Handbook of Philosophy of Mathematics

Inconsistent Mathematics 633not restrict us t o a narrow conception of provability as constructability, as theintuitionists have done: a generous inclusive methodology of proof, which can in-clude model theory, should be our starting point. It does mean, however, that weshould be open both t o the possibility that the intuitionists allowed, that neitherA nor not-A be provable for some A, and the possibility that both A and not-Abe provable, for some A. It also means that we should be less inclined to askhow could an inconsistent proposition be true in mathematics. Rather we shouldbe more inclined to wonder where that might lead. Perhaps later might come anappreciation of mathematical objects with inconsistent properties, as the truth-makers for preferred mathematical propositions, and a basis for model theory. Butthis metaphysical extra is certainly not necessary to make a beginning with. Hence, our starting point is collections of propositions. More precisely, if weare to study structure, we must deal with mathematical theories, that is, setsof propositions closed under a deducibility relation. Deducibility relations arecharacteristic of logics; and it is well-known that there are many deducibilityrelations, since there are many logics. Hence the discussion has t o be generalisedto L-theories, that is, theories of a logic (or deducibility relation) L. An L-theoryTh is inconsistent iff for some proposition A both A E Th and N A E Th, where-- represents the symbol for negation (there are other symbols for special kinds ofnegations). Th is incomplete iff for some A neither A E Th nor A E Th. Th istrivial iff Th is the whole language, i.e. Th contains every proposition; otherwiseTh is nontrivial. The members of any L-theory are also called its theorems, andare said t o hold in the theory. In the end it will be desirable to suppress the logical apparatus provided by Las much as possible. However, for the present, consideration of logic is forced uponus by the logical principle ECQ itself, which, if correct, would ensure chat there isjust one inconsistent theory, the trivial theory. This, in turn, would prevent anydistinctions between kinds of inconsistency, between inconsistent mathematicalstructures. But at this point we are able to exercise some free choice: we candecide t o countenance mathematical theories of logics for which ECQ fails. If thereare none, then invent them. There are plenty of paraconsistent logics around tosupply adequate logical apparatus. Thus there is a sense in which classical logic,regarded as the logic of mathematics, is made false by the existence of inconsistentmathematical theories. To paraphrase Marx, philosophers have hitherto attemptedto understand the nature of contradiction, the point however is to change it. Given a logic, there are two ways to construct theories of that logic: by axiomsor by models. The first intentionally inconsistent arithmetical theory was RobertK. Meyer's RM3(mod 2), which was specified by a model. Its background logic wasthe paraconsistent 3-valued logic RM3. The theory RM3(mod 2) was inconsistentbecause both 0 = 2 and (0 = 2) were theorems. However, Meyer constructedthis theory because he wanted to study the relevant arithmetic R#, which isaxiomatically constructed. The logic for R# is Anderson and Belnap's quantifiedrelevant logic R, axiomatically presented. R# is then given by taking the classicalaxioms for Peano Arithmetic, replacing their classical implication connectives > by

634 Chris Mortensenthe implication connective --+ of R, and closing under the deducibility relation forcR. There is no suggestion that R# is inconsistent. However, by virtue of Meyer'sresult that R# RM3(mod 2), it follows that R# can have 0 = 2 added as anaxiom, the result being an inconsistent axiomatically-presented arithmetic whichis nontrivial. Indeed, RM3(mod 2) itself has an axiomatic presentation: t o R# add0=2 together with all instances of the propositional axiom Mingle, A -4 (A -+ A).See [Meyer 1976; Meyer and Mortensen, 1984; Mortensen, 19951. The question can be asked: given that there are many paraconsistent logics,which is \"best7' for inconsistent mathematics? The answer that emerged was thatit doesn't much matter which: the properties of inconsistent theories tend to beinvariant over a large class of background logics. To be more exact, when theoriesare specified by means of models, their logical properties tend t o take second placebehind mathematical calculations which are performed at the sub-atomic level (sub-atomic relative to the atoms of logic, that is). This suggests an important idea:that mathematics is after all different from logic since logic deals with the generalproperties of propositions, predicates and identity, while mathematics deals withcalculations in particular kinds of structures. We will be developing this theme aswe proceed. Even so, there is one paraconsistent logic which is particularly natural: closedset logic. It is well known that intuitionist logic is the logic of open sets; closedset logic is its topological dual. For many familiar logics, such as tense and modallogics, we can think of propositions as indexed by sets of points in an appropriatespace, such as a set of times, or a set of possible worlds, or a phase-space. This ideacan then be extended by supposing that the index set has a topological structure.If we make the stipulation that propositions only ever hold on open sets of points,we obtain open set logic. It is not difficult to then think of the disjunction oftwo propositions as holding on the union of the sets on which they hold, andconjunction as holding on the intersection. Considering negation however, it isapparent that the negation of a proposition A cannot hold on the set-theoreticcomplement of set of points on which A holds, since the set-theoretic complementof an open set is not in general open. It is thus customary to take for negationthe largest open set contained in the set-theoretic complement. We can then seethe familiar intuitionist property of negation emerging: at the boundary neither Anor not-A holds. Theories of open set logic may thus be incomplete. It is widelyacknowledged that this is a natural-sounding semantics. Applying the topological open-closed duality, we must have closed set logic.Closed set logic is the stipulation that whatever holds, holds on closed sets ofpoints. The interesting case is negation. It is thus customary to take the smallestclosed set containing the set-theoretic complement. We then have the familiarparaconsistent property of negation emerging: at the boundary both A and not-Aholds. It is apparent that this is an equally natural semantics t o that of open setlogic. It is one in which ECQ fails and which supports inconsistent theories. Thisis as natural as the natural transformation: open ++ closed.

Inconsistent Mathematics 635 Unfortunately, it was soon found that inconsistency can spread for reasonsother than ECQ. Curry's paradox generates triviality for nahe set theory evenin the absence of negation, even in the absence of ECQ (see e.g., [Meyer, Rout-ley and Dunn, 19791). All that is necessary is the logical law of Contraction(A --+ (A --+ B)) --+ ( A --+ B ) as well as Modus Ponens, Simplification and Univer-sal Instantiation. Indeed, even weaker principles suffice, as shown by Slaney [I9891and Rogerson [2000]. Thus we must not live in a fool's paradise when constructinginconsistent theories axzomatzcally. Maybe some variant of Curry's paradox canjump up and bite us as a consequence of our axioms, even if we are sure that ECQcannot hurt our theories. Nevertheless, we have a guarantee from model theory that the spread of con-tradictions can be stopped short of triviality, at least for nai've set theory. Thiswas essentially shown by Brady very early on [1971],using a model-theoretic fixed-point method derived from Gilmore [1967]. (It should be noted that Brady's resultwas not explicitly inconsistent, but the latter follows by a trivial manouvre, as helater realised [1989].) Similar work was done independently by Da Costa ( seee.g. [1974)]. The importance of Brady's and Da Costa's result cannot be stressedenough. Brady demonstrated nontriviality in the presence of the Russell Set andthe Curry set; so by brute force, whatever logical principles have to fail for thesesets not to lead to explosion, must fail in Brady's construction. In a further paral-lel to Meyer, Brady developed the method in later papers to show that classicallyfalse ordinal equations are not provable in nai've set theory either (see [1989]).Thus, just as in arithmetic, the contradictions in nai've set theory are far away andcontained, and do not interfere with serious mathematical calculations. Hence, problems for inconsistency arising from logic are not insurmountable.But this is far from being an end to it. Dunn pointed out that if any clqssically falseequation was added to real number theory, then every equation became provable.The proof of this is elementary algebra: from a = b, where a and b are distinctreal numbers, we can subtract a from both sides to get 0 = (b- a). Each side maythen be multiplied by any number we please t o get 0 = r for any real number r .Hence by the principle of the substitutivity of identicals, every real number equalsevery other. We can coin the term mathematically trivial for any (mathematical) theory all ofwhose (logically) atomic propositions are theorems. Now mathematical trivialityimplies full triviality in the presence of the rule ECQ. But in general it does notdo so. Yet, it is mathematical triviality that is catastrophic for mathematics: nocalculation would mean anything. And in Dunn's argument we have an examplewhere mathematical triviality is spread by principles other than ECQ or anythingelse from pure logic. Conversely, if calculations are possible at all, then it isnothing short of crude classical hegemony to insist that a detour through merelogical principles such as ECQ ought to render the theory useless for this purposeor any other. Correspondingly, we can define a theory t o be transparent if it permits fullsubstitutivity of identicals; that is, if tl = t 2 holds then Ftl holds iff Ft2 holds,

636 Chris Mortensenwhere F is any context. A theory is functional if substitutivity of identicals isrestricted to logically atomic contexts; that is, F is any atomic context. In theoriesof classical logic, functionality implies transparency, but this is not so in the generalcase. Furthermore, Dunn's argument requires no more than functionality to work.But now we can see that it is functionality that matters more for mathematics thantransparency, since functionality is what ensures that calculations can proceed.Failure of substitutivity because of logic is not such a weighty matter, while bothfunctionality and its failure are of greater moment for mathematics. 3 PURE MATHEMATICSIt is impossible in this brief account t o survey all the results of inconsistent math-ematics. However, some broad outlines can be touched on. The study has tendedto concentrate on techniques from model theory rather than axiomatics, and wewill take that approach here. Thus we begin with a first-order language containing(i) names for mathematical objects, such as the natural numbers, integers, real numbers, sets, topological spaces;+,(ii) term-forming operations on these objects, such as x, -, + , I (successor) ;c,(iii) atomic predicates and relations, such as =, E;(iv) logical operations such as &, V, N , >,+, @,V, 3.Well-formed formulae are defined in the usual way. A model is a triple (D,L, I),where D is a domain of mathematical objects, L is a many-valued logic, some ofwhose values are designated and the others undesignated, and I is an interpretationwhich maps names to elements of the domain, term-forming operators to (partial)operators on the domain, predicates to subsets of the domain, n-ary relations tosubsets of Dn, and wffs to the values of L in accordance with the interpretations ofparts of the wff to the domain or other values respectively. The theory associatedwith the model is then formed by taking the all those wffs of the model which takea designated value in the interpretation. One device worth mentioning is the use of extensions and anti-extensions foreach predicate and n-ary relation. The idea, due to Dunn and used by Priest, isthat the extension and anti-extension of a predicate can overlap and in that casethe predicate is counted as both true and false of those objects. However, it is notnecessary to use this device, and it is less than fully general when a logic havingnumerous values is being used. The reader is cautioned at this stage from takingmodels with too much ontological seriousness. Models are to be regarded in thefirst instance as devices for controlling the membership of theories. Notice also inpassing the implied distinction between mathematics and logic in that, with theexception of = and perhaps E, logic proper only enters under (iv).

Inconsistent Mathematics 637 To take an example, consider the language to contain names for all natural+,numbers (perhaps constructed in the usual way from 0 and the successor oper-ation), arithmetical operations x , l , and a single binary relation =. Let thedomain D be the natural numbers modulo 2, and the logic L be the 3-valuedparaconsistent logic RM3, with values {T, B , F ) where T and B are designatedvalues (B is understood as \"both\"). Interpret names for the natural numbers astheir counterparts mod 2 and term-forming operators as their corresponding op-erators mod 2. Atomic sentences tl = t2 are interpreted as taking the value B iftlmod 2 = tamod 2, otherwise tl = tz is interpreted as taking the value F. Theset of sentences taking either of the designated values {T,B ) is Meyer's theoryRM3(mod 2). The theory is inconsistent since the equation 0 = 2 takes the valueB while (0 = 2) takes T . Meyer then proved that relevant arithmetic R#W 3 ( m o d 2), which was the basis for his finitary nontriviality proof for R# (see[Meyer, 19761). It is obvious that Meyer's construction can be modified t o produceRM3(mod n) for any number n. Since R# is contained in any of these, we can alsosee that no classically false equation t l = tz can be proved in R#. (See [Meyerand Mortensen, 19841.) Meyer7sproof that R# 2 RM3(mod 2) was finitary in Hilbert7ssense, in thatit relied solely on ordinary mathematical induction over the length of formulae.Since by inspection RM3(mod 2) is nontrivial, it follows that R# can be shown tobe non-trivial by finitary means. By contrast, it follows from Godel's incomplete-ness theorems that there is no finitary proof of the non-triviality (equivalently,consistency) of classical Peano arithmetic. This was viewed with great pessimismby Hilbert, who felt that it spelt the end of his program to demonstrate the consis-tency of mathematics by finitary means. However, Meyer concluded that Hilbert7spessimism is unfounded, as long as we cast aside the shackles of cl-assical logicand ECQ. A further corollary of Meyer's result was not merely that the explosivespread of contradiction in relevant arithmetic is prevented, but that no false atomicpropositions (equations) can be proved in R#. Thus calculation is untouched bycontradiction in relevant arithmetic. This is then a further important consequencefor the philosophy of mathematics. We saw earlier that logicism might be re-habilitated from Russell's paradox by retaining na'ive comprehension, as long asECQ fails. Now we see that the Hilbert program similarly has excellent prospectsfor rehabilitation in logics in which ECQ fails. These include logics only slightlyweaker than classical logic. It is fairly easy to show that extensional part of R# (with logical operators&, V, >, =,3,V, =, but lacking intensional operators +, *) is a subset of clas-sical Peano arithmetic PA.There was for a time the hope that they coincidedexactly. This would of course imply the non-triviality of PA, and hence its consis-tency. That would not of course violate Godel7ssecond incompleteness theorem,since there is no suggestion that the proof method itself be representable in clas-sical arithmetic. But it would be a new proof all the same, perhaps using quitedifferent techniques from the usual. It was eventually discovered by Meyer, adapt-ing Friedman, that R# is strictly weaker than PA, [Meyer-Friedman, 19921. This

638 Chris Mortensendashed the hopes of a consistency proof for PA. Meyer himself expressed pes-simism that R# was thereby shown t o be less than adequate for arithmetic, sincethere are true extensional propositions unprovable in R#. But it seems that thismakes R# all the more interesting: a genuine rival to PA in which all calculationscan be performed; and in which, moreover, all primitive recursive functions arerepresentable so that the incompleteness theorems apply. Moreover, it is hardlysomething that adherents to classical PA can rejoice in, since they, too, have hadto live with the incompleteness theorems ever since they were proved: what is theGijdel sentence if not a true-but-unprovable statement?The class of all mod models, for varying modulus n, has various interestingproperties. Its intersection RMw has the property that its counter-theorems arerecursively enumerable, but it is not known whether it is recursive or not. Thereare also non-standard mod models (see [Mortensen, 1987; 19951). Recently, Priest[1997; 20001 has completely characterised the class of mod models, that is, heshowed that all mod models take a certain form.Of interest is the case of RM3(mod p) where p is prime, since it is known thatthe natural numbers mod p form a field; that is, division is well-defined. Thisraises the question of where Dunn's proof of triviality for the inconsistent realnumber field breaks down in mod p. The answer is that in an inconsistent modarithmetic the equation a = b holds only if the classical difference between a andb differ by an integral multiple of the modulus. Multiplying or dividing both sidesby the same integral number does not disturb that, so the inconsistency does notspread everywhere.It is well known that in the history of the calculus debate raged about whetherone should take seriously the use of \"very small\" real numbers. By the earlynineteenth century it seemed that disputes over the status of infinitesimals wereresolved in favour of real numbers alone by means of the Cauchy-Weierstrass ( E , 6)technique, which quickly became the orthodoxy in mathematics departments. By1960, however, Robinson revived infinitesimals by showing rigorously that onecould develop calculus just as well with them, and that calculus based on in-finitesimal~is in various ways simpler to manipulate, (see [Robinson, 19661). Nowit is notorious that in working out derivatives Newton opportunistically dividedby very small numbers, yet set them to zero when it was convenient to ignorethem. Perhaps then one might be able to make them inconsistently both equal tozero and not equal to zero? However, the prospect that inconsistency in the realnumbers spreads uncontrollably into triviality poses an obvious problem for devel-oping inconsistent differential and integral calculus, and resorting to infinitesimalsdoes not offer an obvious relief since the mathematical triviality proof goes overimmediately to a mathematical triviality proof in the hyperreal field.One way to avoid this is to take as one's domain something with a little lessthan the full structure of fields. This is accomplished by beginning with thenoninfinite hyperreal numbers, that is, the finite real numbers =togbettohemr ewaintht theinfinitesimals. Selecting an infinitesimal number 7, define a hat(a - b)/q is infinitesimal or zero. One may then prove that the equivalence classes

Inconsistent Mathematics 639so generated form a ring under the induced operations. This ring serves as thedomain for an inconsistent model. Taking RM3 as background logic as before, setI ( t l = t2) to be T if tl = t2 as real numbers, set I ( t l = tz) to be B if tl andt2 are distinct real numbers but [vt2l]== [tz], and set I ( t l = t2) to be F otherwise.Then it is easy t o see that both 0 and (v2 = 0) hold, whereas 77 itself isconsistently non-zero. The prospect that infinitesimals smaller that a certain levelin size (i.e. infinitesimals which are even infinitesimal w.r.t. 7) can be equated withzero, allows calculations in which they can be ignored, even though their \"effects\"remain in that division by them is retained in various contexts. Differentiation andintegration can be developed, and Taylor's theorem and the fundamental theoremof the calculus can all be proved.There is more to be said about results from pure mathematics than this. Anal-ysis, topology and category theory have all been studied. For an extended dis-cussion, see [Mortensen, 1995; 2000; 2002al. However, we now proceed with oursurvey by turning t o make some brief remarks on geometry. 4 GEOMETRYConsider the picture below. There are many others. It is notable that the beginnings of inconsistent math-ematics avoided dealing with such pictorial puzzles, though now the situation isslowly being remedied. Interestingly, classical mathematics has also largely avoideddealing with them. In the classical mathematical literature there were to be foundthree approaches. The first, due to Thaddeus Cowan [1974],studied n-sided fig-ures in terms of the properties of their corners, employing the theory of braids.Second, George Francis [I9871 asked what sort of consistent non-Euclidean spacecould be inhabited by such objects. The answer, for the above figure, is R2 x S1.Third, Roger Penrose [I9911 used the theory of cohomology groups to obtain nec-essary and sufficient conditions for a picture to be of a consistent object; a fortiorithe failure of those conditions would mean that the picture was of an inconsistentobject, (see also [Penrose and Penrose, 19581).

