F. A. Muller

One of the reasons provided for the shift away from an ontology for physical reality of material objects & properties towards one of physical structures & relations (Ontological Structural Realism: OntSR) is that the quantum-mechanical description of composite physical systems of similar elementary particles entails they are indiscernible. As material objects, they 'whither away', and when they wither away, structures emerge in their stead. We inquire into the question whether recent results establishing the weak discernibility of elementary particles pose a threat for this quantum-mechanical reason for OntSR, because precisely their newly discovered discernibility prevents them from 'whithering away'. We argue there is a straightforward manner to consider the recent results as a reason in favour of OntSR rather than against it

Alan Musgrave, Michael Friedman, Jeffrey Foss, and Richard Creath raised different objections against the Distinction between observables and unobservables when drawn within the confines of Bas C. van Fraassen's Constructive Empiricism, to the effect that the Distinction cannot be drawn there coherently. Van Fraassen has only responded to Musgrave but Musgrave claimed not to understand van Fraassen's succinct response. I argue that van Fraassen's response is not enough. What remains in the end is an unsolved problem which CE cannot afford to leave unsolved, or so I argue; I then strengthen Musgrave's criticism and indicate that an extension of the epistemic policy of CE is mandatory to solve the problem. I also argue that Friedman's and Foss' objection against the Distinction in CE misses the mark on closer inspection. An objection due to Creath does hit the mark but can be taken care of without too much ado. All these objections seem alive and kicking until the present day; I try to put them all to rest

Once Hilbert asserted that the axioms of a theory `define` theprimitive concepts of its language `implicitly''. Thus whensomeone inquires about the meaning of the set-concept, thestandard response reads that axiomatic set-theory defines itimplicitly and that is the end of it. But can we explainthis assertion in a manner that meets minimum standards ofphilosophical scrutiny? Is Jané (2001) wrong when hesays that implicit definability is ``an obscure notion''''? Doesan explanation of it presuppose any particular view on meaning?Is it not a scandal of the philosophy of mathematics that no answersto these questions are around? We submit affirmative answers to allquestions. We argue that a Wittgensteinian conception of meaninglooms large beneath Hilbert''s conception of implicit definability.Within the specific framework of Horwich''s recent Wittgensteiniantheory of meaning called semantic deflationism, we explain anexplicit conception of implicit definability, and then go on toargue that, indeed, set-theory, defines the set-conceptimplicitly according to this conception. We also defend Horwich''sconception against a recent objection from the Neo-Fregeans Hale and Wright (2001). Further, we employ the philosophicalresources gathered to dissolve all traditional worries about thecoherence of the set-concept, raisedby Frege, Russell and Max Black, and whichrecently have been defended vigorously by Hallett (1984) in hismagisterial monograph Cantorian set-theory and limitationof size. Until this day, scandalously, these worries havebeen ignored too by philosophers of mathematics.

In a recent issue of this journal, M. Frisch claims to have proven that classical electrodynamics is an inconsistent physical theory. We argue that he has applied classical electrodynamics inconsistently. Frisch also claims that all other classical theories of electromagnetic phenomena, when consistent and in some sense an approximation of classical electrodynamics, are haunted by “serious conceptual problems” that defy resolution. We argue that this claim is based on a partisan if not misleading presentation of theoretical research in classical electrodynamics.

The author endeavours to show two things: first, that Schrödingers (and Eckarts) demonstration in March (September) 1926 of the equivalence of matrix mechanics, as created by Heisenberg, Born, Jordan and Dirac in 1925, and wave mechanics, as created by Schrödinger in 1926, is not foolproof; and second, that it could not have been foolproof, because at the time matrix mechanics and wave mechanics were neither mathematically nor empirically equivalent. That they were is the Equivalence Myth. In order to make the theories equivalent and to prove this, one has to leave the historical scene of 1926 and wait until 1932, when von Neumann finished his magisterial edifice. During the period 1926–1932 the original families of mathematical structures of matrix mechanics and of wave mechanics were stretched, parts were chopped off and novel structures were added. To Procrustean places we go, where we can demonstrate the mathematical, empirical and ontological equivalence of ‘the final versions of’ matrix mechanics and wave mechanics. The present paper claims to be a comprehensive analysis of one of the pivotal papers in the history of quantum mechanics: Schrödingers equivalence paper. Since the analysis is performed from the perspective of Suppes structural view (‘semantic view’) of physical theories, the present paper can be regarded not only as a morsel of the internal history of quantum mechanics, but also as a morsel of applied philosophy of science. The paper is self-contained and presupposes only basic knowledge of quantum mechanics. For reasons of length, the paper is published in two parts; Part I appeared in the previous issue of this journal. Section 1 contains, besides an introduction, also the papers five claims and a preview of the arguments supporting these claims; so Part I, Section 1 may serve as a summary of the paper for those readers who are not interested in the detailed arguments.

