Western Ontario Series in Philosophy of Science – serie

The idea to produce the current volume was conceived by Jiirgen Mittelstrass and Robert E. Butts in 1978. Idealist philosophers are wrong about one thing: the temporal gap separating idea and reality can be very long indeed - even ten or so years! Problems of timing were joined by personal problems and by the pressure of other professional commitments. Fortunately, James Brown agreed to cooperate in the editing of the volume; the infusion of his usual energy, good judgement and good-natured promptness saved the volume and made its produc­ tion possible. Despite the delays, the messages of the papers included in the book have not gone stale. An extremely worthwhile exercise in international philosophical cooperation has come to fruition; the German constructivist philosophical position is here represented in papers in English that will make its contemporary importance available to a larger audience. The editors owe thanks to many persons. All involved in the project owe much to the interest and support of Nicholas Rescher, a friend of the undertaking from the time of its inception. My review of the translations was helped immensely by Andrea Purvis' careful copy editing of the typescript. Most of all, however, we owe gratitude and admiration for the tireless efforts on behalf of this enterprise to Jiirgen Mittelstrass.

This collection of papers has its origin in a conference held at the Uni- versity of Toronto in June of 1988. The theme of the conference was Physicalism in Mathematics: Recent Work in the Philosophy of Math- ematics. At the conference, papers were read by Geoffrey Hellman (Minnesota), Yvon Gauthier (Montreal), Michael Hallett (McGill), Hartry Field (USC), Bob Hale (Lancaster & St Andrew's), Alasdair Urquhart (Toronto) and Penelope Maddy (Irvine). This volume supplements updated versions of six of those papers with contributions by Jim Brown (Toronto), John Bigelow (La Trobe), John Burgess (Princeton), Chandler Davis (Toronto), David Papineau (Cambridge), Michael Resnik (North Carolina at Chapel Hill), Peter Simons (Salzburg) and Crispin Wright (St Andrews & Michigan). Together they provide a vivid, expansive snapshot of the exciting work which is currently being carried out in philosophy of mathematics. Generous financial support for the original conference was provided by the Social Sciences & Humanities Research Council of Canada, the British Council, and the Department of Philosophy together with the Office of Internal Relations at the University of Toronto.Additional support for the production of this volume was gratefully received from the Social Sciences & Humanities Research Council of Canada.

Incongruent counterparts are objects that are perfectly similar except for being mirror images of each other, such as left and right human hands. Immanuel Kant was the first great thinker to point out the philosophical significance of such objects. He called them "counter­ parts" because they are similar in nearly every way, "incongruent" because, despite their similarity, one could never be put in the place of the other. Three important discussions of incongruent counterparts occur in Kant's writings. The first is an article published in 1768, 'On the First Ground of the Distinction of Regions in Space', in which Kant con­ tended that incongruent counterparts furnish a refutation of Leibniz's relational theory of space and a proof of Newton's rival theory of absolute space. The second is a section of his Inaugural Dissertation, published two years later in 1770, in which he cited incongruent counterparts as showing that our knowledge of space must rest on intuitions. The third is a section of the Prolegomena to Any Future Metaphysics of 1783, in which he cited incongruent counterparts as a paradox resolvable only by his own theory of space as mind-dependent. A fourth mention in the Metaphysical Foundations of Natural Science of 1786 briefly repeats the Prolegomena point. Curiously, there is no mention of incongruent counterparts in either of the editions (1781 and 1787) of Kant's magnum opus, the Critique of Pure Reason.

