Non-ontological Structuralism

Non-ontological Structuralism ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects (eliminative structuralism) or they maintain that mathematical objects are structures or positions in them (sui generis structuralism). Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view. Structuralism in the philosophy of mathematics has been largely viewed as an ontological doctrine concerning the nature of mathematical objects. Positive versions interpret mathematical objects as places or positions in structures, while the negative versions hold that mathematics has no objects of its own and studies structures abstracted from non-mathematical realms. By contrast non-ontological versions of structuralism leave open questions concerning the existence and nature of mathematical objects. My own view began as ontological in the positive sense, but as it developed I began to doubt the prospects for an ontology in which mathematical objects are positions in structures, and eventually opted for a non-ontological reading of my view. I would like to explain and clarify this further. To this end, I will begin with some familiar history and move to a bit of autobiography. Distinguishing between ontological and non-ontological structuralism makes it harder to classify certain historical anticipations of structuralism. It is usual to interpret Gauss, Hilbert, and Poincaré as anticipating negative versions of ontological structuralism. Thus early in the nineteenth century, when there were still reservations concerning the complex numbers, Gauss wrote: The mathematician abstracts entirely from the quality of the objects and the content of their relations; he just occupies himself with counting and comparing their relations to each other. (Quoted in [Ferreirós, 2007, p. 254]) Around the beginning of the twentieth century Hilbert wrote to Frege along similar lines: ‘It is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes’ [Frege, 1980, p. 40]. A few years later Poincaré echoed this structuralist theme: Mathematicians study not objects, but relations between objects; the replacement of these objects by others is therefore indifferent to them, provided the relations do not change. The matter is for them unimportant, the form alone interests them. [Poincaré, 1913, p. 42] One motivation for structuralism is to avoid having to explain the nature of prima facie mysterious mathematical objects.1 We may assume that these writers were so motivated. But are they merely saying that mathematics is not concerned with the nature of the objects it deals with but only with their structure? Or are they also claiming that there are no proper mathematical objects? If the former, they anticipated non-ontological structuralism; if the latter, they anticipated negative forms of ontological structuralism.2 We do find ontological structuralism — a positive version — in Gauss’s student Dedekind. Writing in the late nineteenth century, he spoke of the human mind creating the system of natural numbers through abstracting its structure. The laws holding here ‘therefore are always the same in all ordered simply infinite systems’.3 Negative ontological structuralism has been dogged by the problem of uninstantiated structures. If a mathematical theory purports to describe a structure which no non-mathematical objects exhibit, then, according to these structuralists, the theory has no models, and its sentences are vacuously true. Hilbert came to appreciate this, and famously claimed that consistency would suffice for non-vacuity. Geoffrey Hellman, a contemporary structuralist, developed a modal version of structuralism to address the problem. Put roughly, so long as it is possible that the structure in question is exhibited, vacuity does not arise.4 During the first half of the twentieth century philosophy of mathematics turned away from structuralist ideas to focus on issues surrounding logicism, intuitionism and formalism. Just about the time these movements began to lose their luster Quine’s writings set the stage for the emergence of contemporary structuralism. His critique of the a priori and his naturalized epistemology turned our attention to the genesis of mathematical knowledge. This was the context for Paul Benacerraf’s instructive and provocative papers. His [1965] paper, ‘What numbers could not be’, observed that there is no compelling mathematical reason to prefer one reduction of the numbers to sets to another or, in fact, to any other progression of objects. He suggested that, in view of this, number theory is not the theory of specific objects, the numbers, but rather the theory of all progressions. This introduced eliminative structuralism to the contemporary philosophical literature. Benacerraf’s [1973] paper, ‘Mathematical truth’, surveyed the failings of the famous twentieth-century philosophies of mathematics. Those that offered a plausible epistemology came with an account of mathematical language that did not jibe with our account of the rest of language. Those that were able to integrate the language of mathematics with that of the rest of science offered no reasonable account of how we might acquire mathematical knowledge or how we might integrate an epistemology for mathematics with that of the rest of science. In oral comments when Benacerraf presented his paper,5 Oswaldo Chataubriand suggested that just as observation can lead us to knowledge of physical properties it might lead us to knowledge of mathematical properties. If so, we could meet Benacerraf’s challenge by taking reference to mathematical objects as a method for describing mathematical properties and basing our epistemology for mathematics on our ability to acquire knowledge of properties. My thought at the time was ‘Why not patterns instead of properties?’ Patterns can be extended. They can be copied; they can be tacked together. They have occurrences in other patterns, and they can be transformed into other patterns. They can be abstracted from every-day experience but also from patterns themselves. Treating mathematics as a science of patterns would explain why it takes the same approach to different problems; for we can see mathematicians as studying patterns as they appear in different settings. And it would explain why it takes multiple approaches to the same problems; for we can see mathematics as finding different patterns or different ways of describing a pattern in one and the same problem area. It would explain why there are multiple foundations for mathematics. Some patterns occur within each other, so that it can be arbitrary as to which we take as the containing or fundamental pattern and which we take as the derivative pattern. This idea would also explain why mathematical knowledge does not seem to be knowledge of single objects. I began to articulate these thoughts in [Resnik, 1975]. Here I wrote of patterns as if they were universals and likened our knowledge of mathematical patterns to our knowledge of musical and linguistic patterns. I thought that this provided hope for an epistemology of mathematics, since in each case our experience leads us to knowledge of something abstract, and that, at least in the cases of language and music, there is every reason to believe that the mechanisms in question are capable of scientific study. Looking over that early paper I see that, although I did mention Quine’s views on reference and ontological relativity and his holist epistemology of science, I was reluctant to make them my own. This reluctance soon passed, and as I developed my philosophical account of patterns I kept his doctrine of ontological relativity in the forefront of my mind. According to Quine, to specify the ontology of a theory is to interpret it in some domain that one has already taken as fixed. There is no saying what the ontology of a theory is absolutely but only relative to such an interpretation [Quine, 1969, p. 50]. This seemed to me to be the right way to respond to the multiple-reductions problem. When one pattern has multiple occurrences within another, as, for example, the letter type ‘p’ has in the word type ‘apple’, it does not make sense to ask, ‘Which of these occurrences is the real one?’ If we think of number theory and set theory as describing patterns, then the existence of multiple reductions of the former to the latter is a matter of the pattern or patterns described by number theory having multiple occurrences within those described by set theory. There is no fact of the matter as to which of these occurrences is the one to which number theory purports to refer. With Quine [1969, p. 55], I claimed something stronger: identity and reference are relative as well. There is no fact of the matter as to the identity of a position in a pattern except with respect to the pattern itself — indeed the occurrence of the pattern — to which it belongs. Given positions $$x$$ and $$y$$ of an occurrence of a pattern $$P$$, $$x = y$$ or $$x \neq y$$. But otherwise there is no fact of the matter as to whether two positions are the same. The way in which I put this in earlier writings was something to the effect that positions are like geometric points: they have no inner structure and have no identity or distinguishing features outside of a structure. Also on this view, reference is doubly relative: relative to taking an occurrence of a structure as fixed, and relative to a reference assignment. Moreover, there may be no fact of the matter as to the constant’s reference to positions in other occurrences of structures — even of congruent occurrences. This version of the relativity of reference allows individual constants to denote specific positions in fixed occurrences of structures without entailing that positions have an identity outside a structure. These ideas apply, of course, to other reductions and associated philosophical problems. On my view, there is no fact of the