An analyse of Quine's 'Ontological Reduction and the World of Numbers'

STEWARD. AN ANALYSIS OF QUINE’S “ONTOLOGICAL REDUCTION AND THE WORLD OF NUMBERS” ABSTRACT. A detailed analysis of Quine's paper on ontological reduction shows that the proxy-function requirement, in his characterization of the concept of ontological reduction, is superfluous for blocking Pythagoreism and inappropriate for a general blockade of ontological monism. l. INTRODUCTION A short answer to the question “What has Quine contributed on the issue of ontological reduction?” could possibly be as follows:' Quine has introduced the demand for the so-called “proxy function” to strengthen a concept of reduction, which asks for a mapping of the sentences of a language £1, which are true in the structure to be reduced, to sentences of a language £2, which are true in the reducing structure, preserving the predicate-logical structure of the sentences. A proxy function is a mapping from the universe to be reduced in the reducing universe such that each basic predicate of the language £; applies to a n-tuple of entities from the universe to be reduced if, and only if, the n-tuple of entities of the reducing universe, assigned by the proxy function, satisfies the £2-formula assigned to the £,-predicate. Thereby Quine blocked the argument for Pythagoreism (the assumption of natural numbers as the universal ontology), which was based on the weaker concept of reduction, and uses the downward Löwenheim-Skolem-Theorem, because such a proxy function cannot be found for every universe. Who has ever read Quine’s paper “Ontological Reduction and the World of Numbers” will readily note that I just gave an enriched (by a few \ \ \D SW.AN, Doso words) and more or less intelligible paraphrasing of what Quine himself formulated, in a much more concise and elegant form, as “the standard of reduction”: The standard of reduction ofa theory (/ to a theory #’ can now be put as follows. We specify a function, not necessarily in the notation of # or 6’, which admits as arguments all objects in the universe of Y and takes values in the universe of 4’. This is the proxy function. Erkenntnis 53: 195-218, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. Mast ed

Author(s): Edward N. Zalta Source: Erkenntnis (1975-), Vol. 53, No. 1/2, Concepts of Reduction in Logic and Philosophy (2000), pp. 219-265 Published by: Springer Stable URL: http://www.jstor.org/stable/20013013 . Accessed: 09/02/2011 06:37 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=springer. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Springer is collaborating with JSTOR to digitize, preserve and extend access to Erkenntnis (1975-). http://www.jstor.org

EDWARD N. ZALTA NEO-LOGICISM? AN ONTOLOGICAL REDUCTION OF Die nat?rlichen der Rest Zahlen hat der liebe Gott Leopold It is now well proper axioms logical notions gemacht, istMenschenwerk. is false. The primitive that logicism accepted theories are not reducible of mathematical Kronecker notions and to primitive the idea underlying and logical axioms. Even logicism to be somewhat problematic, for if the existence claims of math? are to be reducible seem that the to logical truths, then it would appears ematics of objects of some kind. Though a case logicist has to assert the existence can, and has, been made for thinking that logic can contain or imply ex? is at least controversial claims, the matter a as to whether of mathematics reduction question istence and so there is at least a to a logic with existence a reduction to logic. of mathematics thesis which may however, we defend a philosophical some of of the Our thesis is that mathematical preserve spirit logicism. are abstract to) the (reducible objects systematized by a cer? objects just claims would constitute In this paper, tain axiomatic, mathematics-free to be a version of mathematical a certain much simple and intuitive more metaphysical platonism, theory. This thesis appears for if correct, it would make about mathematics position philosophical that mathematics a realm of ab? describes rigorous, namely, are two in there which the present view Nevertheless, ways The first is that the comprehension constitute a kind of neo-logicism. stract objects. might principle for abstract objects that forms part of the metaphysical theory can as a principle be reformulated that 'looks and sounds' like an analytic, if not logical, principle we shall not argue here that the reformulated ana? have argued that principles other philosophers truth. Although is analytic, to it are. The is that the abstract objects systematized by some sense, logical objects. By offering a in theory are, reduction of mathematical objects to logical objects, the present view may us a new with kind of logicism. thereby present logous the metaphysical To establish already in place. second our thesis, we need two elements, The first element is the axiomatic, the first of which metaphysical JUL Erkenntnis 53: 219-265, 2000. W\ ? 2000 Kluwer Academic Publishers. Printed in the Netherlands. theory

EDWARD N. ZALTA of abstract the background described objects. We shall employ ontology of the axiomatic abstract in Zalta (1983) theory by objects developed and (1988).l The axioms of this theory can be stated without appealing or notions of any kind; one of these axioms to mathematical primitives is a comprehension assert explicitly The second for abstract objects the instances of such objects. required for our thesis is this: we must principle the existence element for an arbitrary mathematical of the terms and predicates as denoting a sense theory T, of T which there is a precise (a) analyzes of which show that interpretation these expressions in the background and (b) defines objects ontology, the theorems of T are true. This second element has abstract in which In Zalta in previous work. and (1983) and Linsky only been sketched was de? of the language of mathematics Zalta (1995), a basic analysis the previous work by offering amuch veloped. The present paper advances more detailed account of reference and truth with respect to mathematical with reference, we explicitly identify the steps terms and relation symbols of an the well-defined In connection language. required to interpret arbitrary mathematical denote unique ab? theory so that those expressions stract individuals in our background and abstract relations, respectively, In 4. with truth, in connection Section This task is accomplished ontology. we use under which objects to state the conditions are true. This task is accomplished theories section of the paper (Section 7), we make several the theory of abstract the theorems of mathematical in Section 6. In the final the present the extent to which and we briefly consider a kind of neo-logicism. axiomatic '0', theory of abstract objects, henceforth about our work observations view constitutes Our background has been developed in both a modal and a type-theoretic setting and these two manifestations of the theory have been applied in numerous ways. The and in the applica? abstract individuals, modal version of 0 axiomatizes Platonic the laws governing forms, possible worlds, and natural numbers (among other things) have been the language of fiction and belief has been given a precise tions of this theory, Leibnizian concepts, derived and interpretation. the abstract objects of typed version of 0 axiomatizes asserts that for each logical type logical type. This version The every simple indi? relation among of object (e.g., individual, property of individuals, relation of of individuals, viduals, property among properties properties of individuals, etc.), there are abstract objects of that type (in addition to ordinary objects of that type). ory, such things as the Fregean and relations of rejected of this type the? applications of predicates, the fictional properties In the various senses scientific theories, mathematical properties

relations have all being prime), and mathematical (e.g., set membership) been identified as particular abstract properties and abstract relations.2 Now it might be thought that all of these applications of 0 involve a reduction. But, in fact, this is not the case. It is single kind of ontological that there are two distinct kinds of ontological very important to recognize that can be constructed reduction within 0. We shall call these classical and metaphysical reductions, - we follows establish This distinction is critical to respectively. the main thesis of this paper by developing of the objects of mathematical theories to the what a metaphysical reduction of our metaphysics. objects In order to distinguish text of 0, and metaphysical in the con? reductions to mention first that in classical the reductions, theories the axioms of some theory T are derived as classical it is important reduction is between theorems of 0. In the case of metaphysical it ismore reductions, however, to say that the objects of a theory T are reduced to the objects perspicuous of 0. (We shall precisely define 'object of theory 7" in Section 3.) Now various philosophers have described the basic idea of a classical as a to a theory T just in case all reduction follows: theory S is reducible of the non-logical notions of S can be explicitly defined in T in such a way are theorems that the translations of the theorems of S (via the definitions) of T ? We can recast by saying that S is reducible in language more this definition to T just in case to logicians of S constitute a familiar the theorems extension of T. This definition gives us the basic subtheory of a definitional sense of reduction that the logicist might have used to claim that the proper axioms idea was of mathematics are reducible to the theorems of logic. The logicist by a set of analytically-true logical ax? T is reducible Then, a mathematical theory that a logic L is defined ioms and rules of inference. to L just in case the primitive terms and predicates of T are definable in the language of L and the proper theorems of T (when translated into the language of L) become logical theorems of L. Of course, as mentioned that the primitive notions or proper earlier, probably no one now believes axioms and theorems of mathematical in this way to theories are reducible the primitive notions and axioms of logic. briefly to mention It is important to digress gicians have introduced a variety notions of reduction tions as relative between of much more the fact that modern lo? and fine-grained explicit theories T and S. They have defined such no? reduction, model-theoretic interpretability, proof-theoretic and even axiomatized notions of reduction.4 However, these reduction, more fine-grained notions of reducibility will not play a role in what fol? lows. Although the classical reductions available in 0 are all instances of

EDWARD relative interpretations, of a classical reduction it should become apparent that the above definition for the purposes of this paper. some classical reductions that are avail? should suffice 1, we shall rehearse see that the primitive in 0. We'll In Section able N. ZALTA world theory, language of 0 and Dedekind/Peano of situation notions number and that the proper in terms of these explicit theory theory, possible can be defined in the these notions can be governing definitions and) derived as theorems of axioms (couched as we mentioned to classical 0. However, earlier, we shall not be appealing the main thesis of this paper. reductions of any kind to establish a new kind of onto? Instead, we shall defend our thesis by developing reduction. The additional logical reduction which we call metaphysical resources to develop make it 0 this by ontological provided possible and essentially different kind of reduction. We'll spend of the paper preparing the ground for, and developing preponderance distinctive reductions. amples of, metaphysical arbitrary mathematical The will the ex? that, for an identifies both show examples reduction theory T, a metaphysical terms and predicates the reference of the well-defined of T and preserves a sense in which the theorems of T are true. More 0 will specifically, in which we theoretical framework provide us with a mathematics-free can precisely abstract individuals and abstract relations. Certain specify of abstracta simply objectify the roles that mathematical in? specifications once we T. extend So dividuals and relations play in amathematical theory 0 by adding the terms and predicates of T and by adding the analytic truths which mathematical of T are alleged to play in T, we'll be able objects to theoretically identify those of an arbitrary mathe? their objectified roles. The theorems T will then have theory compositionally specifiable with objects matical on which individuals they (the theorems) and relations of 0 will therefore abstract individuals ical reductions distinctive the reason So neither that every mathematical abstract individuals readings in 0 turn out to be true. Moreover, the abstract figure into these readings. Our metaphys? show that each mathematical and abstract classical grained notions of reduction is that if the view developed show the role that the mathematical articulate theory is about relations. reductions nor the other more fine? role in what follows will play a significant reductions will here is correct, metaphysical own kinds of about its distinctive is theory and/or abstract relations. From the point of view of on? of language, then, there may be no metaphysical tology and the philosophy reason to investigate these other kinds of reduction of one mathematical there are still math? theory to another mathematical theory. Of course, ematical reasons reduction as they apply to investigate classical tomathematical and more theories, notions of fine-grained to assess the for example,

power of certain theories in various ways and to understand the various ways in which one mathematical theory might be distinguished as a foundation But even if one mathematical for the rest of mathematics. mathematical theory from (in) which all other math? theory emerges as the foundational or as that in which all other ematical theories can be derived (interpreted), can be modeled, it doesn't follow that those other are just theories of or about the objects described by theory, at least not if we can show that each mathematical mathematical theories mathematical theories the foundational kind of mathematical individuals and theory is about its own distinctive account of truth and relations. Moreover, 0 would offer a metaphysical for the language and theorems as a foundation for mathematics. reference emerges of any mathematical theory that Although we shall return to these issues in the last section of the paper, one issue to which we shall not return, namely, it is important to mention our assumption that both classical tute genuine ontological reductions. matter, we shall not spend shall assume that the work time and metaphysical Although in this paper carried reductions this may consti? be a controversial on the matter; instead, we some reason to think out here offers that the ontological individual and mathematical categories mathematical are not sui generis but rather subcategories of the more funda? abstract individual and abstract relation, ontological categories relation mental respectively. We turn, then, to a brief description of some classical reductions so that we will be better prepared to appreciate what is distinctive the metaphysical reductions familiar with Readers to Section that establish these applications it will be of object thesis of this paper. theory may skip ahead 2. 1. CLASSICAL In order the main in 0, about to discuss the classical REDUCTIONS reductions for the reader IN 0 that have been effected in 0, to know the language and axioms of important the theory. In what follows, we shall presuppose that the reader is famil? iar with one of the canonical presentations of 0 in other publications. In this section, we shall discuss the version of the theory that has been ex? in a syntactically second-order modal (S5 with Barcan formulas) pressed so as to calculus has been modified (without predicate identity) which include a second kind of atomic formula, namely, formulas of the form 'if1' (individual x encodes property F1). A single theoretical primitive '?!' ('being concrete') is used to define the property of being property abstract (A\x =df -'OElx) and the comprehension principle for abstract