640 Chris Mortensen These were unquestionably all very perceptive approaches. However, as arguedby the present writer in [1997b; 2002b; 2002~1,they all shared a common deficit:they did not explain the sense we have that we are perceiving a n object with im-possible properties. This suggests a different conception of the problem, namely,to think of the brain as encoding a n inconsistent geometrical theory. The problemwould then become to write out such a theory (or rather theories, for there aremany different impossible pictures with different properties). The theory in ques-tion would stand to the pictures in somewhat the way that projective geometrystands to the experience of perspective; and with somewhat the same justification,namely, that projective geometry is important to us because of the experience ofhaving an eye. This kind of justification of the study of inconsistency has been described as theepistemic or cognitive justification. Such justifications appeal to a human capacity,typically the capacity to reason in a logically-anomalous environment, without in-tellectual collapse into triviality. There is of course no suggestion that inconsistentobjects exist in the physical world. Rather, it is that our perceptions construct ageometrical theory while at the same time retaining geometrical principles whichare incompatible with it. It seems that in inconsistent pictures we have a clearexample of the mind's ability to make constructions which are inconsistent andyet persist even when the impossibility is manifest to us. The lack of cognitivepenetrability of the experiences is characteristic of the modularity of perceptualcapacities which has been noted by various authors, e.g. [Fodor, 19831. The details of such mathematical theories are still in an early stage of development. The interested reader is invited to consult the above references for furtherelucidation. 5 APPLIED MATHEMATICSA good antidote to the error that mathematics develops in pristine logical order ist o read the works of Imre Lakatos [1976]. It is particularly in applied mathematics,physics and engineering where mathematical opportunism is most apparent. Herethe lack of classical rigor comes with applications built-in. Hence we can ask, aswith the historical disputes over infinitesimals, whether the \"logically erroneous\"theory might be more accurately described as an inconsistent theory rich enoughto permit useful calculations. A good example is Dirac's Delta \"function\", 6(x). This had the twin properties:(i) 6(x) = 0 for all x # 0 , and (ii) J 6(x)dx = 1, where the integration was over thewhole real line. It is apparent that there is no such function on the real numbers.Yet Dirac perceived a use for it in his version of Quantum Mechanics. In this he wasfollowed by many of the physics community. Quantum theory developed rapidlyand decisively. It was not for some forty years that Laurent Schwartz managedto put things on a consistent footing by using functionals rather than functions.There was a significant cost, however, in that the new theory was considerablymore complicated. There is a fairly obvious construction for the Delta function

Inconsistent Mathematics 641which uses infinitesimals: draw a triangle of infinitesimal base P and infinite height2 / P . This satisfies something close to the condition (i), namely 6(x) = 0 for all realx # 0; and clearly the second condition is satisfied since the area of the triangle is1. This was not Robinson's construction, however, since it requires second-orderprinciples; whereas Robinson restricted himself t o first-order conditions, so that histheory amounted pretty much to a copy of Schwartz'. It turns out, however, thatthere is an inconsistent theory which adapts the construction above of inconsistentinfinitesimals, and which has the property that 6(x) = 0 for all x for which x = 0fails to hold. Since it is the propositions that hold that are relevant to property offunctionality, we can say that the construction recovers the concept of a function,albeit an inconsistent function. It is hardly surprising that Quantum Mechanics lends itself to inconsistent ap-plications, since QM has long been regarded as a source of anomaly and paradox.One more application in this area is quantum measurement. In cases where anoperator has a discrete spectrum, such as the energy levels of the hydrogen atom,elementary QM postulates discontinuous changes in the wave function when ameasurement is made. Now discontinuity is an enemy of causality: it would bedesirable to have a theory in which quantum measurement was reducible t o theother familiar quantum process of unitary evolution. This is the measurementproblem, and it is fair to say that the measurement problem remains unsolved,and is even intensified given the problem of nonlocality, Bell's theorem and As-pect's experiments. An approach using inconsistent continuous functions seemsto allow both for continuity/causality and at the same time discrete spectra. Formore details, see [Mortensen, 1997al. The cognitive justification of paraconsistency, discussed before, is apparent inthe application to information systems. Nuel Belnap [I9771 famously pointedout that any control system with more than one stream of informational inputs,must allow for the possibility that its inputs may be in conflict. Furthermore,it may be impossible to shut the system down until the problem is resolved, aswith an aircraft aloft. Thus there has to be a way of operating in an anomalousinformational environment, which is after all what we humans manage to do. Onetheory taking this approach considers the problem of solving inconsistent systemsof linear equations. Inconsistent systems of linear equations have been knownabout for centuries, and the standard mathematical reaction has been to throwup the hands in despair. However, it proves possible to solve some such systemsof equations in an inconsistent space. Now the classical theory of control systemsmakes heavy use of systems of linear equations. This in turn suggests that if onewere able to model a malfunctioning control system in terms of an inconsistentsystem of linear equations, there might be a way of continuing to exercise somelimited control. The modelling proved not to be so difficult. According to classicalcontrol theory, when a system is functioning correctly, its internal organisation ismodelled by a transfer matrix, which transforms a (column) vector of inputs intoa vector of outputs. When the system is malfunctioning, there is a differencebetween the expected outputs and the observed outputs. By superimposing the

642 Chris Mortensenobserved outputs onto the expected transfer matrix, one obtains an inconsistentsystem of equations which can then be solved. In software modellings this has metwith some limited success. A related approach has been taken by the Braziliangroup around Abe [2000],who have demonstrated a paraconsistent robot, Emmy. It should be noted that it is not being claimed here that the control system isbehaving inconsistently in the real world. It is rather that the discrepancy betweenexpected and observed creates an epistemological gap that has to be resolved.Calculations take place in a virtual space in which all the information availableis used to form a composite picture with the aim of continuing to function untilproper knowledge and control can be fully restored. A final point to be noted is the shift in ontology that takes place between puremathematics and applied mathematics. In rejecting Platonism, we were rejectingabstract truthmakers for pure mathematics. The truthmakers for applied mathe-matics, one would imagine, are its applications. These involve systems of physicalobjects and their physical quantities, the kinds of things which are causally active,changing and producing change. Physical quantities, such as 5 gram, 2 cm, 3 sec,come as a package of a number (\"5\") and a quantity kind or dimension (\"gram\").In the present writer's view, the best account of quantities treats them as causallyrelevant universals. Laws of nature come out as relations between universals (see[Armstrong, 19781). Real numbers then emerge fairly unproblematically as ra-tios (i.e. relations of comparison) between dimensioned quantities having the samedimension (see [Forrest and Armstrong, 1987; Bigelow, 1988; Mortensen, 19981).It is not proposed to develop this account here, the reader is directed to thesereferences. The point being made is that there is not necessarily an equivalencebetween the problem of the truthmakers for pure mathematics, and truthmakersfor applied mathematics. The harder problem seems t o be for pure mathematics,while applied mathematics looks rather more tractable. 6 LOGICISM AND FOUNDATIONALISM REVISITEDWith this all-too-sketchy survey of what is known to date, we return to ourflirtatious quarrel with logicism. The foregoing suggests that we can draw a(rough) line between logics and mathematics in the kinds of reasonings they em-ploy. Logics deal with universally applicable principles of reasoning, centrally(k,k,&, v, -, +, H ,3,V, =), and other constructions arising in natural language(e.g. tense, modality, adverbs). Logic applies to mathematical reasoning, certainly,but it applies to that aspect of mathematical reasoning that applies to other sub-ject matter as well. In contrast, mathematics distinctively deals with concepts likethose of algebra, calculus, differential equations, analysis and geometry. Some-where in the middle between logic and mathematics lie set theory, number theory,recursion theory and parts of algebra. In the case of algebra, logicians' interestshave tended to be confined to structures which can supply a plausible semanticsfor various sets of logical axioms, such as lattices. With only a few exceptions,logicians have not been much interested in groups, for example. This leads to

Inconsistent Mathematics 643the challenge to logicism: in what sense is mathematics no more than logic withdefinitions? It all depends on which definitions. Mathematicians tend to be anti-foundationalist. The previous challenge canalso be directed against foundationalism, and it explains why mathematicians havenot taken logic's attempts a t hegemony too seriously. Claims like \"set theory is afoundation for mathematics\" or \"mathematics reduces to logic\" look like they aresaying that all there i s to mathematics is set theory or logic. But this is preciselyto suppress what is distinctive about mathematics. They give a false sense of whatis the nature of mathematics. The point can be further illustrated by considering the \"reduction\" of geometryto algebra. It is uncontestable that Descartes' discovery of the coordinatisation ofthe plane enabled an immense step forward in geometry. The methods of algebracould then be applied to the study of the plane. Space could be studied by solvingequations involving real numbers and their functions. Nonetheless, it is a mistaket o take this as implying that geometry is nothing but real number theory, as Russellseems t o have thought (see e.g. [Ayer, 1972, 431). The two-dimensional plane isnot R2; space is not a collection of numbers. Its parts are points, lines, curves,and planes, not sets or real numbers. We need only pay attention t o our ownperceptions of space to see this: we perceive areas, lines etc., we do not perceivenumbers. In short, there is no conceptual equivalence possible between geometryand set theory. This is why a mathematician can pursue the study of space payinglittle or no attention to foundations: mathematics has a conceptual autonomy thatfoundations cannot supply. From this point of view, the gap between mathematics and logic is even widerthan that between mathematics and real numbers and set theory. Logicians study\"and\", \"or\", \"not\", \"implies\" and the like. Their discipline begins where mathe-matics leaves off in studying the behaviour of geometry, groups and the like. Thismakes logic look more like a small area in the corpus of mathematics, rather than afoundation for it. Furthermore, it exposes ECQ for what it is: a tool in a takeoverbid to establish the hegemony of logic over mathematics. As a piece of personal reportage I recall years ago explaining to a visiting emi-nent mathematician why I was inclined to reject ECQ. After listening politely, heasked: \"Excuse me, but are you not denying that the null set is a subset of everyset?\" This confusion embodies a subtle reversal, but it is no better motivated. Wemay be inclined to make a limited \"reduction\" of set theory to logic by adoptingnGve set theory and claiming that there is nothing to set theory but logic. Na'ivecomprehension would then be an expression of the reduction. In favour it can besaid that it is certainly less ad hoc than rival comprehension principles. However,our eminent mathematician was reversing the order of explanation: he felt thatthe principles of set theory were sui generis and that the legitimacy of ECQ wasensured by that!

Chris Mortensen 7 REVISIONISM AND DUALITYEarlier, we referred to the topological duality between incomplete theories of openset logic, and inconsistent theories of closed set logic. There is another kind ofduality, Routley-' duality. This applies between theories of logics in which the- -laws of Double Negation A o NN A and De Morgan ( AV B ) ++ (N A & B )and ( A & B) +-+ (N A V B ) hold. Neither open set logic nor closed set logichas these laws unrestrictedly, however many of the logics in the Anderson-Belnapclass of relevant logics have them. For any set of sentences S, define S* to be{A : N A @ S } . Then a simple argument shows that if Th is any theory ofa logic containing Double Negation and De Morgan, then Th is inconsistent iffTh* is incomplete. Since DN ensures that Th** = Th, we also have that Th isincomplete iff Th* is inconsistent. That is, incompleteness and inconsistency as properties of theories are duals ofone another in two senses: they are topological duals of one another, and they areRoutley-* duals of one another. Duality results are of course sources of \"theoremsfor free\". As a quick illustration of free theorems, we note a dualisation of Kripke'smodelling of the truth predicate in an incomplete theory. Kripke [I9751 showed,using a fixed point method deriving from Gilmore [I9671 and Brady [1971], thatthe Liar proposition L and its negation are excluded from a theory satisfying thecondition for a truth predicate: A ++ T(A) where A is any proposition and T(.) isthe truth predicate for the name (Gijdel number) of A. Kripke interpreted this asshowing that the Liar proposition L ought to be regarded as neither true nor false.However, applying the Routley-* to the truth theory, we can immediately concludethat there is a theory satisfying the conditions for a truth theory to which bothL and N L belong. We might also observe that the inconsistent dual theory hascertain advantages over the incomplete theory. Any theory which, like Kripke's,declares that L lacks a value, suffers from a dilemma. Either we say that theinstance of the T-scheme for the Liar sentence has a value (presumably True), orit does not. If it does, then we have the oddity that none of L, L,T(L) andT ( N L) receive a truth value even though L ++ T(L) and L i+ T ( N L) hold inthe theory. If it does not, then it odd to say that the T-scheme holds even thoughsome of its instances fail to hold. Note that while Kripke employed a third logicalvalue in his construction, he was clear that this was a formal device for calculationonly, and that he regarded the liar sentence as lacking a value. This is perfectlyreasonable as a proof device, however it seems strange that a valueless propositioncould yet contribute to making a compound hold. In contrast, in the inconsistentdual, all of L, L, T(L) and T(- L) take contradictory values; which is at leastsome reason to hold that L * T(L) does too. Intuitionism and constructivism are examples of revisionist philosophies of math-ematics, in that they declare that certain principles accepted in classical mathe-matics are incorrect. They aim to revise mathematics by truncating it, basedon a narrower conception of what is an acceptable proof. However, revisionismleaves unanswered an important question: why are the excluded areas yet mathe-

Inconsistent Mathematics 645matics? In their haste t o offer a theory of correct proof, revisionists neglect thecentral question of the philosophy of mathematics: what is mathematics? This ishardly to be answered adequately by declaring those parts of mathematics thatthe theorists don't like, not to be mathematics at all. The classical Hilbertian ideal of a mathematical theory is one which is com-plete and demonstrably consistent. Revisionist theories, by excluding aspects ofclassical theories, render themselves incomplete, a fact which has been long-notedin connection with intuitionism. By contrast, inconsistent mathematics is notrevisionist at all. Taking a lead from the duality results, it aims to extend math-ematics, not weaken it. The duals of incomplete theories are inconsistent, andthey include classical consistent complete theories as subtheories, and consistentincomplete theories as sub-sub-theories. Thus inconsistent mathematics supportsa principle of tolerance about what counts as mathematics, an inclusive approachnot an exclusive one. Both classical mathematics and revisionist mathematicsemerge as special cases of a more generalised conception of mathematics, whichincludes inconsistent mathematics as well. 8 THE ROLE OF TEXTOne further matter needs t o be raised, though dealing with it fully would take muchmore space than we have here. If we ask what makes all of the above examplesmathematics, it is apparent that the answer must have something central to dowith the characteristic use of notation or symbols. That is to say, mathematics istextually distinctive. Importantly, this is something it shares with symbolic logic.It is apparent that the rise of symbolic logic in the twentieth century is attributableto its use of mathematical text. The question is: just how is it that this has beenso efficacious? This dovetails with the broader question of just why it is that thedistinctive textual features of any mathematics do their jobs so well? We are allfamiliar with examples like the advantage of the change from Roman numeralsto Arabic numerals: it is clear that this is a microcosm of the general questionof the distinctive nature and efficacy of mathematical text. There is somethingimportant to be explained about how mathematical meaning is carried by text. There is another observation which is a kind of converse to this one. In hisUniversity of Adelaide PhD thesis TheRole of Notation i n Mathematics [1988],Edwin Coleman pointed out the varieties of mathematical text. He drew attentionthe differences between a page from Euclid, a page from Principia Mathernatica, apage from a text on business mathematics, a page from a standard calculus text,and a page from a mechanical engineering text. Consider for example the varyingrole of diagrams, and the presence or absence of natural language. The differencesare richly textual, and yet the very stuff of mathematics. Thus, the questionof the usefulness of distinctive texts in mathematics, is part of the question ofhow mathematical text generates meaning. It is the interplay of similarity anddifference that needs t o be understood.