matter, for example, whether the real numbers are Dedekind cuts, Cauchy sequences, or sums of infinite series. So far we have been considering what I think of as ontological versions of structuralism. They either identify the ontology of mathematics with structural objects (positions, places, or structures) or deny that mathematics has any ontology at all. In my later papers6 I expressed reservations about attributing a specific ontology to mathematics, whether it is one of positions in patterns or something else. First, the claim that mathematics is really about so and so contravenes ontological relativity. Second, there are technical problems in working out the formal details of a theory of patterns of the sort I had put forth. Presumably a theory of patterns would be describing a pattern itself, one in which all patterns would occur. But then would it not have to occur in itself? And how could there be such a pattern when every pattern should be extendable. These are analogues to the familiar problems facing set theory. Then there was the worry that such an all-inclusive theory would create facts of the matter where I had denied them. So even in my 1981 paper I stated that I was not offering a foundation for mathematics or a reduction of its ontology. A few years later I read Quine’s essay, ‘Things and their place in theories’ [1981] in which he argued that in our overall theory of the world ‘objects ... serve merely as indices’ which ‘we may permute or supplant’ as we please so long as we preserve ‘the sentence-to-sentence structure’ [1981, p. 20]. About a decade after this in a short essay called, ‘Structure and nature’, Quine compared his structuralism with that of David Lewis’s Parts of Classes [1991]: ‘Structuralism for classes, hence, for all abstract objects is undeniably congenial. They are things that are known anyway by their structural role in cognitive discourse ... My own line is a yet more sweeping structuralism, applying to concrete and abstract objects indiscriminately’ [Quine, 2008, p. 402]. He goes on to say that one might wonder how to reconcile ‘this global structuralism’ with his realism or naturalism [2008, p. 405], and then answers that we turn to science for an account of what there is, and that the structural considerations he presented ‘belong not to ontology but to the methodology of ontology, and thus to epistemology’.7 His global structuralism is non-ontological and simply another formulation of ontological relativity. Quine argues for his global structuralism by observing that if we can specify a one-one mapping from the ontology of a theory onto another domain — he calls it a proxy function — then we can re-interpret the theory in the new domain while preserving the original theory’s connection with observation.8 In thinking of the objects of a theory as being nodes he is thinking that they have no properties that preclude applying a proxy-function transformation. Still in presenting these remarks about ontology he does not affirm a structural ontology. To me, a hallmark of structural objects is their incompleteness; that is, their having no identity or distinguishing features independently of the structure to which they belong. But Quine’s systems obey excluded middle; so for any $$x$$ and $$y$$ whatsoever, $$x$$ is or is not identical to $$y$$. For him incompleteness phenomena come up when reducing previously given things to his ontology, when for example, reducing real numbers to sets or electrons to space-time regions; but this is where his doctrines of ontological relativity and explication as elimination come into play. On his view, a reduction need only preserve the structure of the theory being reduced. The old ontology is not resurrected in the new one; rather it is eliminated. For Quine evidence is ultimately observational, a matter of how our sensory receptors are stimulated and whether they are stimulated as we expect. Our talk of external things, our very notion of things, is just a conceptual apparatus that helps us foresee and control the triggering of our sensory receptors in the light of previous triggering of our sensory receptors. The triggering, first and last, is all we have to go on. [Quine, 1981, p. 1]. The apparatus is a sentential structure connected to experience via observation sentences. His global structuralism depends upon the claim that one can preserve this sentential structure and its connections to observation while changing its associated ontology. One might object to Quine’s view and his argument for it on the grounds that it ignores other factors that fix the reference of our terms and ontology. Typically, these objections involve appealing to causal or environmental elements that determine, for example, that our word ‘automobile’ refers to automobiles and not to certain fusions of automobile parts. But these objections have less purchase when it comes to the mathematical version of ontological relativity. For, as Quine puts it [1969, p. 44], ‘Numbers ... are known only by their laws, the laws of arithmetic, so that any constructs obeying those laws — certain sets, for instance — are eligible in turn as explications of number. Sets in turn are known only by their laws, the laws of set theory’. As I was finishing my [1997] book, Mathematics as a Science of Patterns, I read Quine’s remarks on structuralism once again, and it occurred to me that I had not been clear in my own mind as to whether I had been presenting an ontological position or an epistemological one. Given my previous reservations concerning an ontological version of a theory of patterns, and given my intention to follow and explain Quine’s views on ontological relativity, I decided to opt for a Quinean reading of my view. I did this without giving as much thought as I should have to what this actually means. I want to address this now.9 To start, let us consider a passage I wrote in presenting my view in slogan form: The objects of mathematics, that is, the entities which our mathematical constants and quantifiers denote, are [themselves atoms,] structureless points or positions in structures. And as such, they have no identity or [distinguishing] features outside of a structure.10 I do not want to abandon the insights this passage expresses; however, in characterizing mathematical objects, it has too much of an ontological ring to it for representing my current view. I had intended the passage to express the main idea behind ontological relativity. Quine thinks of it in terms of sentences and their logical connections. Reference to objects helps generate these connections, and there is nothing more to them than this. Thus, as he says [1981, p. 20], ‘we may permute or supplant [them] as we please so long as we preserve the sentence-to-sentence structure’ of the sentences in question. I had been thinking in more visual terms: mathematical theories describe patterns; their objects are like positions in a diagram, and so long as their relational structure is preserved, they may be replaced at will. It does not matter whether we represent the pattern using dots, stars, small circles, strokes, or what have you.11 I had been thinking that if we think of mathematical objects as positions in patterns then ontological relativity would not only make sense, it would be required. But this leads one to think of mathematics as concerned with some kind of weird objects. So instead of expressing my view by putting the emphasis on objects I will put the emphasis on theories: Mathematics speaks of objects in order to describe or present structures; from the point of view of a mathematical theory, the denotations of its constants and quantifiers might as well be points or positions in a structure or structures; for the theory attributes to them no identifying features outside of the structure or structures in question. In stating this I do not mean to preclude work in the foundations of mathematics that offers explications of mathematical objects, not even one such as [Shapiro, 1997] that takes them to be places in structures. Such a theory will be subject to ontological relativity too: relative to taking it and its language at face value, mathematical objects are as it says they are, though there is no further perspective by which we can answer the question of whether they are really as it says. Much of my old view survives. Now instead of talking about positions in patterns we talk about theories and singular terms and quantifiers. Instead of saying that there is no fact as to whether the positions of a natural-number sequence are identical to a certain sets, we say that there is no fact as to which of the many interpretations of number theory in set theory is the correct one. This is just a consequence of ontological relativity without the explanation in terms of positions in patterns. On my view mathematics is like fiction in presenting incomplete descriptions of its objects, but that does not commit one to holding that mathematical objects are fictional. I remain a mathematical realist, although I am not going to give a defense of realism here. I want to separate structuralism from mathematical realism. Without committing oneself to realism about mathematical objects one can still hold that the business of mathematics is to study patterns and that talk of mathematical objects is a way of talking of positions in patterns. Talk of positions is a way of depicting how objects might be arranged. It is a way of converting talk of possibilities into talk of actualities. Another feature of my old view that survives is its postulational epistemology. For it is an account of how we come to posit mathematical objects and it never involved informational exchanges between mathematical objects and ourselves. Are numbers objects? I say they are, but not because there is a fact to their being these or those objects given to us in some independent context. Rather I take them to