Russell s Absolutism vs.(?) Structuralism

Transcription

1 Russell s Absolutism vs.(?) Structuralism Geoffrey Hellman Abstract Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind s structuralist views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell s insights. Problems of absolutism plague some versions, and, interestingly, Russell s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modal-structuralism and a category theoretic approach as remaining non-absolutist options. It is suggested that these should be combined. Contents 1 Twists and Turns: Absolutism and Hints of Structuralism in Russell 2 2 Varieties of Modern Structuralism and Their Challenges 6 3 Russellian Critique of Dedekind-Structuralism (with Help from Leibniz and Benacerraf) 10 4 Beyond Absolutism 15 1

2 1 Twists and Turns: Absolutism and Hints of Structuralism in Russell Russell s book, Introduction to Mathematical Philosophy (1919), is almost as rich in philosophical views as it is in information for the general reader about foundations of mathematics. Early on, Russell writes, The question, What is number?, is one which has been often asked but has only been correctly answered in our own time. The answer was given by Frege in 1884, in his Grundlagen der Arithmetik. ([17], 11.) Here we have a good expression of an absolutist stance: There is such a thing as the correct answer to the question, What is number?, and, moreover, it is (essentially) the one Frege gave. (Cardinal) numbers are classes of equinumerous concepts (Frege), or Russell would tolerate this much flexibility equinumerous classes (Russell himself). Interestingly, this is immediately preceded by a brief discussion of an alternative algebraic or structuralist understanding of number concepts: It might be suggested that, instead of setting up 0, number, and successor as terms of which we know the meaning..., we might let them stand for any three terms that verify Peano s five axioms. They will then no longer be terms which have a meaning that is definite though undefined: they will be variables, terms concerning which we make certain hypotheses, namely, those stated in the five axioms, but which are otherwise undetermined....our theorems...will concern all sets of terms having certain properties. ([17], 10.) But no sooner has he described the view than he proceeds to take it off the table. The passage continues, Such a procedure is not fallacious; indeed for certain purposes it represents a valuable generalization. But...in the firstplace,itdoesnotenableus to know whether there are any sets of terms verifying Peano s axioms...in the second place...we want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties. (ibid.) Nowadays, this must be regarded as a rather hasty dismissal. Even to Russell, one may justly claim, it should have appeared so. For, regarding the first reason, why should an algebraic or structuralist reading 2

3 of number theory per se carry with it the assurance demanded, that of mathematical existence of a model? Of course, somewhere in an overall system one would want such assurance (to whatever extent that is possible), but neither definitions nor the use of grammatically proper names ever by themselves guarantee existence, as surely Russell knew, in connection, for example, with the ontological argument! Thus, Dedekind [6] first formulated the Peano postulates but gave them in the form of a definition of simply infinite system ( progression, in Russell s terminology), but then attempted to prove existence separately. (The proof, as we know, went outside mathematics and was not rigorous, assuming the totality of objects of Dedekind s thought, but the need for a separate proof was clear.) Moreover, Russell knew full well that one cannot fall back on the Frege-Russell definition of number to gain the assurance sought, for he had had his own bout with the Axiom of Infinity, to which hehadhadtoyieldbysimplyassuming it (as was done in Zermelo s set theory as well). Answering the second objection requires scarcely more resources, surely not exceeding those at Russell s command. One can account for counting on an algebraico-structuralist reading as providing a bijection between enumerated objects and the relevant initial segment of any progression as indicated by the highest numeral reached. The Frege-Russell solution in which the class enumerated belongs to the number is elegant (in this respect) but hardly privileged. Set theory does it the other way, via bijections (even if one fixes on a particular definition of ordinals, say von Neumann s), and, to us, the elegance of the Frege-Russell account of counting appears as an artifact. This is especially so in light of the heavy price paid elsewhere in the system, e.g. reduplication of numbers at all type levels beyond their first appearance, not to mention their non-existence in the going set theory (ZFC). 1 In fairness to Russell, he had already had raised a serious objection against Dedekind s structuralism (one we shall encounter again below in amoderncontext),andanalternativeversionwasnotathand. Perhaps it only took a mild lack of sympathy with the approach to tempt one into an easy dismissal. In any case, Russell s absolutism was itself not absolute. A few pages after introducing Frege s great discovery, we find the following clue that a touch of imposition is implicated: 1 To be sure, if one lowers one s sights and aims to recover just number theory and classical analysis, the advantages of the Frege-Russell definition can be realized in a demonstrably consistent second-order system invented by Boolos called Frege Arithmetic. [3] (The demonstration is relative to the consistency of second-order Peano Arithmetic, also called classical analysis in a formal sense.) 3