646 Chris Mortensen Coleman argued that the right discipline t o undertake such a study was thetheory of signs, semiotics. Co-discovered by Peirce and Saussure, semiotics aimsto study how text and other signs generate meaning. Saussure in particular hadto rely somewhat more heavily on the internal differences within a code or systemof signs, because, unlike Peirce, his account lacked a theory of extra-linguistic ref-erence. While this is an obvious drawback in any general account of language, itcan be seized on by (we) anti-Platonists as just right for any account of mathe-matical meaning, where (according to us) there are no abstract objects t o be thereferents. This is nothing but an application of Saussure's concept of difference.Ockham's Razor does the rest against Platonism. A certain amount of literaturewhich addresses these issues in the indicated ways has grown up, including NelsonGoodman [1981],Rene Thom [1980],Brian Rotman [1987; 19901, Coleman [1988;1990; 19921, and Mortensen and Roberts [1997]. We sawearlier that Meyer's nontriviality result serves fit to re-habilitate Hilbert'sprogram of demonstrating that mathematics does not have false consequences. Butthere are problems for Hilbert's program of a different sort here. Drawing on theabove, Coleman attacked Hilbert's formalism. Like Brouwer, Hilbert gave way tothe despair of revisionism. In order to demonstrate mathematics to be consistentand complete, or at least without error, mathematical theories must be displayedin canonical form, as formal systems, purely symbolic and devoid of all meaning(save that generated internally). But here, as with revisionisms anywhere, we canagain ask why are the uncanonical parts yet mathematics? Don't get me wrong.I am certainly not against reconstruction of a theory as a first order formal the-ory, if only because then you could automate it! But notice that in producing an\"equivalent\" formal theory we are suppressing a difference that is part of what hasto be explained: in what sense can notationally distinct codes be equivalent, andhow can textual features contribute to distinctness of code, and thus to differencesof meaning? This kind of study cuts across the inconsistency program to some extent. Nonethe-less, it serves t o reinforce the point that an inclusive point of view about math-ematics is necessary if one is to understand what mathematics is. Revisionisminevitably reduces our view of what is possible for mathematics, and thus distortsour understanding of the phenomena. 9 CONCLUSIONSTo summarise, the following propositions have been advanced. 1. Logicism and foundationalism may well be saved if we adopt a logic lacking ECQ. 2. Similarly, part of Hilbert's program, to prove that mathematics has no false consequences, may well be saved if such a logic is adopted. There are many suitable logics, some of them only slightly weaker than classical logic.

Inconsistent Mathematics 647 3. Nonetheless, logicism and foundationalism do not explain the conceptual autonomy of mathematics from logic. In particular, geometry is conceptually separate from logic and set theory, and does not reduce to them. 4. Revisionist philosophies of mathematics, whether they be revisionist about the truths of mathematics (intuitionism) or revisionist about notation (for- malism), are open to the objection that they do not account for the varieties of mathematics outside of approved canonical norms. 5. In contrast to revisionism, we must take an inclusive position, whereby incon- sistent mathematics is seen as extending our conception of what is possible for mathematics rather than rejecting the corpus of existing mathematics. 6. This is just as well, since inconsistent mathematics has numerous applica- tions beyond itself. 7. As part of comprehending the nature of mathematics, the distinctively tex- tual aspects of mathematics, both the similarities and the differencesbetween textual styles, have to be understood; and semiotics seems to be the best theoretical framework for this project. Two related issues of traditional philosophy of mathematics have been placed onthe backburner in this essay. One is the matter of truthmakers for pure mathemat-ics. The other is the distinctive epistemology of mathematics, and in particular themethod of a priori proof. Neither can be neglected in a full account. However, wemight make the very limited suggestion that if the primary phenomenon to be ex-plained for mathematics is textual, then it is not so speculative that the accountought to derive from the features of text, rather than abstract acapsal objects.Certainly, the legitimacy of inconsistency ought to give pause to the Platonist. Itposes the dilemma: either abandon Platonism, or admit inconsistent objects. Onesalient virtue in sheeting home the primary account to the theory of signs, is thatit scores well on the second issue: we have a readily-understandable epistemologyfor signs. It can hardly be denied that getting in contact with signs, such as thoseon your keyboard, is a thoroughly natural activity. The same can't be said forPlatonism. BIBLIOGRAPHY[ A b e et al., 20001 Abe et al. Emmy, a Paraconsistent Robot, Second World Congress on Para- consistency, Juquehy Beach, 2000.[Armstrong,19781 D. Armstrong. Universals and Scientific Realism, Cambridge, CambridgeUP, 1- -97-8[Ayer, 19721 A. J . Ayer. Russell, Fontana Modern Masters, 1972.[ B e h a p , 19771 N. D. Belnap. How a Computer Should T h i n k , in G.Ryle, ed. Contemporary Aspects of Philosophy, Stocksfield, Oriel Press, 30-55, 1977.[Bigelow,19881 J . Bigelow. The Reality of Numbers, Oxford,T h e Clarendon Press, 1988.[ ~ r a d19~71, 1 R. Brady. T h e Consistency o f t h e Axioms o f Abstraction and Extensionality in a Three-Valued Logic, Notre Dame Journal of Formal Logic (NDJFL) 12, 447-453, 1971

648 Chris Mortensen[Brady, 19891 R. Brady. T h e Non-triviality o f Dialectical Set Theory. In [Priest et al, 1989, 437-4711.[ ~ r a adnd~ Routley, 19891 R . Brady and R.Routley. T h e Non-triviality o f Extensional Dialec- tical Set Theory. In [Priest et al., 1989, 415-4361.[Coleman, 19881 E. Coleman. The Role of Notation in Mathematics, PhD thesis, U . o f Adelaide, 1988[Coleman, 19901 E. Coleman. Paragraphy, Information Design Journal, 6, 131-146[Coleman, 19921 E. Coleman. Presenting Mathematical Information. In R.Penman and D.Sless, eds, Designing Information for People, Canberra, ANU Press, 1992.[Cowan, 19741 T h . Cowan. T h e Theory o f Braids and the Analysis o f Impossible Pictures, Jour- nal of Mathematical Psychology, 11, 190-212, 1974.[ ~Caosta, 19741 N. C . A. Da Costa. O n t h e Theory o f Inconsistent Formal Systems, NDJFL, 15, 497-510, 1974[Dunn, 19791 J . M. Dunn. A Theorem in Three-Valued Model Theory with Connections t o Number Theory, T y p e Theory and Relevant Logic, Studia Logica, 38, 149-169, 1979.orr rest[Fodor,19831 J . Fodor. The Modularity of Mind, MIT Press, 1983. and Armstrong, 19871 P. Forrest and D. Armstrong. T h e Nature o f Number, Philosoph- ical Papers, 16, 165-186, 1987.[Francis, 19871 G . Francis. A Topological Picturebook, Springer-Verlag, 1987.[Gilmore, 19671 P. Gilmore. T h e Consistency o f Partial Set Theory Without Extensionality. In D.Scott (ed.) Axiomatic Set Theory, Proceedings of Symposia i n Pure Mathematics, Los Angeles, U . o f California, 1967.[Goodman, 19811 N. Goodman. Languages of Art, Brighton, Harvester, 1981.[ ~ a t c h e r1,9821 W . Hatcher. The Logical Foundations of Mathematics, Amsterdam, North- Holland, Elsevier, 1982.[ ~ r i ~ k19e7,51 S. Kripke. Outline o f a Theory o f Truth, The Journal of Philosophy 72, 690-716,1975[Lakatos, 19761 I . Lakatos. Proofs and Refutations, Cambridge, Cambridge University Press,1976.[Meyer, 19761 R . K . Meyer. Relevant Arithmetic, Bulletin of the Section of the Polish Academyof Science, 5, 133-137, 1976.[ ~ e ~anedrFriedman, 19921 R. K . Meyer and H. Friedman. Whither Relevant Logic? The Jour-nal of Symbolic Logic, 57, 824-831, 1992.[ ~ e ~anedrMortensen, 19841 R. K . Meyer and C. Mortensen. Inconsistent Models for Relevant -Arithmetics, The Journal of Symbolic Logic, 49, 917-929, 1984.[Meyer et al., 19791 R. K . Meyer, R. Routley, and J . M. Dunn. Curry's Paradox, Analysis, 39,124-128, 1979.orte tens en, 19871 C. Mortensen. Inconsistent Nonstandard Arithmetic, The Journal of Sym- bolic Logic, 52 , 512-18, 1987.[Mortensen, 19881 C . Mortensen. Inconsistent Number Systems, Notre Dame Journal of Formal Logic, 29 (1988), 45-60, 1988.orte tens en, 19901 C. Mortensen. Models for Inconsistent and Incomplete Differential Calculus, Notre Dame Journal of Formal Logic, 31, 274-285, 1990.[Mortensen,19951 C . Mortensen. Inconsistent Mathematics, Kluwer Mathematics and Its A p plications Series, 1995.[Mortensen, 1997al C . Mortensen. T h e Leibniz Continuity Condition, Inconsistency and Quan- t u m Dynamics, The Journal of Philosophical Logic, 26, 377-389, 1997.[Mortensen, 1997bl C . Mortensen. Peeking at t h e Impossible, Notre Dame Journal of Formal Logic, Vol 38 No 4 (Fall 1997), 527-534, 1977.[Mortensen, 19981 C . Mortensen. O n t h e Possibility o f Science Without Numbers, The Aus- tralasian Journal of Philosophy, 182-197, 1998.[Mortensen,20001 C. Mortensen. Topological Separation Principles and Logical Theories, Syn- these 125 Nos 1-2, 169-178, 2000.orte tens en, 2002a] C . Mortensen. Prospects for Inconsistency, in D.Batens (et.al.) eds. Fron- tiers of Paraconsistent Logic, London, Research Studies Press, 203-208, 2002.[Mortensen,2002bl C. Mortensen. Towards a Mathematics o f Impossible Pictures. In W . Car-nelli, M . Coniglio, and I . D'Ottaviano, eds, Paraconsistency, The Logical Way t o the Incon-sistent, Lecture Notes in Pure and Applied Mathematics Vol 228, New York, Marcel Dekker,445-454, 2002.

Inconsistent Mathematics 649orte tens en, 2002~1C. Mortensen. Paradoxes Inside and Outside Language, Language and Com- munication, Vol 22, No 3, 301-311, 2002.[Mortensen and Roberts, 19971 C. Mortensen and L. Roberts. Semiotics and the Foundations of Mathematics, Semiotica, 115, 1-25, 1997.[penrose and Penrose, 19581 L. S. Penrose and R. Penrose. Impossible Objects, a Special Kind of Illusion, British Journal of Psychology, 49, 1958.[Penrose, 19911 R. Penrose. On the ~ o h 6 m o l oof~I~mpossible Pictures, Strvctuml Topology, 17, 11-16, 1991.[priest, 19791 G. Priest. The Logic of Paradox The Journal of Philosophical Logic, 8, 219-241, 1979.[priest, 1987) G. Priest. I n Contmdiction, Dordrecht, Hijhoff, 1987.[Priest, 19971 G. Priest. Inconsistent Models of Arithmetic; I, Finite Models, The Journal of Philosophical Logic, 26, 223-235, 1997.[Priest, 20001 G. Priest. Inconsistent Models of Arithmetic; 11, The General Case, The Journal of Symbolic Logic, 65, 1519-29, 2000.[Priest et al., 19891 G . Priest, R. Routley, and J. Norman, eds. Pamconsistent Logic, Essays on the Inconsistent, Philosophia Verlag, 1989.[~obinson1,9661 A. Robinson. Nonstandard Analysis, Amsterdam, North-Holland, 1966.[Rogerson, 20001 S. Rogerson. Curry Paradoxes. AAL Annual Conference, U. of Sunshine Coast, Noosa, 2000.[Slaney, 19891 J. Slaney. RWX is not Curry Paraconsistent. In [Priest et al., 1989, 4721-4821.[Rotman, 19871 B. Rotman. Signifying Nothing: The Semiotics of Zero, London, Macmillan, 1-9.8.7. ~[Rotman, 1990] B. Rotman. Towards a Semiotics of Mathematics, Semiotica, 72, 1-35, 1990[Thom, 19801 R. Thom. L'espace des Signes, Semiotica, 29, 193-208, 1980.

This page intentionally left blank

MATHEMATICS AND THE WORLD Mark ColyvanOne of the most intriguing features of mathematics is its applicability t o empiricalscience. Every branch of science draws upon large and often diverse portions ofmathematics, from the use of Hilbert spaces in quantum mechanics to the use ofdifferential geometry in general relativity. It's not just the physical sciences thatavail themselves of the services of mathematics either. Biology, for instance, makesextensive use of difference equations and statistics. The roles mathematics playsin these theories is also varied. Not only does mathematics help with empiricalpredictions, but it also allows elegant and economical statements of many theories.Indeed, so important is the language of mathematics to science, that it is hard toimagine how theories such as quantum mechanics and general relativity could evenbe stated without employing a substantial amount of mathematics. F'rom the rather remarkable but seemingly uncontroversial fact that mathemat-ics is indispensable t o science, some philosophers have drawn serious metaphysi-cal conclusions. In particular, Quine [1948/1980; 1951/1980; 1981b] and Putnam[1971/1979; 19791 have argued that the indispensability of mathematics to empiri-cal science gives us good reason to believe in the existence of mathematical entities.According to this line of argument, reference to (or quantification over) mathe-matical entities such as sets, numbers, functions and such is indispensable to ourbest scientific theories, and so we ought to be committed to the existence of thesemathematical entities. To do otherwise is to be guilty of what Putnam has called\"intellectual dishonesty\" [Putnam, 1971/1979, p. 3471. Moreover, mathematicalentities are seen to be on an epistemic par with the other theoretical entities ofscience, since belief in both kinds of entities is justified by the same evidence thatconfirms the theory as a whole. This argument is known as the Quine-Putnamindispensability argument for mathematical realism. In this chapter I will discussthis argument and some of the various attempts to defuse it. I will also consider another topic related to mathematics and its applications:the so-called unreasonable effectiveness of mathematics. The problem here is that(pure) mathematical methods are largely a priori and driven by largely aestheticconsiderations, and yet mathematics is in great demand in describing and even ex-plaining the physical world. As Mark Steiner puts it lLhowdoes the mathematician-closer to the artist than the explorer - by turning away from nature, arrive atits most appropriate descriptions?\" [Steiner, 1995, p. 1541. This problem and itsrelationship t o the indispensability argument will also be examined.Handbook of t h e Philosophy of Science. Philosophy of MathematicsVolume editor: Andrew D. Irvine. General editors: Dov M. Gabbay, Paul Thagard and JohnWoods.@ 2009 Elsevier B.V. All rights reserved.

