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1 Entity, but no Identity Décio Krause Department of Philosophy Federal University of Santa Catarina [email protected] Abstract Inspired in Quine's well known slogans “To be is to be the value of a variable” and "No entity without identity", we provide a way of enabling tha t non-individual entities (as characterized below) can also be values of variables of an adequa te "regimented" language, once we consider a possible meaning of the background theory Quine reports to ground his view. In doing that, w e show that there may exist also entities without identity, and emphasize the importance of paying attention to the metalanguage of scientific theorie s, for they may be also fundamental in determining the theory's ontological commitment. Key Words: ontological commitment, quantum objects, quantum o ntology, non-individuals, quasi-set theory. Introduction Inspired in Quine's well known slogan "No entity without identity" (Qui ne 1969, p. 23; see also Chateaubriand 2003), we provide a way of assuming that also non-individua l entities (as characterized below) may be values of variables, once we conside r an adequate metamathematical theory which here plays de role of Quine’s background theory . In other words, we consider the background theory as the metatheory we use to speak about the consider ed (object) theory. Thus, by a convenient change in the metalanguage, we shall be able to s ustain that "To be is to be the value of a variable", and that this holds not only for individuals entities as , but for non-individual well, here understood as entities to which the concept of identity described by classical logic does nguages' of our scientific theories are not apply. This case study suggests that not only the 'object la alanguages we use to formulate them relevant for their ontological commitments, but also the met should deserve a careful attention as well. At a first glanc e, the results achieved here can be said to motivated by Quine's ideas, but we are not aiming at to provide an exegesis of his views, for be background theory. instance, in comparing our use of the metalanguage and Quine’s Indiscernibility and structures Quine's criterion of ontological commitment is sufficiently wel l known and widespread in the literature to be recalled once more here. We shall just make some remarks in order to fix the terminology and the main ideas we would like to emphasize. Speci fying about a theory’s ontological commitment, Quine remarks that "Ontology is indeed doubly relative. Specifying the universe of a theory makes sense only relative to some background theory, and only relati ve to some choice of a manual of translation of the one theory into the other. (... ) We cannot know what something is without knowing how it is marked off from other things. Ide ntity is thus a piece with ontology. Accordingly, it is involved in some relativity, as may be readily illustrated. Imagine a fragment of economic theory. Suppose its universe c omprises persons, but its predicates are incapable of distinguishing between persons whose i ncomes are equal. The interpersonal relation of equality of income enjoys, within the t heory, the substitutivity property of the identity relation itself; the two relations are ind istinguishable. It is only relative to a sonal identity than equality of income, background theory, in which more can be said of per nt of the fragment of economic theory, that we are able even to appreciate the above accou 1

2 hinging as the account does on a contrast between p ersons and incomes." (Quine 1969, pp. 54-5) Then, although people may have the same incomes and even if they cannot be distinguished one are y, they individuals, for (by each other by the predicates of the (considered) economic theor hypothesis) they can be distinguished from one another in the richer ' background theory'. Here, we shall understand the background theory as the metalanguage in which we speak about our object 1 Thus, the sample advanced by Quine looks quite language and describe its semantic concepts. similarly to what happens when we speak about indiscernible objects within a certain mathematical structure A = D, (R riant by the ) , characterizing them as elements of the domain that are inva 〉 〈 I i i ∈ automorphisms of the structure. For instance, a certain set of people can be taken as D and some set of relations on D taken as the relevant relations (operations and distinguished elements can be reduced to relations in the standard way) to fit Quine’s sampl e, so that people with the same income cannot be distinguished by any relation. But, since standard mat hematical structures can in principle be built in a set theory (suppose Zermelo-Fraenkel,ZF for simplici ty), we can take ZF as the 2 l theories. But the ‘whole’ ZF, background language, which seems to