be objects for the same reason Frege did — simply because they fall within the range of the first-order variables of number theory. According to my structuralism, certain questions cannot be resolved independently of any theory, since they presuppose facts of matters where there are none. Here are important examples. What are the numbers? To what objects does number theory refer? Is the natural number 2 the real number 2? Is the number 2 Julius Caesar? Within a theory we can resolve some of these questions, though perhaps arbitrarily.12 In the remainder of this paper I will consider how the latest formulation of my view can respond to criticisms of structuralism, in general, and my view in particular. A long-standing criticism is based upon the assertion that mathematical objects as positions in structures have no features outside of those structures. But, it is claimed, they surely have structurally independent properties such as being abstract or being the subject of intense research. This objection does not touch the current formulation of my view, since it is no longer formulated in terms of objects. And it surely is not a fault of my view that mathematical theories do not address the question of whether numbers are abstract.13 Another criticism concerns patterns with structurally indistinguishable positions, such as, the complex numbers with $$i$$ and $$-i$$ or the points in a Euclidean plane. When directed towards the current version of my view, it amounts to the claim that any mathematical theory that posits several objects without providing a means for distinguishing them is thereby defective. To borrow from David Lewis, dare we tell this to the geometers and complex analysts?14 John Burgess has raised a number of criticisms of views like mine. He thinks it is wrong to think all mathematical objects are (or can be viewed or profitably treated as) points or positions in structures. In the following passage he summarizes his discussion of counter-examples to this view: A very large range of objects familiar from traditional and early modern mathematics and incipient late-modern mathematics can thus without much artificiality be regarded as equivalence types of one kind or another, or as closely related to such equivalence types; and when so regarded their introduction takes the form of connecting them with the items of which they are the equivalence types or to which they are otherwise related. Structuralism, by contrast, considers sorts of objects introduced as elements or points of a structure, and explained only in terms of their relationships to each other within that structure, any external relations being,derivative from these. [Burgess, 2010] Structuralism, at least as I have formulated it, only claims that when a theory quantifies over mathematical objects it treats them as if they are points in a structure. It does not make this claim with respect to symbols that are not replaceable by quantifiable variables. It may be useful to treat some mathematics as dealing with equivalence types. But then I would want to know whether the mathematics in question quantifies over these types. To see what I mean consider Burgess’s example of numbers being treated as answers to how-many questions. We can use the familiar $$NxFx$$ notation to represent them as ‘equivalence types’ governed by Hume’s principle   \[ NxFx =^* NyGy \text{ if and only if there is a 1-1 mapping from the}\,F\,\text{onto the}\,G. \] We do not have to think of the numerical abstracts $$NxFx$$ and the like as denoting objects, and can treat them as Quine treats his virtual classes. In this instance Hume’s principle serves as a contextual definition introducing complexes consisting of virtual numbers flanking the starred identity sign. The identity sign is starred to signal that the numerical abstracts are not (yet) singular terms, and that it does not (yet) signify identity between objects. Even so we can mimic operations with finite cardinal numbers and some of the Peano axioms. For example, in place of ‘every number has a successor’ we have $$\forall F\exists G(NxGx\ S\ NxFx)$$, where $$NxGx\ S\ NxFx$$ is defined in the familiar way. However, this approach does not make sense of $$PNxFx$$ for arbitrary predicates $$P$$, and will give us at most a limited form of induction. We do not even have the principle ‘$$z$$ is a number if and only if $$\exists z(z =^* NxFx)$$, since we have not given a sense to $$=^*$$ when flanked by a variable. To get to number theory we need to quantify over numbers, and, on my view, this is to treat them as positions in a structure. How these numbers relate to the virtual numbers will depend upon the details of the number theory and its language.15 In an argument that applies to all forms of structuralism, Burgess concludes structuralism fails in the case of set theory. He notes that there is no categorical description of iterative hierarchies, and thus no single structure which structuralists can claim as the home structure for sets. So they must think of set theory as concerned with the models of some specific axiom system, such as ZFC. From this he concludes there seems to be no room for the activity, important to many set theorists, of going back to an intuitive notion of set motivating the axioms in order to motivate more axioms to settle questions not settled by the existing axioms. Structuralism here ties set theory to a particular axiom system in a way that seems to block the road of inquiry. [2009, p. 26] But is this situation any different from what it has been since the beginning of axiomatic set theory? One investigates various axiom systems, sometimes using an intuitive understanding of sets, sometimes not. In the course of such investigations various structures may be partially and tentatively described. There is no reason why structuralists should frown on this. We would be the first to admit that both observation and intuition play an important role in mathematical theory building. We are only claiming that from the point of view of a mathematical theory its objects are like positions in a structure. Furthermore, I do not see why structuralists are required to hold that set theory is about a unique structure. It might be like group theory and concerned with a class of structures, as Zermelo [1930] suggested, or even with a class that is too variegated to be characterized by a compact definition. Burgess has another criticism of structuralism, but we need a bit of background to explain it. When codifying mathematics one often has a choice among alternative constructions and definitions. Working mathematicians do not care which choices are made; it is enough that the mathematics in question can be rigorously codified. When the codification concerns the identification of certain mathematical objects with others, as happens with the construction the various number systems, Burgess refers to the mathematical community’s attitude as indifference to identification, and maintains that this has been one of the main motivations for post-Benacerraf structuralism.16 Indifference to identification has led mathematicians to use the symbols for the objects in question as ‘permanent parameters’. For example, being indifferent to the constructions of real closed fields given by Dedekind, Weirerstrass, and Cantor, they in effect, say ‘Let $$\mathbb{R}$$ be one of them’.17 The problem with structuralism, he claims, is that it goes too far by drawing ontological conclusions from this phenomenon and generalizing it to all mathematics.18 According to Burgess, set theory is the exception to structuralism’s generalization, since the permanent-parameter hypothesis cannot apply to it. For not only does it provide the identifications to which mathematics is indifferent, but there is also no background theory which provides different universes of sets. So mathematicians are not in a position to say, ‘Let $$(V, {\in})$$ be one of them.’19 Of course, Burgess is correct to the extent that when one’s theory is taken as foundational then there will be no further theory providing alternative ontological options. However, we know that even in the case of ZF set theory there are alternative iterative hierarchies starting with different ur-elements. Some reconstructions of mathematics even take the natural numbers as given and form sets from these. There are also proposals to found set theory on category theory. Working mathematicians, in so far as they are aware of these alternatives, are probably indifferent between the various foundations for the set theory they actually use. For them ‘$$(V, \in)$$’ would be a permanent parameter.20 The lesson of history is that what is currently seen as foundational might be part of a future superstructure. The lesson of ontological relativity is that there is no fact as to which is correct. In any case, I would deny Burgess’s premise: there is more to structuralism than the permanent-parameter hypothesis. The hypothesis is at best an account of the logical and linguistic role of certain mathematical terms whereas structuralism makes wider claims about mathematical activity. Among other things it advances views about some of the aims of mathematics, and explains why mathematical theories speak of objects, why their descriptions are incomplete, and why certain methods are applied in very different branches of mathematics, and even why mathematicians are indifferent to certain identifications. Footnotes 1Avoiding mysteries motivated David Lewis’s structuralism. See his [1993], pp. 14–15. 2Also during this period G.H. Hardy wrote: ‘What is essential in mathematics is that its symbols should be capable of some interpretation; generally they are capable of many, and then, so far as mathematics is concerned, it does not matter which we adopt’ [1908, p. 15]. While the contrast between structure and content is not as clear in Hardy’s remark as it is in Hilbert’s and Poincaré’s, one might view it as version of structuralism or, as John Burgess (see below) might put it, structuralism enough. 