Mark Colyvan 1 THE INDISPENSABILITY ARGUMENTI . 1 Realism and Anti-realism in MathematicsThere are many different ways to characterise realism and anti-realism in math-ematics. Perhaps the most common way is as a thesis about the existence ornon-existence of mathematical entities. Thus, according to this conception of re-alism, mathematical entities such as functions, numbers, and sets have mind- andlanguage-independent existence or, as it is also commonly expressed, we discoverrather than invent mathematical theories (which are taken to be a body of factsabout the relevant mathematical objects). This is usually called metaphysicalrealism. Anti-realism, then, is the position that mathematical entities do not en-joy mind-independent existence or, alternatively, we invent rather than discovermathematical theories. According to this characterisation, a realist believes thatFermat's Last Theorem was true before Wiles's proof and, indeed, even beforeFermat first thought of his now famous theorem. This is because, according to therealist, the integers exist independently of our knowledge of them and Fermat'stheorem is a fact about them. Of course there are other characterisations of realismand anti-realism but since my interests in this chapter are largely metaphysical,I'll be content with this characterisation of rea1ism.l There are various Platonist and nominalist strategies in the philosophy of math-ematics. Each of these has its own particular strengths and weaknesses. Platonistaccounts of mathematics generally have the problems of providing an adequateepistemology for mathematics [Benacerraf, 1973/1983] and of explaining the ap-parent indeterminacy of number terms [Benacerraf, 1965/1983]. On the otherhand, nominalist accounts generally have trouble providing an adequate treat-ment of the wide and varied applications of mathematics in the empirical sciences.There is also the challenge for nominalism to provide a uniform semantics formathematics and other discourse [Benacerraf, 1973/1983]. Let's consider a fewdifferent strategies encountered in the literature. An important nominalist response t o these arguments is fictionalism. A fiction-alist about mathematics believes that mathematical statements are, by and large,false. According to the fictionalist, mathematical statements are 'true in the storyof mathematics' but this does not amount to truth simpliciter. Fictionalists taketheir lead from some standard semantics for literary fiction. On many accounts ofliterary fiction 'Sherlock Holmes is a detective' is false (because there is no suchperson as Sherlock Holmes), but it is 'true in the stories of Conan Doyle.' Themathematical fictionalist takes sentences such as 'seven is prime' to be false (be- 'While on matters terminological, I should also point out that, in keeping with most ofthe modern literature in the area, I will use the terms 'mathematical realism' and 'Platonism'interchangeably. So I take Platonism t o be the view that mathematical objects exist and, whatis more, that their existence is mind and language independent. I also take it that accordingt o Platonism, mathematical statements are true or false in virtue of the properties of thesemathematical objects. I do not mean t o imply anything more than this. I do not, for instance,intend Platonism to imply that mathematical objects are causally inert, that they are not locatedin space-time, or t h a t they exist necessarily.

Mathematics and the World 653cause there is no such entity as seven) but 'true in the story of mathematics.' Thefictionalist thus provides a distinctive response to the challenge of providing a uni-form semantics - all the usually accepted statements of mathematics are false.2The problem of explaining the applicability of mathematics is more involved, andI will leave a discussion of this until later (see section 4). In recent times many Platonist strategies have responded to the epistemologi-cal challenge by placing mathematical objects firmly in the physical realm. ThusPenelope Maddy in Realism i n Mathematics [1990a] argued that we can see sets.When we see six eggs in a carton we are seeing the set of six eggs. This ac-count provides mathematics with an epistemology consistent with other areas ofknowledge by giving up one of the core doctrines of traditional Platonism -thatmathematical entities are abstract. In response to the apparent indeterminacy ofthe reduction of numbers to sets, one popular Platonist strategy is to identify agiven natural number with a certain position in any w-sequence. Thus, it doesn'tmatter that three can be represented as (((0))) in Zermelo's w-sequence and(0, {g), (0, (0))) in von Neumann's w-sequence. What is important, according tothis account, is that the structural properties are identical. This view is usuallycalled structuralism since it is the structures that are important, not the itemsthat constitute the structures3 These are not meant t o be anything more than cursory sketches of some of theavailable positions. Some of these positions will arise again later, but for now I willbe content with these sketches and move on to discuss indispensability argumentsand how these arguments are supposed to deliver mathematical realism.1.2 Indispensability A r g u m e n t sAn indispensability argument, as Hartry Field points out, \"is an argument thatwe should believe a certain claim . ..because doing so is indispensable for certainpurposes (which the argument then details)\" [Field, 1989, p. 141. Clearly thestrength of the argument depends crucially on what the as yet unspecified purposeis. For instance, few would find the following argument persuasive: We shouldbelieve that whites are morally superior to blacks because doing so is indispensablefor the purpose of justifying black slavery. Similarly, few would be convinced by theargument that we ought to believe that God exists because to do so is indispensableto the purpose of enjoying a healthy religious life. The \"certain purposes\" ofwhich Field speaks must be chosen very carefully. Although the two argumentsjust mentioned count as indispensability arguments, they are implausible because'enjoying a healthy religious life' and 'justifying black slavery' are not the right his is not quite right. Since fictionalists take the domain of quantification t o be empty,they claim that all existentially quantified statements (and statements about what are apparentlydenoting terms) are false, but that all universally quantified sentences are true. So, for example,'there is a n even prime number' is taken t o be false while 'every number has a successor' is takent o be true. 3See, for example, Hellman [1989],Resnik [1997],and Shapiro [1997].

654 Mark Colyvansort of purposes to ensure the cogency of the respective arguments. This raises thevery interesting question: Which purposes are the right sort for cogent arguments? I know of no easy answer to this question, but fortunately an answer is notrequired for a defence of the class of indispensability arguments with which Iam concerned. I will restrict my attention largely t o arguments that addressindispensability to our best scientific theories. I will argue that this is the rightsort of purpose for cogent indispensability arguments. I will also be concernedprimarily with indispensability arguments in which the \"certain claim\" of whichField speaks is an existence claim. We may thus take a scientific indispensabilityargument to rest upon the following major premise:ARGUMENT 1 Scientific Indispensability Argument. If apparent reference t osome entity (or class of entities) 5 is indispensable to our best scientific theories,then we ought to believe in the existence of (. In this formulation, the purpose, if you like, is that of doing science. This is arather ill-defined purpose, and I deliberately leave it ill defined for the moment.But to give an example of one particularly important scientific indispensabilityargument with a well-defined purpose, consider the argument that takes provid-ing explanations of empirical facts as its purpose. I'll call such an argument anexplanatory indispensability argument. Although indispensability arguments are typically associated with realism aboutmathematical objects, it's important to realise that they do have a much widerusage. What is more, this wider usage is fairly uncontroversial. To see this, weneed only consider an example of an explanatory indispensability argument usedfor non-mathematical purposes. Most astronomers are convinced of the existence of so called \"dark matter\"to explain (among other things) certain facts about the rotation curves of spiralgalaxies.4 This is an indispensability argument. Anyone unconvinced of the exis-tence of dark matter is not unconvinced of the cogency of the general form of theargument being used; it's just that they are inclined to think that there are betterexplanations of the facts in question. It's not too hard to see that this form of argument is very common in both sci-entific and everyday usage. Indeed, in these examples, it amounts to no more thanan application of inference to the best explanation. This is not to say, of course,that inference to the best explanation is completely uncontroversial. Philoso-phers of science such as Bas van Fraassen [1980] and Nancy Cartwright [1983]reject unrestricted usage of this style of inference. Typically, rejection of inferenceto the best explanation results in some form of anti-realism (agnosticism, abouttheoretical entities in van Fraassen's case and anti-realism about scientific lawsin Cartwright's case). Such people will have little sympathy for indispensabilityarguments. Scientific realists, on the other hand, are generally committed t o infer-ence to the best explanation, and they are the main target of the indispensability 4 ~ h e s aere graphs of radial angular speed versus mean distance from the centre of the galaxy for stars in a particular galaxy.

Mathematics and the World 655argument.5 Indispensability arguments about mathematics urge scientific realistst o place mathematical entities in the same ontological boat as (other) theoreticalentities. That is, it invites them to embrace P l a t o n i ~ m . ~ The use of indispensability arguments for defending mathematical realism isusually associated with Quine and Putnam. Quine's version of the indispensabilityargument is t o be found in many places. For instance, in 'Success and Limits ofMathematization' he says: Ordinary interpreted scientific discourse is as irredeemably committed to abstract objects - to nations, species, numbers, functions, sets - as it is t o apples and other bodies. All these things figure as values of the variables in our overall system of the world. The numbers and func- tions contribute just as genuinely to physical theory as do hypothetical particles. [Quine, 1981b, pp.149-1501Here he draws attention to the fact that abstract entities, in particular mathe-matical entities, are as indispensable to our scientific theories as the theoreticalentities of our best physical t h e o r i e ~ .E~lsewhere [Quine, 1951/1980] he suggeststhat anyone who is a realist about theoretical entities but anti-realist about math-ematical entities is guilty of holding a \"double standard.\" For instance, Quinepoints out that the position that scientific claims, but not mathematical claims,are supported by empirical data is untenable: The semblance of a difference in this respect is largely due to overem- phasis of departmental boundaries. For a self-contained theory which we can check with experience includes, in point of fact, not only its various theoretical hypotheses of so-called natural science but also such portions of logic and mathematics as it makes use of. [Quine, 1963/1983, p. 3671He is claiming here that those portions of mathematical theories that are employedby empirical science enjoy whatever empirical support the scientific theory as awhole enjoys. (I will have more to say on this matter in section 5.2.) Hilary Putnam also once endorsed this argument: [Qluantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantifica- 51ndeed, one of the most persuasive arguments for scientific realism is generally taken t oappeal t o inference t o the best explanation. This argument is due t o J.J.C. Smart [1963]. 6 ~ ' mnot claiming here that the indispensability argument for mathematical entities is simplyan instance of inference t o the best explanation; I'm just noting that inference t o the bestexplanation is a kind of indispensability argument, so those who accept inference t o the bestexplanation are a t least sympathetic t o this style argument. '1 often speak of certain entities being dispensable or indispensable t o a given theory. Strictlyspeaking it's not the entities themselves that are dispensable or indispensable, but rather it's thepostulation of or reference to t h e entities in question that may be so described. Having said this,though, for the most part I'll continue t o talk about entities being dispensable or indispensable,eliminable or non-eliminable and occurring or not occurring. I d o this for stylistic reasons.

Mark Colyvan tion; but this commits us to accepting the existence of the mathe- matical entities in question. This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishon- esty of denying the existence of what one daily presupposes. [Putnam, 1971/1979, p. 3471Elsewhere he elaborates on this \"intellectual dishonesty\": It is like trying t o maintain that God does not exist and angels do not exist while maintaining at the very same time that it is an objective fact that God has put an angel in charge of each star and the angels in charge of each of a pair of binary stars were always created at the same time! [Putnam, 1979, p. 741 Both Quine and Putnam, in these passages, stress the indispensability of math-ematics t o science. It thus seems reasonable to take science, or at least whateverthe goals of science are, as the purpose for which mathematical entities are indis-pensable. But, as Putnam also points out [1971/1979, p. 3551, it is doubtful thatthere is a single unified goal of science -the goals include explanation, prediction,retrodiction, and so on. Thus, we see that we may construct a variety of indis-pensability arguments, all based on the various goals of science. As we've alreadyseen, the explanatory indispensability argument is one influential argument of thisstyle, but it is important to bear in mind that it is not the only one. To state the Quine-Putnam indispensability argument, we need merely replace'5' in argument 1with 'mathematical entities'. For convenience of future referenceI will state the argument here in an explicit form.ARGUMENT 2 The Quine-Putnam Indispensability Argument. 1. We ought to have ontological commitment to all and only those entities that are indispensable t o our best scientific theories; 2. Mathematical entities are indispensable to our best scientific theories. Therefore: 3. We ought t o have ontological commitment to mathematical entities. A number of questions about this argument need to be addressed. The firstis: The conclusion has normative force and clearly this normative force originatesin the first premise, but why should an argument about ontology be normative?This question is easily answered, for I take most questions about ontology to bereally questions about what we ought to believe to exist. The Quine-Putnam indis-pensability argument, as I've presented it, certainly respects this view of ontology.Indeed, I take it that indispensability arguments are essentially normative. Forexample, if you try to turn the above Quine-Putnam argument into a descriptiveargument, so that the conclusion is that mathematical entities exist, you find you

Mathematics and the World 657must have something like 'All and only those entities that are indispensable to ourbest theories exist' as the crucial first premise. This premise, it seems t o me, ismuch more controversial than the normative one. As we shall see, this normativityarises in the doctrine of naturalism, on which I will have more to say shortly. The next question is: How are we to understand the phrase 'indispensable t oour best scientific theory'? In particular, what does 'indispensable' mean in thiscontext? Much hangs on this question, and I'll need t o treat it in some detail. I'lldo this in the next section. In the meantime, take it to intuitively mean 'couldn'tget by without' or the like. In fact, whatever sense it is in which electrons, neutronstars, and viruses are indispensable to their respective theories will do.8 The final question is: Why believe the first premise? That is, why should webelieve in the existence of entities indispensable to our best scientific explanations?Answering this question is not easy. Briefly, I will argue that the crucial firstpremise follows from the doctrines of naturalism and holism. Before I embark onthis task, I should point out that the first premise, as I've stated it, is a littlestronger than required. In order to gain the given conclusion, all that is reallyrequired in the first premise is the 'all,' not the 'all and only.' I include the 'alland only,' however, for the sake of completeness and also to help highlight theimportant role naturalism plays in questions about ontology, since it is naturalismthat counsels us to look to science and nowhere else for answers to ontologicalquestions. Although 1'11 have more to say about naturalism and holism (in section 3), itwill be useful here to outline the argument from naturalism and holism to thefirst premise of argument 2. Naturalism, for Quine at least, is the philosophi-cal doctrine that there is no first philosophy and the philosophical enterprise iscontinuous with the scientific enterprise. What is more, science, thus construed(i.e., with philosophy as a continuous part) is taken to be the complete story ofthe world. This doctrine arises out of a deep respect for scientific methodologyand an acknowledgment of the undeniable success of this methodology as a way ofanswering fundamental questions about all nature of things. As Quine suggests,its source lies in \"unregenerate realism, the robust state of mind of the naturalscientist who has never felt any qualms beyond the negotiable uncertainties inter-nal to science\" [Quine, 1981a, p. 721. For the metaphysician this means looking toour best scientific theories t o determine what exists, or, perhaps more accurately,what we ought to believe to exist. Naturalism, in short, rules out unscientific waysof determining what exists. For example, I take it that naturalism would rule outbelieving in the transmigration of souls for mystical reasons. It would not, how-ever, rule out belief in the transmigration of souls if this were required by our bestscientific theories. Naturalism, then, gives us a reason for believing in the entities in our best sci-entific theories and no other entities. Depending on exactly how you conceive of s ~ yfou think that there is no sense in which electrons, neutron stars, and viruses are indis-pensable t o their respective theories, then the indispensability argument is unlikely t o have anyappeal.

