Some open problems in the philosophy of space and time

This article is concerned to formulate some open problems in the philosophy of space and time that require methods characteristic of mathematical traditions in the foundations of geometry for their solution. In formulating the problems an effort has been made to fuse the separate traditions of the foundations of physics on the one hand and the foundations of geometry on the other. The first part of the paper deals with two classical problems in the geometry of space, that of giving operationalism an exact foundation in the case of the measurement of spatial relations, and that of providing an adequate theory of approximation and error in a geometrical setting. The second part is concerned with physical space and space-time and deals mainly with topics concerning the axiomatic theory of bodies, the operational foundations of special relativity and the conceptual foundations of elementary physics.     

Statistical manifold as an affine space: A functional equation approach

A statistical manifold M_μ consists of positive functions f such that defines a probability measure. In order to define an atlas on the manifold, it is viewed as an affine space associated with a subspace of the Orlicz space L^Φ. This leads to a functional equation whose solution, after imposing the linearity constrain in line with the vector space assumption, gives rise to a general form of mappings between the affine probability manifold and the vector (Orlicz) space. These results generalize the exponential statistical manifold and clarify some foundational issues in non-parametric information geometry.     

Huygens' Center-of-Mass Space-Time Reference Frame: Constructing a Cartesian Dynamics in The Wake of Newton's “De gravitatione” Argument

This paper explores the possibility of constructing a Cartesian space-time that can resolve the dilemma posed by a famous argument from Newton's early essay, De gravitatione. In particular, Huygens' concept of a center-of-mass reference frame is utilized in an attempt to reconcile Descartes' relationalist theory of space and motion with both the Cartesian analysis of bodily impact and conservation law for quantity of motion. After presenting a modern formulation of a Cartesian space-time employing Huygens' frames, a series of Newtonian counter-replies are developed in order to estimate the viability of this relationalist project.     

Hermann Weyl on Minkowskian Space–Time and Riemannian Geometry

Hermann Weyl as a founding father of field theory in relativistic physics and quantum theory always stressed the internal logic of mathematical and physical theories. In line with his stance in the foundations of mathematics, Weyl advocated a constructivist approach in physics and geometry. An attempt is made here to present a unified picture of Weyl’s conception of space–time theories from Riemann to Minkowski. The emphasis is on the mathematical foundations of physics and the foundational significance of a constructivist philosophical point of view. I conclude with some remarks on Weyl’s broader philosophical views.     

A physicalist reinterpretion of ‘phenomenal’ spaces

This paper argues that phenomenal or internal metrical spaces are redundant posits. It is shown that we need not posit an internal space-time frame, as the physical space-time suffices to explain geometrical perception, memory and planning. More than the internal space-time frame, the idea of a phenomenal colour space has lent credibility to the idea of internal spaces. It is argued that there is no phenomenal colour space that underlies the various psychophysical colour spaces; it is parasitic upon physical and psychophysical colour spaces. The argumentation is further extended to other sensory spaces and generalised quality spaces.     

God as Spirit—and Natural Science

The biblical sentence "God is Spirit" (John 4:24) occasioned the development of the Christian doctrine about God as Spirit. But since patristic times "spirit" was interpreted in the sense of Nus, which rather means "intellect." The biblical concept of spirit (pneuma), however, has its root meaning in referring to "air in movement," as in breath or storm. The similar concept of pneuma in Stoic philosophy has become the "immediate precursor" (Max Jammer) of the field concept in modern physics, so that the conclusion is suggested that God is spirit as something like a field of force rather than as intellect. This essay argues for such a conception by relating the divine eternity and immensity to the concepts of space and time, the basic requirements of any physical field. God's eternity and immensity are interpreted in terms of undivided infinite space (and time) which is presupposed in all concepts of parts of space or time (or space-time), therefore in all mathematical and physical measurement.     

Point-Free Geometry and Verisimilitude of Theories

A metric approach to Popper’s verisimilitude question is proposed which is related to point-free geometry. Indeed, we define the theory of approximate metric spaces whose primitive notions are regions, inclusion relation, minimum distance, and maximum distance between regions. Then, we show that the class of possible scientific theories has the structure of an approximate metric space. So, we can define the verisimilitude of a theory as a function of its (approximate) distance from the truth. This avoids some of the difficulties arising from the known definitions of verisimilitude.     

Husserl on Geometry and Spatial Representation

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl.     

Claudia Card's Concept of Social Death: A New Way of Looking at Genocide

Scholarship in the multidisciplinary field of genocide studies often emphasizes body counts and the number of biological deaths as a way of measuring and comparing the severity and scope of individual genocides. The prevalence of this way of framing genocide is problematic insofar it risks marginalizing the voices and experiences of victims who may not succumb to biological death but nevertheless suffer the loss of family members and other loved ones, and suffer the destruction of relationships, as well as the foundational institutions that give rise to and sustain those relationships. The concept of social death, which Claudia Card offers as the central evil of genocide, marks a radical shift in conceptualizing genocide and provides space for recovering the marginalized voices of many who suffer the evils of genocide but do not suffer biological death. Here her concept of social death is explored, defended, and criticized.     