4 We naturally think that the class of couples (for example) is something different from the number 2. But there is no doubt about the class of couples: it is indubitable and not difficult to define, whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples...than to hunt for a problematical number 2 which must always remain elusive. ([17], 18.) Moreover, as it turned out, the correct definition even had to be adjusted in light of paradoxes of naive class theory: But, for the reasons set forth [above, concerning paradoxes], if for no others, we cannot accept class as a primitive idea...classes cannot be regarded as part of the ultimate furniture of the world. ([17], ) As if this were not taking back enough, we are surprised to find in the end the following proposal regarding the nature of mathematical and logical propositions generally: We may thus lay down, as a necessary (though not sufficient) characteristic of logical or mathematical propositions, that they are to be such as can be obtained from a proposition containing no variables...by turning every constituent into a variable and asserting that the result is always true or sometimes true....logic (or mathematics) is concerned only with forms... ([17], 199.) Now, when we consider that relations (or propositional functions) as well as individuals count as constituents of propositions, then we realize that this criterion is met by construing mathematical propositions as formulated in higher-order logic without constants. Indeed, it suffices to work with second-order logic for number theory, analysis, and even set theory. Applying Russell s criterion to the case of number theory requires replacing 0 and successor with an individual and two-place relation variable, respectively, in any sentence of second-order Peano Arithmetic (in which the other standard functions are explicitly definable). We may also replace the term number with a unary relation variable, X, considered as the domain to which all quantifiers are relativized. But this is just the proposal Russell considered and dismissed at the outset, as quoted above! More precisely, Russell explicitly considered the replacement procedure applied to the Dedekind-Peano axioms. Applying the 4

5 overall procedure to an arbitrary sentence A of the language of number theory naturally leads to the formation of a conditional of the form, R[PA 2 A](S/R), in which PA 2 stands for the conjunction of the axioms and S/R indicates systematic replacement of the successor constant with the relation variable R throughout. (Here, to avoid clutter, we have dropped 0 as it can be introduced by definition from S.) If A is logically implied by the axioms, the result is a truth of second-order logic. (If not, the result of replacing A with its negation yields such a truth, in light of the categoricity of the axioms, as Dedekind established.) Still better, taking the predicate number itself into account as suggested, and generalizing, we obtain X R[PA 2 A] X (S/R), where the superscript indicates relativization of all quantifiers to domain X. Again we have second-order logical truths under the same conditions. Bynowwehavecomefullcircle(atleast uptonegation! ),as this is already an expression of an eliminative structuralism applied to arithmetic, for it quite straightforwardly formalizes the claim that the truths of arithmetic are what hold in any progression whatever. As Russell had phrased it, our theorems...will concern all sets of terms having certain properties. Implicitly, indeed almost explicitly, Russell seems to have endorsed this view after all! Indeed, with but one more step that of treating Russell s phrase, and asserting that the result is always true, still more broadly to include possibilities of progressions, not just actual progressions we arrive at the hypothetical component of a modal-structuralist interpretation, which simply prefixes the above with a necessity operator, as governed by a suitable modal logic (naturally chosen to be S-5, with certain restrictions, see [8], Ch. 1, and [9]). The same procedure generalizes to analysis and many extensions, including set theories. Furthermore, as Russell clearly recognized, the axiom of infinity, however it is stated precisely, while formulable in such a logical notation, is not a logical truth. Existence axioms or possibility axioms are still required. The possibility of a progression, X R[PA 2 ] X (S/R), must be assumed, or it can be derived from a still more elementary possibility assumption, which is not hard to formulate. That forms the categorical component of the interpretation, and it is needed to ward 5