658 Mark Colyvannaturalism, it may or may not tell you whether to believe in all the entities of yourbest scientific theories. I take it that naturalism does give us some (defeasible)reason to believe in all such entities. This is where the holism comes to the fore- in particular, confirmational holism. Confirmational holism is the view thattheories are confirmed or disconfirmed as wholes. So, if a theory is confirmed byempirical findings, the whole theory is confirmed. In particular, whatever math-ematics is made use of in the theory is also confirmed. Furthermore, as Putnam[1971/1979] has stressed, the same evidence that is appealed t o in justifying be-lief in the mathematical components of the theory is appealed to in justifying theempirical portion of the theory (if indeed the empirical can be separated from themathematical). Taking naturalism and holism together, then, we have the firstpremise of argument 2. Before concluding this section, I would like to outline one other indispensabilityargument that appears in the literature: Michael Resnik's [1995] pragmatic indis-pensability argument. This argument focuses on the purpose of 'doing science'and is a response to some problems raised for the Quine-Putnam indispensabilityargument by Penelope Maddy and Elliott Sober. Although I won't discuss theseproblems here (I do so a little later on, in section 5), one point is important inunderstanding Resnik's motivation. Resnik wishes to avoid the Quine-Putnamargument's reliance on confirmational holism. Resnik presents the argument in two parts. The first is an argument for theconditional claim that if we are justified in drawing conclusions from and withinscience, then we are justified in taking mathematics used in science to be true. Hepresents this part of the argument as follows: 1) In stating its laws and conducting its derivations science assumes the existence of many mathematical objects and the truth of much mathematics. 2) These assumptions are indispensable to the pursuit of science; more- over, many of the important conclusions drawn from and within science could not be drawn without taking mathematical claims to be true. 3) So we are justified in drawing conclusions from and within science only if we are justified in taking the mathematics used in science to be true. [ ~ e s n i k1,995, pp. 169-1701He then combines the conclusion of this argument with the argument that we arejustified in drawing conclusions from and within science, since this is the onlyway we know of doing science. And clearly we are justified in doing science. Theconclusion, then, is that we are justified in taking whatever mathematics is usedin science to be true.g This argument clearly fits the mould of the scientific indispensability argumentthat I outlined earlier. It differs from the Quinean argument in that it doesn't 9 ~ fnact, Resnik draws the additional (stronger) conclusion that mathematics is true, arguingt h a t this follows from the weaker conclusion, since t o assent t o the weaker conclusion whiledenying the stronger invites a kind of Moore's paradox. (Moore's paradox is t h e paradox ofasserting ' P but I don't believe P.')

Mathematics and the World 659rely on confirmational holism. Resnik pinpoints the difference rather nicely in thefollowing passage: This argument is similar to the confirmational argument except that instead of claiming that the evidence for science (one body of state- ments) is also evidence for its mathematical components (another body of statements) it claims that the justification for doing science (one act) also justifies our accepting as true such mathematics as science uses (another act). [Resnik, 1995, p. 1711 This argument has some rather attractive features. For instance, since it doesn'trely on confirmational holism, it doesn't require confirmation of any scientific the-ories in order for belief in mathematical objects to be justified. Indeed, even ifall scientific theories were disconfirmed, we would (presumably) still need mathe-matics to do science, and since doing science is justified we would be justified inbelieving in mathematical objects. This is clearly a very powerful argument andone with which I have considerable sympathy. Although I won't have much moreto say about this argument in what follows, it is important to see that a cogentargument in the general spirit of the Quine-Putnam argument can be maintainedwithout recourse to confirmational holism. 2 WHAT IS I T TO BE INDISPENSABLE?The Quine-Putnam indispensability argument may be stated as follows: We havegood reason to believe our best scientific theories and there are no grounds onwhich to differentiate scientificentities from mathematical entities, so we have goodreason to believe in mathematical entities, since they, like the relevant scientificentities, are indispensable to the theories in which they occur. Furthermore, itis exactly the same evidence that confirms the scientific theory as a whole, thatconfirms the mathematical portion of the theory and hence the mathematicalentities contained therein. The concept of indispensability is doing a great deal ofwork in this argument and so we need to have a clear understanding of what ismeant by this term. I've already pointed out that one way an entity can be indispensable is that itcan be indispensable for explanation (in which case the resulting argument is aninstance of inference to the best explanation). But I think there are other waysin which an entity can be indispensable to a theory.10 In order to come to a clearunderstanding of how 'indispensability' is to be understood, I will consider a casewhere there should be no disagreement about the dispensability of the entity inquestion. I shall then analyse this case to see what leads us to conclude that theentity in question is dispensable. 1°Quine actually speaks of entities existentially quantified over in t h e canonical form of ourbest theories, rather than indispensability. (See [Quine, 1948/1980] for details.) Still, the debatecontinues in terms of indispensability,so we would be well served t o clarify this latter term.

660 Mark Colyvan 'c7Consider an empirically adequate and consistent theory r and let be thename of some entity neither mentioned, predicted, nor ruled out by r . Clearly wer+can construct a new theory from J? by simply adding the sentence 'E exists' tor. It is reasonable to suppose that ( plays no role in the theory I?+;'' it is merelypredicted by it. I propose that there should be no disagreement here when I sayethat is dispensable to I?+, but let us investigate why this is so. <On one interpretation of 'dispensable' we could argue that is not dispensablesince its removal from I?+ results in a different theory, namely, r.12 This is nota very helpful interpretation though, since all entities are indispensable to thectheories in which they occur under this reading. Another interpretation of 'dis-pensable' might be that is dispensable to l?+ since there exists another theory,I', with the same empirical consequences as I?+ in which J does not occur.13 Thisinterpretation can also be seen to be inadequate since it may turn out that notheoretical entities are indispensable under this reading. This result follows fromCraig's theorem.14 If the vocabulary of the theory can be partitioned in the waythat Craig's theorem requires (cf. footnote 14), then the theory can be reaxioma-tised so that any given theoretical entity is eliminated.15 I claim, therefore, thatthis interpretation of 'dispensable' is unacceptable since it fails to account for whyJ in particular is dispensable. This leads to the following explication of 'dispensable7: An entity is dispensable to a theory iff the following two conditions hold:(1) There exists a modification of the theory in question resulting in a second theory with exactly the same observational consequences as the first, in which the entity in question is neither mentioned nor predicted.(2) The second theory must be preferable to the first.c cIanndthreipsrepcreefdeirnagbleextaomrpl+e,inthtehna, t is dispensable since r makes no mention of the former has fewer ontological commitments,all other things being equal. (Assuming, of course, that fewer ontological commit-ments is better.16)llThe reason I hedge a bit here is that if r asserts that all entities have positive mass, forinstance, then the aexroislteenincero+f .EI helps account for some of the \"missing mass\" of th e universe.Thus, does play know of no way of ruling out such cases; hence the hedge. 12More correctly, we should say that we can remove all sentences asserting or implying theexistence of ( from r f .13Modulo my concerns in footnote 11.1 4 ~ h iths eorem states that relative t o a partition of the vocabulary of an axiomatisable theoryT into two classes, T and w (theoretical and observational say), there exists an axiomatisabletheory T' in the language whose only non-logical vocabulary is w , of all and only the consequencesof T that are expressible in w alone.15Naturally, t h e question of whether such partitioning is possible is important and somewhatcontroversial. If it is not possible, it will be considerably more difficult t o eliminate theoreticalentities from scientific theories. Let's grant for the sake of argument, a t least, that such apartitioning is possible.160ne way in which you might think that fewer ontological commitments is not better, is if E

Mathematics and the World 661Now, it might be argued that on this reading once again every theoretical entityis dispensable, since by Craig's theorem we can eliminate any reference to anyentity and the resulting theory will be better, since it doesn't have ontologicalcommitment to the entity in question. This is mistaken though, since the reasonfor preferring one theory over another is a complicated question -it is not simply amatter of empirical adequacy combined with a principle of ontological parsimony.We thus need to consider some aspects of confirmation theory and its role inindispensability decisions.Quine clearly had the hypothetico-deductive method in mind as his model ofscientific theory confirmation. Philosophy of science has moved on since then; nowsemantic conceptions of theories and confirmation prevail. But the details of thetheory of confirmation need not concern us. All that really matters for presentpurposes is that in order to decide whether one theory is better than another weappeal to desiderata for good theories and these (for the scientific realist, at least)typically amount to more than mere empirical adequacy.There's no doubt that a good theory should be empirically adequate; that is, itshould agree with (most) observations. Second, all other things being equal, we'dprefer our theories to be consistent, both internally aanlrdeawdiythsoetehne,rrmaanjodr theories.This is not the whole story though. As we have I?+ havethe same degree of empirical adequacy and consistency (by construction), and yetwe are inclined to prefer the former over the latter. Typically such a deadlock issettled by appeal to additional desiderata such as:(1) Simplicity/Parsimony: Given two theories with the same empirical ade- quacy, we generally prefer that theory which is simpler both in its statement and in its ontological commitments.(2) Unificatory/Explanatory Power: Philip Kitcher [1981] argues rather convincingly for scientific explanation being unification; that is, accounting for a maximum of observed phenomena with a minimum of theoretical de- vices. Whether or not you accept Kitcher's account, we still require that a theory not simply predict certain phenomena, but explain why such predic- tions are expected. Furthermore, the best theories do so with a minimum of theoretical devices.(3) Boldness/Fruitfulness: We expect our best theories not to simply predict everyday phenomena, but to make bold predictions of novel entities and phenomena that lead to fruitful future research.actually exists. In this case it seems that F+ is the better theory since it best describes reality.<This, exists. If it will betherehowever, is t o gloss Eo'vseerxtihsteenimcep, otrhteanntrqfuewsitlilonindofeehdowbewteh come t o know t hatis some evidence of e better theory, since E, then it seemsrempirically superior.If the rre +is no such evidence for t he existence of entirely over as I suggest. It is the latter I had in mind when I set up thisreasonable to prefercase. Indeed, the former case is ruled out by construction. I am not concerned with whether Eactually exists or not -just that there be no empirical evidence for it.

662 Mark Colyvan (4) Formal Elegance: This is perhaps the hardest feature to characterise (and no doubt the most contentious). However, there is at least some sense in which our best theories have aesthetic appeal. For instance, it may well be on the grounds of formal elegance that we rule out ad hoc modifications to a failing theory. I will not argue in detail for each of these, except to say that despite the notori-ous difficulties involved in explicating what we mean by terms such as 'simplicity'and 'elegance,' most scientific realists, at least, do look for such virtues in our bestrtheories.17 Otherwise, we could never choose between two theories such as andI?+. I do not claim that this list of desiderata is comprehensive nor do I claim thatit is minimal;'' I merely claim that these sorts of criteria are typically appealedto in the literature to distinguish good theories and I have no objections t o suchappeals. In the light of the preceding discussion then, we see that to claim that an entityis dispensable is to claim that a modification of the theory in which it is positedcan be made in such a way a s to eliminate the entity in question and result in atheory that is better overall (or at least not worse) in terms of simplicity, elegance,and so on. Thus, we see that the argument I presented at the end of the previoussection that any theoretical entity is dispensable does indeed fail, as I claimed.This is because in most cases the benefit of ontological simplicity obtained by theelimination of the entity in question will be more than offset by losses in otherareas. While it seems reasonable to suppose that the elimination from the body ofscientific theory of physical entities such as electrons would result in an overallreduction in the previously described virtues of that theory, it is not so clear thatthe elimination of mathematical entities would have the same impact. Someonemight argue that mathematics is certainly a very effective language for the expres-sion of scientific ideas, in that it simplifies the calculations and statement of muchof science, but to do so at the expense of introducing into one's ontology the wholegamut of mathematical entities simply isn't a good deal. One response to this is to deny that it is a high price at all. After all, apowerful and efficient language is the cornerstone of any good theory. If you haveto introduce a few more entities into the theory to get this power and efficiency,then so be it. Although I have considerable sympathy with this line of thought, amore persuasive response is available. Elsewhere [Colyvan, 1999b; Colyvan, 2001a; Colyvan, 20021 I have argued thatmathematics plays an active role in many of the theories that make use of it. Thatis, mathematics is not just a tool that makes calculations easier or simplifies thestatement of the theory; it makes important contributions to all of the desiderataof good theories I mentioned earlier. Let me give just one brief example here of 17And recall that the main target of the indispensability argument is scientific realists. 18For instance, it may be possible t o explain formal elegance in terms of simplicity and unifi- catory power.

Mathematics and the Worldhow mathematics can help provide unification.lg Consider a physical system described by the differential equation:(where y is a real-valued function of a single real variable). Equations such asthese describe physical systems exhibiting (unconstrained) growth and we cansolve them with a little elementary real algebra. But now consider a strikinglysimilar differential equation that describes certain periodic behaviour:(where, again, y is a real-valued function of a single real variable). Somewhatsurprisingly, the same real algebra cannot be used to solve equations such as (2)-we are pushed to complex methods20 Now since complex algebra is a generalisation of real algebra, we can employ thesame (complex) method for solving both (1) and (2). Thus we see how complexmethods may be said to unify, not only the mathematical theory of differentialequations, but also the various physical theories that employ differential equa-tions. But the unification doesn't stop there. The exponential function, which isa solution to (I), is very closely related to the sine and cosine functions, which aresolutions to (2). This relationship is spelled out via the definitions of the complexsine and cosine functions. Without complex methods, we would be forced t o con-sider phenomena described by (1) and (2) a s completely disparate and, moreover,we would have no unified approach to solving the respective equations. I see thisis a striking example of the unification brought to science by mathematics - bycomplex numbers, in this case. (It is by no means the only such case though;detours into complex analysis are commonplace in modern mathematics - evenfor what are essentially real-valued phenomenon.) 3 NATURALISM AND HOLISMWith a more precise understanding of what indispensability amounts to, let usnow turn to the doctrines required to support the Quine-Putnam indispensabilityargument. Although a great deal of Quine's philosophy is interconnected, makingthe isolation of particular doctrines very difficult, I will argue that the two essen-tial theses for our purposes - confirmational holism and naturalism - can bedisentangled from the rest of the Quinean web. IgAlthough if you are inclined towards the view that explanation is unification that I mentionedearlier, then the following case might be thought t o be one in which the mathematics is playingan explanatory role. 200f course, in this simple case we can solve t h e equations in question by other means (suchas by inspection) but the fact remains that complex methods are needed t o provide a systematicand unified approach to all such differential equations. See [Boyce and DiPrima, 19861 for details.