I would like to record my thanks to Paul Thompson for useful conver­ sations over the years, and also to several generations of students who have helped me develop my ideas on biological theory and on Darwin. My wife has, as usual, been more than helpful; in particular she typed a good portion of the manuscript while I was on leave a few years ago, more now than I like to remember. My parents were both looking forward to holding a final copy of this book. I only regret that my mother did not live long enough to see its completion. I must also thank the publishers and their staff. They have been re­ markably patient about meeting deadlines - promises were repeatedly made and then, owing to family situations, had to be broken - and for this I am considerably in their debt. I would further like to thank the following authors and publishers for permission to use their work: R. C. Lewontin, The Genetic Basis of Evolutionary Change, Figure 1, p. 14; © 1964 Columbia University Press; reprinted here by kind permission of the author and publisher. F. Wilson, 'Goudge's Contribution to the Philosophy of Science', in L. W. Sumner, J. G. Slater, and F. Wilson (eds.), Pragmatism and Purpose: Essays in Honour of T. A. Goudge; © 1964 University of Toronto Press; reproduced here in part by kind permission of all the editors and the publisher.

Deservedly so, Jules Vuillemin is widely respected and greatly admired. It is not simply that he has produced a large body of outstanding work, in many different areas of philosophy. Or that he combines to an unusual degree rigorous standards with a very wide perspective. Or even that in his path-breaking accounts of algebra, of !)escartes, of Kant and of Russell, he showed in new and profound ways how the histories of science and philosophy could be used to illuminate each other. It is also that he has pursued the application of formal techniques and the defense of liberal institutions with a rare singlemindedness and courage. In a time and place where the former were generally ignored and the latter often attacked, he carried on, at some personal cost, embodying a traditional and ideal conception of the philosophical life, bridging national differences. Those who know him also treasure his friendship. Always curious, he delights in new facts and new experiences, and continually heightens the perception of those around him. Almost yearly, at the College de France he introduced brand new courses always with fresh and fruitful inSights. Exceptionally solicitous, he follows the lives of the families around him in great detail. The devotion of his students is legend. His personal energy is also legend. Many of us have followed him bounding up the stairs two at a time or through the gardens of the Luxembourg, his wit and irony apace.

This collection of essays is dedicated to 'Joe' Karel Lambert. The contributors are all personally affected to Joe in some way or other, but they are definitely not the only ones. Whatever excuses there are - there are some -, the editors apologize to whomever they have neglected. But even so the collection displays how influential Karel Lambert has been, personally and through his teaching and his writings. The display is in alphabetical order - with one exception: Bas van Fraassen, being about the earliest student of Karel Lambert, opens the collection with some reminiscences. Naturally, one of the focal points of this volume is Lambert's logical thinking and (or: freed of) ontological thinking. Free logic is intimately connected with description theory. Bas van Fraassen gives a survey of the development of the area, and Charles Daniels points to difficulties with definite descriptions in modal contexts and stories. Peter Woodruff addresses the relation between free logic and supervaluation semantics, presenting a novel condition which recovers desirable metatheoretic properties for free logic under that semantics. Terence Parsons shows how free logic can be utilized in interpreting sentences as purporting to denote events (true ones succeed and false ones fail) and how this helps to understand natural language.

The three spatial characteristics of length, height and depth are used in the same unreflective way by laymen, technicians and scientists alike to describe the forms, positions and measure of bodies and hollow bodies. But how do we know that the space we live in has just three dimensions? This question has occupied philosophers and scientists since antiquity. The answers proposed have become ever more presumptuous and have increasingly lost sight of everyday intuitions and have sacrificed explanatory power. In this work, Peter Janich shows that all explanations of three-dimensionality hinge on an unreflective geometrical language which seems to accept the lack of an alternative for the three sorts of entities - lines, planes and solids. This is a Euclidean heritage in a dual sense. Euclid himself adopted a geometrical language from the art of figure drawing, and left a tradition of doing geometry as planimetry and of doing steremetry by rotating plane figures. The systematic approach offered here starts out from operational definitions of the spatial forms - plane, straight edge and perpendicularity - and proofs that only three planes can intersect pairwise orthogonally.This is the constructive solution in the frame theory of action, providing an unequivocal characterization of spatial relations in the physical world. The traditional order of geometric concepts turns out to be the most important obstacle to the methodical ordering of everyday scientific concepts.

