Science Without Numbers PDF - A Defense of Nominalism (2016, Oxford University Press)

OUP CORRECTED PROOF – FINAL, //, SPi Science Without Numbers, Second Edition OUP CORRECTED PROOF – FINAL, //, SPi OUP CORRECTED PROOF – FINAL, //, SPi Science Without Numbers A Defense of Nominalism Second Edition Hartry Field 1  OUP CORRECTED PROO...

OUP CORRECTED PROOF – FINAL, //, SPi Science Without Numbers, Second Edition OUP CORRECTED PROOF – FINAL, //, SPi OUP CORRECTED PROOF – FINAL, //, SPi Science Without Numbers A Defense of Nominalism Second Edition Hartry Field 1  OUP CORRECTED PROOF – FINAL, //, SPi 3 Great Clarendon Street, Oxford, OX DP, 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 © This edition Hartry Field  The moral rights of the author have been asserted First Edition published in  Impression:  All rights reserved. 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Printed in Great Britain by Clays Ltd, St Ives plc  OUP CORRECTED PROOF – FINAL, //, SPi Contents Contents New to this Edition Preface to Second Edition P- .. Arithmetic and Cardinality Quantifiers P- .. Mereology and Logic P- .. Representation Theorems P- .. Conservativeness P- .. Indispensability P- .. Other Forms of Anti-Platonism P- .. Miscellaneous Technicalia P- Bibliography for Second Preface P- Note P- Letter from W. V. Quine P- Contents of First Edition Preface to First Edition i Preliminary Remarks  . Why the Utility of Mathematical Entities is Unlike the Utility of Theoretical Entities  Appendix: On Conservativeness  . First Illustration of Why Mathematical Entities are Useful: Arithmetic  . Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance  . Nominalism and the Structure of Physical Space  . My Strategy for Nominalizing Physics, and its Advantages  . A Nominalistic Treatment of Newtonian Space-Time  . A Nominalistic Treatment of Quantities, and a Preview of a Nominalistic Treatment of the Laws Involving Them   OUP CORRECTED PROOF – FINAL, //, SPi vi c on t e n t s . Newtonian Gravitational Theory Nominalized  A. Continuity  B. Products and Ratios  C. Signed Products and Ratios  D. Derivatives  E. Second (and Higher) Derivatives  F. Laplaceans  G. Poisson’s Equation  H. Inner Products  I. Gradients  J. Differentiation of Vector Fields  K. The Law of Motion  L. General Remarks  . Logic and Ontology  Bibliography for Original Text  Index (to entire volume)   OUP CORRECTED PROOF – FINAL, //, SPi Preface to Second Edition When I began writing Science Without Numbers in the winter of /, I did not intend to write a book, but a long article; but it grew until it reached a point where publication as a normal journal article did not seem feasible. And the final product didn’t really seem like a book (which is why I called it a monograph, a term I never used before or since): I gave only a cursory motivation for a certain project, that of presenting physical theories in a certain (“nominalistic”) format, and spent the rest of the time trying to overcome skepticism about the feasibility of the project by giving a detailed sketch of how it might be accomplished for a particular non-trivial physical theory.F A “real book” would have required a more detailed philosophical discussion of the motivations for the project, but I wasn’t ready to give that at the time of publication of the monograph. I did attempt more philosophical justification over the ensuing decade or so, in a number of articles, and in the Introduction to a volume (Field /) that contained some of these articles. (I also devoted some attention to an obvious lacuna of the book, the issue of whether the nominalistic position of the book could accommodate metalogic.) By some time in the early s, though, my interests within the philosophy of mathematics had shifted a bit,F in part because of increasing doubts about the Quinean framework that SWN presupposed. In particular, I became increasingly doubtful of the following two suppositions: that the question of what exists has a univocal and non-conventional content; that the right way to answer this question is to look at the existential quantifications of our most fundamental theories; it is “doublethink” to employ a fundamental theory and not literally believe its posits if there is no serious prospect of showing how those posits could be eliminated. F “Defense” in the sub-title was intended literally: I was defending it against an attack, not going on the offense. F See for instance my , a, and b.  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on I still regard myself as anti-platonist in a broad sense, and still regard the work in SWN as relevant to supporting the credentials of anti-platonism; but I no longer want to rest my anti-platonism on the claim that the program of SWN can be completely carried out. ‘Platonism’ can mean a number of things. In the book I took it to be primarily a thesis about what exists: in particular, I took the exis- tence of mathematical entities to suffice for “platonism”. “Nominalism” is the denial of platonism in this sense. But another interpretation of ‘platonism’ is that what distinguishes a good mathematical theory from a bad one is how accurately it describes mathematical reality. Slightly more precisely, the view (said to be “typical of extreme platonism” in Chapter  of the book) is that higher mathematics is objective in the way that the sciences are: not only is it objective what’s a good proof, it’s also objective what’s a correct axiom, so that e.g. there’s an objectively correct answer to the size of the continuum even though this is unset- tled by current axioms. But platonism in this sense can go with anti- platonism in the “ontological” sense: witness the Putnam-Hellman idea (Putnam ; Hellman ) that mathematical questions are thoroughly objective but to be understood modally. Conversely, ontological platon- ism needn’t require platonism in the “objectivity” sense: witness Mark Balaguer’s “plenitudinous platonism” (Balaguer ), or the view that Putnam seems to advocate in the early section of Putnam , or views on which the existence of mathematical entities and the laws they obey is a matter of convention. There is considerable plausibility in the idea that the arithmetic of nat- ural numbers is objective in the sense I’ve described, a fact that I take to be explainable by the close connection between it and the logic of cardinality-quantifiers. I think this is consonant with the viewpoint of Science Without Numbers, though the book ought to have emphasized it more. The book did, as noted, come out against this sort of objectivity for other parts of mathematics that don’t seem so intimately related to logic; and in this I believe it was correct. I now regard this objectivity issue as more important than the existence issue;F indeed, I’m not entirely sure that the question of what exists has a univocal and non-conventional content, though I’ve never been entirely F “ ‘Ontology’, I spoke the word, as if a wedding vow. Ah, but I was so much older then, I’m younger than that now.” (Bob Dylan, approximately.)  OUP CORRECTED PROOF – FINAL, //, SPi pre fac e to se c on d e di t i on P- satisfied by attempts like Carnap’s () to make sense of the view that it doesn’t. But whatever one thinks of the existence issue, I think that the program of SWN bears on the objectivity issue (as it arises for parts of mathematics that go beyond the arithmetic of natural numbers), and tends to support the anti-platonist position on it. Moreover, if one does take the existence issue seriously, one can take SWN as bearing on it with- out buying into the Quinean view presupposed in the book, according to which existence questions are to be settled by reading the answers directly from our most fundamental theories. More on these matters later. There were, actually, two main motivations for the project of SWN. One concerned the platonism issues, most explicitly, the ontological one: the project was to rid us of the need to literally believe in mathemat- ical entities (not just numbers). Normal (“platonistic”) formulations of physical theories contain reference to all sorts of mathematical entities; whereas the “nominalistic” reformulations of theories that I proposed contained no reference to mathematical entities of any kind. The idea was that until you present a physical theory nominalistically, it looks as if literal belief in the theory requires literal belief in mathematical enti- ties; but the possibility of nominalistic formulations shows this not to be the case. The view was that mathematical theories are essentially useful calculation devices. In the context of a given nominalistic theory, some consistent mathematical theories may be more useful for calculation than others, but this doesn’t make them better in any context-independent or non-utilitarian sense. (The arithmetic of natural numbers is a partial exception since its most characteristic use is for dealing with cardinality quantification, a role that it has independently of any particular empirical theory.) SWN is often described as advocating an error theory about mathemat- ics, but I think that this description is highly misleading. The book does assert that mathematical theories aren’t literally true, if taken at face value; but to say that this is an error theory suggests that most ordinary people, or mathematicians, or physicists who use mathematics in their theories, falsely believe the theories true in this sense. I am skeptical that ordinary people or mathematicians or physicists typically have any stable attitude toward that philosophical question. The only error I saw was on the part of platonist philosophers (e.g. my most explicit target, Quine, who often said that mathematical objects are real in just the way that physical objects are—see for instance Quine ).  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on John Burgess (; also Burgess and Rosen ) likes to distinguish between “revolutionary” and “hermeneutic” nominalists, the former trying to correct what they see as widespread error and the latter purporting to elucidate what ordinary people and mathematicians and physicists meant all along. In my view, this is a false dichotomy. I cer- tainly didn’t think that the account I was providing was “hermeneutic”, but it wasn’t “revolutionary” either: I took what I was doing, rather, as providing an account that explains why ordinary mathematical practice is perfectly fine, and doesn’t require a platonist ontology. (The claim that my account accepts all of “ordinary mathematical practice” would have to be qualified if one construed ordinary practice as taking seriously questions like “What is the real cardinality of the continuum?”