Notions of Relative Ubiquity for Invariant Sets of Relational Structures

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers w as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on w. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on w is ubiquitous in the set of linear orderings on w

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Temporal naturalism

Two people may claim both to be naturalists, but have divergent conceptions of basic elements of the natural world which lead them to mean different things when they talk about laws of nature, or states, or the role of mathematics in physics. These disagreements do not much affect the ordinary practice of science which is about small subsystems of the universe, described or explained against a background, idealized to be fixed. But these issues become crucial when we consider including the whole universe within our system, for then there is no fixed background to reference observables to. I argue here that the key issue responsible for divergent versions of naturalism and divergent approaches to cosmology is the conception of time. One version, which I call temporal naturalism, holds that time, in the sense of the succession of present moments, is real, and that laws of nature evolve in that time. This is contrasted with timeless naturalism, which holds that laws are immutable and the present moment and its passage are illusions. I argue that temporal naturalism is empirically more adequate than the alternatives, because it offers testable explanations for puzzles its rivals cannot address, and is likely a better basis for solving major puzzles that presently face cosmology and physics. This essay also addresses the problem of qualia and experience within naturalism and argues that only temporal naturalism can make a place for qualia as intrinsic qualities of matter

Triangleland. I. Classical dynamics with exchange of relative angular momentum

In Euclidean relational particle mechanics, only relative times, relative angles and relative separations are meaningful. Barbour--Bertotti (1982) theory is of this form and can be viewed as a recovery of (a portion of) Newtonian mechanics from relational premises. This is of interest in the absolute versus relative motion debate and also shares a number of features with the geometrodynamical formulation of general relativity, making it suitable for some modelling of the problem of time in quantum gravity. I also study similarity relational particle mechanics (`dynamics of pure shape'), in which only relative times, relative angles and {\sl ratios of} relative separations are meaningful. This I consider firstly as it is simpler, particularly in 1 and 2 d, for which the configuration space geometry turns out to be well-known, e.g. S^2 for the `triangleland' (3-particle) case that I consider in detail. Secondly, the similarity model occurs as a sub-model within the Euclidean model: that admits a shape--scale split. For harmonic oscillator like potentials, similarity triangleland model turns out to have the same mathematics as a family of rigid rotor problems, while the Euclidean case turns out to have parallels with the Kepler--Coulomb problem in spherical and parabolic coordinates. Previous work on relational mechanics covered cases where the constituent subsystems do not exchange relative angular momentum, which is a simplifying (but in some ways undesirable) feature paralleling centrality in ordinary mechanics. In this paper I lift this restriction. In each case I reduce the relational problem to a standard one, thus obtain various exact, asymptotic and numerical solutions, and then recast these into the original mechanical variables for physical interpretation.Comment: Journal Reference added, minor updates to References and Figure

The Affordances of Place: Digital Agency and the Lived Spaces of Information

International audienceAffordances provide a useful frame for understanding how users interact with devices, but applications of the term to digital devices must take into account that human agents operate within a material environment that is distinct from the digital environment in which these devices operate. A more restrictive approach to affordances would focus on the agency of digital devices distinct from the agency of human users. Location-aware mobile devices provide a particularly compelling example of the complex interplay of agents and agencies, and how “augmented affordances†give rise to a lived space of information for human users

The End of Time?

I discuss J. Barbour's Machian theories of dynamics, and his proposal that a Machian perspective enables one to solve the problem of time in quantum geometrodynamics (by saying that there is no time). I concentrate on his recent book 'The End of Time' (1999).Comment: 48 pages Latex. A shortened version will appear in 'The British Journal for Philosophy of Science