Music and Science in the Age of Galileo features twelve new essays by leading specialists in the fields of musicology, history of science, astronomy, philosophy, and instrument building that explore the relations between music and the scientific culture of Galileo's time. The essays take a broad historical approach towards understanding such topics as the role of music in Galileo's experiments and in the scientific revolution, the musical formation of scientists, Galileo's impact on the art and music of his time, the scientific knowledge of instrument builders, and the scientific experiments and cultural context of Galileo's father, Vincenzo Galilei. This volume opens up new areas in both musicology and the history of science, and twists together various strands of parallel work by musicians and scientists on Galileo and his time. This book will be of interest to musicologists, historians of science and those interested in interdisciplinary perspectives of the late Renaissance -- early Baroque. For its variety of approaches, it will be a valuable collection of readings for graduate students, and those seeking a more integrated approach to historical problems. The book will be of interest to historians of science, philosophers, musicologists, astronomers, and mathematicians.

Earlier in this century, many philosophers of science (for example, Rudolf Carnap) drew a fairly sharp distinction between theory and observation, between theoretical terms like 'mass' and 'electron', and observation terms like 'measures three meters in length' and 'is _2° Celsius'. By simply looking at our instruments we can ascertain what numbers our measurements yield. Creatures like mass are different: we determine mass by calculation; we never directly observe a mass. Nor an electron: this term is introduced in order to explain what we observe. This (once standard) distinction between theory and observation was eventually found to be wanting. First, if the distinction holds, it is difficult to see what can characterize the relationship between theory :md observation. How can theoretical terms explain that which is itself in no way theorized? The second point leads out of the first: are not the instruments that provide us with observational material themselves creatures of theory? Is it really possible to have an observation language that is entirely barren of theory? The theory-Iadenness of observation languages is now an accept­ ed feature of the logic of science. Many regard such dependence of observation on theory as a virtue. If our instruments of observation do not derive their meaning from theories, whence comes that meaning? Surely - in science - we have nothing else but theories to tell us what to try to observe.

From the mid-1960s, after the important works by J. Hintikka, S. Körner, W. Sellars and P.F. Strawson, there has been a marked revival of Kantian epistemological thought. Against this background, featuring fruitful exchange between historical research and theoretical prospects, the main point of the book is the discussion of Kantian theory of scientific knowledge from the perspective of present-day analytical philosophy and philosophy of empirical and mathematical sciences. The main topics are the problem of a priori knowledge in logic, mathematics and physics, the distinction between analytic and synthetic judgments, the constitution of physical objectivity and the questions of realism and truth, the Kantian conception of time, causal laws and induction, the relations between Kantian epistemological thought, relativity theory, quantum theory and some recent developments of philosophy of science. The book is addressed to research workers, specialists and scholars in the fields of epistemology, philosophy of science and history of philosophy.

The unusual ambition of this volume is to engage scientists, historians, and philosophers in a common quest to delineate the structure of the creative thinking responsible for major advances in physical theory. The topic does not fit anyone discipline's proprietary interests, and can only be pursued cooperatively. This volume was conceived in the hope that the importance of learning something general about how theories are developed and what makes the difference between productive and abortive directions of theo­ retical inquiry could overcome well-known barriers to such cooperation. The volume originated in a conference held at the University of North Carolina, Greensboro in 1988, as an installment of the annual Greensboro Symposium in Philosophy. Most of the papers descend from papers pre­ sented on that occasion. The authors are well known in their own disciplines, but should be identified to the wider audience for interdisciplinary work in science studies. Rafael Sorkin, of Syracuse University, and Don Page, of the University of Alberta, are theoretical physicists who have done research in quantum gravity and cosmology. John Stachel, a physicist at Boston University, is widely known as the Director of the Einstein Project and editor of Einstein's papers. William Harper, a historian of science and philosopher at the University of Western Ontario, is a Newton scholar and specialist in decision theory.