suffice for almost all physica V = 〈 V, seen as a structure (for instance, the cumulative hierarchy 〉 can be seen as a structure), is ∈ rigid, that is, its only automorphism is the identity function. Fur thermore, since we can prove in ZF that every structure can be extended to a rigid one, it results that although we can speak of two objects x and y that they are indiscernible relatively to a certain struct ure A, if they are not the same can object (that is, if ≠ y ), then they x be distinguished from the outside of A. In Quine’s sample, this means that we will ever find outside the economic theory a c ertain relation which distinguishes bility can be achieved in standard among distinct people. The general rule is as follows: indiscerni mathematics only within (that is, if we remain confined to) a certain mathematical structure, but in the whole mathematics the sense of obeying (read, in the whole ZF), every entity is an individual, in back to this point soon). the classical (Leibnizian) theory of identity (we shall be Objectuation, the primitive act of our mind di Francia says that In virtue of what can an object be said to be an individual? Toraldo , the primitive act of dividing the world in objects, is the f irst act of ours in forming our objectuation knowledge of the world (Toraldo di Francia 1986, p. 23). So, accepting at least partially this view, we can say that we do individualize the things first, but this does not entail that they are individuals in our conception ( ) of the world. Jean Piaget, in describing how a child constructs Weltanchaung the notion of an object (Piaget 1955, chapter 1), says that in the first days or weeks of her life, although a child plays with objects, she has not yet constructed th e notion of object (we prefer to say: the notion, concept, or idea of an individual). Only later, by circa of eighteen months, she has , that notion. Piaget’s view is in certain sense Kantian, elaborated, or constructed but he disagrees with Kant in that our categories of understanding are not a priori, but dependent of several factors, individual determined by our evolution as human beings. An is identified as such only when the child attributes to the object a notion of permanence , being able to recognize it as the very same object in two different opportunities or occurrences of it. In her f irst days, the child plays with the object, but if it leaves her field of attention, and another one t akes its place, she will not realize that that previous object is missing. Although the child individuates the object, for she plays with it in 1 I am not claiming that this interpretation of the am just inspired by background theory fits Quine's. As I said before, I his ideas, and my argumentation is independent of w hatever exegetical analysis. Thus, I am assuming th e so called ‘model theoretical view’, that is, accepting that for every (object) language there exist a metalangu age in which we can talk about the object language itself and express f or instance its semantic concepts. 2 Of course there are various non equivalent set the ories, as Quine’s NF system (in the Rossser’s versi on) is not mentation can be d in the framework of category theory. But our argu equivalent to ZF, so as structures can be considere developed by considering ZF. 2

3 distinction from other of her toys, even from quite similar ones, it is not yet (to her) an individual. The notion of individual is relative. rmanence. Without pursuing a discussion The difficulty here is to make precise some terms like pe on this topic (which we intend to do in another work), we shall assum e that we have an intuitive being permanent (for a certain period of account to the idea of permanence, in the sense that a thing in that period of time (time is of a time) means that it endures in that time, or that it is continuant course another concept that deserves explanation in this context, a nd here is taken as subjective). ussing in virtue of what a certain We shall also leave aside the reasons we could point out in disc thing has permanence, for instance by mentioning the two basic approa ches to the subject, namely thing in this direction, taken the theories of substratum and the so called bundle theories (some quantum physics in mind, can be seen in French and Krause 2006). Thus, an indi vidual is something that to a certain child has permanence as thing, and can always be distinguished that from any other by some quality. Since, as some authors suggest, the se intuitive phenomenological mathematics and classical conceptions originate not only classical logic but also classical mechanics, it seems reasonable (according to us) to postulate th at something is an individual if it obeys the rules of the classical theory of identity (CTI). By CTI we understand either the first order theory of identity or a higher order