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Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 295–323 10.1093/philmat/nkx011 Philosophia Mathematica Advance Access Publication on June 27, 2017 ∗ ∗∗ Johannes Korbmacher and Georg Schiemer Department of Philosophy and Religious Studies, Utrecht University, The Netherlands E-mail: [email protected] ∗∗ Department of Philosophy, University of Vienna, Vienna, Austria E-mail: [email protected]. ABSTRACT Informally, structural properties of mathematical objects are usually character- ized in one of two ways: either as properties expressible purely in terms of the primitive relations of mathematical theories, or as the properties that hold of all structurally similar mathematical objects. We present two formal explications corresponding to these two informal characterizations of structural properties. Based on this, we discuss the relation between the two explications. As will be shown, the two characterizations do not determine the same class of mathemat- ical properties. From this observation we draw some philosophical conclusions about the possibility of a ‘correct’ analysis of structural properties. 1. INTRODUCTION Structural properties play a central role in the contemporary philosophy of mathematics, particularly in the debate about mathematical structuralism. This is the view that mathematics is not concerned with the ‘internal nature’ of its objects, but rather with how these objects ‘relate to each other’ [Shapiro, 1997; Resnik, 1997; Parsons, 1990]. Take the natural numbers as an exam- ple. The standard mathematical theory of the natural numbers is second-order Peano arithmetic. It is well-known that many diﬀerent set-theoretic systems satisfy the axioms of Peano arithmetic, such as the (ﬁnite) von Neumann ordi- nals ∅, {∅}, {∅, {∅}},... and the Zermelo ordinals ∅, {∅}, {{∅}},... for example. According to the structuralists, however, mathematics is not concerned with We would like thank three anonymous referees, Michael Resnik, Hannes Leitgeb, Christopher von Bulo ¨ w, Laurenz Hudetz, Jon Wigglesworth, Øystein Linnebo, and O. Foisch, as well as the participants of the ﬁrst FILMAT conference at the University of San Raﬀaele in Milan, the SotFoM conference at the University of Vienna, the workshop Representation and Axiomatization: Power and Limits at IHPST Paris, and the CLMPS congress at the University of Helsinki for their useful comments and suggestions. c The Authors [2017]. Published by Oxford University Press. This is an Open Access article dis- tributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. 295 Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 296 Korbmacher and Schiemer the concrete set-theoretic structure of these models — second-order Peano arithmetic does not describe the numbers as speciﬁc sets. Rather, arithmetic describes how the numbers add up, how they can be divided, and so on — it describes the structure that the set-theoretic systems satisfying the theory have in common. In other words, according to structuralism, mathematics is concerned with the structural properties of its objects. Despite the importance of structural properties for mathematical structural- ism, there appears to be no formal explication of the concept in the literature. Informally, structuralists usually characterize structural properties of objects in a mathematical system in one of two ways: (i) as properties deﬁnable from the primitive relations of a given system, or (ii) as properties of objects that are shared by structurally similar systems. Compare, for instance, Shapiro on the ﬁrst approach in his account of non-eliminative structuralism: Deﬁne a property to be ‘structural’ if it can be deﬁned in terms of the relations of a given structure. [Shapiro, 2008, p. 286] The central idea here is that structural properties are precisely the proper- ties that are deﬁnable in the language of a mathematical theory. The second approach, in contrast, is based on the notion of structural invariance or abstrac- tion. Compare Linnebo on this account, again in the context of non-eliminative structuralism: A structural property can now be characterized as a property that can be arrived at through this process of abstraction, or, equivalently, a property that is shared by every system that instantiates the structure in question. [Linnebo, 2008,p.64] The main idea here is to specify structural properties of objects based on an act of abstraction from isomorphic systems: a property of objects in a system counts as structural if it also holds of all corresponding objects in isomor- phic systems. So far, however, little work has been done to make these two approaches formally precise. In this paper, we aim to remedy this situation. We will present two formal explications of structural properties corresponding to the two informal charac- terizations above. We will call them the deﬁnability account and the invariance account of structural properties respectively. A central point to be made here is that each of these two accounts comes in two versions based on two ways of It should be noted here that in Shapiro’s account, the mathematical entities consid- ered here are not concrete objects in model-theoretic systems, but ‘places’ in ante rem structures. Compare also [Shapiro, 2006] for a more detailed philosophical discussion of the notion of structural properties. An important exception is [Linnebo and Pettigrew, 2014] which contains a formal discussion of structure-abstraction principles for non-eliminative structuralism and of ‘fundamental properties’ of positions in such structures. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 297 representing mathematical entities: (i) a version for structured mathematical systems and (ii) a version for entities conceived as elements in such systems. The reason for this bifurcation is that being a structural property means dif- ferent things in those two contexts. Take the natural numbers as an example again. A natural-number system is a system of objects that satisﬁes the axioms of second-order Peano arithmetic. Such a system of objects may have addi- tional structure, but in this context all we care about is the structure in virtue of which the system satisﬁes the theory in question. Intuitively, then, a struc- tural property of a natural-number system is a property the system has or does not have in virtue of this structure. The elements within a natural-number system, in contrast, are the numbers from the point of view of the system. A structural property of a number, then, is a property that the number has or does not have in virtue of the relevant structure of the system in which it occurs. The properties of a prime number are examples of structural properties in this sense, while the property of being a von Neumann ordinal is a counterexample. Thus, we have two senses of structural properties depending on two diﬀerent contexts: if we look at structured systems as a whole, we get one sense of structural properties, and if we look at the elements within such structured systems, we get another sense. Present work on mathematical structuralism focuses mainly on structural properties in the latter sense, that is, on the structural properties of elements in mathematical systems. This holds in particular for recent contributions to non-eliminative structuralism, speciﬁcally in the debate on the identity of struc- turally indiscernible places in pure structures. This discussion focuses on the adequacy of structuralist identity principles formulated in terms of structural properties of such places. The principle in question says that two places in a pure structure count as identical here if they share the same structural properties. In turn, when philosophers discuss structural properties of mathematical systems — for instance in work on eliminative or category-theoretic structuralism — then usually no connection is made to these other debates. The present paper wants to bridge these diﬀerent lines of research in philoso- phy of mathematics by giving a uniﬁed account of the notion. In particular, the ﬁrst main goal here will be to show that both the invariance account and the deﬁnability account can be made to work in a precise sense for both types of mathematical properties, that is, for properties of systems as well as properties of elements in such systems. Our focus will consequently not be on a particular See, for instance, [Ker¨ anen, 2001; Shapiro, 2008; Ketland, 2006; Leitgeb and Ladyman, 2008]. Even though the debate centers around this and related principles of structural indiscernibility, structuralists do not universally accept the principle. Indeed, Leitgeb and Ladyman, among others, reject the principle as a criterion of identity for objects in (ante rem) structures. An important exception to this is [Landry and Marquis, 2005] where structured systems (such as groups, topological spaces, etc.) are treated as objects within category- structured systems. See [Awodey, 1996] for a general account of category-theoretic structuralism. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 298 Korbmacher and Schiemer version of mathematical structuralism such as non-eliminative or eliminative structuralism. Neither will we take a stance on standard structuralist claims involving structural properties, such as the identity of structural indiscernibles or the so-called purity thesis, which states that mathematical objects have only structural properties (see [Linnebo and Pettigrew, 2014]). Rather, the aim will be to give a general conceptual and logical analysis of the notion that provides us a better understanding of how structural properties should be used in these philosophical debates. The second main goal in the paper is to get a clearer understanding of the relation between the invariance-based and the deﬁnability-based accounts of structural properties. In particular, we will show that the two accounts do not characterize the same concept. Based on this observation, we propose a tolerant, Carnapian stance with respect to the choice of explication: we argue that neither of the two explications gives us the ‘correct’ notion of structural properties; instead both accounts have their philosophical and mathematical merits. The paper will be organized as follows: In Section 2, we will lay the con- ceptual foundations for the rest of the paper. In particular, we will further explain the distinction between structural properties of systems of mathemati- cal objects and structural properties of elements in such systems. In Section 3, we will present a generalized version of the invariance-based account of struc- tural properties. Section 4 will present the explication of structural properties in terms of their deﬁnability in a mathematical language. Section 5 will then turn to a more general discussion of the conceptual relation between the two approaches. Speciﬁcally, we oﬀer a philosophical assessment of the fact that the two accounts do not determine the same pre-theoretical notion of a structural property. Section 6 will contain a summary and some suggestions for future research. 