3[Dedekind, 1901, pp. 33–34]. The monograph from which this passage is translated first appeared in 1888. For further discussion and interpretation of Dedekind see [Parsons, 2008, pp. 46–47]. 4[Hellman, 1989]. I used to think of Hellman as a negative ontological structuralist because he argued that much of his modal structural interpretation of mathematics can be seen as nominalist. But after he informed me that he thinks of his view as ‘quintessentially’ non-ontological, I looked again at his book and found him characterizing his view as the thesis that ‘mathematics is the free exploration of structural possibilities’ (p. 6). 5At a symposium on Mathematical Truth, sponsored jointly by the American Philosophical Association, Eastern Division, and the Association for Symbolic Logic, December 27, 1973. 6Especially [Resnik, 1981]. 7I have quoted from [Quine, 1981, p. 21] where I think the point is made more clearly. 8See [Quine, 1981, pp. 19–20], and [2008, pp. 404–405]. 9[Resnik, 1997, pp. 268–269]. Although in my book I used the term ‘epistemological’ in characterizing my position, I am using the term ‘non-ontological’ here to separate my view from that of Lisa Lehrer Dive [2003]. Since her ‘epistemic structuralist account of mathematical knowledge’ combines ‘Aristotelian structuralism with modal structuralism’, I would classify it as a form of eliminative ontological structuralism. I would count Marc Gasser [2015] and John Burgess [2009; 2010; 2015] as non-ontological structuralists; though Burgess’s structuralism is probably limited to those parts of mathematics, such as number theory and analysis, that have alternative foundations among which mathematicians are indifferent. 10[Resnik, 1997, p. 201]. The passage occurred first in [Resnik, 1981]. The words in brackets were added in the 1997 version. 11Some of the devices we use for depicting patterns can matter, however. Depending upon the type of transformation we admit, two triangles may count as the same pattern (because similar) or not (because not congruent). I call this ‘structural relativity’. See [1997, pp. 250–254]. 12My account of matters of fact in [1997, pp. 243–246] committed me to holding that classical logic does not apply to a certain portion of our language. Quine [1977, p. 283] takes a different line that saves bi-valence: ‘... there is no difference in matters of fact without a difference in the fulfillment of the physical-state predicates by space-time regions’ In the case of translation manuals the claim that there is no fact as to which is correct amounts to the claim that each is ‘compatible with the fulfillment of just the same elementary physical states by space-time regions’ [1977, p. 284]. Now that I have semantically ascended in stating my view Quine’s way is open to me. My claim that there are no facts as to which sets are numbers can be construed as the claim that there is no fact as to the correct interpretation/translation of number theory in set theory. 13The objection is discussed at length in [Gasser, 2015]. I do not think it applied to the earlier version of my view either. Think of number theory as dealing with a very simple type of pattern — omega sequences. Think of the metaphysics of mathematics as dealing with a more complex type of pattern. Suppose it is one in which omega sequences occur and that their members are somehow represented as abstract or concrete. In response to the objection, I would have denied that there is a fact of the matter as to whether positions in omega sequences that occur outside of the larger pattern are identical to their correspondents within it. And hence there is no fact of the matter as to whether the former are abstract or concrete. 14[Lewis, 1993, p. 15]. Jukka Keränen [2001] argued the objection. In responding, Stewart Shapiro [2008] introduced the permanent parameter hypothesis (see below) to explain mathematical discourse involving indistinguishable objects. See also [Pettigrew, 2008]. 15In [Resnik, 1997, pp. 229–232] I indicate how one might move from numbers treated as patterns of cardinalities to a pattern in which they are treated as positions in progressions. 16Burgess says Hardy’s Principle is operative here. See note 2. 17As Burgess notes, this idea was independently promulgated by Pettigrew [2008] and Shapiro [2008]. I find it earlier in passages from a 1992 essay of Quine’s, ‘We bandy our numbers without caring which classes we are bandying from among the wealth of alternatives’ [2008, p. 401] and ‘Arithmetical structuralism, as expressed in Ramsey sentences, depended upon there being classes in which the required arithmetical structures could be defined and realized, though we were freed of choosing among the alternatives’ [2008, p. 402]. 18See [2009] and [2015, p. 145]. On pp. 166–167 of [2015] he seems to moderate his opposition to a more extensive structuralism. 19Burgess writes: Mathematicians do consider different constructions of real numbers out of more basic entities and display indifference between them, but it cannot be said that they consider different constructions of sets out of more basic entities and display indifference between them. For there are no such constructions to be considered since there are no more basic entities than sets in contemporary mathematics. [2009, p. 34] 20 David Lewis even uses a Ramsey sentence to codify set theory, though he demurs at claiming that mathematicians are structuralists. Thanks to Geoffrey Hellman for reminding me that multiplicities of cumulative hierarchies are available, since any non-set can serve as an initial element of a cumulative hierarchy. REFERENCES Benacerraf Paul [ 1965]: ‘What numbers could not be’, reprinted in [Benacerraf and Putnam, 1983], pp. 272– 294. Benacerraf Paul [ 1973]: ‘Mathematical truth’, reprinted in [Benacerraf and Putnam, 1983], pp. 403– 420. Benacerraf P., and Putnam H. [ 1983]: Philosophy of Mathematics, Selected Readings . Cambridge University Press. Burgess John P. [ 2009]: ‘Putting structuralism in its place’, unpublished. Burgess John P. [ 2010]: ‘Parsons and the structuralist view’, forthcoming. Available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.404.7997. Accessed December 2017. Burgess John P. [ 2015]: Rigor and Structure . Oxford University Press. Dedekind Richard [ 1901]: ‘The nature and meaning of numbers’, in Beman W.W. trans., Essays on the Theory of Numbers . Chicago: Open Court. Dive Lisa Lehrer [ 2003]: An Epistemic Structuralist Account of Mathematical Knowledge . Ph.D. thesis, University of Sydney. Available at web.maths.unsw.edu.au/~jim/lisadivethesis.pdf. Ferreirós José [ 2007]: ‘The rise of pure mathematics as arithmetic with Gauss’, in The Shaping of Arithmetic after C.F. Gauss’s Disquisitiones Arithmeticae.  Catherine Goldstein, Norbert Schappacher, Joachim Schwermer, eds. Springer. Available at http://www.autistici.org/sitosovversivo/w-notes/notes/The_Shaping_of_Arithmetic_after_C.F.Gausss_Disquisitiones_Arithmeticae.pdf. Accessed December 2017. Frege G. [ 1980]: Philosophical and Mathematical Correspondence . Brian McGuinness, ed., Hans Kaal, trans. Oxford: Blackwell. Gasser Marc [ 2015]: ‘Structuralism and its ontology’, Ergo  2. http://quod.lib.umich.edu/e/ergo/12405314.0002.001/- -structuralism-and-its-ontology?rgn=main;view=fulltext. Acessed December 2017. Hardy G.H. [ 1908]: A Course in Pure Mathematics . Cambridge University Press. Hellman Geoffrey [ 1989]: Mathematics Without Numbers . Oxford: Clarendon Press. Keränen Jukka [ 2001]: ‘The identity problem for realist structuralism’, Philosophia Mathematica  ( 3) 9, 308– 330. Google Scholar CrossRef Search ADS   Lewis David [ 1991]: Parts of Classes . Oxford: Blackwell. Lewis David [ 1993]: ‘Mathematics is megethology’, Philosophia Mathematica  ( 3) 1, 3– 23. Google Scholar CrossRef Search ADS   Parsons Charles [ 2008]: Mathematical Thought and Its Objects . Cambridge University Press. Pettigrew Richard [ 2008]: ‘Platonism and Aristotelianism in mathematics’, Philosophia Mathematica  ( 3) 16, 310– 332. Google Scholar CrossRef Search ADS   Poincaré Henri [ 1913]: The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method . George Bruce Halsted, trans. New York: The Science Press. Quine W.V. [ 1969]: Ontological Relativity and Other Essays . New York: Columbia University Press. Quine W.V. [ 1977]: ‘Facts of the matter’ Reprinted in [Quine, 2008], pp. 271– 286. Quine W.V. [ 1981]: ‘Things and their place in theories’, in Theories and Things , pp. 1– 23. Cambridge, Mass.: Harvard University Press. Quine W.V. [ 1992]: ‘Structure and nature’. Reprinted in [Quine, 2008], pp. 401– 406. Quine W.V. [ 2008]: Confessions of a Confirmed Extensionalist and Other Essays . Dagfinn Føllesdal and Douglas B. Quine, eds. Cambridge, Mass.: Harvard University Press. Resnik Michael D. [ 1975]: ‘Mathematical knowledge and pattern cognition’, Canadian Journal of Philosophy  5, 25– 39. Google Scholar CrossRef Search ADS   Resnik Michael D. [ 1981]: ‘Mathematics as a science of patterns: Ontology and reference’, Noûs  15, 529– 550. Google Scholar CrossRef Search ADS   Resnik Michael D. [ 1997]: Mathematics as a Science of Patterns . Oxford: Clarendon Press. Shapiro Stewart, [ 1997]: Philosophy of Mathematics, Structure and Ontology . New York: Oxford. Shapiro Stewart, [ 2008]: ‘Identity, indiscernibility, and ante rem structuralism: The tale of $$i$$ and $$-i$$’, Philosophia Mathematica  ( 3) 16, 285– 309. Google Scholar CrossRef Search ADS   Zermelo Ernst [ 1930]: ‘On boundary numbers and domains of sets’, Hallett M. trans., in Ewald W. ed., From Kant to Hilbert: A Source Journal in Mathematics . Vol. 2. pp. 1208– 1233. Oxford University Press. © The Author [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Philosophia Mathematica Oxford University Press

Non-ontological Structuralism

