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Conventionalism in Reidâs âGeometry of Visiblesâ
The role of conventions in the formulation of Thomas Reidâs theory of the geometry of vision, which he calls the âgeometry of visiblesâ, is the subject of this investigation. In particular, we will examine the work of N. Daniels and R. Angell who have alleged that, respectively, Reidâs âgeometry of visiblesâ and the geometry of the visual field are non-Euclidean. As will be demonstrated, however, the construction of any geometry of vision is subject to a choice of conventions regarding the construction and assignment of its various properties, especially metric properties, and this fact undermines the claim for a unique non-Euclidean status for the geometry of vision. Finally, a suggestion is offered for trying to reconcile Reidâs direct realist theory of perception with his geometry of visibles
Do Spins Have Directions?
The standard Bloch sphere representation was recently generalized to the
'extended Bloch representation' describing not only systems of arbitrary
dimension, but also their measurements. This model solves the measurement
problem and is based on the 'hidden-measurement interpretation', according to
which the Born rule results from our lack of knowledge about the interaction
between measuring apparatus and the measured entity. We present here the
extended Bloch model and use it to investigate the nature of quantum spin and
its relation to our Euclidean space. We show that spin eigenstates cannot
generally be associated with directions in the Euclidean space, but only with
generalized directions in the Blochean space, which generally is a space of
higher dimension. Hence, spin entities have to be considered as genuine
non-spatial entities. We also show, however, that specific vectors can be
identified in the Blochean theater that are isomorphic to the Euclidean space
directions, and therefore representative of them, and that spin eigenstates
always have a predetermined orientation with respect to them. We use the
details of our results to put forward a new view of realism, that we call
'multiplex realism', providing a specific framework with which to interpret the
human observations and understanding of the component parts of the world.
Elements of reality can be represented in different theaters, one being our
customary Euclidean space, and another one the quantum realm, revealed to us
through our sophisticated experiments, whose elements of reality, in the
quantum jargon, are the eigenvalues and eigenstates. Our understanding of the
component parts of the world can then be guided by looking for the possible
connections, in the form of partial morphisms, between the different
representations, which is precisely what we do in this article with regard to
spin entities.Comment: 28 pages, 11 figure
Quantum mechanics as a theory of probability
We develop and defend the thesis that the Hilbert space formalism of quantum
mechanics is a new theory of probability. The theory, like its classical
counterpart, consists of an algebra of events, and the probability measures
defined on it. The construction proceeds in the following steps: (a) Axioms for
the algebra of events are introduced following Birkhoff and von Neumann. All
axioms, except the one that expresses the uncertainty principle, are shared
with the classical event space. The only models for the set of axioms are
lattices of subspaces of inner product spaces over a field K. (b) Another axiom
due to Soler forces K to be the field of real, or complex numbers, or the
quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's
theorem fully characterizes the probability measures on the algebra of events,
so that Born's rule is derived. (d) Gleason's theorem is equivalent to the
existence of a certain finite set of rays, with a particular orthogonality
graph (Wondergraph). Consequently, all aspects of quantum probability can be
derived from rational probability assignments to finite "quantum gambles". We
apply the approach to the analysis of entanglement, Bell inequalities, and the
quantum theory of macroscopic objects. We also discuss the relation of the
present approach to quantum logic, realism and truth, and the measurement
problem.Comment: 37 pages, 3 figures. Forthcoming in a Festschrift for Jeffrey Bub,
ed. W. Demopoulos and the author, Springer (Kluwer): University of Western
Ontario Series in Philosophy of Scienc
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