In my early years I was constituted in the exacting imperatives of philosophical analysis. That stern face is present in the composition of the Newton essays chosen here for republication. It is my hope that potential readers will be patient with the old Adam of analysis, and seize the portrait of Newton's intellec­ tual world presented in these essays. It is gratifying for me to acknowledge the encouragement of Robert Butts and John Nicholas of the University of Western Ontario, intellectual comrades in arms. It was at Western that I began my intellectual journey, and many of the present members of the Philosophy Department remain my friends and mentors. I thank also Marta Spranzi­ Zuber who long ago believed in the merit of my Newton scholarship. But most important to me is the sustaining encouragement of Professor Barbara Tuchanska, who shares my vision of the historicity of scientific thought. It is a pleasure to express my gratitude for membership, over twenty years, in the Department of History and Philosophy of Science at the University of Pittsburgh. It is the mecca for one who seeks to understand. J. E.

But to admit things not visible to the gross creatures that we are is, in my opinion, to show a decent humility, and not just a lamentable addiction to metaphysics. J. S. Bell, Are There Quantum Jumps? ON CANADIAN THANKSGIVING WEEKEND in the autumn of 1994, a lively conference was held at The University of Western Ontario under the title "Conceptual Problems of Relativistic Quantum Mechanics". Most of the eighteen papers in this volume are directly connected with that conference. Articles by both theoretical physicists and philosophers of science are included, and many authors will be recognized immediately for their already substantive work in the foundations of physics. A quarter century ago Howard Stein suggested that relativistic quantum field theory should be 'the contemporary locus of metaphysical research', but there were few takers. Only fairly recently has that changed, with the result that the bulk of the papers here pursue issues that go beyond nonrelativistic quantum mechanics (or at least have serious implications for its relativistic generalization). Nevertheless, problems interpreting the nonrelativistic theory remain a persistent thorn in the side of any such endeavor, and so some of the papers develop innovative approaches to those issues as well.

This volume, honoring the renowned historian of science, Allen G Debus, explores ideas of science - `experiences of nature' - from within a historiographical tradition that Debus has done much to define. As his work shows, the sciences do not develop exclusively as a result of a progressive and inexorable logic of discovery. A wide variety of extra-scientific factors, deriving from changing intellectual contexts and differing social millieus, play crucial roles in the overall development of scientific thought. These essays represent case studies in a broad range of scientific settings - from sixteenth-century astronomy and medicine, through nineteenth-century biology and mathematics, to the social sciences in the twentieth-century - that show the impact of both social settings and the cross-fertilization of ideas on the formation of science. Aimed at a general audience interested in the history of science, this book closes with Debus's personal perspective on the development of the field. Audience: This book will appeal especially to historians of science, of chemistry, and of medicine.

Recent debates in philosophy of science and epistemology are characterized by some strong oppositions, such as those between instrumentalism and scientific realism, methodological anarchism and critical rationalism, historical contextualism and metaphysical realism, naturalism and normativism. By means of a close interaction between historical research and theoretical perspective, this book aims to develop a "third way" between absolutism and radical relativism. This conception is called "positive" because it is linked to the basic features of the esprit positif: empiricism, moderate epistemic relativism, and non-metaphysical objectivism. This book revives the early 20th-century controversy between the immanent and transcendent conception of the object of knowledge, and offers a new interpretation of the relationships between logical positivism, Kantian thought, and conventionalist philosophy of science. According to the author, such a "positive philosophy" is able to counter the "conversational" and post-modern attacks on objectivity and truth.The book is aimed at libraries and research workers in epistemology, philosophy of science, metaphysics, history of philosophy and history of science, and philosophy of language.

According to the modal interpretation, the standard mathematical framework of quantum mechanics specifies the physical magnitudes of a system, which have definite values. Probabilities are assigned to the possible values that these magnitudes may adopt. The interpretation is thus concerned with physical properties rather than with measurement results: it is a realistic interpretation (in the sense of scientific realism). One of the notable achievements of this interpretation is that it dissolves the notorious measurement problem. The papers collected here, together with the introduction and concluding critical appraisal, explain the various forms of the modal interpretation, survey its achievements, and discuss those problems that have yet to be solved.

