actually infinite number of temporal marks. (My account of temporal marks is developed more fully in chapter 5, section 21.2.)

III.a.(ii).9 Objection: KCA thought experiments trade upon our naive understanding of “more” and “less” to reach their conclusion. But what Cantor’s work has demonstrated is that a naive understanding of “more” and “less” must be abandoned when we are confronted with questions concerning the infinite. Response: Our understanding of “more” and “ less” is temporally, conceptually, and logically prior to our understanding of the infinite. To either adjust or jettison our naive understanding of “more” and “less” for the sake of Cantorian mathematics would be inappropriate since we rely upon our intuitive understanding of “more” and “less” to compare and make sense of the different orders of infinity Cantor identifies. Thus, without a prior understanding of “more” and “less,”

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we could not make sense of Cantor’s proof that there are more real numbers than natural numbers, or understand how the power set of ℵ1 requires a higher order of infinity, ℵ2, for its expression.

III.a.(ii).10 Objection: Both “=” and “