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

Our Relationships to Nature and Loss Through Art

Societal forces that push ideas of productivity, individual gains, and material wealth have distorted humankind’s relationship with the natural world. The sooner more of us acknowledge that nature is composed of living entities whose fates are interlaced with our own, the sooner we can alter our lives to focus on a relationship of respect and reciprocity with the natural world. Art is a medium that can communicate threatened aspects of nature due to climate change, while also provoking viewers to become more aware of their connections with nature. Drawing and painting are contemplative processes that can tell a story of times past, loss, and hope for the future. My project consisted of drawings signifying the effects of climate change on local Vermont species to distill such a vast issue into a comprehensible and personal form. These drawings were incorporated into a website allowing for clear explanation beyond the artistic interpretation of the viewer. Finally, my project invited the public to experience the artistic process for themselves by painting with water on “magic” canvases that turn black when wet and fade as the water evaporates. The public’s encounters with these canvases invited people to break from the hustle of everyday life. The outdoor setting and adjoining online graphics encouraged people to reflect on their relationships with nature. As the water evaporated, the fading of people’s paintings represented the metaphorical loss of biodiversity we face worldwide. People responded to this experience in notable ways, including recognizing the importance of nature in their lives and what we stand to lose. Unlike existing forms of climate change art that instill fear in viewers as a call to action, my project allowed people to recognize their appreciation for nature and as a result find more ways to enjoy and protect it. The poignancy of this experience will carry with people and inspire us all to strengthen our relationships with the Earth and consequently safeguard it

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