The Metaphysics of Quantities

This book articulates and defends a new and original answer to two questions: What are physical quantities and what makes them quantitative? This novel position—substantival structuralism—says that quantitativeness is an irreducible feature of particular attributes, and quantitative attributes are best understood as substantival structured spaces. Physical quantities like mass, momentum, or temperature play an important role in formulating laws of nature and in testing scientific theories. It is therefore important to have a clear philosophical understanding of what makes these attributes special. Traditional views of quantities have either suggested that quantities are determinables, that is, attributes that require determination by magnitudes, or that quantities are in some sense numerical, but neither view is satisfactory. The book shows how to use the representational theory of measurement to provide a better, more abstract criterion for quantitativeness: only attributes whose numerical representation has a high degree of uniqueness are quantitative. The best ontology for quantities is offered by a form of sophisticated substantivalism applied to quantities as structured spaces. Substantivalism, because an infinite domain is required to satisfy the formal requirements of quantitativeness; structured spaces, because they contain fundamental relations; sophisticated substantivalism because the identity of positions in such spaces is irrelevant. The resulting view is a form structuralism about quantities. The topic of the book falls squarely in the metaphysics of science, with contributions to general metaphysics and philosophy of science.

Imaginaries and invariant types in existentially closed valued differential fields

We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.

GLOBAL EXISTENCE AND FINITE TIME BLOW-UP OF SOLUTIONS TO A NONLOCAL P-LAPLACE EQUATION

In this paper a class of nonlocal diffusion equations associated with a p-Laplace operator, usually referred to as p-Kirchhoff equations, are studied. By applying Galerkin’s approximation and the modified potential well method, we obtain a threshold result for the solutions to exist globally or to blow up in finite time for subcritical and critical initial energy. The decay rate of the L 2 norm is also obtained for global solutions. When the initial energy is supercritical, an abstract criterion is given for the solutions to exist globally or to blow up in finite time, in terms of two variational numbers. These generalize some recent results obtained in [Y. Han and Q. Li, Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(9):3283–3297, 2018].