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Philosophy of Mathematics

Handbook of the Philosophy of Science General Editors Dov Gabbay Paul Thagard John Woods AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO North Holland is an imprint of Elsevier

Philosophy of Mathematics Edited by Andrew D. Irvine University of British Columbia, Vancouver, Canada AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO North Holland is an imprint of Elsevier

North Holland is an imprint of Elsevier30 Corporate Drive, Suite 400, Burlington, MA 01803, USALinacre House, Jordan Hill, Oxford OX2 8DP, UKRadarweg 29, PO Box 211, 1000 AE Amsterdam, The NetherlandsFirst edition 2009Copyright © 2009 Elsevier B.V. All rights reservedNo part of this publication may be reproduced, stored in a retrieval systemor transmitted in any form or by any means electronic, mechanical, photocopying,recording or otherwise without the prior written permission of the publisherPermissions may be sought directly from Elsevier’s Science & Technology RightsDepartment in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333;email: [email protected]. Alternatively you can submit your request online byvisiting the Elsevier web site at http://elsevier.com/locate/permissions, and selectingObtaining permission to use Elsevier materialNoticeNo responsibility is assumed by the publisher for any injury and/or damage to personsor property as a matter of products liability, negligence or otherwise, or from any useor operation of any methods, products, instructions or ideas contained in the materialherein. Because of rapid advances in the medical sciences, in particular, independentverification of diagnoses and drug dosages should be madeBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryLibrary of Congress Cataloging-in-Publication DataA catalog record for this book is available from the Library of CongressISBN: 978-0-444-51555-1 For information on all North Holland publications visit our web site at books.elsevier.comPrinted and bound in Hungary09 10 11 11 10 9 8 7 6 5 4 3 2 1Cover Art: University of British Columbia Library, Rare Books and Special Collections,from Oliver Byrne, The First Six Books of The Elements of Euclid in which ColouredDiagrams and Symbols are used instead of Letters for the Greater Ease of Learners(London: William Pickering, 1847)

GENERAL PREFACE Dov Gabbay, Paul Thagard and John Woods Whenever science operates at the cutting edge of what is known, it invariablyruns into philosophical issues about the nature of knowledge and reality. Scientificcontroversies raise such questions as the relation of theory and experiment, thenature of explanation, and the extent to which science can approximate to thetruth. Within particular sciences, special concerns arise about what exists andhow it can be known, for example in physics about the nature of space and time,and in psychology about the nature of consciousness. Hence the philosophy ofscience is an essential part of the scientific investigation of the world. In recent decades, philosophy of science has become an increasingly centralpart of philosophy in general. Although there are still philosophers who thinkthat theories of knowledge and reality can be developed by pure reflection, muchcurrent philosophical work finds it necessary and valuable to take into accountrelevant scientific findings. For example, the philosophy of mind is now closelytied t o empirical psychology, and political theory often intersects with economics.Thus philosophy of science provides a valuable bridge between philosophical andscientific inquiry. More and more, the philosophy of science concerns itself not just with generalissues about the nature and validity of science, but especially with particular issuesthat arise in specific sciences. Accordingly, we have organized this Handbook intomany volumes reflecting the full range of current research in the philosophy ofscience. We invited volume editors who are fully involved in the specific sciences,and are delighted that they have solicited contributions by scientifically-informedphilosophers and (in a few cases) philosophically-informed scientists. The resultis the most comprehensive review ever provided of the philosophy of science. Here are the volumes in the Handbook: Philosophy of Science: Focal Issues, edited by Theo Kuipers. Philosophy of Physics, edited by John Earman and Jeremy Butterfield. Philosophy of Biology, edited by Mohan Matthen and Christopher Stephens. Philosophy of Mathematics, edited by Andrew D. Irvine. Philosophy of Logic, edited by Dale Jacquette. Philosophy of Chemistry and Pharmacology, edited by Andrea Woody, Robin Hendry and Paul Needham.

Dov Gabbay, Paul Thagard and John Woods Philosophy of Statistics, edited by Prasanta S. Bandyopadhyay and Malcolm Forster. Philosophy of Information, edited by Pieter Adriaans .and Johan van Ben- them. Philosophy of Technological Sciences, edited by Anthonie Meijers. Philosophy of Complex Systems, edited by Cliff Hooker. Philosophy of Ecology, edited by Bryson Brown, Kent Peacock and Kevin de Laplante. Philosophy of Psychology and Cognitive Science, edited by Pau Thagard. Philosophy of Economics, edited by Uskali Mki. Philosophy of Linguistics, edited by Ruth Kempson, Tim Fernando and Nicholas Asher. Philosophy of Anthropology and Sociology, edited by Stephen Turner and Mark Risjord. Philosophy of Medicine, edited by Fred Gifford. Details about the contents and publishing schedule of the volumes can be foundat http://www.johnwoods.ca/HPS/. As general editors, we are extremely grateful to the volume editors for arrangingsuch a distinguished array of contributors and for managing their contributions.Production of these volumes has been a huge enterprise, and our warmest thanksgo to Jane Spurr and Carol Woods for putting them together. Thanks also toAndy Deelen and Arjen Sevenster at Elsevier for their support and direction.

CONTENTSGeneral PrefaceDov Gabbay, Paul Thagard and John WoodsPrefaceAndrew D. IrvineList of ContributorsLes Liaisons DangereusesW. D. HartRealism and Anti-Realism in MathematicsMark BalaguerAristotelian RealismJames FranklinEmpiricism in the Philosophy of MathematicsDavid BostockA Kantian Perspective on the Philosophy of MathematicsMary TilesLogicismJaakko HintikkaFormalismPeter SimonsConstructivism in MathematicsCharles McCartyFictionalismDaniel BonevacSet Theory from Cantor to CohenAkihiro Kanamori

viii Contents 461Alternative Set TheoriesPeter Apostoli, Roland Hinnion, Akira Kanda andThierry LibertPhilosophies of ProbabilityJon WilliamsonOn ComputabilityWilfried SiegInconsistent MathematicsChris MortensenMathematics and the WorldMark ColyvanIndex

PREFACE One of the most striking features of mathematics is the fact that we are muchmore certain about what mathematical knowledge we have than about what math-ematical knowledge is knowledge of. Mathematical knowledge is generally acceptedto be more certain than any other branch of knowledge; but unlike other scientificdisciplines, the subject matter of mathematics remains controversial. In the sciences we may not be sure our theories are correct, but at least we knowwhat it is we are studying. Physics is the study of matter and its motion withinspace and time. Biology is the study of living organisms and how they react andinteract with their environment. Chemistry is the study of the structure of, andinteractions between, the elements. When man first began speculating about thenature of the Sun and the Moon, he may not have been sure his theories werecorrect, but at least he could point with confidence t o the objects about which hewas theorizing. In all of these cases and others we know that the objects underinvestigation - physical matter, living organisms, the known elements, the Sunand the Moon - exist and that they are objects within the (physical) world. In mathematics we face a different situation. Although we are all quite certainthat the Pythagorean Theorem, the Prime Number Theorem, Cantor's Theoremand innumerable other theorems are true, we are much less confident about whatit is to which these theorems refer. Are triangles, numbers, sets, functions andgroups physical entities of some kind? Are they objectively existing objects insome non-physical, mathematical realm? Are they ideas that are present only inthe mind? Or do mathematical truths not involve referents of any kind? It is thesekinds of questions that force philosophers and mathematicians alike to focus theirattention on issues in the philosophy of mathematics. Over the centuries a number of reasonably well-defined positions have been de-veloped and it is these positions, following a thorough and helpful overview by W.D. HartI1 that are analyzed in the current volume. The realist holds that math-ematical entities exist independently of the human mind or, as Mark Balaguertells us, realism is \"the view that our mathematical theories are true descriptionsof some real part of the world.\"' The anti-realist claims the opposite, namelythat mathematical entities, if they exist at all, are a product of human invention.Hence the long-standing debate about whether mathematical truths are discoveredor invented. Platonic realism (or Platonism) adds to realism the further provisionthat mathematical entities exist independently of the natural (or physical) world. W. D. Hart, \"Les Liaisons Dangereuses\", this volume, pp. 1-33. 2Mark Balaguer, \"Realism and Anti-realism in Mathematics,\" this volume, pp. 35-101.

x PrefaceAristotelian realism (or Aristotelianism) adds the contrary provision, namely thatmathematical entities are somehow a part of the natural (or physical) world or,as James Franklin puts it, that \"mathematics is a science of the real world, justas much as biology or sociology are.\"3 Platonic realists such as G.H. Hardy, KurtGodel and Paul Erdos are thus regularly forced to postulate some form of nonphys-ical mathematical perception, distinct from but analogous to sense perception. Incontrast, as David Bostock reminds us, Aristotelian realists such as John StuartMill typically argue that empiricism - the theory that all knowledge, includingmathematical knowledge, is ultimately derivable from sense experience - \"is per-haps most naturally combined with Aristotelian r e a l i ~ m . \" ~ The main difficulty associated with Platonism is that, if it is correct, mathe-matical perception will appear no longer to be compatible with a purely naturalunderstanding of the world. The main difficulty associated with Aristotelianismis that, if it is correct, a great deal of mathematics (especially those parts ofmathematics that are not purely finitary) will appear to outrun our (purely finite)observations and experiences. Both the Kantian (who holds that mathematicalknowledge is synthetic and a priori) and the logicist (who holds that mathematicsis reducible to logic, and hence that mathematical knowledge is analytic) attemptto resolve these challenges by arguing that mathematical truths are discoverableby reason alone, and hence not tied to any particular subject matter. As MaryTiles tells us, Kant's claim that mathematical knowledge is synthetic a priori hastwo separate components. The first is that mathematics claims to provide a prioriknowledge of certain objects because \"it is the science of the forms of intuition\";the second is that \"the way in which mathematical knowledge is gained is throughthe synthesis (construction) of objects corresponding to its concepts, not by theanalysis of concept^.\"^ Similarly, initial accounts of logicism aimed to show that,like logical truths, mathematical truths are \"truths in every possible structure\"and it is for this reason that they can be discovered a priori, simply because \"theydo not exclude any possibilities.\"6 Exactly how much, if any, of such programscan be salvaged in the face of contemporary meta-theoretical results remains amatter of debate. Constructivism, the view that mathematics studies only enti-ties that (at least in principle) can be explicitly constructed, attempts to resolvethe problem by focusing mathematical theories solely on activities of the humanmind. In Charles McCarty's helpful phrase, constructivism in mathematics ulti-mately boils down t o a commitment to the \"business of practice rather than ofprinciple.\"7 Critics claim that all three positions - Kantianism, logicism andconstructivism - ignore large portions of mathematics' central subject matter.(Constructivism in particular, because of the emphasis it places upon verifiability,is regularly accused of failing to account for the impersonal, mind-independent 3 ~ a m eFs ranklin, \"Aristotelian Realism,\" this volume, pp. 103-155. c avid Bostock, \"Empiricism in the Philosophy of Mathematics,\" this volume, pp. 157-229. 5 ~ a r yTiles, \"A Kantian Perspective on the Philosophy of Mathematics,\" this volume,pp. 231-270. 6Jaakko Hintikka, \"Logicism,\" this volume, pp. 271-290. 7Charles McCarty, \"Constructivism in Mathematics,\" this volume, pp. 311-343.

Prefaceparts of mathematics.) Formalism, the view that mathematics is simply the \"formal manipulations ofessentially meaningless symbols according to strictly prescribed rules,\" goes a stepfurther, arguing that mathematics need not be considered to be about numbersor shapes or sets or probabilities at all since, technically speaking, mathematicsneed not be about anything. But if so, an explanation of how we obtain ournon-formal, intuitive mathematical intuitions, and of how mathematics integratesso effectively with the natural sciences, seems to be wanting. Fictionalism, theview that mathematics is in an important sense dispensable since it is merely aconservative extension of non-mathematical physics (that is, that every physicalfact provable in mathematical physics is already provable in non-mathematicalphysics without the use of mathematics), can be attractive in this context. Butagain, it is a theory that fails to coincide with the intuitions many people -including many working mathematicians -have about the need for a realist-basedsemantics. As Daniel Bonevac tells us, even if fictionalist discourse in mathematicsis largely successful, we are still entitled to ask why \"that discourse, as opposedto other possible competitors, succeeds\"; and as he reminds us in response tosuch a question, any citation of a fact threatens to collapse the fictionalist projectinto either a reductive or modal one, something not easily compatible with thefictionalist's original aimsg The moral appears to be that mathematics sits uncomfortably half way betweenlogic and science. On the one hand, many are drawn to the view that mathematicsis an axiomatic, a priori discipline, a discipline whose knowledge claims are in someway independent of the study of the contingent, physical world. On the other hand,others are struck by how mathematics integrates so seamlessly with the naturalsciences and how it is the world - and not language or reason or anything else- that continually serves as the main intuition pump for advances even in puremathematics. In fact, in spite of its abstract nature, the origins of almost all branches ofmathematics turn out to be intimately related to our innumerable observations of,and interactions with, the ordinary physical world. Counting, measuring, group-ing, gambling and the many other activities and experiences that bring us intocontact with ordinary physical objects and events all play a fundamental role ingenerating new mathematical intuitions. This is so despite the sometimes-madeclaim that mathematical progress has often occurred independently of real-worldapplications. Standardly cited advances such as early Greek discoveries concerningthe parabola, the ellipse and the hyperbola, the advent of Riemannian geometriesand other non-Euclidean geometries well in advance of their application in contem-porary relativistic physics, and the initial development of group theory as long agoas the early 1800s themselves all serve as telling counterexamples to such claims.Group theory, it turns out, was developed as a result of attempts to solve sim-ple polynomial equations, equations that of course have immediate application in 8Peter Simons, \"Formalism,\" this volume, pp. 291-310. gDaniel Bonevac, \"Fictionalism,\" this volume, pp. 345-393.