N. ZALTA EDWARD asserts individuals an abstract that for any condition cp (without the that encodes just properties individual = & VF(xF free jc's), there is <p (i.e., satisfying are said to be identical individuals <p))).5 Abstract same properties, the but to show that encode they necessarily x and y are the same abstract individual, it suffices to show that x and same the is rigid (i.e., encode since the y logic of encoding properties, -> DxF). OxF 3x(A\x whenever of the theory of abstract individuals also term. There is a complex way of denoting of complex of the form ixcp (for any individuals, namely, rigid definite descriptions are in the usual way, These definite axiomatized formula (p). descriptions The canonical formulation two kinds includes analysis of descriptions by a principle which asserts that Russell's a to There is also formula that contains atomic any applies description.6 a complex way of denoting of the form relations, namely, ?-expressions no no and no subformulas free F's, yn cp] (where (p has [Xy\... encoding namely, are axiomatized by the usual principle a ensures that and (i.e., ?-abstraction), by principle which no to the relation denoted difference makes bound variables of descriptions). ?-Conversion These ?-expressions exchange ?-Conversion immediately by the ?-expression.7 The of relations for relations.8 theory principle tion of when relations are said to be identical.9 (whether identical abstract to work in all contexts. yields a comprehension is completed by a defini? of identicals Substitution or identical individuals relations) is stipulated reductions in terms of this language and theory, several classical Now, are achieved in precisely the way one have been effected. These reductions that it is often the case that the target the? would expect with the exception So, for example, although ory has not been given a canonical presentation. we defined situation theory does not have a canonical axiomatization, (in Zalta the following (1993)) of 0 :x is a situation, guage situation of basic s is a part of situation situation the theory theory of possible worlds does of situation s makes t. From are derivable theory in the lan? state of affairs p true, and the usual axioms these definitions, notions situation as theorems not have of 0. Similarly, although a canonical axiomatization, we defined (in Zalta (1983) and (1993)) the following notions of world theory: object x encodes proposition p is true at world w, w is maximal, and w is actual.10 Then we p, x is a possible world, proposition w is modally w is consistent, closed, of world theory: principles is consistent, is maximal, (c) every world (b) every world (a) every world is modally closed, (e) a proposition (d) there is a unique actual world, is true iff it is true in all possible worlds, is necessarily (f) a proposition some have whenever worlds true true it in iff is (g) possible world, possibly derived the usual

true at them, those worlds are identical.11 Although the same propositions a canonical not it seems clear that has had axiomatization, theory world this theory would employ some subset of these as axioms. If this is right, then we have a classical ontological of world theory to 0. to axiomatize any attempt principles reduction theory of (complete applied so as to reduce Leibniz's as an example But of and Plato's forms.12 individual) theory concepts we where the theory being reduced does have a canonical axiomatization, note the following. When 0 is extended with the logic of actuality and two 0 has also been a priori definitions and plausible axioms, Frege's and the Dedekind/Peano and 0 can of Predecessor axioms for number theory become about natural numbers). (in addition to many Fregean principles in the language of 0: the following notions In Zalta (1999), we defined x is a predecessor of y, x is the number of G's, x is a natural number, to axioms and zero. We then added the formal versions of the following be constructed theorems and its weak ancestral are relations, and (b) if there is (a) predecessor a natural number n which numbers the property G, then there might have a that concrete individual distinct from all of the concrete individuals been 0: axioms for the G.13 As a result, the Dedekind/Peano exemplify in 0. become of natural numbers provable theory From a logical point of view, there is nothing unusual or distinctive in 0. They may, reductions available about these classical ontological actually or logician in? interest for the metaphysician however, hold philosophical or concerned the number of ontological terested in minimizing categories are to find a system in which there claims that proofs of metaphysical philosophers typically have to stipulate.14 REDUCTIONS 2. METAPHYSICAL are now to be contrasted with the just discussed reductions that are available in 0. The metaphysical The classical new and distinctive point of these metaphysical reductions logical notions of mathematical formulate of mathematics theories theories denote abstract theories denote abstract theorems of mathematical themselves the primitive of 0 or to derive and that the relation relations, theories and reductions will the axioms but rather to: (1) interpret (i.e., for) the language and axioms of individuals can be identified non of 0, so as to reveal both that the individual theories Our metaphysical in the language as theorems truth conditions denotational mathematical is not to define reductions symbols of those in 0 of the readings (2) develop on which those theorems also show as abstract terms of those turn out true. that the mathematical individuals. In order theories tomake these

EDWARD ideas perfectly nical machinery. and precise, we shall need to introduce some tech? in Section 3. technical machinery will be developed clear This in Section 4, we construct Then, mathematical objects to abstract important to prepare the reader the theoretical that reduce descriptions In the meantime, it is however, objects. for the material the philosophy by briefly outlining is based.15 N. ZALTA in the following on which of mathematics sections this material assumes of mathematics that the primary data that philosophy a are sen? true the analysis requires ordinary mathematical philosophical tences of the form 'Inmathematical for theory T, p (is true)'. So, example, we shall try to systematically as 'In Real Num? interpret such statements Our set theory, ber Theory, n is greater than 3' and 'In Zermelo-Fraenkel no set is a member that these statements, when of the empty set'. Note stripped of the prefix 'Inmathematical in the formal languages of mathematics. of English expressions denote distinctive abstract the formal So, on some fixed English and symbols require in what occasions sentences their standard formal follows, we will refer to both the unpre set is a member 'No of the empty set') and (e.g., as ordinary & xe0)') renditions (e.g., '->3x(Set(x) shall call the ordinary language language. We ' ... the 'theory-operator' and, in what theory T, a formal notion of 0 which will be used to precisely of mathematical 'In mathematical prefix follows, we will define translate this theory operator. The translation truth conditions for the prefixed statements become compositional By contrast, the unprefixed will be resolved in the framework. The the formal of expressions the and (Both objects. English a philosophical and analysis.) interpretation the mathematical mathematics statements theory T', are frequently expressed We shall want to show that both this ambiguity that of mathematics) idea underlying guage (including that is disambiguated thereby yields well-defined, statements of mathematics. subject to an ambiguity that in ordinary lan? is that predication is subject to a structural ambiguity 'Fx' and 'xF\ The by our two modes of predication 'n is irrational' and 'No set is mathematics of (e.g., unprefixed a member of the empty set') are subject to this ambiguity. They have a on a which they turn out reading they turn out true and reading on which statements as sentences will be analyzed of the unprefixed one on statement is the 'n For reading, example, encoding predications. irrational' (made in connection with Real Number Theory 9t) is true if and namely n^ (which can be precisely only if a certain abstract individual, encodes a certain abstract property, namely, being irrational^ identified), false. The true readings identified). But (which can also be precisely is true iff n^ exemplifies is irrational' being on the second irrational^. We reading, take these

statements to second 'exemplification' readings of ordinary mathematical non as such be false, n^ does exemplify abstract, properties being being round, being non-red, being thought about by the reader at this moment, etc. But the present view is that the mathematical of rc^ are properties not exemplifies.16 the standard ex? Although are false, the sentences of mathematical unprefixed emplification readings recover the mathematical and intuition readings encoding philosophical that it encodes, properties that there is a sense We esting issues until Sections sentences are true.17 these unprefixed and defense of the myriad of inter? further discussion in which shall postpone that arise in connection 6 and 7. But the preliminary preciate follow. 3. with PRELIMINARY of mathematics now be in a position to ap? of definitions, rules and which THEORETICAL PRINCIPLES the reader series this philosophy should theorems solely on the language and axioms of 0, leaving the ques? to represent the data from ordinary mathematics to Sections 4 we a reading and 6. Now recall that two paragraphs described back, terms for 'n is irrational' of the abstract property (a theorem of 9?) in We now focus tion of how Just as abstract individuals encode properties, abstract of and abstract encode relations properties properties etc. To we such of claims relations, represent properties employ precisely, the type-theoretic version of 0. This theory is stated in a typed language being irrational^. encode properties by the following governed i is a logical Where t\,..., definition type. tn are any types, of 'logical (t\,..., type': tn) is a logical type. Our includes for each type. (constants and) variables x*, yl,... language so a is / for individuals the and xl will be variable ran? type Intuitively, over individuals. {t\,..., tn) is the type for relations that hold among ging objects quently to make Instead of x{tu -,tn\ we fre? ,tn, respectively. types t\,... having use the variable F^1,~'Jn) to range over relations of this type, so as it clearer that the object in question is a relation. For each type t, '?!(/)' ('concrete' ') that applies to things predicate of type t. In terms of this predicate, we define a predicate that characterizes the ordinary objects ('0!') and abstract objects ('A!') of type t as follows: there is a distinguished 0!(V Al{t)x* =df -0?!(V

EDWARD n may Finally, since ranging over objects N. ZALTA be 0 in (t\,..., tn), we shall use 'p' as a variable of the empty type ( ). Intuitively, this is the type for propositions. With this typing scheme, we may assume that the formulas and complex in the usual way. terms of the language of our type theory can be specified that we Note two kinds still have xt\ p(h,...,tn) of atomic formula: xtn xlF{t) Since it is straightforward to specify the well-formed formulas and com? can be inferred terms, we will omit the definition here. The definition now some which of of the main 0, operate at principles by examining of a term whose each type (we suppress types on the reoccurrences type plex in the formula): has already been specified (2) 0\{t)xl -> D^3F{t)xF (3) 3xt(A\{t)x (4) x< = / 0!(V &VF{t)(xF = <p)), where cphas no free*'s =df & 0\yl & nVF{t)(Fx = Fy) V A!(V & A!/ & DVF(i)(xF ==yF) (5) ... ... xtn = where ytn (p]xtl [?/1 (pxn 'XJ, and no definite descriptions subformulas (6) Ojc'F(?) <phas no encoding -> DjcF type, do not encode proper? (2) asserts that ordinary objects, of whatever for abstract ties. (3) is the comprehension objects and asserts that principle cp is a condition on properties F(i) (i.e., the F's characterize objects of type t), there is an abstract object of type t that encodes all and only the for all objects: objects x% F's satisfying cp. (4) defines identity conditions are are either they both ordinary objects of type identical whenever and yx when or they are both abstract the same properties exemplify same t encode the and of (Substitution properties. necessarily type objects the is ?-Conversion of identicals governs this defined notion.) (5) principle t and necessarily that governs xtn exemplify It asserts that objects xh,..., ?-expressions. relation being ayh,... ,ytn such that cp if and only if x'1,..., complex

is not relative satisfy (p. (6) is a logical axiom which asserts that encoding are rigidly encoded. to any circumstance encoded properties that (3) and (4) jointly guarantee that for any formula Notice <p (with no free x* 's), there is a unique abstract object of type t that encodes all and cp. (There couldn't be two distinct abstract satisfying only the properties the properties cp if distinct abstract exactly satisfying to differ with respect to at least one encoded property.) So that objects that encode objects means have canonical the following of an abstract description object is always well-defined: = ix^Al^x&WF^^xF Moreover, such canonical cp)) are governed descriptions of the logic of descriptions, consequence encodes exactly the properties satisfying satisfies cp: iff G^ in what follows. shall appeal to this theorem on occasion 1 of this paper, we discussed In Section only abstract individuals. How? = ixt(A\{t)x&yF{t)(xF ever, we can now assert the existence and abstract relations that individuals 3x{i)(A\mx & WFm(xF 3x{U)(A\{{U))x&WF{{u))(xF We a property G^ (1) We namely, cpencodes by a straightforward the abstract object x* that plan among (second) Now to show = cp))G{t) <p$ of abstract properties may exemplify: = cp)), where = of individuals q>has no free x{ih cp)), where <phas no free jc^s the abstract can be found (relations) properties are to exist by the first which asserted asserts to have a formula of 0 that follows, is a mathematical suffice theory. It would that mathematical properties of the above principles. in what it will be useful, explicitly for our purposes that something as a primitive relation symbol 'MathTheory' of 0. Such a predicate would allow us to introduce axioms and specify for mathematical theories. However, instead of this ex? identity conditions a new more of it is pedient adding single primitive notion, philosophically to just add to define the notion of 'mathematical perspicuous other primitive, but reasonably well-understood, theory' notions. in terms of two these are, it is important to point out that we our pretheoretic ability to recognize mathematical we say shall ultimately rely on we come theories when across of what them. We do not intend to use the definition Before 'MathTheory(xY

