Mimetic Representation and Abstract Objects 1 Michaela Markham McSweeney

Mimetic Representation and Abstract Objects 1 Michaela Markham McSweeney ABSTRACT: I argue for radical negative and positive theses about certain kinds of abstract objects, such as the equilateral triangle. The negative thesis says that the equilateral triangle is not triangular, nor trilateral; nor does it have an abstract analogue of triangularity. The positive thesis says that the equilateral triangle does, however, have an internal nature, and that that nature is purely structural. I examine the relationships between different representations of the equilateral triangle, and argue that we can learn things about the nature of the triangle from those relationships. Introduction Many philosophers have given accounts of abstract objects. These accounts are very often negative they are accounts of what properties abstract objects lack, or of what they are missing that differentiates them from concrete objects. 2 Few philosophers have given positive accounts of the nature of abstract objects, and when they have, they have given positive accounts that don t seem to tell us much, or anything, about what abstract objects are like. 3 In this paper, I motivate both a negative and (more tentatively) a positive thesis about the nature of two abstract objects: the equilateral triangle and Alegri s choral work Miserere Mei Deus. (I hope, of course, that the theses will generalize to at least abstract objects of the same kinds.) Both theses are radical, and both are argued for by examining the idea that we might be able to learn something about abstract objects from representations of those objects. The negative thesis says that these abstract objects lack particular qualitative properties (and perhaps lack any qualitative properties, and even lack quasi-qualitative 1 Acknowledgements to be inserted. 2 e.g. Frege (sort of), Goodman and Quine (1947) seem to assume a negative account, where abstract objects lack spatiotemporal location, (though they are arguing against the existence of abstracta), as does Quine (1948). Armstrong (1978) discusses them as lacking causal power. 3 I am thinking here of abstractionist and related accounts, e.g. Hale (1987), Wright (1983); Dummett (1973). Plato is clearly an exception to my claims here. Zalta (1983) and Cowling (forthcoming) may also be. The view here has something in common with Zalta, but is not similar enough to warrant a lengthy discussion. 1

properties); and hence the particular abstract objects under consideration don t have qualitative natures. One virtue of this thesis is that it helps explain why it is so hard to give a positive account of what abstract objects are like. The question of what is x like? seems to be a question about what qualities x has; and if (at least some) abstract objects lack qualitative natures, and don t instantiate any qualitative (or quasiqualitative) properties, then they aren t like anything at all. My positive thesis says that, despite lacking qualitative natures, the abstract objects in question do have natures. (So, in some sense, it does answer the question what are abstract objects like?.) They have purely structural natures. What this means is an important question, and one which, I must admit, I will not give a full answer to in this paper, though I hope it will become somewhat more clear. (I will also, at the end of the paper, temper this positive thesis somewhat; if you re allergic to structuralism, you should still be able to accept most of what follows.) For now, let me say two things to begin to situate the view. First, my positive thesis contrasts with each of the following theses: (i) qualitativism: there is some sense in which the abstract objects in question instantiate qualitative properties, or abstract analogues of qualitative properties, or quasiqualitative properties; so, for example, it really is true that there is some sense in which Sherlock Holmes enjoys cocaine; (ii) pointillism: the abstract objects in question are featureless/structureless points in abstract space ; and (iii) substructuralism; the abstract objects in question are mere nodes in an abstract structure that is, in some sense or other, prior to them. (I use the term substructuralism to indicate that the objects in question are nodes in a structure rather than structures themselves.) Second, despite the fact that my view is inconsistent with what I am calling substructuralism, it bears most in common with structuralist theses. Shapiro (1997) and Resnik (1997) advocate forms of mathematical structuralism on which e.g. numbers are nodes in mathematical structures; but those mathematical structures themselves are platonic, abstract, structural objects. They are objects that have complex natures (since they are constituted by, presumably, a large number of internal relational properties). The view I advocate here suggests a similar sort of structuralism about both geometry (point and line are nodes in the structural object the equilateral triangle) and musical works (b# might be a node in Miserere); but it is inconsistent with the view that the equilateral triangle itself is a mere node in a structure. 2