xii Prefacenumerous areas. Non-Euclidian geometries arose in response to logical problemsintimately associated with traditional Euclidean geometry, a geometry that, atthe time, was understood to involve the study of real space. Early Greek workstudying curves resulted from applied work on sundials. Mathematics, it seems,has always been linked to our interactions with the world around us and to thecareful, systematic, scientific investigation of nature. It is in this same context of real-world applications that fundamental ques-tions in the philosophy of mathematics have also arisen. Paradigmatic over thepast century have been questions associated with issues in set theory, probabilitytheory, computability theory, and theories of inconsistent mathematics, all nowfundamentally important branches of mathematics that have grown as much froma dissatisfaction with traditional answers to philosophical questions as from anyother source. In the case of set theory, dissatisfaction with our understandingof the relationship between a predicate's intension and its extension has led tothe development of a remarkably simple but rich theory. As Akihiro Kanamorireminds us, set theory has evolved \"from a web of intensions to a theory of exten-sion par e x ~ e l l e n c e . \"A~t~the same time, striking new developments continue tobe made, as we see in work done by Peter Apostoli, Roland Hinnion, Akira Kandaand Thierry Libert.ll In the case of probability theory, the frustrating issue ofhow best to interpret the basic concepts of the theory has long been recognized.But as Jon Williamson suggests, Bayesianism, the view that understands probabil-ities as \"rational degrees of belief\", may help us bridge the gap between objectivechance and subjective belief.12 Wilfried Sieg13 and Chris Mortensen14 give ussimilarly exciting characterizations of developments in computability theory andin the theory of inconsistent mathematics respectively. Over the centuries the philosophy of mathematics has traditionally centeredupon two types of problem. The first has been problems associated with discover-ing and accounting for the nature of mathematical knowledge. For example, whatkind of explanation should be given of mathematical knowledge? Is all mathemat-ical knowledge justified deductively? Is it all a priori? Is it known independentlyof application? The second type of problem has been associated with discoveringwhether there exists a mathematical reality and, if so, what about its nature canbe discovered? For example, what is a number? How are numbers, sets and othermathematical entities related? Are mathematical entities needed to account formathematical truth? If they exist, are mathematical entities such as numbers andfunctions transcendent and non-material? Or are they in some way a part of, orreducible to, the natural world? During much of the twentieth century it was thefirst of these two types of problem that was assumed to be fundamental. Logicism,formalism and intuitionism all took as their starting point the presupposition that 1°Akihiro Kanamori, \"Set Theory from Cantor t o Cohen,\" this volume, pp. 395-459. \"Peter Apostoli, Roland Hinnion, Akira Kanda and Thierry Libert, \"Alternative Set Theo-ries,\" this volume, pp. 461-491. l2 on Williamson, \"Philosophies of Probability,\" this volume, pp. 493-533. 13Wilfried Sieg, \"Computability,\" this volume, pp. 535-630. 14Chris Mortensen, \"Inconsistent Mathematics,\" this volume, pp. 631-649.

Preface ... x111it was necessary to account for the absolute certainty that was assumed to bepresent in all genuine mathematical knowledge. As a result, all three schools em-phasized that they could account for the resolution of antinomies, such as Russell'sparadox, in a satisfactory way. All three hoped that such a crisis in the foundationsof mathematics could be guaranteed never to happen again. Their disagreementswere over matters of strategy, not over ultimate goals. Only in the latter parts ofthe century was there a shift away from attempting to account for the certaintyof mathematical knowledge towards other areas in the philosophy of mathematics.This leaves us, as Mark Colyvan says, \"with one of the most intriguing featuresof mat he ma tic^,\"^^ its applicability to empirical science, and it on this topic thatthe current volume ends. For their help in preparing this volume, my thanks goes to Jane Spurr and CarolWoods as well as to the series editors, Dov Gabbay, Paul Thagard and John Woods,but most especially to the contributors for their hard work, generosity of spirit,and especially their redoubtable expertise in such a broad range of fascinating andimportant topics. Andrew D. Irvine University of British ColumbiaI5Mark Colyvan, \"Mathematics and the World,\" this volume, pp. 651-702.

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CONTRIBUTORSPeter ApostoliUniversity of Pretoria, RSA.peter-cornerstone@ yahoo.caMark BalaguerCalifornia State University, Los Angeles, [email protected] BonevacUniversity of Texas, Austin, [email protected] BostockMerton College, Oxford, UK.Mark ColyvanUniversity of Sydney, [email protected] FranklinUniversity of New South Wales, Australia.j [email protected]. D. HartUniversity of Illinois at Chicago, [email protected] HinnionUniversitk libre de Bruxelles, [email protected] HintikkaBoston University, [email protected] D. IrvineUniversity of British Columbia, [email protected]

xvi ContributorsAkira KandaOmega Mathematical [email protected] KanamoriBoston University, [email protected] LibertUniversitd Libre de Bruxelles, [email protected] McCartyIndiana University, [email protected] MortensenAdelaide University, [email protected] SiegCarnegie Mellon University, [email protected] SimonsTrinity College, Dublin, [email protected] TilesUniversity of Hawaii at Manoa, [email protected] WilliamsonUniversity of Kent at Canterbury, UK.j [email protected]

LES LIAISONS DANGEREUSES W. D. Hart Mathematics and philosophy are roughly coeval in our historical imagination.Plato's dialogues form the oldest surviving extended body of work in the canon ofwestern philosophy. Euclid's Elements is the oldest surviving intact monument inthe evolution of our mathematics. Plato taught Aristotle, who died in 322 B.C.,and Euclid's floruit is around 300 B.C., so the gap from Plato to Euclid is like thatfrom grandparent t o grandchild, and from nearly two and a half millennia later,that gap looks small. There was of course philosophy before Plato. We have fragments from the pre-socratics, and Plato made his teacher Socrates the star of most of his dialogues.There was mathematics before Euclid. He seems to have been as much an editor asa mathematician, and probably not the first. The Greeks had invented or discov-ered proof centuries before; just think of the Pythagorean theorem or the proof ofthe irrationality of the square root of two. In Plato's day, Theatetus seems to haveproved that there are exactly five regular solids, a gorgeous result that impressedPlato enough to give Theatetus a leading role in a major dialogue. As proofs pro-liferate, patterns start to emerge, and aficionados want to organize the profusionof arguments into a coherent whole developed logically from a minimal stock ofassumptions. Doubtless there were such editions of geometry before Euclid, buthis Elements is the work whose authority lasted through the centuries. Aristotle seems to have been less impressed by mathematics than Plato. ButAristotle did begin the systematic study of logic. His account of syllogisms is nowusually assimilated to our monadic quantification theory (and truth functionallogic is usually credited to the later Stoic philosophers).' An interest in logiccould have arisen from the effort to piece disparate proofs together into a unifiedand coherent system, though syllogistic is a pretty thin. description of the reasoningdeployed in ancient geometry. Still, we should not be impatient, since it was notuntil the nineteenth century that people like de Morgan2 and Peirce began towork out a systematic understanding of relations, which was crucial in the logicistregimentation of mathematics. But besides starting systematic logic, Aristotle also articulated a version, ora vision, of the axiomatic method. In the Posterior Analytics he describes areal body of knowledge as deduced by infallible logic from axioms. The axioms lWilliam and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 1962). 2All horses are animals, from which it follows that all heads of horses are heads of animals.De Morgan observed that Aristotle's syllogistic does not suffice t o certify the validity of thisinference, which turns on relations and polyadic quantification.Handbook of t h e Philosophy of Science. Philosophy of MathematicsVolume editor: Andrew D. Irvine. General editors: Dov M. Gabbay, Paul Thagard and JohnWoods.@ 2009 Elsevier B.V. All rights reserved.

2 W. D. Hartshould have an immediate appeal, and the logic should transmit this appeal tothe theorems. As we said, Aristotle wrote before Euclid. But he might have beentrying to articulate an ideal he saw struggling to emerge from editions of geometryolder than Euclid. And one wonders whether Euclid might have been strugglingto realize an ideal earlier articulated in Aristotle. This is our theme, the dangerous liaisons between mathematics and philosophy.They are not just coevel, like strangers or distant acquaintances who happen tohave been born in the same town within a short span of time. They have alsobeen bedfellows, sometimes strange even if not made so by politics. We will sketchsome of their offspring. Some does not mean all; ours will not be a family tree, buta selection of hybrids. And because I am a philosopher who admires mathematicsbut does not claim to be a mathematician, most of these compounds will havemore philosophical elements than mathematical. In some ways, the axiomatic method can seem like proof writ large. To besure, a proof aims to establish a single theorem, while in an axiomatic systemwe prove a sequence of theorems. In the heat of live mathematics, one does notpractice axiomatically. One does not copy one's premisses out of a constitutionwritten down and approved by the founding mathematical fathers and mothers.One starts instead from what is clear, and clarity here probably means what one'speers will accept without complaint. So one needs to be sensitive t o one's peers,and for pretty much all of us this requires being admitted to the community of peersthrough an education. But once the community has approved a body of proofs,some of the peers may set out to regiment it. This process includes collecting theclear starting premisses that passed muster, selecting from them some from whichthe rest can be derived, and so on until we have axioms from which a sequenceof theorems follow, where of course some later theorems are deduced from earlier.Once such a system is established, incorporability of a new argument in it canbecome a standard for being a proof. Euclid set such a standard in geometry forcenturies, and set theory (usually in Zermelo-Frankel form) did so for mathematicsgenerally in the twentieth century. This is a rather sociological description of axiomatization. Philosophers andmathematicians share a taste for long and abstract chains of reasoning, but theyoften differ in how they get started. Mathematicians seem to like their premissesto be shared, perhaps throughout their community, or as close to that as possible.That way the community can be expected to follow their reasoning. There arephilosophers, like Aristotle and Kant, who seem not to want to frighten the horses,but they may be trying to calm things down after earlier philosophers like Platoand Hume have stirred them up by going where the reasoning led from premissesfor which they may have claimed more popularity than was generally recognized.At any rate, philosophy looks more contentious than mathematics. But howevermuch they disagreed elsewhere, Plato and Aristotle seem to have agreed that aversion of the axiomatic method describes an ideal for knowledge. Even if it is not perfectly clear whether this ideal starts life in mathematicsor in philosophy, the axiomatic method is a mode of exposition that has become

Les Liaisons Dangereuses 3a tried and true device in the mathematical repertoire. It was exaggerated byphilosophers into the ideal of foundations of knowledge, or this or that departmentof knowledge. A vivid example is Spinoza writing his Ethics in more geometrico.It is perhaps ironic t o note that the Latin word \"mos\" from which \"more\" declinesmeans custom or usage, which seems more sociological than Spinoza probably hadin mind. (Anyone trying to formalize Spinoza's system by modern lights is in fora bad time.) The basic philosophical idea seems to be that there is a right wayto organize for justification truths, beliefs, or knowledge. This idea has grippedphilosophical imaginations for centuries. The ideal can be articulated in different ways. Sometimes the right order isthe right order in which to justify our beliefs or knowledge. In Descartes's urbanrenewal of knowledge we are to rebuild from clear and distinct ideas of indubitablecertainty like the cogito. In more empiricist philosophers like Locke, Berkeley,and Hume we are to begin from sense experience, and increasingly their problemis whether we can get beyond our impressions without losing the certainty thatmade perception an appealing foundation. It was this empirical spectre of skepticism that startled the horses and wokeKant from his dogmatic slumbers. To trace out firm foundations for knowledge,he looked to the surest systematic body of knowledge going, and from the Greekson, mathematics had always been the best-developed system of the most absolutetruth known with the greatest certainty. In Kant's day and before, mathematicsmeant first and foremost geometry, and geometry meant Euclid's system not justof planes but also of the space in which we live and move and have our being.The idea of other spaces is later and quite unkantian, and the mathematics ofnumber (beyond elementary number theory like the infinity of the primes) achievesindependence only in the nineteenth century. Kant does of course give sensibilitya basic role in contributing to knowledge. But it is his conception of the characterof geometrical knowledge that not only gets his critical philosophy going, but alsosets an agenda for many later and rather unkantian philosophers. To exposit this conception we need some distinction^.^ Assume the anachro-nistically labeled traditional analysis of knowledge as justified true belief. Episte-mology is much more about justification than knowledge. Kant calls knowledge aposteriori when it is justified, even in part, by appeal to sense experience. Knowl-edge is a priori when it is knowledge but not a posteriori, that is, not justifiedeven in part by experience. Kant thought that mathematics, that is, geometry, andlogic are systematic bodies of a priori knowledge. We will consider an argumentfor this thought in a moment. Consider next judgments. This is Kant's usual term for mental states like beliefs(such as that grass is green or seven is prime) and thoughts. Around the turn of 3Kant draws his distinctions in the introduction t o The Critique of Pure Reason, trans. Nor-man Kemp Smith (London: MacMillan, 1963). Moore discussed propositions in chapter 3 ofSome Main Problems of Philosophy (New York: Collier Books, 1962). T h e basic Tarski pieceis \"The Concept of Truth in Formalized Languages,\" in Logic, Semantics, and Metamathemat-ics, trans. J . H . Woodger (Oxford: Clarendon Press, 1956). For Austin, see his \"Truth,\" inPhilosophical Papers, ed. J . 0 . Urmson and G. J. Warnock (Oxford: Clarendon Press, 1961).

4 W. D. Hartthe twentieth century, G. E. Moore and Russell will replace judgments by propo-sitions, which are platonic abstracta like numbers rather than mental. Frege'sthoughts are more like Russell's propositions than Kant's judgments. During thetwentieth century, philosopher-logicians like Tarski will replace both judgmentsand propositions with sentences. Sentences are linguistic items where judgmentsand propositions were supposed to be independent of language. (Around 1950J. L. Austin will try to replace sentences with statements thought of as actionsperformed using sentences.) Kant divides judgments into analytic and synthetic.Analytic judgments are reminiscent of Locke's trifling propositions (not to be con-fused with russellian propositions) and Hume's relations of ideas. One way to move in on analyticity is through examples. An example of Moore'sis the claim that all bachelors are unmarried. The Social Science Research Councilwould be ill advised to fund a door-to-door survey in which bachelors are askedwhether they are married, the results are tallied, and finally the bold hypothesisthat all of them are unmarried is advanced. This would be a waste because, sothe story goes, being unmarried is part of what it means to be a bachelor. It seems clear that there is some sort of difference between the claim that bach-elors are unmarried and the claim that bachelors are more flush financially thanhusbands. Controversy sets in when we try to articulate the difference. Kantgave two accounts of analyticity. On one, the predicate of an analytic judgment iscontained in its subject. Note three points about this account. First, it seems topresuppose that all judgments are of subject-predicate form. Whatever grammar-ians may say, Russell was excited by the revelation in the logic reforming aroundhim of other forms, especially quantificational, of judgment. We follow Russell,so Kant's account may seem too narrow to us. Second, his account presupposesthat judgments have subjects and predicates. That is, Kant seems to be readingsentence structure back into judgments. One role in which Kant's judgments orRussell's propositions or Tarski's sentences are cast is as bearers of the truth val-ues; these are the things that are true or false. Whether it is true that Socrates wassnub-nosed depends in part on the man Socrates and what his nose was like. Thatis, the truth bearers (or vehicles, as Austin called them) need to be articulated intobits smaller than whole truth vehicles. Sentences wear such an articulation intosmaller bits, words, on their inscribed faces. It seems all but irresistible t o readthis articulation back into the judgments or propositions expressed by sentences.But then Tarski's choice of sentences as truth vehicles seems more up front thanKant's judgments or Russell's propositions. Third, Kant's trope of the predicateof a judgment being contained in its subject is clearly a metaphor, and this leavesus without a literal account of analyticity. On Kant's other account of analytic-ity, the denial of an analytic judgment cannot be thought without contradiction.Never mind that thought here seems to assume the analyticity of logic withoutargument. What might be worth noting here is the relativity of this account towhich premisses we are allowed. If, for example, it is one of our premisses thatbachelors are richer than husbands, we will not be able to think the denial of thejudgment that bachelors are richer than husbands without contradiction. That

Les Liaisons Dangereuseswould make the judgment analytic contrary to the motivation for the notion. For much of the twentieth century, the quick gloss on analyticity was that asentence (proposition, judgment) is analytic if it is true by virtue of the meaningsof the words in the sentence (used t o express the proposition or judgment). Thisgloss seems confused. Analyticity is a t best a mode of justification, not of truth. Itis an ancient and honorable view that truth is correspondence t o fact; for a sentenceto be true, the world should be as the sentence says it is. Moreover, truth isunivocal. That is why the conjunction of two truths from areas however disparateis nonetheless true. The truth that bachelors are unmarried (or, for that matter,bachelors) is as much about bachelors as the claim that bachelors are richer thanhusbands, and if true, they are so because bachelors are unmarried, and richer thanhusbands. What was distinctive about the claims attracting the label analytic wasepistemic, a matter of justification rather than the nature of a kind of truth. Thejudgment that bachelors are unmarried would then be analytic if knowledge of themeanings of the words used to express the judgment sufficed without experienceof its subject matter (bachelors) to justify belief in the judgment. This provisional story is no better than our grasp of what knowledge of themeanings of words comes to, and that grasp is at best pretty shaky. How muchit makes analytic is unclear. Kant said it is synthetic that all bodies are heavy.(The synthetic judgments are those that are not analytic. Analyticity wears thetrousers in its distinction, as being a posteriori does in its; they get a positiveaccount, and their opposites are defined by negation.) This example is plausibleif we are reluctant to build gravitational attraction into the meaning of the word\"body.\" But it would be embarrassing to have to give a justification for thisreluctance. Kant said it is analytic that all bodies are extended. This is ratheran odd example for Kant to give. Like any eighteenth-century intellectual, Kantadmired Newton. Indeed, part of Kant's objective was to secure certainty for muchof Newton's physics, and Kant was not innocent of that science. Anyone familiarwith Newton will remember how much he makes of mass-points. Does Kant meanto exile the mass-points from the bodies by definition? Is he defining mass-pointsout of existence? Analyticity is often a cloak for arbitrary legislation. With two binary distinctions, we get four compounds. Kant ruled out the ana-lytic a posteriori. A survey could amass evidence that .all bachelors are unmarried,but such cases do not seem worth fretting over. The synthetic a posteriori wouldinclude most of the natural science, the physics, astronomy, and chemistry comingto be in Kant's day. Since a posteriori knowledge is justified by experience, wehave at least the beginnings of a story about how such science is known. But sincea priori knowledge is defined negatively as not a posteriori, there is a questionhow it could be justified. Kant thought logic is known a priori, but is analytic, andso justified from the meanings of logical words like \"if\" and \"all\" and \"is.\" Kantwrote during the low-water mark of the history of logic. The achievements of theSchoolmen had been largely rejected during the enthusiasm of the Renaissance,and what was left was Aristotle's syllogistic and some Stoic truth function theory.It is hardly a blunder to think Barbara (If all cats are vertebrates and all verte-

6 W. D. Hartbrates are animals, then all cats are animals) can be certified by an elaboration ofthe meanings of \"all\" and \"are,\" but logic did not stick a t its kantian low-watermark. The moneybox was a priori knowledge of synthetic truths. Such truths are notknown from experience, nor are they justified from the meanings of the words usedto express them. So how are they known? How, Kant asks, is synthetic a prioriknowledge possible? This question is the pretext for the critical philosophy, andKant7sanswer is transcendental idealism. But Kant's question has purchase onlyif there is synthetic a priori knowledge. To appreciate one example whose consid-eration goes back to the nineteenth century, we should make a third distinction,this time between necessary and contingent truths. People don't usually read forlong standing up, so you are probably not standing as you read this. If so, it is truethat you are not standing. But you could have been, so that truth is contingent.You are also identical with yourself, and that is not something you could fail t obe, so that truth is necessary. The contingent truths could be otherwise, but thenecessary ones could not. At B3 in the first Critique, Kant says that experienceteaches us that a thing is so and so, but not that it cannot be otherwise. In theways we can see color and shape, or feel shape and texture, we have no experi-ence of necessity or (nonactual) possibility, only of actuality. So, Kant thought,knowledge of necessity is a priori. Kripke4 later observed that we sometimes caninfer a necessary truth from two premisses, one known a posteriori, so if a pri-OTZ knowledge rules out all justification by experience, some necessary truths areknown a posteriori. Fair enough; these examples Kant missed out, and even inthem the necessity in the conclusion comes from the premiss known a priori, whichvindicates Kant somewhat. The nineteenth-century example is that nothing couldbe red all over and green all over a t the same time and place. That these colorsexclude each other does seem t o be a necessary truth; in general, determinants(for example, being six feet tall and being seven feet tall) of a determinable (forexample, height) exclude one another necessarily. We cannot imagine an objecthaving both, and the imagination is the royal road to knowledge of possibilityand necessity (even if it does not always accord with, say, materialist prejudices).Granted that it is necessary that nothing is a t once red all over and green all over,knowledge of it would be a priori. But, the story continues, the colors red andgreen are simple and basic enough that no definitions of them in more basic termsare available, so there are no definitions of them from which t o show that thisnecessity is analytic. It is then a synthetic necessity known a priori. This examplehas been much discussed and no plausible definitions of the colors that would showit analytic have been generally a ~ c e p t e d .O~n the other hand, i t bucks the trend 4Saul A. Kripke, Naming and Necessity (Cambridge: Harvard University Press, 1972). 5See, for example, Arthur Pap, \"Logical Nonsense,\" Philosophy and Phenomenological Re-search 9 (1948), 269-83; \"Are All Necessary Propositions Analytic?\" Philosophical Review 50 (1949), 299-320; Elements of Analytic Philosophy (New York: MacMillan, 1949), chap. 16b;Hilary Putnam, ''Reds, Greens, and Logical Analysis,\" Philosophical Review 65 (1956), 206-17;Pap, \"Once More: Colors and the Synthetic a Priori,\" Philosophical Review 66 (1957), 94-99;Putnam, \"Red and Green All Over Again: A Rejoinder t o Arthur Pap,\" Philosophical Review

Les Liaisons Dangereuses 7in most twentieth-century analytic philosophy that necessity is our creature; thereis no necessity out there in nature independent of us. That determinants of the same determinable exclude one another necessarilyyields a relatively scattered fund of examples, so if that were the only synthetica priori knowledge, it would have been less front and center on the postkantianphilosophical agenda. But Kant thought that mathematics is synthetic a priori.That it is a priori might seem evident, since mathematicians do not performexperiments on prime numbers, nor do they make expeditions to examine exoticones; they just sit around and think, and what could be more a priori? Hisargument for the synthetic is more dubious. His example is that 7 f 5 = 12.He says that the concept of the sum of 7 and 5 contains nothing save the unionof the two numbers into one, and in this no thought is being taken as to whatthat single number may be which combines both. He says that the concept of12 is by no means already thought in merely thinking this union of 7 and 5, andanalyze our concept of such a sum as we please, still we will never find the 12in it. Instead, we must go outside the concepts of 7, 5, and addition, and callin examples (like intuitions of fingers or points) to see the number 12 come intobeing. It is probably anachronistic to be too fussy about whether Kant is herediscussing psychology (ideas of 7, 5, sum, and 12) or semantics (the meanings ofnumerals and function signs). But in the way a dictionary definition for \"bachelor\"seems forthcoming and uncontroversial, definitions for the numerals and functionsigns are more problematic. We can, to be sure, label some claims in which theyare used definitions, and from these deduce some conventional arithmetic wisdomlogically, but even if all this is necessarily true, where is the semantics independentof the philosophy at issue to settle whether these claims are analytic or synthetic?No one has stated such a semantics that convinces many others. But maybe Hume can help Kant out here. In part IX of his Dialogues Concern-ing Natural Relig~onH,~ume turns to the a priori arguments for the existence ofGod, like Anselm's ontological argument. Usually Hume is as patient as all get outat criticism; he likes to give his opponent all the rope he wants with which to hanghimself. But here Hume is brisk. We might try to articulate what is eating Humeby saying that no existence proposition is analytic; you cannot make things existhowever you define terms purporting to denote them. Of course Hume did notuse Kant's term of art L'analytic.\" In Hume's vocabulary we could say existenceis always a matter of fact, never a relation of ideas. But the thesis that no exis-tence proposition is analytic seems to be one of the few constants in philosophicalconsciences. For almost any philosophical view, one can find a stretch in Russell'slife, for example, where he believed that view; nevertheless, not even Wittgensteincould con Russell into analytic existence. Let us offer Kant Hume's thesis in kantian terms: no existence judgment is ana-lytic. Now note that there are many existence claims in mathematics, witness theinfinity of primes, the five regular solids, and undecidable propositions of Princzpia66 (1957), 100-03. 6David Hume, Dialogues Concerning Natural Religion (New York: Hofner Press, 1948).

8 W. D. HartMathernatica and related systems. It would follow that some mathematical truthsare synthetic, and so granting that they are known a priori, mathematics providesa fund of synthetic a priori knowledge. How extensive a fund would remain to beseen, but if it requires making meaning out, the prospects are dim. Frege agreed with Kant that geometry is synthetic a priori. Note here Euclid'sfifth and most famous postulate. In the form made familiar by the eighteenth-century editor Playfair,7 this postulate says that given a line L in a plane and apoint P in that plane not on L, there is one and only one line in the plane through Pparallel to L. This is clearly an existence claim, and so by our earlier argument notanalytic but synthetic. But that argument would not have persuaded Frege. Fregessaid that the distinctions between the a priori and a posteriori and between theanalytic and synthetic concern not the content of the judgment but the justificationfor making the judgment. (In a footnote to this remark, Frege adds that he doesnot mean to assign new senses t o these terms, but only to state accurately whatearlier writers, Kant in particular, meant by them. The reader may wish to becareful about whether Kant would have agreed with Frege on this point.) Fregesays that when a proposition is called a posteriori or analytic in his sense, this isnot a judgment about how we might form the content of the proposition in ourconsciousness or about how one might come to believe it, but about the ultimateground upon which rests the justification for holding it t o be true. This is an in-your-face remark because it assumes flat out that there is a unique and ultimateground on which the justification for a proposition rests. Frege perhaps takesthe figure of foundations of knowledge more seriously than any other philosopher.F'rege initiated the analytic style of philosophy, and an important trend in itsdevelopment is a loss of confidence in the idea that knowledge has foundations;in this process we diminished our exaggeration of the mathematician's axiomaticmethod. Frege says that to settle whether a truth is analytic or synthetic, a priori or aposteriori, we must find the proof of the proposition and follow it right back t o theprimitive truths. So a truth has a single, unique proof that begins from a uniquestock of primitive truths (Unuahrheiten). The presuppositions are so pronouncedthat one wonders whether they are as metaphysical as epistemic, whether the rightorder of truths is their logical order in being as much as the order in which wemay justify them. If the primitive truths from which the proof of the propositionproceeds are nothing but general logical laws and definitions, then the truth isanalytic. (Here he reminds us that when a singular term, for example, is definedby a definite description, the definition is admissible only if the predicate in thedefinite description is true of at least and at most one thing, and we will needto find the primitive truths on which rest these conditions for the admissibility ofthe definition.) If the primitive truths belong to the sphere of a special science, 7Heath says Playfair's axiom is stated by Proclus. See Euclid, The Thirteen Books of theElements, trans. and intro. Sir Thoms Heath, vol. 1, 2nd ed. (New York: Dover), p. 220. sGottlob Frege, The Foundations of Arithmetic, trans. J. L. Austin (Oxford: Blackwell, 1959), sec. 3.

Les Liaisons Dangereuses 9then the truth is synthetic. This account makes analyticity turn on generalityrather than meaning. But the generality cannot be that all the quantifiers inthe primitive truths are universal, so the primitive truths would be true even ifthere were nothing at all, and so cannot prove any existence claims. For Frege isout t o show that the truths of elementary number theory are analytic, and theserequire the existence of infinitely many natural numbers. For a truth t o be aposteriori it must be impossible to prove it without appeal to facts, that is, truthsthat cannot be proved and are not general because they contain assertions aboutparticular objects. Note the absence here of any mention of justification by senseexperience; the focus is instead on singular truths about particular objects that insome absolute sense cannot be proved, so it is almost as metaphysical as epistemic.For a truth to be a priori, its proof should proceed exclusively from general lawsthat neither need nor admit of proof. (Here Frege adds in a footnote an argumentfor general primitive truths.) So on Frege's view, the analytic and the a prioriboth descend from utter generality, and we expect a substantial overlap betweenthem. He does say that the general law underlying the analytic should be logical,while those underlying the a priori should neither need nor admit of proof. He mayhave meant there are general primitive truths besides those of logic underlying,say, geometry. The intriguing notion here is of general laws that neither need noradmit of proof. Need here might mean that we can know, and so be justified inbelieving, these general laws without proof, and then the interesting question ishow we know them. To say that they do not admit of proof raises the questionhow we would know of a general truth that it cannot be proved, and there is nostandard way to prove such an absolute claim. Frege's account makes it analytic that logic is analytic. His strategy was totake over ad hominem from Kant the premiss that logic is analytic. Then hewould reduce the mathematics of elementary number theory to logic. While Euclidhad axiomatized plane geometry millennia before, Dedekind had only recentlyaxiomatized the natural numbemg Frege had to define Dedekind7sprimitive terms(zero, the successor function, and being a number) in purely logical terms, andthen deduce Dedekind's axioms from logic and these definitions. This reductionmay be called logicism in a narrow sense. Given the reduction and the analyticityof logic, the analyticity of elementary number theory follows, and that may becalled logicism in a broader sense. Logicism in a broader sense would depriveKant's critical philosophy of some of its presuppositions. From time to time, aphilosopher claims that thus-and-suches (material objects, say) are reducible t oso-and-so (like sense experience). What was new with Frege, and still impressiveto this day, is that instead of blustering, he gets on with it. Thereby he begana constructional tradition in analytic philosophy that includes Russell, Carnap,Tarski, Goodman, and David Lewis.'' gRichard Dedekind, \"The Nature and Meaning of Numbers,\" in Essays on the Theory ofNumbers, trans. W . W . Beman (New York: Dover, 1963). 1°Besides Principia Mathematica, see also Russell's Our Knowledge of the External 'world(New York: Mentor, 1960); Rudolf Carnap, The Logical Stmcture of the World, trans. Rolf

10 W . D. Hart Frege viewed his reduction as an extension of the arithmetization of analysis,which was a central focus of nineteenth-century mathematics. Analysis is the partof mathematics with the calculus (differentiation and integration) at its core. New-ton and Leibniz had invented or discovered the calculus in the seventeenth century,and Newton put it to work in his physics. That physics was perhaps the singlemost notable event in the emergence of natural science, which after the FrenchRevolution replaced theology as the most prestigious part of knowledge. But inPrinczpia, Newton did not express himself in terms of the new calculus. Instead hisarguments are mostly novel but recognizable extensions of Euclid, whose authoritywas unabated.'* The public face of the calculus was geometrical. The only num-bers the Greeks were comfortable recognizing as such were positive whole numbers(and even one was maybe insufficientlyplural). They talked about proportions butnot fractions or rationals, and they did not have decimals for irrationals. Wherewe now talk about functions, like the square of a number, mathematicians fromthe Greeks until Kant's day might talk about curves, by which they meant literalcurves like the parabola. Differentiation was about tangents to a curve, and inte-gration was about area under a curve, and area was less a number than a patchof the plane. Euclid7sgeometry is mostly finitary. Angles get bisected and polygons havefinitely many sides. Archimedes had given some gorgeous limiting arguments,but their infinitary aspect was much of what made them so striking. But asthe calculus developed during the eighteenth century, its infinitary aspect becameinescapable. In our expositions, for example, one of the earliest concepts distinctiveof the calculus is the notion of, say, a point being the limit of an infinite sequenceof points. The problem is that while in finitary Euclidean geometry, intuition(visual imagination) by and large did not lead people recognizably astray, intuitionhad begun to founder on paradox among infinitary processes by the turn of thenineteenth century.12 Here is an example that Henri Lebesque says was currentamong schoolboys near the turn of the twentieth. In triangle ABC, let D be themidpoint of side AB, E of side BC, and F of side AC. Join D and E, and join Eand F.A. George (Berkeley: University o f California Press, 1969); Tarski on truth cited above for thenotion o f material adequacy conditions; Nelson Goodman, The Structure of Appearance (Indi-anapolis: Bobbs-Merrill, 1966); David Lewis, Counterfactuals (Cambridge: Harvard UniversityPress, 1973). llFrancois De Gandt, Force and Geometry i n Newton's Princzpia, trans. Curtis Wilson(Princeton: Princeton University Press, 1995). 12Henri Lebesgue, Measure and the Integral, ed. Kenneth 0. May (San Francisco, London,Amsterdam: Holden-Day, 1966).

Les Liaisons Dangereuses It is familiar from Euclid that quadrilateral A D E F is a parallelogram, and A Fis as long as D E , and EF is as long as DA. Hence, the broken line C F E D B is aslong as the two sides CA and AB together. Now repeat in the two little trianglesFEC and D B E the argument just carried our in ABC to get a four tooth jaggedline equal to CA and AB together. Carry this repetition out ad infinitum, andthe length never changes from that of CA and AB together. But the limit of thesejagged lines certainly looks to be side BC. So the sum of two sides of a triangleis not greater than the third, and a straight segment is not the shortest distancebetween two points. One wonders how the author of the antinomies of pure reasonwould have reacted to this infinitary paradox. There may be some relief to be hadin noting that the angles of the teeth never flatten but are always equal to theangle at A. So the limit of the infinite sequence of jagged lines, assuming we canfigure out what this limit is, is not the straight segment B C , but rather a linethat all too often has no single unique direction. Analysis came to dote on suchexamples during the nineteenth century. A line each of whose points is arbitrarily close to B C but mostly has no uniquedirection (say from B to C or vice versa) is not easy to visualize. To understandsuch things, mathematicians turned from intuition to understanding. The calculuswas transposed from geometry to number. With hindsight we can see Descartespointing a way to do this. Analytic geometry is usually credited to him. Weare taught to begin cartesian coordinates with the number line. This hybrid isrooted in a one-to-one correspondence between the points on an infinite Euclideanstraight line and (what we now call) the real numbers that preserves order, thatis, such that point p is left of point q if and only if the number assigned to p isless than that assigned to q. It is not patent how much of our conception of thereals Descartes shared, so it is probably unfair to ask how he knew there is sucha one-to-one correspondence. Where we are taught analytic geometry as a way touse numbers to answer geometrical questions, Descartes used it as a way to answernumerical questions with geometry. But once we lose confidence in the capacityof geometrical intuition to answer infinitary questions, one strategy would be toreverse direction and take real numbers more seriously. To do so, we want a non-geometrical account of the reals, and that is wherethe arithmetization of analysis comes from. Let us briefly rehearse the familiarsaga. Start with the natural numbers, the whole numbers 0, 1, and so on. We getnegative numbers by arranging to subtract 5 from 3. So let us see how to handlethe ordered pair (n, k ) of natural numbers as if it were the difference n - k nomatter whether n is bigger than k or not. A little algebra shows that (n,k ) and

12 W. D. Hart+ +(p,q) have the same difference when n q is p k, addition being familiar on the+ +natural numbers. So we specify a relation R to hold between a pair (n,k) and apair (p,q) just in case n q = p k. Then R is reflexive (any pair is R to itself),symmetric (if (n,k) is R to (p,q), (p,q) is R to (n, k)), and transitive (if (n, k) isR to (p,q) and (p,q) is R to (r,s), then (n, k) is R to (r,s ) ) . Such a relation iscalled an equivalence relation. Being as tall as is an equivalence relation, and itpartitions people into exclusive and exhaustive groups of people equally tall. Thephilosophy starts to creep in if we think of these groups as the heights, like sixfoot six. Our R partitions the ordered pairs of natural numbers into groups, orequivalence classes as they are called, some think of as the integers. Write [(n,k)]for the equivalence class of the pair (n,k). We should raise addition from thenatural numbers to the integers, and the evident way to do so is to set the sum of+ +[(n,k)l and [(P,q)l equal to [(n P, k 41. Then for any [(P,dl,the sum of [(P,q)]and [(n,n)] is [(p,q)],so [(n,n)] is the zero of the integers. For any [(p,q)], the sumof [(p,q)] and [(q,p)] is the zero of the integers, so [(q,p)] is the negative of [(p,q)].To subtract an integer [(p,q)] from [(n,k)], add [(q,p)]to [(n,k)]. The integers[(n,o)] are a copy of the naturals, and the integers [(o,n)] are their negatives. Ina flush of enthusiasm, some call this constructing the integers from the naturals. Constructing the rationals is similar except that this time we start wanting tobe able to divide any integer i, even 3, by any non-zero integer j, even 5. We takeordered pairs (i,j ) of integers, and we say (i,j ) is Q to (p,q) if and only if theproduct iq equals the product jp, which should hold when i is to j as p is to q. Q isan equivalence relation, and its equivalence classes are the rational numbers. Theequivalence relation is as old as Eudoxus's theory of proportions, but we probablydid not step all the way to its equivalence classes, the rational numbers, until thenineteenth century. The next step is to the real numbers. One way to see the needis through the square root of two. By Pythagoras's theorem, this should be thelength of the diagonal of a square whose side is of length one, but one can findin Euclid a proof that no ratio of, in effect, rational numbers can have 2 as itssquare. There is a story that the Pythagoreans knew this proof long before Euclidbut hushed it up because it put the root of 2 beyond their reach. Here are twoways to flesh out the rationals with irrationals. The first is Dedekind cuts.13 ADedekind cut is a pair (L, R) of sets of rationals such that every rational is in Lor R, no rational is in both, and every member of L is less than every member ofR. Picture L as the left part, and R the right, produced by cutting the rationalnumber line in two somewhere. If L is the rationals less than or equal to one butR is those greater, this cut corresponds to the rational real one. Note that L hasa greatest member, namely one. Now let L be the rationals whose squares areless than or equal to 2, while R is the rationals whose squares are greater than 2.This time L has no greatest member, and R, no least. This cut corresponds tothe irrational real, root 2. But square roots are an unrepresentative ground for irrationals. Decimals give a 13Dedekind, \"Continuity and Irrational Numbers,\" in Essays on the Theory of Numbers, trans.W. W. Beman (New York: Dover, 1963).

Les Liaisons Dangereuses 13better picture. It is not difficult to show that the decimal for any rational numberis either finite (like .25 for 114) or repeating (like .333 . .. for 1/3), and conversely.So the decimal for the root of 2 is neither finite nor repeating. But we can compute+its decimal as far as we like. Its first n digits are those in the Arabic numeral for theleast natural number k such that (k 1)' > 2(1OZn).So it starts off 1.414213 ....We can break this decimal up into the sequence 1,1.4,1.41,1.414,1.4142, .... Allthe members of this sequence are finite decimals and so represent rationals. Theserationals are all less than, say, 1.5, but they never decrease. So they get squeezedcloser and closer together, which makes us expect them to be pushing up againsta limit, a least number greater than or equal to all of them. But there is no suchrational number, so we need an irrational limit. A more general version of thisissue is that a non-decreasing sequence bounded above should have a least upperbound. This property is called completeness, and the rationals do not have it. Butnow suppose L1, L2,... are the left halves of an infinite sequence of Dedekind cutssuch that for each n there is a member Ln+l greater than every member of L, (sothe sequence is increasing), but there is a rational greater than every member ofevery L, (so the sequence is bounded above). Then the union of all the L1, Lz, ...fixes the left half of the cut that is the desired least upper bound, or limit, of thesequence of cuts. The reals are complete, and their completeness under limits ispart of their centrality in analysis. We can also approach the reals and completeness by thinking about squeezing.14Let S be a non-empty set we will call a space. A metric on S is a binary functiond that assigns to any two points x and y in S a number intended t o representthe distance between x and y. Usually these numbers are real, but since we areconstructing the reals, let us start off with rational distances. It is required thatthe values of d be non-negative, that the distance between x and y be zero if andonly if x is y, that the distance between x and y be the distance between y and x,and that the sum of the distance from x t o y and that from y to z be less than orequal to that fcom y to z . This last is called the triangle inequality; the sum of twosides of a triangle is greater than the third. For any number x, its absolute value1x1 is x if x is positive, but -x if x is negative. (Intuitively the absolute value ofx is its distance from 0.) If for any rationals x and y we set d(x,y) = Ix - yI, thend is a metric on the space of rationals. In our sequence 1,1.4,1.41,1.414,... ofrationals, the distance between successive members gets smaller. In fact for any nhowever big there comes a stage in the sequence after which any two members ofthe sequence are less than l l n apart. Such a sequence is called a Cauchy sequence.Cauchy sequences get squeezed. Let pl,p2,. .. be a sequence of points in a metricspace. A point p is a limit of the sequence if for every n however big there is a ksuch that for m greater than or equal to k, the distance between p, and p is lessthan l l n . Another version of completeness requires that every Cauchy sequencehave (or converge to) a limit. Our sequence 1,1.4,1.41,... is Cauchy but does notconverge. But now let al,a2,. . . and bl, b2, . .. be two sequences. Say that these 14See, for example, Patrick Suppes, Axiomatic Set Theory (Princeton: D. Van Nostrand,1960), chap. 6.

14 W. D. Hartsequences are C t o each other if for every n however big there is a I; such that form greater than or equal to k, the distance from a, to b, is less than l l n . Whensequences are C they come together. C is an equivalence relation, and we canconstruct the reals as the C equivalence classes of the Cauchy sequences. Thenour sequence has a limit, namely, its C equivalence class. So we can construct integers from natural numbers, rationals from integers, andreals from rationals. Kronecker said God made the natural numbers; all the rest isthe work of man.15 But construction and work are metaphors here. We construedintegers, for example, as equivalence classes of ordered pairs of natural numbers.Nowadays we take equivalence classes and ordered pairs as sets. This is also trueof sequences. We do not build sets; there are too many of them, and they are tooabstract, for that. Instead, we assume them. Besides prying the various sorts ofnumbers out of geometry, we also unify them as applications of set theory to thenatural numbers. If we go on to qualify, we might say both Frege and Cantor complete the processby reducing natural numbers to sets. The first qualification is that F'rege worksnot with sets but with functions.16 He recognizes two truth values, truth andfalsity, and he defines a concept as a function whose value is always a truth value.So the concept of humanity is the function whose value is truth for the argumentSocrates but falsity for the number seven as argument. The value range of afunction is roughly its graph, so the value range of humanity is the curve passingthrough truth over people but falsity over everything else. Though Frege wouldnot like it, we can recover a set as the part of the domain of a concept on which ittakes the value truth. All Frege's functions have the same domain, the universe ofabsolutely everything, so our device attributes to Frege what Godel describes asthe conception of sets as all ways to divide the universe in two.'' Frege describesnumbers as objects belonging to concepts, and if we replace his concepts by sets,we could say both Frege and Cantor think of cardinal numbers as answers to thequestion how many members does a set have. For both the central notion is ofa function f that maps a set A one-to-one onto a set B. If f assigns differentvalues in B to different arguments in A, it is one-to-one, and since there is thenno collapsing of different arguments into a single value, B is at least as big as A.If every member of B is also a value of f for some argument in A, the f mapsA onto B , and since everything in B is hit by f at least once, A is a t least asbig as B. So if f is both one-to-one and onto, B is at least as big as A, andA is at least as big as A, and they are of the same size. We can use one-to-one 1 5 ~ e r m a nWeyl, Philosophy of Mathematics and Natural Science (New York: Atheneum, 1963), p. 33, cites Kronecker as saying God created the integers, but Kronecker does not giveus enough credit. Constructing the integers is so like constructing the rationals that we shouldextend Kronecker's trope. 16Gottlob Frege, \"Function and Concept,\" in Translations from the Philosophical Writings of Gottlob Frege, trans. Peter Geach and Max Black (Oxford: Blackwell, 1960). 17Kurt Godel, \"What is Cantor's Continuum Problem?\" in Collected Works, vol. 11, ed. Solomon Federman, John Dawson, Stephen Kleene, Gregory Moore, Robert Solovay, and Jeanvan Heijenoort (New York: Oxford University Press, 1990), p. 180.

Les Liaisons Dangereuses 15and onto functions t o explain having the same number of members without aprior account of number. You can exhibit such a function by putting your fingertips together, thereby showing you have as many digits on one hand as the otherwithout counting either. The as-many-as relation is an equivalence relation between sets, and we couldtry the equivalence class of A under it as the (cardinal) number of (members of) A.That was roughly Frege's approach, but Cantor construes the number of membersof A as something we abstract from the sets the same size as A.18 Abstraction inthis traditional sense is metaphysically and epistemically less sophisticated thanthe equivalence class construction. (There is an interesting critique of abstractionin Peter Geach's Mental Acts.)lg Frege and Cantor go in different directions from the number of members ofa set. Cantor was after infinite numbers. Euclid took it as an axiom that awhole is always greater than any of its (proper) parts. When Galileo noted thatdoubling maps the natural numbers one-to-one onto the even numbers, so thereare as many even numbers as natural numbers even though the even numbers donot exhaust the natural numbers, Euclid's authority sufficed for Galileo to inferthat there is no completed totality comprised of either the even numbers or thenaturals, and Leibniz further concluded that there are no infinite numbers either.20Such views are of a piece with Aristotle's doctrine that the infinite can only everbe potential (though a possibility that cannot be actual seems contradictory).Dedekind inverted this conventional wisdom. He defined an infinite set as onethe same size as one of its proper subset^.'^ (One set is a subset of another ifall members of the first are members of the second, and it is proper if there aremembers of the second not in the first.) Dedekind also gave a truly dodgy proofthat there is an infinite set: we have, he said, an idea of each of our ideas, butwe also have an idea of ourselves, who are not ideas, so the set of our ideas isinfinite.\" Russell agonized over this argument23 instead of just denying that wehave an idea of each of our ideas. Nowadays we usually just assume the existenceof an infinite set. A number is infinite when it is the number of members of an infinite set. It is amarked advantage of constructing numbers from sets that it makes sense of infinitenumbers. Cantor is perhaps most famous for proving that there are differentinfinite numbers. He proved, for example, that there are more real numbers thannatural numbers. A set A is smaller than or equal in size to a set B if there is a lsGeorg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans.Philip Jourdain (New York: Dover, n.d.; originally published 1915), p. 86. lgPeter Geach, Mental Acts (London: Routledge and Kegan Paul, 1957). 20Herman Weyl, T h e Philosophy of Mathematics and Natural Science (New York: Atheneum,1963), pp. 47-48. 21Dedekind, \"The Nature and Meaning of Numbers,\" in Essays o n the Theory of Numbers,trans. W. W. Beman (New York: Dover, 1963). 22Dedekind, in Essays o n the Theory of Numbers, article 66. 23Bertrand Russell, Introduction t o Mathematical Philosophy (London: George Allen andUnwin, 1919), pp. 138-40.

16 W. D. Hartone-to-one function that assigns to each member of A a member of B. B is biggerthan A if A is smaller than or equal in size t o B but B is neither smaller thannor equal in size to A. A number n is less than or equal to a number k if thereare sets A and B such that n is the number of members of A, k is the number ofmembers of B, and A is smaller than or equal in size to B , and k is larger than nif n is less than or equal to k but k is neither less than nor equal to n. So Cantorhad a t least two infinite numbers. But he had more. He defined the power set of a set as the set of all its subsets.It is called the power set because if it is finite and has, say, n members, then informing an arbitrary subset of it, there are only two things to be done with anymember of the set -either put it in the subset or leave it out. So it has 2, subsets.An induction shows that 2\" is always bigger than n , so the power set of a finite setis always larger than the set. Cantor showed that this holds for infinite sets too.He proved this by what is now called a diagonal argument, a mode of reasoningCantor discovered, though there seem to be many more diagonal arguments inrecursion theory than in set theory. Diagonal arguments remind some people ofthe liar paradox and can be controversial. But Cantor's theorem that the powerset of a given set is always strictly larger than the given set is now as receivedas any other theorem. It gives us a wealth of infinite sizes. Let A. be the set ofnatural numbers and for each n, let A,+l be the power set of A,. Then Ao,Al, ...give us as many infinite numbers as natural numbers. Next let A be the union ofall of Ao,Al, . ... If A were smaller than or equal in size to some A,, it would besmaller than A,+1, but since A,+l is a subset of A, A,+l is smaller than or equalin size to A, and thus A cannot be smaller than A,+1. Hence, A is larger than allof Ao,Al, .... Now we iterate power set from A as we did from Ao. Indeed we goon to iterating power set and union into an indefinite distance. For each infinitenumber n, there are more than n infinite numbers (just as for each natural numberk, there are more than 5 natural numbers). David Hilbert called this wealth ofinfinite numbers, and its mathematics, Cantor's paradise.24 Suppose we took F'rege's truth values, truth and falsity, to be the numbers 0 and1. The characteristic function of a set A of natural numbers is the function thatassigns 0 (truth) to members and 1 (falsity) to non-members, so the characteristicfunction of A is reminiscent of Frege's concept of A-ness. We are used t o decimalswritten with the ten Arabic numerals, but we could as well write them with thebinary numerals 0 and 1 favored by computers. Then the binary decimal for anon-negative real less than 1 is a list of the values of the characteristic functionfor a set of natural numbers, and conversely. So there are as many reals in thatinterval as there are members of the power set of natural numbers. Sending each+non-negative real x to x/(x 1) and each negative x to -(x/(x - 1))shows thereare as many reals as there are reals between -1 and 1,and sending point p to f (p)as in 24David Hilbert, \"On the Infinite,\" trans. Stefan Bauer-Mengelberg, in Jean van Heijenoort,From Frege t o Godel: A Sourcebook in Mathematical Logic, 1879-1931 (Cambridge: HarvardUniversity Press, 1967), p. 376.

Les Liaisons Dangereusesshows that any two bounded intervals contain the same number of points. So Can-tor's theorem generalizes his result that there are more real than natural numbers.He conjectured that there is no infinite number between the size of the naturalsand the size of the reals. This is called the Continuum Hypothesis, and Hilbert putit first on the agenda for mathematics in the twentieth century. In the 1930s Godelshowed that the continuum hypothesis is consistent with set theory (if set theoryis consistent), and in the 1960s Paul Cohen showed that its negation is consistentwith set theory (if set theory is consistent). We will not settle the continuumhypothesis without agreeing on new axioms. Where Cantor was after the infinite, F'rege was after the finite. To reduce thenatural numbers, Frege had to define Dedekind's primitive notions. Zero is thenumber of things not identical with themselves, and (for present purposes) thesuccessor of n is the number of numbers less than or equal to n. The infinitenumbers show that the natural numbers do not exhaust the cardinal numbers,so Frege needed to separate the natural numbers from the cardinals. He definesthem as the members of all sets of which zero is a member and the successor of amember is always a member. This definition makes the analyticity of mathematicalinduction go down all too smoothly. F'rege's and Cantor's projects both founder in paradox.25 To reduce the mathe-matics of natural numbers to logic, Frege had to beef up logic. Axiom V of his logicsays that all and only the objects falling under one concept fall under another justin case the value ranges of those concepts are identical. This requires that everyconcept have a value range, or in more familiar terms, that every predicate havean extension, the set of all and only the things of which the predicate is true. Thislast is often called comprehension, as if a predicate comprehends, or collects, anextension. Traditional logic was shot through with talk of extensions of concepts,so Frege's V could easily seem at home in logic. But thinking about Cantor's diag-onal argument, Russell wondered about the extension of the non-self-membershippredicate. The set of all lions is not a lion, but the set of all sets is a set. Rus-sell's set R collects all sets like the first, that is, all those that are not membersof themselves. But then R is a member of R if and only if R is not a memberof R. This reasoning is known as Russell's paradox. Russell sent it to Frege in aletter in 1902. Frege lived until 1925, but to my mind nothing he did after 1902 25All three original presentations of the paradoxes of set theory occur in van Heijenoort'sSourcebook, cited in the previous note.

18 W. D. Hartmeasures up to what he did before. In Cantor's case, consider the universe, theset U of absolutely anything. The power set of U is, by Cantor's theorem, largerthan U, but it is also a subset of U , and so smaller than or equal in size to U.This reasoning is known as Cantor's paradox, and Cantor sent it t o Dedekind in aletter in 1895. There is also a third paradox of set theory about ordinal numbersand known as the Burali-Forth paradox after the man who published it in 1897. To those who take the figure of foundations of mathematics seriously, the para-doxes of set theory are a crisis. But any crisis was philosophical. Frege's axiomV is not analytic. It is not necessarily true, and it is not known a priori. It isnone of these things because it is just plain false. Russell's paradox is a proof byreductio ad absurdum that it is false. If someone says more or less out of the blue that the prime minister of Estoniais at this very moment seated rather than standing, it seems sensible to withholdjudgment at least for a bit. Evidence may be forthcoming, but it hasn't yet. Butthere are claims where, as it were, saying is believing. Playfair's version of theaxiom of parallels says that given a line L in a plane and a point P in the planenot on L, there is one and only one line in the plane through P parallel to L.People largely innocent of geometry typically agree to Playfair's axiom withoutargument. The same acquiescence typically greets Euclid's axiom that the wholeis larger than any of its (proper) parts. Back when set theory was only na'ive, thesame acquiescence was also accorded assuming that for every predicate there is aset of all and only those things of which the predicate is true. We might try torehabilitate an abused notion by taking intuitions as beliefs for which we perhapsshould have justification but do not and yet nevertheless hold.26 On this account,intuition is not some faculty whose exercisejustifies belief and yields knowledge inmysterious ways. On the contrary, intuitions are beliefs held without justificationbut held anyway. Suppose a person holds a belief without justification. Suppose he is asked notwhether this belief is true but why he believes it. Often such a person responds bydigging in his heels and repeating his intuition with increasing assurance. Supposenow that everyone else (or almost everyone else) who considers the issue agreeswith him, and no one (or almost no one) disagrees or withholds judgment. What isemerging may be the sociology of analyticity and a priori knowledge. If so, thesenotions are table pounding in fancy dress: Necessary truth may differ somewhatbecause claims that a truth is necessary can be tested against fertile imaginations.With claims that a truth is analytic one may get nothing but question-beggingsemantic claims, and with claims that a truth is known a priori one usually getsnothing at all. The case is worse when we are expected to believe that a claim is true becauseit is analytic, necessary, or known a priori. Axioms lack proof, at least in thesystems of which they are axioms. If the axiom is not independent of the othersbut deducible from them, then the question just shifts over to the axioms from 26Compare W. V. Quine, Word and Object (Cambridge: Technology Press of the MassachusettsInstitute of Technology, and London and New York: John Wiley and Sons, 1960), p. 36 fn.

