Quine, Structure, and Ontology 0198864280, 9780198864288

Table of contents :
Contents
Contributors
1. Introduction • Frederique Janssen-Lauret
2. Quine’s Non-ontological Structuralism (and Mine) • Michael D. Resnik
3. W. V. Quine and David Lewis: Structural (Epistemological) Humility • Frederique Janssen-Lauret and Fraser MacBride
4. Quine on Ontological Commitment in Light of Predicate-Functor Logic • John Collins
5. Inscrutability of Reference as a Result of Quine’s Structuralism • Jaroslav Peregrin
6. Quine’s Deflationary Structuralism • Paul A. Gregory
7. The Ontogeny of Quine’s Ontology: Pythagoreanism, Nominalism, and the Role of Clarity • Greg Frost-Arnold
8. Quine’s Structural Holism and the Constitutive A Priori • Robert Sinclair
9. Quine on Ontology: Chapter 7 of Word and Object • Gary Kemp and Andrew Lugg
10. On What Exists • Nathan Salmón
11. Quine vs. Quine: Abstract Knowledge and Ontology • Gila Sher
12. A New Look at Quine on Set Theory • Marianna Antonutti Marfori
13. What Quine (and Carnap) Might Say about Contemporary Metaphysics of Time • Natalja Deng
Index

Citation preview

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Quine, Structure, and Ontology

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M I N D A S S O C IAT I O N O C C A S I O NA L S E R I E S This series consists of carefully selected volumes of significant original papers on predefined themes, normally growing out of a conference supported by a Mind Association Major Conference Grant. The Association nominates an editor or editors for each collection, and may cooperate with other bodies in promoting conferences or other scholarly activities in connection with the preparation of particular volumes. Director, Mind Association: Julian Dodd Publications Officer: Sarah Sawyer Recently published in the series In the Light of Experience Edited by Johan Gersel, Rasmus Thybo Jensen, Morten S. Thaning, and Søren Overgaard Evaluative Perception Edited by Anna Bergqvist and Robert Cowan Perceptual Ephemera Edited by Thomas Crowther and Clare Mac Cumhaill Common Sense in the Scottish Enlightenment Edited by C. B. Bow Art and Belief Edited by Ema Sullivan-Bissett, Helen Bradley, and Paul Noordhof The Actual and the Possible Edited by Mark Sinclair Thinking about the Emotions Edited by Alix Cohen and Robert Stern Art, Mind, and Narrative Edited by Julian Dodd The Social and Political Philosophy of Mary Wollstonecraft Edited by Sandrine Bergès and Alan Coffee The Epistemic Life of Groups Edited by Michael S. Brady and Miranda Fricker Reality Making Edited by Mark Jago The Metaphysics of Relations Edited by Anna Marmodoro and David Yates

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Quine, Structure, and Ontology edited by

Frederique Janssen-Lauret

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Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © the several contributors 2020 The moral rights of the authors have been asserted First Edition published in 2020 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020937532 ISBN 978–0–19–886428–8 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Contents Contributors

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1. Introduction Frederique Janssen-Lauret

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2. Quine’s Non-ontological Structuralism (and Mine) Michael D. Resnik

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3. W. V. Quine and David Lewis: Structural (Epistemological) Humility Frederique Janssen-Lauret and Fraser MacBride

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4. Quine on Ontological Commitment in Light of Predicate-Functor Logic John Collins

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5. Inscrutability of Reference as a Result of Quine’s Structuralism Jaroslav Peregrin

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6. Quine’s Deflationary Structuralism Paul A. Gregory

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7. The Ontogeny of Quine’s Ontology: Pythagoreanism, Nominalism, and the Role of Clarity Greg Frost-Arnold

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8. Quine’s Structural Holism and the Constitutive A Priori Robert Sinclair

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9. Quine on Ontology: Chapter 7 of Word and Object Gary Kemp and Andrew Lugg

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10. On What Exists Nathan Salmón

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11. Quine vs. Quine: Abstract Knowledge and Ontology Gila Sher

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12. A New Look at Quine on Set Theory Marianna Antonutti Marfori

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13. What Quine (and Carnap) Might Say about Contemporary Metaphysics of Time Natalja Deng

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Index

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Contributors Marianna Antonutti Marfori, Munich Centre for Mathematical Philosophy, Ludwig-Maximilians-Universität München John Collins, University of East Anglia Natalja Deng, Underwood International College, Yonsei University Greg Frost-Arnold, Hobart and William Smith Colleges Paul A. Gregory, W&L University Frederique Janssen-Lauret, University of Manchester Gary Kemp, University of Glasgow Andrew Lugg, University of Ottawa Fraser MacBride, University of Manchester Jaroslav Peregrin, Institute of Philosophy, Czech Academy of Sciences and Faculty of Philosophy, University of Hradec Kràlové Michael D. Resnik, University of North Carolina Nathan Salmón, University of California, Santa Barbara Gila Sher, University of California, San Diego Robert Sinclair, Soka University, Tokyo