. But of course the invention of new axioms and the investigation of their impli- cations for the cardinality of the continuum is a part of mathematical practice that comes out perfectly fine on my account; it’s only the ques- tion of which such competing axioms are “really true” that my account rejects.) The other motivation for SWN was based on a feeling that though formulating an empirical theory using a high-powered mathematical apparatus can in many ways be illuminating (especially when it comes to comparing that theory with others), it can sometimes make it hard to see what is really going on in the theory. Formulating physical the- ories without the high-powered mathematical apparatus, and in what I called an “intrinsic” manner, was intended to illuminate those theories; and in combination with representation theorems, to give an account of the application of mathematics that would be appealing even to the platonist. The demands for a nominalistic formulation and for an “intrin- sic” one aren’t the same: a formulation of the theory of gravitation that made reference to a numerical inscription of . × − to represent the gravitational constant in m /kg− /s− might count as nominalistic, but would seem far less “intrinsic” than some formulations that aren’t fully nominalistic. This second motivation for my program started out as about as important to me as the first, but I never managed to make it or the idea of intrinsic explanation very precise, and the philosophical justification that I embarked on in the ensuing decade was devoted almost exclusively to the first motivation. (A partial exception is the discussion of the distinction between “heavy duty platonism”, “moderate platonism”, and “very moderate platonism” in Field b.)  OUP CORRECTED PROOF – FINAL, //, SPi pre fac e to se c on d e di t i on P- Besides containing only cursory philosophical discussion, SWN was incomplete as even a sketch of how to avoid literally believing in math- ematical entities. While it suggested a method of formulating (certain kinds of) fundamental physical theories in a way free of reference to mathematical entities, it said nothing about the use of mathematical enti- ties in other areas, in particular in using metalogic to facilitate logical reasoning. What the nominalist should say about metalogic is a matter I turned to in the next decade or so, in particular in Field , , and . And that work is indirectly relevant to the case of fundamen- tal physical theories. For as David Malament (: –) noted in an excellent critical review of SWN, much discussion of such theories con- cerns the existence of models with certain features, or (as in the case of determinism) whether all models with certain features have certain other features, and so on: in short, it concerns claims about what is consistent with the theory, or what follows from it given certain assumptions. I certainly think such questions are sensible; my view of them is that the model-theoretic formulations are “abstract counterparts” of formulations in terms of logical possibility, which I argued to be intelligible indepen- dent of mathematical entities (and to be a clearer and more austere notion than “metaphysical possibility”). SWN also said nothing about the use of mathematics in sciences other than fundamental physics, e.g. economics or psychology. My view was that these theories are heavily idealized any- way, so not candidates for literal belief, so that if mathematical entities were indispensable in them it would be of no ontological significance. Still, I thought that even for idealized theories there is strong motivation for finding an “intrinsic” formulation over an “extrinsic” one (e.g. in Bayesian psychology, using relations of comparative credence rather than numerical credence functions); while I did not in any way contribute to that program, I had an interest in it. The book dealt only with Newtonian gravitation theory, but it was per- fectly clear that the basic ideas extend to other field theories in flat space- time, such as classical electrodynamics (viewed special-relativistically). A number of people have suggested that there would be a problem extend- ing it to general relativity. This surprised me: the methods that I used to handle quantities were clearly modeled after the treatment of co- and contra-variant tensors in differential geometry, and in the footnote at the end of Chapter , Section E I outlined a way in which one could apparently carry them over to curved space-times with an affine connection (i.e. with  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on a notion of geodesic), or at least with a metric. As I said there, there are details that would need to be worked out (I give a slightly more complete sketch at the end of this Preface), and maybe there is a problem, but to my knowledge, none of the people who have expressed skepticism have given any hints as to where such problems might lie. There do seem to be problems extending to certain other kinds of physical theory. In his aforementioned review, David Malament raised the problem of theories formulated in terms of configuration space or phase space: Lagrangian and Hamiltonian formulations of classical mechanics, Gibbs-style statistical mechanics, and quantum theory.F My initial reaction to the examples of Lagrangian and Hamiltonian formulations was that these formulations are mathematically convenient and have heuristic value but can be thought of as instrumental. Perhaps one could say something like this of statistical mechanics too, without giving up the idea that, for truly basic theories, everything needed in the formulation of the theory must genuinely represent; though this doesn’t strike me as entirely comfortable. But in the case of (even non-relativistic) quantum theory the problem seems very hard to escape: while there is presumably little difficulty in describing the wave function “intrinsically” in terms of predicates of comparative amplitude and comparative phase-difference,F these would be predicates on configuration space (which is naturally viewed as the space of possible configurations of particles or possible deco- rations of space), or rather, on a space that adds to configuration space an extra dimension for time. There are ad hoc tricks that might allow recasting this in terms of predicates on ordinary space-time, but I don’t know any that clearly work, and they would seem bound to make the description far more complicated.F F A fuller discussion of the problem, with special reference to classical statistical mechanics, is in Meyer . F Cian Dorr has pointed out to me that phase differences in quantum mechanics aren’t invariant under Galilean transformations; as a result, the phase-difference comparison pred- icates need two extra places, for non-simultaneous points representing a state of motion. F Some of the discussion in the literature of the problem of carrying out the program of nominalization for quantum mechanics suggests that the problem is even worse, on the grounds that we would need to have a nominalistic analog of the algebra of Hermitian operators on Hilbert space. This is, at the very least, contentious: for instance, it doesn’t arise on the view that “the operator observables of quantum mechanics are [merely] book- keeping devices for effective wave function statistics” (Dürr and Teufel : ), a view which I find compelling independent of issues about nominalism. But even if the problem isn’t worse, it is bad enough!  OUP CORRECTED PROOF – FINAL, //, SPi pre fac e to se c on d e di t i on P- Independent of a commitment to phase space or configuration space, and indeed independent of issues of nominalism in general, there are other features of both classical statistical mechanics and quantum mechanics that suggest a partially instrumentalist treatment: I have in mind especially the notion of chance, whose role as simultaneously describing reality and directing our degrees of belief raises many philo- sophical perplexities. These are among the pressures that eventually led me away from the strict Quinean standard that one’s ontological commitments are to be read off one’s views about the best ultimate theory, and toward the more relaxed ontological attitude to be suggested here. I should mention that a number of people have made technical contri- butions to the positive program, most notably Frank Arntzenius and Cian Dorr in chapter  of Arntzenius . (Among other things, they sketch an approach to general relativity that doesn’t depend on the differential manifold having an affine connection, and so is presumably generalizable to other space-time theories in a way that my approach isn’t; they also sketch an approach to fiber-bundle theories. In addition, they provide a good discussion of the philosophical significance of the program.) Brent Mundy has also contributed ( and elsewhere); his program allows quantification over physical properties, which I avoided, but I did say in Chapter  that quantification over physical properties would be “at least arguably nominalistic”. John Burgess’s  also deserves mention, though the philosophical concerns behind it are rather different from mine: not only was he thoroughly opposed to nominalism, he also didn’t share the (admittedly somewhat vague) desire to formulate laws “intrinsically”. This is illustrated, for instance, by his heavy use of coding devices, and by his willingness to “quantify over arbitrary choices” of coordinate systems rather than avoiding the use of coordinate systems in the first place.F (His claim in Burgess  that space-time is every bit as dispensable as numbers depends on this.) F Like me, Burgess uses standard work in geometry and measurement theory to present the geometry and the “spaces” of physical quantities, and this is all as “intrinsic” in his case as in mine. He uses the coding devices to define the mathematics within the physics. My reservation about his approach is that when it comes to formulating the laws, he simply uses the standard mathematized formulations, understanding the mathematics as defined in this way; the formulations of the laws thus inherit the arbitrary coding built into his geometric definition of the mathematics. Similarly, the mathematized formulations depend on arbitrary choices, even if it is then shown that the choices don’t matter.  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on SWN generated quite a lot of critical discussion. In part to avoid obviating any of this (but mainly because I’m far too lazy), I’m repub- lishing the book without any substantive changes: I’ve merely corrected typographical errors and minor slips, and made a few other changes to improve clarity. For instance, I’ve broken up some overlong sentences and paragraphs, and improved the formatting for added readability; and I’ve switched from endnotes to footnotes, and put the references into a separate bibliography. Also I’ve done a bit of rewriting in a couple of proofs in Chapter  to make them easier to follow; and I’ve added two clarificatory footnotes, which I’ve marked as new to this edition. There are two slightly less minor slips in Chapter  concerning the definition of conservativeness (one in a footnote and the other in the Appendix) that I’ve treated specially: I’ve corrected them, but also noted the original wording. (I don’t think anyone could have seriously been misled as to what was intended in either case, but I do know of one published discus- sion that rests on the original wording of the footnote, and it’s possible that this is also so for the claim in the Appendix.) Similarly in Chapter , in formulating the first order nominalistic theory I omitted a needed axiom; I have added it, with a note on the addition, and done some rewording to accommodate its inclusion. I have also decided against explicitly commenting in this preface on the critical discussions of the book, except in passing: it would be hard for me to do a good job after so many years away from these issues, and I fear that if I got in I could never get out. But I will make a few general remarks about ways in which my thinking has evolved since I wrote the book. It should go without saying that my thoughts have doubtless been