664 Mark Colyvan3.1 Introducing NaturalismNaturalism, in its most general form, is the doctrine that we ought to seek accountsof the nature of reality that are not \"other-worldly\" or \"unscientific,\" but t o bemore precise than this is to immediately encounter trouble. For instance, DavidPapineau points out that \"nearly everybody nowdays wants to be a 'naturalist',but the aspirants t o the term nevertheless disagree widely on substantial questionsof philosophical doctrine\" [Papineau, 1993, p. 11. In one way this is not at allsurprising, for, after all, there is no compulsion for all naturalists t o agree on otherphilosophical stances, distinct from naturalism, and such stances, when combinedwith naturalism, presumably yield different results. It all depends on what youmix your naturalism with. There is, however, another reason for disagreement among naturalistic philoso-phers: Different philosophers use the word 'naturalism' to mean different things.Naturalism involves a certain respect for the scientific enterprise - that much iscommon ground - but exactly how this is cashed out is a matter of considerabledebate. For instance, for David Armstrong naturalism is the doctrine that \"noth-ing but Nature, the single, all-embracing spatio-temporal system exists\" [Arm-strong, 1978, Vol. 1,p. 1381,whereas, for Quine, naturalism is the \"abandonmentof the goal of a first philosophy\" [Quine, 1981a, p. 721. One issue on which naturalistic philosophers disagree, and which is of fundamen-tal importance for our purposes, is the ontological status of mathematical entities.We've already seen how the Quine-Putnam indispensability argument legitimatesbelief in mind-independent mathematical objects, and that this argument dependson naturalism. On the other hand, philosophers such as David Armstrong citenaturalism as grounds for rejecting belief in any such mind-independent abstractobjects. While there is no way of preventing philosophers from mixing their naturalismwith other philosophical doctrines (so long as the mix is coherent), there is goodreason for requiring that the various, often contrary, positions that fly under thebanner of naturalism be disentangled, from one another. This is a very large taskbut we can at least try t o identify the difference between the varieties of naturalismthat may be used to undermine mathematical realism and the Quinean variety.3.2 Quinean NaturalismQuine's aphoristic characterisations of naturalism are well known. In 'Five Mile-stones of Empiricism' he tells us that naturalism is the abandonment of the goal of a first philosophy. It sees natural science as an inquiry into reality, fallible and corrigible but not answerable to any supra-scientific tribunal, and not in need of any justification beyond observation and the hypothetico-deductive method. [Quine, 1981a, p. 721And that:

Mathematics and the World 665 [tlhe naturalistic philosopher begins his reasoning within the inherited world theory as a going concern. He tentatively believes all of it, but believes also that some unidentified portions are wrong. He tries to improve, clarify, and understand the system from within. He is the busy sailor adrift on Neurath's boat. [Quine, 1981a, p. 721The aphorisms are useful, but they also mask a great deal of the subtlety andcomplexity of Quinean naturalism. Indeed, the subtleties and complexities of nat-uralism are far greater than one would expect for such a widely held and intuitivelyplausible doctrine. We would do well t o spend a little time attempting to betterunderstand Quinean naturalism. As I see it, there are two strands to Quinean naturalism. The first is a normativethesis concerning how philosophy ought to approach certain fundamental questionsabout our knowledge of the world. The advice here is clear: look to science(and nowhere else) for the answers. Science, although incomplete and fallible, istaken to be the best guide to answering all such questions. In particular, \"firstphilosophy\" is rejected. That is, Quine rejects the view that philosophy precedesscience or oversees science. This thesis has implications for the way we shouldanswer metaphysical questions: We should determine our ontological commitmentsby looking to see which entities our best scientific theories are committed to. Thus,I take it that naturalism tells us (1) we ought to grant real status only to theentities of our best scientific theories and (2) we ought to (provisionally) grantreal status to all the entities of our best scientific theories. For future referenceI'll call this first strand of Quinean naturalism the no-first-philosophy thesis andits application to metaphysics the Quinean ontic thesis. It is worth pointing out that the Quinean ontic thesis is distinct from a thesisabout how we determine the ontological commitments of theories. According tothis latter thesis, the ontological commitments of theories are determined on thebasis of the domain of quantification of the theory in question.21 Call this thesisthe ontological commitments of theories thesis. One could quite reasonably believethe ontological commitments of theories thesis without accepting the Quineanontic thesis. For instance, I take it that Bas van Fraassen [1980] accepts thatour current physics is committed to entities such as electrons and the like, butit does not follow that he believes that it is rational to believe in these entitiesin order to believe the theory. The ontological commitments of theories thesis ispurely descriptive, whereas the Quinean ontic thesis is normative. From here onI shall be concerned only with the Quinean ontic thesis, but it is worth bearingin mind the difference, because I don't think that the ontological commitments oftheories thesis rightfully belongs to the doctrine of naturalism. It is an answer t othe question of how we determine the ontological commitments of theories, but itis not the only naturalistic way such questions can be answered. The second strand of Quinean naturalism is a descriptive thesis concerningthe subject matter and methodology of philosophy and science. Here naturalism \"See [Quine, 1948/1980, pp. 12-13] for details.

666 Mark Colyvantells us that philosophy is continuous with science and that together they aim toinvestigate and explain the world around us. What is more, it is supposed thatthis science-philosophy coalition is up to the task. That is, all phenomena arein principle explicable by science. For future reference 1'11 call this strand thecontinuity thesis. Although it is instructive to distinguish the two strands of Quinean naturalismin this way, it is also important to see how intimately intertwined they are. First,there is the intriguing interplay between the two strands. The no-first-philosophythesis tells us that we ought to believe our best scientifictheories and yet, accordingt o the continuity thesis, philosophy is part of these theories. This raises a questionabout priority: In the case of a conflict between philosophy and science, which getspriority? Philosophy does not occupy a privileged position. That much is clear.But it also appears, from the fact that philosophy is seen as part of the scientificenterprise, that science (in the narrow sense -i.e., excluding philosophy) occupiesno privileged position either. The second important connection between the two strands is the way in whichthe continuity thesis lends support to no-first-philosophy thesis. The traditionalway in which first philosophy is conceived is as an enterprise that is prior anddistinct from science. Philosophical methods are seen to be a priori while those ofscience are a posteriori. But accepting the continuity thesis rules out such a viewof the relationship between philosophy and empirical science. Once philosophy islocated within the scientific enterprise, it is more difficult to endorse the view thatphilosophy oversees science. I'm not claiming that the continuity thesis entails theno-first-philosophy thesis, just that it gives it a certain plausibility.22 Now to the question of why one ought to embrace naturalism. I won't embarkon a general defence of naturalism - that would be far too ambitious. I take itthat almost everyone accepts some suitably broad sense of this doctrine.23 Butsubscribing to some form or another does not entail subscribing to Quinian nat-uralism. Again, I won't try anything so ambitious as defend Quinian naturalismhere.24 Still it is useful t o see what's at issue. Let's start by marking out the common ground. Naturalists of all ilks-agreethat we should look only to science when answering questions about the natureof reality. What is more, they all agree that there is at least prima facie reasonto accept all the entities of our best scientific theories. That is, they all agreethat there is a metaphysical component to naturalism. So they are inclined toaccept the first part of the Quinean ontic thesis (the 'only' part) and are inclinedto, at least provisionally, accept the second part (the 'all' part). (Most naturalistsbelieve that naturalism entails scientific realism but they are inclined to be a little 221ndeed, the continuity thesis cannot entail the no-first-philosophy thesis since the former isdescriptive and the latter normative. 23Again it is worth bearing in mind that the primary targets of the indispensability argumentare scientific realists disinclined t o believe in mathematical entities. These scientific realiststypically subscribe t o some form of naturalism. 24See [Colyvan, 2001a, chap. 2 and 31 for a limited defence.

Mathematics and the World 667reluctant to embrace all the entities of our best scientific theories.)25 What Itake to be the distinctive feature of Quinean naturalism is the view that our bestscientific theories are continuous with philosophy and are not to be overturned byfirst philosophy. It is this feature that blocks any first-philosophy critique of theontological commitments of science. Consequently, it is this feature of Quineannaturalism that is of fundamental importance to the indispensability argument.3.3 HolismHolism comes in many forms. Even in Quine's philosophy there are at least twodifferent holist theses. The first is what is usually called semantic holism (althoughQuine calls it moderate holism [1981a, p. 711) and is usually stated, somewhatmetaphorically, as the thesis that the unit of meaning is the whole of the language.As Quine puts it: The idea of defining a symbol in use was ... a n advance over the im- possible term-by-term empiricism of Locke and Hume. The statement, rather than the term, came with Bentham to be recognized as the unit accountable to an empiricist critique. But what I am now urging is that even in taking the statement as unit we have drawn our grid too finely. The unit of empirical significance is the whole of science. [Quine, 1951j1980, p. 421Semantic holism is closely related to Quine's denial of the analyticjsynthetic dis-tinction and his thesis of indeterminacy of translation. He argues for the former ina few places, but most notably in 'Two Dogmas of Empiricism' [Quine, 1951/1980],while the latter is presented in Word and Object [ ~ u i n e1,9601. The other holist thesis found in Quine's writings is confirmatzonal holism (alsocommonly referred to as the QuinejDuhem thesis). As Fodor and Lepore pointout [Fodor and Lepore, 1992, pp. 39-40], the QuinejDuhem thesis receives manydifferent formulations by Quine and it is not clear that all these formulations areequivalent. For example, in Pursuit of Truth Quine writes:26 [Tlhe falsity of the observation ~ a t e g o r i c adl ~oe~s not conclusively re- fute the hypothesis. What it refutes is the conjunction of sentences that was needed to imply the observation categorical. In order to retract that conjunction we do not have to retract the hypothesis in question; we could retract some other sentence of the conjunction instead. This is the important insight called holism. [Quine, 1992, pp. 13-14]And in a much quoted passage from 'Two Dogmas of Empiricism', he suggests that\"our statements about the external world face the tribunal of sense experience not 25For example, Keith Campbell (19941 advocates \"selective realism\", and Quine restricts com-mitment t o indispensable entities. 26Cf. Duhem [1962, p. 1871 for a similar statement of the thesis. 2 7 ~ 'yobservation categorical' Quine simply means a statement of the form 'whenever P, thenQ.' For example, 'where there's smoke, there's fire.'

668 Mark Colyvanindividually but only as a corporate body'' [Quine, 1951/1980, p. 411. Elsewhere,in a similar vein, he tells us: As Pierre Duhem urged, it is the system as a whole that is keyed t o experience. It is taught by exploitation of its heterogeneous and sporadic links with experience, and it stands or falls, is retained or modified, according as it continues to serve us well or ill in the face of continuing experience. [Quine, 195311976, p. 2221In the last two of these three passages Quine emphasizes the confirmational aspectsof holism - it's the whole body of theory that is tested, not isolated hypotheses.In the first passage he emphasizes dzsconfirmational aspects of holism -when ourtheory conflicts with observation, any number of alterations t o the theory can bemade to resolve the conflict. Despite the difference in emphasis, I take it that thesetheses are equivalent (or near enough). Moreover, I take it that they are all true,modulo some quibbles about how much theory is required to face the tribunal atany time. Although Quine was inclined to argue for confirmational holism from (the morecontroversial) semantic holism, this is not the only way to establish the former.Both Duhem [1962]and Lakatos [1970]have argued for confirmational holism with-out any (obvious) recourse to semantic considerations. They emphasize the simpleyet undeniable point that there is more than one way in which a theory, faced withrecalcitrant data, can be modified to conform with that data. Consequently, cer-tain core doctrines of a theory may be held onto in the face of recalcitrant data bymaking suitable alterations to auxiliary hypotheses. Indeed, in its most generalform, confirmational holism is little more than a point about logic. Before leaving the doctrine of holism, I wish to consider one last question: Mightone accept confirmational holism as stated, but reject the claim that mathematicalpropositions are one with the rest of science? That is, might it not be possibleto pinpoint some semantic difference between the mathematical propositions em-ployed by science and the rest, with empirical confirmation and disconfirmationreserved for the latter? Carnap [1937], with his appeal t o \"truth by conven-tion,\" suggested precisely this. Quine, of course, denies that this can be done[1936/1983; 195111980; 1963/1983], but exploring the reasons for his denial wouldtake us deep into issues in the philosophy of language. For our purposes, it willsuffice to note that there is no obvious way of disentangling the purely mathemat-ical propositions from the main body of science. Our empirical theories have theso-called empirical parts intimately intertwined with the mathematical. A cur-sory glance at any physics book will confirm this, where one is likely to find mixedstatements such as: 'planets travel in elliptical orbits'; 'the curvature of space-time bis not zero'; 'the work done by the force on the particle is given by W = Ja F . dr.' Thus, even if you reject Quine's semantic holism and you think that mathemat-ical and logical language is different in kind from empirical language, you neednot reject confirmational holism. In order to reject confirmational holism, youwould need (at the very least) to separate the mathematical vocabulary from the

Mathematics and the World 669empirical in all of our best scientific theories. Clearly this task is not trivial.28If you still feel some qualms about confirmational holism, though, you may restassured -this doctrine will be called into question when we consider some of theobjections to the indispensability argument.3.4 The First Premise RevisitedLet's return t o the question of how confirmational holism and Quinean natu-ralism combine t o yield the first premise of the Quine-Putnam indispensabilityargument. First, you might wonder whether holism is required for the argument.After all, (Quinean) naturalism alone delivers something very close to the crucialfirst premise. (More specifically, the Quinean ontic thesis is very suggestive of therequired premise.) As a matter of fact, I think that the argument can be made t ostand without confirmational holism: It's just that it is more secure with holism.The problem is that naturalism is somewhat vague about ontological commitmentto the entities of our best scientific theories. It quite clearly rules out entities notin our best scientific theories, but there seems room for dispute about commitmentto some of the entities that are in these theories. Holism helps to block such amove since, according t o holism, it is the whole theory that is granted empiricalsupport. So, naturalism tells us to look to our best scientific theories for our ontologi-cal commitments. We thus have provisional support for all the entities in thesetheories and no support for entities not in these theories. For reasons of parsi-mony, however, we may wish to grant real status to only those entities that areindispensable to these theories. However, we are unable to pare down our onto-logical commitments further by appealing to some distinction based on empiricalsupport because, according t o holism, all the entities in a confirmed theory receivesuch support. In short, holism blocks the withdrawal of the provisional supportsupplied by naturalism. And that gives us the first premise of the Quine-Putnamindispensability argument. 4 THE HARD ROAD TO NOMINALISM: FIELD'S PROJECTIn the last twenty five years, the indispensability argument has suffered attacksfrom seemingly all directions. Charles Chihara [1973] and Hartry Field [1980]raised doubts about the indispensability of mathematics to science, then ElliotSober [1993],Penelope Maddy 11992; 1995; 19971 and others have expressed con-cerns about whether we really ought to be committed to the indispensable entitiesof our best scientific theories. These attacks can be divided into two kinds: hard-road strategies and easy-road strategies. The hard-road strategies seek to show that mathematics, despite 28As we shall see, Hartry Field [1980] undertakes this task for reasons not unrelated t o thoseI've aired here.

670 Mark Colyvaninitial appearances, is in fact dispensable to science. That is, the hard road tonominalism is to attempt to demonstrate the falsity of the second premise of theindispensability argument. As we shall see in this section, there is a great dealof quite technical work associated with this enterprise - much of which is yetto be carried out. The alternative, the easy road, tackles the first premise andattempts to show that we need not be committed to all the indispensable entitiesof our best scientific theories. This latter strategy, if successful, would avoid themany difficulties associated with the hard road. I begin this section by consideringHartry Field's hard-road strategy, then in the next I consider a couple of attemptsat finding an easy road t o nominalism. Field's distinctive fictionalist philosophy of mathematics has been very influ-ential in the 25 years since the publication of Science Without Numbers [Field,19801. This influence is no accident; it's a tribute to the plausibility of the accountof mathematics offered by Field and his unwillingness t o dodge the issues associ-ated with the applications of mathematics. Furthermore, unlike other nominalistphilosophies of r n a t h e m a t i ~ s ,F~ie~ld's nominalism is not revisionist: I do not propose t o reinterpret any part of classical mathematics; in- stead, I propose to show that the mathematics needed for application to the physical world does not include anything which even prima fa- cie contains references to (or quantifications over) abstract entities like numbers, functions, or sets. Towards that part of mathematics which does contain references t o (or quantification over) abstract entities - and this includes virtually all of conventional mathematics - I adopt a fictional attitude: that is, I see no reason to regard this part of mathematics as true. [Field, 1980, pp. 1-21He accepts the Quinean backdrop discussed in section 3 and agrees that if math-ematics were indispensable t o our best scientific theories, we would have goodreason to grant mathematical entities real status. Field, however, denies thatmathematics is indispensable to science. In effect he accepts the burden of proofin this debate. That is, he accepts that he must show (1) how it is that mathe-matical discourse may be used in its various applications in physical science and(2) that it is possible to do science without reference to mathematical entities.This is indeed an ambitious project and certainly one deserving careful attention,for if it succeeds, the indispensability argument is no longer a way of motivatingmathematical realism.4.I Science without NumbersBefore discussing the details of Field's project, it is important t o understand some-thing of its motivation. Field is driven by two things. First, there are well known 29For example, see [Chihara, 19731, where mathematical discourse is reinterpreted so a s t o beabout linguistic entities rather than mathematical entities.

