1,780 research outputs found
A Topos Foundation for Theories of Physics: IV. Categories of Systems
This paper is the fourth in a series whose goal is to develop a fundamentally
new way of building theories of physics. The motivation comes from a desire to
address certain deep issues that arise in the quantum theory of gravity. Our
basic contention is that constructing a theory of physics is equivalent to
finding a representation in a topos of a certain formal language that is
attached to the system. Classical physics arises when the topos is the category
of sets. Other types of theory employ a different topos. The previous papers in
this series are concerned with implementing this programme for a single system.
In the present paper, we turn to considering a collection of systems: in
particular, we are interested in the relation between the topos representation
for a composite system, and the representations for its constituents. We also
study this problem for the disjoint sum of two systems. Our approach to these
matters is to construct a category of systems and to find a topos
representation of the entire category.Comment: 38 pages, no figure
Tangled closure algebras
The tangled closure of a collection of subsets of a topological space is the largest subset in which each member of the collection is dense. This operation models a logical ‘tangle modality’ connective, of significance in finite model theory. Here we study an abstract equational algebraic formulation of the operation which generalises the McKinsey-Tarski theory of closure algebras. We show that any dissectable tangled closure algebra, such as the algebra of subsets of any metric space without isolated points, contains copies of every finite tangled closure algebra. We then exhibit an example of a tangled closure algebra that cannot be embedded into any complete tangled closure algebra, so it has no MacNeille completion and no spatial representation
Spatial logic of tangled closure operators and modal mu-calculus
There has been renewed interest in recent years in McKinsey and Tarski’s interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces. We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame. This yields completeness theorems over X for a number of languages: (i) the modal mucalculus with the closure operator ; (ii) and the tangled closure operators (in fact can express ); (iii) , ∀; (iv) , ∀, ; (v) the derivative operator ; (vi) and the associated tangled closure operators ; (vii) , ∀; (viii) , ∀,. Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with , X validates the well-known axiom G1. For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connecte
A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
This paper is the second in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper, we study in depth the topos representation of the
propositional language, PL(S), for the case of quantum theory. In doing so, we
make a direct link with, and clarify, the earlier work on applying topos theory
to quantum physics. The key step is a process we term `daseinisation' by which
a projection operator is mapped to a sub-object of the spectral presheaf--the
topos quantum analogue of a classical state space. In the second part of the
paper we change gear with the introduction of the more sophisticated local
language L(S). From this point forward, throughout the rest of the series of
papers, our attention will be devoted almost entirely to this language. In the
present paper, we use L(S) to study `truth objects' in the topos. These are
objects in the topos that play the role of states: a necessary development as
the spectral presheaf has no global elements, and hence there are no
microstates in the sense of classical physics. Truth objects therefore play a
crucial role in our formalism.Comment: 34 pages, no figure
Contextual logic for quantum systems
In this work we build a quantum logic that allows us to refer to physical
magnitudes pertaining to different contexts from a fixed one without the
contradictions with quantum mechanics expressed in no-go theorems. This logic
arises from considering a sheaf over a topological space associated to the
Boolean sublattices of the ortholattice of closed subspaces of the Hilbert
space of the physical system. Differently to standard quantum logics, the
contextual logic maintains a distributive lattice structure and a good
definition of implication as a residue of the conjunction.Comment: 16 pages, no figure
Building of the global movement for health equity: from Santiago to Rio and beyond.
Health inequalities are present throughout the world, both within and between countries. The Commission on Social Determinants of Health drew attention to dramatic social gradients in health within most countries and made proposals for action. These inequalities are not inevitable. The purpose of this article is to report on activity that has taken place worldwide after the report by the Commission on Social Determinants of Health. First, we summarise the global situation. Second, we summarise an interim report of the emerging findings from an independent review of social determinants and the health divide, which was commissioned by the WHO European region. The world conference on social determinants of health will be held in Rio de Janeiro, Brazil, in October, 2011. This summit provides an opportunity to galvanise support, prioritise action, and respond to the call by the Commission on Social Determinants of Health for social justice as a route to a fair distribution of health
A Topological Study of Contextuality and Modality in Quantum Mechanics
Kochen-Specker theorem rules out the non-contextual assignment of values to
physical magnitudes. Here we enrich the usual orthomodular structure of quantum
mechanical propositions with modal operators. This enlargement allows to refer
consistently to actual and possible properties of the system. By means of a
topological argument, more precisely in terms of the existence of sections of
sheaves, we give an extended version of Kochen-Specker theorem over this new
structure. This allows us to prove that contextuality remains a central feature
even in the enriched propositional system.Comment: 10 pages, no figures, submitted to I. J. Th. Phy
- …