There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.

Beginning with Zagzebski (The Philosophical Quarterly 44:65–73, 1994), some philosophers have argued that there can be no solution to the Gettier counterexamples within the framework of a fallibilist theory of knowledge. If true, this would be devastating, since it is believed on good grounds that infallibilism leads to scepticism. But I argue here that these purported proofs are mistaken and that the truthmaker solution to the Gettier problems is both cogent and fallibilist in nature. To show this I develop the (...) notion of evidence of a state of affairs, a crucial concept in the truthmaker theory. I also argue that a common principle of the transmission of evidence through entailment is false, and the cause of much of the trouble. (shrink)

The significance of Max Black’s indistinguishable spheres for the nature of particles in quantum mechanics is discussed, focusing in particular on the use of the idea of weak indiscernibility. It is argued that there can be four such Black spheres but that five are impossible. It follows from this that Black’s example cannot serve as a model for indistinguishability in physics. But Black’s discussion of his spheres gave rise to the idea of weak discernibility and it is argued that such (...) predicates are unsatisfiable in the way intended. The underlying problem with weak discernibility spreads out to also undermine the whole notion that indistinguishability rests on a notion of the permutation invariance of particles. A better foundation is indicated. (shrink)

The indistinguishability of bosons and fermions has been an essential part of our ideas of quantum mechanics since the 1920s. But what is the mathematical basis for this indistinguishability? An answer was provided in the group representation theory that developed alongside quantum theory and quickly became a major part of its mathematical structure. In the 1930s such a complex and seemingly abstract theory came to be rejected by physicists as the standard functional analysis picture presented by John von Neumann took (...) hold. The purpose of the present account is to show how indistinguishability is explained within representation theory. (shrink)

It’s a truism of philosophy that Realists must not postulate more than we could reasonably hope to know, while Anti-Realists must not leave us with so little that all knowledge is impossible. But balance is not easily come by—and even less in philosophy than in life. So philosophy continues to struggle over the hard cases, with neither the Realist nor the Anti-Realist able to score an easy victory.

Since 1969, when Bas van Fraassen wrote 'Facts and Tautological Entailments', it has been assumed that if facts, or states of affairs, exist at all, they can only play the role of truthmakers for propositions if the truthmaker relation is defined in a relevantist revision of classical logic. Greg Restall revived this notion in 1996, and it has since been discussed positively by Stephen Read. I argue in this paper that this was always a mistake. The truthmaking relation between facts (...) and propositions can indeed be made sense of-but no relevantist revision of classical logic is required. The correspondence theory of truth can thus be shown to have been essentially correct all along. (shrink)

A proof is presented that a form of incompleteness in Quantum Mechanics follows directly from the use of unbounded operators. It is then shown that the problems that arise for such operators are not connected to the non- commutativity of many pairs of operators in Quantum Mechanics and hence are an additional source of incompleteness to that which allegedly flows from the..

It is argued that the part-whole account of the relation between evidence and the larger state of affairs the evidence is evidence of—an account that was elucidated in the paper ‘Truthmaking, Evidence of, and Impossibility Proofs’ —provides a better basis for epistemology than causal relations between events. I apply this to a well-known phenomenon in physics which suggests that causal connectedness is not necessary for knowledge.

There is a large literature on the issue of the lack of properties (i.e. accidents) in quantum mechanics (the problem of “hidden variables”) and also on the indistinguishability of particles. Both issues were discussed as far back as the late 1920’s. However, the implications of these challenges to classical ontology were taken up rather late, in part in the ‘quantum set theory’ of Takeuti (Curr Issues Quant Logic 303–322, 1981), Finkelstein (in Beltrametti EG, Van Fraassen BC (eds) Current issues in (...) quantum logic. Plenum, New York, 1981) and the work of Décio Krause (1992)—and subsequent publications). But the problems created by quantum mechanics go far beyond set theory or the identity of indiscernibles (another subject that has been often discussed)—it extends, I argue, to our accounts of truth. To solve this problem, i.e. to have an approach to truth that facilitates a transition from a classical to a quantum ontology one must have a unified framework for them both. This is done within the context of a pluralist view of truthmaking, where the truthmakers are unified in having a monoidal structure. The structure of the paper is as follows. After a brief introduction, the idea of a monoid is outlined (in Sect. 1) followed by a standard set of axioms that govern the truthmaker relation from elements of the monoid to the set of propositions. This is followed, in Sect. 2, by a discussion of how to have truthmakers for two kinds of necessities: tautologies and analytic truths. The next Sect. 3, then applies these ideas to quantum mechanics. It gives an account of quantum states and shows how these form a monoid. The final section then argues that quantum logic does not, despite what one might initially suspect, stand in the way of an account of quantum truth. (shrink)

