Re‐framing systemic paradigms for the art of learning

Thoughtful people are increasingly concerned that the current paradigms for social, corporate, and educational activities are in disgraceful disarray. The “problem‐solving” or analytical model, the competitive or game model, the commercial or consumer model, the bureaucratic or institutional model, and the disease or illness model which prevail in public discourse are proving to be especially unwholesome. We cannot, however, educate ourselves without paradigms. A credible educational paradigm must be generally accessible without being simplistic, informative without being monothematic, and accommodating as well as discriminating. Given our disquiet with the current cognitive situation, a renewing paradigm must be somehow novel; given the character of human nature, a sustaining paradigm must be somehow familiar.

For a very long time now, professional Sciences have committed themselves to paradigms about “reality out there,” while professional Arts have devoted themselves to expressing “imagination from within here.” The more these two worldviews polarize in opposition to one another, the more room there is—and the more human heed there becomes—for mediation by an applied philosophy which accommodates the “real” as well as the “imaginary” in a complementary way. Such a philosophy would address not only “what do you know?” and “how do you do?” but also “how do you know?” and “why do you do?” In earlier times, people would have been considered neither educated nor wise unless they appreciated the Sciences and the Arts whole. In our time, we may not survive unless we can re‐integrate our fractured perceptions. How might we proceed to do so? There may be a systemological way.     

Dependence of normal contact stress on the surface morphology of solid surfaces

Contact problems between two solids are essential to interpret the loading-unloading behaviour in indentation tests. Using linear elastic theory, a general relationship between the normal stress and the surface deformation for elastic contact of two solids was derived, in which the normal stress is a function of the mean curvature of the undeformed surface.     

In the Vicinity of the Human

Abstract

Beginning with the situated character of the question concerning the human, this paper argues that the problem of the human is itself inextricably bound to the problem of situation or place. Consequently, any genuine philosophical anthropology must take the form of a philosophical topology. This line of argument is developed through the work Abraham Heschel, Martin Heidegger, Martin Buber, and also Helmut Plessner.     

Vagueness,Kant and Topology: a Study of Formal Epistemology

In this paper we propose an approach to vagueness characterised by two features. The first one is philosophical: we move along a Kantian path emphasizing the knowing subject’s conceptual apparatus. The second one is formal: to face vagueness, and our philosophical view on it, we propose to use topology and formal topology. We show that the Kantian and the topological features joined together allow us an atypical, but promising, way of considering vagueness.     

Towards a topological philosophy

This article examines the use of mathematical concepts in philosophy, focusing on topology, which may be viewed as a modern supplement to geometry. We show that Plato and Parmenides were already employing geometric ideas in their research, and discuss three examples of the application of topology to philosophical problems: the first concerns the analysis of the Cartesian distinction between res extensa and res cogitans, the second the ontology of possible worlds of Wittgenstein's Tractatus, and the third Leibniz's monadology. We also consider the role of topology in mathematical explanations of the sort found in science, arguing that it can perform a role in philosophy that is of comparable importance.     

On Dynamic Topological and Metric Logics

We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems (W.f), fi, where W is a topological space and f a homeomorphism on W. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits {w, f(w), f² (w), ˙˙˙} of points w ε W. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and so not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems (W, f), where W is a metric space and f and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius a, ‘for a ε Q ⁺. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function.     

Whole and Part in Mathematics

The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.This revised version was published online in October 2005 with corrections to the Cover Date.     

Categorical Ontology of Levels and Emergent Complexity: An Introduction

An overview of the following three related papers in this issue presents the Emergence of Highly Complex Systems such as living organisms, man, society and the human mind from the viewpoint of the current Ontological Theory of Levels. The ontology of spacetime structures in the Universe is discussed beginning with the quantum level; then, the striking emergence of the higher levels of reality is examined from a categorical—relational and logical viewpoint. The ontological problems and methodology aspects discussed in the first two papers are followed by a rigorous paper based on Category Theory, Algebraic Topology and Logic that provides a conceptual and mathematical basis for a Categorical Ontology Theory of Levels. The essential links and relationships between the following three papers of this issue are pointed out, and further possible developments are being considered.     

Place and Displacement: Towards a Distopological Approach

Most recently, debates on decolonization, transformation, and Africanization raise, again, critical questions about the continuous dominance of the Western practice of philosophy in an African place. Such debates bear particular reference to colonization; however, they are relevant to any place where displacement is an issue and transformation demanded. Yet, the concept of displacement receives surprisingly little attention in these debates or in literature on place. I argue that place and displacement are inherently related, and explore some implications of this relation with regard to the practice of philosophy. My contention is that while displacement is an issue that raises specific challenges to the practice of Western philosophy in an African context, it is in fact a defining feature of philosophy as such in search of its place in the world. In conclusion, I propose what I call a distopological approach towards a philosophical practice that takes seriously its place and inherent relation to displacement.     

A Complete Axiom System for Polygonal Mereotopology of the Real Plane

This paper presents a calculus for mereotopological reasoning in which two-dimensional spatial regions are treated as primitive entities. A first order predicate language with a distinguished unary predicate c(x), function-symbols , · and – and constants 0 and 1 is defined. An interpretation for is provided in which polygonal open subsets of the real plane serve as elements of the domain. Under this interpretation the predicate c(x) is read as region x is connected and the function-symbols and constants are given their meaning in terms of a Boolean algebra of polygons. We give an alternative interpretation based on the real closed plane which turns out to be isomorphic to A set of axioms and a rule of inference are introduced. We prove the soundness and completeness of the calculus with respect to the given interpretation.