6 off the plague of if-then ism. (Of course, as is to be expected from strong medicines, there are side-effects.) Clearly, Russell s absolutist stance was at odds with some of his actual proposals concerning the nature of mathematics. His logicism had structuralist elements within it, and, as we shall see, his critique of Dedekind s views is of special relevance in assessing more recent efforts to articulate structuralism. 2 Varieties of Modern Structuralism and Their Challenges Four main varieties of modern structuralism are readily identified: (1) set-theoretic ( STS ), based on model theory, (2) structures as sui generis universals (the ante rem structuralism of Shapiro [18] and Resnik [15]) ( SGS ), (3) modal-structuralism ( MS ), and (4) an approach based on category theory ( CTS ). These have been described in some detail elsewhere. (See, e.g., [10] and [11].) Here we recall briefly their leading characteristics. STS goes back to the Bourbaki and today would appeal to model theory, with ZF as the background, as providing general concepts of mathematical structures as well as a theory of their interrelations and existence. Regarding number systems, although fixed set-theoretic interpretations are familiar (e.g. the finite von-neumann ordinals for arithmetic, Dedekind cuts in the rationals for real analysis, etc.), these are seen as convenient ways to fix ideas. Arithmetic truths are taken as truths in the language of arithmetic true in any (standard) model of PA (whether first or second order); similarly for the reals, the complexes, etc. With respect to these theories, STS is a version of eliminative structuralism, in that numbers as definite objects, referents of numerals and other singular terms, are eliminated in favor of multiple structures. This accords with the insight that the nature of the individuals is irrelevant; what matters are structural relationships, realizable in many isomorphic ways. More generally, all the spaces and structures familiar from the branches of ordinary mathematics are understood as set-theoretic structures, set domains together with functions and relations on them, whether or not the associated theories are categorical. Notably, however, set theory itself is not treated structurally: although one investigates various set-models of ZF and extensions, the ZF axioms themselves are not read algebraically or structurally, i.e. as mere defining conditions on structures of interest; rather they are taken as assertions of truths outright, truths about the cumulative hierarchy, itself too big to be a set. SGS treats mathematical structures as universals, patterns in Resnik s 6

7 terminology, answering to what all particular systems (realizing key axioms)haveincommon,whetherthesystemsaremadeupofconcrete items or sets. One speaks literally of the natural number structure, for example, made up of positions or places treated as abstract objects, not merely schematically as place-holders or offices to be filled by particulars. (Hence Shapiro s term, ante rem, to contrast these structures from the in re systems, such as those of mereology (part-whole theory) or set theory.) It is such positions that are taken as the literal referents of numerals, etc., and the structural relations among them are taken as directly expressed by relation constants of our language, e.g. successor, addition, etc. Thus, this is a non-eliminative structuralism, closely akin to Dedekind s preferred conception. Axiom systems in mathematics are at once defining conditions on structures and assertions about the ideal types they are taken to be about. On Shapiro s version, in the background is second-order logic and a list of (assertory) axioms directly governing the existence of ante rem structures; these resemble the axioms of second-order ZFC but with the addition of a Coherence Axiom, guaranteeing a structure answering to any coherent set of second-order conditions, where this is a new primitive corresponding to realizable as amodel intheframeworkofsts.(cf.[18],95.) MS has already been introduced above. Clearly, it is an eliminative structuralism, more so than STS in that it applies to set theories as well as those of ordinary mathematics. In its use of second-order logic, however, MS appears to require classes or Fregean concepts in the background. While such interpretations are possible, a class-free interpretation is available through a combination of mereology and plural quantification. [9] The objects of any entertained structures now in the in re sense are unspecified. So long as it makes sense to speak of wholes or combinations of them, they can occur in structures. Unlike SGS, no special structural objects are involved. Indeed, through the useofmereologyandplurals,structuresthemselvesasobjectsneednot be recognized. One simply speaks of some, or these, or those, objects related in relevant ways among themselves or to other objects. A full theory of relations (and functions) is recovered, so that there need be no recognition of these as objects either. In this sense, MS can be understood as a (modal) nominalistic reconstruction. (However, it does not have to be taken that way. Its machinery is available to a wide variety of ontological frameworks, about which, officially, it can remain quite neutral.) Finally, it is an important feature of MS, in contrast to both STS and SGS, that it recognizes no absolutely maximal universe or domain for mathematics. It incorporates instead an Extendability Principle, that any domain whatever can be extended. Comprehension 7

8 principles for wholes or for pluralities are restricted to be extensional: collection-like operations are confined, as it were, to within a world. (Officially, worlds or possibilia of course are not recognized; modal operators are primitive in the system, and are not explained literally as quantifiers.) CTS is somewhat harder to describe because, although category theory and topos theory are well developed as branches of mathematics, a structuralist interpretation of mathematics in categorical terms remains somewhat inchoate. Sometimes it seems to be suggested that merely formulating mathematics categorically is enough to express a structuralist philosophy, since, indeed, CT has its own characteristic way of getting at mathematical structure, via morphisms between structures as pointlike objects and via functorial relations among categories. That view is problematic, however, as it leaves unaddressed fundamental foundational issues, such as What is the external, background logic?, What existence axioms govern categories and topoi themselves?, Are modal notions involved?, Is an extendability principle recognized, as in MS, or should we take seriously an all-embracing category of categories? (see [13]), etc. A full-fledged CT version of structuralism should address such questions. On one proposal, ordinary mathematics can be carried out relative to any number of topoi as universes of discourse, and this can be done without a set-theoretic background. [1] The result is a kind of relativity of ordinary mathematical concepts, and a distinction then arises between invariant mathematics (e.g. obeying intuitionistic logic, arising naturally inside a topos) and essentially relative mathematics, e.g. theory of classical continua for which special conditions on a topos are required (e.g. a choice principle, well-pointedness, etc.). Topoi are then viewed as possible universes for mathematics; there is no privileged, unique one. Is CTS then eliminative? With respect to number systems, surely it is; one may stipulate, for example, that a topos have a natural number object, but this is not a unique structure in either Dedekind s or Frege s or Russell s sense. The case of set theory is less clear, as one speaks of the category of sets, and uses boldface type to name it. How literally should this be taken? (Add this to the list of questions above.) As will be evident already, none of these versions is problem-free. Vestiges of absolutism, present in STS and SGS, are not easily avoided. The goal of giving a structuralist interpretation of set theory (better theories) is sufficient motivation to look beyond STS. As will be brought out in the next section, however, SGS with its absolute structural objects is subject to a reinforced variant of Russell s critique of Dedekind s structuralism. Moreover, both STS and SGS suffer from commitment to 8

9 a fixed, maximal universe for mathematics, violating the extendability principle, which is firmly rooted in mathematical thought and practice. Both these problems are overcome in MS and, pending further clarification, in CTS as well. But these approaches face their own characteristic difficulties. The various trade-offs are summarized in the following table. 2 The remainder of this paper will be devoted to explanation and discussion of some of the key boxes and relationships that emerge. 3 Let us begin by explaining further the left-most column of problems. The first, maximal universe is clear, but it is worth quoting Mac Lane on the matter: Understanding Mathematical operations leads repeatedly to the formation of totalities: the collection of all prime numbers, the set of all points on an ellipse...the set of all subsets of a set..., or the category of all topological spaces. There are no upper limits; it is useful to consider the universe of all sets (as a class) or the category Cat of all small categories as well as CAT, the category of all big categories. This is the idea of a totality...after each careful delimitation, bigger totalities appear. No set theory and no category theory can encompass them all and they are needed to grasp what Mathematics does. ([14], 390.) That STS violates this open-endedness is familiar from its commitment to a fixed real world of sets. But SGS also appears to violate it in it commitment to a totality of all positions in structures (at least, this is so on Shapiro s formulation in second-order logic). However, SGS improves on STS at the next row, as it does apply its structural interpretation to set theories generally, and it need recognize no maximal totality of all sets. It automatically avoids questions such as, Are the sets really well-founded?, Do they satisfy Choice?, etc.; as long as the relevant axiom systems are coherent, there are structures answering to them, and none is ontologically privileged (although of course some may be of greater mathematical interest than others). The third row, lack of equivalence types, concerns the status of objects such as Frege-Russell numbers, or equivalence classes under isomorphisms preserving structure such as ω-sequence, countable, dense 2 In this table, a check-mark indicates that the problem in question does affect the version of structuralism, so that the aim of the game is to draw a blank (unless otherwise noted, as in the third row). Concerning CTS, despite what we have said, we have taken our best guess as to how to fill in the boxes (trying to give the view the benefit of the doubt from a structuralist perspective). 3 For a fuller discussion of the contents of this table, see my [10] and [11]. 9

10 linear ordering without endpoints, separable continuum, complex plane, etc. Formation of equivalence classes is a natural way of passing from particular instances of a type of structure to the type itself, but, as we know, this breaks down in set theory unless the instances are restricted to some level of the cumulative hierarchy (otherwise, proper classes are needed, raising problems of their own, especially from a structuralist perspective). SGS overcomes all this, as its abstract structures are supposed to be platonic archetypes, ideal exemplars answering directly to what all instances have in common. MS, of course, recognizes no such animals, but counts this as a virtue, making sense of shared structure in more modest terms, e.g. by reference to satisfaction of the same axioms or via structure preserving maps between structures, spelled out in the peculiar, MS fashion. CTS, at least on the many topoi relativist view suggested by Bell, also makes a virtue of the absence of absolute equivalence types. Again, shared structure is explained via external relations, i.e. morphisms and functorial relations, and one does not miss maximal or absolute archetypes. The fourth row concerns impertinent questions occasioned by a literal, set-theoretical realism, such as How do you know that the realworld power set of N is full? Maybe some subsets are missing! Such questions do not arise on the other approaches. Each can tell its own story as to just how coherence of the notion of full power set is sufficient for mathematics (which leaves open the possibility also of denying coherence, which is a matter prior to structuralist interpretation). The fifth and sixth rows bring as back to Russell. 3 Russellian Critique of Dedekind-Structuralism (with Help from Leibniz and Benacerraf) As SGS illustrates, structuralism has sometimes been conceived in absolutist terms. The trick has been to posit special structural, nonspatio-temporal objects, pure places in archetypal structures abstracted from all particular cases, whether themselves concrete or abstract. Shapiro calls the particular cases systems, a more general term than structure in the ante rem sense. Structures qualify also as systems, but they can be self-exemplifying so that no third man regress is generated. (Since, however, plenty of systems are commonly thought of as already abstract, e.g. those built out of pure sets, or properties, etc., if recognized, I have called SGS a hyperplatonist view of mathematical structure.) This procedure has its most distinguished antecedent in Dedekind 10

11 ([5] and [6]), as is made clear from correspondence with Weber. 4 The natural numbers (or the reals, the topic of the Weber correspondence) form a unique, particular system over and above all other simply infinite systems (continua), exemplifying what they all share in common and supposedly lacking irrelevant features, i.e. those beyond distinctness from other objects and their roles as defined by structural relationships specified in the mathematics itself. (Tait calls the passage from particular realizations, e.g. set theoretic, to such pure archetypes Dedekind abstraction. [19]) Now one may immediately question the idea of purity. How can any objects fail to have mathematically irrelevant properties, such as arise from adventitious relations to external things, e.g. being thought of by Dedekind, being designated by the English monosyllable, nine, being the number of planets, etc. Efforts have been made to salvage purity by distinguishing essential or intrinsic properties from the rest, but this is also problematic: surely on the conception being advanced, being abstract, being non-spatio-temporal, lacking color or mass, etc., qualify as essential, but the mathematics is silent on such matters. In the Principles, Russell voiced misgivings along these lines: It is impossible that the ordinals should be, as Dedekind suggests, nothing but the terms of such relations as constitute progressions. If they are to be anything at all, they must be intrinsically something; they must differ from other entities as points from instants, or colors from sounds...dedekind does not show us what it is that all progressions have in common, nor give any reason for supposing it to be the ordinal numbers, except that all progressions obey the same laws as ordinals do, which would prove equally that any assigned progression is what all progressions have in common...his demonstrations nowhere not even when he comes to cardinals involve any property distinguishing numbers from other progressions. ([16], 249.) Nevertheless, ante rem structures are posited in SGS, in apparent defiance of these strictures. But perhaps it can be conceded that places in structures must be intrinsically something, but that all that need 4 See [7], vol. 3, There is also, however, a letter to Lipschitz (dated June 10, 1876 [7] 65, cf. [18], 173) saying, if one does not want to introduce new numbers, I have nothing against this... While he may have preferred to think of (natural and real) numbers as special, newly introduced objects, Dedekind recognized that mathematics does not require this, and that what matters is the holding of the right properties, which is compatible with an eliminative structuralism. 11

12 amount to is that they just are different from the items Russell lists (and a lot more), that they depend essentially on intra-structural relations to other places, and that they are grasped by a kind of abstraction from systems as suggested. This is a stand-off, and further argument is needed to break it, one way or the other. Two related objections have in fact emerged recently which challenge the very coherence of ante rem structure and Dedekind-abstraction, in support of Russell. Both have a Leibnizian flavor, and the second comes with a topping of Benacerraf (retaining a similar flavor). The first, due independently to Keränen and Burgess, concerns the notion of identity of places in an ante rem structure. In a nutshell the argument goes like this: it seems that these structural objects should be individuated sufficiently by intra-structural relations (including functions) alone, without help from outside or from individual constants. (The reasons for this are given in detail in [12].) This yields a version of Leibniz identity of indiscernibles : any items bearing exactly the same intra-structural relations to other items must be not many but one. But this immediately implies that there can be no non-trivial automorphisms of the structure, i.e. that the structure is rigid. Now, while some structures central to mathematics are rigid, such as the natural numbers, the reals (as a field), and segments of the cumulative hierarchy of sets, many are not. The complex field admits an automorphism interchanging i and -i; the additive group of integers admits an automorphism interchanging +1 and -1, etc. Indeed, such structures abound in mathematics, as Keränen observes: every group of order other than 1 or 2, any geometric figure with a reflectional symmetry, homogeneous Euclidean n-space (Burgess), etc. The rigid structures, though fundamental, are also rather special. Keränen explores possible responses the ante rem structuralist might give, including invoking haecceities (the property of being just this thing and no other ), or giving no account of identity of structural objects at all, and finds them all unsatisfactory. Shapiro has replied that an account of identity is not really required. The debate over this continues. Another response on behalf of SGS might be to appeal to reduction of all structures to rigid ones, perhaps technically possible if every structure can be adequately modelled set-theoretically. This seems alien to the whole SGS program and more in line with STS. In any case, the proposal is impotent against the second objection. This objection, which I developed in [10], challenges the very intelligibility of ante rem structures altogether, rigid as well as non-rigid. Whereas the Keränen-Burgess objection takes structural relations as given as goes on to question the distinguishability of the structural ob- 12

13 jects as relata, this objection goes further in questioning the intelligibility of purely structural relations, in the context of putatively structural objects as relata. This is very much in the spirit of Russell s critique: if we do not appeal to relata as independently given somehow, but instead think of them as determined only through structural relations as surely seems part and parcel of the ante rem structuralist view then what have we to go on in specifying structural relations other than the axiomsthemselvesasdefining conditions? What, for example, can it mean to speak of the ordering of the natural numbers, as objects of an ante rem structure, if, in turn we have no independent grasp of what these numbers are apart from their position in that ordering? As mathematical functions (treated extensionally in classical mathematics), the successor function of any given system or model is distinct from that of any other. So the archetypal one we re after isn t literally the same as any other we may be familiar with (such as the next numeral in our counting system, or the next item in some spatio-temporal sequence). Does it make sense to speak of nextness itself, as a platonic abstraction from all instances, when anything whatever can be next after anything else in some system or other? It seems we can only make sense of next in the way that Dedekind himself did, namely, as relative to a given function ϕ, or arrangement of some sort, which is to say, in this case, relative to a given simply infinite system. Thus, we have a vicious circularity in the very notion of ante rem structure: such a structure is supposed to consist of purely structural relations among purely structural objects, but understanding either of these depends on already understanding the other. Given the objects, you can get the relations, and (if it weren t for the Keränen-Burgess objection) vice versa. Butyoucan tgetbothatonce! One response to this (given by Shapiro in correspondence) is to deny that structural relations need be prior to places of an ante rem structure. Forexample, wehave finite cardinal structures as degenerate cases of ante rem structures, in which there are a finite number of places but no relations at all (other than identity)! The view must be that a structure is determined by its places and relations. Neither is prior to the other. I agree that if we can somehow independently pick out the places, then we can speak of relations among them. This is a standard platonist procedure, falling back on, say, designation by certain singular terms in our language. But then the objects are not given purely structurally, but by reference to external, andinthiscase, evencontingent relations. But if we do not help ourselves to such extra-structural means of identifying the objects (which, I would suggest, is what we usually 13