In his 2009 PSA Recent Ph.D. Award winning contribution to the bi-annual PSA Conference at Pittsburgh in 2008, C. Wu ̈thrich mounted an argument against struc- turalism about space-time in the context of the General Theory of Relativity, to the effect that structuralists cannot discern space-time points. An “abysmal embarrass- ment” for the structuralist, Wu ̈thrich judged. Wu ̈thrich’s characterisation of space-time structuralism is however incorrect. We demonstrate how, on the basis of a correct char- acterisation of space-time structuralism, it is possible to discern space-time points in the GTR-structures under consideration. Thus Wu ̈thrich’s argument crumbles

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

The quest for finding the right interpretation of Quantum Mechanics is as old as QM and still has not ended, and may never end. The question what an interpretation of QM is has hardly ever been raised explicitly, let alone answered. We raise it and answer it. Then the quest for the right interpretation can continue self-consciously, for we then know exactly what we are after. We present a list of minimal requirements that something has to meet in order to qualify as an interpretation of QM. We also raise, as a side issue, the question how the discourse on the interpretation of QM relates to hermeneutics in Continental Philosophy.

This paper, accessible for a general philosophical audience having only some fleeting acquaintance with set-theory and category-theory, concerns the philosophy of mathematics, specifically the bearing of category-theory on the foundations of mathematics. We argue for six claims. (I) A founding theory for category-theory based on the primitive concept of a set or a class is worthwile to pursue. (II) The extant set-theoretical founding theories for category-theory are conceptually flawed. (III) The conceptual distinction between a set and a class can be seen to be formally codified in Ackermann's axiomatisation of set-theory. (IV) A slight but significant deductive extension of Ackermann's theory of sets and classes founds Cantorian set-theory as well as category-theory, and therefore can pass as a founding theory of the whole of mathematics. (V) The extended theory does not suffer from the conceptual flaws of the extant set-theoretical founding theories. (VI) The extended theory is not only conceptually but also logically superior to the competing set-theories because its consistency can be proved on the basis of weaker assumptions than the consistency of the competition.

In a recent issue of this journal, P.E. Vermaas ([2005]) claims to have demonstrated that standard quantum mechanics is technologically inadequate in that it violates the 'technical functions condition'. We argue that this claim is false because based on a 'narrow' interpretation of this technical functions condition that Vermaas can only accept on pain of contradiction. We also argue that if, in order to avoid this contradiction, the technical functions condition is interpreted 'widely' rather than 'narrowly', then Vermaas, argument for his claim collapses. The conclusion is that Vermaas' claim that standard quantum mechanics is technologically inadequate evaporates

We maximally extend the quantum‐mechanical results of Muller and Saunders ( 2008 ) establishing the ‘weak discernibility’ of an arbitrary number of similar fermions in finite‐dimensional Hilbert spaces. This confutes the currently dominant view that ( A ) the quantum‐mechanical description of similar particles conflicts with Leibniz’s Principle of the Identity of Indiscernibles (PII); and that ( B ) the only way to save PII is by adopting some heavy metaphysical notion such as Scotusian haecceitas or Adamsian primitive thisness. We take sides with Muller and Saunders ( 2008 ) against this currently dominant view, which has been expounded and defended by many. *Received July 2008; revised May 2009. †To contact the authors, please write to: F. A. Muller, Faculty of Philosophy, Erasmus University Rotterdam, Burg. Oudlaan 50, H5–16, 3062 PA Rotterdam, The Netherlands; e‐mail: [email protected] , and Institute for the History and Foundations of Science, Utrecht University, Budapestlaan 6, IGG–3.08, 3584 CD Utrecht, The Netherlands; e‐mail: [email protected] . M. P. Seevinck, Institute for the History and Foundations of Science, Utrecht University, Budapestlaan 6, IGG–3.08, 3584 CD Utrecht, The Netherlands; e‐mail: [email protected].

The author endeavours to show two things: first, that Schrödingers (and Eckarts) demonstration in March (September) 1926 of the equivalence of matrix mechanics, as created by Heisenberg, Born, Jordan and Dirac in 1925, and wave mechanics, as created by Schrödinger in 1926, is not foolproof; and second, that it could not have been foolproof, because at the time matrix mechanics and wave mechanics were neither mathematically nor empirically equivalent. That they were is the Equivalence Myth. In order to make the theories equivalent and to prove this, one has to leave the historical scene of 1926 and wait until 1932, when von Neumann finished his magisterial edifice. During the period 1926–1932 the original families of mathematical structures of matrix mechanics and of wave mechanics were stretched, parts were chopped off and novel structures were added. To Procrustean places we go, where we can demonstrate the mathematical, empirical and ontological equivalence of ‘the final versions of’ matrix mechanics and wave mechanics. The present paper claims to be a comprehensive analysis of one of the pivotal papers in the history of quantum mechanics: Schrödingers equivalence paper. Since the analysis is performed from the perspective of Suppes structural view (‘semantic view’) of physical theories, the present paper can be regarded not only as a morsel of the internal history of quantum mechanics, but also as a morsel of applied philosophy of science. The paper is self-contained and presupposes only basic knowledge of quantum mechanics. For reasons of length, the paper is published in two parts; Part I appeared in the previous issue of this journal. Section 1 contains, besides an introduction, also the papers five claims and a preview of the arguments supporting these claims; so Part I, Section 1 may serve as a summary of the paper for those readers who are not interested in the detailed arguments.