In section one, I define mimetic representation, which is, very roughly, representation that mimics or resembles its target. I argue that a principle, MIMESIS, falls out of the definition of mimetic representation. 4 This principle tells us (very roughly) that if two representations of a given target are equally good at representing that target s properties, then that target can t have any properties that one but not the other representation has. In section two, I begin to motivate the claim that we often have such ties of mimetic representation when it comes to abstract objects. I give an account of mimetic translation, and propose some constraints on what it takes to be a strong mimetic translation. I then suggest that, in cases in which we have strong mimetic translations that hold between representations, we have a strong but defeasible reason to think that those representations are equally good mimetic representations. In section three, I argue that there are no defeaters for that reason. I show that it follows that we can t believe that these abstract objects have, in any sense, the properties or qualities that their representations do. I also show that the considerations presented in the paper should lead us to believe that the abstract objects I consider here have internal structure, and hence have natures, and that it follows that substructuralism and pointillism are false of these objects. But these considerations should also lead us to believe that these abstract objects have no qualitative or even quasi-qualitative properties; and so qualitativism is false. So, I conclude, Miserere and the equilateral triangle have purely structural internal natures. (I also mention some ways one could resist going all the way to this conclusion.) Before I continue, I want to mark two background assumptions I am making. First, I am interested in what the Platonist should say about the nature of abstract objects; so, I assume (some form of) Platonism in what follows. But I don t assume anything other than that about what abstract objects, generally, are; that is what I am trying to figure out! The methodology, in other words, is to start out by asking what are two particular abstract objects like? What are they? and to hope, but not assume, that there is something that we can learn more generally from doing so. Second, I assume a very generous view of what counts as a representation. So, e.g., if there is an abstract object the equilateral triangle, then both an algebraic 4 I use mimetic and mimesis as reappropriated technical terms (to be defined in section one); evoking Plato is intentional, but evoking anything post-plato (including Adorno) is certainly not. 3

formula and a drawing of an equilateral triangle count as representions of that object (the equilateral triangle is the target of both representations). I don t really care about the boundaries of what counts as a representation, so long as those boundaries allow in the things I am concerned with. But I do deny that in order for something to count as a representation, there must be an intention (either on the part of the perceiver or the representer ) that it do so. I further claim (as will become clear) that resemblance is necessary for at least some kinds of representations, which contrasts with Goodman (1976). I don t defend these claims here, but not much turns on them: if you don t like my use of the word representation, simply substitute shrepresentation. 1. Mimetic Representation In this section of the paper, I introduce and define mimetic representation. I show that a principle, MIMESIS, falls out of this definition. In the next section, I argue that the antecedent of this principle is indeed satisfied for certain abstract objects. Mimetic representation: The most mimetic representation of a given target is the representation that predicate-shares the most with the intrinsic properties of the target. 5 To get an initial feel for mimetic representation, ignore predicate-sharing and think of this as simply the representation that shares the most properties with its target, or the representation that best mimics its target, though momentarily I will explain predicate-sharing. 6 5 Here I am invoking a notion of being an intrinsic property without fleshing out what that really means. While the general idea is clear to be an intrinsic property is to be, as David Lewis puts it, a property which things have in virtue of the way they themselves are (1986 p. 61), It is notoriously difficult to define intrinsicness (see e.g. Sider 1996, Lewis and Langton 1998, Cameron 2009, Francescotti 2014) and, while I suspect that different definitions will get us different results when it comes to the problem at hand, I doubt that those results will differ enough that I need to choose a definition rather than simply appealing to Lewis pretheoretic gloss. 6 Mimetic representation is supposed to be very much like what Dretske (1981, ch. 1) appropriated the term analog representation for; I don t use this term here for three reasons: first, it suggests that an analogy is being drawn between the representation and the target; but in some instances, mimetic representations will literally share properties with their targets. Second, I will later argue that, when it comes to abstract objects at least, representations that we would typically take to be digital can actually be mimetic; this will become confusing if I use Dretske s terminology. And third, I am using the term 4

