Results 1 
5 of
5
PLURALISM IN MATHEMATICS
, 2004
"... We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We defend pluralism in mathematics, and in particular Errett Bishop’s constructive approach to mathematics, on pragmatic grounds, avoiding the philosophical issues which have dissuaded many mathematicians from taking it seriously. We also explain the computational value of interval arithmetic.
Quotient spaces and coequalisers in formal topology
 J. UCS
"... Abstract: We give a construction of coequalisers in formal topology, a predicative version of locale theory. This allows for construction of quotient spaces and identification spaces in constructive topology. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract: We give a construction of coequalisers in formal topology, a predicative version of locale theory. This allows for construction of quotient spaces and identification spaces in constructive topology.
Apartness and formal topology
 New Zealand Journal of Mathematics
, 2005
"... The theory of formal spaces and the more recent theory of apartness spaces have a priori not much more in common than that each of them was initiated as a constructive approach to general topology. We nonetheless try to do the first steps in relating these competing theories to each other. Formal to ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The theory of formal spaces and the more recent theory of apartness spaces have a priori not much more in common than that each of them was initiated as a constructive approach to general topology. We nonetheless try to do the first steps in relating these competing theories to each other. Formal topology was put forward in the mid 1980s by Sambin [8] in order to make available to Martin–Löf’s type theory [7] the concepts of classical topology that are worth keeping to such a constructive and predicative framework. In the meantime formal topology has proved a fairly universal setting for doing topology in a point–free way. We refer to [9] for a recent and exhaustive survey of formal topology. The theory of apartness spaces was started by Bridges and Vîță [4] nearly twenty years later to reformulate set–theoretic topology as an extension of Bishop’s constructive analysis [2, 3]. The subsequent development of the theory of apartness spaces has also shed some light on its classical counterpart, the theory of proximity or nearness spaces. A comprehensive overview will be available soon [5]. In formal topology ‘basic neighbourhood ’ is a primitive concept, whereas ‘point ’ is a derived notion; as sets of basic neighbourhoods, points have to be handled with particular care to meet the needs of a predicative framework like Martin–Löf type theory. In the theory of apartness spaces, it is the other way round: as in classical topology, points are given as such, and (basic) neighbourhoods are sets of points. Since, however, it is hard to detect any truly impredicative move in the practice of Bishop’s constructive mathematics in general, we dare to undertake the following attempt to link formal topology and the theory of apartness spaces to each other. 1 Basic definitions We recall the standard definitions associated with formal topologies and morphisms between them (approximable mappings).
Bl $X\mathrm{N}\emptyset$
"... Let $X $ be a nonempty set. We assume that there is a setset apartness relation $*3$ ..."
Abstract
 Add to MetaCart
Let $X $ be a nonempty set. We assume that there is a setset apartness relation $*3$
www.cosc.brocku.ca A discrete duality between apartness algebras and apartness frames
, 2008
"... Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apa ..."
Abstract
 Add to MetaCart
Apartness spaces were introduced as a constructive counterpart to proximity spaces which, in turn, aimed to model the concept of nearness of sets in a metric or topological environment. In this paper we introduce apartness algebras and apartness frames intended to be abstract counterparts to the apartness spaces of Bridges and Vîta [3], and we prove a discrete duality for them. 1