Intuition and the Axiomatic Method PDF
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INTUITION AND THE AXIOMATIC METHOD THE WESTERN ONTARIO SERIES IN PHILOSOPHYOF SCIENCE ASERIES OF BOOKS IN PHILOSOPHYOF SCIENCE, METHODOLOGY, EPISTEMOLOGY, LOGIC, HISTORYOF SCIENCE, AND RELATED FIELDS Managing Editor WILLIAM DEMOPOULOS Department of Philosophy, University of Western Ontario, Canada Department of Logic and Philosophy of Science, University of Californina/Irvine Managing Editor1980–1997 ROBERTE. BUTTS Late, Department of Philosophy, University of Western Ontario, Canada Editorial Board JOHN L. BELL,University of Western Ontario JEFFREY BUB,University of Maryland PETER CLARK,St Andrews University DAVID DEVIDI,University of Waterloo ROBERT DiSALLE,University of Western Ontario MICHAEL FRIEDMAN,Indiana University MICHAEL HALLETT,McGill University WILLIAM HARPER,University of Western Ontario CLIFFORD A. HOOKER,University of Newcastle AUSONIO MARRAS,University of Western Ontario JÜRGEN MITTELSTRASS,Universität Konstanz JOHN M. NICHOLAS,University of Western Ontario ITAMAR PITOWSKY,Hebrew University VOLUME 70 INTUITION AND THE AXIOMATIC METHOD Edited by EMILY CARSON McGill University, Montreal, Canada and RENATE HUBER University of Dortmund, Germany AC.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4039-3 (HB) ISBN-13 978-1-4020-4039-9 (HB) ISBN-10 1-4020-4040-7 (e-book) ISBN-13 978-1-4020-4040-5 (e-book) Published by Springer, P.O. Box 17, 3300 AADordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved © 2006 Springer No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Printed in the Netherlands. Contents Preface vii Acknowledgements xiii PartI MathematicalAspects LockeandKantonMathematicalKnowledge 3 ’ EmilyCarson TheViewfrom1763: KantontheArithmeticalMethodbeforeIntuition 21 OfraRechter TheRelationofLogicandIntuitioninKant’sPhilosophyofScience, ParticularlyGeometry 47 UlrichMajer EdmundHusserlontheApplicabilityofFormalGeometry 67 Rene´ Jagnow TheNeo-FregeanPrograminthePhilosophyofArithmetic 87 WilliamDemopoulos Go¨del,RealismandMathematical‘Intuition’ 113 MichaelHallett Intuition,ObjectivityandStructure 133 ElaineLandry PartII PhysicalAspects IntuitionandCosmology: ThePuzzleofIncongruentCounterparts 157 BrigitteFalkenburg ConventionalismandModernPhysics: aRe-Assessment 181 RobertDiSalle v vi IntuitionandtheAxiomaticMethod IntuitionandtheAxiomaticMethodinHilbert’sFoundationofPhysics 213 UlrichMajer,TilmanSauer SoftAxiomatisation: JohnvonNeumannonMethod andvonNeumann’sMethodinthePhysicalSciences 235 Miklo´sRe´dei,MichaelSto¨ltzner TheIntuitivenessandTruthofModernPhysics 251 PeterMittelstaedt FunctionsofIntuitioninQuantumPhysics 267 BrigitteFalkenburg IntuitiveCognitionandtheFormationofTheories 293 RenateHuber Preface Followingdevelopmentsinmoderngeometry,logicandphysics,manysci- entists and philosophers in the modern era considered Kant’s theory of in- tuition to be obsolete. Frege’s and Russell’s logicism seemed to go against Kant’sclaimthatmathematicsisbasedonsyntheticapriorijudgments. Inany case, according to Russell, the sole reason Kant introduced intuition into his philosophy of mathematics and physics was that he did not have available to him modern logic, i.e., the quantified logic of relations. Following this view, the articulation of the new logic thus rendered Kantian intuition redundant. Moreover, the enormous expansion of modern mathematics in the nineteenth century, an expansion accelerated by the development of the modern abstract theoryofsets,appearedtotakemathematicsoutofthereachofKantian‘sen- sible’intuition. Agooddealofthismathematicswasalsousedinvariousways inmathematicalphysics. Eveninarelativelyconcretecase,namelygeometry, sensible intuition seemed incapable of deciding between alternative geomet- rical systems. Many mathematicians and physicists also adopted a version of the ‘axiomatic method’, derived from Hilbert’s work on the foundations of geometry, which allowed that mathematical theories need not have a unique ‘content’foundedinlogicorintuitionorempiricaltheories,butthataxiomati- callypresentedsystemsarefreetobeinterpretedasmathematicalorscientific requirementsdictate. Thus,thereisnouniquelycorrectgeometricaltheoryof space. Moreover, Reichenbach and Carnap in particular argued that special and general relativity disprove Kant’s theories of space and time. Reichen- bach and Carnap were two of the most prominent representatives of logical empiricism, which owed much to the work of Frege and Russell. The rise of logicalempiricismsawtheentrenchmentoftheseviewsofKant’sphilosophy ofmathematicsandscience. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physi- cists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics withsatisfactoryfoundations. AllofHilbert,Go¨del,Poincare´,WeylandBohr thought that intuition was an indispensable element in describing the founda- tionsofscience. Theyhadverydifferentreasonsforthinkingthis,andtheyhad verydifferentaccountsofwhattheycalledintuition. Buttheyhadincommon vii viii IntuitionandtheAxiomaticMethod that their views of mathematics and physics were significantly influenced by theirreadingsofKant. Inthepresentvolume,variousviewsofintuitionandtheaxiomaticmethod (and their combination) are explored, beginning with Kant’s own approach. Bywayoftheseinvestigations,wehopetounderstandbettertherationalebe- hindKant’stheoryofintuition,aswellastograspmanyfacetsoftherelations betweentheoriesofintuitionandtheaxiomaticmethod,dealingwithboththe strengths and the limitations of the latter; in short, the volume covers logi- cal and non-logical, historical and systematic issues in both mathematics and physics,andalsoviewsbothsympatheticto,aswellascriticalof,Kant’sown