Mimetic representation is relatively easy to cash out when it comes to concrete objects: suppose we are wondering which representation of a particular beech tree is most mimetic. And suppose our options are: the linguistic description a beech tree ; a child s drawing of the beech tree; and a highly accurate three-dimensional clay model of the beech tree. The model is likely the most mimetic representation among these; it directly models many of the properties of the beech tree, e.g. having green leaves, having leaves of a very specific shape, having a trunk, having a certain pattern of bark, being three-dimensional, etc. Of course, some of these properties (e.g. being three-dimensional) it literally shares with the beech tree; many others (e.g. having green leaves) it does not, since its leaves are not really leaves. What should we make of this kind of pseudo-propertysharing? I am going to make a fairly controversial claim here, but I don t think too much substantively turns on it I make it largely for convenience s sake. The beech tree has green leaves, and there is at least some sense in which the model has green leaves, and the italicized predicates pick out two distinct properties (this much, I hope, is not controversial). What makes it appropriate for us to use the same predicate to pick out both properties is that the property of the model accurately mimics or represents the property of the tree. And, at least when it comes to concrete objects, we are pretty good judges of when this predicate-sharing is appropriate; all we have to do is look at the target, and look at the representation, and see whether the property of the representation does a good job looking like (or smelling like, or tasting like, or sounding like) the property of the target. 7 for such different purposes than Dretske. Mimetic representation also bears some similarity to Quine s notion of canonical representation, or Shapiro s (e.g. 1985); but again, I use the term differently; while the spirit is similar, Quine and Shapiro are targeting largely linguistic (or mathematical-linguistic) representations, whereas in the cases I m concerned with the most mimetic representations will typically not be linguistic at all. 7 There is another controversial claim lurking in the background here: I think that the best way to make sense of Plato s puzzling discussions about self-predication of the Forms in The Republic and The Phaedo Plato s view is to attribute a similar view to him: that he thinks that it is appropriate for a concrete object and a Form to predicate-share when the concrete object is a good model or representation of the Form; but he doesn t think that the Form and the concrete object literally share a property in these cases. On this view, properties are more fine-grained than predicates; it is appropriate to predicate largeness of the Form of the Large, but this is really to attribute a property, large*, to the 5

Again, nothing deep in this paper turns on this controversial claim; if, for example, one wanted to say that the model really does have green leaves, one could reconstruct the argument accordingly. I am less confident that one could reconstruct the argument if one wanted to deny that the model either property-shares or predicateshares having green leaves with the tree; but it is also hard for me to see the motivation for such a view, since it would require giving a systematic error theory for claims like that model has green leaves. How should we think about MIMESIS when it comes to abstract objects? The following example will help illustrate, but I want to be clear that I am going to argue against the supposition later in the paper. Suppose that the abstract object the equilateral triangle is spatial or quasi-spatial (spatial in the sense that it has a location, and is not a mere point, in the abstract correlate of physical space, or something like that). It will follow that pictorial representations of the equilateral triangle are more mimetic than algebraic ones, despite the fact that they don t share properties with the equilateral triangle. They have a property being concretely twodimensional that is a predicate-sharer with the property being abstractly twodimensional that the equilateral triangle has. Pictorial representations have the physical correlate of the abstract property that the abstract object has in this case, for example, they have three equal angles and three equal sides. So if we think that the equilateral triangle has three equal angles and three equal sides, then we should think a pictorial representation is more mimetic than an algebraic one, which doesn't itself have any angles or sides. (Again, I will argue against this kind of picture later, so this is just an example.) Questions of mimetic representation are not questions about what the best language to write the book of the world is, or what representations are most jointcarving ; nor do they have anything to do with questions of grounding or fundamentality. For the most part (unless, perhaps, the target of our representation is a word, or a sentence, or a bit of language itself), linguistic representations will not be most mimetic, but they may well be what belongs in the book of the world. To see Form of the Large, that is distinct from the property large. But that is highly controversial, and a digression, though one which will hopefully help explain why I have adopted this set-up. 6

this, just return to the beech tree example. Whatever linguistic description of a beech tree carves nature at its joints the best is irrelevant to what mimetically represents the beech tree. Call the most joint-carving description of a beech tree b. There are almost no predications we can make of both b and the beech tree. b doesn t mimic the beech tree; there is no sense in which b has green leaves, or a trunk, or a particular pattern of bark. Relatedly, questions of mimetic representation are not questions about which representation is best, or which communicates or encodes the most information. Notice that a twenty-page account of what properties a beech tree has, written by an expert in beech trees, will communicate more information about the beech tree than our threedimensional model does. But it is in no sense mimetic: it doesn t represent by showing in the same way that the model does. 8 It does not matter to my argument whether mimetic representations are in fact best representations, or whether descriptions might do better, or whether, as I suspect, there is no non-contextual notion of a best representation. So one needn t think that mimetic representation is anything special to follow the argument past this point; I am not claiming it is a metaphysically privileged kind of representation, or anything like that. 1.2 Epistemology of Mimetic Representation: Motivating MIMESIS How can we know, or even have evidence, that a representation of an object is more mimetic than another? When it comes to concrete objects, this is in some sense very easy: each object most mimetically represents itself, so we can, in one sense, always know what the most mimetic representation of a given object is. Even in the concrete realm, we might find it harder to identify the second-mostmimetic representations of an object. If we have a grasp of which properties of an object are intrinsic, and also of when predicate sharing is appropriate, then we will be able to identify mimetic representations. But of course, sometimes we won t know these things. Further, some representations will seem mimetic along certain dimensions but not others (consider a drawing of the DNA structure of a lemon vs. an 8 Again, I mean show here to be inclusive of not just looking like, but smelling like, tasting like, sounding like, etc. 7