influenced by the literature and by com- ments I have heard over the years. I’m sure that were I to try to cite those who have influenced me I would come up with only a tiny proportion of those who belong on the list, so I will not try. .. Arithmetic and Cardinality Quantifiers One of the parts of the book I’m least satisfied with is the second chapter, on the arithmetic of the natural numbers. The view allowed us to regard sentences of form There are exactly  Fs  OUP CORRECTED PROOF – FINAL, //, SPi a ri t h m et i c a n d c a rdi nal i t y qua n t i f i e r s P- as literally true, since they could be paraphrased in terms of first-order logic with identity, quantifying over nothing but Fs (and in particular, not quantifying over numbers). But it did not allow us to regard sentences of form There is a prime number of Fs as literally true, because it could not be so paraphrased. I now think the right response to this is that ‘There is a prime number of ’ is a perfectly respectable quantifier in its own right. In the final chapter I contemplated adding some other quantifiers as primitive, e.g. the binary quantifier There are fewer Fs than Gs. But I now think it clear that a satisfactory theory must systematically add the means for defining a vast array of such quantifiers. Like Hodes , I’m inclined to the following combination of views: () Logic shouldn’t postulate numbers. () Logic should contain a rich theory of cardinality quantifiers— much richer than available in first-order logic. () The primary point of the arithmetic of natural numbers is to encode these cardinality quantifiers, or at least some of them, and to encode the logical relations among them. Arithmetic gives us a simple means for formulating the logic of such quantifiers; but at least when taken at face value, it does so by means of a fiction, or at least in a way that goes beyond the bounds of logic. The underlying idea is that because of the obvious ties between natural numbers and numerical quantifiers, the arithmetic of natural numbers is much closer to logic than other parts of mathematics are. In carrying out this idea, it would be nice to formulate the logic of such cardinality quantifiers directly, without going beyond the bounds of logic. Exactly how this ought to go, I’m not sure. (I don’t much like Hodes’ way of doing it, which involves impredicative higher-order logic; I don’t think the impredicativity sits well with his insistence that the “concepts” that the second order quantifiers range over are in some non-mathematical sense “predicative entities”.F ) A first question is how vast an array of F If properties are “predicative entities” in Hodes’ loose sense of being somehow like predicates, then the relation of instantiating or falling under should be “predicative” in a  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on quantifiers one wants to come out of it. I’d be inclined to want, at the very least, for each k ≥  and each primitive recursive k-place predicate , a k-ary quantifier Q , where (Q x)[F (x)... Fk (x)] means intuitively that (ηxF (x),... , ηxFk (x)) (where ηxFi (x) is the number of x such that Fi (x)).F And one might well think that the scope should include more than the primitive recursive. But whatever the decision on this, a second question is how best to formulate the theory so that it develops these quantifiers systematically from a small definitional base, without looking like it’s relying on an ontology of numbers (or of anything else). This program strikes me as well worth pursuing.F One option, which seems more appealing to me now than it would have when I wrote SWN, is to partially mimic Hodes, but with some sort of predicative higher-order logic. (That’s in a somewhat loose sense of ‘predicative’ that includes non-stratified approaches, and in particular includes the sort of nonclassical non-stratified approach to properties mentioned in note F. These latter approaches don’t actually restrict property-comprehension, but they are broadly predicative in that they corresponding sense of being like satisfaction. And in that case, whatever precautions you think are needed to avoid semantic paradoxes are needed for properties too. And on almost every view of the semantic paradoxes, this gives rise to something like predicativity in the mathematical sense. (The most familiar classical view of the semantic paradoxes is Tarski’s, according to which you have to stratify linguistic predicates; this corresponds almost exactly to the original technical notion of predicativity. Variant solutions tend to avoid the strati- fication, but something of the flavor of predicativity remains: e.g. on the property analog of the account of satisfaction in Kripke  or its extension in Field , while there are no predicativity restrictions on comprehension, still the law of excluded middle may fail for certain properties that are defined only impredicatively.) F As stated, the specification doesn’t handle the case where for one or more of the i there are infinitely many Fi. What we really need to do is first extend the domain of application of  from the natural numbers to the natural numbers plus a number “infinity” (which does not differentiate among infinite cardinalities); the intuitive meaning above is for the extended . (The notion of primitive recursiveness could be extended to the natural numbers plus infinity by means of the bijection taking infinity to  and each natural number to its successor. This would have the consequence that the finiteness quantifier counts as primitive recursive, which raises a question about how to understand the extended logic; I’ll say a bit about this starting in the paragraph after next.) F I believe that one commentator on SWN, I can’t remember who, suggested that I interpret such cardinality quantifiers in terms of numeral-shaped regions of space-time. I imagine that he or she was joking, but in case not: it is totally alien to the methodology of SWN to invoke space-time regions except in the context of theories that have space-time as their subject matter; and even there, it is totally alien to the methodology to use constructs that are “extrinsic” in a sense hard to precisely define but of which the proposal here would be a gross exemplar.  OUP CORRECTED PROOF – FINAL, //, SPi a ri t h m et i c a n d c a rdi nal i t y qua n t i f i e r s P- don’t generally guarantee classical logic for impredicative properties.) Such a broadly predicativist approach wouldn’t yield the logicist inter- pretation of arithmetic that Hodes wants (it wouldn’t yield a translation of arithmetic language in which you can prove mathematical induction), but it would suffice for defining a vast array of numerical quantifiers and proving many properties of them. But it’s also worth exploring the possibility of developing a theory that gives you a large array of cardinality quantifiers without invoking even predicative higher-order quantification, by using some sort of inductive procedure for generating new cardinality quantifiers from some basic ones. (This would probably involve the use of schematic variables in the quantifier subscripts.) There are a number of prima facie possible ways to proceed here; I don’t know whether any of them would ultimately yield a satisfactory logic. The final chapter actually did make a suggestion in the direction of extending logic to allow for more cardinality quantification: it suggested the addition of a binary quantifer ‘fewer than’ (interpreted in such a way as to not make distinctions among infinite cardinalities). To this I expressed a somewhat ambivalent attitude. On the one hand I took it to be far more attractive than the introduction of second-order devices (including the “complete logic of Goodmanian sums”, i.e. impredicative second-order mereology, on which more presently). On the other hand, after pointing out that a ‘fewer than’ quantifier enables us to define a finiteness quantifier Qfin (or ∃fin as I somewhat inappropriately called it), I pointed out that if one defines logical consequence in the usual model-theoretic way this will lead to a consequence relation that is neither compact nor recursively enumerable, which may seem to go against the idea of logic. (The same would be true if one invoked a finiteness predicate of regions and took it to be, like ‘=’, a logical notion.) The latter consideration led me to tentatively propose dispensing with the finiteness quantifier, in the applications I put it to in Newtonian grav- itation theory,F by a predicate of regions, taken as non-logical. But there are better courses (that don’t involve avoiding the use of the notion in the F As we’ll see in .(D), I could have avoided the use of the notion of finiteness there by an independently motivated expansion of the primitives. But the burden of this section is that even aside from the needs of the Newtonian theory, a treatment of cardinality quan- tifiers is highly desirable.  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on theory—see note F). One is to keep the quantifier and come up with some kind of deductive system for it; exactly what that deductive system should involve depends on what general framework for defining cardi- nality quantifiers is best. But in any case, such a deductive system would presumably involve some sort of induction schema, and an adequate treatment of Qfin as logical requires the extensibility of the schemas as the language expands (see subsection ...). A variant procedure would be to use a predicate of regions (say a finiteness predicate) instead of the special quantifier, and axiomatize it with an induction schema, but treat the predicate as logical by understanding the schema to be extensible as the language expands. Either way, we could regard conclusions obtained by such a deductive system as valid, without commitment one way or the other to the question of whether conclusions involving Qfin or the finiteness predicate that can’t be established in this manner but which are valid according to the most obvious model-theoretic account ought to be given the honorific ‘valid’. A smaller point about this chapter is that, contrary to what it suggests, you can’t really make it all that much easier to assess nominalistic infer- ences like the aardvaark inference simply by using arithmetic to code the quantifiers: to make the inferences manageable, you do need that, but you need some metalogic as well. This is one of several points in the book where work I did later, on how a nominalist should understand metalogic, is relevant to giving the full picture. But this doesn’t affect the main point I was trying to make, which is that the use of arithmetic to code the quan- tifiers is perfectly legitimate from the nominalist viewpoint, given the conservativeness of impure number theory (to be discussed presently). .. Mereology and Logic The other way (besides the finiteness quantifier) that the middle chapters of SWN go beyond first-order logic is in connection with mereology, and I was far less happy with this. Indeed, I recognized that mereology as I was understanding it simply isn’t part of logic if we understand logic as topic- neutral, for mereology as I understood it deals with special entities, space- time regions.F For that reason, it has no use except when space-time is F There is an alternative attitude toward mereology, according to which any entities can be “summed” in a way that needn’t have a spatial interpretation. I