Mathematics and the World 671prima facie difficultieswith Platonism -namely, the two Benacerraf problems [Be-nacerraf, 1965/1983; Benacerraf, 1973/1983] - which nominalism avoids [Field,1989, p. 6].30 Second, he is motivated by certain rather attractive principles inthe philosophy of science: (1) we ought to seek intrinsic explanations wheneverthis is possible and (2) we ought to eliminate arbitrariness from theories [Field,1980, p. ix]. In relation to (I), Field says, \"one wants to be able to explain thebehaviour of the physical system in terms of the intrinsic features of that system,without invoking extrinsic entities (whether non-mathematical or mathematical)whose properties are irrelevant to the behaviour of the system being explained\"(emphasis in original) [ ~ i e l d 1, 984/1989, p. 1931. He also points out that thisconcern is orthogonal t o nominalism [Field, 1980, p. 441. As for (2), this too is in-dependent of nominalism. Coordinate-independent (tensor) methods used in mostfield theories are considered more attractive by Platonists and nominalists alike.These motivations are important for a full understanding of Field's project; theproject is driven by more than just nominalist sympathies. Now t o the details of Field's project. There are two parts to the project. Thefirst is t o justify the use of mathematics in its various applications in empirical sci-ence. If one is to present a believable, fictional account of mathematics, one mustpresent some account of how mathematics may be used with such effectiveness inits various applications in physical theories. To do this, Field argues that math-ematical theories don't have t o be true to be useful in applications; they merelyneed t o be conservative. Conservativeness is, roughly, that if a mathematical the-ory is added t o a nominalist scientific theory, no nominalist consequences followthat wouldn't follow from the scientific theory alone. I'll have more to say aboutthis shortly. The second part of Field's project is to demonstrate that our bestscientific theories can be suitably nominalised. To do this, he is content to nomi-nalise a large fragment of Newtonian gravitational theory. Although this is a farcry from showing that all our current best scientific theories can be nominalised,it is certainly not trivial. The hope is that once one sees how the elimination ofreference to mathematical entities can be achieved for a typical physical theory, itwill seem plausible that the project could be completed for the rest of science. One further point that is important to bear in mind is that Field is interestedin undermining what he takes t o be the only good argument for Platonism. Heis thus justified in using Platonistic methods. His strategy is to show Platonisti-cally that abstract entities are not needed in order to do empirical science. If hisproject is successful, L'[P]latonismis left in an unstable position: it entails its ownunjustifiability\" [Field, 1980, p. 61. I'll now discuss the first part of his project. Field's account of how mathematical theories might be used in scientific theories,even when the mathematical theory in question is false, is crucial to his fictional-ism about mathematics. Field, of course, does provide such an account, the keyto which is the concept of conservativeness, which may be defined (roughly) asfollows: 300r, rather, nominalism trades these problems for a different set of problems -most notably,t o disarm the indispensability argument.

672 Mark Colyvan A mathematical theory M is said t o be conservative if, for any body of nomi-+nalistic assertions S and any particular nominalistic assertion C, then C is not aconsequence of M S unless it is a consequence of S. A few comments are warranted here in relation to definition 4.1. First, as itstands, the definition is not quite right; it needs refinement in order to avoid cer-tain technical difficulties. For example, we need to exclude the possibility of the+nominalistic theory containing the assertion that there are no abstract entities.Such a situation would render M S inconsistent. There are natural ways ofperforming the refinements required, but the details aren't important here. (See[Field, 1980, pp. 11-12] for details.)31 Second, 'nominalistic assertion' is takento mean an assertion in which all the variables are explicitly restricted to non-mathematical entities (for reasons I suggested earlier). Third, Field is at timesa little unclear about whether he is speaking of semantic entailment or syntacticentailment (e.g., [Field, 1980, pp. 16-19]; in other places (e.g., [Field, 1980, p. 401,and [ ~ i e l d1,985/1989]) he is explicit that it is semantic entailment he is concerned Finally, the key concept of conservativeness is closely related to (seman-tic) consistency.33 Field, however, cannot (and does not) cash out consistencyin model-theoretic terms (as is usually the case), for obviously such a construaldepends on models, and these are not available to a nominalist. Instead, Fieldappeals t o a primitive sense of possibility. Now if it could be proved that all of mathematics were conservative, then itstruth or falsity would be irrelevant to its use in empirical science. More specifically,if some mathematical theory were false but conservative, it would not lead to falsenominalistic assertions when conjoined with some nominalist, empirical theory,unless such false assertions were consequences of the empirical theory alone. AsField puts it, ''mathematics does not need to be true to be good\" [Field, 1985/1989,p. 1251. Put figuratively, conservativeness ensures that the alleged falsity of themathematical theory does not \"infect\" the whole theory. Field provides a number of reasons for thinking that mathematical theories areconservative. These reasons include several formal proofs of the conservativenessof set theory.34 Here I just wish to demonstrate the plausibility of the conserva-tiveness claim by showing how closely related conservativeness is t o consistency.First, for pure set theory (i.e., set theory without u r e l e m e n t ~ c~o~n)servativenessfollows from consistency alone [Field, 1980, p. 131. In the case of impure settheory, the conservativeness claim is a little stronger than consistency. An impureset theory could be consistent but fail to be conservative because it implied con- - 3 1 ~ h e r eare, however, more serious worries about Field's formulation of t h e conservativenessclaim. See [Urquhart, 19901 for details. 3 2 0 f course, this is irrelevant if the logic in question is first-order. But since Field was a t onestage committed t o second-order logic, the semanticsyntactic issue is non-trivial. See [Shapiro,19831 and [Field, 1985/1989] for further details. See also footnote 39 of this chapter. 33Conservativeness entails consistency and, in fact, conservativeness can be defined in termsof consistency. 34See [Field, 1980, pp. 16-19] and [Field, 19921 for details. 35A urelement is a n element of a set that is not itself a set.

Mathematics and the World 673clusions about concrete entities that were not logically true. Field sums up thesituation (emphasis in original): [Sltandard mathematics might turn out not t o be conservative ..., for it might conceivably turn out to be inconsistent, and if it is inconsis- tent it certainly isn't conservative. We would however regard a proof that standard mathematics was inconsistent as extremely surprising, and as showing that standard mathematics needed revision. Equally, it would be extremely surprising if it were t o be discovered that standard mathematics implied that there are at least lo6 non-mathematical ob- jects in the universe, or that the Paris Commune was defeated; and were such a discovery t o be made, all but the most unregenerate ratio- nalists would take this as showing that standard mathematics needed revision. Good mathematics is conservative; a discovery that accepted mathematics isn't conservative would be a discovery that it isn't good. [Field, 1980, p. 131 It is also worth noting that Field claims that there is a disanalogy between math-ematical theories and theories about unobservable physical entities. The latter hesuggests do facilitate new conclusions about observables and hence are not conser-vative [Field, 1980, p. lo]. The disanalogy is due to the fact that conservativenessis also closely related to necessary truth. In fact, conservativeness follows fromnecessary truth. Field remarks that \"[c]onservativeness might loosely be thoughtof as 'necessary truth without the truth'\" [Field, 1988/1989, p. 2411. With conservativeness established, it is permissible for a fictionalist about math-ematics t o use mathematics in a nominalistic scientific theory, despite the falsityof the former. It remains to show that our current best scientific theories can bepurged of their references to abstract objects. Field's strategy for eliminating allreferences t o mathematical objects from empirical science is to appeal to the repre-sentation theorems of measurement theory. Although the details of this are fairlytechnical, no account of Field's project is complete without at least an indicationof how this is done. It is also of considerable interest in its own right. Further-more, as Michael Resnik points out, this part of his project provides a very niceaccount of applied mathematics, which should be of interest to all philosophers ofmathematics, realists and anti-realists alike [Resnik, 1983, p. 5151. In light of allthis, it would be remiss of me not to at least outline this part of Field's project. Field's project is modelled on Hilbert's axiomatisation of Euclidean geometry[Hilbert, 1899/1971]. The central idea is to replace all talk of distance and loca-tion, which require quantification over real numbers, with the comparative predi-cates 'between' and 'congruent,' which require only quantification over space-timepoints. It will be instructive to present this case in a little more detail. Mytreatment here follows [Field, 1980, pp. 24-29]. For present purposes, the important feature of Hilbert's theory is that it containsthe following relations: 1. The three-place between relation (where 'y' is between 'x'and '2' is written

Mark Colyvan 'y Bet xz'), which is intuitively understood t o mean that x is a point on the line segment with endpoints y and z. 2. The four-place segment-congruence relation (where 'x and y are congruent to z and w' is written 'xy Congzw'), which is intuitively understood to mean that the distance from point x to point y is the same as the distance from point z to point w.The notion of (Euclidean) distance appealed to in the segment-congruence relationis not part of Hilbert's theory; in fact, it cannot even be defined in the theory. Butthis does not mean that Hilbert's theory is deficient in any sense, for he proved ina broader mathematical theory the following representation theorem:THEOREM 3 Hilbert's Representation Theorem. For any model of Hilbert's ax-i o m system for space S , there exists a function d : S x S -+ iT%+U( 0 ) which satisfiesthe following two homomorphism conditions: (a) For any four points x, y, z, and w, xy Cong zw iff d(xy) = d(zw); +(b) For any three points x, y, and z, y Bet xz iff d(xy) d(yz) = d(xz).F'rom this it is easy t o show that any Euclidean theorem about length would be trueif restated as a theorem about any function d satisfying the conditions of theorem 3.In this way we can replace quantification over numbers with quantification overpoints. As Field puts it (emphasis in original):So i n the geometry itself w..e.;cbaunt'twtealhkavaeboaumt entuamthbeeorrse,tiacnpdrohoefnwcehiwchecan't talk about distancesassociates claims about distances ...with what we can say in the the-ory. Numerical claims then, are abstract counterparts of purely geo-metric claims, and the equivalence of the abstract .counter-part withwhat it is an abstract counterpart of is established in the broader math-ematical theory. [Field, 1980, pp. 271 Hilbert also proved a uniqueness theorem corresponding to theorem 3. Thistheorem states that if there are two functions dl and da satisfying the conditionsof theorem 3, then dl = kdz where k is some arbitrary positive constant. This,claims Field, provides a satisfying explanation of why geometric laws formulatedin terms of distance are invariant under multiplication by a positive constant (andthat this is the only transformation under which they are invariant). Field claimsthat this is one of the advantages of this approach: The invariance is given anexplanation in terms of the intrinsic facts about space [ ~ i e l d1, 980, pp. 271. With the example of Hilbert's axiomatisation of Euclidean space in hand, Fieldthen does for Newtonian space-time what Hilbert did for IB2. This in itself isnon-trivial, but Field is required to do much more, since he must dispense withall mention of physical quantities. He does this by appeal to relational properties,which compare space-time points with respect to the quantity in question. For ex-ample, rather than saying that some space-time point has a certain gravitational

Mathematics and the World 675potential, Field compares space-time points with respect to the 'greater gravita-tional potential' relation.36 The details of this and the more technical task of howto formulate differential equations involvingscalar quantities (such as gravitationalpotential) in terms of the spatial and scalar relational primitives need not concernus here. (The details can be found in [Field, 1980, pp. 55-91].) The importantpoint is that Field is able to derive an extended representation theorem:37THEOREM 4 Field's Extended Representation Theorem. For any model of atheory N with space-time S that uses comparative predicates but not numericalfunctors there are: (a) a 1-1 spatio-temporal co-ordinate function : S -+ It4, which is unique up to generalised Galilean transformation, (b) a mass density function p : S --+ R+ U {0), which is unique up to a positive multiplicative transformation, and (c) a gravitation potential function Q : S + R, which is unique up to positive linear transformation,all of which are structure preserving (in the sense that the comparative relationsdefined i n terms of these functions coincide with the comparative relations used i nN ) ; moreover, the laws of Newtonian gravitational theory i n their functorial formhold i f a, p, and Q are taken as denotations of the relevant functors. There are many complaints against Field's project, ranging from the complaintthat it is not genuinely nominalist [Resnik, 1985a; Resnik, 1985b] since it makesuse of space-time points, to technical difficulties such as the complaint that it ishard to see how Field's project can be made to work for general relativity where thespace-time manifold has non-constant curvature [Urquhart, 1990, p. 1511 and fortheories where the represented objects are not space-time points, but mathematicalobjects [Malament, 19821.38 Other complaints revolve around issues concerningthe appropriate logic for the project -should it be first- or second-order? - andvarious problems associated with each option.39 Finally, Field's project has been 360f course there is the task of getting t h e axiomatisation of the gravitational potential relationsuch that the desired representation and uniqueness theorems are forthcoming. But much ofField's work has, in effect, been done for him by workers in measurement theory [Field, 1980,pp. 57-58]. 3 7 ~ h setatement of the theorem here is from [Field, 1985/1989, pp. 13@131]. 38For example, in classical Hamiltonian mechanics the represented objects are possible dynam-ical states. Similar problems, it seems, will arise in any phase-space theory, and the prospectslook even dimmer for quantum mechanics [ ~ a l a m e n t 1, 982, pp. 533-5341. See also [ ~ a l a ~ u e r ,1998, chap. 61 for an indication of how the nominalisation of quantum mechanics might proceed. 3gSee, for example, [Shapiro, 1983; Urquhart, 1990; Maddy, 1990b; Maddy, 1990~1in thisregard. See also [Field, 19901, where Field seemingly retreats from his earlier endorsement ofsecond-order logic as a result of subsequent debate. The interested reader is also referred t o[Burgess and Rosen, 19971, (especially pp. 118-123 and pp. 19@196) for a nice survey anddiscussion of criticisms of Field's project.

676 Mark Colyvancriticised because it seems unlikely that his nominalised science is able to properlyaccount for progress [Baker, 2001; Burgess, 19831 and unification [Colyvan, 1999b;Colyvan, 2001a] in science. While such debates are of considerable interest, I will not pursue them here. Itwould seem that the consensus of informed opinion on Field's project is that thevarious technical difficulties it faces leaves a serious question over its likely success. Although I am not yet convinced that Field's project will be successful, I haveno doubt about the importance of his project. Indeed, I, like Field, believe thatthe correct philosophical stance with regard to the realismlanti-realism debate inmathematics hangs on the outcome of his project. However, not everyone takesthis view. In the next section I turn to some criticisms of the first premise ofindispensability argument which are in some ways more fundamental than Field's.The authors I discuss in the next section argue that even if mathematics turnsout to be indispensable to our best scientific theories, that does not mean weneed t o treat mathematics realistically (or as having been confirmed). If they areright about this, then Field's project is irrelevant t o whether mathematical objectsought to be considered real or not. 5 THE EASY ROAD TO NOMINALISM: REJECTING HOLISMNow I turn to some of the attacks on the first premise. There are many suchattacks and I don't have space to do justice to them all here. Instead, 1'11 focuson just two influential ones that give the flavour of this style of critique of theindispensability argument.40 What is common to the following critiques of theindispensability argument is that, in different ways, each rejects holism. That isthey offer arguments against the Quinean thesis that we ought t o be committedto all the indispensable entities of our best scientific theories.5.1 MaddyOne-time mathematical realist Penelope Maddy has advanced some serious objec-tions t o the indispensability argument. Indeed, so serious are these objections,that she has renounced the realism she so enthusiastically argued for in [Maddy,1990a].~l That realism crucially depended on indispensability arguments. Al-though her objections to indispensability arguments are largely independent ofone another, there is a common thread that runs through each of them. Maddy'sarguments draw attention to problems of reconciling the naturalism and confir-mational holism required for the Quine-Putnam indispensability argument. Inparticular, she points out how a holistic view of scientific theories has problems *'Jody Azzouni [2004], Mark Balaguer [1998, chap. 71, Colin Cheyne [2001] and Joseph Melia[2000] are others who have recently argued against the first premise of the indispensability argu-ment. 41She implicitly renounces the set theoretic realism of Realism i n Mathematics in many places,but she explicitly renounces it in [Maddy, 19971.