Les Liaisons Dangereuses 19which it is deduced. Lacking proof means lacking proof, not having some superbut ineffable proof. So even if, like everyone else, I believe it, I still am better off ifI acknowledge that I just do not have a proof of it. Kant thought Playfair's axiomwas synthetic a priori, Frege thought comprehension was analytic, and Galileoappealed to Euclid's axiom that the whole is greater than the part to deny theactual infinite. Our best bet nowadays is that all three axioms are false. A shrewdphilosopher is wary when an interesting claim is called analytic, necessary, or apriori, since often the really interesting stuff is to be broached by working out howthe claim could be false. When set theory was nayve, it assumed only comprehension (every predicatehas an extension) and extensionality (sets with the same members are identi-~ a l ) .O~nc~e comprehension had been refuted, set theory needed reform. It isstriking that Russell, who refuted comprehension, never quite gave it up. Insteadhe denied meaning to the claim that the set of all lions is not a member of itself.He converted the two-way split between truth and falsity into a three-way splitbetween these two and meaninglessness. For any predicate counted as meaningfulby his theory of types, he allowed himself to form a singular term denoting its ex-tension, but his allegiance to comprehension was so implicit that he never statedthat all meaningful predicates have extensions. Russell's focus on meaning, andespecially his denial of meaning to claims we have no trouble understanding, is inno small way responsible for the coming of philosophy of language. Comparingthe claim that the set of all lions is not a member of itself with Lewis Carroll's linethat 'twas brillig and the slithy toves did gyre and gimbel in the wabe, we mightwant to ask an analogue of Prichard's questionz8 whether moral philosophy restson a mistake. When Russell was working out Principia Mathernatica with Whitehead, hismain opponent in the journals was Henri PoincarB. PoincarB7swell-deserved pres-tige in France coupled with his hostility to the new logic made that logic unpopularin France for decades, which was a real loss to logic. But however much Russelland PoincarQ differed, they agreed in assimilating what we now in hindsight dis-tinguish as the paradoxes of set theory to the semantic paradoxes. These last areillustrated by a sentence which says of itself that it is false, for if it is false, it hasthe property it ascribes to itself and so is true, while if it is true, then because itsays it is false, it must be false. The analogy was between self-membership andself-reference in Russell's paradox and the liar paradox. We owe the distinctionbetween the two sorts of paradox to R a m ~ e ~ .H' ~e said logic need not addressthe semantic paradoxes, and can solve the other by arranging things in types withnon-sets on the bottom, and sets of things of type n in type n f1. It is strikingthat only a few years later Tarski3' proposed to solve the semantic paradoxes with \"See Paul R. Halmos, Naive Set theory (Princeton: D. Van Nostrand, 1960). '*H. A. Prichard, \"Does Moral Philosophy Rest on a Mistake?\" in Moral Obligation (Oxford:Clarendon Press, 1949). 2gFrank Ramsey, \"The Foundations of Mathematics,\" in The Foundations of Mathematics, ed.R. B. Braithewaite (Paterson, N.J.: Littlefield, Adams, 1960). 30Alfred Tarski, \"The Concept of Truth in Formalized Languages\" cited in note 3 above.

20 W. D. Hartlevels of language in which talk not about language is on the bottom, and talk+about language of level n is conducted in language of level n 1. From a suffi-ciently abstract point of view, the similarity between Ramsey's simplified typesand Tarski's levels of language makes one wonder whether Russell and Poincar6were on to a good thing. It is a basic tenet of the theory of types that no proposition may say anythingabout itself. Nor could there be a claim, say C, about absolutely everything, sincethen C would have to be about C too, and that would violate the first tenet.Let K be the claim that no claim is about absolutely everything. It follows fromK that K is not about absolutely everything, so it certainly looks as though thetheory of types violates itself. Such self-destruction is surprisingly frequent amongphilosophical theories, and it is certainly a weapon that we want to remain in thephilosopher's critical armory. We want, for example, to be able t o fault a theoryT of theories that gives a good account of all theories except T , and any line likethe theory of types that purports to resist such self-applications should for thatreason be resisted. In Principia Russell had to reconcile two conflicting objectives. On the onehand he had to weaken logic enough to prevent the derivation of paradox, but onthe other hand he had to keep logic strong enough for the mathematics of numberto be reducible to it and to keep open the gates to Cantor's paradise. To do so, heneeded four assumptions that bothered him, or should have. He recognized thathe needed axioms of choice, infinity and reducibility, and he used his version ofcomprehension even if he did not quite recognize it explicitly. For the philosopher itmatters right now less what these assumptions say than that they are assumptionsof existence. Russell was not comfortable with the line that existence propositionscan be analytic. Indeed he looks to toy with inverting Frege's syllogism: sincemathematics is synthetic and mathematics is reducible to logic, logic is synthetic.In the preface to Principia he said that the chief reason in favor of a theoryon the principles of mathematics must always be i n d ~ c t i v e .B~y~this he meantthat instead of the conventional mathematical wisdom being justified by deductionfrom the so-called foundations of mathematics, the foundation is justified only if itsuffices for the deduction of the conventional mathematical wisdom. This inversionis an important moment in the critique of the philosopher's exaggeration of themathematician's axiomatic method, even if it was not properly appreciated for awhile. But eventually Q ~ i n weil~l s~ay that no statement is any more intrinsicallya postulate than is a point in Ohio intrinsically a starting point. From Playfair'saxiom we can show that if two angles of one triangle equal two of another, thetriangles are similar. But if we replace Playfair's axiom with this theorem, then 31Bertrand Russell and Alfred North Whitehead, Principia Mathernatica, 2nd ed. (Cambridge:Cambridge University Press, 1925), vol. I, p. v. 3 2 ~V.. Quine, \"Two Dogmas of Empiricism,\" in From a Logical Point of View, 2nd ed. (Cambridge: Harvard University Press, 1961), p. 35. Frege and Quine express extreme viewsabout foundations of knowledge. There are probably pairs of claims on which there would be aconsensus as t o which is epistemically more basic. But we do not have a settled body of principlesof epistemic priority.

Les Liaisons Dangereuses 21we can deduce Playfair's axiom.33 While Russell's theory of types was the reform of set theory best known amongphilosophers, it was not much favored among mathematicians. It is hideous andprobably immune to thorough understanding. Hilbert's student Zermelo inaugu-rated in 1908 what has become pretty much received set theory.34 He had torestrict comprehension, so he assumed instead that for every definite property Pand any set x, there is a set of all those members of x having P. This is calledseparation since it separates out the members of x with P. There is no settledmathematical consensus on properties, and about a decade later Frankel refor-mulated separation in terms of the predicates of a first-order formal language.35Separation is weak enough that it needs supplementing by existence assumptions.The more interesting of these include infinity (there is an infinite set), union (givena set, there is a union of all its members), and the big one, power set (every sethas a power set). Skolem added replacement, which says that if the domain of afunction is a set, so is its range.36 Kripke pictures sets as corrals, and we mightsimilarly picture them as lassos, except that extensionality would forbid two lassosroping the same set. Functions can be pictured as collections of arrows from thedomain to the range. If we have lassoed the domain, slide the lasso along thearrows to lasso the range. Zermelo-Frankel set theory, or ZF, had emerged by the early 1920s, thoughin justice it should be called ZFS set theory to give credit to Skolem. In 1929von Neumann worked with a structure in which there are no infinite sequencesX I , 52, ... of sets such that x,+l is a member of x,, that is, there are noinfinite descending membership chains.37 By around 1960 this assumption hadbeen incorporated into ZF. Indeed in those days some set theorists leaned on thisassumption, called foundation, and some remarks of Godel's in the 1 9 4 0 ~t,o~ ~disparage some of Quine's work on comparative set theory.39 Consensus createsbullies. Foundation does make set theoretic life easier, but if we explore its denialwe can model self-referential propositions.40 If the proposition that Socrates isbald is the ordered pair whose first member is Socrates and whose second memberis the set of bald men, we can say the proposition is true if its first member is 33George David Birkhoff and Ralph Beatley, Basic Geometry, 3Td ed. (New York: ChelseaPublishing Company, 1959). 3 4 ~ r n s tZermelo, \"Investigations in the Foundations of Set Theory I,\" in van Heijenoort'sSourcebook, cited in note 24. See note 36 also. 35Abraham A. Fraenkel, \"The Notion 'Definite' and the Independence of the Axiom of Choice,\"in van Heijenoort's Sourcebook, cited in note 24. 3 6 ~ h o r a l fSkolem, \"Some Remarks on Axiomatized Set Theory,\" in van Heijenoort's Source-book, cited in note 24. This address is a major contribution. 3 7 ~ o h nvon Neumann, \"Ueber eine Widerspruchsfreiheitsfrage in der axiomatischen Mengen-lehre,\" Journal fur reine und angewandte Mathematik 160 (1929), 227-41. 3sSee note 17. 39W. V. Quine, Set Theory and Its Logic, rev. ed. (Cambridge, Mass.: Belknap Press, 1969). 40See Peter Aczel, Non-Well-Founded Sets (Stanford, Cal.: Center for t h e Study of Languageand Information, 1988,) Lecture Notes No. 14, and Jon Barwise and John Etchemendy, T h eLiar (Oxford: Oxford University Press, 1987).

22 W . D. Harta member of its second. A proposition that says of itself that it is a propositioncould then be an ordered pair p whose first member is p and whose second is theset of propositions. Foundation is neither analytic nor known a priori. Zermelo's axiom of separation turns Russell's paradox into a proof that theuniverse does not exist. For if there were a set of which absolutely everything, oreven just every set, were a member, the predicate for non-self-membership wouldseparate Russell's paradoxical set from it. So \"is self-identical\" and \"is a set\"are predicates that do not have extensions. This does not mean that they haveempty extensions, since then they would have extensions, albeit empty ones. Itmeans that there is no set of all those things identical with themselves, and thereis no set of all sets. Nor does any set have a complement whose members are thethings not in the set, for otherwise the union of a set with its complement wouldgive us back the universe. Where Russell and PoincarC blamed the paradoxes onself-membership and its ilk like self-reference, ZF shies away from collections itdeems too big to exist. One might call the loss of the universe the revenge of thepotential infinite. By replacement, no collection as big as the universe can exist,and this rules out a set of all (cardinal) numbers. Suppose there were a set N ofall cardinal numbers. For every cardinal k, the set N(k) of ordinals with fewerthan k predecessors is a set of cardinality k, so we have a function that maps thecardinals one-to-one into the sets. On the other hand, for every set A there is acardinal n(A) which is the number of members of A. (In ZF we use the axiom ofchoice t o show this, but set theory would be in a sorry state without it, so often weassume ZFC, which is ZF with the axiom of choice. Choice says that for any set ofnon-empty sets there is a function whose value at each of these sets is a memberof it. Russell observed that while from each of an infinity of pairs of shoes we canpick the right shoe, we need choice to pick from an infinity of pairs of socks.41Since Zermelo first articulated choice in 1904, there has been controversy aboutit. In the 1920s, for example, Tarski and Banach used choice to show that anysphere the size of a pea can be cut into no more than four pieces that, without theinflation of topology, can be reassembled into a sphere the size of the sun.)42 Giventhat every set has a unique number of members, we have a function mapping thesets one-to-one into the cardinals. It follows (by the Schroder-Bernstein theorem)that there are exactly as many sets as cardinals, so \"is a cardinal number\" hasno extension either. Note also that if N existed, it should have a number n ofmembers. Since N(k) is always a subset of N , each k is less than or equal t o n.But by Cantor's theorem, the power set of N would have a number of membersbigger than n. ZF dodges (the usual proofs of) Cantor's paradox by not assertingthe existence of the universe or N . Not accepting the universe has consequences for logic. Frege and Peirce liber-ated logic by isolating the quantifiers, all and some, from the applied quantifiers, all 41Bertrand Russell, Introduction to Mathematical Philosophy (London: George Allen andUnwin, 1919),p. 126. Frankel's proof o f t h e independence o f the axiom o f choice in t h e papermentioned in note 35 above is a formalization o f Russell's observation. 42Stan Wagon, T h e Banch-Tarski Paradox (Cambridge: Cambridge University Press, 1985).