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1 Introduction Frederique Janssen-Lauret

Ontology, epistemology, and the philosophy of logic and language in current analytic philosophy still owe much to the influence of Quine. Quine urged, against the positivists and especially Carnap, that language touches upon the world as a whole, not being separable into a conventional part and a referential part, and that consequently existence questions are all of a piece with empirical ones. Ontological questions are thus real, answerable questions with empirical import, not unanswerable, mere matters of stipulation, or answerable easily and trivially. Quine’s arguments for this position continue to hold the attention of many prominent philosophers, some of whom are concerned to defend them, others to rebut them. In the recent past, work in these areas has begun to take an increasingly dismissive attitude towards Quine. Commentators frequently portray him as having an overly narrow conception of metaphysics and an out-of-date perspective on the philosophy of language. But such dismissals are often themselves liable to oversimplify Quinean views. This volume, grown out of a Mind Association-funded international conference hosted by the Forum for Quine and the History of Analytic Philosophy, aims to fill the resulting gaps in the literature by collecting together a range of philosophers expert in Quine to investigate questions on the intersection of language, epistemology, and ontology, from both Quinean and well-informed critical points of view. Its chapters are centred around the rather underrated Quinean doctrine of structuralism. One source of prevalent misconceptions about Quine is a treatment of Quinean thought as a monolith, rather than a body of work which reveals substantial growth and development and often provides detailed answers to, or deftly rebuts, objections currently presented as novel and daring. Quine’s later views, especially his attempts to resolve certain tensions in his earlier works, have gone unremarked by contemporary critics. Contemporary philosophers often take for granted that certain Quinean crowd-pleasers, such as ‘On What There Is’, ‘Two Dogmas of Empiricism’, chapter 2 of Word and Object, or ‘Ontological Relativity’, represent a full statement of his final views on ontology, semantic holism, or language, interpretation, and reference. Well-worn lines from these classics are frequently quoted out of context and held Frederique Janssen-Lauret, Introduction In: Quine, Structure, and Ontology. Edited by: Frederique Janssen-Lauret, Oxford University Press (2020). © Frederique Janssen-Lauret. DOI: 10.1093/oso/9780198864288.003.0001

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2 frederique janssen-lauret up against the yardstick of a set of assumptions widely accepted among twenty-firstcentury philosophers. Such commentators forget that Quine’s approach to philosophy was always holistic and systematic, and often, openly or implicitly, set against the commonplace philosophical assumptions of his own day as well as ours. In his middle and later periods, answering his critics or modifying his positions under their influence, Quine made moves which cast his thought and his influence on analytic philosophy in a light not usually appreciated by current anti-Quineans. His holist story of theory formation, according to which we learn to understand the world only through the medium of an overall theory grown from observations, led him to examine closely the contribution of observation sentences, and to confront questions of translation, interpretation, and how our words denote objects in the world. By reflecting on these issues, Quine came increasingly to recognize that his account implies that the structure of a theory’s ontology is paramount. Hence ‘Structure and Ontology’; but although Quine can be called a ‘structuralist’ in this regard, his position differs from what is often nowadays called ‘structuralism’ in the philosophy of science and philosophy of mathematics. Mathematical structuralism is the view that what comprises the subject matter of mathematics is an ontology of abstract structures, structural theoretical objects whose presence we discern by considering their application to concrete observable objects. Quine, ever the holist, resisted such sharp distinctions between the theoretical and the observable. A more thoroughgoing structuralism is proposed by ontological structuralists in the philosophy of science, who maintain that everything there is is a structure, that all things are relations rather than relata. Quine’s reservations about the existence of attributes made him wary of the doctrine that all things are relations. Still he partially agreed with such structuralists, claiming that science is concerned with the network of interlocking roles fulfilled by objects according to our best theory, not with the individual identities of roleoccupants. Quine did not infer from that thesis that only structures exist. Exactly what form his structuralism takes, how it relates to other aspects of his thought, and how it may influence or inform twenty-first-century debates on ontology, epistemology, language, and the philosophy of science and mathematics is the subject of the chapters in this volume, which express a variety of perspectives on the issue. Michael Resnik, one of the main contemporary advocates of mathematical structuralism whose work inspired other prominent figures in the field such as Shapiro, re-evaluates both his own contributions to the field and the influence of Quine in his chapter in this volume, ‘Quine’s Non-ontological Structuralism (and Mine)’. Having previously defended an ontological structuralism of the type outlined above, Resnik here embraces a kind of structuralism similar to Quine’s. He makes the case in this chapter that it is misleading to characterize Quine’s structuralism as ontological at all, tracing the development of Quine’s structuralism and the influence it bore upon his own evolution from an influential proponent of an ontological form of structuralism to a non-ontological structuralist. Resnik attributes to Quine an epistemological version of structuralism according to which all our theory-mediated knowledge of the