Zeeman argued that the Euclidean (i. e. manifold) topology of Minkowski space-time should be replaced by a strictly finer topology that was to have a closer connection with the indefinite metric. This proposal was extended in 1976 by Rudiger Göbel and Hawking, King and McCarthy to the space-times of General Relativity. It is the purpose of this paper to argue that these suggestions for replacement misrepresent the significance of the manifold topology and overstate the necessity for a finer topology. The (...) motivation behind such arguments is a realist view of space-time topology as against (what can be construed to be) the instrumentalist position underlying some of the suggestions for replacement. (shrink)

That there is an edge at all is, of course, philosophically controversial; it would be denied by anti-realists of a verificationist stripe. However, we accept, since G¨odel, that there are true propositions of elementary arithmetic that are unprovable in arithmetic; just so, we should accept—by analogy—that there are true statements that are unknowable. An argument called the Fitch Argument tells us that it is so. Williamson has long argued that the Fitch Argument cannot by itself refute antirealism—because the anti-realist is (...) already committed to the denial of some of the principles of classical logic required to derive the anti-realist conclusion. The point is well made.1 In Knowledge and its Limits, however, Williamson looks at what the Fitch argument tells us if we adhere to classical logic: and that is that there are unknowable truths. (shrink)

The geometric mean is also called the mean proportional. This is how the mathematicians of the √ −1. 19th Century, such as Gauss, understood..

The Received View of particles in quantum mechanics is that they are indistinguishable entities within their kinds and that, as a consequence, they are not individuals in the metaphysical sense and self-identity does not meaningfully apply to them. Nevertheless cardinality does apply, in that one can have n> 1 such particles. A number of authors have recently argued that this cluster of claims is internally contradictory: roughly, that having more than one such particle requires that the concepts of distinctness and (...) identity must apply after all. A common thread here is that the notion of identity is too fundamental to forego in any metaphysical account. I argue that this argument fails. I then argue that the failure of individuality and identity applies also to macroscopic physical objects, that the problems cannot be constrained to apply only within the microscopic realm. (shrink)

In his essay ‘Laws and States in Quantum Mechanics’, John Forge presents a case for considering laws of nature to be privileged sets of states, trajectories in the quantum mechanical analogue of phase space. Having presented an argument to show that states have to be taken with full ontological seriousness, Forge then uses those states to undergird his favourite account of laws and explanation — called the Instance View. On this view laws are a special sort of pattern, a certain (...) kind of regularity, and explanations are instances of these regularities (so that you explain some phenomenon A by indicating that it is an instance of some regularity R). Forge then uses this account of laws and explanation to urge the acceptance of van Fraassen’s Modal Interpretation of Quantum Mechanics. (shrink)

In this paper I put forward a suggestion for identifying causality in micro-systems with the specific quantum field theoretic interactions that occur in such systems. I first argue — along the lines of general transference theories — that such a physicalistic account is essential to an understanding of causation; I then proceed to sketch the concept of interaction as it occurs in quantum field theory and I do so from both a formal and an informal point of view. Finally, I (...) present reasons for thinking that only a quantum field theoretic account can do the job — in particular I rely on a theorem by D. Currie and to the effect that interaction cannot be described in (a Hamiltonian formulation of) Classical Mechanics. Throughout the paper I attempt to suggest that the widespread scepticism about the ability of quantum theory to support a theory of causality is mistaken and rests on several misunderstandings. (shrink)