Les Liaisons Dangereuses 23As and some As, of Aristotle's syllogistic. This made quantification into polyadicpredicates possible (everything is R to something), and that is the life's blood ofthe new logic. One should expect some fraternization between the universe andthe universal quantifier. For example, in the (admittedly peculiar) set theory in\"New Foundations,\" Quine accepts the universe and explains universal quantifica-tion by saying that everything is F if and only if the set of F s is the universe.43Without the universe, universal quantification will be limited. We mostly followTarski in using models (or structures) to interpret languages.44 A model is a settheoretic object with a domain required to be a (non-empty) set. The predicatesof the language are interpreted by assigning them extensions (of suitable polyadic-ity) in the domain, the constants, by assigning them members of the domain, andmutatis mutandis for the function signs. The universal quantifier is interpretedas all members of the domain. (Some go so far as to infer from this dependenceof quantification on the choice of domain that the quantifiers are not logical con-stants.) Suppose we want to say that everything is self-identical. Not acceptingthe universe means that we are restricted to saying that every member of thisor that set taken as domain of a model is self-identical. An axiom of ZF assuresus that everything is a member of a set, namely, the set whose only member isthat thing. This is called its unit set or singleton. So for everything there is amodel whose domain has that thing as a member, and we can use that model tosay that thing is self-identical. Tarski offers us a notion of validity under whicha claim is valid if it comes out true under all interpretations in all models, so bysaying the claim that everything is self-identical is valid we might seem to havean indirect way of saying what we originally wanted to say. But then we wonderhow t o interpret the universal quantifiers in \"Everything is a member of a set\"or \"Such and such a claim comes out true under all interpretations in all mod-els.\" We meant absolutely everything by \"everything,\" and we are not allowed to.There are as many models as sets, so we are not allowed to quantify over them allat once. Neither Russell's theory of types nor ZF accepts the universe, and thisrestraint trips logic up enough to make us look for ways to dodge the paradoxesof set theory while hanging on to the universe.45 Frege took zero as the number of things not identical with themselves, andon our set theoretic construal of his reduction, this turns out to be the unit setof the empty set, the set with no members. Then one, the successor of zero,is the number of numbers less than or equal t o zero, so since only zero is lessthan or equal to zero, one is the set of all unit sets. But since everything hasa unit set, the set of unit sets is the same size as the universe. So the numberone does not exist, and neither do any other numbers except zero. It would bea distinct embarrassment for a reduction of mathematics to logic if it turned out 43W. V. Quine, \"New Foundations for Mathematical Logic,\" in From a Logical Point of View,2nd ed. (Cambridge: Harvard University Press, 1961), p. 94. 44Alfred Tarski, \"On the Concept of Logical Consequence,\" in Logic, Semantics, and Meta-mathematics, trans. J . H . Woodger (Oxford: Clarendon Press, 1956). 45The late Radl Orayen often stressed this difficulty.

24 W. D. Hartthat zero is the only cardinal number. Remember Russell's insight that insteadof the conventional mathematical wisdom being justified by deduction from theso-called foundations of mathematics, the foundation is justified only if it sufficesfor the deduction of the conventional mathematical wisdom. So how are we tosave the numbers? John von Neumann did so by selecting from each of Frege'sequivalence classes a standard representative.46 Since Frege's zero is the unit setof the empty set, von Neumann's zero is the empty set. Then von Neumanntakes the successor of n to be the union of n with its unit set. Thus each n hasn members. So where Frege took a set A's having n members as A's being amember of n, von Neumann takes it as A's being the same size as n. Zermeloalso took zero as the empty set, but he took the successor of n as its unit set.This works for the natural numbers too, and it has been argued that becausethere is no unique right way to reduce numbers to sets, numbers are not sets afterall. Well, maybe. But von Neumann's finite cardinals generalize beautifully intothe infinite while Zermelo's do not. Enthusiasm for von Neumann's constructiontempts one to think he discovered which sets the cardinal numbers really are. Itis as if Cantor's nineteenth-century theory of transfinite cardinals reveals whichsets the prime numbers Euclid proved to be infinite in number really are (thoughit has not yet helped to figure out the distribution of the primes). Logicism like Frege's is a kind of platonism. Platonism is the metaphysical viewthat there are non-mental, non-physical abstract objects like numbers that do notdepend on us or anything mental or physical for their existence. Plato's theory offorms was the first recorded metaphysical platonism. Very abstract objects likenumbers or platonic forms are utterly inert, so since, as G r i ~ aerg~ue~s, perceptionis by its nature causal, there is an epistemological problem for empiricists abouthow we might justify belief in and about abstract objects. Plato struggles with thisproblem in the Symposium, the Meno, and the Republic. Set theory can be viewedas a contemporary version of the theory of forms. Both forms and sets answer topredicates, and comparing Russell's paradox with the third man argument showsthat as set theory has a problem with self-membership, the theory of forms hasa problem with self-participation. One difference may be that set theory hasan explicit commitment to extensionality, an issue Plato does not seem to haveaddressed. We should be clear that even sets of concrete objects are not to beidentified with the physical aggregates of' the members. An example of Quine'spoints out that while the set of states in the US and the set of counties in the UShave the same physical aggregates, they have no members in common and so aredifferent sets.48 It may seem curious that while logicism is a form of platonism, Frege held thatlogic and mathematics are analytic. The curiosity is most acute for those content 46J. Donald Monk, Introduction to Set Theory (New York: McGraw-Hill, 1969), has a goodexposition of von Neumann's theory. 4 7 ~P. Grice, \"The Causal Theory of Perception,\" reprinted in Perceiving, Sensing and Know- ing, ed. Robert J . Swartz (New York: Doubleday, 1965). 4 8 ~ V.. Quine, Mathematical Logic, rev. ed. (New York: Harper Torchbooks, 1962), p. 120.

Les Liaisons Dangereuses 25with the quick gloss of analyticity as truth by virtue of meaning. If one takes truthby meaning as an alternative to truth by correspondence to fact, the analyticityof mathematics looks like a way to deny that the theorem that there are infinitelymany primes is true only if there are infinitely many primes, and so numbers, andso t o avoid platonism. For Ayer in Language, Truth and Logic, a virtue of theanalyticity of the a priori is the exit it offers to p l a t ~ n i s m . ~ ~ For Frege, of course, analyticity is not truth by meaning but derivability fromlogic, and his logic is pretty platonic. But platonism makes some people nervous;they may be letting their ontological (what are the basic sorts of things?) viewbe governed tacitly by the prevailing naturalist cosmology (how are the basic bitsunified into the world?) put by Hume as the idea that causation is the cement ofthe univer~e.~F' or if abstract objects like numbers are utterly inert, how couldcausation cement them t o the mental and physical stuff in the system of the world?Maybe membership is another cement of the world, not only binding sets in asystem, but also binding the concrete to the abstract as in sets of concrete things.Then reducing mathematical objects to sets would have a unifying cosmologicalad~antage.~' But suppose one favors naturalism. Medieval nominalists like William of Occamrejected platonic forms and universals in favor of linguistic items like predicates,and in the philosophy of mathematics those who favor signs like numerals over ab-stracta like numbers are sometimes called formalists. (After the paradoxes, Hilbertinaugurated a program to prove the consistency of reformed mathematical systems.This was also called formalism since it took these systems as systems of notation.But Hilbert thought consistency proofs would help justify belief in abstracta. Hewas not a metaphysical formalist.) If numerals are to have the epistemic appeal ofconcreta, they should be actual physical inscriptions and utterances, and as Fregeobserved long ago,52there will never be enough inscriptions and utterances to doduty for the infinity of natural numbers. One move often made at this point isto replace actual inscriptions with possible ones, or with set theoretically definedsequences of actual inscriptions, and then the epistemic appeal of numerals beginsto fade.53 But truth is the crux. Formalists often take truth in mathematics to be notcorrespondence t o fact but provability. Proof looks t o be a linguistic activity andthe favored way mathematicians settle mathematical questions. Of course, if proofis deduction from axioms, there will arise a question about how axioms are settled, 49Alfred Jules Ayer, Language, T m t h and Logic, 2nd ed. (New York: Dover, 1946),chap. IV. 5 0 ~ h pehrase comes from David Hume's abstract of A Deatise of Human Nature, ed. L. A.Selby-Bigge, 2nd ed., ed. P. H. Nidditch (Oxford: Oxford University Press, 1978), p. 662, asJohn L. Mackie made plain in The Cement of the Universe: A Study of Causation (Oxford:Oxford University Press, 1974). 5 1 ~ h Q~usine sees set theory a s unifying rather than founding mathematics. 52Gottlob Frege, \"Frege Against the Formalists,\" in Translations from the Philosophical Writ-ings of Gottlob FI-ege,trans. Peter Geach and Max Black (Oxford: Blackwell, 1960),p. 222. 53See W. V. Quine, \"Ontological Relativity,\" in Ontological Relativity and Other Essays (NewYork: Columbia University Press, 1969),pp. 41-42.

26 W. D. Hartbut let us bracket that question. For there is a juicier confrontation between theformalist conception of mathematical truth as provability and Godel's Incomplete-ness Theorem (1931). Let us sketch what the theorem says informally. Considerany system of proof with three properties. First, it should be consistent. (Ac-tually, Godel used a stronger property in 1931, but in 1936 Rosser showed thatconsistency suffices.) Given that contradictions are false, any system in whichproof suffices for truth should be consistent. Second, it should encode calculationson natural numbers, like addition and multiplication, for which there are algo-+rithms. A due respect for the conventional mathematical wisdom urges encoding,for example, 7 5 = 12. Third, there should be an algorithm for whether a patchof discourse in the language of the system is or is not a proof. Church argues thatthe point of proof is to settle mathematical questions decisively and it cannot doso unless there is such an algorithm.54 Then, for any such system we can writedown a sentence in the language of the system such that neither it nor its negationis provable in the system. Since, by the law of the excluded middle, one of theseis true, there is an unprovable truth of the system. Philosophy is probably supple enough for any of its interesting theses t o dodgeknock-down refutation by a precise scientific result. Godel's first incompletenesstheorem may knock down formalism, but perhaps the formalist can get up again.One maneuver might be to stretch the requirement for an algorithm for whatcounts as a proof by allowing some infinitely large proofs. If we allow inferences ofa universally quantified sentence from all its instances, then any truth with enoughinstances will be provable but proofs will in general be infinitely large and ourepistemic access to them will be as mathematical a s our grasp of other infinities.A palliative here might be to allow ourselves only infinite patches of proofs thatcan be grasped by finite algorithms or acceptable generalizations of such. JonBarwise wrote an excellent exposition of some ways of doing this in Admissible Setsand S t r ~ c t u r e sb, ~ut~as yet there is no received systematic philosophical critiqueof such expanded proof in print. Going second order is another stratagem. Ina first-order system one quantifies only into subject position or other positionsin a sentence occupied by a singular term, while in a second-order system onemay also quantify into predicate position. Dedekind wrote down a single second-order sentence from which every truth of elementary number theory follows, soa formalist might try saying that being a truth of elementary number theory isfollowing from Dedekind's sentence. The rub is in the logic. For first order systems,Godel (and others) showed that we can write down rules of inference such that forany premisses, anything that follows from those premisses may actually be deducedfrom them by the rules, and conversely. In a first-order system any sentencethat follows from a set of premises can be deduced from them; this is what thecompleteness of first-order logic says. But completeness fails second-order. So even 54Alonzo Church, Introduction to Mathematical Logic, vol. I (Princeton: Princeton UniversityPress, 1956), pp. 50 R. 55Jon Barwise, Admissable Sets and Strzlctures (Berlin, Heidelberg, New York: Springer-Verlag, 1975).

Les Liaisons Dangereuses 27though a truth of elementary number theory follows from Dedekind's sentence,showing that it does is just as much a mathematical problem as any other. Weare not guaranteed proofs to which we have epistemic access that would obviateplatonism. Godel himself suggested another maneuver.56 His theorem is about formalizeddeductive theories. Maybe his theorem really shows that genuine proof in mediasmathematical res eludes thoroughgoing formalization. Consider our favorite suchformal system S and add to the language of S a unary sentence operator D (withgrammar like negation) such that for any sentence p in the language of S, D p isintended to mean that it is provable (in the full-blooded mathematical way) thatp. We should have enough confidence in S t o add to it a rule allowing us to inferfrom any theorem p of S to Dp, and we should have enough confidence in proofto add to S an axiom saying that if Dp, then p. If we write the dash for negationand the ampersand for conjunction, it is then not difficult to show thatis a theorem of S. This says that it is absolutely provable that a contradiction isnot absolutely provable, that is, absolute provability proves its own consistency.But Godel's second incompleteness theorem shows that no system of proof sat-isfying the three conditions stated above proves its own consistency. From thisone might infer that the absolute provability the formalist wants is not proof ina formalized system. On the other hand one might also wonder just what suchabsolute provability looks like, and how to be sure one has recognized it correctly.Such uncertainty may drain some of the epistemic appeal from formalism. Godel'ssecond incompleteness theorem also looks to impede Hilbert's program for provingthe consistency of reformed mathematical systems, since especially for strong sys-tems like set theory, the second incompleteness theorem seems to say that provingthe consistency of the system requires methods stronger than those of the system,thus depriving the consistency proof of its epistemic purpose. For now, formalismis still down, though for all we know it will rise again. The history of philosophyis not of solving problems but of extending the dialectic they generate. Godel's first incompleteness theorem shows that mathematical truth outstripsprovability. But by how much? The third condition on systems to which it appliesrequires that the relation that holds when and only when a stretch of discourse isa proof of a statement be decidable, that is, that there is an algorithm for whethersome discourse is or is not a proof of a sentence. The property of theoremhoodarises from this relation by existential quantification since a sentence is a theorem ifand only if there is a proof of it. Suppose the system S in question is a typical first-order formalization of elementary number theory, and assume S consistent. Wecan show S is not decidable, that is, there is no algorithm for theoremhood in S. Soprovability is one quantifier away from decidability. Godel proved incompleteness 5 6 K ~ r Gt odel, \"An Interpretation of the Intuitionistic Propositional Calculus,\" in CollectedWorks, vol. I., ed. Solomon Federman, John Dawson, Stephen Kleene, Gregory Moore, RobertSolovay, and Jean van Heijenoort (New York: Oxford University Press, 1986), pp. 301-03.