Mathematics and the World 677explaining the legitimacy of certain aspects of scientific and mathematical prac-tices -practices that presumably ought to be legitimate given the high regard forscientific methodology that naturalism endorses.42 The first objection to the indispensability argument, and in particular t o con-firmational holism, is that the actual attitudes of working scientists towards thecomponents of well-confirmed theories vary \"from belief to grudging toleranceto outright rejection\" [Maddy, 1992, p. 2801. In 'Taking Naturalism Seriously'[Maddy, 19941 Maddy presents a detailed and concrete example that illustratesthese various attitudes. The example is the history of atomic theory from earlylast century, when the (modern) theory was first introduced, until early this cen-tury, when atoms were finally universally accepted as real. The puzzle for theQuinean \"is to distinguish between the situation in 1860, when the atom became'the fundamental unit of chemistry', and that in 1913, when it was accepted asreal\" [Maddy, 1994, p. 3941. After all, if the Quinean ontic thesis is correct, thenscientists ought to have accepted atoms as real once they became indispensableto their theories (presumably around 1860), and yet renowned scientists such asPoincark and Ostwald remained sceptical of the reality of atoms until as late as1904. For Maddy the moral to be drawn from this episode in the history of scienceis that \"the scientist's attitude toward contemporary scientific practice is rarelyso simple as uniform belief in some overall theory\" [Maddy, 1994, p. 3951. Fur-thermore, she claims that \"[slome philosophers might be tempted to discount thisbehavior of actual scientists on the grounds that experimental confirmation isenough, but such a move is not open to the naturalist\" [Maddy, 1992, p. 2811,presumably because \"naturalism counsels us t o second the ontological conclusionsof natural science\" [Maddy, 1995, p. 2511. She concludes: If we remain true to our naturalistic principles, we must allow a dis- tinction t o be drawn between parts of a theory that are true and parts that are merely useful. We must even allow that the merely useful parts might in fact be indispensable, in the sense that no equally good theory of the same phenomena does without them. Granting all this, the indispensability of mathematics in well-confirmed scientifictheories no longer serves to establish its truth. [Maddy, 1992, p. 2811 The next problem for indispensability, Maddy suggests, follows on from the last.Once one rejects the picture of a scientific theory as a homogeneous unit, there'sa need to address the question of whether the mathematical portions of theories 421 should mention that Maddy does not claim t o be advancing a nominalist philosophy ofmathematics; her official position is neither Platonist nor nominalist. Instead, she rejects thismetaphysical approach t o the philosophy of mathematics in favour of a more methodologically-based approach. This results in a position she calls set theoretic naturalism. See [Maddy, 19971for details. Despite her official stance on the realismlanti-realism issue, I include her here amongthe \"easy roaders\" because she, like the others in this camp, rejects t h e first premise of theindispensability argument. It is because of this t h a t she in turn rejects Platonism. This isenough t o make her an easy roader, or at least a travelling companion of the easy roaders.

678 Mark Colyvanfall within the true elements of the confirmed theories. To answer this question,Maddy points out first that much mathematics is used in theories that make use ofhypotheses that are explicitly false, such as the assumption that water is infinitelydeep in the analysis of water waves or that matter is continuous in fluid dynamics.Furthermore, she argues that these hypotheses are indispensable to the relevanttheory, since the theory would be unworkable without them. It would be foolish,however, to argue for the reality of the infinite simply because it appears in ourbest theory of water waves [Maddy, 1995, p. 2541. Next she looks at instances of mathematics appearing in theories not known tocontain explicitly false simplifying assumptions and she claims that \"[s]cientistsseem willing to use strong mathematics whenever it is useful or convenient to doso, without regard to the addition of new abstracta t o their ontologies, and indeed,even more surprisingly, without regard to the additional physical structure presup-posed by that mathematics\" [Maddy, 1995, p. 2551. In support of this claim shelooks at the use of continuum mathematics in physics. It seems the real numbersare used purely for convenience. No regard is given to the addition of uncountablymany extra entities (from the rationals, say) or t o the seemingly important ques-tion of whether space and time (which the reals are frequently used to model) arein fact continuous or even dense. Nor is anyone interested in devising experimentsto test the density or continuity of space and time. She concludes that \"[tlhisstrongly suggests that abstracta and mathematically-induced structural assump-tions are not, after all, on an epistemic par with physical hypotheses\" [Maddy,1995, p. 2561. Maddy begins her third line of objection by noting what she takes to be ananomaly in Quinean naturalism, namely, that it seems t o respect the methodologyof empirical science but not that of mathematics. It seems that, by the indispens-ability argument, mathematical ontology is legitimised only insofar as it is useful toempirical science. This, claims Maddy, is at odds with actual mathematical prac-tice, where theorems of mathematics are believed because they are proved fromthe relevant axioms, not because such theorems are useful in applications [Maddy,1992, p. 2791. Furthermore, she claims that such a \"simple\" indispensability argu-ment leaves too much mathematics unaccounted for. Any mathematics that doesnot find applications in empirical science is apparently without ontological com-mitment. Quine himself suggests that we need some unapplied mathematics inorder to provide a simplificatory rounding out of the mathematics that is applied,but \"[m]agnitudesin excess of such demands, e.g. Iu or inaccessible numbers\"43should be looked upon as \"mathematical recreation and without ontological rights\"433w = Uac, L,where & = 2'a-1, a is an ordinal and Zlo = No. See [ ~ n d e r t o n1, 9 7 7 ,pp. 214-2151 for further details. (a) K > No A cardinal number n is said to be VinXac<cesnsi2b' le iff the following conditions hold: represent K(some texts omit this condition) (b) n and (c) It is not possible t o <Las the supremum of fewer than rc smaller ordinals (i.e., n is regular). For example, satisfies(a) and (b) but not (c). N o satisfies (b) and (c) but obviously not (a). Inaccessible numbershave t o be postulated (by large cardinal axioms) in much the same way as t h e axiom of infinitypostulates (a set of cardinality) No.

Mathematics and the World[Quine, 1986, p. 4001.~~ Maddy claims that this is a mistake, as it is at odds with Quine's own nat-uralism. Quine is suggesting we reject some portions of accepted mathematicaltheory on non-mathematical grounds. Instead, she suggests the following modifiedindispensability argument:45 [Tlhe successful application of mathematics gives us good reason to believe that there are mathematical things. Then, given that mathe- matical things exist, we ask: By what methods can we best determine precisely what mathematical things there are and what properties these things enjoy? To this, our experience to date resoundingly answers: by mathematical methods, the very methods mathematicians use; these methods have effectively produced all of mathematics, including the part so far applied in physical science. [Maddy, 1992, p. 2801This modified indispensability argument and, in particular, the respect it pays t omathematical practice, she finds more in keeping with the spirit, if not the letter,of Quinean naturalism. She then goes on to consider how this modified indispensability argument squareswith mathematical practice. She is particularly interested in some of the indepen-dent questions of set theory such as Cantor's famous continuum hypothesis: Does2N0 = N1? and the question of the Lebesgue measurability of C i sets.46 One aspect 4 4 ~ a t e Qr uine refined his position on the higher reaches of set theory and other parts ofmathematics, which are not, nor are ever likely t o be, applicable t o natural science. For instance,in his last book, he suggested: They are couched in the same vocabulary and grammar as applicable mathematics, so we cannot simply dismiss them a s gibberish, unless by imposing an absurdly awkward gerrymandering of our grammar. Tolerating them, then, we are faced with the question of their truth or falsehood. Many of these sentences can be dealt with by the laws t h a t hold for applicable mathematics. Cases arise, however (notably the axiom of choice and t h e continuum hypothesis), that are demonstrably independent of prior theory. It seems natural a t this point t o follow t h e same maxim that natural scientists habitually follow in framing new hypotheses, namely, simplicity: economy of structure and ontology. [Quine, 1995, p. 561A little later, after considering the possibility of declaring such sentences meaningful but neithertrue nor false, he suggests: I see nothing for it but t o make our peace with this situation. We may simply concede that every statement in our language is true or false, but recognize that in these cases the choice between truth and falsity is indifferent both t o our working conceptual apparatus and t o nature a s reflected in observation categoricals. [Quine, 1995, p. 571Elsewhere [Quine, 1992, pp. 94-95] he expresses similar sentiments. 45This suggestion was in fact made earlier by Hartry Field [1980, pp. 4-51, but of course hedenies that any portion of mathematics is indispensable t o science so he had no reason t o developthe idea. 4 6 ~ 5sets are part of t h e projective hierarchy of sets, obtained by repeated operations ofprojection and complementation on open sets. The C i sets, in particular, are obtained from theopen sets (denoted CA) by taking complements t o obtain the IIA sets, taking projections of theset o obtain the C l sets, taking complements of these t o obtain the rIi sets and finally, taking the

680 Mark Colyvanof mathematical realism that Maddy finds appealing is that independent questionssuch as these ought t o have determinate answers, despite their independence fromthe usual ZFC axioms. The problem though, for indispensability-motivated math-ematical realism, is that it is hard to make sense of what working mathematiciansare doing when they try to settle such questions, or so Maddy claims.For example, in order to settle the question of the Lebesgue measurability ofthe C: sets, new axioms have been proposed as supplements to the standardZFC axioms. Two of these competing axiom candidates are Godel's axiom ofconstructibility, V = L, and large cardinal axioms, such as MC (there exists ameasurable cardinal). These two candidates both settle the question at hand, butwith different answers. MC implies that all Ci sets are Lebesgue measurable,whereas V = L impliesconsensus of informed that there exists a#nLona-nLdebthesagtuseommeealsaurrgaeblcearCd;insael ta. xTiohme opinion is that Vor other is true,47 but the reasons for this verdict seem to have nothing t o dowith applications in physical science. Indeed, much of the appeal of large cardinalaxioms is that they are less restrictive than V =L, so t o oppose such axioms wouldbe \"mathematically counterproductive\" [Maddy, 1995, p. 2651. These are clearlyintra-mathematical arguments that make no appeal to applications.Furthermore, if the indispensability argument is cogent, it is not unreasonableto expect that physical theories would have some bearing on developments in settheory, since they are both part of the same overall theory. For example, Maddyclaims that if space-time is not continuous, as some physicists are suggesting,48this could undermine much of the need for set theory (at least in contexts whereit is interpreted literally) beyond cardinality No. Questions about the existenceof large cardinals would be harder to answer in the positive if it seemed thatindispensability considerations failed t o deliver cardinalities as low as XI. Maddythus suggests that indispensability-motivated mathematical realism advocates settheorists looking a t developments in physics (e.g., theories of quantum gravity) inorder to tailor set theory to best accord with such developments.49 Given that settheorists in general do not do this, a serious revision of mathematical practice isbeing advocated by supporters of the indispensability argument, and this, Maddyclaims, is a violation of naturalism [Maddy, 1992, p. 2891. She concludes:In short, legitimate choice of method in the foundations of set theorydoes not seem to depend on physical facts in the way indispensabilitytheory requires. [Maddy, 1992, p. 2891 Maddy's sustained critique of the indispensability argument is a serious chal-lenge for any defender of the indispensability argument. And I think it's fair to sayprojections of these to obtain the C i sets. See a add^, 1990a, chap. 41 (and references containedtherein) for further details and an interesting discussion of the history of the question of theLebesgue measurability of these sets. 4 7 ~ h e raere, of course, some notable supporters of V = L, in particular, Quine [1992, p. 951and Keith Devlin 119771. 48For example, Richard Feynman [1965, pp. 166-1671 suggests this. 49Cf. [Chihara, 1990, p. 151 for similar sentiments.

Mathematics and the World 681that a defence of the indispensability argument in the light of Maddy's argumentswill need t o address issues about the role of naturalism and the precise role ofmathematics in specific episodes in the history of science. Maddy quite rightlydraws attention t o the diverse roles mathematics plays in science and the differentattitudes scientists can have towards the mathematics they use. Independently ofwhether Maddy's critique of the indispensability argument is deemed successful,this move to a more careful attitude towards both the history and the particulardetails of mathematics in applications is a welcome one. Let me note one issue that Maddy's critique raises: the role of naturalism indebates about ontology and scientific practice. An important part of Maddy'sstrategy for undermining the indispensability argument is to show that confirma-tional holism flies in the face of naturalism. For instance, in her case study of earlyatomic theory, she shows how prominent scientists such as Poincark and Ostwalddid not take the indispensability of atoms to the theory in question to imply thereality of atoms. That is, Maddy takes it that working scientists do not take theholistic attitude t o confirmation that Quine would like. This, claims Maddy, showsthat naturalism and holism are in conflict. But what is the conception of natural-ism being invoked here? At times Maddy suggests that naturalism implies that \"ifphilosophy confiicts with [scientific] practice, it is the philosophy that must give\"[Maddy, 1998a7p. 1761. And, indeed, much of Maddy's case against Quine seemsto rely on such a reading. But this is certainly not Quine's conception of natural-ism. There is much ground between first philosophy, which Quine rejects, and thisphilosophy-last style naturalisms0 that Maddy seems to endorse. For instance,there is the position that science and philosophy are continuous with one anotherand as such there is no high court of appeal. On this view, the philosopher ofscience has much t o contribute t o discussions of both scientific methodology andontological conclusions, as does the scientific community. It may be that you'reinclined t o give more credence to the views of the scientific community in the even-tuality of disagreement between scientists and philosophers, but even this does notimply that it is philosophy that must always give. I take it that this view of scienceand philosophy as continuous, without either having the role of \"high court,\" isin fact the view that Quine intends. As it turns out, this is also the version ofnaturalism that Maddy subscribes to (as she points out in more careful statementsof her position [Maddy, 1998a7p. 1781). Rather than 'philosophy must give' in theearlier passage, she really just means that first philosophy must give. Now returning t o the issue of prominent scientists not adhering to holism. If weunderstand naturalism as 'philosophy last', then the naturalistic philosopher must,with the scientists in question, reject holism. But if we take naturalism to be therejection of first philosophy, then there is room to mount a naturalistic critiqueof the scientists in question. One needs to take care not to attract the chargeof practicing first philosophy whilst mounting this critique, but there is at least 50Elsewhere [Colyvan, 2001a] I've referred to this variety of naturalism as \"rubber stampnaturalism\", since the only role it gives to philosophy is that of rubber stamping approval of allscientific practice.

682 Mark Colyvanroom for a critique. Moreover, the question of whether the Poincark and Ostwaldwere correct in their instrumentalism about atomic theory will not be decidedby appeal t o any general principle that tells us to always side with prominentscientists. Maddy is quite right to focus attention on the historical details andon the role of naturalism here. In the end, I don't think that Maddy's objectionsare as telling against the indispensability argument as may first appear.51 Butirrespective of what Maddy's arguments mean for the fate of the indispensabilityargument, the debate has certainly been shifted in very interesting and fruitfuldirections.5.2 SoberElliott Sober's [1993] objection to the indispensability argument is framed fromthe viewpoint of contrastive empiricism, so it will be necessary to first considersome of the details of this theory in order to evaluate the force of Sober's objection.As will become apparent, though, contrastive empiricism has some difficultiesthatI'm inclined to think cannot be overcome. This robs Sober's objection of much- but not all - of its force. Finally, I will recast the objection without thecontrastive empiricism framework and show that this version of the objection alsofaces significant difficulties. Contrastive empiricism is best understood as a position between scientific re-alism and Bas van Fraassen's [1980] constructive empiricism. The central ideaof contrastive empiricism is the appeal t o the Likelihood Principle as a means ofchoosing between theories. The Likelihood Principle Observation O favours hypothesis H I over hypothesisHz iff P(OIH1) > P(O(H2).It's clear from principle 5.2 that the support a hypothesis receives is a relativematter. As Sober puts it (emphasis in original): The Likelihood Principle entails that the degree of support a theory enjoys should be understood relatively, not absolutely. A theory com- petes with other theories; observations reduce our uncertainty about this competition by discriminating among alternatives. The evidence we have for the theories we accept is evidence that favours those the- ories over others. [Sober, 1993, p. 391According to Sober, though, evidence can never favour one theory over all possiblecompetitors since \"[olur evidence is far less powerful, the range of alternatives thatwe consider far more modest\" [Sober, 1993, p. 391. Another consequence of principle 5.2 is that some observational data may failto discriminate between two theories. For instance, contrastive empiricism can-not discriminate between standard geological and evolutionary theory, and Gosse'stheory that the earth was created about 4,000 years ago with all the fossil records 51See [Colyvan, 1998a; Colyvan, 2001a; Resnik, 1995;Resnik, 19971 for some replies to Maddyon these issues.