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introduction 3 world, i.e. all our knowledge, is structural rather than object-based. Resnik continues to explain how he arrived at a similar destination to Quine via considerations in the philosophy of mathematics rather than the philosophy of language. He defends his view, inter alia, against Burgess’s objections to structuralism in the philosophy of mathematics. A different source of support for ascribing a non-ontological structuralism to Quine is presented in Frederique Janssen-Lauret and Fraser MacBride’s chapter ‘W. V. Quine and David Lewis: Structural (Epistemological) Humility’. Expounding Quine’s mature conception of scientific knowledge as coming to an understanding of objects through grasping their contributions to the law-like regularities captured by our best theories, they show how this approach obviates singular reference and confirms inscrutability. Objects are not structures or structural nodes, but cognized only through their places in a structural network of scientific explanation; all objects are theoretical, none directly observed. Such absence of direct reference need not worry us, Quine maintained. In the mathematical realm we don’t expect a single correct analysis of what it is to be an ordered tuple or natural number, instead accepting a RamseyanCarnapian line which states that sets and numbers should be known by their laws alone, and any satisfier of the laws will do. As good holists, Quine concludes, we should expect no more of the posits of the natural sciences, which are known by their laws just as mathematical posits are. Janssen-Lauret and MacBride go on to reveal, drawing upon unpublished correspondence, that Quine’s student David Lewis, initially just a mathematical structuralist, eventually came to embrace Quine-style structuralism, and argue that such a reading makes better sense of Lewis’s much-debated late paper ‘Ramseyan Humility’. So Quine’s structuralism is seen to be highly relevant even to the many Lewisians on the contemporary scene. John Collins interrogates Quine’s views on ontological commitment from the point of view of contemporary philosophy of language. Collins’s chapter, ‘Quine on Ontological Commitment in Light of Predicate-Functor Logic’, addresses the question why first-order logic should, as Quine claimed, have any priority as a regimented language. Why should ontological commitments be discovered exclusively by means of a language involving quantifiers and bound variables? As Quine emphasized, there are alternative predicate-functor languages that are expressively adequate without the apparatus of quantifiers and variable-binding. Collins explores different avenues of response suggested by Quine’s various remarks on this topic. He comes down in favour of the suggestion that in fact there is a common criterion of ontological commitment that stretches across languages, namely one in terms of predicate satisfiability. To make sense of the resulting picture, Collins suggests that we should divorce the question of the logical form of a sentence from the question of what is responsible for making a sentence true. Jaroslav Peregrin, in his chapter ‘Inscrutability of Reference and Quine’s Structuralism’, argues that Quine was committed to a form of structuralism, but that such structuralism is neither ontological nor epistemological as envisaged above by Resnik

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4 frederique janssen-lauret and Janssen-Lauret and MacBride. Peregrin takes Quine, rather, as a structuralist about meaning itself. According to Peregrin, words (for Quine) refer to nodes in a structure, where these nodes may themselves be conceived as objects, such that the structure which constitutes the semantics of a language admits of non-trivial automorphisms. In this way, Peregrin argues, Quine’s doctrine of inscrutability of reference can be seen to flow from his structuralism about meaning. Horwich and Searle have both offered arguments critical of Quine’s doctrine of inscrutability of reference. But from this structuralist perspective, Peregrin seeks to debunk Horwich and Searle’s criticisms. By contrast, Paul Gregory argues that Quine’s structuralism is indeed ontological. Gregory’s chapter, ‘Quine’s Deflationary Structuralism’, claims that Quine’s ontological structuralism is nevertheless distinct from contemporary versions of mathematical or ontological structuralism, but is to be read in a deflationary spirit, as Carnap famously conceived ontology. This poses a problem of interpretation because Quine famously rejected Carnap’s deflationism about ontology when he advocated making metaphysics continuous with science. Gregory confronts this problem head-on by tracing the gradual development of structuralist ideas from Quine’s views in the late 1940s through to what in Quine in the 1980s characterized as the ‘humiliating demotion’ of ontology as a consequence of his proxy function argument, which preserves only the structure of our theories. Greg Frost-Arnold, Robert Sinclair, and Gary Kemp and Andrew Lugg pursue themes of structure and ontology from a historical point of view, investigating Quine’s development from early to late and drawing upon of the history of analytic philosophy. Frost-Arnold’s chapter, ‘The Ontogeny of Quine’s Ontology: Pythagoreanism, Nominalism, and the Role of Clarity’ investigates the increasingly robust existential assumptions in Quine’s philosophy of mathematics beginning with a previously unknown early source: the young Quine’s logic notebook. In that notebook Quine toyed with a Pythagorean ontology, according to which only numbers exist, as well as with the nominalism which we more commonly associate with the early Quine, and which is wholly at odds with Pythagoreanism. Thus, in these notes we find very early foreshadowings of Quine’s moderately sympathetic treatment of Pythagorean ontologies later on in the context of ontological relativity. Frost-Arnold argues that Quine’s later non-nominalistic ontology, which takes mathematical as well as physical objects as equally theoretical, is motivated in part by Quine’s re-evaluating his earlier quest for ‘clarity’ and coming to a more nuanced understanding of what, in a scientific context, counts as clarity. Robert Sinclair, in his chapter ‘Quine’s Structural Holism and the Constitutive A Priori’ focuses on another respect in which Quine’s philosophy may be deemed structuralist, namely the holistic principle that mathematics and logic face the tribunal of experience not directly, but only via their contribution to the web of belief. Friedman, drawing upon Kantian insights, has criticized Quine on this score, arguing that Quine’s holism is unable to explain the asymmetric dependencies between logical,