extremely realistic looking plastic lemon), and it might be hard to determine how to weigh those dimensions against one another (if, as I suggested, we restrict our attention to intrinsic properties, this might become slightly easier, except that now we need be very concerned with what exactly counts as an intrinsic property). Still, identifying mimetic representations is far easier in the concrete realm than when the target of representation is an abstract object. Mental objects will also pose a challenge for the epistemology of mimetic representation, but again, it won't be the same sort of challenge that abstract objects pose. Again, each mental object will most mimetically represent itself. But what about second-most-mimetic representations? Perhaps we can examine our own mental objects and see what they are like (easier said than done, but we do have some access to this). If they are in picture-form, then perhaps pictorial representations of them are more mimetic. If they are mental analogues of words, or sentences, then perhaps linguistic representations of them are more mimetic; if they are propositions then which representations of them are most mimetic are going to depend on what propositions are. So while identifying (second-most) mimetic representations of both mental and concrete objects might be difficult, neither poses the same sort of seemingly intractable epistemic challenge as abstract objects do, for both involve targets that we have some independent access to. If the target is abstract, the only access we have to the nature of the target is through its concrete representation. Given the way I ve defined mimetic representation, we can generate the following principle. MIMESIS: For any abstract object a, if we have a set of Rs that are candidates for being most mimetic representations of a, and no good reason to think any of the Rs is more mimetic than the others, then we should not believe that a predicate-shares with any properties that some but not all of the Rs in question have. Because every object most mimetically represents itself, this principle will always be trivially satisfied. We only get interesting results when we restrict it to the representations of a target that we have access to. And this is what my focus is here: how can MIMESIS help us in cases where we are ignorant of the target in question? I 8

set aside self-representation in the remainder of the discussion, in order to maintain clarity. MIMESIS is, I hope, close to analytic, given the definition of mimetic representation and a generous notion of representation. To see this, notice that there are only two ways for the consequent to be false, and notice that if either of them obtains, the antecedent will be false too. First, the consequent could be false if we thought that a should predicate-share a particular predicate with, say, R, but not R or R, and that otherwise, for every property a has, it either predicate-shares with all of the Rs or with none of them. But if this were so, R would, by definition, be a more mimetic representation than R or R, so the antecedent would be false. Second, the consequent could be false if each of some subset of the Rs has a distinct property that a predicate-shares with, and the rest of a s properties are either predicate-shared with all of the Rs, or with none of them. In such a case, we should simply be able to construct a new R that has each of the properties that are predicateshared by a and just one of the original Rs. (In addition to the property base shared by all of R, R, R it will simply have any additional properties that some but not all of the original Rs predicate-share with a.) R by definition will be more mimetic than the original Rs. So the antecedent will be false, since none of the Rs it is considering will be candidates for being most mimetic. In short: if the consequent of MIMESIS is false, then either there is a good reason to think that one of R, R', R'' is more mimetic than the others, or we don't have good reason to think that any of R, R', R'' are among the most mimetic representations of a. A few notes about this: first, my claim depends on the idea that we can always construct more mimetic representations if we have two representations that are equally mimetic, each of which has a property that the other lacks, but that the target has. We do so by constructing a representation that has both properties. Of course, we might be currently incapable of doing this, for some particular target. But there is nothing in theory that stops us from doing so we won t end up in situations where we need to represent something as having logically or metaphysically incompatible properties. For recall that the target itself is going to have the properties all the properties in question (or at least, will predicate-share them). And nothing can have logically or 9

metaphysically incompatible properties. So constructing a representation will never require that we do the impossible. Second, MIMESIS isn't a principle that tells us anything about whether or not any particular R represents a particular target. This is as it should be, given that it trivially falls out of the concept of mimetic representation. Rather, MIMESIS applies only once we already think that some Rs represent a, and have no way to choose which most mimetically represent a. But how do we get ourselves in such a situation? And why think that this is often our epistemic situation, with respect to abstract objects? In the next section, I will introduce mimetic translation, that tells us when two representations represent the same object. I will argue that, when certain conditions are met, mimetic translations give us reason to think that each of the two representations is an equally good candidate for being the most mimetic representation of their target. If this is right, then there are multiple competing representations that are all equally mimetic; so we have a privileged set of competitors for most mimetically representing abstract objects, and MIMESIS applies. Third, one might worry about the following: all of our representations of abstract objects will be concrete, and will hence share a property, concreteness, that abstract objects lack, and will fail to predicate-share with abstract objects along this dimension (there is no predicate sharing the properties of concreteness and abstractness!). So, one might think, clearly no concrete representation of an abstract object could be most mimetic. (Indeed, if objects can represent themselves, then every object is likely the most mimetic representation of itself.) But, as I said before, MIMESIS is really only of interest to us if we restrict it to thinking about representations that we have access to. I think this worry dissolves if we do so, since we lack the relevant kind of access to any abstract objects. 2. Mimetic Translation In this section, I introduce a notion of mimetic translation. I argue that mimetic translations can play a similar role, for abstract objects, to that that representation theorems do for quantities. I then go on to suggest that certain mimetic translations give us reason to think that MIMESIS antecedent is satisfied. Representation theorems are used in measurement theory along with 10