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ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects (eliminative structuralism) or they maintain that mathematical objects are structures or positions in them (sui generis structuralism). Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view. Structuralism in the philosophy of mathematics has been largely viewed as an ontological doctrine concerning the nature of mathematical objects. Positive versions interpret mathematical objects as places or positions in structures, while the negative versions hold that mathematics has no objects of its own and studies structures abstracted from non-mathematical realms. By contrast non-ontological versions of structuralism leave open questions concerning the existence and nature of mathematical objects. My own view began as ontological in the positive sense, but as it developed I began to doubt the prospects for an ontology in which mathematical objects are positions in structures, and eventually opted for a non-ontological reading of my view. I would like to explain and clarify this further. To this end, I will begin with some familiar history and move to a bit of autobiography. Distinguishing between ontological and non-ontological structuralism makes it harder to classify certain historical anticipations of structuralism. It is usual to interpret Gauss, Hilbert, and Poincaré as anticipating negative versions of ontological structuralism. Thus early in the nineteenth century, when there were still reservations concerning the complex numbers, Gauss wrote: The mathematician abstracts entirely from the quality of the objects and the content of their relations; he just occupies himself with counting and comparing their relations to each other. (Quoted in [Ferreirós, 2007, p. 254]) Around the beginning of the twentieth century Hilbert wrote to Frege along similar lines: ‘It is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes’ [Frege, 1980, p. 40]. A few years later Poincaré echoed this structuralist theme: Mathematicians study not objects, but relations between objects; the replacement of these objects by others is therefore indifferent to them, provided the relations do not change. The matter is for them unimportant, the form alone interests them. [Poincaré, 1913, p. 42] One motivation for structuralism is to avoid having to explain the nature of prima facie mysterious mathematical objects.1 We may assume that these writers were so motivated. But are they merely saying that mathematics is not concerned with the nature of the objects it deals with but only with their structure? Or are they also claiming that there are no proper mathematical objects? If the former, they anticipated non-ontological structuralism; if the latter, they anticipated negative forms of ontological structuralism.2 We do find ontological structuralism — a positive version — in Gauss’s student Dedekind. Writing in the late nineteenth century, he spoke of the human mind creating the system of natural numbers through abstracting its structure. The laws holding here ‘therefore are always the same in all ordered simply infinite systems’.3 Negative ontological structuralism has been dogged by the problem of uninstantiated structures. If a mathematical theory purports to describe a structure which no non-mathematical objects exhibit, then, according to these structuralists, the theory has no models, and its sentences are vacuously true. Hilbert came to appreciate this, and famously claimed that consistency would suffice for non-vacuity. Geoffrey Hellman, a contemporary structuralist, developed a modal version of structuralism to address the problem. Put roughly, so long as it is possible that the structure in question is exhibited, vacuity does not arise.4 During the first half of the twentieth century philosophy of mathematics turned away from structuralist ideas to focus on issues surrounding logicism, intuitionism and formalism. Just about the time these movements began to lose their luster Quine’s writings set the stage for the emergence of contemporary structuralism. His critique of the a priori and his naturalized epistemology turned our attention to the genesis of mathematical knowledge. This was the context for Paul Benacerraf’s instructive and provocative papers. His [1965] paper, ‘What numbers could not be’, observed that there is no compelling mathematical reason to prefer one reduction of the numbers to sets to another or, in fact, to any other progression of objects. He suggested that, in view of this, number theory is not the theory of specific objects, the numbers, but rather the theory of all progressions. This introduced eliminative structuralism to the contemporary philosophical literature. Benacerraf’s [1973] paper, ‘Mathematical truth’, surveyed the failings of the famous twentieth-century philosophies of mathematics. Those that offered a plausible epistemology came with an account of mathematical language that did not jibe with our account of the rest of language. Those that were able to integrate the language of mathematics with that of the rest of science offered no reasonable account of how we might acquire mathematical knowledge or how we might integrate an epistemology for mathematics with that of the rest of science. In oral comments when Benacerraf presented his paper,5 Oswaldo Chataubriand suggested that just as observation can lead us to knowledge of physical properties it might lead us to knowledge of mathematical properties. If so, we could meet Benacerraf’s challenge by taking reference to mathematical objects as a method for describing mathematical properties and basing our epistemology for mathematics on our ability to acquire knowledge of properties. My thought at the time was ‘Why not patterns instead of properties?’ Patterns can be extended. They can be copied; they can be tacked together. They have occurrences in other patterns, and they can be transformed into other patterns. They can be abstracted from every-day experience but also from patterns themselves. Treating mathematics as a science of patterns would explain why it takes the same approach to different problems; for we can see mathematicians as studying patterns as they appear in different settings. And it would explain why it takes multiple approaches to the same problems; for we can see mathematics as finding different patterns or different ways of describing a pattern in one and the same problem area. It would explain why there are multiple foundations for mathematics. Some patterns occur within each other, so that it can be arbitrary as to which we take as the containing or fundamental pattern and which we take as the derivative pattern. This idea would also explain why mathematical knowledge does not seem to be knowledge of single objects. I began to articulate these thoughts in [Resnik, 1975]. Here I wrote of patterns as if they were universals and likened our knowledge of mathematical patterns to our knowledge of musical and linguistic patterns. I thought that this provided hope for an epistemology of mathematics, since in each case our experience leads us to knowledge of something abstract, and that, at least in the cases of language and music, there is every reason to believe that the mechanisms in question are capable of scientific study. Looking over that early paper I see that, although I did mention Quine’s views on reference and ontological relativity and his holist epistemology of science, I was reluctant to make them my own. This reluctance soon passed, and as I developed my philosophical account of patterns I kept his doctrine of ontological relativity in the forefront of my mind. According to Quine, to specify the ontology of a theory is to interpret it in some domain that one has already taken as fixed. There is no saying what the ontology of a theory is absolutely but only relative to such an interpretation [Quine, 1969, p. 50]. This seemed to me to be the right way to respond to the multiple-reductions problem. When one pattern has multiple occurrences within another, as, for example, the letter type ‘p’ has in the word type ‘apple’, it does not make sense to ask, ‘Which of these occurrences is the real one?’ If we think of number theory and set theory as describing patterns, then the existence of multiple reductions of the former to the latter is a matter of the pattern or patterns described by number theory having multiple occurrences within those described by set theory. There is no fact of the matter as to which of these occurrences is the one to which number theory purports to refer. With Quine [1969, p. 55], I claimed something stronger: identity and reference are relative as well. There is no fact of the matter as to the identity of a position in a pattern except with respect to the pattern itself — indeed the occurrence of the pattern — to which it belongs. Given positions $$x$$ and $$y$$ of an occurrence of a pattern $$P$$, $$x = y$$ or $$x \neq y$$. But otherwise there is no fact of the matter as to whether two positions are the same. The way in which I put this in earlier writings was something to the effect that positions are like geometric points: they have no inner structure and have no identity or distinguishing features outside of a structure. Also on this view, reference is doubly relative: relative to taking an occurrence of a structure as fixed, and relative to a reference assignment. Moreover, there may be no fact of the matter as to the constant’s reference to positions in other occurrences of structures — even of congruent occurrences. This version of the relativity of reference allows individual constants to denote specific positions in fixed occurrences of structures without entailing that positions have an identity outside a structure. These ideas apply, of course, to other reductions and associated philosophical problems. On my view, there is no fact of