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introduction 5 mathematical, and empirical principles because Quine has only the cumbersome notion of entrenchment to bring to bear. Sinclair adduces detailed evidence from the Quine corpus to support his claim that Quine had a far more subtle approach to the logic of science than Friedman recognizes. But still, Sinclair argues, Quine has difficulty addressing the co-ordination of abstract mathematical principles with concrete sensible events. Nonetheless, Sinclair concludes, Quine and Friedman’s approaches to scientific theories can be seen as complementary, roughly speaking working out a common Kantian thought that empirical theories require formal constraints, but from different viewpoints, from logical and historical standpoints. Kemp and Lugg’s chapter ‘Quine on Ontology: Chapter 7 of Word and Object’ reconsiders Quine’s views on ontology expressed in his magnum opus, Word and Object, which, they argue, have been unduly neglected. They reconstruct Quine’s reflections in the final chapter of that work, drawing connections with both Russell and Wittgenstein. They argue that Quine has no meta-ontology but an immanent view of ontology continuous with science. They conclude that Quine moved away from the methodological perspective of Word and Object, which emphasizes the centrality of existence questions, to his later emphasis upon structure, because of the development of his views about reference. Nathan Salmon, Gila Sher, Marianna Antonutti Marfori, and Natalja Deng consider Quine’s views on the structure of logic, language, and theories in relation to contemporary philosophy. They relate his work to debates in ontology, the philosophy of logic and mathematical realism, philosophy of set theory, and philosophy of time respectively. Salmon, in ‘On What Exists’, takes a critical line. His chapter subjects Quine’s criterion of ontological commitment to extended and exacting scrutiny. Salmon undertakes a number of refinements of the criterion on the behalf of Quine under fire from the criticisms of Church, Kripke, and Cartwright. According to Salmon, the resulting conception of ontological commitment makes unavoidable appeal to two notions against whose utility Quine had long campaigned: analyticity and modality. Ontological commitment, Salmon argues, covertly involves analyticity because it presupposes the notion of a meta-truth of pure semantics. What’s more, it covertly involves modality because of its reliance on set-theoretic models. Set theory (as well as Quinean proxy functions) yields many unintended models. Quine’s holist epistemology makes it extremely difficult to single out an intended model without relying either on direct cognitive access to intended referents—something which Quine claims to oppose—or some form of modality. This leads Salmon to formulate a dilemma for Quine and his followers. Either Quine’s criterion is construed without reference to analyticity or modality, in which case it is fatally flawed, or it isn’t flawed, because it incorporates reference to analyticity and modality, but then Quine is relying upon exactly what he disavows. Gila Sher investigates, in her chapter ‘Quine vs. Quine’, how Quine’s ideas may illuminate twenty-first-century debates, especially those about the status of logic and a kind of mathematical realism not beholden to either Platonism or empiricism. She

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6 frederique janssen-lauret makes the case that Quine’s opposition to foundationalism, his scepticism about the analytic–synthetic and a priori–a posteriori dichotomies, and his views on posits, jointly contribute to a picture which allows for open-minded philosophical investigation of topics like the status of logic and mathematical realism. They allow us, she argues, to sidestep the peculiarly twentieth-century pitfalls of redefining philosophical questions as questions of language, neuroscience, or psychology. Marianna Antonutti Marfori applies a contemporary philosopher of mathematics’ perspective to Quine’s position on set theory. Antonutti Marfori first argues that while there are solid objections to be raised to Quine’s view, certain widespread arguments against result from overly crude and uncharitable interpretations of Quine. She then turns to the question what kind of evidence it would take for a Quinean naturalist to change their mind about certain theses, such as the size of the set-theoretic universe. She argues that Quineans might be moved to embrace further set-theoretic ontology in the light of the mathematical utility of large cardinals, and potentially even the ‘multiverse’ position on set theory. Natalja Deng’s chapter, ‘What Quine (and Carnap) Might Say about Contemporary Metaphysics of Time’ tackles the Quinean versus the Carnapian approach to ontology. Deng traces the roots of recent debates on the philosophy of time to Quine’s disagreement with Carnap over ontology, and to Quine’s arguments pertaining to fourdimensionalism and quantification without built-in tense. She goes on to explain how twenty-first-century philosophers of time have put distance between themselves and Quine, and considers how Quine and Carnap might have responded to contemporary work on time, tense, and quantifiers. We gratefully acknowledge the support of the Mind Association, the Scots Philosophical Association, and the Aristotelian Society for the original conference, and to Sarah Sawyer and Peter Momtchiloff for making this volume in the Mind Occasional Series a reality.