suppose that if one  OUP CORRECTED PROOF – FINAL, //, SPi m e re ol o gy an d l o g i c P- the subject matter. (There is some precedent for use of the term ‘logic’ in connection with special subject matters—witness “temporal logic”—but on the whole I think that the extended use is unfortunate.) But if mere- ology is simply part of the theory of space-time structure, it doesn’t seem legitimate to invoke a special and powerful consequence relation in con- nection with it. While I’m sure that I was aware of this, I’m embarrassed to say that the book doesn’t address it head on. The last half of Chapter  does however express a preference for avoiding the special consequence relation, and making do with first-order logic, or perhaps first-order logic supplemented with the logic of the cardinality quantifier Qfin. But this was left somewhat programmatic. The last chapter does not sufficiently emphasize the difference between going beyond first-order logic in connection with mereology and going beyond it in connection with cardinality quantification. I’ve already men- tioned one point: cardinality quantification is topic-neutral, whereas mereology (as I construe it) isn’t. But there’s a related point, which is how we understand schemas. If a physical theory says something about regions definable in the language of that theory, there is no obvious reason why it needs to say the corresponding thing about regions not definable in that language but definable in broader languages; because of this, when one posits regions via a comprehension schema, there is no reason to think that this involves an implicit commitment that when the language expands, the same comprehension schema together with the same assumptions about regions will be unrestrictedly valid. With cardinality quantifiers, the situation seems quite different I think that our understanding of finiteness does involve a commitment that the rules here hold not just in the case of language as it is currently but to any expansion of the language. (Certainly other logical schemas behave like this: we don’t normally think that the deMorgan laws are perfectly good logical laws for our current language but don’t automatically hold for expansions of the language.) accepted such a view, one could regard regions of space-time as atoms, and regard the “sum” of two perhaps overlapping regions r and r as an entity distinct from the sum of any other regions, and from the smallest region that includes all spatial parts of r and r. That would yield far more expressive power, but it strikes me (and struck me then) as philosophically unpalatable: it would just be monadic second-order logic (of a presumably impredicative sort) by another name.  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on I’ll say more about these issues after I’ve discussed why extensions of first-order logic might seem needed, which is in connection with repre- sentation theorems. .. Representation Theorems The reason I found it necessary (at least temporarily) to employ the “com- plete logic of mereology” was for the Hilbert representation theorem that I discussed in Chapters  and  and extended in the chapters that follow it. And these later extensions of the representation theorem in this form seem enough to satisfy a reasonable demand for “intrinsicness”, even if the “complete logic” is thought of as depending on set theory. But of course I wanted a formulation that was genuinely nominalistic, and for that purpose any dependence on set theory was verboten. However, as I mentioned in passing in Chapter , more general representation theorems are available, where the representation isn’t nec- essarily by the real numbers but by some real-closed field. Tarski  gives a representation theorem for a purely geometric theory in which there is quantification only over points; it allows representation by an arbitrary real-closed field. If one extends the underlying geometry to include regions, with a first-order axiomatization of them that doesn’t rely on sets, then only rather special real-closed fields are representing fields. (That’s because the comprehension schema for regions will guarantee the existence of regions that are specified via impredicative quantification over regions, and because Dedekind completeness (which can now be given in a single axiom instead of a schema) will then guarantee that if bounded these have closest bounds.) This more general sort of repre- sentation theorem can be used for physics more generally; I’m not sure that I was fully aware of this in the book (though perhaps there are hints in Chapter ), but this was a major theme of a follow-up paper (Field a). The extended representation theorem for physics given in that paper says that for any model M of the nominalistic physical theory N, there is a real-closed field FM , depending on M, whose “real numbers” can represent distance and the various physical quantities. (The same representing field is used for both.) Again only rather special real-closed fields are representing fields, not only for the reason given above but also because the physical vocabulary of the theory can be used to specify  OUP CORRECTED PROOF – FINAL, //, SPi re pre se n tat i on t h e ore m s P- bounded sub-regions of lines, and the Dedekind completeness axiom for regions then guarantees that these too have closest bounds. So FM will be “close to the real numbers”. (If the quantifier Qfin is used in N, and the model M is assumed to treat it standardly, then FM is Archimedean, so can be taken to be a subfield of the reals.) I think I ought to have used representation theorems of this more gen- eral sort in the book. (Mundy  also advocates their use.) It wouldn’t have changed a whole lot in the position advocated, but it would have made the first-order option discussed in Chapter  more appealing. (Or the “almost first-order” option also mentioned there, of using Qfin but nothing beyond that.) Of course, as with any version of the first-order (or “almost first-order”) option, it would mean that the proposed nominal- ized physics doesn’t capture quite the full content of standard platonistic physics. I doubt that the loss would be of much significance to physics: work on subsystems of second-order arithmetic (see Simpson  for an excellent survey) suggest that far less than the full content of even Henkin second-order arithmetic plays a role in physics. (In the unlikely event that some aspect of standard platonistic physics not captured in the proposed nominalistic axiomatization proved important, we could look for a richer nominalistic theory that did capture it: it’s not as if this part of mathematics is in principle inapplicable.) It would be nice to know just where the mathematics used in the platonistic theories that one gets from the representation theorems mentioned (the one without Qfin and the one with it) fit with the hierarchy of subsystems of second-order arithmetic, but that is better addressed by others. (The answer might well be affected by the need, noted in .(A), to include a nominalistic analog of integration theory in the nominalistic system.) We could also avoid the use of representation theorems altogether. This is what is proposed in Burgess  and Burgess and Rosen : instead of representation theorems, we develop a piece of mathematics (a subsystem of second-order number theory) within a very simple conservative exten- sion of the synthetic theory, one obtained by adding certain equivalence classes. (For instance, we identify real numbers with equivalence classes of ratios of line-lengths.) I’m not sure that this is ultimately very different from the approach of the previous paragraph, but it gives a different emphasis in three respects. First is their de-emphasis of intrinsicness in the formulation of laws, a matter I’ve already discussed. Second, their approach encourages the view that we’re in a fairly literal sense defining  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on the mathematics geometrically, and that once so defined we’re free to use it in applications (say in the theory of credences); whereas part of my picture was that if ratios of line-lengths aren’t “intrinsic to” credences we shouldn’t use real numbers so defined for credences. Third, their approach doesn’t emphasize that the mathematics that can’t be so defined, such as higher set theory, can be legitimately employed in geometry and else- where, by the conservativeness property that I now turn to.F .. Conservativeness One of the points on which the book was widely criticized, and to some degree justifiedly, was the discussion of “conservativeness” in the opening chapter. I stand by the basic idea, but there are some things I wish I had done differently. ... The Basic Idea The idea was to state a criterion of goodness for mathematical theo- ries that doesn’t involve truth. In some ways conservativeness was to be stronger than truth: it was intended to capture the idea that whether a mathematical theory is good is independent of what the physical world is like. It is often assumed that the way to capture this idea is to say that mathematics is necessarily true, but my criterion of conservativeness was to be “like necessary truth but without the truth”. An informal character- ization, close to what I gave, would be this: A mathematical theory S is conservative iff for any nominalistic assertion A, and any body N of such assertions, A isn’t a consequence of N + S unless A is a consequence of N alone.F (“N + S” simply meant the union of N and S, considered as sets of theo- rems. In subsection .. I’ll look at the charge that this gives too anemic a reading of what it is to “add” a mathematical theory to a nominalistic one.) The claim was that good mathematical theories are conservative in this sense. F As we’ll see, its use in empirical comprehension principles goes beyond conservativeness. F Both this definition and the variant in the book are in platonistic terms: I left the task of “nominalizing” metalogic to later work.  