INTUITION AND THE AXIOMATIC METHOD

Transcription

1 INTUITION AND THE AXIOMATIC METHOD

2 THE WESTERN ONTARIO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS IN PHILOSOPHY OF SCIENCE, METHODOLOGY, EPISTEMOLOGY, LOGIC, HISTORY OF 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 Editor ROBERT E. 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

3 INTUITION AND THE AXIOMATIC METHOD Edited by EMILY CARSON McGill University, Montreal, Canada and RENATE HUBER University of Dortmund, Germany

4 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN (HB) ISBN (HB) ISBN (e-book) ISBN (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. 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.

5 Contents Preface vii Acknowledgements xiii Part I Mathematical Aspects Locke and Kant on Mathematical Knowledge 3 Emily Carson The View from 1763: Kant on the Arithmetical Method before Intuition 21 Ofra Rechter The Relation of Logic and Intuition in Kant s Philosophy of Science, Particularly Geometry 47 Ulrich Majer Edmund Husserl on the Applicability of Formal Geometry 67 René Jagnow The Neo-Fregean Program in the Philosophy of Arithmetic 87 William Demopoulos Gödel, Realism and Mathematical Intuition 113 Michael Hallett Intuition, Objectivity and Structure 133 Elaine Landry Part II Physical Aspects Intuition and Cosmology: The Puzzle of Incongruent Counterparts 157 Brigitte Falkenburg Conventionalism and Modern Physics: a Re-Assessment 181 Robert DiSalle v

6 vi Intuition and the Axiomatic Method Intuition and the Axiomatic Method in Hilbert s Foundation of Physics 213 Ulrich Majer, Tilman Sauer Soft Axiomatisation: John von Neumann on Method and von Neumann s Method in the Physical Sciences 235 Miklós Rédei, Michael Stöltzner The Intuitiveness and Truth of Modern Physics 251 Peter Mittelstaedt Functions of Intuition in Quantum Physics 267 Brigitte Falkenburg Intuitive Cognition and the Formation of Theories 293 Renate Huber

7 Preface Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kant s theory of intuition to be obsolete. Frege s and Russell s logicism seemed to go against Kant s claim that mathematics is based on synthetic a priori judgments. In any 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 theory of sets, appeared to take mathematics out of the reach of Kantian sensible intuition. A good deal of this mathematics was also used in various ways in mathematical physics. Even in a relatively concrete case, namely geometry, sensible intuition seemed incapable of deciding between alternative geometrical 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 founded in logic or intuition or empirical theories, but that axiomatically presented systems are free to be interpreted as mathematical or scientific requirements dictate. Thus, there is no uniquely correct geometrical theory of space. Moreover, Reichenbach and Carnap in particular argued that special and general relativity disprove Kant s theories of space and time. Reichenbach 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 logical empiricism saw the entrenchment of these views of Kant s philosophy of mathematics and science. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, Gödel, Poincaré, Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common vii

8 viii Intuition and the Axiomatic Method that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method (and their combination) are explored, beginning with Kant s own approach. By way of these investigations, we hope to understand better the rationale behind Kant s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both the strengths and the limitations of the latter; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics, and also views both sympathetic to, as well as critical of, Kant s own account and use of intuition. It goes without saying that this collection represents only a modest step in understanding the full impact which Kant s theory of intuition had on the development of the exact sciences. Part I of this volume deals with the mathematical aspects of the relations between intuition and the axiomatic method, Part II with the physical aspects; the volume thus falls naturally into two parts, although the separation cannot be exact. Both parts begin with detailed investigations of Kant s own views and of the limitations of what we now call the axiomatic method, and of the ways 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, mathematics and physics. The contributions to Part I deal with Kant s theory of geometry and arithmetic, with modern interpretations of Kant s reasons for introducing intuition into the foundations of mathematics, with Husserl s and Gödel 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 being uncontroversial. Emily Carson sets the stage by considering a traditional correlate of the axiomatic method as it is presented by Locke and the pre-critical Kant in order to show how Kant s central notion of pure intuition fills in certain epistemological gaps in Locke s and the early Kant s accounts of mathematical knowledge. Ofra Rechter undertakes to shed light on Kant s Critical philosophy of arithmetic by examining his discussion of the symbolic method of arithmetic in the Prize Essay of This elucidation of the connection between arithmetical definitions and arithmetical symbolisms serves to clarify the notion of symbolic construction which Kant distinguishes from the ostensive constructions of geometry, and thus to clarify the notion of construction in intuition in general. Ulrich Majer makes the bridge to modern interpretations of Kant s philosophy of mathematics. In particular, he distinguishes the logical and phenomenological approaches to interpreting Kant s theory of intuition. According to the former, Kant must appeal to intuition in his account of mathematics because of his restricted conception of logic. According to the latter, however, the appeal to intuition as a non-logical source of knowledge is necessary

