Results 1 
4 of
4
The Dedekind reals in abstract Stone duality
 Mathematical Structures in Computer Science
, 2008
"... Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherentl ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos. ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the algebra of open subspaces. ASD is presented as a lambdacalculus, of which we provide a selfcontained summary, as the foundational background has been investigated in earlier work. The core of the paper constructs the real line using twosided Dedekind cuts. We show that the closed interval is compact and overt, where these concepts are defined using quantifiers. Further topics, such as the Intermediate Value Theorem, are presented in a separate paper that builds on this one. The interval domain plays an important foundational role. However, we see intervals as generalised Dedekind cuts, which underly the construction of the real line, not as sets or pairs of real numbers. We make a thorough study of arithmetic, in which our operations are more complicated than Moore’s, because we work constructively, and we also consider backtofront (Kaucher) intervals. Finally, we compare ASD with other systems of constructive and computable topology and analysis.
A DEFENCE OF MATHEMATICAL PLURALISM
, 2004
"... We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context. ..."
Abstract
 Add to MetaCart
We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.
Constructive Solutions of Continuous Equations
, 2003
"... We modify some seminal notions from constructive analysis, by providing witnesses for (strictly) positive quantifiers occurring in their definitions. For instance, we understand... ..."
Abstract
 Add to MetaCart
We modify some seminal notions from constructive analysis, by providing witnesses for (strictly) positive quantifiers occurring in their definitions. For instance, we understand...
Near convexity, metric convexity, and convexity
, 2005
"... It is shown that a subset of a uniformly convex normed space is nearly convex if and only if its closure is convex. Also, a normed space satisfying a mild completeness property is strictly convex if and only if every metrically convex subset is convex. 1 Classical and constructive mathematics The ar ..."
Abstract
 Add to MetaCart
It is shown that a subset of a uniformly convex normed space is nearly convex if and only if its closure is convex. Also, a normed space satisfying a mild completeness property is strictly convex if and only if every metrically convex subset is convex. 1 Classical and constructive mathematics The arguments in this paper conform to constructive mathematics in the sense of Errett Bishop. This means roughly that they do not depend on the general law of excluded middle. More precisely, the arguments take place in the context of intuitionistic logic. Arguments in the context of ordinary logic will be referred to as classical. As intuitionistic logic is a fragment of ordinary logic, our arguments should be valid from a classical point of view, although some of the maneuvers to avoid invoking the law of excluded middle may seem puzzling. I had intended to write this paper primarily to be read classically, at