OUP CORRECTED PROOF – FINAL, //, SPi c on se rvat i v e n e s s P- An equivalent formulation invokes the notion of consistency according to which a theory is consistent iff not everything is a consequence of it (or if no contradiction is a consequence of it): then a conservative theory is one that is consistent with every consistent nominalistic theory, i.e. A mathematical theory S is conservative iff for any consistent body N of nominalistic assertions, N + S is also consistent. There is a boring issue to be addressed about the reading of ‘nominalis- tic assertion’. In the book I counted claims like ‘Everything has mass’ and  ‘There are fewer than  things’ as nominalistic, on the grounds that they don’t imply the existence of mathematical objects as normally con- ceived. But not only don’t they imply the existence of mathematical enti- ties, they pretty much rule them out: more precisely, the second of the two claims is inconsistent with the existence claims of standard mathematics, and the first is inconsistent with the usual conception of mathematical entities and hence with what would seem to be a harmless supplementa- tion of standard mathematics. And because of this, conservativeness as formulated here (with N + S taken as simply the union of N and S) would come out trivially false for standard mathematical S, on that understand- ing of ‘nominalistic assertion’. There are two ways to avoid the problem. One way keeps this formulation of conservativeness but takes ‘nominal- istic assertion’ more narrowly: it requires a nominalistic assertion to be neutral to the existence of sets or numbers or other sorts of mathematical entities, i.e. they not only can’t imply their existence but also cannot rule them out. The obvious way to achieve this is to suppose that, in a nomi- nalistic assertion, all quantifiers must be restricted to non-mathematical objects; and I’ve used that understanding in various papers subsequent to the book. The second way to avoid the problem, the one adopted in the book, keeps the broader understanding of ‘nominalistic assertion’, but replaces the formulation of conservativeness with something more com- plicated that makes appropriate quantifier restrictions to achieve neutral- ity (and handles a complication about an existence assumption built into standard logic). There is nothing of any philosophical significance in the divergence between these approaches, they’re simply two different ways of doing the same thing. (They give rise to a further minor difference in the technical formulation of conservativeness, to be addressed later.) Of more importance is what is meant by ‘mathematical theory’. Obvi- ously I didn’t mean ‘theory that uses mathematics (or mathematical  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on entities) in its formulation’, for standard formulations of physical the- ory do that, and they have nominalistic consequences on their own and hence certainly aren’t conservative. Rather, I meant ‘theory that should be regarded as part of mathematics’. This includes impure mathematical theories as well as pure ones. A pure mathematical theory doesn’t speak at all about non- mathematical entities, and its only non-logical vocabulary is special to mathematics; such theories (for which conservativeness reduces to con- sistency) are of interest to applications only as parts of larger mathemati- cal theories. The larger mathematical theories are impure, like impure set theory or impure number theory; and it is these for which conservative- ness strengthens consistency. I take impure set theory to () Not only posit “pure” sets but conditionally posit sets of physical objects: the theory says that for any physical objects there may be, there are lots of sets that have them as members—including a set of all of them. (Also of course, sets of sets of physical objects, sets that include both pure sets and physical objects, and much more.) () Allow the vocabulary for physical objects to be used in specifying what sets there are. For instance, in impure set theory based on a language that includes ‘star’, we may speak of the set of all stars. (Without allowing words such as ‘star’ to appear in the compre- hension principle, this is a set which wouldn’t be definable even with parameters, if there are infinitely many stars.) Despite these features, impure set theory is part of mathematics: a pla- tonist would regard it as true by mathematical necessity. A somewhat similar impure theory (though simpler because it doesn’t posit objects not in the corresponding pure theory) is impure number theory. Impure number theory includes an operator ‘the number of ’ which can apply to formulas that include non-mathematical vocabulary, so that we can say such things as “For any star x, if there are exactly two planets of x then the number of planets of x is prime”. I didn’t specifically talk about impure number theory in the book—there was no real need to, since it is a consequence of impure set theory and  OUP CORRECTED PROOF – FINAL, //, SPi c on se rvat i v e n e s s P- so the latter can serve all the purposes that the former can—but it is another example of the kind of impure theory I had in mind. Theories like these, even though impure, are the sort of things that a platonist would regard as true by mathematical necessity; and they are the kind of theory I held to be conservative. By contrast, a claim like ‘Every object is located in Euclidean -space’ isn’t part of mathematics but of physics: no platonist would regard it as true of mathematical necessity, and the conservativeness condition quite properly rules it as not part of what we should regard as good mathematics.F There seems to be some confusion in the literature between on the one hand my conservativeness claims (that good mathematics is conservative, and that standard impure number theory and impure set theory are good in this sense), which I expected would be uncontroversial once pointed out, and on the other hand certain claims related to the dispensability of mathematical entities, which I took as far less obvious. These less obvious claims concern the existence of suitable nominalistic theories. Once one has an interesting nominalistic theory T , the conservativeness of impure set theory tells us that the result of adding impure set theory to it adds no new nominalistic consequences. So if it can also be shown that the result of adding impure set theory to it has pretty much the same nominalistic consequences as a platonistic theory T, then conservativeness entails that T and T have pretty much the same nominalistic consequences, which is what one needs for the dispensabilty of mathematical entities for T. But the conservativeness claim does nothing to show the existence of the interesting nominalistic theories.F My apologies if I’m belaboring the obvious, but even as astute a philosopher of mathematics as Michael Dummett seems to have been confused by the point. In Dummett  he writes, Field envisages the justification of his conservative extension thesis as being accomplished only piecemeal. For each mathematical theory, and each theory to F A more “mathematical-looking” theory that is ruled out of good mathematics by the conservativeness criterion is the modification of impure set theory obtained by replacing the axiom of infinity by its negation (keeping the replacement schema and the existence of a set of all non-sets): if it includes the axiom of choice it rules out all theories in which the physical world is infinite, and even without choice it rules out most, e.g. those with a definable infinite linear order relation. F Of course, each platonistic theory T conservatively extends the “theory” consisting of the Craigian transcription of the set of nominalistic consequences of T; but I assume that everyone agrees that such Craigian theories aren’t interesting.  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on which it is to be applied, the demonstration is to be carried out specifically for those two theories; no presumption is created by the successful execution of the programme for one case that it will work in others. (p. ) This is incorrect: I gave reasons (indeed, platonistic proofs of a sort) for thinking that impure set theory is generally conservative, from which it follows that weaker theories are too. For instance, consider the theory obtained by taking the theory of real numbers and adding to it just enough impure set theory to speak of functions from physical objects to the real numbers, so that we can apply the real numbers to the physical objects. This is guaranteed to apply conservatively to any nominalistic theory whatsoever, as a result of the fact that impure set theory does and that this theory is a subtheory of that. Dummett does have a point here, but it isn’t about conservativeness: rather, it is that finding an interesting nominal- ization of one physical theory by no means guarantees that one can find it for another. (Even so modified, his ‘no presumption’ claim is a bit strong: to the extent that the theories are similar in mathematical structure, I’d think that nominalization of one is grounds for expecting that we could nominalize the other. But it is right that there is no guarantee.)F I’ve been spelling out the intuitive idea of conservativeness; but there are some issues that need to be clarified. I will discuss the main issues in the next subsection. But first I should make more explicit one other technicality in the definition of ‘nominalistic assertion’, especially since the definition as given in footnote  of the original edition included a slip. If we want to use the simple formulation of conservativeness, we need that a nominalistic assertion (in a language with no singular terms other than variables, to make things simple) is one: F It’s possible that rather than confusing conservativeness with nominalizability, Dummett was misunderstanding how conservativeness was supposed to work. Perhaps he thought that the impure mathematical theory containing real numbers that we add to certain nominalistic theories didn’t consist in simply the theory of real numbers plus a theory of functions from objects to numbers that follows from impure set theory, but instead was a theory with substantial physical presuppositions that might be met for some physical theories but not for others. But I’m not sure how to fill out such an alternative interpretation in detail, and I think it should have been clear from the book that whatever substantial physical presuppositions are needed for an application of the real numbers need to be built into the nominalistic theory rather than be taken as part of the mathematics. So again, whatever objection there might be in the vicinity of Dummett’s remarks, it is not to the conservativeness claim I was defending.  OUP CORRECTED PROOF – FINAL, //, SPi c on se rvat i v e n e s s P- (A) in which all quantifiers are restricted to non-mathematical objects, and (B) which employs no specifically mathematical vocabulary. (Logical vocabulary, including ‘=’, is of course allowed; so is a special term ‘mathematical’, used to make the restriction to non-mathematical objects explicit.) [(B) is important: without it, the claim (∗ ) “there are non-mathematical x and y such that x ∈ y” would count as “nominalistic”. But impure set theory says that anything that has a member is a set, and it was essen- tial to build into impure set theory that sets are mathematical (in order that the quantifier restrictions to non-mathematical objects serve their purpose). So (∗ ) is inconsistent with impure set theory, and that would violate conservativeness if (∗ ) were counted nominalistic. This is for the simple formulation of conservativeness. For the one in the book, (A) can be dropped, since the restriction of variables is done by other means; but (B) is still needed, and indeed must be strengthened to preclude “mathematical” from appearing in nominalistic statements, or at least to include restrictions on the kind of occurrences it can have in them.F ] Unfortunately in that footnote  I slipped in the formulation of (B), and said that to be nominalistic, a statement must not employ any non-logical vocabulary that appears in our mathematical theories. This made no sense in the context, where I’d stressed that all non-mathematical vocabulary is part of the impure mathematical theories of interest; and none of the subsequent discussion depended on it. I’m sure no one was misled, but I’ve corrected it in the current edition, with a statement about the change to the original text. F Thanks to Marko Malink for the observation that it must be strengthened in this way. Illustration: In Principle C of Chapter , take N to be the rather trivial theory ∀x(x = x), and A to be ‘Everything is non-mathematical’; A doesn’t follow from N, but A∗ is vacuous. (The last sentence of the derivation of Principle C from Principle C in the first paragraph of note  of the text relied on the assumption that nominalistic claims don’t include ‘mathematical’.) The blanket refusal to let any sentence containing ‘mathematical’ count as nominalistic is slightly counterintuitive, in that no statement of form A∗ where A contains quantifiers would count as nominalistic on this criterion. Perhaps it would be more natural to disallow the use of ‘mathematical’ in nominalistic claims except in the contexts ‘for all non-mathematical x...’ and ‘for some non-mathematical x...’