2. MATHEMATICAL OBJECTS AND THEIR PROPERTIES As mentioned in the introduction, structural properties will be speciﬁed here in accordance with two diﬀerent ways to represent mathematical objects in partic- ular contexts: (i) as elements in structured systems, and (ii) as the structured systems themselves. When we speak about mathematical entities in the latter sense, we usually refer to them in the context of certain axioms that describe the relevant structure of the system: systems are thus considered as models of these axioms. For instance, when the natural numbers are treated as a struc- tured number system, this system is understood as a model of the axioms of second-order Peano arithmetic. Notice that mathematicians may talk diﬀer- ently about the same system in diﬀerent contexts: for example, they may treat the natural numbers as a model of Peano arithmetic or as a monoid, i.e.,asa system satisfying the monoid axioms. Moreover, such systems can themselves be elements in larger mathematical systems: for example, the natural numbers viewed as a monoid are themselves elements in the system of submonoids of the natural numbers. Thus, how we conceptualize the mathematical objects we talk about is highly context sensitive. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 299 In what follows, we will present both ways to think about mathematical entities in a standard model-theoretic setting. Structured systems always belong to a particular mathematical type, for instance the type of groups, graphs, or number systems. A speciﬁc type is usually deﬁned by a set of axioms and formulated in an associated language. This is often a ﬁrst-order language, for instance a ﬁrst-order language describing abstract groups. Nevertheless, in the remainder of this paper, we will also consider higher-order languages for the description of mathematical systems and their properties. The second-order formulation of Peano arithmetic is a well-known case in point here. In the following account of mathematical types, systems, and objects, we deliberately leave the logical strength of the mathematical languages unspeciﬁed for the moment. If T is a type of structured systems of mathematical objects, we denote the associated set of axioms by Λ and the language in which these axioms are formulated by L . Typically, the non-logical vocabulary of such a language contains a set of function symbols F , a set of relation symbols R, and a set of individual constants C. For example, the vocabulary of a (ﬁrst- or second-order) language L of Peano arithmetic contains the function symbols S, +, and · for PA the successor function, addition, and multiplication, as well as the individual constant 0 for zero. One can view a structured system canonically as a model of L which satisﬁes the axioms Λ — i.e., we can view it as a model-theoretic T T system of the form M M M M M = D ,f ,R ,c : f ∈F,R ∈R,c ∈C that consists of a non-empty domain D , a number of functions and relations over the domain that interpret the function and predicate symbols, as well as of a number of distinguished elements that interpret the individual constants of the language such that all of these components behave as Λ says. Given this, we can simply view elements of systems as individuals in the domain of a model- N N N theoretic structure. So, if N = N , 0 ,S is a natural-number system, then an element in N , i.e., a natural number in the system, is simply an individual number n ∈ N . In this model-theoretic framework, the important notion of structural similarity between systems can be made precise in terms of the notion of iso- morphism for models of the relevant language. This relation is deﬁned in the usual way: Deﬁnition 1 (T-isomorphism). Two L -systems M and N are isomorphic M N if there exists a bijection λ : D → D such that: M N (i) λ(c )= c , for all individual constants c ∈L M N (ii) (a ,...,a ) ∈ R ⇔ (λ(a ),...,λ(a )) ∈ R , for all n-ary relation 1 n 1 n symbols R ∈L and a ,...,a ∈ D . T 1 n M N (iii) λ(f (a ,...,a )) = f (λ(a ),...,λ(a )), for all n-ary function sym- 1 n 1 n bols f ∈L and a ,...,a ∈ D . T 1 n Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 300 Korbmacher and Schiemer We say that two systems M and N of type T are isomorphic, in symbols M N,iﬀ M and N are isomorphic as models of language L . We next turn to the properties of mathematical objects. A lot can be said about the nature and existence of properties and philosophical positions on these matters abound. We shall try to remain as neutral as possible on the subject. For the most part, an intuitive understanding of properties suﬃces: properties are simply those things that we can attribute to or predicate of things. Informally, one usually refers to properties using gerunds of the form ‘being ...’ or ‘having ...’. For technical purposes, we distinguish between properties of elements in systems and properties of systems. For example, the property of being prime is a property of numbers in a number system and the property of having an inﬁnite domain is a property of systems of a certain type. For both elements and systems, we write P(x) to indicate that the object x has the property P. In the following, we will treat properties of mathematical systems as classes of such systems. For instance, the property of being a commutative group will be understood here as the class of Abelian groups. In contrast, properties of elements in such systems will be understood (in a Lewisian sense) as functions from systems to sets of elements in these systems. For example, the property of being an even number will be treated as a function that maps to each number system satisfying second-order PA the set of even numbers in its domain. Mathematical properties of both types can thus be expressed more formally in the following way: Deﬁnition 2 (Mathematical Properties). (1) A property P of T-systems is the class {S ∈ T | P(S)} of all and only the systems of type T that have the property P. (2) Let P be a property of elements of T-systems. The local extension of P in a system S is the class P = {x ∈D | P(x)} of all and only those things in the domain of S that have the property. Property P is the function ιP : S → P that assigns to every system S of type T the local extension P of P in system S. Given this account, three points should be noted here. First, since most types of mathematical systems (e.g., the type of Abelian groups) are not sets but, For an overview see, for instance, [Oliver, 1996]. Strictly speaking, extensions of mathematical properties so construed are often proper classes. An axiomatic class theory such as von Neumann-Bernays-G¨ odel set theory (NBG) would therefore be a suitable theoretical framework to express the present account of mathematical properties more formally. This functional account of mathematical properties essentially conforms with Lewis’s [1986] possible-worlds approach to properties. In the remainder of the paper, we will also use the following notation to speak about the local extensions of properties of elements in particular systems: if M is a mathematical system, d ∈ D an object in this system, and P a property, then we also write P (d)to say that d is in the extension of P in M. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 301 strictly speaking, proper classes, it follows that the mathematical properties of elements in such systems, if understood in the above sense, cannot be presented by set-theoretic functions. Instead, they have to be thought of as proper-class- sized functions, i.e., as functions between proper classes. Second, given the present account, one can think of a mathematical prop- erty of elements as being instantiated in diﬀerent systems of a given type. Consider again of the property of being even in the context of number sys- tems: this number-theoretic property is understood here as a function from systems satisfying second-order PA to particular number sets, namely the sets of even numbers in the particular model considered. As will be shown in the next section, these local extensions of the property can diﬀer from system to system. For instance, given two diﬀerent set-theoretic models of PA, the one based on the von Neumann construction of the natural numbers and the other on Zermelo’s construction of the natural numbers, the respective sets of even numbers will clearly be distinct. Nevertheless, the present account allows us to treat these local extensions as belonging to the general arithmetical property that can be instantiated in all of these systems. This functional understand- ing of properties of elements is clearly motivated by a structuralist account of mathematics. In particular, a structuralist would think of ‘being even’ as a general number-theoretic property that is independent of any particular system satisfying the theory. Finally, following Quine’s famous dictum of ‘no entity without identity’, the present account of mathematical properties also calls for a speciﬁcation of their identity conditions. When should we be committed to saying that two math- ematical properties are identical? The philosophical literature on properties discusses a number of possible criteria of property individuation suitable for this task. Generally speaking, one can think of these identity criteria as laws of the following form: For all properties P and Q: P = Q iﬀ C(P, Q), where C(P, Q) is a condition, usually expressed in the form of an equivalence relation between the properties P and Q. If we restrict our attention to the case The notion of classes is treated informally in standard set theory; that is, classes are not described by the axioms of (ﬁrst-order) ZF, but treated as deﬁnable predicates in the language of set theory. A proper-class-sized function F can also be dealt with in ZF by representing it as a ﬁrst-order deﬁnable formula of the form Φ(x, y) such that Φ(x, y) holds in V if and only if F (x)= y, where by V we mean the cumulative von Neumann hierarchy of sets. Proper-class-sized functions can also be characterized more explicitly in an axiomatic class theory such as NBG or Morse-Kelley set theory. Compare [Jech, 2002, pp. 5–6] for details. We would like to thank an anonymous reviewer for emphasizing this point in his report. From the perspective of non-eliminative structuralism, it is tempting to say that such general mathematical properties hold primarily of pure positions in abstract structures. Since these structures can be exempliﬁed by more concrete systems, it follows that the properties also apply to elements in systems that instantiate these pure positions. In the present paper, we choose to treat mathematical properties of elements functionally instead of adopting such a non-eliminative account of structures. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 302 Korbmacher and Schiemer of properties of elements in systems, one natural approach would be to specify identity in terms of co-extensionality in all systems: For all properties P and Q of objects in systems of type T: P = Q iﬀ for S S all systems S of T and for all objects x ∈D : {x ∈D | P(x)} = {x ∈ D | Q(x)}. For most applications discussed in the following, this identity criterion will be adequate. However, one might argue that it is still too coarse-grained for the individuation of mathematical properties. In particular, given the functional account of properties outlined above, one will be forced to identify properties that are intuitively distinct. Consider again of the case of arithmetic: the prop- erties of ‘being the square of 2’ and of ‘being the fourth successor of 0’ will turn out as identical according to the above criterion since they share the same extension in every system satisfying PA. Given these reasons, one might consider other, so-called hyperintensional identity criteria for mathematical properties. These criteria are ﬁner-grained than co-extensiveness in all systems. Recent work on the metaphysics of prop- erties has focused on diﬀerent versions of structured -property theories. Very roughly, properties so conceived are either primitive or equipped with some internal propositional structure. The identity of complex properties is then determined by reference to this internal structural composition. We will not be able to discuss such hyperintensional identity criteria and their possible rel- evance in the context of mathematical properties any further here. Instead, we simply acknowledge the fact that thinking about identity of mathematical properties in terms of co-extensiveness can lead to problematic results. A more deﬁnitive theory of identity conditions for mathematical properties will have to be developed elsewhere. Given this general model-theoretic picture of mathematical systems, objects, and their properties, when do mathematical properties qualify as structural? Intuitively, a structural property is a property a mathematical object has in virtue of or because of its structure. As should be clear, this means diﬀerent things for systems and elements of such systems: a structural property of a system is a property the system has because of its internal structure — it tells us something about the structural composition of the system. In the case of elements in structured systems, in turn, structural properties are properties that express information about the role of the elements in the overall structure of the system. Put diﬀerently, these are properties a particular element has because of its contextual structure, i.e., the relations in which it stands with the other See, for instance, [Menzel, 1993] for an algebraic approach and [King, 2007] for a ‘quasi-syntactic’ approach to structured properties. The conception of quasi-syntactically structured properties in the context of mathematical properties can already be found in [Lewis, 1986, pp. 56–58]. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 303 elements of the system it belongs to. The aim in the next two sections will be to see how these two informal ways of thinking about structural properties can be made formally precise in terms of the notions of invariance and deﬁnability. 3. THE INVARIANCE ACCOUNT One way to specify structural properties is based on the notion of invariance under structure-preserving transformations. This notion has a long mathemat- ical history, tracing back to nineteenth-century work in algebra and algebraic geometry. The ﬁrst attempt to deﬁne the notion of structural properties explic- itly in terms of invariance can be found in Carnap’s early work on axiomatics. In his manuscript Untersuchungen zur allgemeinen Axiomatik, Carnap gives us the following deﬁnition: Deﬁnition 1.7.1 The property fP of relations is called a ‘structural property’ if, in case it applies to a relation P , it also applies to any other relation isomorphic to P . To say that fP is a structural prop- erty is expressed in a formula as follows: (P, Q)[(fP &Ism(Q, P )) → fQ]. The structural properties are so to speak the invariants under isomorphic transformation. They are of central importance for axiomatics. [Carnap, 2000,p.74] Carnap deﬁnes structural properties of relations here as those properties that remain invariant under isomorphisms. How can we generalize this invariance- based approach to apply to properties of arbitrary mathematical objects? To address this, it should be emphasized that something Carnap’s account does not give us is an understanding of structural properties in terms of more basic or non-structure-related notions. His deﬁnition can be taken to rely in one way or another on some primitive structural facts that determine whether two objects count as structurally equivalent or isomorphic. To see this, consider again how the relation of isomorphism for a given class of mathematical objects is usually deﬁned. The ‘traditional way’ speciﬁed in Section 2 — and already anticipated by Carnap — is by ﬁrst identifying a set of primitive properties and relations and then to deﬁning two objects as being isomorphic if and only if there is a bijection between them that preserves these primitive properties. Carnap’s deﬁnition is thus based conceptually on a prior axiomatic stipula- tion of certain facts that are taken as our basis for speaking about structure and structural invariance. What the above deﬁnition does is that it extends these axiomatically speciﬁed properties to arbitrary properties. This said, it is Examples of such properties can be traced back to Benacerraf’s famous paper ‘What number could not be’ from 1965: ‘Being prime’ or ‘being even’ are structural properties of individual natural numbers whereas ‘being a speciﬁc Zermelo number’ is not. Take groups as a concrete example. A group isomorphism between groups G and G G G is a bijective function from G to G , such that g ◦ h = j iﬀ f (g) ◦ f (h)= f (j). Thus, the relevant primitive property is having a group operation ◦. A group isomorphism is a bijection that preserves this property. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 304 Korbmacher and Schiemer explicitly left open how we identify the primitive properties of mathematical objects in the ﬁrst place. Moreover, these properties clearly diﬀer for diﬀer- ent classes of mathematical objects. For instance, an object N is isomorphic to the natural numbers N iﬀ there is a bijection from N to N that preserves the successor function and the distinguished object 0 ∈ N. Similarly, an object R is isomorphic to the reals R iﬀ there is a bijection from R to R that pre- serves the distinguished objects 0 ∈ R and 1 ∈ R, the addition function +, the multiplication function ·, and the order ≤ of the reals. Thus, for diﬀerent kinds of mathematical objects we have diﬀerent kinds of axiomatically deﬁned constitutive properties. Based on these considerations, we can outline a modernized account of Car- nap’s deﬁnition of structural properties. Speciﬁcally, we will assume that a deﬁnition has to be relativized to a particular type of mathematical objects. As mentioned above, this type comes with a distinguished ‘structural vocabu- lary’, i.e., a set of primitive terms in the formal language of the theory. The explication of structural properties of structured objects or systems of a given mathematical type then looks as follows: Explication 1. A property P is a structural property of systems of type T iﬀ ∀S, S ∈ T : P(S)& S S ⇒ P(S ). Structural properties of systems of a given type are thus properties that remain invariant under the isomorphisms between systems of that type. For instance, being a graph of a certain order or clique number clearly turns out as a struc- tural property of graphs under this condition. Similarly, being an inﬁnite cyclic group turns out as a structural property of groups. Our discussion of the invariance-based account has so far focused on prop- erties of systems, that is, mathematical objects that already possess some kind of internal structural composition. What about properties of elements in such a system? Interestingly, several analogous deﬁnitions of structural prop- erties of individuals in a structured system can be given in terms of invariance conditions. One possible approach here is to restrict attention to the structure- preserving permutations of a given domain in which such objects occur. Let A be a particular model-theoretic system. The objects considered now are the elements in D. Properties of such elements can be treated extensionally as subsets of the system’s domain. Given this, a possible invariance condition for such properties is based on the automorphisms of A, that is, isomorphisms of the form f : D → D. Speciﬁcally, we can say that a property of elements in An enticing idea in this context would be simply to consider structures as limit cases of structural properties of mathematical systems. For instance, we might understand the natural-number structure as the structural property of being isomorphic to the canoni- cal natural-number system N. We would like to thank a referee for pointing us to this possibility. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 305 A is structural if and only if it remains invariant under every automorphism on A. The symmetry between this approach and Explication 1 should be clear enough: in both versions, the ‘structural’ character of a property is deﬁned in terms of its invariance under certain structure-preserving transformations. In the ﬁrst case, this is invariance under isomorphisms, in the second case it is invariance under automorphisms, i.e., the inner isomorphisms of a given system. In spite of this nice symmetry, the automorphism-based account does not give us a materially adequate account of structural properties of objects in a given system. One can easily construct counterexamples, namely properties that turn out to be structural in this sense but fail to be structural from a pre- theoretical understanding of the term. Typical cases in point here are properties of elements in rigid systems, i.e., systems without non-trivial automorphisms. Examples of this are the natural-number systems satisfying second-order PA or the real-number ﬁeld. Given that the only automorphisms on these systems are the respective identity mappings, it follows that every property of elements in their domains is by deﬁnition invariant in the above sense. This includes properties such as ‘being someone’s favorite numbers’ which clearly fail to count as structural from an intuitive point of view. Fortunately, there exists an alternative explication of structural properties of elements in mathematical systems that is also in direct symmetry with Expli- cation 1. The underlying idea here is to specify the notion not in terms of invariance under automorphisms of a single given system, but — as in the above case — in terms of the invariance under isomorphisms between diﬀerent systems: Explication 2. Let S be a system of type T. Then a property P is a structural property of the objects in the domain of S iﬀ for all systems S (also of type T) S S and for all isomorphisms λ : D → D : ∀x ∈ D : P (x) ⇒ P (λ(x)) S S Structural properties of objects in a system S are speciﬁed here as those prop- erties that the objects ‘keep’ when making isomorphic copies of S. Consider, for instance, the property of having no predecessor in the context of systems More formally, we say that a property P of objects in A is a structural property iﬀ it is invariant in A, i.e., for every element in the automorphism class f ∈ Aut(A), we have f (P)= P. The idea of characterizing structural properties in terms of invariance under automorphism has also been discussed in non-eliminative mathematical structuralism, in particular, in [Ker¨ anen, 2001]. It is interesting to see that the distinction between rigid and non-rigid systems also plays a key role in recent work on non-eliminative structuralism, in particular, on the identity of indiscernible positions in a pure structure. See, e.g., [Ker¨ anen, 2001; Leitgeb and Ladyman, 2008; Shapiro, 2008]. The notion of structural properties is deﬁned here as a binary relation between sys- tems and properties of elements in these systems. It could also be deﬁned as a ternary Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 306 Korbmacher and Schiemer satisfying PA. In the standard system of natural numbers, this property is true only of the number zero. Furthermore, the property turns out as structural in the above sense since it also applies to all isomorphic copies of zero, i.e.,tothe ‘base element’ in any other model of PA. To illustrate this isomorphism-based account further, let us look at another example. Take the arithmetical property of being an even number discussed already in the previous section. For present purposes, we will focus on two particular number systems of PA. The ﬁrst one can be called the von Neu- mann system: natural numbers are represented here by sets in the following vN vN vN vN vN vN way: 0 = ∅, 1 = {∅}, 2 = {∅, {∅}},..., and N = {0 , 1 ,...}. The vN vN successor function is speciﬁed as S (n)= n ∪{n}, for any n ∈ N . The vN vN vN vN system N = N , 0 ,S satisﬁes second-order PA. The second number system is the Zermelo system: natural numbers are identiﬁed here with sets in Z Z Z Z Z Z the following way: 0 = ∅, 1 = {∅}, 2 = {{∅}},..., and N = {0 , 1 ,...}. Z Z The successor function is speciﬁed as S (n)= {n}, for any n ∈ N . The system Z Z Z Z N = N , 0 ,S is also a model of second-order PA. Benacerraf’s central insight in [Benacerraf, 1965] was that, from a purely structural point of view, it does not matter which of the two set-theoretic systems is taken to represent the natural-number structure. Neither of them should therefore be singled out as the preferred model of arithmetic. Put diﬀerently, both systems are suited equally well for this task since the elements in them share the same structural, i.e., purely relational properties. Based on this insight, we suggested treating arithmetical properties such as ‘being an even number’ as functions from PA- systems to subsets of elements in them. In the particular example, the function vN vN vN presenting the property of being even will pick out the set {2 , 4 , 6 ,... } Z Z Z relative to the von Neumann system, the set {2 , 4 , 6 ,... } relative to the Zermelo system, and corresponding sets for any other model of PA. This prop- vN Z erty is clearly structural according to Explication 2: systems N and N are vN isomorphic and any isomorphism between them maps the object 2 to object Z vN Z vN Z 2 , object 4 to 4 , object 6 to 6 , and so on. Thus, the property of being even is preserved under isomorphisms in the sense speciﬁed above. Does Explication 2 rule out the kind of counterexamples to the automorphism-based approach mentioned above? Consider again the case of accidental properties such as being someone’s favorite numbers. How such prop- erties are to be evaluated depends on how they are understood given the present framework. In our view, the most natural way to interpret them is to say that such properties apply to objects of a particular number system, for instance, of a particular model of PA. Thus, we take it that such properties are about a particular set of natural numbers in a particular number system. To give an example, consider the property ‘being one of Zermelo’s favorite numbers’ (henceforth Zer) and let Zermelo’s favorite numbers be the numbers in system Z Z Z Z N which form his birth date {1871 , 7 , 27 }. As can easily be shown, Zer relation between systems, properties, and particular elements in the following way: a prop- erty P is a structural property of an object a in system S iﬀ for all systems S (also of type T) and for all isomorphisms λ : D → D : P (a) ⇒ P (λ(a)). S S Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 307 turns out to be invariant and thus structural (relative to N ) according to the automorphism-based approach. However, it fails to be structural according to the isomorphism-based account. One way to think of Zer more formally is as a Z Z Z Z function that gives us the set {1871 , 7 , 27 } for system N and the empty set for any other PA-system. This treatment reﬂects the fact the Zer applies only to certain elements of system N and to no objects in any other number system. As a consequence, Zer is not preserved by isomorphisms between PA-systems and thus fails to be structural. In this respect, the present approach is clearly preferable to the automorphism-based account. Note that Explication 2 of the notion of structural properties presupposes the functional understanding of mathematical properties of elements in systems outlined in Section 2. Thus, in contrast to the automorphism-based approach, it makes little sense to conceive of properties purely locally as sets of elements in a particular system. Rather, we have to assume that a property can recur in diﬀerent mathematical systems and have diﬀerent interpretations (or local extensions) in them. It is therefore natural to think of properties as functions from systems to such local extensions, viz., as sets of individuals in a given system, in the sense speciﬁed in Section 2. Viewed in this way, a property qualiﬁes as structural if there exists, between its local extensions in any two systems, a bijective correlation that is induced by an isomorphism between the systems. This functional treatment of mathematical properties (of elements in sys- tems) also allows us to address a possible objection regarding the descriptive adequacy of Explication 2. Consider again the example of ‘being one of Zer- melo’s favorite numbers’ but let this now be the property of being a prime number in a PA-system. Is this property structural? This again depends on whether the property is interpreted only locally, i.e., as applying exclusively to It should be noted here that there exists a second way to interpret ‘being one of Zermelo’s favorite numbers’. According to this view, the property applies not to elements of a particular PA-system, but rather to abstract number positions that can be exempliﬁed by such set-theoretic objects. In particular, given the present framework, we could think of Zer as a function that assigns the respective interpretations of the numerals ‘1871’, ‘7’, and ‘27’ for each PA-system. Understood in this way, Zer is a structural property. How are the two explications of structural properties suggested here related to the diﬀerent versions of mathematical structuralism discussed in the literature? The automorphism-based approach sides well with non-eliminative structuralism and its focus on properties of positions in pure structures. In contrast, Explication 2 seems closer in spirit to eliminative structuralism given that structural properties are speciﬁed here for objects in systems and in terms of the generalization over systems and isomorphisms between them. That said, it is interesting to compare Explication 2 with the treatment of structural properties in the context of non-eliminative structuralism given in [Linnebo and Pettigrew, 2014]. Linnebo and Pettigrew specify pure structures as well as the positions and relations in such structures in terms of Fregean abstraction principles. ‘Fundamental relations’ of such positions are constructed by abstraction from relations on the domain of a concrete system. Linnebo and Pettigrew further show that fundamental properties so construed are structural in the sense speciﬁed in Explication 2. We would like to thank one of the anonymous reviewers for pointing out this objection to us and for very helpful suggestions. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 308 Korbmacher and Schiemer objects of a particular system or more generally, as a property applying to the prime numbers in all PA-systems. Considering the ﬁrst — in our view more natural — interpretation, one could say that the property of being Zermelo’s ∗ Z Z Z Z Z favorite numbers (henceforth Zer ) is given by the set {2 , 3 , 5 , 7 , 11 ,... } of objects in the Zermelo system. Alternatively, we could understand Zer as the function that selects this set relative to system N and the empty set for any other PA-system. So construed, the property is not isomorphism invariant and thus not structural. Considering the second option, one could also understand ‘being one of Zer- melo’s favorite numbers’ more generally as the property of prime numbers in all PA systems (henceforth Zer ). Thus, again adopting our functional treat- ment of properties of elements, the property can then be viewed as the function that selects the set of prime numbers in each PA system. As mentioned above, accidental properties of elements in systems of a given type (such as the prop- erty of belonging to person X’s favorite numbers) should intuitively not count as structural, simply because they are not about the internal structure of the systems in question. However, Zer clearly turns out to be structural according to the isomorphism-based account since it is invariant under any isomorphic transformation of the standard natural-number system. Does the property ‘being one of Zermelo’s favorite numbers’ present a coun- terexample to Explication 2? In the local version of it, i.e., understood as Zer , the property clearly presents no problem to the isomorphism-based account. In the general version Zer , the situation is less obvious. In particular, an answer to the question depends on the choice of a particular criterion of identity that one wants to adopt for mathematical properties. Recall from Section 2 that diﬀer- ent criteria can be considered here. As we saw, one natural approach is to take two properties to be identical if they are co-extensive in all possible systems considered. Applied to the present example, this would imply that the number- theoretic properties ‘being one of Zermelo’s favorite numbers’ (interpreted as Zer ) and ‘being a prime number’ are in fact identical. Following a reviewer’s suggestion, we might say that being one of Zermelo’s favorite numbers (under- stood in this way) is in fact the property of being a prime presented in an exotic disguise. Moreover, as we saw, it also turns out to be a structural property, at least if one adopts Explication 2 as a way to precisify this notion. Therefore, at least if identity of properties is captured in terms of co-extensiveness in all systems, property Zer does not present a counterexample to Explication 2. A diﬀerent assessment may be needed, however, if other, ﬁner-grained criteria of identity for properties are considered. A more detailed discussion of the intri- cate relationship between diﬀerent identity criteria for mathematical properties and the notion of structural properties (as speciﬁed in Explication 2) would Consider again the two systems of PA, namely the Zermelo system and the von Neu- mann system. Relative to these two systems, property Zer picks out two inﬁnite sets, Z Z Z Z Z vN vN vN vN vN namely {2 , 3 , 5 , 7 , 11 ,... } and {2 , 3 , 5 , 7 , 11 ,... } respectively. Since Z vN any isomorphism between N and N will preserve this property, it turns out to be structural. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 309 go beyond the scope of this paper. We therefore leave this issue for possible future work. Given the two invariance-based explications of structural properties, two further points of commentary are in order here. Notice ﬁrst that in contrast to Carnap’s original account, the modernized versions only give a partial def- inition of structural properties relative to a given mathematical type or to particular objects of systems of a given type. Moreover, the deﬁnitions are non-reductive in the sense that they assume a prior speciﬁcation of the primi- tive structural terminology used to describe the objects of this type. Thus, the notion of structure-preserving mappings is presupposed in the invariance-based account. As a consequence of this, the approach is sensitive to how we present objects and systems, that is what types and languages we associate with them. Second, as already indicated above, both accounts are materially adequate in the sense that they agree with many of our intuitions about what counts as a structural property. In particular, the standard examples of such properties mentioned in [Benacerraf, 1965] all turn out to be structural according to Expli- cations 1 and 2: having inﬁnitely many prime numbers is a structural property of PA-systems. Being prime is a structural property of numbers in such sys- tems. In contrast, having the set of von Neumann ordinals as the domain is not a structural property of PA-systems. Being a speciﬁc set is also not a structural property of numbers in such systems. Thus, both invariance-based explications seem to reﬂect closely our pre-theoretical understanding of structural properties in mathematics. 4. THE DEFINABILITY ACCOUNT The second approach to deﬁning structural properties is based on the notion of logical deﬁnability. As with the invariance-based approach, it has a long his- tory, tracing back to early work on formal philosophy and the logic of science. In Russell’s monograph The Analysis of Matter, one can ﬁnd the view that a relation’s ‘structure is what can be expressed by mathematical logic’ [Russell, 1927, p. 254]. The same view is elaborated more fully in Carnap’s Der Logis- che Aufbau der Welt [1928]. The position defended there is that the structure of a relation is presented by ‘the totality of its formal properties’. Formal or structural properties in turn are speciﬁed in the following way: By formal properties of a relation, we mean those that can be formulated without reference to the meaning of the relation and the type of objects between which it holds. They are the subject of the theory of relations. The formal properties of relations can be deﬁned exclusively with the aid of logistic symbols, i.e., ultimately with the aid of the few fundamental symbols which form the basis of logistics (symbolic logic). [Carnap, 1928, p. 21] Thus, unlike in [Carnap, 2000], structural properties are not deﬁned here in terms of invariance under isomorphisms, but in terms of their logical deﬁn- ability. How can we make this second approach more precise? As in the above Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 310 Korbmacher and Schiemer case, it makes sense to highlight some facts about Carnap’s account that need further consideration. First, properties are deﬁned here not only for mathematical relations, but for all possible relations, including physical or empirical ones. In the present context, the approach of specifying structural properties in terms of their deﬁn- ability can be restricted to the case of mathematical properties. The second point concerns what is meant by ‘logical’ in talk about logical deﬁnability here. In Carnap’s case, deﬁnability in logic means expressibility in a pure logical lan- guage, i.e., a language without non-logical vocabulary. Thus, according to this account, structural properties are precisely those properties expressible in pure higher-order logic. In the context of mathematics, this approach seems insuf- ﬁcient. When mathematicians speak of deﬁnable properties here, they usually have in mind formal languages that additionally contain some primitive math- ematical vocabulary. Any modern reconstruction of the present account will thus have to be explicit about both the logical resources of a given language and its mathematical signature. The ﬁnal point concerns the metatheoretic notion of ‘deﬁnability’ in use here. For Carnap, being deﬁnable means that a property can be explicitly deﬁned in the background language. Such deﬁnitions are conceived of purely syntactically by him, i.e., as expressions ‘formulated without reference to the meaning’ of the relations and their relata [Carnap, 1928, p. 21]. In contrast to this, our present account of mathematical properties will be based on a model-theoretic understanding of deﬁnability. With these points in mind, we can turn to an explication of structural proper- ties in terms of logical deﬁnability. We say that structural properties of systems can be speciﬁed in terms of the notion of deﬁnable classes of models of type T in the following sense: Explication 3. A property P is a structural property of systems of type T iﬀ there is a closed formula ϕ ∈L such that ϕ deﬁnes P, i.e.: {S ∈ T | P(S)} = {S ∈ T |S ϕ}. Structural properties of systems in this account are properties whose extensions — conceived here as classes of models of a given type — are deﬁnable in the associated language of that type. Consider some examples of mathematical properties that turn out to be structural in the above sense: the property of a group G to have κ elements in the underlying set for κ a ﬁnite cardinal number is deﬁnable in the language of groups. The property of a graph to have n edges is deﬁnable in the language of graph theory. This explication applies to properties of mathematical systems with some kind of internal structural composition. As in the case of the invariance See, e.g., [Marker, 2000, §1.3] for a detailed discussion of the notion of deﬁnability in model theory. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 311 approach, an analogous explication can be given for properties of elements in such systems. One possible way to go here is to specify structural properties relative to a single system. In this case, L -deﬁnability does not concern classes of models of a certain signature, but sets or relations in such a given system. Structural properties of elements in a system S of type T are thus properties whose extensions in S are deﬁnable in the associated language L . This account corresponds closely to the automorphism-based approach stated in Section 3. As we saw, invariant properties of objects can also be speciﬁed relative to a particular system, namely in terms of the invariance under all automorphisms of that system. We have mentioned, however, that these ‘local’ approaches are not fully satisfactory for present purposes. In particular, they fail to capture our general ‘structuralist’ motivation for these deﬁnitions, namely to describe properties as entities that apply to individual objects across systems. Thus, according to this view, the property of ‘being prime’ is not considered here as a property of natural numbers of a particular number system, but as an arithmetical property that applies to objects in all systems satisfying PA. For this reason, we propose a more general explication. First, we make sure that whenever we have a system S of type T, we have a name for every member of the domain D of S.We achieve this by adding to the language L an individual constant d for every member d ∈ D. This gives us the extended language L . We then move to the extended system S , which is deﬁned just like S, except that we stipulate that d = d. Then we say: Explication 4. A property P is a structural property of elements in T-systems iﬀ for every system S of type T there is a formula ϕ(x) ∈L such that: {d ∈ D | P (d)} = {d ∈ D |S ϕ(d)}. According to this account, structural properties of elements in the systems of a given mathematical type are thus properties whose local extension in each system is deﬁnable in the associated language. Some points of commentary are again in order here. First, notice that both deﬁnability-based explications closely capture our informal ways of thinking about structural properties (as outlined in Section 2). Recall that a structural property of elements in a system is usually understood as a property expressing some piece of information about their relational or contextual structure, i.e., about the relations in which these objects stand to other objects in the system. As mentioned before, such local approaches to invariance and deﬁnability are com- patible with a structuralist account of mathematics if one adopts a non-eliminative understanding of structures. Structures are then conceived as universals or patterns with a domain of abstract positions that can be instantiated by objects of concrete mathematical systems. Deﬁnable properties in this context do not concern objects in systems but rather their abstract placeholders in such structures. See, e.g., [Ker¨ anen, 2001]. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 312 Korbmacher and Schiemer Exactly this understanding of structural properties is made precise in Expli- cation 4. Similarly, we can say that the informal understanding of structural properties of structured systems is captured closely by Explication 3: structural properties are conceived here as those properties expressing a fact about the structural composition of the systems considered. Here again, deﬁnability in terms of the primitive terminology of a mathematical theory secures that the properties so expressed are about these internal or intrinsic structural facts. Second, it is insightful to see how the present approach diﬀers conceptually from the invariance-based approach outlined above. Notice that Explications 3 and 4 also give us a partial and non-reductive account of structural properties. The account is partial because structurality is always speciﬁed relative to a particular mathematical context or type. It is non-reductive in the sense that a prior identiﬁcation of primitive terminology is assumed that allows us to identify what we mean by structure in this particular context in the ﬁrst place. Moreover, just as in the invariance approach, these explications are sensitive to how we represent objects and systems, that is how we set up the languages to describe them. The central diﬀerence from the invariance approach is that the present account is also highly sensitive to the expressive power or logical strength of the associated language. Thus, it makes a diﬀerence whether the mathematical language L in use is ﬁrst-order or higher-order. To give just one example: some fairly simple graph properties — for instance, properties concerning the clique number of a graph — are not expressible in a ﬁrst-order framework and are thus not structural according to Explication 3. They do become deﬁnable and thus structural, however, if one adopts a second-order language that allows one to quantify over subsets of a graph’s vertex set. More generally, we can say that what counts as a structural property according to the deﬁnability approach is strongly dependent on the choice of the background language. It depends both on the mathematical (viz., non-logical) signature as well as the logical resources (viz., the types of variables and types of quantiﬁers that are part of the language) in use. 5. COMPARISON Two diﬀerent explications of the notion of structural properties were presented here, both of which seem to capture our pre-theoretical understanding of such properties in mathematics. In light of this, a natural question to ask is whether the invariance-based and the deﬁnability-based accounts are in fact equivalent. Put diﬀerently, do they determine the same collection of properties for a given type of mathematical objects? Prima facie, this seems a plausible assumption to make. The symmetry between invariance and deﬁnability has long been inves- tigated in model theory. Thus, the approaches can be considered as two sides of This does not mean that the deﬁnitions expressible in a formal language necessarily capture all structural facts about the mathematical systems considered. As will be pointed out below, the logical strength of a language in use also plays a signiﬁcant role here. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 313 the same coin, that is, as two ways of describing the structure of mathematical objects. This fact has also been acknowledged in the more recent literature on math- ematical structuralism. For instance, a close variant of the deﬁnability-based explication of structural properties of objects in a system is discussed at length in the context of non-eliminative structuralism in [Ker¨ anen, 2001]. Ker¨ anen introduces there the notion of ‘inter-structural’ relational properties of places in a pure structure that can be instantiated by the elements of concrete systems. These properties are, he points out, deﬁnable in the relevant mathematical lan- guage (without individual constants) [2001, pp. 315–317]. Ker¨ anen goes on to argue that . . . a property is guaranteed to be invariant under the automorphisms of S if and only if it can be speciﬁed by formulae in one free variable and without individual constants. [2001, p. 318] Unfortunately, the relation between the invariance-based and the deﬁnability- based accounts is not as clear as is suggested there. In fact, Ker¨ anen’s observation does not hold in general. It is not always the case that the expli- cations of structural properties in terms of invariance match those given in terms of deﬁnability in a formal language. To see this, a simple cardinality- based argument can be given. Consider again the standard system of natural N N N numbers N = N , 0 ,S (which, for the sake of concreteness, we may take Z vN either to be N or N ). As pointed out above, N is rigid, i.e., there are no automorphisms on N other than the identity mapping. It follows from this that any property (conceived as a set) of the elements in N is invariant in the above sense. Hence, there are 2 invariant properties of natural numbers. How- ever, assuming that our language or arithmetic L is ﬁnite, there are at most PA countably many deﬁnable properties of natural numbers. In order to clarify the relation between invariance and deﬁnability in the context of our discussion, we need to move back from Ker¨ anen’s non-eliminative account to a more neutral understanding of structuralism and translate his claim into our preferred way of thinking about invariance across systems, that is, invariance with respect to the isomorphisms between systems. Given the previous distinction between properties of structured systems and properties of the elements of such systems, his assumption can be reformulated in terms of two equivalence claims, namely: (α) A property of systems of type T is invariant under isomorphisms between T-systems iﬀ it is L -deﬁnable. (α ) A property of elements in a T-system S is invariant under isomorphisms between T-systems iﬀ it is L -deﬁnable in S. See, for instance, [Hodges, 1997, p. 93]. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 314 Korbmacher and Schiemer Do these equivalences hold? The respective right-to-left directions can easily be conﬁrmed. Put diﬀerently, the deﬁnability account subsumes the invariance account. This is a simple consequence of the isomorphism lemma from elemen- tary model theory, viz., the result that isomorphic systems are semantically equivalent. Thus, for case (α), the following holds: Observation 1. Let P be a property of systems of type T.If P is L -deﬁnable, then it is invariant. Proof sketch: Intuitively, what needs to be shown here is that any class of systems deﬁned in L consists of full isomorphism classes of Ts. Let the extension of P be deﬁned by sentence Φ and let S∈ T such that S|=Φ . P P Then, by the isomorphism lemma, for any X ∈ T:if X S we also have X |=Φ . A formal proof of this can be given by induction on the complexity of formulas. An analogous argument can be given in support of the claim that L-deﬁnable properties of elements in a system S are invariant under all isomorphic copies of S. While this direction of the equivalence claims is straightforward, the left- to-right direction is not. Speciﬁcally, it is not generally the case that invariant properties are also deﬁnable in the particular language of the theory in question. To see this, consider some counterexamples against claim (α): the property of ﬁrst-order PA-systems of ‘being isomorphic to the standard number system N’ is not deﬁnable in ﬁrst-order L , but is clearly invariant under isomorphisms PA of PA-systems. As is well known, the property becomes characterizable if one works within a second-order language of arithmetic, simply due to the fact that the second-order formulation of PA (with the second-order axiom of induction) gives a categorical axiomatization of this number system. One could thus think that the equivalence stated in (α) could be restored if a notion of second- order deﬁnability were presupposed. However, this is also not the case. One Lemma (‘Isomorphism Lemma’): Let L be a language. Then for all models M and N of L and all ϕ ∈L: M N⇒ M ϕ ⇔N ϕ. A full proof of the isomorphism lemma for ﬁrst-order languages is given in [Marker, 2000, pp. 13–14]. This proof can easily be generalized to apply also to higher-order languages. PA is categorical if one assumes a standard semantics for the second-order language in which the theory is formulated. See [Shapiro, 1991, §4.2] for a detailed discussion of dif- ferent semantics for second-order languages as well as for a proof sketch of the categoricity of arithmetic. It should be noted here that using second-order languages (and higher-order lan- guages more generally) in this context might be seen as problematic since this would qualify certain properties of mathematical objects as structural that we would intuitively not count as such. Consider, for instance, the second-order language for real-number ﬁelds. One can easily formulate statements in this language which are true if and only if the con- tinuum hypothesis is true. However, few number theoretists would take such a statement as expressing a structural property of the real-number systems. We would like to thank an anonymous reviewer for drawing our attention to this fact. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 315 can construct examples of invariant properties that fail to be second-order deﬁnable. To give one example: Observation 2. Let L := L ∪{ }, where is a unary predicate that is PA 2 2 PA intended to apply exactly to the codes of all the second-order validities. Then the N N N 2 2 property of ‘being isomorphic to N , 0 ,S , Val ’, where Val is the set of codes of second-order validities, is not second-order deﬁnable, but is invariant 2 30 under isomorphisms between L -models. PA Proof. By Theorem 41C of [Enderton, 2001, p. 268] the set Val is not deﬁnable N N N in N , 0 ,S by any formula of second-order logic. Suppose that we could N N N 2 deﬁne the property of being isomorphic to N , 0 ,S , Val by a formula N N N 2 2 ϕ, i.e., M N , 0 ,S , Val iﬀ M ϕ. Then n ∈ Val iﬀ ϕ → (n) = 2 N N N 2 is valid. To see this, note that if M (n) and M N , 0 ,S , Val , 2 = then n ∈ Val . And so if we could deﬁne the property of being isomorphic N N N 2 2 N N N to N , 0 ,S , Val , Val would be deﬁnable in N , 0 ,S after all, in contradiction to Enderton’s Theorem 41C. So the property of being isomorphic N N N 2 to N , 0 ,S , Val is not deﬁnable by any formula in second-order logic. The latter example suggests that however strong one chooses one’s back- ground language to be, there is always a way to construct examples of mathematical properties that are invariant but fail to be deﬁnable in that language. An important exception to this, at least for the case of properties of elements in systems, is the limit case where deﬁnability is speciﬁed relative to an inﬁnitary language. Let L be a language of pure inﬁnitary logic, ∞,∞ i.e., an extension of a ﬁrst-order language that allows formulas with inﬁnitely long sequences of conjunctions, disjunctions, and quantiﬁers. There are a num- ber of well-known mathematical results — discussed mainly in the debate on invariance-criteria for logical notions — that show that the notions deﬁnable in such a language in fact coincide with those meeting certain invariance condi- tions. The central result in this debate, presented in [McGee, 1996], is based on Tarski’s [1986] suggestion of characterizing logical notions in terms of invari- ance under permutations of a given domain of objects. McGee’s theorem states, As a consequence of a result of Hintikka [1955], this observation generalizes to higher- order logics beyond second-order logic. Consider the language L := L ∪{ }, where PA n PA th is intended to apply to exactly the codes of the validities of n -order logic, for n a N N N n natural number. Then the property of ‘being isomorphic to N , 0 ,S , Val ’, where n th Val is the set of codes of second-order validities, is not n -order deﬁnable. The reason n 2 is that by Hintikka’s result Val is (computably) reducible to Val . More speciﬁcally, for th n 2 every sentence ϕ ∈L we can ﬁnd a sentence ψ ∈L such that ϕ is valid in n -order PA PA N N N n logic iﬀ ψ is valid in second-order logic. Thus, if ‘being isomorphic to N , 0 ,S , Val N N N 2 were deﬁnable in L , then ‘being isomorphic to N , 0 ,S , Val ’ would be deﬁnable PA in L , which we have already observed is not the case. Montague [1965] has shown that PA Hintikka’s result even extends to inﬁnitary-order languages, and thus our observation also applies there. See, in particular, [McGee, 1996; Bonnay, 2008; Bonnay and Engstr¨ om, 2015]. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 316 Korbmacher and Schiemer roughly, that a logical operation is invariant under all permutations of a given domain if and only it is deﬁnable in L . Note that this result is more general ∞,∞ than the invariance and deﬁnability conditions discussed in the present paper. Invariance in the context of logical operations means invariance under all per- mutations of a given domain, not invariance under permutations that preserve some additional (mathematical) structure. Similarly, deﬁnability means deﬁn- ability in a pure logical language without non-logical constants for McGee. In contrast, our focus in the present context is on mathematical properties that are deﬁnable in languages with a non-empty signature. That said, one can easily present a relativized version of McGee’s theorem that connects deﬁnability in a mathematical language with invariance under the automorphisms of a given model of that language. Let L be an inﬁnitary ∞,∞,T language with a signature of mathematical type T and let M be an interpreta- tion of it. In order to keep the following discussion simple, we assume here that M is a purely relational system of the form D, R ,...,R with R ,...,R 1 n 1 n ﬁrst-order relations (of a given arity) on an inﬁnite domain D. Observation 3. For all systems M of type T, the extension of a property P in M is invariant under all automorphisms of M; in symbols, P ∈ Inv(Aut(M)), iﬀ the extension of P in M is deﬁnable in L . ∞,∞,T Proof. (⇐) Assume that the extension of P in M is deﬁnable by a formula ϕ(x) ∈L , i.e., {d ∈ D | P (d)} = {d ∈ D |M ϕ(d)}. Let f be an ∞,∞,T M automorphism of M. By deﬁnition, for any R (of arity n)in M, we have: M M (d ,...,d ) ∈ R ⇔ (f (d ),...,f (d )) ∈ R . One can show by straightfor- 1 n 1 n ward induction on the complexity of formulas that M|= ϕ(d)iﬀ M|= ϕ(f (d)). Hence, P (d)iﬀ P (f (d)), and thus P ∈ Inv(Aut(M)). M M (⇒) Let P ∈ Inv(Aut(M)). We construct a sentence ϕ ∈L that ∞,∞,T deﬁnes it. Take an enumeration I → D of the elements in D, and pick a set of variables V of cardinality |D| enumerated by I →V. Let J = {j ∈ I | P(d )} be the set of indices in I that enumerate the members of the extension of P in M. Next, for every n-ary relation R and every K ⊆ I with |K| = n, R M let δ be R(x ,...,x )if(d ,...,d ) ∈ R and ¬R(x ,...,x )if K k k k k k k 1 n 1 n 1 n M M (d ,...,d ) ∈ R . Then the description Δ M of R in L is the k k ∞,∞,T 1 n R conjunction: Δ M = δ . K⊆I Now, we can let ϕ be: (∃x ) ( (x = x ) ∧∀y (y = x ) ∧ Δ M ∧ x = x ). i i∈I i j i j i,j∈I,i=j i∈I R∈R j∈J A version of this result with second-order relations is presented in [Bonnay and Engstr¨ om, 2015]. Our proof of this result (for ﬁrst-order relations) is based on the proof in [Rogers, 1967]. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 317 It remains to show that ϕ deﬁnes the extension of P in M, i.e., P(d)iﬀ M|= ϕ(d): From left to right, let d be in the extension of P in M. Then d = d , for some j ∈ J . Now consider any assignment σ in M. Then simply pick some assignment σ , such that σ (x )= d , which will be a V-variant of σ. Then the i i formula: (x = x ) ∧∀y (y = x ) ∧ Δ M ∧ d = x (∗) i j i j i,j∈I,i=j i∈I R∈R j∈J is clearly satisﬁed relative to σ . Hence ϕ is satisﬁed relative to σ, which is what we needed to show. From right to left, let σ be an assignment such that ϕ(d) is satisﬁed in M relative to σ. Hence there is a V-variant σ of σ, such that (∗) is satisﬁed relative to σ . Now consider the values σ (x ) ∈ D of the variables x ∈V for i i i ∈ I. Since σ satisﬁes the ﬁrst and second conjuncts of (∗), we know that D = {σ (x ) | i ∈ I}. Thus, we can ﬁnd a permutation f : D → D such that f (σ (x )) = d for i ∈ I. Now note that since σ satisﬁes the third conjunct of i i (∗), f ◦ σ is an automorphism of M (simply inspect the deﬁnition of δ ). Finally, since σ satisﬁes the fourth conjunct of (∗), we know that d = σ (x ) for some j ∈ J . Then for this j ∈ Jf (σ (x )) = d is in the extension of P in M. But since f ◦ σ is an automorphism and P is invariant under automorphisms by assumption, this means that d is in the extension of P in M, which is what we needed to show. This result shows that if a suﬃciently strong inﬁnitary language is adopted, invariant properties always turn out to be deﬁnable. Notice, however, that the invariance condition used here, namely invariance under automorphisms of a given system, has been rejected by us as a proper way to explicate the notion of structural properties. Thus, as it stands, Observation 3 is not directly relevant to our present investigation. Nevertheless, there exists an interesting generalization of McGee’s theorem that turns out to be applicable to the present discussion. The so-called Tarski- Sher thesis, also formulated in the debate on the nature of logical constants, states that an operation is logical if and only if it is invariant under bijections across domains. Thus, in contrast to Tarski’s original thesis, logicality is deﬁned here not in terms of invariance relative to a given domain, but in terms of invariance across domains. Given this approach, McGee [1996] has formulated an interesting corollary of his result about logical notions (understood now as operations across domains) that are invariant under bijections. The theorem states that an operation P is invariant under all bijections between domains iﬀ for each cardinal κ = 0 there is a formula ϕ ∈L which describes the κ ∞,∞ action of P on domains of cardinality κ. Applied to the present discussion of mathematical properties, this result can be easily transformed into the following relativized result. Let P be a property of elements in systems of mathematical type T and let language L be ∞,∞,T Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 318 Korbmacher and Schiemer speciﬁed as above. Let us say that the extension of P is invariant under isomor- phisms of T-systems iﬀ for all T-systems S, T , all isomorphisms λ : S→ T , and all elements d ∈ D , P (d) ⇒ P (λ(d)). We can then show the following: S T Observation 4. The extension of P is invariant under isomorphisms of T- systems iﬀ for each equivalence class κ =[M] of isomorphic T-systems there is a formula ϕ ∈L which deﬁnes the extensions of P in all systems κ ∞,∞,T M∈ κ. Proof. (⇐) Assume that the extension of P is deﬁnable in all systems in all κ =[M] by formula ϕ (x). Let S, T be two T-systems. If they are not isomorphic, then the claim trivially holds; so assume without loss of generality that S T and λ : S→ T is an isomorphism between the two. Since the two systems are isomorphic, we know that [S] =[T ] , and so the extension of P in both systems is deﬁned by the same