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2 Quine’s Non-ontological Structuralism (and Mine) Michael D. Resnik

2.1 Introduction Structuralism in the philosophy of mathematics developed through attempts to understand the subject matter of mathematics and how we come to know it. But it was highly influenced by developments in mathematics itself, in particular, by the emergence of abstract algebraic theories, such as, group theory, that characterize the structure shared by different mathematical examples. To be a group—or a ring or a lattice or a topological space, etc.—is just to be a set with relations defined on it that satisfy certain conditions. In mathematical jargon it is just to be a set carrying a certain structure. The nature of the members of the set is irrelevant to its being a group, ring, lattice, etc. One form of structuralism, eliminative structuralism, generalizes the abstract approach to all of mathematics by viewing each mathematical theory as an abstract theory whose axioms define a type of system. Thus, on the eliminativist construal, just as group theory is the theory of all groups and its sentences are tacitly of the form ‘for any group G, ( . . . G. . . )’, so is theory of natural numbers the theory of all progressions and its sentences are tacitly of the form ‘for any progression P, ( . . . P . . . )’. The early group theorists found examples of groups in mathematics itself; important scientific applications came later. But number theory is as basic mathematics as one can get. So where are we to find the progressions? If there are none, then all the sentences of number theory are vacuously true. We cannot look elsewhere in mathematics, e.g. to geometry, since eliminative structuralism is supposed to apply to all of mathematics. Some philosophers respond by suggesting that there are progressions of physical objects or that it is at least possible that there are such. Avoiding vacuity complicates quickly as one turns to branches of mathematics, such as set theory, that posit

Michael D. Resnik, Quine’s Non-ontological Structuralism (and Mine) In: Quine, Structure, and Ontology. Edited by: Frederique Janssen-Lauret, Oxford University Press (2020). © Michael D. Resnik. DOI: 10.1093/oso/9780198864288.003.0002

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8 michael d. resnik infinities upon infinities. But for this chapter we need not go into the interesting moves eliminative structuralists make.1 Another brand of structuralism, sui generis structuralism, deals with the problem of vacuity by distinguishing between ‘abstract’ mathematical theories, such as group theory, that are not intended to have a specific subject matter and ‘concrete’ ones that are, such as number theory, analysis and set theory. On this view, each concrete mathematical theory is supposed to be about a structure or pattern with its singular terms and quantifiers referring to places or positions in the structure in question. These theories provide examples for the abstract ones, and guarantee their nonvacuity. The nature of these patterns or structures is not entirely clear, but people often think of them as universals.2 Both eliminative and sui generis structuralisms are forms of ontological structuralism. They take a position on the ontology of mathematics by maintaining that either it has no ontology of its own or that it has one of structures. In ‘Structure and Nature’3 Quine advocates a ‘global ontological structuralism’, one that sees all objects, even nonmathematical ones, as positions or nodes in a structure. The term ‘ontological’ here is misleading and would be better deleted. For Quine is no ontological structuralist; in the same paper he makes it clear that he is not proposing a structuralist ontology or even an ontological doctrine in which structures are eliminated in favour of systems instantiating them. My own mathematical structuralism evolved from a sui generis form to a Quinean non-ontological form. In this chapter I will discuss the evolution of Quine’s structuralism and how it shaped my own.

2.2 Structuralism To start, I will propose a general characterization of structuralism. Structuralism depends upon the fact that no theory has the means within itself to distinguish between its various models. For example, if we have a model of natural number theory, we can distinguish within the model its ‘even numbers’. But then stepping outside the model, so to speak, we can use these ‘even numbers’ to form another model of the theory in which they play the role of the entire number sequence. We just take the domain of the new model as the even numbers of the first model and take our new successor function as the double successor of the first one. Moreover, we can use this trick again and again to form indefinitely many models of the theory. We can do this with the models of other theories as well, though the constructions are not likely to be as easy and as obvious as those for natural number theory. It happens that all the models of natural number theory that I have mentioned are isomorphic, and most structuralists tend to think of isomorphisms as structurepreserving transformations, and of each type of isomorphism as defining a type of

1

For details see Hellman (1989) and Parsons (2008).

2

Shapiro (1997).

3

Quine (1992).