. Or we could avoid the whole issue by using the simpler sort of formulation of conservativeness given above.)  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on ... Consequence, Proof, and ω-Conservativeness In the Preliminary Remarks and the opening chapter I tacitly assumed that we were working in first-order logic. I wrote these parts of the book early on, but as the book developed I seriously contemplated expanding the logic of the theory. One expansion was to something akin to “full” monadic second- order logic, since this was used in the Hilbert representation theorem. I rejected ordinary second-order logic on the grounds that the second- order variables are required to range over sets or Fregean concepts or some such things. (This understanding of second-order variables has recently been challenged—see Rayo and Yablo —though in my opin- ion the worries about impredicativity that I voiced earlier arise as well for the Rayo-Yablo view, and predicative second-order logic wouldn’t suffice for the representation theorem.) In a desperate attempt to keep the Hilbert representation theorem nonetheless, I suggested a less gen- eral version of monadic second-order logic that I called “the complete logic of mereology” [or “the complete logic of Goodmanian sums”, which was supposed to make it sound more nominalistically respectable]. The idea was to suppose, as a matter of logic(!), that the regions of space- time form a complete atomic Boolean algebra except for the absence of an empty region. Then quantifying over regions is a surrogate for quantifying over sets of atoms of the algebra, i.e. over sets of space- time points. This is less than one would get with monadic second- order reasoning generally: since regions were taken as basic entities, monadic second-order logic would have variables ranging over sets or concepts of arbitrary regions, not just of the atomic regions that are points. But I didn’t need that added power for any representation theo- rems (or anything else), so I thought the “complete logic of mereology” would do. But as I said in section ., the idea of a “logic” of mereology seems misguided. I regret having considered it. I also still tend to resist the more genuinely second-order option, especially in the impredicative and indeed non-axiomatic form that would be required for Hilbert’s version of the representation theorem. The other expansion I contemplated, which played a far more limited role though which I was much happier with, was to include the cardinality  OUP CORRECTED PROOF – FINAL, //, SPi c on se rvat i v e n e s s P- quantifier ‘fewer’ or ‘finitely many’, conceived as logical.F Given second- order logic, the cardinality quantifiers are definable (and given the “com- plete logic of mereological sums” they are definable as applied to points of space-time); nonetheless in Chapter  I considered the two expan- sions independently, so as to allow for the option of using the cardinality quantifiers without second-order logic or its mereological analogue. (This option was emphasized more emphatically in Field a.) As I’ve noted, this would allow the use of representation theorems in which the repre- senting field is guaranteed to be Archimedean. As I said, the early parts of the book were written before I appreci- ated the pressures toward going beyond first-order logic. (When those pressures did become evident, instead of rewriting the early material as I should have, I simply made a few inadequate remarks at the end of Chapter  and in Chapter  about how what was said earlier ought to be adjusted.) In a more-than-first-order setting, there is room to ask what is meant by ‘consequence’ as it occurs in the definition of conservativeness. If logic is taken to be thoroughly first order, there is no real issue about the extension of the consequence relation: we all agree that first-order consequence is syntactically characterizable. (At least, it is as long as we put issues of achieving a nominalistic metalogic aside, as I did in the book and I will continue to do here.) As long as we’re happy with the purely first- order extended representation theorem, there isn’t even a prima facie issue about what ‘consequence’ means in the definition of conservativeness. But what if we consider the cardinality quantifier ‘fewer’ or ‘finitely many’, or second-order quantifiers, or mereology, as logical? In each case, the obvious semantic definition of consequence (in terms of standard models) gives a relation that is neither compact nor recursively enumer- able; it would thus diverge from a syntactic definition in terms of proofs. Of course as a nominalist I wasn’t going to define consequence either in terms of models or in terms of proofs, but there was still the question F I didn’t consider the option of adding such quantifiers as new syntactic operators without considering them as logical: I assumed that if you didn’t want them as logical you would dispense with them in favor of predicates. (Conversely, I didn’t consider the issue of adding a predicate but treating it as logical, in the manner mentioned above in section . and discussed further below.) In retrospect the issue of operators vs. predicates seems inessential to the points at issue.  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on of whether on my understanding it lines up in extension with (a) the extension that it would have on a semantic construal were platonism true; (b) the extension it would have on a syntactic construal were platonism true; or (c) something else, presumably something in between. (Or maybe its extension is somewhat indeterminate.) In note  of Chapter  I took a stand for its lining up with the semantic (and explained the apparent syntactic remarks in the opening chapter as due to a temporary tacit assumption of first-order logic, where the two coincide). But at least in the case of full second-order logic or the “logic” of mereology, this stand makes much of the rhetoric in the opening chapter quite misleading. The reason is that, as many people correctly complained later (e.g. John Burgess and Stewart Shapiro), semantic conservativeness simply amounts to truth in this second-order context. Indeed I now think that we have no clear grasp of second-order consequence (or its mereological analog) in the sense intended in the book, i.e. the semantic consequence relation of “full” impredicative second-order logic; I think the book takes this notion entirely too uncritically at a number of points. On the other hand, the idea of a logic of cardinality quantifiers such as Qfin seems much more appealing. For it too, the obvious semantic char- acterization diverges from any syntactic one, and so the issue of how ‘con- sequence’ is to be understood arises in this context as well. And it seems to me that if we’re to regard this as a logical notion, we can’t suppose that our grasp of the induction principle is adequately captured by the totality of instances of the induction schema in any fixed first-order language. The usual alternative to such a syntactic characterization is a semantic characterization in terms of standard models. Of course, a nominalist can’t literally embrace a model-theoretic characterization, but a nominal- ist can’t literally embrace a proof-theoretic characterization either; again, I want to put aside here the issue of how exactly a nominalist ought to deal with metalogic. My claim is just that the semantic characterization is a better platonist approximation to how we should think about the logic of Qfin than is the syntactic characterization based on a fixed language. If we do want to employ cardinality quantifiers in our nominalistic physics and think they have a logic that can’t be syntactically character- ized, to what extent would that undermine the general picture of the role of mathematics I tried to paint in Chapter ? I don’t think it undermines the main idea: conservativeness in this sense (which to avoid any confusion might be called  OUP CORRECTED PROOF – FINAL, //, SPi c on se rvat i v e n e s s P- ω-conservativeness)F serves much of the function that necessary truth was supposed to serve, but doesn’t imply truth. A mathematical theory that (ω-)implied that the physical universe was finite, or that it was infinite, or (ω-)implied any conclusion about the number of planets or the fate of the Paris Commune even if that answer were true, would be bad mathematics. But two mathematical theories that both met this condition might disagree about many purely mathematical claims such as the size of the continuum; short of positing an ambiguity they couldn’t both be true, but on the ω-conservativeness criterion they could each be good mathematics. (Of course, there might be reasons why in some applications one was more useful; but which was the more useful might vary from one application to the next.) The lack of synthetic characterizability may however undermine some of my rhetoric. For instance, in drawing the contrast between mathemat- ics and physics I say that the conclusions we arrive at by applying mathe- matics to nominalistic premises “are not genuinely new, they are already derivable in a more long-winded fashion from the premises, without recourse to mathematical entities”. This claim can be “saved” by replacing “they are derivable from” by “they are ω-consequences of ”, but I admit that this pulls at least some of their punch.F As mentioned, conservativeness is a slight generalization of consis- tency. Of course, if it’s ω-conservativeness that’s in question, this is con- sistency in (essentially) ω-logic, and thus not explainable in terms of any deductive system in a fixed language. I don’t think this would seriously undermine the main point of Chapter . In the Appendix to that chapter I offer a (rather obvious) model-theoretic argument for why a platonist should believe that impure set theory (and hence any mathematics that F Incidentally, though I only came to realize it much later, the idea of ω-conservativeness is important in another context: in Kripke’s theory of truth. The part of Kripke  based on the Kleene evaluation schemes argues in effect that any theory adequate to basic syntax can be extended in an ω-conservative fashion to include a truth predicate in which (i) “True()” is everywhere intersubstitutable with “p” in non-intentional contexts, (ii) the usual composition rules hold, and (iii) ‘True’ is allowed in mathematical inductions. It’s well known that the extension isn’t deductively conservative (when the base theory is consistent and recursively enumerable): for the extended theory can prove the consistency of the base theory, whereas the base theory can’t. So the focus on ω-conservativeness in the book wasn’t really novel, though the use I put it to has a rather different flavor from what we have in Kripke’s theory. F Even in the fully first-order case, “long-windedness” isn’t really what’s at issue; rather, the mathematical formulation is an aid to seeing what the consequences are.  