the matter, for example, whether the real numbers are Dedekind cuts, Cauchy sequences, or sums of infinite series. So far we have been considering what I think of as ontological versions of structuralism. They either identify the ontology of mathematics with structural objects (positions, places, or structures) or deny that mathematics has any ontology at all. In my later papers6 I expressed reservations about attributing a specific ontology to mathematics, whether it is one of positions in patterns or something else. First, the claim that mathematics is really about so and so contravenes ontological relativity. Second, there are technical problems in working out the formal details of a theory of patterns of the sort I had put forth. Presumably a theory of patterns would be describing a pattern itself, one in which all patterns would occur. But then would it not have to occur in itself? And how could there be such a pattern when every pattern should be extendable. These are analogues to the familiar problems facing set theory. Then there was the worry that such an all-inclusive theory would create facts of the matter where I had denied them. So even in my 1981 paper I stated that I was not offering a foundation for mathematics or a reduction of its ontology. A few years later I read Quine’s essay, ‘Things and their place in theories’ [1981] in which he argued that in our overall theory of the world ‘objects ... serve merely as indices’ which ‘we may permute or supplant’ as we please so long as we preserve ‘the sentence-to-sentence structure’ [1981, p. 20]. About a decade after this in a short essay called, ‘Structure and nature’, Quine compared his structuralism with that of David Lewis’s Parts of Classes [1991]: ‘Structuralism for classes, hence, for all abstract objects is undeniably congenial. They are things that are known anyway by their structural role in cognitive discourse ... My own line is a yet more sweeping structuralism, applying to concrete and abstract objects indiscriminately’ [Quine, 2008, p. 402]. He goes on to say that one might wonder how to reconcile ‘this global structuralism’ with his realism or naturalism [2008, p. 405], and then answers that we turn to science for an account of what there is, and that the structural considerations he presented ‘belong not to ontology but to the methodology of ontology, and thus to epistemology’.7 His global structuralism is non-ontological and simply another formulation of ontological relativity. Quine argues for his global structuralism by observing that if we can specify a one-one mapping from the ontology of a theory onto another domain — he calls it a proxy function — then we can re-interpret the theory in the new domain while preserving the original theory’s connection with observation.8 In thinking of the objects of a theory as being nodes he is thinking that they have no properties that preclude applying a proxy-function transformation. Still in presenting these remarks about ontology he does not affirm a structural ontology. To me, a hallmark of structural objects is their incompleteness; that is, their having no identity or distinguishing features independently of the structure to which they belong. But Quine’s systems obey excluded middle; so for any $$x$$ and $$y$$ whatsoever, $$x$$ is or is not identical to $$y$$. For him incompleteness phenomena come up when reducing previously given things to his ontology, when for example, reducing real numbers to sets or electrons to space-time regions; but this is where his doctrines of ontological relativity and explication as elimination come into play. On his view, a reduction need only preserve the structure of the theory being reduced. The old ontology is not resurrected in the new one; rather it is eliminated. For Quine evidence is ultimately observational, a matter of how our sensory receptors are stimulated and whether they are stimulated as we expect. Our talk of external things, our very notion of things, is just a conceptual apparatus that helps us foresee and control the triggering of our sensory receptors in the light of previous triggering of our sensory receptors. The triggering, first and last, is all we have to go on. [Quine, 1981, p. 1]. The apparatus is a sentential structure connected to experience via observation sentences. His global structuralism depends upon the claim that one can preserve this sentential structure and its connections to observation while changing its associated ontology. One might object to Quine’s view and his argument for it on the grounds that it ignores other factors that fix the reference of our terms and ontology. Typically, these objections involve appealing to causal or environmental elements that determine, for example, that our word ‘automobile’ refers to automobiles and not to certain fusions of automobile parts. But these objections have less purchase when it comes to the mathematical version of ontological relativity. For, as Quine puts it [1969, p. 44], ‘Numbers ... are known only by their laws, the laws of arithmetic, so that any constructs obeying those laws — certain sets, for instance — are eligible in turn as explications of number. Sets in turn are known only by their laws, the laws of set theory’. As I was finishing my [1997] book, Mathematics as a Science of Patterns, I read Quine’s remarks on structuralism once again, and it occurred to me that I had not been clear in my own mind as to whether I had been presenting an ontological position or an epistemological one. Given my previous reservations concerning an ontological version of a theory of patterns, and given my intention to follow and explain Quine’s views on ontological relativity, I decided to opt for a Quinean reading of my view. I did this without giving as much thought as I should have to what this actually means. I want to address this now.9 To start, let us consider a passage I wrote in presenting my view in slogan form: The objects of mathematics, that is, the entities which our mathematical constants and quantifiers denote, are [themselves atoms,] structureless points or positions in structures. And as such, they have no identity or [distinguishing] features outside of a structure.10 I do not want to abandon the insights this passage expresses; however, in characterizing mathematical objects, it has too much of an ontological ring to it for representing my current view. I had intended the passage to express the main idea behind ontological relativity. Quine thinks of it in terms of sentences and their logical connections. Reference to objects helps generate these connections, and there is nothing more to them than this. Thus, as he says [1981, p. 20], ‘we may permute or supplant [them] as we please so long as we preserve the sentence-to-sentence structure’ of the sentences in question. I had been thinking in more visual terms: mathematical theories describe patterns; their objects are like positions in a diagram, and so long as their relational structure is preserved, they may be replaced at will. It does not matter whether we represent the pattern using dots, stars, small circles, strokes, or what have you.11 I had been thinking that if we think of mathematical objects as positions in patterns then ontological relativity would not only make sense, it would be required. But this leads one to think of mathematics as concerned with some kind of weird objects. So instead of expressing my view by putting the emphasis on objects I will put the emphasis on theories: Mathematics speaks of objects in order to describe or present structures; from the point of view of a mathematical theory, the denotations of its constants and quantifiers might as well be points or positions in a structure or structures; for the theory attributes to them no identifying features outside of the structure or structures in question. In stating this I do not mean to preclude work in the foundations of mathematics that offers explications of mathematical objects, not even one such as [Shapiro, 1997] that takes them to be places in structures. Such a theory will be subject to ontological relativity too: relative to taking it and its language at face value, mathematical objects are as it says they are, though there is no further perspective by which we can answer the question of whether they are really as it says. Much of my old view survives. Now instead of talking about positions in patterns we talk about theories and singular terms and quantifiers. Instead of saying that there is no fact as to whether the positions of a natural-number sequence are identical to a certain sets, we say that there is no fact as to which of the many interpretations of number theory in set theory is the correct one. This is just a consequence of ontological relativity without the explanation in terms of positions in patterns. On my view mathematics is like fiction in presenting incomplete descriptions of its objects, but that does not commit one to holding that mathematical objects are fictional. I remain a mathematical realist, although I am not going to give a defense of realism here. I want to separate structuralism from mathematical realism. Without committing oneself to realism about mathematical objects one can still hold that the business of mathematics is to study patterns and that talk of mathematical objects is a way of talking of positions in patterns. Talk of positions is a way of depicting how objects might be arranged. It is a way of converting talk of possibilities into talk of actualities. Another feature of my old view that survives is its postulational epistemology. For it is an account of how we come to posit mathematical objects and it never involved informational exchanges between mathematical objects and ourselves. Are numbers objects? I say they are, but not because there is a fact to their being these or those objects given to us in some independent context. Rather I take