Online research in philosophy

The classical space-time structure is derived from the structure of an abstract infinite dimensional separable Hilbert space S. For this S is first realized as a Hilbert space H* of functions of abstract parameters. Such a realization is associated with the process of measuring position of macroscopic particles naturally occurring in the universe. The process of decoherence and collapse induced by the measurement is in return associated with the choice of a "decohered" submanifold M of realization H*. The submanifold M is then identified with the classical space-time. The mathematical formalism is developed which permits to recover the usual Riemannian geometry on space-time in terms of the Hilbert structure on S. The specific functional realizations of S are shown to produce space-times of different geometry and topology.

In the chapter “The Geometry of Visibles” in his ‘Inquiry into the Human Mind’, Thomas Reid constructs a special space, develops a special geometry for that space, and offers a natural model for this geometry. In doing so, Reid “discovers” non-Euclidean Geometry sixty years before the mathematicians. This paper examines this “discovery” and the philosophical motivations underlying it. By reviewing Reid’s ideas on visible space and confronting him with Kant and Berkeley, I hope, moreover, to resolve an alleged impasse in Reid’s philosophy concerning the contradictory characteristics of Reid’s tangible and visible space.

Cognitive impenetrability (CI) of a large part of visual perception is taken for granted by those of us in the field of computational vision who attempt to recover descriptions of space using geometry and statistics as tools. These tools clearly point out, however, that CI cannot extend to the level of structured descriptions of object surfaces, as Pylyshyn suggests. The reason is that visual space – the description of the world inside our heads – is a nonEuclidean curved space. As a consequence, the only alternative for a vision system is to develop several descriptions of space–time; these are representations of reduced intricacy and capture partial aspects of objective reality. As such, they make sense in the context of a class of tasks/actions/plans/purposes, and thus cannot be cognitively impenetrable.

Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds.

Visual space can be distinguished from physical space. The ?rst is found in visual experi- ence, while the second is de?ned independently of perception. Theorists have wondered about the relation between the two. Some investigators have concluded that visual space is non- Euclidean, and that it does not have a single metric structure. Here it is argued (1) that visual space exhibits contraction in all three dimensions with increasing distance from the observer, (2) that experienced features of this contraction (including the apparent convergence of lines in visual experience that are produced from physically parallel stimuli in ordinary viewing con- ditions) are not the same as would be the experience of a perspective projection onto a fronto- parallel plane, and (3) that such contraction is consistent with size constancy. These properties of visual space are di?erent from those that would be predicted if spatial perception resulted from the successful solution of the inverse problem. They are consistent with the notion that optical constraints have been internalized. More generally, they are also consistent with the notion that visual spatial structures bear a resemblance relation to physical spatial structures. This notion supports a type of representational relation that is distinct from mere causal cor- respondence. The reticence of some philosophers and psychologists to discuss the structure of phenomenal space is diagnosed in terms of the simple materialism and the functionalism of the 1970s and 1980s.

Visual space can be distinguished from physical space. The first is found in visual experience, while the second is defined independently of perception. Theorists have wondered about the relation between the two. Some investigators have concluded that visual space is non- Euclidean, and that it does not have a single metric structure. Here it is argued (1) that visual space exhibits contraction in all three dimensions with increasing distance from the observer, (2) that experienced features of this contraction (including the apparent convergence of lines in visual experience that are produced from physically parallel stimuli in ordinary viewing conditions) are not the same as would be the experience of a perspective projection onto a frontoparallel plane, and (3) that such contraction is consistent with size constancy. These properties of visual space are different from those that would be predicted if spatial perception resulted from the successful solution of the inverse problem. They are consistent with the notion that optical constraints have been internalized. More generally, they are also consistent with the notion that visual spatial structures bear a resemblance relation to physical spatial structures. This notion supports a type of representational relation that is distinct from mere causal correspondence. The reticence of some philosophers and psychologists to discuss the structure of phenomenal space is diagnosed in terms of the simple materialism and the functionalism of the 1970s and 1980s. Ó 2003 Elsevier B.V. All rights reserved.

My concern in this paper is with the aspect of the phenomenal character of visual experience that pertains to its spatial dimension. I shall refer to this aspect as visual space. Kant famously claimed that the representation of space is the a priori form of the faculty of sensibility. This claim is sometimes interpreted as the view that visual space is a pictorial canvas contributed by the mind on which sensations such as color experiences are organized. This reading, whether it is correct or not, consists of at least two independent claims. First, visual space is contributed by the mind. Second, visual space is a pictorial canvas. Both of theses claims, of course, demand further explication. However.

Parallels between the mathematics of tiling, which describes geometries of visual space, and neo-Riemannian theory, which describes geometries of musical space, make it possible to show that certain paradoxes featured in the visual artworks of M. C. Escher also appear in the pitch space modelled by the neo-Riemannian Tonnetz . This article makes these paradoxes visually apparent by constructing an embodied model of triadic pitch space in accordance with principles drawn from the mathematics of tiling, on the one hand, and from cognitive science, on the other – specifically, the notion that our experience of pitch relationships is governed in part by the metaphorical projection of patterns abstracted from embodied experience known as image schemas. These paradoxes are illustrated with reference to passages drawn from four compositions to whose expressive character such paradoxes contribute: the fifteenth-century motet 'Absalon fili mi'; the finale of Haydn's String Quartet in G major, Op. 76 No. 1; Brahms's Intermezzo in B minor, Op. 119 No. 1; and Wagner's Parsifal.

The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it, 2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense.