Bertrand Russell is concerned in this book with the foundations of knowledge. He approaches his subject through a discussion of language, the relationships of truth to experience and an investigation into how knowledge of the structure of language helps our understanding of the structure of the world. This edition includes a new introduction by Thomas Baldwin, Clare College, Cambridge.

We show that the notion of cardinality of a set is independent from that of wellordering, and that reasonable total notions of cardinality exist in every model of ZF where the axiom of choice fails. Such notions are either definable in a simple and natural way, or non-definable, produced by forcing. Analogous cardinality notions exist in nonstandard models of arithmetic admitting nontrivial automorphisms. Certain motivating phenomena from quantum mechanics are also discussed in the Appendix.

We demonstrate that the quantum-mechanical description of composite physical systems of an arbitrary number of similar fermions in all their admissible states, mixed or pure, for all finite-dimensional Hilbert spaces, is not in conflict with Leibniz's Principle of the Identity of Indiscernibles (PII). We discern the fermions by means of physically meaningful, permutation-invariant categorical relations, i.e. relations independent of the quantum-mechanical probabilities. If, indeed, probabilistic relations are permitted as well, we argue that similar bosons can also be discerned in all (...) their admissible states; but their categorical discernibility turns out to be a state-dependent matter. In all demonstrated cases of discernibility, the fermions and the bosons are discerned (i) with only minimal assumptions on the interpretation of quantum mechanics; (ii) without appealing to metaphysical notions, such as Scotusian haecceitas, Lockean substrata, Postian transcendental individuality or Adamsian primitive thisness; and (iii) without revising the general framework of classical elementary predicate logic and standard set theory, thus without revising standard mathematics. This confutes: (a) the currently dominant view that, provided (i) and (ii), the quantum-mechanical description of such composite physical systems always conflicts with PII; and (b) that if PII can be saved at all, the only way to do it is by adopting one or other of the thick metaphysical notions mentioned above. Among the most general and influential arguments for the currently dominant view are those due to Schrodinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Giuntini, Mittelstaedt, Castellani, Krause and Huggett. We review them succinctly and critically as well as related arguments by van Fraassen and Massimi. (shrink)

'Object' and 'criterion of identity' are philosophical terms of art whose application lies at a considerable theoretical remove from the surface phenomena of everyday linguistic usage. This partly explains their highly controversial status, for their point of application lies precisely where the concerns of linguists and philosophers of language merge with those of metaphysicians. This chapter explains the possession of determinate identity‐conditions. It argues that the distinction between 'abstract' and 'concrete' objects is itself a highly controversial one, and although it (...) has indeed been argued that this distinction turns ultimately upon differences between the criteria of identity governing objects of these two broad categories. It examines the role which criteria of identity play in our talk about objects of the least controversial varieties. The chapter concerns the problem of identity over time and the paradoxes to which identity criteria often appear to give rise when time is brought into the picture. (shrink)

In a naïve realist approach to reading an ontology off the models of a physical theory, the invariance of a given theory under permutations of its property-bearing objects entails the existence of distinct possible worlds from amongst which the theory cannot choose. A brand of Ontic Structural Realism attempts to avoid this consequence by denying that objects possess primitive identity, and thus worlds with property values permuted amongst those objects are really one and the same world. Assuming that any successful (...) ontology of objects is able to describe a universe containing a determinate number of them, I argue that no version of OSR which both retains objects and understands ‘structure’ in terms of relations can be successful. This follows from the fact that no set of relational facts is sufficient to fix the cardinality of the collection of objects implied by those facts. More broadly, I offer reasons to believe that no version of OSR is compatible with the existence of objects, no matter how ontologically derivative they are taken to be. Any such account would have to attribute a definite cardinality to a collection of objects while denying that those objects are possessed of a primitive identity. With no compelling reason to abandon the classical conception of cardinality, such an attribution is incoherent. (shrink)

Quasi-set theory has been proposed as a means of handling collections of indiscernible objects. Although the most direct application of the theory is quantum physics, it can be seen per se as a non-classical logic (a non-reflexive logic). In this paper we revise and correct some aspects of quasi-set theory as presented in [12], so as to avoid some misunderstandings and possible misinterpretations about the results achieved by the theory. Some further ideas with regard to quantum field theory are also (...) advanced in this paper. (shrink)

The concept of individuality in quantum mechanics shows radical differences from the concept of individuality in classical physics, as E. Schrödinger pointed out in the early steps of the theory. Regarding this fact, some authors suggested that quantum mechanics does not possess its own language, and therefore, quantum indistinguishability is not incorporated in the theory from the beginning. Nevertheless, it is possible to represent the idea of quantum indistinguishability with a first-order language using quasiset theory (Q). In this work, we (...) show that Q cannot capture one of the most important features of quantum non-individuality, which is the fact that there are quantum systems for which particle number is not well defined. An axiomatic variant of Q, in which quasicardinal is not a primitive concept (for a kind of quasisets called finite quasisets), is also given. This result encourages the searching of theories in which the quasicardinal, being a secondary concept, stands undefined for some quasisets, besides showing explicitly that in a set theory about collections of truly indistinguishable entities, the quasicardinal needs not necessarily be a primitive concept. (shrink)

The symmetrization postulates of quantum mechanics (symmetry for bosons, antisymmetry for fermions) are usually taken to entail that quantum particles of the same kind (e.g., electrons) are all in exactly the same state and therefore indistinguishable in the strongest possible sense. These symmetrization postulates possess a general validity that survives the classical limit, and the conclusion seems therefore unavoidable that even classical particles of the same kind must all be in the same state—in clear conflict with what we know about (...) classical particles. In this article we analyze the origin of this paradox. We shall argue that in the classical limit classical particles emerge, as new entities that do not correspond to the “particle indices” defined in quantum mechanics. Put differently, we show that the quantum mechanical symmetrization postulates do not pertain to particles, as we know them from classical physics, but rather to indices that have a merely formal significance. This conclusion raises the question of whether many discussions in the literature about the status of identical quantum particles have not been misguided. (shrink)

Quasi-set theory Q is an alternative set-theory designed to deal mathematically with collections of indistinguishable objects. The intended interpretation for those objects is the indistinguishable particles of non-relativistic quantum mechanics, under one specific interpretation of that theory. The notion of cardinal of a collection in Q is treated by the concept of quasi-cardinal, which in the usual formulations of the theory is introduced as a primitive symbol, since the usual means of cardinal definition fail for collections of indistinguishable objects. In (...) a recent work, Domenech and Holik have proposed a definition of quasi-cardinality in Q. They claimed their definition of quasi-cardinal not only avoids the introduction of that notion as a primitive one, but also that it may be seen as a first step in the search for a version of Q that allows for a greater representative power. According to them, some physical systems can not be represented in the usual formulations of the theory, when the quasi-cardinal is considered as primitive. In this paper, we discuss their proposal and aims, and also, it is presented a modification from their definition we believe is simpler and more general. (shrink)

Steven French and Decio Krause examine the metaphysical foundations of quantum physics. They draw together historical, logical, and philosophical perspectives on the fundamental nature of quantum particles and offer new insights on a range of important issues. Focusing on the concepts of identity and individuality, the authors explore two alternative metaphysical views; according to one, quantum particles are no different from books, tables, and people in this respect; according to the other, they most certainly are. Each view comes with certain (...) costs attached and after describing their origins in the history of quantum theory, the authors carefully consider whether these costs are worth bearing. Recent contributions to these discussions are analyzed in detail and the authors present their own original perspective on the issues. The final chapter suggests how this perspective can be taken forward in the context of quantum field theory. (shrink)