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on can be formalized in it) is conservative. It goes through as before, except that we must start the construction from an ω-model of M of T; what the construction then produces is an ω-model of ZFUV(T) + T∗. (Other arguments in the Appendix are more dependent on the logic being fully first order.) ... Logic and the Extensibility of Schemas It is built into my explanation of impure mathematical theories that any schemas that appear in such theories are to be interpreted as apply- ing not just to instances containing only mathematical vocabulary and quantifying only over mathematical entities, but to instances contain- ing physical vocabulary and/or quantifying over physical entities as well. (I don’t rule out that in special circumstances there might be reason to introduce restricted schemas that impose limits on the vocabulary or quantifier-ranges of the instances, but one doesn’t normally do so: the comprehension schema of impure set theory and the induction schema of impure number theory are supposed to apply to instances containing physical vocabulary, and it would cripple the application of mathematics to alter this.) What about the reverse? Should schemas in empirical theories be deemed to extend to instances containing mathematical vocabulary? In the book I tacitly made a two-fold assumption about this. The first half of the assumption was that, if the schemas appear in those theories as part of the logic, then they should be deemed to extend to all vocabulary including the mathematical. Logic, after all, is supposed to be topic-neutral. The second half of the assumption was that any schemas that appear in physical theories that aren’t based on logical principles (or on mathemat- ical principles, in the case of platonistic theories) should not be extended to new vocabulary, without special justification for doing so. This seemed natural since such extension of the schemas adds empirical content. There could of course be special justification for extending the schemas. Most obviously, there might be empirical evidence for the added empirical content one gains from such an extension. Even without empirical evi- dence, considerations of simplicity or naturalness might perhaps favor the extension, though in the case of extension to mathematical vocabulary these considerations are likely to look different to the platonist than to the nominalist. (If one already has mathematical entities, then a theory  OUP CORRECTED PROOF – FINAL, //, SPi c on se rvat i v e n e s s P- that extends an empirical schema to instances that quantify over them or that contain special vocabulary for them might seem somewhat simpler or more natural than theories with a more restricted schema, though without empirical support for the excess content this wouldn’t seem to me terribly weighty; but if one doesn’t already have the mathematical entities, there is no such added simplicity, indeed there is added complexity in introducing the mathematical apparatus to extend the schemas.) But the view was that, without any such special justification, there is no reason to extend the schemas that appear in non-logical and non-mathematical theories. To illustrate this two-fold assumption, and relate it to conservativeness: () If one were to regard mereology as logic, or take N as a second- order theory, then we should view the schematic letters in the Dedekind-completeness schema or the comprehension schema for regions as in effect second-order variables, and as such, indef- initely extensible. In that case, if we add to N a set theory S that postulates lots of sets, then we should understand “N + S” as expanding the schema to include such sets; this would lead to things being provable in a deductive system for N + S that aren’t provable in N without the expanded schema. This is no violation of conservativeness as I’ve explained it, because it depends on the assumption of second-order mereology or second-order logic more generally, where consequence exceeds provability. On this conception, unless S is a mathematically bad set theory, e.g. an inconsistent one, then N already implies whatever “N + S” does about the existence of regions; S merely serves to elucidate the meaning of the second-order quantifier and hence elucidate what were already consequences of N, in the non-axiomatizable sense of consequence in question. () On the other hand, if N is taken as a merely first-order theory, I took it that there was no reason to expand the Dedekind-completeness schema when adding S to it: N is a theory about regions (i.e. parts of space-time) alone, saying nothing about sets, and there is no reason on this picture why sets are in any way relevant to the notion of region. For instance, even if the mathematics pos- tulates non-measurable sets of points, there is no obvious reason why the physical theory must postulate regions corresponding to  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on them. (For instance, there is no obvious reason to regard a Banach- Tarski decomposition as physically meaningful.) So why should adding a claim about sets have any bearing on the understanding of the schema? (Of course if there were empirical support for the excess content that the platonistic theory with the extended schema has over the nominalistic one, the situation would be somewhat changed: the nominalist would want to try to attain this excess content without the platonism. But as I emphasized in the book, the excess content is quite recherché and currently has no empirical support.) () Similarly, if N is taken as an “almost first-order” theory with- out second-order devices, but employing a quantifier Qfin viewed as genuinely logical (or a finiteness predicate of regions, viewed as a logical predicate like ‘=’), then the induction schema used to axiomatize this quantifier (or predicate) should be taken as indef- initely extensible, including to mathematical predicates when S is added to it. In that case, in analogy to , the addition of S to the theory will enable one to prove more from instances of the com- pleteness schema that contain the quantifier (or predicate) than one can prove without S; but this is no violation of conservativeness, since it is merely elucidating what are already consequences of N. But since on this view there are no logical devices that go beyond first-order devices plus finiteness, there is no reason to expand the completeness schema to set-theoretic instances, absent empirical support for the expansion; in this respect the situation is as with . In all three cases, “N + S” is just the union of N and S; it’s just that N is conceived differently in the three cases. Could any of this sensibly be denied? I doubt that anyone would want to deny that schemas for notions viewed as logical are indefinitely extensible, but I suppose that one might question the implicit assumption in  (and the first part of ) that this extensibility goes all the way to instances that quantify over fictional objects or predicates appropriate to them. And if that assumption is wrong, then a nominalist should think that the indef- inite extensibility doesn’t extend to platonistic instances. If  were denied on this ground, then the “second-order nominalist” would presumably not be able to deduce anything that the first-order nominalist can’t. The second order option would simply be irrelevant: we ’ d have deductive conservativeness even in the second-order case, but no representation  OUP CORRECTED PROOF – FINAL, //, SPi c on se rvat i v e n e s s P- theorems stronger than those in the first-order theory. I’m not advocating this viewpoint, but perhaps it isn’t out of the question. Alternatively, I suppose that one could think that schematic letters even in empirical theories are required to be indefinitely extensible (even to vocabulary employed only in extensions of the original theory that the advocate of the original theory rejects). That would involve giving up  (and the last sentence of .) On such a view, the claim of conservativeness might be deemed misleading: it would still be correct if N + S is understood as simply the union of N and S, but on this viewpoint that might seem too anemic a reading of “adding” S to N. But I don’t see that this viewpoint has much appeal. (Indeed, I’m inclined to stipulate that to regard a comprehension schema for regions as indefinitely extensible in this way just is to treat the schematic letters for predicates as logical variables, and thus to invoke at least a fragment of second-order logic (the  fragment). Similarly, I’m inclined to stipulate that to regard an induction schema for a finiteness predicate as indefinitely extensible just is to treat finiteness as a logical notion.) ... Summary: Conservativeness and Representation Theorems To summarize, I concede to the book’s critics that Chapter , in combina- tion with the middle chapters, was rather misleading: together, they could well be taken to suggest that it is possible to simultaneously maintain both the syntactic conservativeness of mathematics and the full representation theorems of those middle chapters. As Shapiro (a) rightly says, you can’t have both. I noted this myself in Chapter  of the book. I didn’t there rule out keeping the full Hilbert-like representation theorems and taking the rele- vant conservativeness to be semantic conservativeness for the “complete logic of Goodmanian sums”; but I did make two not-very-developed alternative suggestions, one for going fully first order and the other for (in the terminology just used) going “almost first order” but using a primitive finiteness quantifier. In the book it probably appeared that this required giving up on representation theorems, but I made clear in the a paper that this is not so, it merely requires generalizing them by allowing for representing fields that are not quite the real numbers but are extremely similar. In the purely first-order case, we need no notion of conservative- ness beyond the syntactic, but a price is that the representing fields may be non-Archimedean. In the “almost first-order” case the representing  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on fields are all Archimedean, but we require a notion of ω-conservativeness, which goes beyond the syntactic but isn’t nearly as strong as the kind of semantic notion of conservativeness contemplated at some points of the book. I now think that both the purely first-order and “almost first-order” options are vastly more attractive than the option with the “complete logic of Goodmanian sums”. So, while the early part of the book is rather misleading, I still think that the basic line of Chapter  is right: the only obvious requirement on a good mathematics is that it be conservative, not that it be true. .. Indispensability Platonism is usually construed as requiring both the existence of mathe- matical entities and the objectivity of mathematics (in the sense described earlier in this Preface); and the “Quine-Putnam indispensability argu- ment” (Putnam ) is usually taken as an argument for platonism in this strong sense. At least when platonism is so construed, then I stand by the view that there is no serious argument for it other than the indispensabil- ity of mathematical entities to things outside mathematics (though this could include more than basic physics, e.g. it could include the metathe- ory for logic). But how good is even that argument? Of course if the nom- inalization program of SWN could be carried out, it would not ultimately be a good argument for any form of platonism. But in this section I will take for granted the premise that the program of nominalizing theories is likely to fail. And my question is, how good is the argument for platonism on that assumption of indispensability? We need to separate the question of how good the indispensability argument (taking that indispensability premise for granted) is as an argu- ment for mathematical objects and how good it is as an argument for objectivity. The extensive recent literature is focused much more on the first than on the second,F and I will start out with that. F An exception is Hellman , who advocates using modality in connection with higher-order logic to eliminate mathematical objects from physical theory and elsewhere, but thinks that the indispensability argument is still important for objectivity. (Putnam’s own view wasn’t all that different: see Putnam .) I’ll say a bit about his view in section .F.  