uniqueness theorems to show that a given kind of representation of a quantitative structure preserves the structural features of that quantitative structure. Very roughly, representation theorems establish that that there is some mapping from the target domain (the quantitative structure) to the domain posited by the representation (numbers), and uniqueness theorems tell us what kinds of transformations preserve that mapping. In a bit more detail (but still roughly): First, in order to establish the right kind of relationship between the structure of our target domain (D) and the structure of our surrogate (representing) domain (D'), we need to show that there exist homomorphisms from the relations on the target domain to the relations on the surrogate domain. A homomorphism is just a structure-preserving function that maps every member of D to a member of D'. If our D' contains the real numbers, and our D contains physical objects, the idea is that we can select, e.g., a homomorphism that preserves all the mass ratios that hold between objects in D by mapping those ratios to the equivalent ratios on D'. So if the coffee in my glass gets mapped to the real number 3.7, and the water in my water glass is in fact twice as massive as the coffee in my coffee cup, then the water in my water glass will get mapped to the real number 7.4. Representation theorems show that there exist such homomorphisms from D to D'. A mimetic translation is like a representation theorem in that it is structure preserving in a certain way. It, like a normal representation theorem, shows us that there is a certain kind of structural isomorphism between two entities. But in the case of a normal representation theorem, we have a clear, empirically observable target, and what we are showing is that a certain kind of structure that target has is preserved by a given representation of it. In the cases of mimetic representation that we care about, we lack any access to that target that is independent of the very representations we are evaluating. So, for example, the target might be an abstract object, or some other kind of unobservable entity. In such cases, we can still establish a kind of structural isomorphism between representations. We can do so by showing that there is a translation procedure for going back and forth between two representations, and that that translation procedure preserves important structural features of each representation. I should note, however, that we won't be able to show that there are homomorphic functions in (at least some) cases of mimetic translation, precisely 11

because it is going to be far less clear what counts as the domain of the representation. It is important to note this limitation of the analogy. But I still think the analogy is apt, and useful for getting a handle on what these things are supposed to be. As I said in the introduction, I will focus on two abstract objects in what follows. First, the equilateral triangle; second, Allegri s choral work Miserere Mei Deus. First, consider what we take to be a very good representation of the equilateral triangle: a two-dimensional, very thin-outlined, drawing of an equilateral triangle, produced with a compass. What kinds of representations preserve the structure of the drawing? It s clear that, e.g., a small green sphere does not. But an algebraic representation of an equilateral triangle seems to. Second, consider what we take to be a very good representation of Miserere: a near-perfect performance of the work. What kinds of representations preserve the structure of that performance? It s clear that, e.g., a 4-year-old s performance of Row, Row, Row Your Boat does not. But a particular copy of the written score of Miserere does. What explains these facts, I claim, is that there are structure-preserving translations between the drawing and the algebraic formula, and between the score and the perfect performance, but no such translations between the drawing and the sphere, or the performance and Row, Row, Row Your Boat. These translations serve to demonstrate a kind of equivalence between two different kinds of representations: they are (loosely) structurally isomorphic, and hence can represent the same target. Now, what it is to be a translation that preserves x, for any x, is a subject of great consternation. My recipe for being a structure-preserving translation begins with the following: We need (or the most ideal among us need) to be able to reproduce representations of one type from representations of the other type. In the mathematical case, this is fairly simple: anyone who knows enough algebra and geometry can produce a token of an algebraic formula representing the equilateral triangle after looking at a pictorial representation of one, and vice-versa. In the case of Miserere, this is less straightforward. It is easy enough for (some among us) to go in one direction: to look at a copy of the score, and produce a particular performance. It is harder to go in the other direction: to listen to a particular performance, and to produce a written score. Note, though, that I picked this example 12

