[FIRST PARAGRAPHS] \n \nIs vagueness a feature of the world or merely of our representations \nof the world? Of course, one might respond to this question by asserting \nthat insofar as our knowledge of the world is mediated by our \nrepresentations of it, any attribution of vagueness must attach to the latter. \nHowever, this is to trivialize the issue: even granted the point that all \nknowledge is representational, the question can be re-posed by asking \nwhether vague features of our representations are ultimately eliminable or \nnot. It is the answer to this question which distinguishes those who believe \nthat vagueness is essentially epistemic from those who believe that it is, \nequally essentially, ontic. The eliminability of vague features according to \nthe epistemic view can be expressed in terms of the supervenience of \n‘vaguely described facts’ on ‘precisely describable facts’: \n \nIf two possible situations are alike as precisely described in terms of \nphysical measurements, for example, then they are alike as vaguely \ndescribed with words like ‘thin’. It may therefore be concluded that the facts \nthemselves are not vague, for all the facts supervene on precisely \ndescribable facts. (Williamson 1994, p. 248; see also pp. 201- \n204) \n \n \nIt is the putative vagueness of certain identity statements in \nparticular that has been the central focus of claims that there is vagueness \n‘in’ the world (Parfit 1984, pp. 238-241; Kripke 1972, p. 345 n. 18). Thus, \nit may be vague as to who is identical to whom after a brain-swap, to give \na much discussed example. Such claims have been dealt a forceful blow \nby the famous Evans-Salmon argument which runs as follows: suppose for \nreductio that it is indeterminate whether a = b. Then b definitely possesses \nthe property that it is indeterminate whether it is identical with a, but a \ndefinitely does not possess this property since it is surely not \nindeterminate whether a=a. Therefore, by Leibniz’s Law, it cannot be the \ncase that a=b and so the identity cannot be indeterminate (Evans 1978; \nSalmon 1982).
[FIRST PARAGRAPHS] \n \nIs vagueness a feature of the world or merely of our representations \nof the world? Of course, one might respond to this question by asserting \nthat insofar as our knowledge of the world is mediated by our \nrepresentations of it, any attribution of vagueness must attach to the latter. \nHowever, this is to trivialize the issue: even granted the point that all \nknowledge is representational, the question can be re-posed by asking \nwhether vague features of our representations are ultimately eliminable or \nnot. It is the answer to this question which distinguishes those who believe \nthat vagueness is essentially epistemic from those who believe that it is, \nequally essentially, ontic. The eliminability of vague features according to \nthe epistemic view can be expressed in terms of the supervenience of \n‘vaguely described facts’ on ‘precisely describable facts’: \n \nIf two possible situations are alike as precisely described in terms of \nphysical measurements, for example, then they are alike as vaguely \ndescribed with words like ‘thin’. It may therefore be concluded that the facts \nthemselves are not vague, for all the facts supervene on precisely \ndescribable facts. (Williamson 1994, p. 248; see also pp. 201- \n204) \n \n \nIt is the putative vagueness of certain identity statements in \nparticular that has been the central focus of claims that there is vagueness \n‘in’ the world (Parfit 1984, pp. 238-241; Kripke 1972, p. 345 n. 18). Thus, \nit may be vague as to who is identical to whom after a brain-swap, to give \na much discussed example. Such claims have been dealt a forceful blow \nby the famous Evans-Salmon argument which runs as follows: suppose for \nreductio that it is indeterminate whether a = b. Then b definitely possesses \nthe property that it is indeterminate whether it is identical with a, but a \ndefinitely does not possess this property since it is surely not \nindeterminate whether a=a. Therefore, by Leibniz’s Law, it cannot be the \ncase that a=b and so the identity cannot be indeterminate (Evans 1978; \nSalmon 1982).
According to the standard interpretation of quantum mechanics (QM), no meaning can be assigned to the statement that a particle has a precise value of any one of the variables describing its physical propertes before having interacted with a suitable measuring instrument. On the other hand, it is well known that QM tends to classical statistical mechanics (CSM) when a suitable classical limit is performed. One may ask therefore how is it that in this limit, the statement, meaningless in QM, that a given variable has always a precise value independently of having been measured, gradually becomes meaningful. In other words, one may ask how can it be that QM, which is a theory describing the intrinsically probabilistic properties of a quantum object, becomes a statistical theory describing a probabilistic knowledge of intrinsically well determined properties of classical objects.
This thesis examines the concepts of identity and individuality via scientifically-informed philosophical analysis. It has two parts. The first part deals with metaphysical claims that turn out to be, in effect, very general empirical claims extracted from our (supposed, but rationally-accredited) knowledge of the world. I here compare two approaches: a) the Leibniz-Quine view of identity as a derivative relation, and, relatedly, of individuality as dependent on the qualities of things; and b) the view that identity is a non-analysable primitive, and, relatedly, that the individuality of things is not reducible to anything else. The former position, based on the Principle of the Identity of the Indiscernibles as a criterion of individuation, might appear prima facie more plausible. However, I argue that it runs into difficulties both at the level of a priori analysis and in terms of 'fit' with the evidence described by our best science. It is, in fact, not even as compelling from the empiricist point of view as is commonly believed. I therefore argue that the position that identity and individuality are primitive may be preferred. In the second part of the thesis - under the assumption that the proper role of metaphysics is to characterise the best solutions to issues that are left open by current science - I deal with the question regarding whether and how the ultimate constituents of reality can actually be conceived of as primitive individuals. I argue in favour of an ontology of tropes, develop the view in detail and defend it against various criticisms. The fundamental tropes that constitute the basic 'building blocks' of reality are identified with the state-independent properties of elementary particles. The way in which these constitute complex particulars is described, and certain peculiarities having to do with quantum statistics are accounted for. I conclude by suggesting possible avenues for further research.