OUP CORRECTED PROOF – FINAL, //, SPi i n di spe n s a b i l i t y P- ... Objects I now think that the indispensability argument for the existence of math- ematical objects is somewhat overblown. There’s a great deal of recent literature in this direction (e.g. Yablo  and , Sober , Melia , Maddy , Leng  and ). There is also some interesting work in partial defense of the indispensability argument (e.g. Colyvan , , , and Baker ); and one obvious challenge to those who would deny the relevance of the argument is that doing so seems to remove the means of deciding ontological questions. I don’t have a worked out opinion on all of the issues involved in this literature, but I will make some tentative remarks on some of these issues and on some related ones. First, on my understanding of the indispensability argument: discus- sion of this argument (especially by those who emphasize its roots in Quine as opposed to Putnam) is very often tied to some doctrine of “confirmational holism”, whose content tends to be left unclear. (It is sometimes formulated as the claim that all parts of our best theories are equally confirmed; that formulation has the advantage of being fairly clear, but the disadvantage of being totally preposterous.F ) In stating the indispensability argument it’s better to avoid any explicit talk of con- firmation, and talk instead of what to believe. As a crude first try, we might say: We should believe (e.g.) quantum electrodynamics, since it explains so many things and there’s no decent competitor that does so. Quantum electrodynamics entails that there are mathematical entities. So we should believe that there are. (A somewhat more nuanced version would take into account that there are bound to be competing theories we don’t know about, and involve as a premise that the competitors are likely to also entail that there are math- ematical entities.) I don’t see that anything worth calling “confirmational holism” is implicit in this. Quine advocated the indispensability argument in the context of the view that mathematics is fairly straightforwardly empirical—empirical in just the way that entrenched physical principles like the conservation F Does anyone really think that, in the early years of general relativity, the existence of gravitational waves and of black holes was as well-confirmed as the equivalence of inertial and gravitational mass or the gravitational redshift?  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on of energy-momentum are. (Some sort of “confirmational holism” may be relevant to this.) I was inclined to resist that view: indeed I took the conservativeness of mathematics, according to which good mathematics is compatible with any internally consistent theory of the physical world, to tell against the idea that mathematics has any empirical content in any ordinary sense.F Quine’s view seems to have been that if a certain math- ematical theory S proves indispensable to (e.g.) a basic physical theory T, then the empirical evidence for T would be evidence for S, and so S would in a straightforward sense be empirically confirmed. But as Sober has emphasized, the assumption that evidence for T is evidence for S appears to rely on a discredited view of evidence and confirmation (the “conse- quence condition”). Indeed, my focus on conservativeness was supposed to indicate that S was already completely acceptable as mathematics, so no evidence could raise its status. But there is another way of thinking about the bearing of indispens- ability arguments on empiricality, that doesn’t rely on the consequence condition. For if we assume that there are possible physical theories in which mathematical entities are indispensable as well as possible ones where they aren’t, the indispensability argument seems to require us to think that empirical evidence that favors theories of the first sort over theories of the second should enhance the extent to which we have literal belief in mathematical entities (as opposed to merely accepting them on conservativeness grounds).F So if, for instance, classical physics were completely nominalizable while quantum physics isn’t, then the early th century would have provided massive empirical evidence for the existence of mathematical objects (even though our mathematical theories would have already been fully acceptable for use in many con- texts on conservativeness grounds). I don’t think I really faced up to this unappealing consequence of taking indispensability arguments as F I’d occasionally flirted with, though never advocated, the view that logic is empirical. If it is, then since mathematics like every other discipline uses logic, maybe that’s enough to make mathematics in some sense empirical. But I took the conservativeness of math- ematics to show that, on any reasonable measure of “empirical content”, mathematics has no empirical content beyond that of the logic it employs. F More carefully: the evidence favors theories of the first sort over “atheistic variants” of theories of the second sort: variants which dispense with the mathematical entities and add the claim that there are no such things. These atheistic variants have the same empirical contents as the originals.  OUP CORRECTED PROOF – FINAL, //, SPi i n di spe n s a b i l i t y P- decisive for mathematical ontology. My hope was that mathematical enti- ties would prove indispensable to any reasonable fundamental physical theory, whether correct or not, so that insofar as evidence is limited to the selection of one “reasonable” fundamental theory over another it would prove irrelevant to the belief in mathematical entities. Of course this would make the task of establishing the requisite dispensability all the harder. Indeed, to some extent my hope wasn’t limited to fundamental the- ories. Of course, non-fundamental theories (including those in physics, such as thermodynamics and continuum mechanics) are accepted only as approximations, so any nominalistic formulation of them would obviously have to make use of false idealizing assumptions. That doesn’t undermine the interest of trying to provide intrinsic nominalistic formu- lations of such theories based on such false assumptions. But it isn’t clear that a reasonable response to an indispensability argument would require this: see the remarks on “intellectual doublethink” to come. In any case, a view with obvious appeal is that indispensability argu- ments in mathematics simply have no force: that mathematics is just the framework in which physical theories (and other theories, includ- ing e.g. theories in metalogic) are stated, and that there’s no reason to literally believe the framework. Since the physical theory entails aspects of the framework, then if we don’t believe the framework we can’t literally believe the theory, but can still believe it relative to the framework. Despite its obvious appeal, this view worried me, because I didn’t know how to respond in any detail to someone who took the analogous view of subatomic particles: that talk of them is just the framework in which modern physical theories are presented, and that there’s no reason to literally believe in them. Of course I was aware of a possible response: that we should regard as framework only those aspects of a theory that play no causal role. But that seems unhelpful: a claim about a function from configuration space to the complex numbers does enter into explanations of how a physical system behaves, and if such a claim can’t be expressed nominalistically then it’s hard to deny that it plays a causal role (or at any rate, hard to see the content of such a denial). Why then shouldn’t it be literally believed? Perhaps the claim would be that what are to be literally believed are only facts about causally relevant entities, and this excludes the complex numbers (and the configuration space, and the functions  OUP CORRECTED PROOF – FINAL, //, SPi P- pre fac e to se c on d e di t i on from the latter to the former). But even aside from general unclarity in the notion of causation, the notion of a causally relevant entity is especially unclear, and a platonist might well respond to the suggestion (as I did on behalf of the platonist in Field b) by saying that if mathematical entities are theoretically indispensable parts of causal explanations... , there seems to be an obvious sense in which they are causally involved in producing physical effects; the sense in which they are not causally involved would at least appear to need some explanation (preferably one that gives insight as to why it is reasonable to restrict [indispensability arguments] to entities that are “causally involved” in the posited sense). (p. ) Lacking a good response to the apparent analogy between mathemat- ical objects and subatomic particles, I took a hard line: that it is “intel- lectual doublethink” to fully advocate a physical theory that postulates mathematical entities while at the same time denying the existence of mathematical entities. (‘Fully advocate’ means something like ‘advocate as literal truth, not as mere approximation’. So the “intellectual double- think” charge has no direct application to non-fundamental theories, which inherently involve approximations.) Of course one might with- out doublethink accept the physical theory as a temporary expedient that one would have to make do with until a program for eliminating the mathematical entities was carried out; and even after the “nominal- ization” had been carried out, one might without doublethink accept the original “un-nominalized” theory as a convenient calculational or heuristic device whose legitimacy turns on the nominalistic theory that underpins it. Many of the opponents of indispensability arguments mentioned early in this section have stressed that in theories like continuum mechanics—non-fundamental theories which presumably are accepted only as approximations—we make a lot of use of knowingly false assump- tions about the physical world. They seem to take this as showing that it’s perfectly OK to use mathematical assumptions that we believe false in physical theories that are assumed to precisely describe the physical world. It’s hard to see their argument here, and indeed the argument could well seem to go the other way. For a great deal of effort in physics is dev