precisely because the fourteen-year-old Mozart did just this: listened to a single performance of Miserere, and afterwards produced a near-perfect transcription. The reproductive structure-preserving translations at work here are what I m calling mimetic translations. But in order to count as mimetic translations, they need to satisfy further conditions. Our ability to reproduce representations of one type from representations of another type is neither necessary nor sufficient for giving us a mimetic translation. First, to see why it is not sufficient: consider the ease with which we can write down the words equilateral triangle, or speak them, when we see images of equilateral triangles. Likewise, consider the ease with which those of us familiar with Miserere can say that is Miserere when we hear a particular performance of the work. But note that the phrase equilateral triangle doesn t preserve any structural features of a drawing of an equilateral triangle; nor does Miserere preserve any of the structural features of a particular performance of Miserere. And note that, in general, it is easy enough to produce translation manuals that allow us to translate between, say, bottles of various different brands of tasteless beer and pictorial depictions of geometric shapes. We simply give someone a dictionary that looks like this: = Budweiser = Miller Lite = Michelob Ultra And so on. The kind of reproductive translations I ve described here are easy to come by. Even systematic ones are easy to come by, because we can simply stipulate the translation manual into existence. So we need to say something about what makes the translations in question structure-preserving. I of course won t have anything to say here that is fully satisfying to those skeptical of this sort of project. But I do want to say something about what I take it distinguishes good and bad cases. It is clear that a copy of a written score of a piece of music and a particular 13

performance of that piece of music are structurally isomorphic in some sense; the leftto-right dimension of the staff in the written score corresponds to the moving-forwardin-time dimension of the performance; the height of a given representation of a note on the staff corresponds to the pitch of a note in the performance; the shape of a given representation of a note corresponds to how long it is held in the performance; and so on. This preservation of structure is what enables us to reproduce representations of one type from another, and vice-versa: there is a structural core that is preserved by each. 9 I propose two constraints for ensuring that mimetic translations are structurepreserving. Neither is particularly precise take them as first passes. The first constraint is that there can be no particular stipulation: the translations shouldn t be products of stipulation or memorization of particular matches between representations. For example, we can t just use a translation process which requires us to memorize that when we see the label Michelob Ultra, we should produce the blue circle, and when we see the label Miller Lite, we should produce the equilateral triangle, and so on. Good structure-preserving translations do require both memorization (e.g. memorizing which placement of a note on the staff corresponds to which tone produced on an instrument) and stipulation (e.g. without us knowing for at least a few nodes on each structure that they correspond to one another, we can t reproduce one from the other compare representation of mass: there is a sense in which we stipulate that an object in the world has mass of one gram, and without doing so we can t apply the gram scale to mass-in-the-world). But the kinds of memorization and stipulation involved here are general and not specific. In good cases, we might stipulate some minimal things about the relationship between nodes in a structural representation of kind K and nodes in a structural 9 I should admit here that there won t be a perfect correspondence of structure here; the score contains certain marks that correspond in no way to any feature of the performance, and perhaps vice-versa. Still, we can easily imagine an idealized version of the score, where such things are edited out. I will return briefly to this issue in section four. 14

representation of kind K ; but doing so allows us to translate between any representation of kind K and the corresponding representation of kind K, and viceversa. So whatever stipulation and memorization the translation involves, it must be stipulation and memorization of general frameworks for translation and not specific cases. The translation process itself must generate the right results when it is applied to any representations of kind K. It must give us a method for translating a random representation of kind K into one of kind K, and vice-versa, where experts recognize the resulting translation as correct. The second constraint is that there be no (or minimal) deletion and reintroduction of structure in the metalanguage in which the translation is being performed. What does this mean? One way we might generate a translation scheme between bottles of beers and geometric shapes is by merely stipulating that this beer goes with that shape, for each case. The first constraint rules that out. But a second way we might generate a translation scheme is to make the translation procedure itself so complicated that we can actually succeed in making it systematic and general enough: we can make the translation procedure delete and then reintroduce structure to a massive degree. Imagine, for example, a translation procedure that tells you that if you have a representation that has the property of being red, you should ignore all of its properties, add nineteen distinct properties, and translate the resulting twenty properties via some elaborate code into musical notation. One can imagine that we could create systematic and perfectly general procedures like this (where there was a system for determining which colors should result in the adding of which properties), and such a procedure wouldn t be ruled out by the first constraint. So we need a second condition on what kinds of translations give us mimetic translations: it can t be that the translation procedure itself involves the insertion of lots of new structure into the picture (and then translation of that new structure into the second representation); nor can it involve the deletion of a lot of the structure from the first representation when translating to the second. We want our translation procedure itself to preserve everything that we take to matter about the target, once the relevant background is in place. The obvious objection here is that, if we don t know anything about the nature of the target of our representations, then we can t know what structure in a given 15

representation is mere noise and should be ignored or deleted in the translation procedure. This seems like a problem, in some sense (though it s not obvious that there isn t a similar problem for e.g. representation theorems that apply to physical quantities). Still, I claim, there are facts of the matter facts that, in many cases, we know about which representations are more mimetic: better represent the structure of their target. Consider the examples I gave at the beginning of this section: we know that a small green sphere is a less mimetic representation of the equilateral triangle than a child s drawing of a triangle is; similarly, we know that an adult s drawing, with a careful hand and a compass, of an equilateral triangle is a more mimetic representation than the child s drawing is. This is all made sense of by our two constraints: it is hard to imagine how we could have a translation procedure from the small green sphere to the adult s drawing that respects both constraints. A second version of this objection is a more familiar one: even if everything I am saying is right, how do we come to know that any representation of an abstract object actually successfully represents that abstract object? This standard epistemic objection to Platonism is largely outside of the scope of this paper (though I will say something more about it in the next section). I want to begin with the idea that we have a pretty good idea that the steady-handed drawing, made with a compass, is a good candidate for being the most mimetic representation of the equilateral triangle; and see what follows from that. It seems to me that we can rank representations in terms of MIMESIS once we fix the facts about the MIMESIS of a single representation; whether it is possible to do that in the first place is an important question, but one which I don t discuss in detail here. Where does all this leave us, and what does this discussion of mimetic translation have to do with MIMESIS? First, note that there are very strong mimetic translations--that respect both the constraints I ve proposed between, on the one hand, the perfect drawing of the equilateral triangle and the algebraic formula; and, on the other hand, the performance of Miserere and the copy of the written score. Suppose that I define a mass unit, the shram, which is equivalent to 1.4 grams. I take it that we think that gram representations of mass facts and shram representions of mass facts are at least decent candidates for being equally good ways of representing these facts; part of how we determine that they are equally good is by seeing that there 16

are structure-preserving translations between them. Of course, the fact that there are structure-preserving translations between gram facts and shram facts does not entail that mass structure in the world is not either gram-like or shram-like; but what it does do, I think, is shift the burden: we need to be able to provide a reason for thinking that (e.g.) gram talk is more metaphysically perspicuous than shram talk is. Otherwise, what these structure-preserving translations show is that it is arbitrary to claim that gram talk does (metaphysically) better. Likewise, the fact that we can produce very clean structure-preserving translations between the formula and the drawing, and the score and the performance, puts the burden on the philosopher who wants to claim that the targets of those representations are more mimetically represented by (e.g.) the performance than the score. So, suppose that we start out thinking that a perfect performance of Miserere is most mimetic. We know that there is a structure-preserving translation that respects our two constraints from that performance to a copy of the written score. Then it follows we now need some sort of motivation for thinking that one is more metaphysically perspicuous, or that it more mimetically represents the abstract object in question. And here is where the epistemic difficulty with Platonism does come into play: I don t see where we could get such motivation. I will discuss this in the next section. For now, my claim is the following: in the case of both Miserere and the equilateral triangle, there are good mimetic translations (that respect the two constraints) that do this burden-shifting work. So if we cannot come up with compelling reasons for rejecting that they are equally mimetic, it will follow that, e.g., a written copy of Miserere s score and a particular performance of Miserere satisfy the antecedent of MIMESIS; and that means that we are entitled to conclude that there is no sense in which it Miserere predicate-shares with any of the properties that the performance has but the written score lacks. (So, for example, it will turn out that Miserere is not sonic in even an abstract sense; and it will turn out that the equilateral triangle doesn t have three sides.) 3. What should we conclude? In the remainder of this paper, I will argue for three things: first, that there are no reasons for us to believe that one or the other of these candidate representations is 17

