The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.

This dissertation pursues two tightly interwoven goals: to bring out the relevance of questions for the field of logic, and to establish a solid theory of the logic of questions within a classical logical setting. These enterprises feed into each other: on the one hand, the development of our formal systems is motivated by our considerations concerning the role to be played by questions; on the other hand, it is via the development of concrete, workable logical systems that the potential of questions in logic is made clear and tangible.

In this paper I put forward an approach to the problem of describing a particle in a field without assuming the space-time continuum. I deduce as much as possible from very simple assumptions concerning interactions between the elements of a « bootstrap >) type assemblage in which each particle in the assemblage is built out of the interactions of all the others. Interaction either exists (a situation denoted by the digit (1) or else it does not exist (denoted by the digit (0) and there is no other possibility. No dynamical properties are assumed for the particles beyond the discrete, all-or-none interactions, and dynamics, therefore, including the momentum concept has to be built later. The theory being proposed differs vitally in this respect from the bootstrap theories that are based on the S-matrix technique.

Nuevos experimentos apuntan a su naturaleza discreta, no de continuo. Un equipo de físicos ha detectado que la escala mínima de tiempo medible tiene varios órdenes de magnitud mayor que el tiempo de Planck, el mínimo establecido hasta la fecha. Esto, aplicado a las ecuaciones básicas de la mecánica cuántica, señalaría que la estructura del tiempo podría ser como la de un cristal, consistente en segmentos discretos que se repiten periódicamente.

Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum logic enable expression of the state geometry in Hilbert space. Quantum and classical mechanics are then elaborated and applied to subsystems and the measurement process. Consideration is also given to space-time geometry and the constraints this places on the dynamics. Physics and Mathematics, it is argued, are growing apart; the inadequate treatment of approximations in general and localisation in quantum mechanics in particular are seen as contributing factors. In the description of systems, the link between localisation and lack of knowledge shows that quantum mechanics should reflect the domain of applicability. Restricting the class of states provides a means of achieving this goal. Localisation is then shown to have a mathematical expression in terms of compactness, which in turn is applied to yield a topological theory of bound and scattering states.

Mahler has written many papers on the geometry of numbers. Arguably, his most influential achievements in this area are his compactness theorem for lattices, his work on star bodies and their critical lattices, and his estimates for the successive minima of reciprocal convex bodies and compound convex bodies. We give a, by far not complete, overview of Mahler’s work on these topics and their impact.

Whereas for many truths, truthmaker theory offers a plausible account, there are certain kinds of truths for which the theory seems less helpful: principally (though not exclusively) analytic truths. I argue that an augmentation of the usual idea of truthmakers can solve this problem. Moreover that once solved we are able to look afresh at the nature of mathematics, whether conceived as analytic or synthetic, necessary or contingent, and reduce the ontological options. I also argue that it was Quine’s reformulation of analyticity, deployed in place of the correct Leibniz-Wolff-Kant account, which led to his holistic account of knowledge and pragmatist account of scientific revisability.

Our collective views regarding the question “what is fundamental?” are continually evolving. These ontological shifts in what we regard as fundamental are largely driven by theoretical advances (“what can we calculate?”), and experimental advances (“what can we measure?”). Rarely (in my view) is epistemology the fundamental driver; more commonly epistemology reacts (after a few decades) to what is going on in the theoretical and experimental zeitgeist.

In The Hitchhiker’s Guide to the Galaxy, by Douglas Adams, the Answer to the Ultimate Question of Life, the Universe, and Everything is found to be 42 — but the meaning of this is left open to interpretation. We take it to mean that there are 42 fundamental questions which must be answered on the road to full enlightenment, and we attempt a first draft (or personal selection) of these ultimate questions, on topics ranging from the cosmological constant and origin of the universe to the origin of life and consciousness.

We argue that dualities offer new possibilities for relating fundamentality, levels, and emergence. Namely, dualities often relate two theories whose hierarchies of levels are inverted relative to each other, and so allow for new fundamentality relations, as well as for epistemic emergence. We find that the direction of emergence typically found in these cases is opposite to the direction of emergence followed in the standard accounts. Namely, the standard emergence direction is that of decreasing fundamentality: there is emergence of less fundamental, high-level entities, out of more fundamental, low-level entities. But in cases of duality, a more fundamental entity can emerge out of a less fundamental one. This possibility can be traced back to the existence of different classical limits in quantum field theories and string theories.