The essays in this volume cover a wide array of topics related to the work of Merrilee and Wesley Salmon. It includes discussions of scientific explanation, rationality, inference and induction, social behaviour, philosophy of physics and biology. Some of the essays have been written with an historical perspective. The volume is completed by a long commentary on the papers, written by Merrilee and Wesley Salmon. The book should appeal to scholars and researchers in such fields as philosophy of science, epistemology, philosophy of the natural and social sciences, history of contemporary philosophy, and inductive logic.

Philosophy of mind has been one of the most active fields in philosophy for the past three decades. One of the most significant factors in the development of this discipline has been the emergence of cognitive science and the interest philosophers have taken in the empirical study of mind. Another equally important factor has been the "naturalistic tum" brought about by W. V. Quine. His proposal that normative epistemology be replaced by empirical psychology marked a radical departure from the Fregean "anti­ psychologism" and "apriorism" that had characterized much of the analytic tradition in philosophy. But while Quine's program of naturalization called the attention of philosophers to empirical psychology, his conception of psychology was inspired by an austere behaviorism which shunned the mentalism of intentional psychology in the Brentanian and phenomenological tradition. Thus, while agreeing with Brentano that the "intentional idiom" could not be reduced to that of the natural sciences, Quine argued that it is of a piece with the indeterminacy of translation. Most contributors of this col­ lection share the cognitivist stance and believe that the mind needs to be explained rather than eliminated. Three main questions are actually confronting current philosophers of mind, each addressed by one or another of the contributors to the present collection.

A compact survey, at the elementary level, of some of the most important concepts of mathematics. Attention is paid to their technical features, historical development and broader philosophical significance. Each of the various branches of mathematics is discussed separately, but their interdependence is emphasised throughout. Certain topics - such as Greek mathematics, abstract algebra, set theory, geometry and the philosophy of mathematics - are discussed in detail. Appendices outline from scratch the proofs of two of the most celebrated limitative results of mathematics: the insolubility of the problem of doubling the cube and trisecting an arbitrary angle, and the Gödel incompleteness theorems. Additional appendices contain brief accounts of smooth infinitesimal analysis - a new approach to the use of infinitesimals in the calculus - and of the philosophical thought of the great 20th century mathematician Hermann Weyl. Readership: Students and teachers of mathematics, science and philosophy. The greater part of the book can be read and enjoyed by anyone possessing a good high school mathematics background.

Number 10 Sound: The Musical Way 10 the Scientific Revolution is a collection of twelve essays by writers from the fields of musicology and the history of science. The essays show the idea of music held by Euro­ th pean intellectuals who lived from the second half of the 15 century to the th early 17 : physicians (e. g. Marsilio Ficino), scholars of musical theory (e. g. Gioseffo Zarlino, Vincenzo Galilei), natural philosophers (e. g. Fran­ cis Bacon, Isaac Beeckman, Marin Mersenne), astronomers and mathema­ ticians (e. g. Johannes Kepler, Galileo Galilei ). Together with other people of the time, whom the Reader will meet in the course of the book, these intellectuals share an idea of music that is far removed from the way it is commonly conceived nowadays: it is the idea of music as a science whose object-musical sound--can be quantified and demonstrated, or enquired into experimentally with the methods and instruments of modem scientific enquiry. In this conception, music to be heard is a complex, variable structure based on few simple elements--e. g. musical intervals-, com­ bined according to rules and criteria which vary along with the different ages. However, the varieties of music created by men would not exist if they were not based on certain musical models--e. g. the consonances-, which exist in the mind of God or are hidden in the womb of Nature, which man discovers and demonstrates, and finally translates into the lan­ guage of sounds.

Robert E. Butts (1928-1997) was a philosopher and historian of science whose central concerns were the distinction between the rational and the irrational. He viewed scientific rationality as our major defence against the various conditions that encourage witch hunts and similar outbursts of irrationality, with all their attendant pain and terror. Butts saw himself as a pragmatic realist, combining what he took to be the best aspects of logical empiricism with a historically informed pragmatism, deeply appreciative of the methods of science, trying to describe a kind of rationality essential in the struggle to preserve human values. This volume gathers previously unpublished essays and lectures with some previously published, thematically related essays. It includes essays and lectures on philosophical aspects of the European witch hunt, on scientific rationality and methodology, and on the relationships between science and philosophy exhibited in the writings of such historically significant figures as Leibniz, D'Alembert, Hume, Kant, Carnap and Kuhn.