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quine’s non-ontological structuralism (and mine) 9 structure. But there is another, broader notion of structure according to which all models of a theory have the same structure. There are, for example, non-isomorphic groups. All of these groups share the same broad structure qua groups, but some differ with respect to the finer structure associated with their isomorphism types. Group theory encompasses non-isomorphic groups by design. However, some theories are intended to be categorical, i.e. to have only models that are isomorphic to each other. The second-order theory of the natural numbers is such an example. It is essential, however, that its underlying second-order logic have variables that range over the full power set of its first-order universe. If we relax this requirement, as we do in the case of first-order number theory, then we obtain models that are not isomorphic to their ‘standard’ or ‘intended’ ones. Thus first-order number theory involves a broader notion of structure than full second-order the number theory. The models of the latter are included in the models of the former. It can also happen that one and the same system exhibits different structures. Thus a non-standard model of first-order number theory will have the first-order structure of number theory by virtue of modelling it, but unlike standard models it will not be a progression. The richer perspective of set theory which enables us to define the concept of a progression allows us to see the distinction between standard and nonstandard models of first-order number theory.4 I am assuming that in specifying a theory we also specify its underlying logic. Thus I have distinguished first-order number theory from second-order number theory. Thinking of a theory T as specifying the structure shared by all its models (in the sense appropriate to its logic), we can speak of a T-structure. I take this to be a broader conception of structure, and contrast it with a narrower conception according to which systems of the same structure are isomorphic. One can also introduce even narrower conceptions of structure by requiring that the isomorphisms involved preserve certain properties that some isomorphisms do not. For example, certain similar triangles are isomorphic with respect to transformations that preserve their respective angles, but not with respect to ones that preserve the lengths of their respective sides. Now suppose that all our knowledge of a given subject S is through our theory of S. Then the most that we know about the objects, properties, and relations of S is that they form a model of our theory. In this sense our knowledge of S is structural. (I say ‘the most’, because it might be that we don’t even know that our so-called theory of S has a model.) Nothing that S affirms distinguishes one of its models from another. Structuralism with respect to a subject S, then, is the thesis that our knowledge of S is structural. This formulation takes no position on the objects of S. One can be a structuralist is this sense while maintaining that S has no objects of its own, as eliminative structuralists do with respect to mathematics. Or one might combine one’s structuralism with an explanation according to which the objects are positions 4

This illustrates what I call structural relativity: the structures we discern are relative to the devices, including logical ones, we use for depicting structures. See Resnik (1997).

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10 michael d. resnik in a pattern or patterns as sui generis structuralists do. Or one might even hold that the objects have an unknowable nature. And, of course, one might combine it with a quiescence concerning the objects of S. It excludes only the position that there is more to be known about the objects of S than our theory of S tells us.5 It is generally agreed that our knowledge of the natural numbers is only structural.6 The usual grounds given for this are that any progression will work as model for number theory and its applications in counting. Suppose we add to number theory the definition of ‘the number of F = n’ as ‘the Fs are in one–one correlation with the numbers less than n’ and appropriate laws of one–one correlation. Then it is easily shown that so long as our ‘numbers’ form a progression, they will satisfy the axioms of number theory and the laws of counting. In this context numbers function as counters, a job, which any progression of objections, can fulfil—at least in theory. In view of this one might pose the following objection to my characterization of structuralism: on your proposal the structural nature of our knowledge of the numbers should entail that our knowledge of the natural numbers is limited to what is expressed in number theory supplemented with a theory of counting, be it first or second order. But, the objection continues, surely we know much more about the natural numbers than this. We have empirical knowledge involving them—that human hands normally have five fingers, for example, or that some people regard 13 as unlucky. This objection wrongly assumes that everything we know about the numbers can be codified in number theory. This is certainly not true, as the examples show. However, without claiming that anyone has produced such a theory, let N+ be a theory codifying all our knowledge involving numbers. This theory too will be incapable of distinguishing between its models. Hence, unless one wants to claim that we have some theory-independent way of identifying the objects that really are the numbers and singling them out from the impostures who shadow them in modelling N+, our knowledge via N+, and thus, by assumption, of our knowledge of the numbers, will remain structural. Furthermore, suppose that we are given a model of N+ with the domain D containing a subset Dn and a relation Sn (for successor) on Dn that form a model of number theory. (This is all we need for second-order number theory; a model of the usual versions of first-order number theory will have relations for addition and multiplication.) We can introduce a new model of N+ preserving the truth-values of sentences of N+ as follows: 5 Structuralism with respect to one subject does not commit one to structuralism with respect to other subjects. Below we will note that John Burgess thinks that structuralism applies only to parts of mathematics. 6 The classic case for this is Benacerraf (1965). However, as we shall see, Quine had already argued as much in criticizing Russell.