mimetic; second, that given how radically qualitatively different the score and the performance are (and the drawing and the formula are), we are pushed, by applying MIMESIS, towards the claim that the abstract objects in question have no qualitative or quasi-qualitative (abstract analogues of qualitative) properties; and third, that these abstract objects have internal natures. I suggest that all this gives us reason to think that abstract objects like the ones in question have internal, non-qualitative, structural natures. What possible reasons could we have for maintaining, in the face of a mimetic translation, that the drawing of the equilateral triangle was a more mimetic representation than the algebraic formula? Given our general lack of access to abstract objects, I can only think of one: that we intend to be representing something spatially extended, or quasi-spatially extended, with both the drawing and the formula. In other words, that we intend for the object that we are representing to have three equal sides and three equal angles; that we intend for it to be appropriate for it to predicate-share with the drawing but not with the formula. And that somehow, this intention makes it so. This won t do, for familiar reasons. As Balaguer (1998) points out: It would be massively, and bizarrely, coincidental (for the Platonist) if the very abstract objects that existed were all and only those which we intended to pick out with our representations. It seems to follow that the only way our intentions could be relevant is if Plenitudinous Platonism were true: if there was an abstract object out there for every concept we could possibly have; that would remove the arbitrariness concern (though leave us, perhaps, with a lingering concern about magical reference). 10 Perhaps, though, the Plenitudinous Platonist (PP) has resources to resist the arguments in this paper. Plenitudinous Platonism does, I think, at least initially seem to resolve the question of how we could know which of our representations were mimetic (for the ones that best represented what we intended to pick out would be most mimetic). But the problem gets recreated at a different level for the PP. It might seem, at first, that for the PP, a good mimetic translation can t show that two radically 10 Balaguer argues on these sorts of grounds that, if Platonism is true, then Plenitudinous Platonism must be true. I take no stand on this issue here. 18

different representations equally mimetically represent a single abstract object, because the PP is, or should be, committed to there being a distinct object for each representation (every representation of an abstract object is a mimetic representation of the abstract object that it does the most predicate-sharing with). But I think what a good mimetic translation can show, for the PP, is that in addition to the distinct abstract objects that each representation most mimetically represents, there is a third abstract object that they equally mimetically represent! Consider a simple example of a sentence, the candle is bent. One way to represent this sentence is by writing it down. Another is to utter it. The PP might think that the written sentence is a token of one abstract object (the written sentence type) which has some essential or intrinsic property that can only be represented by writing, and the utterance is a representation of another, which has some essential or intrinsic property that can only be represented by saying-out-loud. But if so, her view seems to dictate that she now needs to grant that there is a third abstract object which has neither of these properties: both the written sentence token and the said-out-loud sentence token represent some more general object, the sentence, that is not more or less mimetically represented by either the utterance or the written sentence token. It hence seems to me that what the PP should think is the following: that what a good mimetic translation shows is that, while two representations might represent distinct targets, they also both represent a third target, and do so equally well. The third target is simply the object that corresponds to the concept which is represented equally well by both representations. So, if she were considering the case of the drawing vs. the formula, she might think: there is an abstract object that the drawing most mimetically represents; there is another abstract object that the formula most mimetically represents; and there is a third object that they equally well represent. But now our question is recreated in a slightly different way: which of these three objects is the equilateral triangle? (I don t simply mean: which one do we intend to pick out with the words the equilateral triangle.) Perhaps the answer is indeterminate. It turns out that each of our representations represents many abstract objects, despite our intentions. And in each case, one of those objects for example, the one that is equally mimetically represented by the formula and the drawing has all the same features that, in the rest of this section, I will try to show that the equilateral triangle does. In some sense, then, the PP needn t accept the argument I ve 19

given here; but my argument does suggest that she needs to posit even more abstract objects than she might have initially thought; and that which of those objects are represented by which representations will be more indeterminate than she might have thought. What was the point of this digression? To show that I don t think that going down the route of claiming that our intentions determine which representations are most mimetic completely resolves the problem here. Only the PP can take our intentions to matter; but the PP, I claim, must also believe in the kinds of abstract objects that I claim the equilateral triangle and Miserere are (and further, she must be accidentally representing an angleless, sideless abstract object even when she intends to represent something else!). (You might think that Miserere is special here, because it is a creative artifact, and perhaps there are things to say about the nature of creative artifacts that don t apply to other abstract objects. I am skeptical that anything we could say would avoid invoking intentions in a way that would be susceptible to the problems I have just outlined. But I recognize that not everyone will share this skepticism. I am happy, if you think that creative artifacts are special, with you only accepting my conclusion about the equilateral triangle.) Perhaps there are other ways, besides intentions, to decide between mimeticness of representations. I am open to this. But I do not know what they would be. It seems to me relatively clear that the only kind of evidence we could have for which representation was more mimetic would have to come from the abstract object itself, independent of its representations; but we don t have any access to the abstract object independent of its representations; so it seems to me that we will never have a way to decide between mimeticness of representations. So, for now, my claim is: if we start out with the belief that the drawing is a mimetic representation of the equilateral triangle, we are forced to accept, because of the existence of a mimetic translation that respects both the constraints I outlined in section 2, that, for all we know, the formula is an equally mimetic representation of the equilateral triangle. (Or, for the PP, is an equally mimetic representation of some abstract object, call it the shmequilateral triangle.) If this is right, what follows? Consider MIMESIS again: 20