EDWARD is a mathematical prove that anything is to tell us, in theoretical definition we identify pretheoretically assert this in 0. So to define following mathematical proposition' theory. Instead, the purpose of the terms, what it is that we know when something as a mathematical of a 'mathematical the notion two primitive N. ZALTA notions. First, we theory and then we theory' in 0, need the notion ('Math(pY). Fortunately, on what this notion amounts we need of a have the 'purely a pretty to. It is reasonably grasp good pretheoretic are mathematical. constants and predicates Mathe? clear which primitive con? maticians and logicians certainly have no trouble identifying which stants and predicates in order calculus have to be added to a predicate to state the proper axioms of some mathematical some theory. Though sets are logical objects19 or that (primitive) have supposed philosophers is a logical rather than a mathematical that set membership relation,20 now agrees that when taken as a primitive notion, pretty much everyone notion and theoretical is a non-logical, (i.e., mathematical) 'membership' axioms. So, I'll assume that that the axioms of set theory are non-logical we can judge pretheoretically and constants are mathem? which predicates as a primitive notion.21 atical, and that this ability justifies taking Math(p) The other notion we'll need to define a mathematical theory is the notion two individuals is a relation that holds between 'authorship'. Authorship ' we to assert use that xl authors and xl and yl yl. We'll explain 'A{l'l)xlyl a moment. why this notion is important in just of two primitive notions, we may define a mathe? to individual be abstract (i.e., object of type 0 which any theory out of mathematical constructed encodes (i) properties only propositional Now in terms of our matical and (ii) is authored propositions, (8) by some concrete individual: MathTheory(x[) =df WF{i)(xF -> 3p(Math(p) & F = [?/ p])) & 3y(E\{i)y & A{iJ)yx) about the The authorship relation is used so that we can talk primarily Of course, we theories that have actually been constructed. mathematical shall want our analysis to apply to any possible mathematical theory, and - we in front this is easily done 'possibly' simply add the modal operator of the second conjunct of (8) (i.e., so that it reads: itmight be the case that in what follows, individual). However, by some concrete we use the we need not concern ourselves with this subtlety. Henceforth, variables T and S to range over actual theories. is authored xl We only next say that a proposition p is true in theory T if T encodes the property [Xy p]: (9) T^p =df T[ky p] (=/?') if and

In the next use we will section, this defined to translate the ordin? T, ... '. But for now, notion Tn mathematical ary language theory-prefix theory let us note that given this definition, we may extend our central notion, x* encodes F{t), so that we may say that certain individuals (in particu? We will say that a theory T encodes propositions. p just in case p is true in T, i.e., just in case T encodes [Xy p]. proposition It is important next to stipulate that mathematical theories are closed encode lar, theories) under proof-theoretic is defined \jr,which so defined has been If we consequence. in 0 proof-theoretically in previous work), <pn \ (and which the notion utilize <pi,..., in the usual way stipulate that whenever then we may of propositions p\,... q is a proof-theoretic consequence proposition are true in mathematical and the pt all theory T, then q is true in T: Rule (10) of Closure If pi,..., pn for Mathematical \- q and T f= p\ Theories and ... ,pn T: and T \= pn, then T\=q. In what follows, closed that mathematical theories are therefore, we assume to this rule. In addition, we shall often refer to the proof according as its 'logical' consequences. of a proposition theoretic consequences So to not the reader is hereby cautioned remember both (a) that 'T\=p' does that p is a logical consequence of T but rather is defined as in (9), and (b) that 6p\-q9 asserts that q is a logical (proof-theoretic) consequence of p. assert we are in a position to appreciate the significance of two simple the first of which gives us a theoretical description of the abstract T to with which a mathematical is be identified. It is a theory our of if T is definition of that 'mathematical consequence theory' Now theorems, individual simple a mathematical theory, then T is identical with encodes all and only (the properties that are true in T. In formal terms:22 (11) There MathTheory(T) -> T = ix?(Alx&VF(xF are two important the definite constructed description = individual out of) the propositions about (11). The first is that in (11), namely: ixl (A\x & VF(jcF = 3p(T\=p & F=[ky is well-defined. We established of an abstractum is well-defined, that 3p(T\=p & F=[ky p]))) to make observations that appears the abstract earlier /?]))), that any such canonical description as a consequence of our comprehension principle (3) and identity principle (4) for abstracta.

EDWARD N. ZALTA does not thing to note about (11) is that its consequent de? of mathematical theory T, but rather a theoretical we next T. extend the of 0 of Once the (in section) by language scription theories, the above theorem can introducing names of actual mathematical to those names, resulting in specific identity claims which be instantiated The second a definition offer the named identify important mathematical theories as well-defined individuals. But any names of actual into our in the next section, where we show how to translate the data into At this point, we have simply defined some technical system. and have produced of these notions. of 0 a simple identity simple theorem we for mathematical conditions identity conditions The it is can't yet instantiate in (11) in any interesting way. Nor do we have available true sentences of the form 'r (= <p\ Such sentences will become our formal notions abstract yet introduced language and so we that we haven't F the variable specific available theories to remember second for abstracta theorem are in a position that is statable in terms to appreciate tells us It is a consequence of our the definition of mathematical theories. (4) and theories (8) that theories T and S are identical if and only if all and only the propositions true in T are true in S: T = S = Vp(T?=p = S?=p) This, presumably, for theories. is exactly what one would expect as identity conditions can complete of the machinery the specification needed for meta? of with definitions. the series First we say reductions following physical that an object x* (of type 0 is an object of theory S iff there is a property We such that it is true in S that xt exemplifies F{t) (12) ObjectOf(xt, S) =df F: 3F{t)(S[=Fxt) = i, we individual of follows, when t say that x* is amathematical = we a t of is that mathematical x* S; and when S; when say (/), property = t relation of S. (/, /), we say that xl is a mathematical In what the objects We can now formulate a quite general axiom for identifying of theory S as abstract objects. This statement of the axiom is simplified theories.23 We only pure mathematical by the fact that we are considering assert the following (13) as axiomatic: Reduction Axiom: = ObjectOf(xt, S) -> x ly^Al^y&WF^^yF = S\=Fx)) if xl is a type-i object of theory S, then x is the abstract In other words, true of x in S. object that encodes exactly the properties

it is an immedi? canonical descriptions, By our theorem (7) governing ate Corollary that if xt is an object of S, then x of the Reduction Axiom F: iff it is true in S that x exemplifies encodes a property F^ (14) We ObjectOf(x\ see shall some in the next rollary allow us to identify S) -> (xF{t) specific section. instances = S\=Fx) of our Reduction we In the meantime, a the objects of mathematical Axiom and Co? note that the principles theory no matter what the logical type of the object. It is to be stressed here that from there is no distinguished of view of a foundational metaphysics, the point 'model to tell us what are the 'objects of a theory T. From the objects of a theory are the ones described theoretic' perspective a metaphysical point of view, re to objects. Note its de that the claims, for these attribute properties by re as a statement counts '3xl P{l)xJ de claim about the property F, but that as a de re claim count it doesn't about mathematical individuals. From we can validly infer 3F^(T\=3xlFx), but we can't validly T\=3xl P^x, infer 3xl(T\=Px). This tells us that from a logical standpoint, we cannot inside the scope of the theory operator.24 The validly export the quantifiers of this fact will implications 4. become clearer REDUCING MATHEMATICS in Section 6. TO METAPHYSICS reduce the objects of an arbitrary mathematical theory of our formal metaphysics. We'll the begin by introducing our of ordinary mathematical into the of language language In this section we'll to the objects expressions formal system. it should be clear what the resulting Though the statements of ordinary mathematical language is between relationship and those of our system, we will leave the explicit discussion of the interesting features of this relationship until the next section. let r range over Now, to actually carry out our analysis and reduction, names of mathematical theories and suppose that we pretheoretically judge that a group of sentences constitute the proper axioms of a mathematical r. us to these sentences as 'the axioms of named Let refer simply theory r'. We shall assume that the axioms of r have been, or can be, formalized in a first- or second-order (with identity). So whenever predicate calculus sentence s is an axiom an axiom of r. So as to reduce we make the axioms of r, we will the amount also of work we say that its formal shall have rendition to do in what <p is follows, three simplifying metatheoretical (I) assumptions: of r that are instances of a first-order axiom schema can all the following

EDWARD N. ZALTA axiom which employs replaced by a single second-order quantifiers over relations, the axioms of r involve a primitive rc-place (II) whenever an n + 1 function involving they can be replaced by axioms symbol, '=' r involve the axioms of symbol, and (III) whenever place predicate '=' as a as a logical primitive, involving they can be replaced by axioms be relation symbol (so that the standard two logical non-logical distinguished discuss axioms for identity become proper axioms). We'll simplification it should help if we remind (III) in the next section. For the meantime, of r asserts a sentence an identity, the individuals ex? same the to be identical exemplify (or conditions) properties in the language of r. (For each mathematical theory r, we shall the reader asserted pressible that when '=T' by which one employ a relation symbol same the individuals r-expressible exemplify that nothing 5, we'll demonstrate to assumption (III). Section 4.1. can assert that r-identical in properties.) Moreover, important is lost by appealing 0 Extending to the language In this subsection, we extend 0 by adding new expressions and by adding certain analytic truths and certain obvious facts as new ax? we pretheoretically ioms. Now, whenever judge that a group of sentences constitute the axioms for a mathematical theory named r, we extend the name as we r as a new constant of type add the follows: of 0 (a) language k (if there are any) that constant for / to our language, each (b) primitive of r, we add kt as a new constant of type / to our relation symbol II appearing language, and (c) for each n -place primitive in an axiom of r, we add nr as a new relation symbol of type (/,...,/) in an axiom appears (with n occurrences that clause (c) and sim? '/') to our language. Note ensure an if axiom of r involves that '=', (III) together '=T' as a new 2-place relation symbol of 0. of plifying assumption we shall be adding the formula Next, for each proper axiom cpof theory r, let <p*designate of 0 that results when each primitive constant k in <pis replaced by /cTand each primitive predicate n in ^ is replaced by nT. Then, for each proper axiom (p of r, we take the following r (whenever (15) |= <p* displayed no matter convention possible line of how cp is an axiom of this essay, we In the remainder the form complex (described scope within r\=\//, as a new axiom of 0: of r) In any convention. adopt the following the scope of '|=' will extend over \?r, the \?rmay be. So this is a special case where narrowest in note 10 of Section the of 1) '|=' giving a formula is to be overridden.

of 0 are, in a real sense, analytic truths. Tn the claims of the form They explicitly represent language ' ordinary ... our mathematical in terms of formal machinery. Note also theory r, that these new axioms Notice of r involve '=', then given simplifying assumption us as new axioms of 0 (in which to add the following (15) requires are variables of type / and F is a variable of type (/)): that if the axioms (16) r |= x=zx (17) r \= x=Ty Notice -> VF(Fx = (III), x, y Fy) that the quantifier 'r operator 'VF' in (17) lies within the scope of the theory the indiscernibility of x and y (with respect to on the r -identity of x and y only relative is conditioned So ...'. \= exemplification) to r itself. From our Rule of Closure (17) and the fact that r |= x=Ty, = lets us conclude only that r |= VF(Fx Fy) and so we have to show that t \= Px if we want to conclude 'VF' that r \= Py. The quantifier governs the r-relative properties of x and y. determined) mathematical Finally, for each (pretheoretically therefore we also add the following (18) Notice fact as a new assumption of 0 theory : r, MathTheory(x) (15) and (18), we know metatheoretically ensures that the translation <p*of every ordinary that given of Closure becomes (19) derivable r Reducing in 0 f= cp* behind Reasoning theory in question 4.2. obvious as an explicit (whenever r-relative cpof r truth: \-T cp) the theory operator, therefore, employs classical logic.25 the Objects that our Rule theorem is classical whenever the of Set Theories now examine how the foregoing a metaphysical facilitates reduction a of any set described set As by any theory. particular example, we shall consider the sets of a simple set 'adjunctive' theory. This simple theory is representative and it should be clear how the techniques used can be We set theory and other set applied to reduce the objects of Zermelo-Fraenkel as the axioms of the theory named theories. Let us designate the following 'ST': (ST1) Sets which have the same members are identical.

EDWARD The empty (ST3) No (ST4) For (ST5) For N. ZALTA set is a set. set is is a member of the empty set.26 any two sets, there is a set having members of the second set as members. the first any property F and set x, there members all and only those members is a set which set and of x which the has as exemplify F. These are familiar - axioms is the Axiom (ST1) of Extensionality, (ST2) and (ST3) describe the empty set, (ST4) is theAxiom of Adjunction, and (ST5) is the Axiom in classical axioms of Separation. to formalize It is straightforward these that the primitive, logic by assuming exemplification are term the 'the empty set' ('0') and the expressions singular non-logical 'is a set' OS'), 'is a member of non-logical predicates same as' ('=') The formalization would go as follows: (STf ) VjcVj[5jc &Sy^> [Vz(z e x ('e') and 'is the = ze = y) -> x y]] (ST20 S0 G0) (ST3') -*3x(Sx&x (ST4') & Sy -> VxVy[Sx ez=w=x 3ziw(w (ST50 VFVjc[5jc -* 3y(Sy & Vz(z ey=zex& that we v w e y)] Fz))] (I) so as to formulate assumption simplifying (ST5') in its second-order guise. Since we have such in the language of 0, we need not bother with the instances of Note the Separation quantifiers the first-order We now According of 0 with have invoked Axiom Schema. Separation translate these axioms to the procedure outlined the new non-logical relation symbols non-logical 'ST' and '0ST' are expressions (/), and that 'eST' and '=st' x, y, z, w as variables the following analytic axioms (20) into analytic truths of 0 as follows. above, we first extend the language constants 'ST' and '0ST' and with the new (It should be clear that 'SST', 'eST\ anc* '=st'of type /, that '5ST' is an expression of type are expressions of type {i, i).) Then, using of type / and F as a variable of type (/), we add truths as new for identity discussed axioms of 0 (in addition to the new above): ST h VxVy[SsT* & SSTy -+ [Vz(z eST x = z eST y)

ST h Sst0st (22) ST \=^3x(SSTx & x GST0st) (23) ST f=VxVy[Ssjx & SSTy -* 237 Gst Z = W =ST X V W Gst )0] 3Z?W(W ST \=VFVx[SSTx -> 3y(SSTy & Vz(zGSTy= zeSjx & Fz))] we Finally, also add the assumption: MathTheory (ST) (24) given this last fact, we know from (19) that the translation in 0 as an explicit ordinary theorem (pof ST becomes derivable Now <p*of every ST-relative truth: ST |= cp* With reduce (whenever hST (p) in 0, we are now ready to metaphysically this group of theorems the sets described by ST. However, it is to be emphasized that there is nothing in what follows. We are implied by the order of presentation are not constructing 'in stages'. We the objects simply showing which in 0 constitute proofs. sequences of formulas First we identify the theory ST as a particular abstract individual. To do of the result instantiate (11) to ST and then derive the consequent The result is: from our assumption (24) thatMathTheory(ST). this, we ST = ix[ (A\x & VF(jcF = 3/?(ST \= p & F = [ky p]))) (25) to recognize that this is not a definition of 'ST', but rather of a particular abstract individual. Given theoretical description that sentences of the form 'ST f= /?' are well-defined and that the ordinary It is essential an exact theorems which of ST appear abstract Second, we in this form as theorems individual identify 0ST. To do this, recall 0: ST \= SST0ST (21) It therefore (26) follows of 0, we know in principle ST is. that: 3F{i)(ST\=F0ST) that (21) is a new axiom

EDWARDN. ZALTA (27) ObjectOf X0ST,ST) may therefore instantiate to yield consequent We that 0ST is an object it follows So, by (12), our Reduction the following theorem of ST: Axiom of 0 (13) and detach (28) = ST 0sT = iJcf(A!jc& VF(jcF |=F0ST)) have therefore the Corollary (14) the : 0st as an abstract individual. Notice to the Reduction it also follows from Axiom, identified that by the fact that 0sT is an object of ST that0st encodes a property F if and only if it is a truthof ST that 0st exemplifies F: 0STF= ST \= F0ST (29) in Section 6, when we look at the rela? prove instrumental our formal theorems and their counterparts in ordinary tionship between mathematical language. This fact will To complete our metaphysical reduction, we identify the mathematical relations of ST. We need to identify SST, gst, and =ST. It will suffice to to identify the first two (since the reduction of =r is carried out like that for gst). Now recall that (22) is a new axiom of 0: show how exactly ST \=^3x(SSTx & x Gst 0st) (22) we can 'abstract out' a and ?-Conversion, this, our Rule of Closure, a that of set$j exemplifies (in ST) (the property) being property properties relation exemplifies and a property of relations that the membership^ (in From ST): (30) ST h [kF{i) -*3x(Fx & jcgst0st)]Sst (31) ST h [kF{u) ->3x(SSjx & Fjc0st)] st on each of the above ?-expressions, can now generalize remembering of type ((/)) and that that expressions of the form [?F^ x/s] are expressions We of the form [?F(M) (32) 3Fm(ST\= FSST) (33) 3F{{iJ))(ST^= FeST) expressions x//] are expressions of type

So, by the definition We (34) ObjectOf(SST, ST) (35) ObjectOf(eST, ST) may therefore instantiate to yield consequent we may of ObjectOf, derive our Reduction Axiom Sst = ix{i)(A\mx & VFm(xF (37) Gst= ix{u)(Al{{u))x&WFm))(xF (36) and (37) identify Sst abstract (13) and detach = ST |=FeST)) and Gst as an abstract property that if we had begun with axiom relation, respectively. (Note or (23) instead of (22), we could have reconstructed to prove that =ST is an abstract relation.) Moreover, to the Reduction Axiom, that ?st and Gst are objects (38) SSjFm the following the = ST \= FSST)) (36) Theorems two facts: : of 0 theorems the following the following the above and (20) deduction (14) by our Corollary are also consequences of the fact of ST: = ST h FSst Gst F{{u)) = ST h FeST In other words, in ST. These the properties SST and gst encode precisely they exemplify in Section 6, when we look at facts will prove instrumental the relationship theorems of 0. between ordinary mathematical language and the formal as soon as we analyze By analogy, given the foregoing derivations, set theory (ZF) or ZF + Axiom mathematical theories Zermelo-Fraenkel the of (ZFC) and supplement 0 with new terms and analytic truths in the manner prescribed theorems which become above, the following identify the primitive individuals and relations of these two theories: Choice 0zf = ixi(A\x&WF(xF 0ZFC= ixi(A\x&VF(xF GZF= = ZF f=F0ZF)) = ZFC ix{Ui)(Alx&VF{{iJ))(xF eZFC= ix{iJ)(A\x&VF{{U))(xF h F0ZFC)) = ZF \= Fgzf)) = ZFC h /^zfc)) this analysis, the empty sets and membership relations of ZF and ZFC are analyzed as distinct abstract individuals and distinct abstract relations,

EDWARD N. ZALTA are defined by their which respectively, defined by the truths of ZF and ZFC.27 4.3. Reducing the Objects of Number theoretical roles, i.e., they are Theories (In this subsection, we describe how the above procedure would be applied to the primitive objects of the Dedekind/Peano axioms for number theory. to the one described Since the procedure is almost exactly analogous in the previous subsection, such readers some readers might should note However, why it is important to distinguish the primitive of Dedekind/Peano subsection objects and the classical to skip this subsection. two paragraphs discuss the metaphysical reduction of wish that the final between reduction number theory as outlined in this of the natural numbers we described in Section We 1.) now metaphysically ber theories. As reduce the numbers a representative theory. Let us designate we'll by various num? on a classic number described focus example, as the axioms the following of 'Number Theory' ('NT'): Zero Zero is a number. doesn't succeed two numbers No have any number. the same successor. Every number has a successor. num? If (a) 0 exemplifies the property F and (b) every two successive F then y exemplifies bers x and y are such that if x exemplifies F, F. then every number exemplifies Now suppose these axioms have Zero ('0'), logical expressions 'is the same as' ('='). Then we been formalized in terms of the non 'is a number' extend 'succeeds' (W), ('5"), and the language of 0 with the expres? '=nt'(It should be clear that 'NT' 'Ont', 'Nnt\ 'Snt\ and are of type /, that Wnt' is an expression of type 'Ont' expressions now of type (/, i).) We add (i), and that 'Snt' and '=nt' are expressions new as new to the addition axioms of 0 truths the following (in analytic sions 'NT', and axioms for identity discussed (39) NT h A?ntOnt (40) NT \=^x(Nmx above): & SntOnt*) NT |=VxVy[Nmx & Nmy & x ^NT y -> ~^3z(Nmz & SNjzx & SNTzy)] -* 3y(Nmy & Smyx)) NT \=Vjc(JVNT;c

NT |=VF[F0NT & VxVy(Nmx & Nmy & Smyx & Fx -> (41) Fy) that we Note have the Induction -> Vx(Nmx -? Fjc)] invoked assumption simplifying (41) in its second-order guise. also add the assumption: Finally, we (42) MathTheory(NT) this last fact, we know from theorem cp of NT becomes ordinary Now given every NT-relative (19) that the translation cp* of in 0 as an explicit derivable truth: NT |= cp* With (I) so as to formulate axiom we of theorems in 0, as a particular abstract this group the theory NT (whenever hNT cp) begin our reduction by identifying individual. To do this, we instantiate (11) toNT and appeal to (42) to conclude: NT = ixl (A\x & VF(xF = 3p(NT\=p & F=[ky p]))) (43) we Second, identify Ont- Beginning with (39), we follow the same steps thatwe followed inmoving from (21) through (26) and (27) to reach (28). is, beginning with (39), we to NT, generalize fies according That object of NT, (44) abstract out a property that Ont exempli? on that property, conclude that Ont is an to conclude: and then instantiate our Reduction Axiom = NT 0NT= ixl (A\x & VF(xF \= F0NT)) have therefore identified Ont as an abstract individual. To complete our metaphysical reduction, we identify the mathematical relations of NT. We need to identify Nm, it will Snt> and =Nt- Again, suffice to show how to identify the first two. Beginning with (40), we just We follow in moving the same steps that we followed from (22) via (30)-(35) and That with is, beginning (36) (37). (40), we abstract out to reach both various properties, tiate the Reduction and instan? generalize, apply the definition of ObjectOf two theorems Axiom. We thereby prove the following of0: (45) Nm = ix?(A\mx&.VFm(xF (46) = NT SNT= ix{iJ)(A\{{iJ))x& VF{{u))(xF = NT |= FNm))

EDWARD have We abstract therefore N. ZALTA the mathematical identified relations 7VNt and SNt as relations. we turn our attention to the objects of arbitrary mathematical an is in order. It is subsection, important observation that the objects identified by the above metaphys? important to recognize from the objects ical reduction of NT are completely different identified Before theories in the next in the classical tion reduction 1, we 1. In Section the natural cardinal This of the natural described 0 and to define classical in 0 numbers how has 0 of the concepts was reduction in Sec? described the resources to define 'natural number' the subject of Zalta and and (1999) 'predecessor'. in that paper, the natural cardinals were defined so as to encode ordinary the natural cardinal 0 encodes non-mathematical properties. For example, all and only those properties which are exemplified by no ordinary objects a in of the Arctic Circle), and 0 encodes the property being (e.g., giraffe that are exemplified by just those properties nine ordinary objects (e.g., 9 encodes the property of being a planet in our the predecessor relation and its weak ancestral solar system). Moreover, are asserted to be ordinary relations (since they aren't abstract relations, the natural cardinal they don't encode using a definition was then defined 0 bears the weak 9 encodes any properties). They were defined in terms of encoding, similar to Frege's.28 And the concept of 'natural number' to be any abstract object to which the natural cardinal relation. Given such defini? ancestral of the predecessor axioms become unprefixed theorems tions, the unprefixed Dedekind/Peano of 0. So 0 rules that the basic laws of number theory are true simpliciter. Consequently, are defined which to distinguish it is important in terms of the application objects of the natural world, from the theoretical numbers from these natural of counting the theoretical numbers numbers, the ordinary of NT (and of every other mathematical theory of num? are different because the individual numbers bers). These number systems roles in their respective theories. The theoretical numbers play different to them by their theoretical of NT encode only the properties assigned such as being role in NT. As such, they do not encode ordinary properties a giraffe in the Arctic Circle and numberNT predecessorNT can be identified as abstract relations that NT as formulated or being a planet. Similarly, are both primitive (not defined) relations that encode to them. The standard assigns in exemplification logic the relations in NT. They the properties of only laws of number theory, notions of in terms of the primitive remain true, when translated into 0, zeroNT> numberNT? and predecessor^, theory operator. So although 0 has only when prefixed by the appropriate a to imply of 'natural number') definition mathematical power (via enough the basic laws of number theory as unprefixed (i.e., objective) truths, we

power rather than its mathematical power to give rely on its philosophical a metaphysical reduction of the objects of arbitrary mathematical theories, as we shall now see. 4.4. Reducing the Objects Mathematical of Arbitrary Theories of an arbitrary mathematical theory, we first identify to (25) and (43). Sup? by proving a we that that sentences of the constitute pose group pretheoretically judge a axioms of mathematical theory named r. Suppose further that the axioms of r have been given some standard first- or second-order formalization in we accordance with our simplifying Now (III). (I) assumptions suppose have extended 0 in the way described above. We can then theoretically To reduce the objects the theories identify a theorem themselves similar theory r as follows: the mathematical = r = ixi(A\x&WF(xF 3p(r\=p & F=[ky /?]))) Recall that this is provable from (11) and (18). We now have to identify the objects of r. Consider any primit? k1 that appears in (a sentence pretheoretically expression kx is judged to be) an axiom of r. Then, given our simplifying assumptions, either an individual constant of type i or an n -place relation symbol of type of i).We therefore add k\ to the language of {i,..., i) (with n occurrences 0. Now suppose cp is (a sentence pretheoretically judged to be) an axiom of r and that cpcontains k1\ Then where is the translation of cp into the (p* we a new of the know is axiom that of 0: 0, language following simply ive non-logical From Let this, we can 'abstract out' a property that k* exemplifies the new variable y* for cp~ be the result of substituting we may use ?-Conversion r And and our Rule of Closure k[ in (p*. Then (10) to prove that: |= [ky* (p~]tcT, by generalizing on the ?-expression, 3F{t)(r \=Fk[) So, by (12), k1x is a type-i ObjectOf^,!) object of r: it follows in theory r. that:

EDWARD Then, by our Reduction follows: (47) The k\ (13), we Axiom cannot this theorem of can provably identify k[ in 0 as It offers a = r h Ek{)) = ixt(A\x&WF(xF significance N. ZALTA be overemphasized. of mathematical reduction ontological objects (individuals lations) to the abstract objects of our background ontology. Given is an immediate consequence of (47) that: general (48) K[F{t) In other words, exemplifies are some There deserve a mathematical it object k\ encodes exactly the properties it r. SOME CONSEQUENCES OF THE REDUCTION with the foregoing that that arise in connection we our treatment In shall of this discuss section, commentary. some of the theor? and describe (?5.1) consequences interesting issues identity ems just proved (?5.2). The an important role in Section in a way ical language mathematical arbitrary 5.1. (14), = r \= Fkt in theory 5. and re? When Identity consequences 6, where we that reveals theories is Primitive discussed in Section 5.2 play mathemat? analyze (ordinary) a correlation the theorems between and theorems of (extended) of 0. in x amathematical (III) was that whenever assumption a r can with be reformulated in identity, language theory '=' a becomes in a language without (a) the symbol identity in which a of the that is relation non-logical primitive binary symbol distinguished (re? theory, and (b) the standard (two) logical axioms for identity become formulated as) proper axioms which govern the primitive binary relation Recall that simplifying r is formulated of model '='. Now one might argue, from considerations theory, symbol in this way would not have the same expressive that theories reformulated as the original. The argument would be that the proper '=' in the reformulated only guarantee theory would governing relation and not identity. denotes an equivalence capacity axioms that '=' is straightforward, If the it? to this argument however. response use at the end of the previous of the word alicized paragraph 'identity' in model is supposed to denote some relation that is primitive theory, then we simply point out that from the point of view of the present metaphysics,

there is no such primitive relation. 0 uses both a defined notion of identity theory of identity.29 To defend 0, we get to assume that it is true and that, consequently, the facts about identity are as the theory says. and a proper So, unless in some way, its theory of that theory is shown to be defective The argument from model-theoretic notion. the trumps primitive identity to the object theorist. model-theory unpersuasive simply becomes to de? if the italicized use of the word However, 'identity' is supposed note the notion of identity defined as 'exemplifying the same properties', that definition is correct (i.e., consistent then the question becomes whether substitution of identicals). From the point of view of with the unrestricted of identity is not correct; the identity of indis? 0, this standard definition It cernibles (i.e., ordinary) objects. correctly applies only to non-abstract x are are a and y that distinct is abstract objects theorem of 0 that there (in the sense that they encode different properties) There are so many abstract the same properties!30 tional mode of predication, namely exemplification, but which objects cannot exemplify that the tradi? always discern objects that encode different properties. concern to rest by showing how We may also put the model-theoretic our notion of identity for abstract objects, defined in (4) as 'encoding the same properties', the work it should do. We show that, using does precisely abstract (III), whenever simplification our metaphysics guarantees x and y are objects of x and x \= x=Ty, then that x and y are identical in the sense defined by (4). in the this, recall that when we have a theory r expressed to that has been of reformulated identity language according simplifying the new relation (III), the proper axioms of x that govern assumption symbol =r become added to 0 as the axioms (16) and (17): To see x \= x=Tx (16) r \= x=Ty -> VF(Fjc = Fy) (17) that (17) tells us that it is a truth in x that: if x and y are x -identical, x then and y exemplify the same properties. (Note also that the quantifica? concern tion over properties might be sufficient to put the model-theoretic Note to rest.) Now then x=y. we want to show So assume that if x and y are objects the claims ObjectOf(x, x) ObjectOf(y, x) required of r and x by the antecedent:

EDWARD N. ZALTA * ?=x=Ty Then by (17) and our Rule (?) of Closure, it follows that: x \=VF(Fjc = Fy) to show x=y, we have to show that x and y encode the same proper? ties. Without loss of generality, we simply prove that if x encodes P, then Now P, since the converse uses the same reasoning. P. Since x is an object of r, we may appeal our to conclude: Axiom of Reduction (14) y encodes x encodes xF = So suppose that to the Corollary x \= Fx So since x encodes that r (= Px. We can now appeal to (?) P, it follows to infer that x \= Py. But y is also an object of r, of Closure and our Rule and so the Corollary yF = to the Reduction Axiom implies: x \= Fy is what we had to show. P, which x and y are r-objects that are So, from the point of view of 0, whenever x are same our concludes that and the abstract r-identifical, y metaphysics object. Thus, anything true of the one is true of the other. So y encodes 5.2. Some Further Theorems some interesting consequences of the the? In this subsection, we describe in the last section. Consider the ordinary axiom of ST proved orems that 0 is a set. This theory-prefixed simple axiom exemplifies (49) of Closure introduced into 0 as the analytic, (21): ST h 5st0st The Rule claim was (21) and ?-Conversion the property immediately a property that 0nt of being yield that in ST, Sst exemplifies: ST f= [kG{i)G0st]5st now proceed to show that from (21) and (49), we can derive two further the higher-order that 0St encodes Sst, and that Sst encodes facts, namely We property (50) [?G G0stL 0st5st In formal terms, we prove the following:

247 SST[kG{i)G0ST] to derive (50), recall that we proved (29) in the previous section: 0STF = ST \=F0ST (29) In light of this, (50) is an immediate consequence section: that we proved (38) in the previous = SST/r??? ST \=FSst of (21). Recall also (38) of (49). in virtue of this, (51) is an immediate consequence on We might the fact that if there is an atomic reflect for a moment Now relational represented of theory as the following r of axiom the form Uk\K2, of 0: the following as consequences: axiom not only would it be X \= Y\TK\xKlx, but it would also have KU[kx nTXK2r] K2r[kx T\TK\TX] nT[?F All three encoding Fk1tk2t] claims would therefore be theorems of 0. we the consequences of those theorems of amath? Finally, contemplate ematical theory which are expressed by molecular and quantified formulas. For example, consider the axiom of ST which asserts that no set is a mem? ber of 0. As we noted above, this becomes the following axiom 0: ST \=^3x(SSTx & x gst 0st) By now familiar reasoning, this axiom of 0 (22) implies (52) 0ST[V ->3x(SSTx & x Gst y)] (53) SST[?F(/) -*3x(Fx & x GST0ST)] (54) GST [kG{iJ)-3jc(Sst* & GxOST)] the following:

EDWARD With these in mind, consequences N. ZALTA we now the language and theorems of ordinary and the theorems of our extended 0 on the other. between hand, the relationship on the one mathematics reconsider 6. ANALYSIS OF (ORDINARY) MATHEMATICAL LANGUAGE 2, we divided the true statements of ordinary math? ematical language into the basic ones, which begin with the theory operator of the basic statements are: and the non-basic ones, which don't. Examples Recall that in Section set is a set. (A) In ST, the empty (B) In ST, no set is a member of the empty set. correct truth conditions) of these Clearly, the analyses (i.e., philosophically into our formal system as (21) claims are given by their direct translations and (22), respectively: ST |= Sst0st (21) ST \=-3x(SST* & x Gst 0st) These analyses reveal (22) that the truth conditions for these statements Note that our truth conditions determined.31 compositionally a account the abstract individuals of philosophical panied by relations member The as asserted biguous are accom? and abstract of the expressions that serve as the denotations 'ST', as occur in (A) and (B). and set' of, they 'empty these same statements without However, when we consider first: prefix, we have several interpretative options. Consider empty in the context in the present exemplification set is a set. reading are 'set', 'is a the theory (ST2) am? becomes simple predication one atomic the the hand, following of ST. This theory. On of this claim is false: (ST2*) Sst0st add Though 0 doesn't assert that this formula is false, we may consistently in is grounded that this assumption the assumption that it is false. Recall on our philosophy it is asserted that (i) 0st encodes which of mathematics, rather than exemplifies its mathematical properties, (ii) 0st exemplifies

as being non-red, being non-round, being thought about by the reader now, etc., and (iii) 0st is complete with respect to the exem? of properties but not with respect to the encoding of properties. plification such properties these philosophical ideas, on its sleeve, do not obtain. Given wears its traditional formal 0 However, is an atomic offers encoding saw in the previous we rendition a reading of (ST2*), which it read as (ST2*), both (ST2) and turn out to be false. the truth conditions So when as (ST2') for (ST2) on which claim which it turns out true. (50) true but a theorem of 0, as is not only subsection: 0stSst Given (50) the ambiguity in language described So we have legitimate reading for (ST2). is true. (This preserves the intuition true.32) In this sense, in Section a 2, (50) becomes a sense in which (ST2) recovered that they are saying that says (ST2) is about the of mathematicians our analysis something set 0st. Note that our analysis that we might suggests equally well have re? as a statement about the abstract garded (ST2) property of being an ST set. Our work in the previous a true reading asserts suggests that theorem statement (ST2): of the ordinary Sst[aGG0st] This subsection (51) also offers (51) that the property of being an ST-set encodes the property of that 0st exemplifies. On this reading, our analysis says is about the abstract property of being an ST-set. a property being that (ST2) We'll moment. discuss No We assert the fact that (ST2) has alternative But first, consider set is a member that (ST3) true readings in just a (ST3): of the empty is false when represented set. (ST3) as the formal claim: (ST3*)-3x(Sst*&xGst0st) However, any of the formal representations (52), (53), or (54), which turned up as theorems in the previous subsection, provide us with a true reading of (ST3): 0ST?V -3*(Sst* & x Gst y)] (52) SST[?F(/) ^3x(Fx & x GST0st)1

EDWARD N. ZALTA GST [kG{u) ^3x(SSTx & Gjc0st)] (54) (ST3) is not only about 0St but also about the property of being a setsT and about the relation of memberships^ In some sense, it doesn't matter which of theorems (52)-(54) we assign to (ST3) as the disambiguated condition it is true. From under which the other two, by appealing any one of these statements, we to the Corollary to the Reduction can recover Axiom and ?-Conversion. we can take our analysis of (ST3) one step further. Let us However, define an extended sense of 'encodes' in terms of which we can say that the 0st, Sst, and Gst encode the following complex an individual /, property F{l), and relation G(m) abstract objects namely, being -^3x(Fx ScGxy). The intuitive idea here is to define 'x, y, and z encode /?' as the conjunction of x[ku Ruyz], and z[ku Rxyu]. y[ku Rxuz], ('xyzR') To employ this idea in the case at hand, we can let the following relation, such that notation be defined as the conjunction of (52), (53), and (54): (55) 0StSst eST [?/F(/)G(M> -^3x(Fx & Gxy)] is a relational expression the ?-expression of the (In the above notation, are can use F and G form [kyFG in which all bound We the y, ?.) x/r], by this newly defined statement of 0, and the truth conditions it encapsulates, it is true. Similarly, whereas we take the of (ST3) on which translation (ST3*) of (ST3r) to be false, (55) offers a way straightforward a truth. as to understand (ST30 representing as the reading an for constructing course, this leads to a very general technique a complex that expresses the reading under which condition encoding a x true. sentence is that r S of mathematical theory ordinary Suppose Of has been formulated traditional formal in classical exemplification exemplification logic and that cp is the statement which renders S. precisely (As an example, let S be (ST3) and let cpbe (ST3').) Let the primitive constants of <pbe listed as tch,..., and predicates Ktn. Then nonlogical let of where k{1 ,..., K{n are the new corresponding 0, (p* be the symbols < we sentence of 0 which results when i < n). substitute k** for k?x in cp (1 cp is (ST3'), (p* is (ST3*).) Then, as (Continuing with our example, when are or not and S theorems of x, cp* is to be regarded we've whether seen, <p as false (though if (p is a theorem of r, we know that r |= ^>* is true). However, and only there is a statement let cp~ be generality, occurrences for all the tively, a metaphysical truth if expresses of x. To specify this statement in complete new variables the result of substituting ytl,..., ytn of 0 which if cpand S are theorems expressions k^ , respec? k[x ,..., be the result of substituting the variable ah of the non-logical in (p*, and let i/rfa**/kh)

of the (constant or predicate) symbol ku in \//.We as the reading in the following definition for all the occurrences may then use the definiendum which captures the mathematical ... [Ay" truth underlying q>: y'? 0T] =df K J ... & K*Tl[kytl(p*(ytl/K?)] & Kf?[kytn<p*"(ytn !'<")] *?...<? It should be clear that when (55) is taken as an example of the definiendum in (56), the conjunction of (52), (53), and (54) is an example of the definiens. us allows to represent the truth conditions for ordinary or of (ST3')) (e.g., (ST3) arbitrary complexity in terms of a single defined formula of 0. It should also be clear that the definiens of (56) is derivable as a theorem of 0 whenever cp is a theorem of x. This sets up a correlation - each theorem of an arbitrary mathematical can a be correlated with of theorem 0 that is unprefixed by the theory (56) statements mathematical theory operator! At this point, itmay be that enough has been said to give the reader a sense of how the (ordinary) language of mathematics is to be analyzed. Before we objection that might metaphysics. perspective. turn to the final section, it is important to address our reduction of mathematics be raised against The objection criticizes the reduction Itmight go as follows:33 one to from a model-theoretic In various mathematical are not uniquely of the objects identifiable theories, many by in the language of the theory. An example might be real number descriptions expressible reals are nameable in a standard in many theory, where only countably Indeed, language. some mathematical none of the objects are identifiable, for reasons of symmetry. theories, are classical and Cantor's linear orderings without Examples geometries theory of dense In models of endpoints. sesses exactly the same distinct objects There are actually about these of the domain every element pos? for T. the So you can't reduce all the meaningful theory to distinct abstract objects. homogeneous properties of these theories theories, two separate questions theories which assert the existence identifiable theories raised here, namely, what to do of objects that are not uniquely in the theory, and what to do about by descriptions expressible assert the existence which of distinct symmetrical objects. The two questions are related, however. They both arise because the model theoretic conception of the 'objects of a theory is rather different from the which we defined above as (12). On the model metaphysical conception, an 'object of a theory is any element of any domain conception, of quantification that is part of the intended model of the theory. The uses this definition to claim that there can be objects model-theoretician of a mathematical since the theory has no theory that are inexpressible, terms to denote them. What well-defined analysis does 0 offer when this is the case? theoretic

EDWARD N. ZALTA I think there are two parts to an effective response to this objection. The first part of the response is to point out that the model-theoretic objection, in some deep sense, begs the question as an argument against our founda? assumes the model-theoretic definition objection set of 'object of and so uses mathematical the theoretic (e.g., language notions of domain, model, that satisfaction, etc.), thereby presupposing the semantics of that language is clear. But the semantics of mathematical tional metaphysics. The language is precisely what an account of mathematical is in question. reference Our ontological project is to give and truth in terms of a more basic, which mathematics-free presuppose language and theory. So objections an account of reference and truth in terms of mathematical (the language lose their force. Model of which, after all, is in question) theory a to is mathematical that becomes another theory subject metaphysical just tells us what the terms of model reduction. Our metaphysics theory refer semantics to and tells us the sense The second claim sophical are to account in which its claims are true. is to reiterate the metaphilo response part of an effective tasks of a philosophy of mathematics that the two principal and truth. A philosophy the referents of the well-defined for reference of mathematics must terms and predicates describe the conditions under but also precisely turn out to be true. We now have ac? which the theorems of mathematics reference is given by the analysis of mathematical both tasks complished truth is given by (56). The model (47) and the analysis of mathematical not only identify of mathematical theories conditions can be put to rest by the facts that we have objection in 0 for every sentence of an arbitrary mathematical that we can correlate theoretic stated truth theory and an mathematical of every theory arbitrary with a theorem of (extended) 0 that is unprefixed by the theory operator. a mistake to suppose that in order to answer the on? It therefore becomes the about what theory are, objects of a mathematical tological question theorem has to be able to give metaphysics or model of that theory.34 interpretation, a foundational relative 7. PHILOSOPHICAL a classical reduction, OBSERVATIONS and relations of arbitrary mathematical that the individuals By showing found in the and abstract relations theories are just abstract individuals case a the main for of we've 0, prima facie produced ontology typed thesis of this paper. Of course, 0 has to be extended with new primitive truths of the form r |= <p*, and with the analyses symbols, with analytic to be true, namely, MathTheory(x), of statements which everyone assumes theories x. But the resulting system allows for recognizable mathematical

us to prove what many other philosophers atical objects are abstract objects. Since 0 of fall logic and metaphysics,35 into a more fundamental reduction we of one kind of entity stipulate, namely, that mathem? includes only primitive notions can conclude that mathematical objects ontological category. to another. This is an ontological Of course, there are numerous philosophical issues that arise in connec? our metaphysical reduction. Many of those issues were addressed tion with (1995) and we shall not rehearse them in any detail here. We shall, however, consider the question of mathematical objectivity, but before we do so, it is important to consider the extent to which the in Linsky and Zalta a kind of neo-logicism. present theory constitutes we have presented 0 as a proper metaphysical sion principle be a synthetic there is a way more this essay, Throughout theory. The comprehen? most reasonable by lights, to for abstract objects appears, truth and not an analytic truth of logic. However, to restate the comprehension so that it looks much principle a priori like a truth of like an analytic truth. I shall logic, or at least more that this reformulated version of comprehension is a truth of or some but it will be that would logic analytic, recognized philosophers conclude that it is. Our theory of abstract objects could have been presented not claim the comprehension by replacing canonical descriptions (7): & VF?(xF ijc'(A!<'>jc By elevating this theorem principle (3) by the theorem governing = <p))G? = (7) <pf?l to the status of an axiom, with that all canonical the understanding (stipulation) descriptions objects denote, we have an equivalent formulation of 0. Moreover, if one considers what (7) then it clearly has 'the ring' of an analytic truth: the abstract object asserts, that encodes such that (p encodes property G iff G is just the properties of abstract such that (p. So is (7) an analytic truth? If so, does 0, when reformulated in this way, become a part of logic?36 If the answers to these two questions are 'Yes', then our ontological reduction of mathematical objects might constitute a kind of neo-logicism. As mentioned earlier, I do not claim that (7) is an analytic or a lo? truth. to the following At 'abstraction' principle best, it is analogous gical 'set abstracts') as a substitute that might be employed for (governing (ST5): z G {y Iy G x & Fy} = this also has Although a contextual definition, the ring I doubt zex&Fz of an analytic as truth when introduced that it is analytic when introduced

basic EDWARD N. ZALTA notation that governs the primitive '{y some philosophers have argued that axioms axiom However, and the above G x & Fy}\ to (7) analogous \y are analytic. Using abstraction (1884) Frege's principle as a guide, Wright Context Principle (1983) argues that Hume's Principle = #G = F ? considers the truth.37 If Wright G) is an analytic (#F a to to second-order be result of adding Hume's Principle logic logical then it would system ('number-theoretic logicism'), to have regard the result of adding (7) to the logic gical there are, in the literature, the analytic character concerning Of course, system. Boolos calls that he would as a lo? of encoding of trenchant criticisms of Hume's Principle.38 at length on another occasion, I shall to between that the relationship (7) and (3) is analogous claim that and the existence between Hume's Principle Wright's position Since I hope to discuss simply observe the relationship seem these issues in his (1987).39 The conclusion I wish to draw at constitutes that the above treatment of mathematics 'Numbers' is simply this point a kind of neo-logicism ifWright's can be sustained.40 claim about of Hume's the analyticity Principle next, the question of mathematical Consider, objectivity. No doubt, it will be argued that if every mathematical theory is about a distinctive to account for mathe? then there is no way group of abstract objects, our modulo of the natural classical reduction matical But, objectivity.41 in is correctly described if mathematical numbers, Linsky and objectivity Zalta (1995) and Field (1998b) as being limited to the objectivity of logical then there is no special problem of mathematical theory.42 These works deny that there is a single, consequence, for the above true set theory, that there is an objective fact of the matter are perfectly true there the axiom of foundation is (for good set theories), founded objectivity objectively as to whether and that there is a fact of the matter non-well as to the size of (there are perfectly good set theories which differ answer to the size to the continuum). Each set theory is simply relation. different membership the continuum in their about a (p. 401) that an account of mathemat? Though Field (1998b) concludes is more important than an account of mathematical ical objectivity objects, to say that philosophers need a correct account itmay be more perspicuous of both objects mentioned of mathematics: if they are to have a comprehensive philosophy are to objectivity what reference is to truth. In addition to the ways in notes 17, 27, 31, and 32, the present analysis supplements as follows: (a) it gives a correct account of mathematical Field's work objects that is consistent (b) it explains develops,43 in Field discussed cepts, he the view of mathematical objectivity con? in our mathematical the indeterminacy the idea that our (1994), without abandoning with

mathematical denote particular mathematical relations, and (c) predicates an account of the meaningfulness of the language of inconsistent mathematical theories.44 This last fact deserves a brief discussion. it offers The analysis of mathematical above extends even language described to inconsistent mathematical theories. To take a classic example, consider der Arithmetik. there has been Frege's Grundgesetze Recently, sance of interest in this work and it has become the subject of many there are logical investigations. of pages of formulas in Frege's special script, and des? the inconsistency of the system, these formulas are meaningful! How philosophical hundreds many In the Grundgesetze, and a renais? pite are we to describe of this language? The answer given by terms is that the and predicates of Frege's theory language abstract objects that encode properties that are inconsistent with the semantics the present denote one another.45 Of course, the objects of an inconsistent theory r will be for will encode all in r).46 That (formulable they uninteresting, properties to mathematicians avoid inconsistent theories. try explains why postulating But note that we now have a unified semantics of mathematical language.47 It is now important to reflect on the features of metaphysical reduc? tions that contrast with other forms of reduction. Clearly, our metaphysical reductions ematical are not classical for the theorems of arbitrary math? reductions, a theories T do not constitute extension subtheory of a definitional our metaphysical of 0. Moreover, reductions of the objects theories do not show that those mathematical mathematical of arbitrary theories are in 0. Nor are we using 0 to build models for arbit? relatively interpretable theories. We are not claiming that mathematical notions rary mathematical can be defined in terms of the notions of pure logic and metaphysics. Nor are we that we can get along without the proper axioms of math? creative the with axioms, by being logical axioms, non-logical and definitions a new kind of reduction, of 0. Instead, we've developed suggesting ematics which a sense, yields a precise philosophical our metaphysical reductions account of mathematical constitute a distinctive In objects. new kind of for every theorem cp of an arbitrary mathematical interpretation, r can be correlated with a (specially-identified) theorem of extended theory the definiens we should draw of (56). I think one conclusion 0, namely, from all of this is that no matter how mathematicians carry on with their relative work and no matter how the mathematics turn out, might and something metaphysically precise circumspect the subject matter of the resulting mathematics and about the semantic analysis of the language used to express it.48 philosophers to say about proper There will is one final observation it is fascinating they produce have (and possibly tomake insightful) before we conclude, namely, that to consider that many of the ideas

EDWARD N. ZALTA the metaphysical reduction of mathematical a certain far presuppose 'platonist' interpretation about In the present the quantifiers paper, we have of 0, in which the predicate 'F!" is read thus objects expressed of the formalism of 0. the 'Quinean' understanding employed '3' is read 'there exists' the quantifier 'is concrete". On this understanding, 0 of and asserts and relations) that couldn't possibly (individuals objects of the comprehension is just a consequence (3) principle one can give the formalism of and the definition (1) of 'abstract'. However, that there exist be concrete. This a 'fictionalist' reading, by using the 'Meinongian' reading of the quan? '3' as 'there is' (with no implication of existence) and by reading the as asserts 'F!' that there are objects 'exists'. On this reading, 0 predicate 0 tifier exist. On such a fictionalist (and couldn't possibly) reading of can are one abstract since don't exist. So that 0, say fictions, objects they more to mathematical become reduced the objects general metaphysically that don't category of fiction.49 The fact that 0 has these two fundamental readings of the platonist and fic? is, in our opinion, what grounds the 'equivalence' in Balaguer of mathematics described tionalist philosophies (1998).50 The to how it consider of the remarks reader might find worthwhile many just made in this last section described by this alternative readjustment, apply, with minor of the formalism of 0. reading to the fictions ACKNOWLEDGEMENTS I am indebted its Director, to the Center John Perry, and Information and for the Study of Language like to thank for supporting my research. I would and Linsky, Chris Menzel, Karl-Georg Niebergall, the all of whom draft read carefully penultimate Colyvan, Bernard an anonymous referee, for improvement. Thanks also goes to Sol and offered many suggestions for valuable and Brent Mundy Feferman, Allen Hazen, Thomas Hofweber, Mark discussions Godehard Logic about Link like to thank herein. Finally, Iwould to the workshop in of Reduction Concepts was this material first 1997), where (in September the ideas contained for inviting me and Philosophy presented. NOTES 1 See also Zalta (1993) or (1999) for briefer sketches of the theory and specific applications. 2 See the final chapters of Zalta (1983) and (1988). 3 See Carnap (p. 6), Quine (1976) (p. 218), and Jubien (1969) (p. 534).

4 For relative interpretability, see Tarski et al. ( 1953), Feferman ( 1960), andVisser ( 1998). See Feferman (1988) (or (1998a)) for the definition and discussion of 'proof-theoretic see Niebergall in which of model-theoretic the notions (this volume), Finally, are critically are proposed. for the reducibility and axioms reduction discussed relation 5 x that might concrete We call have been In formal individuals 'ordinary objects'. terms: 0\x that ordinary It is axiomatic fail to encode OE\x. individuals necessarily =df reduction'. properties. 6 More specifically, is an axiom: the following = -* z = x) & tfr*), & x//lyX(p 3x(<p Vz(r? or identity for any atomic To accomodate be 'free' with governing formula \// (y) in which y is free. so as to the classical is modified quantification only descriptions, theory to formulas the above axiom Moreover, containing descriptions. respect is a logical descriptions that is not truth a necessary truth (for the descriptions denote rigidly what they denote at the actual world). So the classical S5 modal logic is of contingent only to admit the presence logical on the above not be applied to any line that depends modified truths (the Rule of Necessitation may 7 More axiom governing descriptions). ...xn= [ky\ ...yn [Xyi where = ...yn(p] .y'n <p'l fry[. are alphabetic the two ?-expressions an axiom It is also are axioms: the following specifically, <p]x\ (PylZlyn that: = [Xyi...ynFnyi...yn] an Thus, specifically, a relation course, can be intersubstitutable for the relation symbol that is a theorem: the following . ..Vxn(Fnxi 3FnnWxi where (p has no free Fs, Of Fn is 'elementary' ?-expression in that expression. appears 8 More variants. ...xn=(p), no encoding in terms specified subformulas, of a formula and no descriptions. a definite cp containing = description ixy if it is first proved that 3y(y ix<p). 9 More the definition of specifically, identity proceeds by first defining identity for properties F1 and G1: F1 = G1 =df Oix{xFl In terms define of identity this definition, we for propositions. p = q=df V^yp] employ Using = = xGX) with ?-expressions vacuously and instead of 'F0' and '/?' V fry q] bound variables define:

25 8 EDWARDN. ZALTA a definition in of relation for n -place relations (n 2) is constructible identity Finally, one of the presentations terms of identity for properties. readers may consult of Interested (9 cited in the text. 10 this work. the formal definitions here for those readers unfamiliar with We provide we give E and In these definitions, the symbols (For |= the narrowest scope. possible = = 'uj f= p -+ p' is to be read p' and '?xp example, p' is to be read as '(H*/?) = \= p) -> p'.) We define: (1) x encodes p ('Ejcp') x[ky p]; (2) World(x) = = = w true at is ('w |= p') E^p; (4)Maximal(w) = Vp(w \= p p); (3) p OVp(SJcp as \w vio (5) Consistent -> -> \= p & D(p #)] [= ~,p); [w (w) u; = -?3p(w; |= p & w |= #; and (7) Actual(w) = \= ->/?); (6) Modally-closediw) = -> |= p Vp(u; p). See Zalta (1993) for further discussion. 11 Using the formal definitions supplied in the previous footnote, principles (a) (g) in the text become the following theorems of 0: (a) VwMaximal(w)\ (b) Vu;Consistent(w)\ (c) = 3w(w \= p); VwModally-closed(w); (d) 3\wActual(w)', (e) Dp = Vw(w \= p); (f) Op and = \= p (g) Wp(w w' \= p) -^ w = See Zalta w'. for the proofs. (1993) 12See Zalta (1983) for the initial sketches. However, Zalta (2000) has a comprehensive treatment Plato. 13 We these (x, y), (its weak 3FVxVy(Fxy 3FVxVy(Fxy by Precedes^ of ?. Then first defining (x, y)9 #p (1983) = of metaphysics, and Zalta (1995) has of mathematics was something to offer first exemplifies therefore be F nor encodes in the sense the negation 'fictionalist' 'platonist' 5. to justify of abstract in Zalta form guise and platonist and will neither of in its y)) however. this project, in a kind sketched 7. the theory in Section. 16 its mathematical encodes 7T^ properties only F such that n^ sense that there are properties is complete However, 7r^, like all other objects, or n^ and Fs'), Section (1999), especially in some detail VI) and then articulated will be said about the fictionalist (1995). More F See Zalta objects. the epistemologist concerned if they assert the existence axioms 7% exemplifies of notions of & Wu(AGu -+ u ?E #G) -> 03y(E\y (Chapter and Zalta technical ('the number are axioms: the following concrete The variable 'w' here ranges over possibly 14 also hold some interest for The reductions may individuals. Linsky 15 This philosophy the treatment = Precedes(x, y)) = Precedes^(x, y)) 3x(NaturalNumber(x)&x the foundational a comprehensive has (2000) formally ancestral) in the language NaturalNumber(x) and Zalta axioms rendered Precedes Pelletier and of Leibniz, in Linsky of interpretations in the 'incomplete' the negation that for any property of F. F, either of F. 17Note the similarities and differences with Field (1980) and (1989). We agree with Field that the standard However, unlike sentences are (he accepts positional true! We that also 'In number truth conditions play predicates 18 It is important relations complex 19 (exemplification) Field, we shall agree are false. sentences mathematical of unprefixed readings on which mathematical these unprefixed offer a reading with Field theory, 2+2=4' for these claims that statements is true), the theory-prefixed but unlike Field, we in which the denotations of are true offer com? the constants and not guarantee the existence subformulas. of encoding of shall a role! to remember and propositions that ?-Conversion definable See, for example, Frege (1893/1903). in terms does

See, for example, Hempel This pretheoretic that we might ability define gests by a sentence 259 (1945). are mathematical which and constants sug? predicates as denoted mathematical any proposition proposition' 'purely consists and, vocabulary predicates only of mathematical non-logical whose to judge But such a definition involves both (a) devices constants. mathematical obviously possibly, introduce and (b) semantic notions. We won't for mentioning of language officially pieces have to worry about semantic into (9, for then we would these devices and notions para? doxes. it may Moreover, be of a purely the notion that mathematical can proposition be defined without introducing these devices and notions into ?. But I will not pursue the here. question The proof on theorem depends Suppose MathTheory{T). canonical (7) governing are rigidly the logical axiom (6) that encoded properties encoded the properties that T encodes exactly ixl (A\x & VF(jcF = 3p(T (? ) So assume descriptions of abstract objects: Then, by the definition of identity for abstract objects (4) and P that T encodes |= p & F = (to show we encoded, simply have to show by: [ky p]))). that the individual just described encodes P). It then follows from the definition of amathematical theory (8) that 3q(P = [ky q]). So suppose P = [ky q\\. Then T encodes [ky q\\. So, by the definition of truth in a theory (9), we therefore know: = T\=qi&P [kyqi] From which it follows: 3p(T So by the theorem = \=p&P [ky p]) canonical governing descriptions ixl (A\x & VF(xF = 3p(T which is what we to show. had (to show follows that T encodes P). Then, it follows that: that: \= p & F = [ky p]))) encodes P. the theorem about q2. So we know: canonical descriptions (7), it that: 3p(T Y=p8cP = So, by (7), \= p & F = [ky p]))) encodes P, (??) Assume ixl (A\x & VF(jcF = 3p(T of abstracta let us call an arbitrary T\=q2&P [ky p]) such proposition = [kyq2\ But, by (9), the first conjunct just means that T encodes [ky qj}. So, it follows that T encodes is what F, which 3 to consider If we were between the abstract, we had applied mathematical to show. mathematical we objects the ordinary, theories, of a theory and would to distinguish non-mathematical have

EDWARDN. ZALTA since ordinary in our defined objects, theory, and individuals sense. When ordinary with dealing be objects of may properties mathematical theories, applied the abstract axiom only to identify apply the following we need not worry here. about this subtlety 24 Of course, if it is true in theory T that there of objects the theory. the applied we would Fortunately, exists a unique sort, the object of a certain term which to include a well-defined denotes the object in question. theory can be extended to existential be subject Such a term would outside the scope generalization by a quantifer of the theory operator. 5 non-classical For mathematical theories of involving logic, we have to adjust our Rule so that we add to ? only those claims derivable the non-classical Closure, using logic in question. Of so that it asserts this would truth if we were course, axiom simplify involves that When Field the statement and the above 'e^817', urelements of (ST3), the primitive in what follows. of ST, we would in the formulation revise is an element whatsoever all of to be useful proves to allow that nothing we would no set. But of the empty longer have a reasonably notions non-logical use such expressions Balaguer can tell us exactly which theorems as of ST. Having this though simple a sentence such and 'e^i', of abstracta 'sets^', 'sets^817', in the plenitude objects that these expressions refer to. See Field (1994), (pp. 420-22), Field (1998a), (p. 293), Balaguer (1995) (pp. 316-17), and Balaguer (1998), (p. 59). 28 these Since assert relations were that are defined in terms relations, of encoding and prove they (ordinary) plicitly to ?, consistent. added it remains See Zalta (1999). 29 in (4) of It is derivable that x* = xl, from the definition as a proper asserted we subformulas, that when those '='. Substitution had to ex? are assertions of is identicals axiom. 30This theorem is proved and explained inZalta (1999) (Section 2), but we will not take the time to repeat the proof 31 As soon as a philosopher of mathematics damental, it then becomes important statements. We can provide therefore here. and explanation takes claims such to specify (compositional) the truth conditions needed described in Field (1989) (p. 3). Without as (A) (B) to be fun? truth conditions for these to complete the position and such truth conditions, Field's position has an a reading on which we shall offer, in just a moment, gap. Moreover, important explanatory of (A) and (B) turn out true. the unprefixed portion 32 offer Here again, this fills another gap in Field's theory, for the latter doesn't important are true. Without such a reading, of mathematics theorems any reading on which unprefixed of a mystery. become of mathematicians the beliefs something 33 am an unpublished I think here from I and paper by Brent Mundy. paraphrasing quoting a concern at in oral raised similar Allen Hazen he states the objection presentations nicely. Hofweber of Alberta. And Thomas and the University the Australian National University about the present in a recent conversation raised a variant of the objection paper. 34 the present without linear orderings In the case of the theory of dense (DLO), endpoints out the properties of relations relation the ordering <dlo by abstracting theory analyzes encode in order tells us the sense in which that <dlo must theory no further 0 individuals "there are ontological which infinitely the present of DLO. And the axioms according are true. But then there are simply of DLO the sentences in particular, to answer; there are no specifiable (type to behave questions constitute objects many points which of DLO that need to be are and can be such such" identified. true Sentences in a mathematical like

even theory though that can everything 35 The there are no names actually be said and no witnesses for the points in a mathematical theory to the claim. Thus, an account. gets implementation of ? deployed in this paper has the following primitives: individual relation and encoding of predication), the usual (i.e., modes (type), exemplification and modal the non-logical and the F!, (->, ->, V, D, ?, 0, primitives primitive of 'MathipY notions and 'Authorship'. None of these are mathematical notions nonlogical constants there are no mathematical like 0, 0, etc., and no mathematical such predicates (type), logical as membership, 3 This of O maps, successor, assumes that we can also question can be understood and properties), and the definition of functions, identicals as is correctly of the theory interpretations that the principles in question 37 For the uninitiated, Hume's of G s iff F in second-order of identicals. are that the two other proper axioms (abstract don't encode (2) so that these does are turn out axioms not automatically truths. objects Since identity the principle for to be in (9, the substitution one could constrain though true in every interpretation, an argument constitute for thinking logical asserts that the number of F's Principle are equinumerous has (where 'equinumerous' See the discussion of 'number-theoretic logicism' and G logic). is defined is identical to the its usual definition inWright (1983), 153-54. pp. 38 39 justify truths. These our non-logical notion '?!*', as a proper asserted axiom. Even interpretations an appeal to such number our primitives. among the claim logical for the substitution the axiom involves etc., See, in particular, Field (1984) and Boolos (1997). Boolos in terms 'Frege Arithmetic' formulates Numbers: VF3\xVG(Gnx = G ^ of the axiom: F) See (1987), p. 5 (or the reprint (1998), p. 186). Boolos discusses how Hume's Principle is grounded r] relation in Numbers. and At the notion some point, of encoding. to discuss I hope the similarities between Boolos' To anticipate, the paradoxes of encoding compare described inZalta (1983), Appendix A (pp. 158-59) with the paradoxes of r?described in Boolos (1987), p. 17 (Boolos (1998), p. 198). 40 41 See Rosen I think for an interesting (1993) one way to defend more account well-developed 'mathematical (substitute theory' its own stants) group of objects. of each framework least gives an account of Carnap come of this question. to suggest that each that it simply 'linguistic in some sense, for framework'), 'linguistic presupposes to explain how the language and con? (predicates to denote the right relations and objects, and our theory at of mathematical the following. (1) that is axiomatized frameworks. In the present framework, in the very specification con? logical of ?. (2) The range over proper? acterize of the second order variables of O is that they understanding as set-theoretic where these are not construed entities. The difference sets and properties sets merely is vast char? classify objects, whereas properties we are not presupposing a definition, based on standard (3) Therefore, objects. models of second-order ties and between relations, 43As mentioned is all a offers framework' failed this in the case It is important to remember is a primitive notion sequence intended discussion the theory here would be of Carnap's view (1950) language, of the second-order logical consequence relation. above, Field defends the view that logical objectivity (suitably qualified) the objectivity that there is in mathematics. He clearly the idea that there is rejects true set theory or one correct answer to such questions as the Continuum Hypothesis. Similar claims were in Linsky defended and Zalta the specific kind of (1995). Moreover, mathematical inherent in number that he would is validated in objectivity accept theory

EDWARDN. ZALTA by the fact that the theory in Zalta and described (1999) of natural 0 to some other view Kronecker's extent, can numbers be in this paper! Here that the natural numbers earlier given is where as a classical reduction, our work substantiates, by God but that all the are made are man-made. numbers 44 It is interesting that in (1998b), Field (p. 398) seems to identify something like the present account with the structuralism of Resnik (1981) and Shapiro (1989). (These works have been superceded by Resnik (1997) and Shapiro (1997), respectively.) I believe that the present mathematics 45 Whereas account of that encode objects it is a theorem as well P property a more offers account of the structuralist fine-grained philosophy but I shall not argue for that here. in these works, than that found of 0 that ->(xF inconsistent as its negation P (9 asserts & -aF), properties. (where P There are the existence abstract of of all kinds that encode objects a [ky ->Py]). =df a contradiction for some and that k* is an object of r. Then, that theory r yields now allow (= <pand r f= -*(p will be true in (9. Not only does our Rule of Closure Suppose <p,both r the new variable to infer r ^= \?r (for any i/r), but where \jr' is the result of substituting we r we know that encodes in infer for k* k1 So, (48), every |= [ky1 x/f^K*. may x/r, by yt r-formulable property [kyt yfr']. us terms and our semantics accounts of mathematical for the denotation At this point, also have a 'sense'. This Fregean of mathematical But the expressions language predicates. sense can also be modeled in 6. See Zalta (1988), Chapters (1983), Chapter VI, and Zalta 9-12. the assignment that whereas Note is independent analysis, that the sense of a mathematical of above of jc's conception the object philosophical to extend ZF. best way 49 a view seems Such of mathematical states of mathematicians, x encodes for person expression This is how we denoted by that expression. beliefs. in mathematical and ignorance 48 therefore also Our work may account on the expressions, we might suppose involved in the properties of denotations the mental of the account of Maddy the conclusion (1997) supplement of any theory that the mathematicians language with for error a decide precise is the of Wagner to be consistent the ontological views with (1982). exist (since couldn't abstract the that However, consequence objects possibly as 'not the kind of thing that could exist' on this interpretation). So al? is defined 'abstract' a large part of Field's it is inconsistent does preserve fictionalism, though this interpretation are fictions fail to exist. See Field that contingently with his view that numbers (1993). 50 he takes to what thesis reaches this interesting by sketching 'equivalence' Balaguer and his versions of platonism and fictionalism. of platonism be the best version Though it does have are not axiomatized, fictionalism de re mathematical of platonism his version the plenitude However, principle. account for our de re mathematical of his principle beliefs. (Indeed, Balaguer in such a way for that one can prove, So FBP is not articulated beliefs.) of fiction? Nor is Balaguer's version abstract individual. is a particular that 0zp example, in such alism developed a way fictionalism, philosophies mathematical for theory-prefixed and 'naive' versions of platonism using these two between kind of equivalence that the truth conditions can be precisely sentences based on a plenitude (FBP) cannot platonism' rejects the claim that we have is, like ours, 'full-blooded But despite specified. on a deep puts his finger Balaguer of mathematics. Indeed, in the last chapter of (1998), Balaguer concludes both (a) that the only point of disagreement exist objects suggest between FBP and that (b) that the present is on and fictionalism there work, is no in some fact of sense, the question whether the matter validates this of whether idea by mathematical I exist. objects the fact that one (and abstract

with and nationalists in which platonists the same formalism is if they both adopted or interpretations of the formalism's readings fictionalism, then fictionalist, is better, the only) substance possibly issue of on only disagree to express their theory is the most if there formulation articulate no is really then one might, fact 'existential' but If the quantifier. of both and plenitudinous platonism as to which or reading, platonist that there reach the conclusion descent', the matter of is no fact of the matter as to whether by 'semantic abstract objects understand conclusions (a) and Balaguer's one could way different formalism of O 263 I think exist. (b) in the last chapter is the proper this of his to way (1998). REFERENCES Balaguer, M.: Press, Oxford. Balaguer, M.: G.: Boolos, Platonism 1995, 'A Platonist P.: Benacerraf, and Anti-Platonism 1998, Epistemology', 'What Numbers Could Not 1965, of Frege's 'The Consistency 1987, in Mathematics, 103, Synthese Oxford University 303-325. 74, 47-73. Be', Review, Philosophical Foundations of Arithmetic', in J. Thomson (ed.), On Being and Saying, MIT Press, Cambridge, MA; reprinted in Boolos (1998), 183-201. pp. in R. Heck G.: 1997, 'Is Hume's and (ed.), Logic, Language, Analytic?', Principle Oxford in Boolos Press, Oxford; (1998), pp. 301-314. Thought, University reprinted G.: and Logic, Harvard MA. Boolos, 1998, Logic, Press, Cambridge, Logic, University R.: Structure R. George of 1967, The Logical (trans.), University of the World, Carnap, California Press, Berkeley. Boolos, R.: Carnap, 'Empiricism, 4, 20-40. Philosophie, Feferman, S.: 1998, Feferman, S.: 1998a, 'What in Feferman (1998), 'Hilbert's matics', S.: Feferman, Journal Feferman, S.: menta Mathematicae H.: Field, Press, 'Which 1998b, Blackwell, H.: Field, P. French, T. Uehling, H. Oxford. of Mathe? Analysis Proof-Theoretical Mathematical Truth (eds), and Foundational in a General Setting', Funda? Sentences Have Determinate Truth and Mathematics, and Mathematical Objectivity Contemporary Objects', in the Foundations Readings Oxford University in S. Laurence of Metaphysics, 387^03. and Mathematical Logical H. Wettstein (eds), Midwest Dame Notre Field, pp. 'Are Our Relativized: 53, 364-384. Logic of Metamathematics Undecidable (eds), de 35-92. pp. 291-310. 'Mathematical Oxford, 1994, Program and G. Olivieri and C. Macdonald Internationale 187-208. pp. XLIX, in H. Dales Oxford, H.: Field, Press, of Logic, Oxford University on What? The Proof-Theoretic Rests of Symbolic 'Arithmetization 1960, 1998a, Values?', Revue Ontology', In the Light 1988, Reductions', and Semantics, 1950, Press, University 'The Conceptual 1993, Nortre Dame, pp. Concepts Studies in 19, 391-429. of Mathematical Contingency Indeterminate?', Highly in Philosophy, Volume Objects', Mind, 102(406), 285-299. Field, H.: 1989, Realism, Field, H.: 1984, 'Critical Mathematics, Notice and Modality, of Crispin Wright: Objects', Canadian Journal of Philosophy, pp. 147-170, Principle'. with the new title 'Platonism Blackwell, Frege's 14, 637-662; for Cheap? Crispin Oxford. Conception of Numbers as reprinted in Field (1989), Wright on Frege's Context

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Zalta, 1983, Abstract Objects: An Introduction Dordrecht. Center for the Study Stanford University of Language Stanford, CA 94305-4115 U.S.A. E-mail: zalta@mally.Stanford.edu and Information to Axiomatic Metaphysics, Reidel,