14 do in platonist mathematical talk), what sense does it make to speak of places, if not relative to given orderings or relations? Withante rem structures as proposed by SGS, we seem to be in the situation of having to succeed in referring to B (the places) as a precondition of referring to A (the relations), and vice versa. But then you can t do both with neither. You have to do both with both, and that seems impossible. What, then, of the finite cardinal structures? On reflection, these seem utterly baffling, indeed the ultimate offense against Leibnizian scruples. The 4 cardinal structure, for example, is supposed to consist of four distinct abstract things period. No structural relations at all are involved; it makes no sense to speak of one of them as first or as occupying any position in any intuitive sense. How is it that any is distinct from any other? Indeed, how can we make sense of referring to any one of them as opposed to any other, or mapping any one of them to or from anything else (which is essential if they are to exemplify cardinality)? Non-identity is not sufficient for this, and unlike the natural numbers on an objects-platonist construe, according to which at least we have standard names for the objects, and unlike identical bosons of the particle physicist which at least make a difference by contributing to the total mass-energy, even if we cannot label them nothing else is available. With these purported structures, as we shall see momentarily, the Keränen-Burgess and Hellman objections come together, in that every permutation of such objects would be a non-trivial automorphism ( trivially, so to speak, as there are no relations to preserve)! This brings us to a final consideration. Suppose, for the sake of argument, that, despite the above argument, we could make sense of introducing an ante rem structure,arigidone,say,forthenatural numbers. Call it < N, ϕ, 1 >, whereϕ is the privileged successor relation and 1 the initial element of N. Then we immediately see that indefinitely many other progressions, explicitly definable in terms of this one, qualify equally well as candidates to serve as the referents of our numerals, even if we require freedom from irrelevant features, to whatever extent < N, ϕ, 1 > itself fulfills this. For we need only consider the result of permuting any (finite, say) number of items of N and adjusting ϕ and 1 accordingly. Any such structure, call it < N,ϕ 0, 1 0 >,isable to serve as the archetypal ante rem progression every bit as well as < N, ϕ, 1 >. Indeed, on what conceivable grounds can we call one but not any other the result of Dedekind-abstraction. We cannot say, e.g., that really 1, not 1 0, is first (if the permutation considered moves 1), for first makes no sense except relative to a successor function, and, relative to ϕ 0, 1 0, not 1, is first. The reader may well be reminded of Benacerraf s famous argument, that numbers cannot really be sets, 14

15 since many progressions of sets are equally available to serve as natural numbers, and it would be absurd to say we are really speaking of one as opposed to any other.[2] Indeed, he generalized the argument to conclude that natural numbers cannot really be objects at all, and here, with ante rem structures, we have another example of why not. Hyperplatonist abstraction, far from transcending the problem, leads straight back to it. This reminds us again of Russell s remark, quoted in the opening section above, on the prudence of settling for the class of couples rather than hunt[ing] for a problematical number 2 which must always remain elusive. He was overly sanguine about the class of couples, but his worry about a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down seems to us exactly on the mark. 4 Beyond Absolutism The fundamental informal intuition behind structuralism is that, in mathematics, what matters are structural interrelationships and not the nature of individual objects. STS runs afoul of this in the case of set theory itself, normally taken as a theory about a fixed universe of absolute objects, the sets. ( absolute in the senses of definite, unconditional, and even pure.) SGS, while even-handed in its application to set theories along with all other mathematical theories, nevertheless violates the intuition by positing special absolute objects, intended to stand in complete abstraction from any irrelevant, non-mathematical features, yet guaranteed by the mere coherence of suitable mathematical axioms implicitly characterizing them. Surely Russell was right to question the intelligibility of such things and to point to the futility of attempting to transcend the nature of individual objects by positing objects without any nature. And we have seen that deeper objections refining Russell s cut against the very notions of ante rem structure and Dedekind abstraction. Moreover, as the table above makes clear, STS and SGS are absolutist in a second sense, in commitment to a maximal universe of mathematical objects, violating extendability, as already described. We are naturally led to seek non-absolute alternatives. As the table also brings out, both MS and CTS bypass these problems of absolutism affecting either STS or SGS, although in very different ways. MS abstracts from the irrelevant features of objects of structures, not by positing special abstract objects, but by generalizing over whatever (suitably interrelated things) there might be, which can remain open-ended. Ordinary constants (e.g. numerals) are construed as representing places in structures, not in the literal sense of SGS, but 15

16 indirectly by indicating the relevant generalization: 2, in context, is paraphrased by reference to the third item of any progression (starting with 0 ), and also as a convenient device in modal-existential instantiation on the assumption that progressions are possible in the first place (no pun here). CTS, on the other hand, avoids direct talk of items internal to mathematical structures entirely in favor of morphisms to and from other structures, these treated as point-like, the objects of a category. In favorable circumstances (e.g. existence of terminals), suitable substitutes for members of a structure can be found, and, with ingenuity, CT paraphrases of ordinary set-theoretic conditions can be expressed. Like MS, CTS transcends the nature of individual objects (in the ordinary sense) not by postulation of special entities but by simply remaining silent about any such matters. MS, recall, enforces silence by sticking to mathematical conditions in its general statements concerning structures, and further by the thorough substitution of first- and second-order variables, as Russell ultimately suggested (cf. 1, above). CTS, on the other hand, enforces such silence in its own clever way, by confining itself to properties got at via morphisms in a category and/or functorial relations among categories. Moreover, both MS and CTS avoid absolutism with regard to the mathematical universe, again in their distinctive ways, MS by directly adopting principles of extendability and restricting comprehension to within a world, CTS by not adopting any official ontology and proceeding informally in matters of (external) logic. As the table also indicates, however, MS and CTS both confront problems of their own, in the case of MS problems relating to primitive modality, and, in the case of CTS, problems arising from failure to address fundamental questions concerning background logic, universes of discourse, and mathematical existence. If CTS avoids modality, that may be merely a manifestation of this stance. Interestingly, the many topoi view of Bell does address the question of universes of discourse without falling back on a domain of sets, and it hints at a modal formulation in which topoi are thought of as possible worlds for mathematics. Moreover, there are independent reasons for thinking that some appeal to modal notions is unavoidable, especially if one is to do justice to the non-contingency of mathematics and to its open-endedness, the extendability of its universes and its freedom to introduce ever new totalities, as described by Mac Lane above. This suggests that we should seek a way of synthesizing MS and CTS, preserving their respective advantages while minimizing their liabilities. It turns out that this is indeed possible: the details cannot be presented 16