9 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 approach. René Jagnow extends the analysis of intuition to Husserl s account of geometry, which distinguishes between formal, geometric, and intuitive space (the latter being the space of everyday experience). The task here, given these 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 different notions of space. The applicability of formal inquiry to intuitive space ensures that the former expresses a genuine concept of space. William Demopoulos addresses the idea of founding arithmetic on second-order logic and Hume s Principle. In the modern neo-fregean programme, Hume s Principle is represented as true by stipulation. Demopoulos argues that we should reject 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 numbers 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. Among other things, it avoids the assumption that the numbers are given in intuition. Michael Hallett discusses the respects in which Gödel s famous appeal to intuition is based on ideas in Kant. Gödel thought Kant s notion of sensible intuition too restrictive for understanding modern mathematical knowledge. What he takes from Kant is rather the idea that there is an underlying conception of physical object through which perception is interpreted. Using this analogy, Gödel claims that there is a notion of mathematical object through which we interpret mathematical facts ; this notion is given by the iterative concept of set, described by axiomatic set theory. The fundamental incompletenesses of mathematics means the description is essentially open-ended. But Gödel thinks that the discovery of large cardinal axioms extending the iterative hierarchy (thus closing mathematical incompletenesses) represent an unfolding of this concept. Finally, Elaine Landry sketches category theory as a modern mathematical tool that might replace the functions which Kant attributed to intuition in our knowledge of objects, namely the schemata of formal concepts. Part II complements Part I with investigations on the role of intuition in modern physics. The contributions focus on Kant s pre-critical worries about the foundations of physical geometry and cosmology, on general relativity and quantum theory, on conceptual analysis and conventionalism, on Hilbert s and von Neumann s views of the axiomatic foundations of physics, and on recent views of the role of intuition in modern physics. 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 counterparts are at odds with Leibniz s relational account of space, although he always

10 x Intuition and the Axiomatic Method adhered to Leibniz s criticism of Newton s notion of Absolute Space. Thus, Kant faced a problem, since neither the relational nor the absolute view of space seemed tenable. It is this dilemma which the theory of spatial intuition is used to overcome. The paper also points out that the problem Kant saw with incongruent counterparts reemerges with parity violation and PCT-invariance. Robert DiSalle analyses the conventionalist view of physical geometry. This arose from the confrontation between the Kantian synthetic a priori and two nineteenth century insights: (i) there is no unique set of a priori conditions of the possibility of spatio-temporal experience; and (ii) geometric principles encapsulate the meanings of fundamental concepts, having thus the character of definitions rather than of synthetic claims about nature. Conventionalists inferred that such principles are matters for free choice, guided by pragmatic rather than by epistemic considerations. DiSalle rejects this inference; he suggests that the definitions expressed by a priori principles in physics are not chosen for convenience, but arise from conceptual analysis of empirical knowledge and practice. This partially explains why such concepts appeared (e.g., to Kant) to be founded in intuition, and why the separation of physical geometry from intuition in modern physics did not separate its fundamental concepts from empirical knowledge. DiSalle concludes that the conventionalism of the logical empiricists, while rightly emphasizing the role of a priori constitutive principles in science, failed to appreciate the empiricist motivations that such principles embody. Ulrich Majer and Tilman Sauer investigate Hilbert s view of the axiomatic method in physics and his understanding of Kant s a priori; they interpret the relation between the a priori and the historical development of physics through a recursive epistemology. According to Hilbert s account of scientific knowledge, our a priori assumptions about the world are inevitably permeated by anthropomorphic elements which have to be reduced as much as possible; as a consequence, the proportion of the a priori in our overall knowledge of the world shrinks in the course of development of natural science. Miklós Rédei and Michael Stöltzner examine the way in which von Neumann applied the axiomatic method to physics, especially in his famous axiomatization of quantum mechanics. Among some physicists, von Neumann s method is highly regarded, and is often compared to a kind of mathematical perception. Others, however, consider it to be too pedantic. Indeed von Neumann himself distinguished between the strict, formal axiomatics of mathematics and a less formal, soft axiomatization which is required in physics. Brigitte Falkenburg applies central ideas of Kant s theory of intuition to Bohr s and Heisenberg s views of quantum theory. There is no unique axiomatic basis for quantum physics; formal quantum theory has to be supplemented with appeal to semi-classical models and measurement theories. Heisenberg s generalized correspondence principle establishes semantic bridges between the classical and the quantum domains, and these bridges can be seen as based on intuition in a Kantian sense. Peter Mittelstaedt argues that our Kantian intuition of physical objects, which is tailored to a classical, Newtonian world, is in fact

