According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)

In this paper I consider the relationship between the spatial properties of visual perceptual experience and the sense-datum inference. I argue that the sense datum inference should be accepted if spatial properties are not merely intentionally present in such experiences. This result serves to underline the seriousness of the difficulties that are presented to direct realism by a particular class of illusory spatial experiences based on the geometry of visual perceptual experience. In light of these considerations I argue that it (...) is a constraint on direct realism that the underlying spatial character of visual perceptual experiences must be the same as that of the physical world. (shrink)

An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be found already (...) in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (shrink)

Provided we agree about the thing, it is needless to dispute about the terms. —David Hume, A treatise of human nature, Book 1, section VIIMap-like representations are frequently invoked as an alternative type of representational vehicle to a language of thought. This view presupposes that map-systems and languages form legitimate natural kinds of cognitive representational systems. I argue that they do not, because the collections of features that might be taken as characteristic of maps or languages do not themselves provide (...) scientifically useful information above and beyond what the individual features provide. To bring this out, I sketch several allegedly distinctive features of maps, and show how they can easily be grafted onto a simple logical language, resulting in a hybrid “manguage.” The ease with which these linguistic and map-like properties can be integrated into a single representational system raises the question of what work broader categories like language and map are doing. I maintain that, as an empirical matter of fact, they serve no particular purpose in cognitive science, and thus do not characterize distinct kinds of cognitive architectures. (shrink)

Poincare’s arguments for his thesis of the conventionality of metric depend on a relationalist program for dynamics, not on any general philosophical interpretation of science. I will sketch Poincare’s development of the relationalist program and show that his arguments for the conventionality of metric do not depend on any global strategies such as a general empiricism or Duhemian underdetermination arguments. Poincare’s theory of space, while empirically false, is more philosophically sophisticated than his critics have claimed.

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 that visual space exhibits contraction in all three dimensions with increasing distance from the observer, that experienced features of this contraction are not the same as (...) would be the experience of a perspective projection onto a fronto-parallel plane, and 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. (shrink)

Our aim is to propose several itineraries which follow the scenario having as a first step Euclid’s Fifth Postulate; as a second step the Bolyai–Lobachevsky’s non-Euclidean geometries and as a third step Einstein’s relativity theory. The role of Euclid’s fifth postulate is successively assumed by Archimedes’ axiom; Zermelo’s choice axiom; Cantor’s continuum hypothesis; von Neumann’s foundation axiom for set theory; Church–Turing thesis and Turing’s computability; the validity of classical logic under the form of the principles of identity, non-contradiction and excluded (...) middle; the macroscopic universe assumed as form of existence, including the fact that humans are macroscopic beings; human language and semiosis assumed as the unique way to approach understanding and communication; the old Greek ideal of order, symmetry and harmony, adopted for representation of the world in Art and Poetry. The common denominator of these starting steps is the fact that they all correspond to the sensorial-empirical perception of the world. We point out the strong interaction among the steps corresponding, in these different itineraries, to Einstein’s relativity theory. We infer in this way the need of a transdisciplinary approach. (shrink)

The empirical study of visual space has centered on determining its geometry, whether it is a perspective projection, flat or curved, Euclidean or non-Euclidean, whereas the topology of space consists of those properties that remain invariant under stretching but not tearing. For that reason distance is a property not preserved in topological space whereas the property of spatial order is preserved. Specifically the topological properties of dimensionality, orientability, continuity, and connectivity define “real” space as studied by physics and are the (...) spatial properties that characterize the physical universe as being an integral whole. By contrast the geometrical analysis of VS has taken little cognizance of its topology. Instead such properties have been presupposed a priori rather than being established a posteriori by empirical means, perhaps because these properties are self-evident. Applying the method of coordinative definition expounded by Hans Reichenbach for determining geometrical and topological properties of physical space, it can be shown that VS fulfills the topological criteria of being a “real” space sui generis. Though theorized to be produced by the brain, the topology of VS is not topologically equivalent with the structure and activity of the brain because, as will be shown, the topology of VS cannot be formed from the topology of the brain without tearing and/or cutting and pasting. (shrink)

Abstract: In this paper I consider recent attempts to establish that the geometry of visual experience is a spherical geometry. These attempts, offered by Gideon Yaffe, James van Cleve and Gordon Belot, follow Thomas Reid in arguing for an equivalency of a geometry of ‘visibles’ and spherical geometry. I argue that although the proposed equivalency is successfully established by the strongest form of the argument, this does not warrant any conclusion about the geometry of visual experience. I argue, firstly, that (...) the resistance of this contemporary argument to empirical considerations counts against its plausibility. Moreover, I argue that the contemporary approach provides no compelling reason for supposing that the geometry offered as the geometry of ‘visibles’ is the correct geometrical description of visual experience. (shrink)

Neste artigo, tentamos mostrar que um problema comum aos capítulos XVI e XX das Observações filosóficas é o da aplicabilidade dos conceitos e «proposições» da geometria à realidade física e perceptiva (visual, em particular), e que o modo pelo qual Wittgenstein aborda esse problema nessa obra difere radicalmente, a despeito de aparentes similitudes, daquele que caracteriza as teorias semânticas do a priori visual em termos de estipulações (notadamente, o de Carnap em 1922). O esclarecimento do estatuto dos enunciados sobre os (...) objetos do espaço visual como regras de sintaxe e a abordagem do problema da aplicação de nossos conceitos geométricos à realidade perceptiva em termos de condições práticas de emprego permite a Wittgenstein dissolver, não sem alguma ambiguidade no vocabulário empregado, as falsas aparências ontológicas suscitadas pelas teorias que postulam a existência de um universo de discurso intermediário entre os corpos físicos e os números. (shrink)