17 here, 5 but the upshot is that the general, neutral apparatus of MS mereology and plurals, without set-membership or classes is sufficient to describe large domains (corresponding to set universes of inaccessible cardinality), and one can postulate these as mathematical possibilities, serving as sufficient backgrounds for categories and topoi, and for models of set theory as well. Of course, far, far less is required for ordinary mathematics, but this gives a unifying, structuralist framework for extraordinary mathematics, one in which topos theory and set theory can be developed side-by-side, neither taken as prior to the other. In this way, the respective structuralist insights of set theory and of category theory can be simultaneously preserved, without ever introducing absolute objects. The relativity of mathematics to choice of topos as suggested by Bell is incorporated in this synthesis, while at the same time the door is left open to classical mathematics as an objective enterprise. References [1] Bell, J.L. From Absolute to Local Mathematics, Synthese 69 (1986): [2] Benacerraf, P. What Numbers Could Not Be, Philosophical Review 74 (1965): [3] Boolos, G. The Consistency of Frege s Foundations of Arithmetic, (1987), in Boolos, G. Logic, Logic, and Logic (Cambridge, MA: Harvard University Press, 1998), [4] Burgess, J. Review of Stewart Shapiro (1997), NotreDameJounral of Formal Logic 40 (1999): [5] Dedekind, R. Stetigkeit und irrationale Zahlen (Brunswick: Vieweg, 1872), tr. as Contintuity and Irrational Numbers in Essays on the Theory of Numbers, W.W. Beman, ed. (New York: Dover, 1963), [6] Dedekind, R. WassindundwassollendieZahlen? (Brunswick: Vieweg, 1888), tr. as TheNatureandMeaningofNumbers,W.W. Beman, ed. (New York: Dover, 1963), [7] Dedekind, R. Gesammelte mathematische Werke 3, R. Fricke, E. Noether, and O. Ore, eds. (Brunswick: Vieweg, 1932). [8] Hellman, G. Mathematics without Numbers: Towards a Modal- Structural Interpretation (Oxford: Oxford University Press, 1989). [9] Hellman, G. Structuralism without Structures, Philosophia Mathematica (3) 4 (1996): [10] Hellman, G. Three Varieties of Mathematical Structuralism, Philosophia Mathematica (3) 9 (2001): For a fuller presentation, see [11]. 17

18 [11] Hellman, G. Does Category Theory Provide a Framework for Mathematical Structuralism, Philosophia Mathematica (forthcoming). [12] Keränen, J. The Identity Problem for Realist Structuralism, Philosophia Mathematica (3) 9 (2001): [13] Lawvere, F.W. The Category of Categories as a Foundation for Mathematics, in S. Eilenberg, et al., eds. Proceedings of the Conference on Categorical Algebra, La Jolla 1965 (Springer, 1966), [14] Mac Lane, S. Mathematics: Form and Function (New York: Springer, 1986). [15] Resnik, M. MathematicsasaScienceofPatterns(Oxford: Oxford University Press, 1997). [16] Russell, B. The Principles of Mathematics (London: Allen and Unwin, 1903). [17] Russell, B. Introduction to Mathematical Philosophy (New York: Simon and Shuster), first published [18] Shapiro, S. Philosophy of Mathematics: Structure and Ontology (New York: Oxford University Press, 1997). [19] Tait, W.W. Truth and Proof: The Platonism of Mathematics, Synthese 69 (1986):