11 PREFACE xi too highly structured with respect to the notions of simultaneity and substance. He suggests a weaker, non-kantian account of intuition according to which neither relativity theory nor quantum theory appear as unintuitive. He also emphasizes the point that classical mechanics itself is very unintuitive when seen from the perspective of everyday experience. The collection is completed by a new, consciously non-kantian theory of intuition developed by Renate Huber. Her theory of intuition is based on an empirically grounded theory of knowledge supported by neuroscience. It sheds new light on various traditional philosophical views of intuition, and it permits the distinction between several kinds of direct or indirect intuitions. These distinctions then give rise to distinctions between directly and indirectly intuitive theories, the latter in contradistinction to non-intuitive theories. The new approach is illustrated with several examples from physics and is also then applied to Poincaré s use of intuition in mathematics. The contributions to this volume emerged from a three-year collaboration between Emily Carson, Robert DiSalle, Brigitte Falkenburg, Michael Hallett, Renate Huber and Ulrich Majer. Our project of investigating various facets of intuition and the axiomatic method as they appear in Kant s work, in mathematics and in physics was generously supported by the GAAC (German- American Academic Council, TransCoop Program) from 1998 to 2000, and in various ways by the Social Sciences and Humanities Research Council of Canada. In addition to individual research contacts between the collaborators, a series of workshops and conferences was held, in Dortmund (1998 and 2000), Göttingen (1999) and Montreal (1999). Through these in particular, we learned a great deal, not just from each other, but also from the other invited participants. This volume represents reworkings of some of the papers presented. The Editors of the volume, Renate Huber and Emily Carson, were responsible for its final realization, and exhibited much patience and tolerance with the unwritten, written, and then rewritten versions of the papers. William Demopoulos proposed the publication of the book in the Western Ontario Series in the Philosophy of Science, and has shown great support throughout. Tobias Fox was responsible for most of the technical editing work using LATEX, and Dirk Schlimm dealt with the final round of corrections. Enormous thanks are due to all of them. Dortmund and Montreal, February 2004 Brigitte Falkenburg and Michael Hallett

12 Acknowledgements The Editors wish to thank the following: The Journal of the History of Philosophy and the British Journal for the History of Philosophy for permission to use material from the papers by Emily Carson, Kant on the Method of Mathematics, Journal of the History of Philosophy, Volume 37 (1999), pp , and Locke s Account of Certain and Demonstrative Knowledge, British Journal for the History of Philosophy, Volume 10 (2002), pp [ The Notre Dame Journal of Formal Logic and Philosophical Books for permission to use material from the papers by William Demopoulos, On the Origin and Status of our Conception of Number, in Notre Dame Journal of Formal Logic, Volume 41 (2000), pp , and On the Philosophical Interest of Frege Arithmetic, in Philosophical Books, Volume 44 (2003), pp The journal Topoi for permission to use material from the paper by Elaine Landry, Logicism, Structuralism and Objectivity, Topoi, Volume 20 (2001), pp The journal Noûs and Blackwell Publishing for permission to reprint the article Conventionalism and Modern Physics: a Re-Assessment by Robert DiSalle, which originally appeared in Noûs, Volume 36 (2002), pp xiii