A compact survey, at the elementary level, of some of the most important concepts of mathematics. Attention is paid to their technical features, historical development and broader philosophical significance. Each of the various branches of mathematics is discussed separately, but their interdependence is emphasised throughout. Certain topics - such as Greek mathematics, abstract algebra, set theory, geometry and the philosophy of mathematics - are discussed in detail. Appendices outline from scratch the proofs of two of the most celebrated limitative results of mathematics: the insolubility of the problem of doubling the cube and trisecting an arbitrary angle, and the Gödel incompleteness theorems. Additional appendices contain brief accounts of smooth infinitesimal analysis - a new approach to the use of infinitesimals in the calculus - and of the philosophical thought of the great 20th century mathematician Hermann Weyl. Readership: Students and teachers of mathematics, science and philosophy. The greater part of the book can be read and enjoyed by anyone possessing a good high school mathematics background.

According to platonists, entities such as numbers, sets, propositions and properties are abstract objects. But abstract objects lack causal powers and a location in space and time, so how could we ever come to know of the existence of such impotent and remote objects? In Knowledge, Cause, and Abstract Objects, Colin Cheyne presents the first systematic and detailed account of this epistemological objection to the platonist doctrine that abstract objects exist and can be known. Since mathematics has such a central role in the acquisition of scientific knowledge, he concentrates on mathematical platonism. He also concentrates on our knowledge of what exists, and argues for a causal constraint on such existential knowledge. Finally, he exposes the weaknesses of recent attempts by platonists to account for our supposed platonic knowledge.This book will be of particular interest to researchers and advanced students of epistemology and of the philosophy of mathematics and science. It will also be of interest to all philosophers with a general interest in metaphysics and ontology.

Philosophers have studied geometry since ancient times. Geometrical knowledge has often played the role of a laboratory for the philosopher's conceptual experiments dedicated to the ideation of powerful theories of knowledge. Lorenzo Magnani's new book Philosophy and Geometry illustrates the rich intrigue of this fascinating story of human knowledge, providing a new analysis of the ideas of many scholars (including Plato, Proclus, Kant, and Poincaré), and discussing conventionalist and neopositivist perspectives and the problem of the origins of geometry. The book also ties together the concerns of philosophers of science and cognitive scientists, showing, for example, the connections between geometrical reasoning and cognition as well as the results of recent logical and computational models of geometrical reasoning. All the topics are dealt with using a novel combination of both historical and contemporary perspectives. Philosophy and Geometry is a valuable contribution to the renaissance of research in the field.

Ernst Mach (1838-1916) was a seminal philosopher-scientist and a deserving member of the canon of major twentieth-century thinkers. Yet, despite a healthy resurgence in Mach studies, he is still widely thought to represent a simplistic positivist, even sensationalist, position that does not at all reflect the depth of Mach's interests and subtlety as a philosopher. By exploring Mach's views on science as well as philosophy, this book attempts to wrest him free from his customary association with logical positivism and to reinterpret him on his own terms as a natural philosopher and naturalist about human knowledge. Mach's development and his influences from 19th century German philosophy and science are probed in great conceptual and historical detail, and attention is paid to his unpublished Nachlaß as well as to the affinities between Mach's thought and that of other major philosopher-scientists such as Einstein, Bertrand Russell, William James, Helmholtz, Riemann, Herbart and Kant. In particular, the book strives to set forth the true nature of Mach's sensation-elements, the motivations for his critique of the concepts of space and time in physics, and the real meaning of his famous critique of metaphysics. The author's work has appeared in Synthese, Kant-Studien, Studies in History and Philosophy of Modern Physics and the Journal of the History of the Behavioral Sciences, but here these inquiries are gathered into a unified historico-critical treatment that follows Mach's conceptual development and the culmination of his work in a unique and intriguing natural philosophy. Physicists, psychologists, philosophers of science, historians of twentieth-century thought and culture, and educators will find this volume a valuable help in interpreting Mach's ideas in a context that includes philosophy and science and the bridge between them.