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quine’s non-ontological structuralism (and mine) 11 The domain of the new model, D∗ , will be D less the zero of Dn , i.e. D∗ = D \ {0n }. We now define a proxy function (to use Quine’s term) prox as follows: prox(z) = z if z is in D but not in Dn ; prox(z) = z +n 1n if z is in Dn . In other words, the members of D that are not in Dn are their own proxies, while ‘adding one’ to a member of Dn yields its proxy in D∗ . If R is a relation Rab . . . k of the model D, then we associate it with the relation R∗ on D∗ such that : R∗ prox(a)prox(b) . . . prox(k) if and only if Rab . . . k. Consider the sentence: ‘my fingers number 10’. In the old model this is interpreted as D-my fingers D-number 10n . In the new model it becomes D∗ -my fingers D∗ -number prox(10n ). Under the proxy function this is equivalent to D-my fingers D∗ -number 11n . But the D∗ -number relation is defined so that D-my fingers D∗ -number 11n if and only if they D-number 10n . So the truth value of the sentence remains unchanged. Or consider the sentence ‘IV’ is the Roman numeral for the number four, where: prox(D-‘IV’) = D-‘IV’; prox(4n ) = 5n ; Prox(z) D∗ -is the Roman numeral for prox(w) if and only z D-is the Roman numeral for w. Then D∗ -‘IV’ D∗ -is the Roman numeral for 5n if and only if D-‘IV’ D-is the Roman numeral for 4n . Again the truth value of the sentence remains unchanged. Quine applied this trick to more extensive theories than N+. If we are able to specify a one–one mapping between the domain D of a model of a theory T and some other domain D∗ , then we can project the classes and relations of the model onto classes and relations among the members of D∗ to form a new model of T. As models of T both will have the same T-structure. This will be true even if T is a theory that contains its observational consequences. The same sentences will be true (false) in both models. Next suppose, as Quine did, that all our knowledge of any subject is through our theory of that subject, that all objects are theoretical posits known only through our theories, and that our evidence for our theories is given through their connection with experience via observation sentences. Then in Quine’s words, ‘the conclusion is that there can be no evidence for one ontology as over against another so long anyway as we can express a one-to-one correlation between them. Save

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12 michael d. resnik the structure and you save all.’7 Quine’s global structuralism is the thesis that our knowledge of the world is structural. It is non-ontological in denying that there is a fact of the matter as to what has the structure in question.

2.3 A Path to Ontological Relativity How might Quine have arrived at this surprising position? One can trace a possible path to it by looking at his work on reference and reduction.8 Early in his career, when his interest was in nominalism and minimizing ontology, he developed a tool that is essential for assessing progress in this area; namely, his criterion of ontological commitment, ‘to be is to be a value of a variable.’ This brings questions of reference to the forefront, because proper application of his criterion to a theory requires that it be ‘regimented’ so that its only devices for reference are first-order quantifiers and variables.9 In ‘Steps toward Constructive Nominalism’ Quine and Nelson Goodman tacitly used Quine’s criterion in assessing the prospects for a finitary, nominalist foundation for mathematics.10 Although they wrote that they renounced abstract entities, later Goodman, while still calling himself a nominalist, was willing to admit universals as individuals—to quantify over them, so to speak. Since he thought that there is more to nominalism than the type of individuals one admits, in Structure of Appearance he proposed (what one might call) structural characterizations of nominalism: in a nominalist system there must be no distinction of entities without a distinction of their contents, and no generation of distinct entities from the same atom.11 This prompted Hao Wang to write a short piece in which he argued that Goodman’s nominalism and his notion of an individual were unclear.12 Wang suggested that instead of assessing a theory using Quine’s criterion, and asking whether it was committed to abstract entities, we should consider whether it was committed to infinitely many things or used impredicative specifications. Although Wang did not say so explicitly in this article, I take him to be arguing that our emphasis should be on structural assumptions—ones that delimit the type of models (e.g. finite or infinite) a theory may have—instead of ontology. Quine’s Word and Object is famous for its ‘Gavagai’ example where the whole sentence can be translated as ‘Rabbit’, whether or not the term ‘gavagai’ refers to rabbits, fusions of rabbit parts, instances of rabbithood, etc. This is because translating the sentence as ‘There’s a rabbit’, ‘Wow, rabbithood instanced here’, or as ‘It’s a 7 (1992: 405). The idea that I illustrated with N+ and that Quine had in mind can be applied to more than the purely extensional languages he countenances, but it is probably limited to languages with an extensional semantics. See Resnik (1996). 8 DISCLAIMER: the following is not intended as an account of Quine’s reasoning or how various ideas occurred to him. 9 10 11 Quine (1948). Goodman and Quine (1947). Goodman (1951). 12 Wang (1953).