28 W. D. Hartby arithmetizing (or godelizing) syntax. This involves a function g with threeproperties. First, g maps expressions in the language of S (like sentences andproofs) one-to-one into the natural numbers. Next, g is computable, that it, thereis an algorithm that given an expression e , yields the number g assigns t o e. Last,g is effective backwards. This requires two algorithms, one that given a naturalnumber n settles whether g assigns an expression to n, and a second that, if gassigns an expression to n, finds it. There are infinitely many different ways t ogodelize theories like S. The number g assigns e is called its godel number. Sincethe set of theorems of S is not decidable, neither is the set of godel numbers oftheorems. So an undecidable set of numbers is one quantifier from decidability. Let A and B be sets of natural numbers. Turing showed how to reconstruemathematically the subjunctive that if there were an algorithm for membershipin A, then there would be one for membership in B. In that case we say A is(Turing) reducible to B. If A is reducible to B and B is reducible to A, then theirmembership problems are of equal difficulty, and they are (Turing) equivalent.This is an equivalence relation, and it partitions the sets of natural numbers intothe degrees of (Turing) unsolvabilty. One degree is higher than another if a problemin the second is reducible to one in the first but not vice versa. Let do be the degreeof godel numbers of theorems of S. We can show that there is a sequence do,d l . . .of degrees such that for all n,dn+l is higher than d, and there is in d , a relationamong natural numbers that is the extension of a formula from the language ofS in the usual model of S and has n quantifiers in its prenex. That is, eachadditional quantifier we allow increases the degree of unsolvability of relations wecan express with formulae in the language of S . We can also show that the setof godel numbers of sentences of S true in the usual model is undecidable. Let dbe its degree. Then d is higher than all of do,d l . . .. So truth in number theoryis infinitely many degrees, or quantifiers, higher than provability in that theory.Truth outstrips provability by a good bit.57 A first-order deductive theory is axiomatic if there is an algorithm for whethera formula in the language of the theory is or is not an axiom of it. At leastprima facie, only the axiomatic theories are of epistemic interest, for only thereare we sure of algorithms for whether a patch of discourse is or is not a proof ofa sentence, and that, as Church argued, is necessary for proof to satisfy its raisond'6tre of settling questions decisively. By 1944, many axiomatic theories had beenshown undecidable, and Post noted that any two of them were of the same degreeof unsolvability (the one we called do). He asked whether any two undecidableaxiomatic theories are of the same degree.58 This is called Post's Problem, and itgave the part of logic called recursion theory a life of its own. In the mid-1950s, ayoung American, Friedberg, and a young Russian, Muchnik, discovered indepen- 57Compare Hartley Rogers Jr., Theory of Recursive Functions and Effective Computability(New York: McGraw-Hill, 1967), sec. 14.7. 58Emil Post, \"Recursively Enumerable Sets of Positive Integers and their Decision Problems,\"reprinted in T h e Undecidable, ed. Martin Davis (Hewlitt, N.Y.: Raven Press, 1965), pp. 304-37.This address is another major contribution.

Les Liaisons Dangereuses 29dently a method called the priority method that suffices t o solve Post's Problem.There are undecidable axiomatic theories of different degrees of unsolvability. Sothe structure of these degrees becomes of at least mathematical interest. Sacks, forexample, showed that they are dense, that is, there are sequences of such degreessuch that between any two there is a third.5g But such degrees could be of more than mathematical interest only. A miner-alogist can learn some things just by looking at rocks, but to figure out chemicalformulae, for example, he will probably use more indirect methods; how does therock interact with this or that acid, for example. When a cosmologist or astro-physicist is looking into the curvature of spacetime, he does not pick up a handfulof spacetime and measure how bent it is. He looks instead to discriminating ob-servable effects of such curvature. In psychology, introspection is not much trustedas a way to examine reason. So one strategy would be t o look to discriminatingobservable products of reason. Since the Greeks invented or discovered reason, de-ductive theories have been among its most salient products. A critique of reasonmight seek structures occupied by such theories. A critique of pure reason mightlook to structures of all possible such theories. It would be fun if recursion theorycould provide a denotation for Kant's title. On the mathematical side, logicism developed out of the arithmetization ofanalysis, and analytic geometry was a crucial link in arithmetizing the curves ofanalysis. It was basic to analytic geometry that there be a one-to-one correspon-dence between the real numbers and the points of a line that preserves order (soless-to-greater goes to left-to-right). But even if the space around us is flat, howdo we know such a correspondence exists? Could the lines in space actually needmore numbers as coordinates? Think back to learning what a derivative is. You are looking at the parabolathat is the curve for the square function, x2, and you want to know the slope ofthis curve above the abscissa x. Slope is rise over run, so you imagine increasing x+ +by tiny amount dx. The new square is x2 2xdx ( d ~ )so~th, e rise is 2xdx+ ( d ~ ) ~+when the run is dx. Over means division so the slope is 22 dx, and to get theslope at x itself you want this increment dx to be so small it can be ignored, whichwould put the slope at 22, the passing answer on your calculus test. This argument feels dodgy. To divide by dx it should not be zero, but t o ignore+it in 22 dx is to treat it as indistinguishable from zero. So dx is one of the noto-rious infinitesimals of the calculus invented or discovered by Newton and Leibnizindependently. Berkeley made fun of infinitesimals in The Analyst as ghosts ofdeparted quantities. For anything we have said so far, infinitesimals do seem likean embarrassment, and yet the calculus is crucial in making Newton's physics thefirst real natural science. Part of what people like Cauchy and Weierstrass didduring the arithmetization of analysis was to replace infinitesimals with the E - 6methods notorious in elementary calculus classes. Then a number a is the deriva-tive of a function f at an argument x if for any positive E however small there is a 59See Robert I. Soare, Recursively Enumerable Sets and Degrees (New York: Springer-Verlag,1987).

30 W. D. Hartpositive b small enough that for arguments y within distance S of x, f (y) is withindistance E of f (x). So we can avoid infinitesimals. But should we? In the 1960s Abraham Robinsondiscovered that we need not.60 Let M be a structure, or model, whose domainis the set R of the (familiar) real numbers. The distinguished individuals of Mare all the members of R, the relations, all those of R, and mutatis mutandis forfunctions. A language of which M is an interpretation is pretty big, but it canstill be first order. One unary predicate N(x) of such a language would have asextension the set of all natural numbers, and a binary predicate x = y would havethe identity relation as extension. Let T' be the set of all sentences of this languagetrue in M . Add to the language a single new constant a , and form T by addingto T all ofwhere Q, 1,... are constants of the language denoting the natural numbers in R.For any finite subset of T', let n be the largest natural number such that the+sentence a # 24 is a member of this set. Then letting a denote n 1 makes all ofthis subset true in M . So by the compactness of first-order languages, T' has amodel M' whose domain R' is the same size as R. Let b be the denotation of a in+R'. Then b has to obey all the laws of natural numbers in T , so b cannot be lessthan zero, nor can it lie between n and n 1, nor can it be any of O,1,2,. . .. Butit must be comparable in size to all of O,1,2,. .., so it can only be greater than allof them. Nor is it alone out there since each natural number other than zero has apredecessor, and each natural number has a successor. But none of b - n can be+a natural k since then b = k n, contrary t o a sentence in T' true in M'. So b is+ +surrounded by a clump . ..,b - 2, b - 1,b, b 1,b 2,. . . all of whose members lieabove 0,1,2,. . .. Nor is there a least clump above 0, 1,.. .. For b is odd or even.Suppose it is even; a similar argument works if it is odd. Then b = 2k for somek. As before if k = b - n for some n , k + n + b = 2k, so k = n and then b = 2n,contrary to a sentence in T' true in MI; nor is k among O,1,. . .. Moreover, given+a clump around c and another around d, assume c d is even, and the clump+around c d/2 lies wholly between the two given clumps. So the natural numbersof M' start with a copy of the usual, or standard, natural numbers. These arefollowed by the clumps, any one of which looks like the integers (negative, zero,positive) in usual order, and the clumps are ordered like the rationals or reals (adense linear order without top or bottom). The standard negative numbers looklike the standard positive integers in reverse order, so the integers of M' look likea dense linear order without endpoints, but of clumps. Since b is not zero, it has a reciprocal lib, and since b is greater than allof 1,2,3,.. ., its reciprocal l / b is less than all of 1,1/2,1/3,. ... But since b is GoseeRobert Goldblatt, Lectures o n the Hyperreals (New York: Springer-Verlag, 1998). Hisexposition is more in the mathematical mainstream than ours, and both differ from AbrahamRobinson, Non-standard Analysis (Princeton: Princeton University Press, 1996).

Les Liaisons Dangereuses 31positive, so is l / b , so it is greater than 0. Hence l / b is greater than 0 but lessthan all of 1,1/2,1/3,. . .. So l / b is infinitesimal. We found a ghost of departedfractions. In secondary school most of us thought of real numbers in terms of decimals.Let a digit be one of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Picture the set Iof integers shrinking in size to the left and increasing to the right. A sequence ofdigits is a function f whose domain is I, whose values are digits, and such that forsome integer n, f (m) = 0 for all m < n, and such that for no integer n is f (m) = 9for all m > n (this last so we do not have two decimals, like .250.. . and ,249.. ., fori).any real, like Then for any real r there is a unique sequence f of digits such thatr is the sum of f(n)/lOn for all integers n. This can be formalized as a sentenceof T true in M.\" So it is also true in MI, but in M' there are more integers, andsequences of digits are longer. The members of R' are called the hyperreals. Letr be a hyperreal and let f be its sequence of digits. If f ( n ) is positive for somenon-standard negative integer n, then r is infinite. If for all negative n and allstandard non-negative n, f (n) is zero, then r is infinitesimal. For a standard realr in R, let T be its name in the language M interprets, and let r*be the hyperrealin R' denoted by in MI. Then we think of r as the standard part of r*. Twohyperreals r and s are close if their difference is infinitesimal. Being close is anequivalence relation, so it partitions the hyperreals into equivalence classes. Theequivalence class of a given hyperreal is called its halo; it is a set of hyperrealsclose to the given hyperreal. In familiar expositions of the calculus we say that a standard real a is the limitof a sequence an of standard reals if for any positive E however small there is a ksuch that for n greater than or equal to k, the distance between a, and a is lessthan E. This is equivalent to saying that for infinite n, a, is within the halo of a,or a, is close to a. (Here we slip back and forth between a and a*.) To say that ais the derivative of f at x is equivalent to saying that when y and x are close butdiffer, then the rise from f (x)to f (y) divided by the run from x to y is in the haloof a. When hyperreals are close but differ, they have the same standard part, andthe standard part of their difference is zero, but their difference is not zero, andso is an acceptable divisor. That is how the hyperreals avoid the embarrassmentof older arguments with infinitesimals. For any standard reals r and s such that r is less than s , the halo of r* is boundedabove by s* but has no least upper bound among the hyperreals. So no one-to-one correspondence between the standard reals and the hyperreals preserves order.Hence, the standard reals and the hyperreals cannot both be cartesian coordinatesfor the points on a line. What would space be like if the hyperreals were the rightcoordinates? Pick a standard measure. This could be a meter or a yard stick, but it couldalso be an angstrom, or a lightyear stick. If we lay off our stick a first time to the 611n the language of M we can name the unique binary function that assigns t o each real Tand each integer n the digit f (n), as we called it, in the decimal for T . So we need not say thatfor each T there is such a sequence f of digits, which would be second order.

32 W. D. Hartleft from here, then a second, and so on through all and only the standard naturalnumbers, then do the same to the right, in front, in back, up and down, then wewill have laid off axes that suffice to locate any point in space as we supposed itback in school solid geometry before we heard of curvature or hyperreals. But ifthe hyperreals are the right coordinates, we will only have measured out a clumpof hyperreals along each axis, and this blob will hardly be all of space. So if the hyperreals are the right coordinates, space partitions into infinitelylarge blobs each with three clumps of hyperreals as its coordinate axes. A goldnugget that exactly fills out one such blob would use up all of our familiar space inschool solid geometry. But if the hyperreals are the right coordinates, there mayfor all we know be such a nugget some way off (in fact an infinite number of miles)to the west. Could we tell? On Newton's theory of gravity, bodies attract oneanother instantaneously, so if two bodies popped into being simultaneously and farapart, there would be no lag time before each attracted the other gravitationally.(If gravity acted by sending a pull to the attracted body, then on Newton's theorythe pull would have to be in more than one place at once, so if nothing can bein more than one place at a time, Newton's gravity has to act at a distance, thatis, without transmission.) On Newton's theory, then, we would have been pulledhard to the west long ago by that nugget in a super gold rush. But on Einstein'stheory, gravitational attraction travels at the speed of light, which is finite. Let cbe the speed of light in miles per hour. If that nugget lies d miles to the west, buthas existed for fewer than d / c hours, then the nugget's gravitational pull has notyet had time t o seize us. Since d is infinite and the age of the universe is finite,we are safe for a while. Infinitesimal, and infinite hyperreals, are probably too small in the first case andtoo big in the second for spatiotemporal and physical phenomena of those sizes tobe obvious to middle-sized wet goods like us. But it would be too parochial andpositivist t o infer from that alone that the standard reals are the right coordinatesfor space. Besides, ingenious experiments and observations may detect what is notobvious. Whether the standard reals or the hyperreals (or some other numbers)are the right coordinates for space is settled by the nature of space. Space, notpeople, picks which numbers work as its coordinates, and space is a matter of fact,not a relation of ideas. It is nether analytic nor known a priori how to arithmetizespace. Both the standard reals and the hyperreals exist, or at least a clear-headedPlatonist should be happy to grant that both exist. A mathematician should befree to study either, as he is free to study complex numbers or quaternions.62 Butconsider how our study of the standard reals emerged historically from our studyof geometry. One wonders whether settling whether the standard reals or the 6 2 ~ h e ries a n interesting sense in which the reals, the complex numbers and t h e quaternionsexhaust an important subject. See I. N. Herstein, Tbpics in Algebra (New York: BlaisdellPublishing Company, 1964), chap. 7, sec. 3. But the reals are applied all over t h e place, andthe complex numbers are applied in many places, while the quaternions are pretty much just acuriosity and comparatively neglected.

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