Is there a geometric module for spatial orientation? squaring theory and evidence

There is evidence, beginning with Cheng (1986), that mobile animals may use the geometry of surrounding areas to reorient following disorientation. Gallistel (1990) proposed that geometry is used to compute the major or minor axes of space and suggested that such information might form an encapsulated cognitive module. Research reviewed here, conducted on a wide variety of species since the initial discovery of the use of geometry and the formulation of the modularity claim, has supported some aspects of the approach, while casting doubt on others. Three possible processing models are presented that vary in the way in which (and the extent to which) they instantiate the modularity claim. The extant data do not permit us to discriminate among them. We propose a modified concept of modularity for which an empirical program of research is more tractable.     

Logic for physical space

Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the major milestones in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces.     

Guru Nanak is not at the White House: An essay on the idea of Sikh-American redemption

This article examines the rights-based discourse deployed by Sikh advocacy organizations, the Sikh Coalition and Sikh American Legal Defense & Education Fund, in order to carve an inclusive space within the United States. We interrogate the deep violence of forgetting embedded within this politics that not only sanctions American values and their regulatory might globally, but also integrates the foundational anti-blackness of Western subjectivity into the conceptual structure of Sikhism. Reconsidering these attachments to the American political project and the white-settler state, we argue Sikh organizations fasten Sikhs to ways of life that are inimical to their own flourishing.     

Cognitive Metaphor Theory and the Metaphysics of Immediacy

One of the core tenets of cognitive metaphor theory is the claim that metaphors ground abstract knowledge in concrete, first‐hand experience. In this paper, I argue that this grounding hypothesis contains some problematic conceptual ambiguities and, under many reasonable interpretations, empirical difficulties. I present evidence that there are foundational obstacles to defining a coherent and cognitively valid concept of “metaphor” and “concrete meaning,” and some general problems with singling out certain domains of experience as more immediate than others. I conclude from these considerations that whatever the facts are about the comprehension of individual metaphors, the available evidence is incompatible with the notion of an underlying conceptual structure organized according to the immediacy of experience.     

The Meaning of Category Theory for 21st Century Philosophy

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. Text of a talk given at the Workshop and Symposium on the Ramifications of Category Theory, Florence, November 18–22, 2003. For further documentation on the conference, see     

Is perceptual space inherently non-Euclidean?

It is often assumed that the space we perceive is Euclidean, although this idea has been challenged by many authors. Here we show that if spatial cues are combined as described by Maximum Likelihood Estimation, Bayesian, or equivalent models, as appears to be the case, then Euclidean geometry cannot describe our perceptual experience. Rather, our perceptual spatial structure would be better described as belonging to an arbitrarily curved Riemannian space.     

Nondirected light signals and the structure of time

Temporal betweenness in space-time is defined solely in terms of light signals, using a signalling relation that does not distinguish between the sender and the receiver of a light signal. Special relativity and general relativity are considered separately, because the latter can be treated only locally. We conclude that the (local) coherence of time can be described if we know only which pairs of space-time points are light-connected. Other consequences in the case of special relativity: (1) a categorical axiom system exists in terms of nondirected light connection alone, with neither particle nor time order as a primitive concept, though we do not actually present the axioms; (2) any concept definable by coordinates is also definable in terms of nondirected light signals if and only if it is invariant under Lorentz transformations, translations, dilations, space reflections, and time reflections; and (3) any transformation of space-time (not necessarily continuous) which preserves nondirected light connection is a product of transformations just listed above. The bulk of the paper is devoted to proving that the definitions we give correspond to their intended interpretations in the usual space-time continua.     

On the Affine Structure of Perceptual Space

Affine geometry is a generalization of Euclidean geometry in which distance can be scaled along parallel directions, though relative distances in different directions may be incommensurable. This article presents a new procedure for testing the intrinsic affine structure of a psychological space by having subjects perform bisection judgments over multiple directions. If those judgments are internally consistent with one another, they must satisfy a theorem first proved by Pierre Varignon around 300 years ago. In the experiment reported here, this procedure was employed to measure the perceived structure of a visual ground surface. The results revealed that observers' judgments were systematically distorted relative to the physical environment, but that the judged bisections in different directions had an internally consistent affine structure. Implications of these findings for other possible response tasks are considered.     

Infinity and the foundations of linguistics

Synthese - The concept of linguistic infinity has had a central role to play in foundational debates within theoretical linguistics since its more formal inception in the mid-twentieth century. The...     

Remarks on the Geometry of Visibles

I offer an explication of Reid's claim (discussed recently by Yaffe and others) that the geometry of the visual field is spherical geometry. I show that the sphere is the only surface whose geometry coincides, in a certain strong sense, with the geometry of visibles.