Graham Solomon, to whom this collection is dedicated, went into hospital for antibiotic treatment of pneumonia in Oc- ber, 2001. Three days later, on Nov. 1, he died of a massive stroke, at the age of 44. Solomon was well liked by those who got the chance to know him—it was a revelation to ?nd out, when helping to sort out his a?airs after his death, how many “friends” he had whom he had actually never met, as his email included correspondence with philosophers around the world running sometimes to hundreds of messages. He was well respected in the philosophical community more broadly. He was for several years a member of the editorial board for the Western Ontario Series in Philosophy of Science. While he was employed at Wilfrid Laurier University in Waterloo, Ontario, several of us at the University of Wat- loo always regarded our own department as a sort of second academic home for him. We therefore decided that it would be appropriate to hold a memorial conference in his honour. Thanks to the generous ?nancial support of the Humphrey Conference Fund, we were able to do so in May 2003. Many of the papers in this volume were presented at that conf- ence.

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. 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 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 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 their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.

InSeptember2000theArchéCentrelauncheda ve-yearresearchprojecten- tled the Logical and Metaphysical Foundations of Classical Mathematics. Its goal was to study the prospects, philosophical and technical, for abstractionist foundations for the classical mathematical theories of the natural, real and complex numbers and standard set theory. Funding was provided by the then Arts and Humanities Research Board (now the Arts and Humanities Research Council) for the appointment of full-time postdoctoral research fellows and PhD students to collaborate with more senior colleagues in the project, and at the same time the British Academy awarded the Centre additional resources to establish an International Network of scholars to be associated with the work. This was the beginning of the serial ‘Abstraction workshops’ of which the Centre had staged no less than eleven by December 2006. We gra- fully acknowledge the generous support of the Academy and Council, sine qua non. The project seminars and Network meetings generated—and continue to generate—a large number of leading-edge research papers on all aspects of the project agenda. The present volume is the rst of what we hope will be a number of anthologies of these researches. With two exceptions,—the contribution by the late George Boolos and the co-authored paper by Gabriel Uzquiano and Ignacio Jané,—the papers that Roy Cook has collected in the present volume are all authored by sometime members of the project team or of the British Academy Network.

JeffandImetwhenIwasagraduatestudentattheUniversityofMinnesotaandhewas a post doctoral fellow, first in the Chemistry Department, and then in the Center for Philosophy of Science. Later we were colleagues atWestern Ontario. Our friendship and collaboration owe a great deal to both these institutions. In the mid-1960s the Center enjoyed great success under Feigl’s directorship. The history of the Center has been only very partially documented. Feyerabend’s recollections,reportedinhisAutobiography,andsomeyearsearlierinhisremarksfor Feigl’s Festschrift, possess an immediacy that makes them particularly noteworthy, even if all too brief. The Center was the first American institution of its kind and a bastion of positivist and neo-positivist thought. At the time Jeff and I were there, the staff included, in addition to Feigl and Maxwell, Paul Meehl, Roger Steuwer and Keith Gunderson. There were many enthusiastic graduate students, and there was participation, on occasion, from the members of the Philosophy Department, as well as the departments of physics, psychology, mathematics and chemistry. The extent to which this (to us ideal) environment was held together by the force of Feigl’s personality became evident only many years later. The political liberalism of the Viennese Positivists was very much reflected in the philosophicalatmosphereFeiglcreated,anatmospherethatwasmarkedbyopenness, collegiality and intellectual freedom. Combined with its excellent permanent faculty and steady stream of distinguished visitors, the Center was especially well-suited to Jeff’s and my early friendship, our analytic and speculative interests, and our early collaboration.This collaboration was continued when we were members of the Philosophy Department at Western Ontario.