account and use of intuition. It goes without saying that this collection repre- sentsonlyamodeststepinunderstandingthefullimpactwhichKant’stheory ofintuitionhadonthedevelopmentoftheexactsciences. Part I of this volume deals with the mathematical aspects of the relations betweenintuitionandtheaxiomaticmethod,PartIIwiththephysicalaspects; the volume thus falls naturally into two parts, although the separation cannot beexact. BothpartsbeginwithdetailedinvestigationsofKant’sownviewsand ofthelimitationsofwhatwenowcalltheaxiomaticmethod, andoftheways in which intuition is needed to overcome such limitations. The contributions shed light on modern views of these Kantian topics in the context of modern logic,mathematicsandphysics. The contributions to Part I deal with Kant’s theory of geometry and arith- metic, with modern interpretations of Kant’s reasons for introducing intuition into the foundations of mathematics, with Husserl’s and Go¨del’s views of the role of intuition in mathematics, and with neo-Fregean logicism and category theory as programmes to replace intuition by the use of formal tools. The distinct approaches show that the role of intuition in mathematics is far from beinguncontroversial. EmilyCarsonsetsthestagebyconsideringatraditionalcorrelateoftheax- iomaticmethodasitispresentedbyLockeandthepre-CriticalKantinorderto showhowKant’scentralnotionofpureintuitionfillsincertainepistemologi- calgapsinLocke’sandtheearlyKant’saccountsofmathematicalknowledge. Ofra Rechter undertakes to shed light on Kant’s Critical philosophy of arith- metic by examining his discussion of the symbolic method of arithmetic in the Prize Essay of 1763. This elucidation of the connection between arith- meticaldefinitionsandarithmeticalsymbolismsservestoclarifythenotionof symbolic construction which Kant distinguishes from the ostensive construc- tionsofgeometry,andthustoclarifythenotionofconstructioninintuitionin general. Ulrich Majer makes the bridge to modern interpretations of Kant’s philosophyofmathematics. Inparticular,hedistinguishesthelogicalandphe- nomenological approaches to interpreting Kant’s theory of intuition. Accord- ingtotheformer,Kantmustappealtointuitioninhisaccountofmathematics because of his restricted conception of logic. According to the latter, how- ever,theappealtointuitionasanon-logicalsourceofknowledgeisnecessary PREFACE ix regardless of the type of logic available. Majer appeals to Hilbert’s work on the foundations of geometry to argue in favour of the phenomenological ap- proach. Rene´ JagnowextendstheanalysisofintuitiontoHusserl’saccountof geometry,whichdistinguishesbetweenformal,geometric,andintuitivespace (thelatterbeingthespaceofeverydayexperience). Thetaskhere,giventhese distinctions, is to account for the possibility that the results of the analysis of formal space apply to the space of geometry, and that the results of geometry apply to intuitive space. Jagnow outlines Husserl’s account and argues that its central aim was to guarantee the conceptual continuity between these dif- ferent notions of space. The applicability of formal inquiry to intuitive space ensures that the former expresses a genuine concept of space. William De- mopoulosaddressestheideaoffoundingarithmeticonsecond-orderlogicand ‘Hume’sPrinciple’. Inthemodernneo-Fregeanprogramme,Hume’sPrinciple isrepresentedastruebystipulation. Demopoulosarguesthatweshouldreject this claim, that it is rather a substantive truth, one that provides the basis for a successful conceptual analysis of our notion of number, deriving the num- bers’ theoretically most salient property from the principle underlying their application. This is a partial (but only partial) vindication of Frege’s original project of showing that our knowledge of number has the character of logical knowledge. Amongotherthings,itavoidstheassumptionthatthenumbersare ‘given in intuition’. Michael Hallett discusses the respects in which Go¨del’s famous appeal to intuition is based on ideas in Kant. Go¨del thought Kant’s notion of sensible intuition too restrictive for understanding modern mathe- matical knowledge. What he takes from Kant is rather the idea that there is anunderlyingconceptionofphysicalobjectthroughwhichperceptionisinter- preted. Usingthisanalogy,Go¨delclaimsthatthereisanotionofmathematical object through which we interpret mathematical ‘facts’; this notion is given by the iterative concept of set, described by axiomatic set theory. The funda- mental incompletenesses of mathematics means the description is essentially open-ended. ButGo¨delthinksthatthe‘discovery’oflargecardinalaxiomsex- tending the iterative hierarchy (thus closing mathematical incompletenesses) represent an unfolding of this concept. Finally, Elaine Landry sketches cat- egory theory as a modern mathematical tool that might replace the functions which Kant attributed to intuition in our knowledge of objects, namely the schemataofformalconcepts. Part II complements Part I with investigations on the role of intuition in modernphysics. ThecontributionsfocusonKant’spre-Criticalworriesabout thefoundationsofphysicalgeometryandcosmology,ongeneralrelativityand quantumtheory,onconceptualanalysisandconventionalism,onHilbert’sand von Neumann’s views of the axiomatic foundations of physics, and on recent viewsoftheroleofintuitioninmodernphysics. Brigitte Falkenburg investigates how Kant’s theory of intuition emerged from his use of the analytical method in cosmology. Conceptual analysis of space convinced Kant that the geometrical properties of incongruent counter- partsareatoddswithLeibniz’srelationalaccountofspace,althoughhealways
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