formula ϕ . It then follows by [S] a simple inductive argument on the complexity of ϕ that for all d ∈ D , [S] S S P (d) ⇒ P (λ(d)), since {d ∈ D | P (d)} = {d ∈ D |S ϕ (d)} and S T S [S] T T {d ∈ D | P (d)} = {x ∈ D |T ϕ (d)} by assumption. [S] (⇒) Assume that the extension of P is invariant under isomorphisms of T- systems and let κ =[M] be some equivalence class of isomorphic T-systems. Consider the formula (∃x ) ( (x = x ) ∧∀y (y = x ) ∧ Δ ∧ x = x ) i i∈I i j i R j i,j∈I,i=j i∈I R∈R j∈J from the proof of Observation 3 now deﬁned for the representative M of κ.We can take this formula to be ϕ . To establish this, we need to show that in all N∈ κ, P (d)iﬀ N ϕ (d). The proof of this fact goes along the same lines N κ as the proof of Observation 3. For the left-to-right direction, assume that P (d) and let λ : N→ M be an isomorphism. Since P is invariant under isomorphisms of T-systems, this means that P (λ(d)) and thus λ(d)= d for some j ∈ J . By the same argument M j as in the proof of Observation 3, we get that M ϕ (λ(d)). But since λ is an isomorphism, it follows by the isomorphism lemma of model theory that N ϕ (d), as desired. For the right-to-left direction, assume that N ϕ (d). Hence, again by the isomorphism lemma, we get that M ϕ (λ(d)) and by the same argument as in the proof of Observation 3 that P (λ(d)). But since P is assumed to be invariant under isomorphisms of T-systems, we get that P (d) as desired. This adaptation of McGee’s second theorem shows that the local extension of a property is always deﬁnable if the property is invariant under isomorphisms. More generally, these results indicate that the invariance-based and deﬁnability- based explications of structural properties tend to converge if one increases the logical strength of the language in use. Moreover, they determine the same class of properties of a given mathematical type if we choose an inﬁnitary language Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 319 as our mathematical language. Leaving aside such limit cases, however, the examples presented above show that the equivalence claims (α) and (α ) are not generally true. In particular, it is not the case that all invariant properties of mathematical objects are also deﬁnable in the languages in which these objects are usually described. It follows that the two explications suggested here do not determine the same pre-theoretical notion of structural properties. Where does this leave us in our philosophical assessment of the paper’s main question? More generally, two diﬀerent philosophical morals can be drawn from the above observations. The ﬁrst one is to uphold the view that there exists a unique pre-theoretical notion of structural properties that can be captured formally by one of the two accounts presented above. What needs to be con- sidered then is which one of the two explications is more adequate or, in Carnap’s terminology, which one bears more similarity to the pre-theoretic explicandum. The problem this approach faces is that neither of the two expli- cations fully captures our informal understanding of the notion. Speciﬁcally, the invariance account tends to overgenerate, that is, it carves out more prop- erties as structural than we would do so intuitively. In contrast, the deﬁnability account (speciﬁed relative to a particular language) tends to undergenerate: It fails to specify properties as structural that we would intuitively charac- terize in this way. Starting with the ﬁrst problem, we can consider two types of properties that qualify as structural according to the invariance-based def- inition, but that do not plausibly qualify as structural pre-theoretically. We will dub them propositional and parasitic properties. Roughly, a propositional property is a property such that there is a proposition that, for everything, having the property is equivalent to the proposition’s being true, i.e., Prop(P) iﬀ ∃p∀x(P(x) ↔ p). Propositional properties are problematic for the invariance-based explica- tion. Since for every proposition p it is a logical truth that p → p, it follows immediately that if P is a propositional property, then P is invariant under every equivalence relation, i.e., about every subject matter. Hence, in par- ticular, any propositional property counts as structural under Explication 1. But it clearly seems unnatural to take all such properties as structural given that they express no information about the structural composition of partic- ular objects or their relations to other objects in a system. To illustrate this point, consider the propositional property of groups such that a group of order two exists. By the previous reasoning this property is a structural property of groups. But pre-theoretically we should only say of groups of order two that they have the property of being such that a group of order two exists in virtue of their internal structure and not the structure of something else. Hence the problem. In turn, we call a property ‘parasitic’ if its formulation is based on the isomor- phism relation for a given mathematical type T. It turns out that no matter how we deﬁne Iso (x, y), the property λx(∃y¬Iso (x, y)) is always trivially invari- T T ant under the isomorphisms in T. However, pre-theoretically λx(∃y¬Iso (x, y)) should not count as a structural property of the Ts as it says that there is Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 320 Korbmacher and Schiemer something with a diﬀerent structure and a system has that property not in virtue of its own intrinsic structure but in virtue of this external fact. Para- sitic and propositional properties of this sort can be viewed as philosophical counterexamples against the view that the invariance-based explications fully capture our pre-theoretical understanding of structural properties. What is typ- ical of these counterexamples is that the properties are invariant, however, not in virtue of the internal structural composition of the systems they hold of. The properties in question have as their extension either the full type T or the empty set ∅. Thus, statements referring to them are non-informative about the structure of the objects in question. Turning to the undergeneration problem, it was already shown above that the deﬁnability account fails to provide an intuitively satisfying demarcation between structural and nonstructural properties. This has to do with the fact that the speciﬁcation of structural properties based on Explications 3 and 4 is strongly dependent on the choice of a particular logical background language. In particular, a property may fail to be structural according to the deﬁnability approach purely because of the limitations of the logical resources of the formal language in use. Depending on the expressive strength of a particular language, some intuitively structural properties can turn out to be indeﬁnable: ‘being iso- N N N morphic to N = N , 0 ,S ’ was a case in point here for ﬁrst-order languages of arithmetic. This language relativity makes it diﬃcult to consider Explica- tions 3 and 4 as the most adequate ways to formalize structural properties in mathematics. The alternative and in our view more promising approach is to take a Car- napian or tolerant view on this matter. This is to embrace the fact that there are diﬀerent ways to formalize our informal understanding of the notion. In contrast to the above view, the invariance and the deﬁnability accounts should thus be seen as two equally legitimate ways to make precise what we mean by structural properties of mathematical objects in informal discourse. Moreover, there are, as we saw, diﬀerent accounts of deﬁnable properties corresponding to the spectrum of possible logical background languages. The choice of one formalization over the other should thus be based on purely practical con- siderations in a particular theoretical context and not on a general ideal of correctness or truth. For instance, for certain mathematical or foundational questions, the invariance-based approach seems preferable because it provides us with a very comprehensive account of structural properties. In other con- texts, the deﬁnability-based approach might seem more useful, in particular in cases where one is looking for a more tractable or constructive criterion for structurality in mathematics. A related form of language relativity has been discussed in debates on mathematical structuralism. See, in particular, Resnik [1997] on this point. He describes in detail the ‘structural relativity’ underlying his non-eliminative theory of patterns or structures there. This is the fact that ‘the structures we can discern and describe are a function of the background devices we have available for depicting structures’ [1997, p. 250]. Downloaded from https://academic.oup.com/philmat/article/26/3/295/3895509 by DeepDyve user on 13 July 2022 What Are Structural Properties 321 6. CONCLUSION The paper presented several explications of the notion of structural properties of mathematical systems as well as of the elements of such systems. Accord- ing to the invariance-based approach, properties are structural if they remain invariant under all isomorphic transformations. According to the deﬁnability- based approach, structural properties are those deﬁnable in the language of a given mathematical theory. It was shown that the two ways of explicat- ing structural properties usually do not determine the same pre-theoretical notion. In what sense are the results given here relevant for present discussions in philosophy of mathematics and for mathematical structuralism in particular? As mentioned in the introduction, the aim in the paper was not to argue for a particular version of mathematical structuralism (such as ‘ante rem’ or ‘uni- versal’ structuralism). Rather, it was to give a general logical analysis of the notion of structural properties that allows one to see how the notion could be understood (in a precise sense) in diﬀerent philosophical debates. Such debates concern, for instance, the ontological dependence between places and pure structures in which they occur [Linnebo, 2008], the identity of structurally indiscernible places [Leitgeb and Ladyman, 2008; Ker¨ anen, 2001; Shapiro, 2008] in non-eliminative structuralism, or the formulation of consistent ‘structural’ abstraction principles [Linnebo and Pettigrew, 2014]. 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Philosophia Mathematica – Oxford University Press
Published: Oct 1, 2018
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