theory, encompassing set theory. The first order theory, as it is we ll known, is characterized by the axioms of reflexivity and substitutivity of equality (the symbol of equali ty, or identity, is taken as a primitive binary predicate symbol), as in Mendelson’s book (1997, p. 95). Semantically, we aim at ation, namely, the set ∆ = { 〈 x , x 〉 : x ∈ to interpret this predicate in the diagonal of the given interpret alized that the above axioms do not D}, being D the domain of the interpretation. But it would be re “characterize” ∆ without ambiguity, that is, no first order language can individuali ze the elements of the domain up to an equivalence relation (Mendelson op.cit. p.100). Al ternatively, we can think of a “classical” second order logic (or a higher order logic), where Leibniz Law (LL) can be taken as the and definition of identity, namely, y = is ∀ F ( Fx ↔ Fy ), where x = y are individual variables and F x D s a first order theory, we add to the a variable for properties of individuals. In set theory, taken a axioms of reflexivity and substitutivity the axiom of extensional ity. Roughly speaking, this is CTI. The objects that obey such a theory are individuals in our sense, t hat is, they can always be distinguished from one another either for having a certain peculiar pr operty or by the existence of a set to which it belongs to but the others do not. Characterizing non-individuality Our use of the expression follows a tradition that came from the seminal work of non-individual In deriving that law, Max Planck in 1900, when he derived his law of the black body radiation. Planck assumed that the way of counting in how many ways P energy e lements can be distributed in 3 N linear oscillators, arriving at his well known formula (Plan ck 1901): + (N 1) - P ! = R (N ! P ! 1) - Later, Ehrenfest realized that such a hypothesis (namely, th e division by P!) conduces to the indiscernibility of the energy elements (the quanta), for the di vision by P! entails that permutations of indiscernible quanta are not regarded as giving raise to differ ent arrangements. Continuing with our analogy concerning the way a child “constructs” the world around her , the situation involving quanta is something like the child in her first weeks of life, who does not make distinctions between 3 For historical details not referred to here, see F rench and Krause 2006, chapter 3. 3

4 two situations originated by the permutation of two distinct but similar objects of her stock of toys t we (it should be realized that at least in principle we assume tha , having “constructed” the notion of teristic mark or scratch). The object, are able to distinguish two of them, say by some charac difference between our sample and the quantum case (and of course there are many) is that the child will evolve to elaborate the notion of object, and this cannot be sa id about quanta. Perhaps we will 4 them as individuals, once we assume them to be non-individuals. never construct have spoken about “non- Heisenberg, Weyl, and Schrödinger are among those who explicitly e additional insights, individuals” (see French and Krause op.cit.). Schrödinger provided som d to elementary particles. suggesting that even the concept of identity has no meaning when applie In order to approach non-individuals, we follow Schrödinger’s intuitions and r efuse the theory of identity as applied to them, although in our case we cannot restrict ourselves in assuming a “particle view”, for the quanta we are considering may be whatever entity a quantum theory makes (implicit) reference to. Thus, there are at least two main aspects of non-individuality to be explained. Firstly, we need a metaphysical account to non-individuals, that is, to dev elop a metaphysics of non- . Secondly, we need a formal description of them. We may assume individuality here the first point informally, due to our characterization of individuals. Really , as put long time ago by Wittgenstein and Ramsey, the traditional concept of identity, that we ca n assumed to be summed up by Leibniz Law (the second order formula shown above), is not a logical truth, so there is no apparent contradiction in assuming that it can be rejected. The objects which violate LL in the sense of sharing their properties without turning to be the very same object, that is, those objects which violate the Principle of the Identity of Indiscernibles (if th ere are some), are (formally) our non- individuals. In particular, we may assume, inspired in Schrödinge r’s ideas, that the relation of equality cannot be applied to them, in the sense that expressions o f the form x = y are not formulas of the considered language. Thus, in particular the property “being identi cal with a ”, for a certain term x a, which we can write as P . ( ) = a x = a , cannot be considered among the properties of the object D a theory of