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quine’s non-ontological structuralism (and mine) 13 fusion of rabbit parts’ preserves its stimulus meaning; for ‘the stimulus situations that prompt assent to “Gavagai” would be the same as for “Rabbit”’.13 This idea applies to sentences in our home language too. Take the sentence ‘There goes a rabbit’. Changing the reference of ‘rabbit’ to rabbit-part-fusion will preserve its truth-value and observational evidence. So too will changing the reference of ‘rabbit’ to ‘rabbit parts’ while also changing ‘there goes’ to ‘there goes a fusion of’. This illustrates how reference can be inscrutable; no observable evidence decides whether the term ‘rabbit’ refers to rabbits or fusions of rabbit parts. Quine’s discussion of this early illustration of the inscrutability of reference, especially his point that changes in the reference of one part of a sentence can be compensated for by changing the reference of other parts, foreshadows the argument from proxy functions that he used to underwrite his global structuralism. Although Word and Object is more concerned with reference than structuralism, the latter appears in Quine’s discussion of numbers. Here he writes that contrary to Russell: “The condition upon all acceptable explications of number . . . can be put: any progression . . . will do nicely . . . . The situation is unlike matrimony. Frege’s version, von Neumann’s and Zermelo’s are three progressions of classes . . . available for selective use as convenient. That all are adequate as explications of natural number means that natural numbers, in any distinctive sense, do not need to be reckoned into our universe in addition.”14 Later in ‘Ontological Relativity’ structuralism is more explicit: ‘Numbers . . . are known only by their laws . . . so any constructs obeying these laws—certain sets, for instance—are eligible as explications of number. Sets in turn are known only by their laws, the laws of set theory.’ And later he repeats his criticism of Russell: ‘Always if the structure is there, the applications will fall into place.’15 Questions of structure come to the forefront in Quine’s ‘Ontological Reduction and the World of Numbers’16 Here he addresses the problem of pseudo ontological reductions via the Lowenheim-Skolem Theorem (LST). According to LST every firstorder theory that has a model in an infinite domain also has one in the universe of natural numbers. However, as Quine observed, ‘any interpretable theory . . . can be modeled in the natural numbers, yes; but does this entitle us to say that it is once and for all reducible to that domain, in a sense that would allow us thenceforward to repudiate the old objects for all purposes and recognize just the new ones, the natural numbers?’17 How are we to avoid blanket and trivial reductions to a numerical ontology?

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14 Quine (1960: 51–2). Quine (1960: 262–63). Quine (1969: 44–5). I assume from the context that he meant this remark to apply to at least the rational, real, and complex numbers, though it also applies to mathematics more generally. 16 Quine (1964). 17 Quine (1964: 201–2). This issue is related to Skolem’s Paradox: ZF ‘says’ that there are uncountably many sets, but there are only countably many sets/numbers in its LST model. Some have argued that this means that there really are only countably many sets. 15

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14 michael d. resnik The problem, as Quine sees it, is that the LST reduction does not preserve the ‘relevant structure.’ On the broad characterization I proposed earlier two models of a theory do share a structure, but it need not be the ‘relevant structure’ in whatever sense Quine has in mind. After some analysis of examples and consideration of other ways of converting one theory into another, Quine proposes that a method M for converting a theory T to T ∗ will preserve the ‘relevant structure’ if it is based upon a proxy function, that is, a function that ‘we specify’ that associates every object in T’s universe with one in T ∗ ’s. Furthermore, as a means for converting T into T ∗ , M will associate an n-place open sentence P∗ of T ∗ with each primitive n-place predicate P of T such at P is true of an n-tuple of arguments of the proxy function just in case P∗ is true of the n-tuple of the corresponding values.18 It is important to note that we are talking about reductions that we can carry out— at least in principle. Quine requires that we ‘specify’ the proxy function and explains how we can then use it to associate open sentences in the reducing theory with ones in the theory being reduced. This is what entitles us ‘to say that [universe] is once and for all reducible to [the new universe], in a sense that would allow us thenceforward to repudiate the old objects for all purposes and recognize just the new ones.’ The proxy function requirement avoids unwanted reductions via LST simply because none of the proofs of LST provide the means for specifying a proxy function. But in some cases we can show that there can be no proxy function reducing T to T ∗ , simply because there can be no function at all that assigns to each member of T’s universe one in T ∗ ’s universe. In particular, there can be no proxy function reducing set theory with its uncountable universe to natural number theory with its countable universe. For suppose that f is a proxy function reduction of an uncountable universe of sets U to the natural numbers N. Then the reduction must supply a two-place open sentence, Exy, such that a) for any sets x and y in U, x ∈ y if and only if Ef (x)f (y). Now suppose that for x and y in U, f (x) = f (y). Then, by the substitutivity of identity, b) for all f (z) in N, Ef (z)f (x) if and only if Ef (z)f (y). Next let w be any set in U. Suppose w ∈ x. Then by (a) Ef (w)f (x), so by (b) Ef (w)f (y) and by (a), w ∈ y. Similarly, w ∈ y

18 Quine (1964: 205). It is important to keep in mind that for Quine a theory is an interpreted set of sentences, a set of sentences cum model. Richard Grandy (1969) exploits this to show that proxy function reduction is not the only method that might be required for an ontological reduction. Consider a theory with the non-logical notation ‘N’, ‘0’, ‘S’, and ‘