Are there quantum objects? What are they if they are neither given nor resemble anything familiar? To answer these questions we have to abstract from the substantive features of familiar things, delineate the pure logical forms by which we acknowledge objects and show how the forms are fulfilled in quantum theories. We have to explicate, in general terms and without resorting to the given, what we mean by objects. The clear criteria will enable us to affirm the objectivity of quantum theories. Auyang 1995, p. 5 Introduction Cantor, famously, defined a set as ‘… collections into a whole of definite, distinct objects of our intuition or of our thought’ (Cantor 1955, p. 85, our emphasis). On this basis the standard formulations of set theory, and consequently much of contemporary mathematics, are erected. Reflecting upon this definition, and the underlying ontological presuppositions, the question immediately arises, how are we to treat, mathematically, collections of objects which are not distinct individuals? This question becomes particularly acute in the quantum mechanical context, of course. As Yu. I. Manin noted at the 1974 American Mathematical Society Congress on the Hilbert Problems, We should consider possibilities of developing a totally new language to speak about infinity [that is, axioms for set theory]. Classical critics of Cantor (Brouwer et al. ) argue that, say, the general choice is an illicit extrapolation of the finite case.
In a deterministic theory, one can deduce the state of the universe at any one time from the dynamical laws that govern its evolution and its state at any other time.In particular, one can deduce from the conditions that obtained at some early boundary condition for the universe, its state at all subsequent times.It is a piece of well-received philosophical common sense that there is no room in a deterministic theory for objective probabilities.Arguments will be given below that this piece of philosophical common sense is mistaken, and that a probability measure over the space of physically possible trajectories is an indispensable component of any theory -deterministic or otherwise -that can be used as a basis for prediction or receive confirmation from the evidence.Phase space is the central representational space in physics.It is the space within which all of the states pertaining to a system are represented, and it encodes all of the general information that we have about those states.To take an obvious example (less obvious examples will be given below) the fact that a pair of quantities are represented by different dimensions in phase space encodes the fact that they represent separate degrees of freedom.Physically possible histories are represented by trajectories through phase space and in a deterministic theory, when dealing with a closed system, there is (no more than) one trajectory through each point.Since the universe as a whole is the only truly closed system, this means there is one trajectory through each point in the phase space of the universe.Probabilities may be invoked in applications of theory, when the state of a system or the values of external variables are not known with full precision, but they are eliminable given complete knowledge.And they don't appear in the statement of the laws.The laws serve simply to identify the physically allowable trajectories.About Newtonian dynamics, Laplace famously said "given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it -an intelligence sufficiently vast to submit these data to analysis -it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, and the past, would be present to its eyes."ii It is a short step from this to the idea that probabilities invoked in classical contexts are subjective rather than objective.If they represent anything, it is facts about our epistemic states, not facts about the world.iii Popper, and Lewis, and other influential figures have endorsed this view, categorically rejecting any role for objective probabilities indeterministic contexts.In Popper's words:
What makes quantum theory so special? Everyone has an opinion on that: one can point at entanglement, non-locality, complementarity, uncertainty, and many other features. In a trivial sense, every answer is legitimate: every feature in which quantum theory differs from other familiar theories can be taken as the distinctive feature, because it allows you to tell quantum theory apart from what you already know. But this would not take you very far. The real question is not how to distinguish quantum theory from theories that you already know, but how to distinguish it from all theories that you could possibly imagine. Most features are not good at that – for example, physicists have invented many toy theories that exhibit entanglement, non-locality, and complementarity and still are not quantum theory. In this chapter we put forward our answer to the question of what makes quantum theory special: purification. In the following, we will explore some of the most fundamental facts about quantum theory, like entanglement and teleportation, connecting purification with a deep stream of thought that goes back to Schrödinger and von Neumann. The results presented in this chapter are also a first example of how one can reason on quantum theory starting from first principles. A Distinctive and Fundamental Trait The purification principle stipulates that, whenever you are ignorant about the state of a system A, you can always claim that your ignorance comes from the fact that A is a part of a larger system AB, of which you have full knowledge. When you do this, the pure state that you have to assign to the composite system AB is determined by the state of A in an essentially unique way. The purification of mixed states is a peculiar feature – surely, not one that we experience in our everyday life. How can you claim that you know A and B if you do not know A alone? This counterintuitive feature had been noted already in the early days of quantum theory, when Erwin Schrödinger famously wrote: “Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts .” And, in the same paper: “I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”