quasi-sets , we have developed to cope with collections of These ideas lie in the core of the non-individuals and which we use to answer the second question posed above. We shall not present this theory here for limitations of space (but see French and Krause op.cit., chapter 7). Of course the metaphysics of non-individuality needs to be further de veloped, but we shall not do it objects which do not obey the classical here. Instead, we shall assume that non-individuals, that is, theory of identity, could exist. Thus, in assuming that, we shal l show how we can provide the grounds for saying that non-individuals in the sense that they can be values of variables. In exist doing that, non-individuals can be assumed in the ontology of suitable t heories if we assume that s indistinguishable in the the background theory is quasi-set theory. Thus, non-individuals, taken a object theory, cannot be distinguished even in the background theory, for the y lack the concept of identity. Let us be more explicit on this point. Non-individuals do exist In order to show that non-individuals can “exist” in the sense of be ing values of variables, we may consider quasi-set theory Q as our metamathematical framework (the “background theory”). Si nce this theory encompasses standard Zermelo-Fraenkel set theory as a sub-theory –really, there is a Q , “copy” of it in Q . Thus, we have at our all standard mathematics can be constructed within disposal all standard set theoretical machinery for considering the relevant structures of our theories. But since Q is compatible with the existence of non-individuals, we have more machinery to deal with, namely, all mathematical constructions (struct ures) that can be achieved with non- individuals. Thus, we may suppose a certain mathematical struct ure (which can stand for our object 4 There is of course the possibility of considering quanta as individuals, but this assumption requires restrictions either on the states the objects may be in or in the obser vable to be allowed to them. We will not pursue thi s case here, since –but see French and Krause 2006, chapter 6. non-individuals alternative we are exploring the 4

5 theory) that involves non-individuals in its domain (in the terminolog y of , they are called m - Q Urelemente M -atoms, atoms, and constitute one of the two kinds of of the theory, the other ones, the rmelo-Fraenkel with Urelemente ). have the standard properties of the atoms of the theory ZFU, Ze since they may share properties, Since the classical identity theory (CTI) does not hold to them, but they can be completely indiscernible not only within the object the ory, but in the background theory as well. Furthermore, since the background theory is Q , adequate indiscernible objects which are values of the variables of the object language cannot be discer ned even in the background theory. Thus, in considering a suitable background theory, we can say, cont rary to Quine, that there can be entities without identity. This result has a corollary which may have interesting philosophic al consequences: in Q , there may be structures which cannot be extended to rigid structures. This kind of structures enable us to deal with indiscernible objects, as non-rigid structures do in standard mathematics (ZF), but contrarily to what happens in ZF, these structures cannot be extended to rigid ones, that is, the indiscernible objects of the structure cannot be taken as individuals in any way. A structure of this kind, we guess, would be of interest to quantum physics, for it would map more accurately the idea of non- individual quanta. But to pursue this possibility is something to be deve loped as a research program, and we leave this topic to future works. References Principia 7 (1/2), 2003, 41-74. CHATEAUBRIAND, O., "Quine and ontology", FRENCH, S. and KRAUSE, D., Identity in Physics: a Historical, Philosophical, and Formal Analysis , Oxford, Oxford Un. Press, 2006. MENDELSON, E., Introduction to Mathematical Logic, London, Chapman & Hall, 4th. ed., 1997. QUINE, W. V., "Ontological relativity", J. of Philosophy 65 (7), 1968, 185-212. QUINE, W.V., Ont ological Relativity and Other Essays , New York, Columbia Un. Press, 1969. QUINE, W. V., "Ontology and ideology revisited", Journal of Philosophy , 80 (9), 1983, 499-502. QUINE, W. V., The Philosophy of W. V. Quine , 2nd.expanded edition, The Library of Living Philosohers, XVIII, edited by L. E. Hahan and P.A. Schilpp, Ope n Court, Second Expanded Edition, 1998. PIAGET, J., The Construction of Reality in the Child , translated by Margaret Cook, 1955, Routledge and Kegan Paul. PLANCK, M., “On the Law of Distribution of Energy in the Normal S pectrum”, Annalen der Physik vol. 4, p. 553 ff (1901). , Bari, Laterza, 1986. Le Cose e I Loro Nomi TORALDO DI FRANCIA, G., 5

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