them to be objects for the same reason Frege did — simply because they fall within the range of the first-order variables of number theory. According to my structuralism, certain questions cannot be resolved independently of any theory, since they presuppose facts of matters where there are none. Here are important examples. What are the numbers? To what objects does number theory refer? Is the natural number 2 the real number 2? Is the number 2 Julius Caesar? Within a theory we can resolve some of these questions, though perhaps arbitrarily.12 In the remainder of this paper I will consider how the latest formulation of my view can respond to criticisms of structuralism, in general, and my view in particular. A long-standing criticism is based upon the assertion that mathematical objects as positions in structures have no features outside of those structures. But, it is claimed, they surely have structurally independent properties such as being abstract or being the subject of intense research. This objection does not touch the current formulation of my view, since it is no longer formulated in terms of objects. And it surely is not a fault of my view that mathematical theories do not address the question of whether numbers are abstract.13 Another criticism concerns patterns with structurally indistinguishable positions, such as, the complex numbers with $$i$$ and $$-i$$ or the points in a Euclidean plane. When directed towards the current version of my view, it amounts to the claim that any mathematical theory that posits several objects without providing a means for distinguishing them is thereby defective. To borrow from David Lewis, dare we tell this to the geometers and complex analysts?14 John Burgess has raised a number of criticisms of views like mine. He thinks it is wrong to think all mathematical objects are (or can be viewed or profitably treated as) points or positions in structures. In the following passage he summarizes his discussion of counter-examples to this view: A very large range of objects familiar from traditional and early modern mathematics and incipient late-modern mathematics can thus without much artificiality be regarded as equivalence types of one kind or another, or as closely related to such equivalence types; and when so regarded their introduction takes the form of connecting them with the items of which they are the equivalence types or to which they are otherwise related. Structuralism, by contrast, considers sorts of objects introduced as elements or points of a structure, and explained only in terms of their relationships to each other within that structure, any external relations being,derivative from these. [Burgess, 2010] Structuralism, at least as I have formulated it, only claims that when a theory quantifies over mathematical objects it treats them as if they are points in a structure. It does not make this claim with respect to symbols that are not replaceable by quantifiable variables. It may be useful to treat some mathematics as dealing with equivalence types. But then I would want to know whether the mathematics in question quantifies over these types. To see what I mean consider Burgess’s example of numbers being treated as answers to how-many questions. We can use the familiar $$NxFx$$ notation to represent them as ‘equivalence types’ governed by Hume’s principle   \[ NxFx =^* NyGy \text{ if and only if there is a 1-1 mapping from the}\,F\,\text{onto the}\,G. \] We do not have to think of the numerical abstracts $$NxFx$$ and the like as denoting objects, and can treat them as Quine treats his virtual classes. In this instance Hume’s principle serves as a contextual definition introducing complexes consisting of virtual numbers flanking the starred identity sign. The identity sign is starred to signal that the numerical abstracts are not (yet) singular terms, and that it does not (yet) signify identity between objects. Even so we can mimic operations with finite cardinal numbers and some of the Peano axioms. For example, in place of ‘every number has a successor’ we have $$\forall F\exists G(NxGx\ S\ NxFx)$$, where $$NxGx\ S\ NxFx$$ is defined in the familiar way. However, this approach does not make sense of $$PNxFx$$ for arbitrary predicates $$P$$, and will give us at most a limited form of induction. We do not even have the principle ‘$$z$$ is a number if and only if $$\exists z(z =^* NxFx)$$, since we have not given a sense to $$=^*$$ when flanked by a variable. To get to number theory we need to quantify over numbers, and, on my view, this is to treat them as positions in a structure. How these numbers relate to the virtual numbers will depend upon the details of the number theory and its language.15 In an argument that applies to all forms of structuralism, Burgess concludes structuralism fails in the case of set theory. He notes that there is no categorical description of iterative hierarchies, and thus no single structure which structuralists can claim as the home structure for sets. So they must think of set theory as concerned with the models of some specific axiom system, such as ZFC. From this he concludes there seems to be no room for the activity, important to many set theorists, of going back to an intuitive notion of set motivating the axioms in order to motivate more axioms to settle questions not settled by the existing axioms. Structuralism here ties set theory to a particular axiom system in a way that seems to block the road of inquiry. [2009, p. 26] But is this situation any different from what it has been since the beginning of axiomatic set theory? One investigates various axiom systems, sometimes using an intuitive understanding of sets, sometimes not. In the course of such investigations various structures may be partially and tentatively described. There is no reason why structuralists should frown on this. We would be the first to admit that both observation and intuition play an important role in mathematical theory building. We are only claiming that from the point of view of a mathematical theory its objects are like positions in a structure. Furthermore, I do not see why structuralists are required to hold that set theory is about a unique structure. It might be like group theory and concerned with a class of structures, as Zermelo [1930] suggested, or even with a class that is too variegated to be characterized by a compact definition. Burgess has another criticism of structuralism, but we need a bit of background to explain it. When codifying mathematics one often has a choice among alternative constructions and definitions. Working mathematicians do not care which choices are made; it is enough that the mathematics in question can be rigorously codified. When the codification concerns the identification of certain mathematical objects with others, as happens with the construction the various number systems, Burgess refers to the mathematical community’s attitude as indifference to identification, and maintains that this has been one of the main motivations for post-Benacerraf structuralism.16 Indifference to identification has led mathematicians to use the symbols for the objects in question as ‘permanent parameters’. For example, being indifferent to the constructions of real closed fields given by Dedekind, Weirerstrass, and Cantor, they in effect, say ‘Let $$\mathbb{R}$$ be one of them’.17 The problem with structuralism, he claims, is that it goes too far by drawing ontological conclusions from this phenomenon and generalizing it to all mathematics.18 According to Burgess, set theory is the exception to structuralism’s generalization, since the permanent-parameter hypothesis cannot apply to it. For not only does it provide the identifications to which mathematics is indifferent, but there is also no background theory which provides different universes of sets. So mathematicians are not in a position to say, ‘Let $$(V, {\in})$$ be one of them.’19 Of course, Burgess is correct to the extent that when one’s theory is taken as foundational then there will be no further theory providing alternative ontological options. However, we know that even in the case of ZF set theory there are alternative iterative hierarchies starting with different ur-elements. Some reconstructions of mathematics even take the natural numbers as given and form sets from these. There are also proposals to found set theory on category theory. Working mathematicians, in so far as they are aware of these alternatives, are probably indifferent between the various foundations for the set theory they actually use. For them ‘$$(V, \in)$$’ would be a permanent parameter.20 The lesson of history is that what is currently seen as foundational might be part of a future superstructure. The lesson of ontological relativity is that there is no fact as to which is correct. In any case, I would deny Burgess’s premise: there is more to structuralism than the permanent-parameter hypothesis. The hypothesis is at best an account of the logical and linguistic role of certain mathematical terms whereas structuralism makes wider claims about mathematical activity. Among other things it advances views about some of the aims of mathematics, and explains why mathematical theories speak of objects, why their descriptions are incomplete, and why certain methods are applied in very different branches of mathematics, and even why mathematicians are indifferent to certain identifications. Footnotes 1Avoiding mysteries motivated David Lewis’s structuralism. See his [1993], pp. 14–15. 2Also during this period G.H. Hardy wrote: ‘What is essential in mathematics is that its symbols should be capable of some interpretation; generally they are capable of many, and then, so far as mathematics is concerned, it does not matter which we adopt’ [1908, p. 15]. While the contrast between structure and content is not as clear in Hardy’s remark as it is in Hilbert’s and Poincaré’s, one might view it as version of structuralism or, as John Burgess (see below) might put it, structuralism enough. 3[Dedekind, 1901, pp. 33–34]. The monograph from which this passage is translated first appeared in 1888. For further discussion and interpretation of Dedekind see [Parsons, 2008, pp. 46–47]. 4[Hellman, 1989]. I used to think of Hellman as a negative ontological structuralist because he argued that much of his modal structural interpretation of mathematics can be seen as nominalist. But after he informed me that he thinks of his view as ‘quintessentially’ non-ontological, I looked again at his book and found him characterizing his view as the thesis that ‘mathematics is the free exploration of structural possibilities’ (p. 6). 5At a symposium on Mathematical Truth, sponsored jointly by the American Philosophical Association, Eastern Division, and the Association for Symbolic Logic, December 27, 1973. 6Especially [Resnik, 1981]. 7I have quoted from [Quine, 1981, p. 21] where I think the point is made more clearly. 8See [Quine, 1981, pp. 19–20], and [2008, pp. 404–405]. 9[Resnik, 1997, pp. 268–269]. Although in my book I used the term ‘epistemological’ in characterizing my position, I am using the term ‘non-ontological’ here to separate my view from that of Lisa Lehrer Dive [2003]. Since her ‘epistemic structuralist account of mathematical knowledge’ combines ‘Aristotelian structuralism with modal structuralism’, I would classify it as a form of eliminative ontological structuralism. I would count Marc Gasser [2015] and John Burgess [2009; 2010; 2015] as non-ontological structuralists; though Burgess’s structuralism is probably limited to those parts of mathematics, such as number theory and analysis, that have alternative foundations among which mathematicians are indifferent. 10[Resnik, 1997, p. 201]. The passage occurred first in [Resnik, 1981]. The words in brackets were added in the 1997 version. 11Some of the devices we use for depicting patterns can matter, however. Depending upon the type of transformation we admit, two triangles may count as the same pattern (because similar) or not (because not congruent). I call this ‘structural relativity’. See [1997, pp. 250–254]. 12My account of matters of fact in [1997, pp. 243–246] committed me to holding that classical logic does not apply to a certain portion of our language. Quine [1977, p. 283] takes a different line that saves bi-valence: ‘... there is no difference in matters of fact without a difference in the fulfillment of the physical-state predicates by space-time regions’ In the case of translation manuals the claim that there is no fact as to which is correct amounts to the claim that each is ‘compatible with the fulfillment of just the same elementary physical states by space-time regions’ [1977, p. 284]. Now that I have semantically ascended in stating my view Quine’s way is open to me. My claim that there are no facts as to which sets are numbers can be construed as the claim that there is no fact as to the correct interpretation/translation of number theory in set theory. 13The objection is discussed at length in [Gasser, 2015]. I do not think it applied to the earlier version of my view either. Think of number theory as dealing with a very simple type of pattern — omega sequences. Think of the metaphysics of mathematics as dealing with a more complex type of pattern. Suppose it is one in which omega sequences occur and that their members are somehow represented as abstract or concrete. In response to the objection, I would have denied that there is a fact of the matter as to whether positions in omega sequences that occur outside of the larger pattern are identical to their correspondents within it. And hence there is no fact of the matter as to whether the former are abstract or concrete. 14[Lewis, 1993, p. 15]. Jukka Keränen [2001] argued the objection. In responding, Stewart Shapiro [2008] introduced the permanent parameter hypothesis (see below) to explain mathematical discourse involving indistinguishable objects. See also [Pettigrew, 2008]. 15In [Resnik, 1997, pp. 229–232] I indicate how one might move from numbers treated as patterns of cardinalities to a pattern in which they are treated as positions in progressions. 16Burgess says Hardy’s Principle is operative here. See note 2. 17As Burgess notes, this idea was independently promulgated by Pettigrew [2008] and Shapiro [2008]. I find it earlier in passages from a 1992 essay of Quine’s, ‘We bandy our numbers without caring which classes we are bandying from among the wealth of alternatives’ [2008, p. 401] and ‘Arithmetical structuralism, as expressed in Ramsey sentences, depended upon there being classes in which the required arithmetical structures could be defined and realized, though we were freed of choosing among the alternatives’ [2008, p. 402]. 18See [2009] and [2015, p. 145]. On pp. 166–167 of [2015] he seems to moderate his opposition to a more extensive structuralism. 19Burgess writes: Mathematicians do consider different constructions of real numbers out of more basic entities and display indifference between them, but it cannot be said that they consider different constructions of sets out of more basic entities and display indifference between them. For there are no such constructions to be considered since there are no more basic entities than sets in contemporary mathematics. [2009, p. 34] 20 David Lewis even uses a Ramsey sentence to codify set theory, though he demurs at claiming that mathematicians are structuralists. Thanks to Geoffrey Hellman for reminding me that multiplicities of cumulative hierarchies are available, since any non-set can serve as an initial element of a cumulative hierarchy. REFERENCES Benacerraf Paul [ 1965]: ‘What numbers could not be’, reprinted in [Benacerraf and Putnam, 1983], pp. 272– 294. Benacerraf Paul [ 1973]: ‘Mathematical truth’, reprinted in [Benacerraf and Putnam, 1983], pp. 403– 420. Benacerraf P., and Putnam H. [ 1983]: Philosophy of Mathematics, Selected Readings . Cambridge University Press. Burgess John P. [ 2009]: ‘Putting structuralism in its place’, unpublished. Burgess John P. [ 2010]: ‘Parsons and the structuralist view’, forthcoming. Available at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.404.7997. Accessed December 2017. Burgess John P. [ 2015]: Rigor and Structure . Oxford University Press. Dedekind Richard [ 1901]: ‘The nature and meaning of numbers’, in Beman W.W. trans., Essays on the Theory of Numbers . Chicago: Open Court. Dive Lisa Lehrer [ 2003]: An Epistemic Structuralist Account of Mathematical Knowledge . Ph.D. thesis, University of Sydney. Available at web.maths.unsw.edu.au/~jim/lisadivethesis.pdf. Ferreirós José [ 2007]: ‘The rise of pure mathematics as arithmetic with Gauss’, in The Shaping of Arithmetic after C.F. Gauss’s Disquisitiones Arithmeticae.  Catherine Goldstein, Norbert Schappacher, Joachim Schwermer, eds. Springer. Available at http://www.autistici.org/sitosovversivo/w-notes/notes/The_Shaping_of_Arithmetic_after_C.F.Gausss_Disquisitiones_Arithmeticae.pdf. Accessed December 2017. Frege G. [ 1980]: Philosophical and Mathematical Correspondence . Brian McGuinness, ed., Hans Kaal, trans. Oxford: Blackwell. Gasser Marc [ 2015]: ‘Structuralism and its ontology’, Ergo  2. http://quod.lib.umich.edu/e/ergo/12405314.0002.001/- -structuralism-and-its-ontology?rgn=main;view=fulltext. Acessed December 2017. Hardy G.H. [ 1908]: A Course in Pure Mathematics . Cambridge University Press. Hellman Geoffrey [ 1989]: Mathematics Without Numbers . Oxford: Clarendon Press. Keränen Jukka [ 2001]: ‘The identity problem for realist structuralism’, Philosophia Mathematica  ( 3) 9, 308– 330. Google Scholar CrossRef Search ADS   Lewis David [ 1991]: Parts of Classes . Oxford: Blackwell. Lewis David [ 1993]: ‘Mathematics is megethology’, Philosophia Mathematica  ( 3) 1, 3– 23. Google Scholar CrossRef Search ADS   Parsons Charles [ 2008]: Mathematical Thought and Its Objects . Cambridge University Press. Pettigrew Richard [ 2008]: ‘Platonism and Aristotelianism in mathematics’, Philosophia Mathematica  ( 3) 16, 310– 332. Google Scholar CrossRef Search ADS   Poincaré Henri [ 1913]: The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method . George Bruce Halsted, trans. New York: The Science Press. Quine W.V. [ 1969]: Ontological Relativity and Other Essays . New York: Columbia University Press. Quine W.V. [ 1977]: ‘Facts of the matter’ Reprinted in [Quine, 2008], pp. 271– 286. Quine W.V. [ 1981]: ‘Things and their place in theories’, in Theories and Things , pp. 1– 23. Cambridge, Mass.: Harvard University Press. Quine W.V. [ 1992]: ‘Structure and nature’. Reprinted in [Quine, 2008], pp. 401– 406. Quine W.V. [ 2008]: Confessions of a Confirmed Extensionalist and Other Essays . Dagfinn Føllesdal and Douglas B. Quine, eds. Cambridge, Mass.: Harvard University Press. Resnik Michael D. [ 1975]: ‘Mathematical knowledge and pattern cognition’, Canadian Journal of Philosophy  5, 25– 39. Google Scholar CrossRef Search ADS   Resnik Michael D. [ 1981]: ‘Mathematics as a science of patterns: Ontology and reference’, Noûs  15, 529– 550. Google Scholar CrossRef Search ADS   Resnik Michael D. [ 1997]: Mathematics as a Science of Patterns . Oxford: Clarendon Press. Shapiro Stewart, [ 1997]: Philosophy of Mathematics, Structure and Ontology . New York: Oxford. Shapiro Stewart, [ 2008]: ‘Identity, indiscernibility, and ante rem structuralism: The tale of $$i$$ and $$-i$$’, Philosophia Mathematica  ( 3) 16, 285– 309. Google Scholar CrossRef Search ADS   Zermelo Ernst [ 1930]: ‘On boundary numbers and domains of sets’, Hallett M. trans., in Ewald W. ed., From Kant to Hilbert: A Source Journal in Mathematics . Vol. 2. pp. 1208– 1233. Oxford University Press. © The Author [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

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