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**1 - 3**of**3**### 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. ..."

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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.

### Do Noetherian Modules Have Noetherian Basis Functions?

"... Abstract. In Bishop-style constructive algebra it is known that if a module over a commutative ring has a Noetherian basis function, then it is Noetherian. Using countable choice we prove the reverse implication for countable and strongly discrete modules. The Hilbert basis theorem for this specific ..."

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Abstract. In Bishop-style constructive algebra it is known that if a module over a commutative ring has a Noetherian basis function, then it is Noetherian. Using countable choice we prove the reverse implication for countable and strongly discrete modules. The Hilbert basis theorem for this specific class of Noetherian modules, and polynomials in a single variable, follows with Tennenbaum’s celebrated version for modules with a Noetherian basis function. In particular, the usual hypothesis that the modules under consideration are coherent need not be made. We further identify situations in which countable choice is dispensable. 1

### What is Continuity, Constructively? 1

"... Abstract: The concept of continuity for mappings between metric spaces should coincide with that of uniform continuity in the case of a compact domain, and still give rise to a category. In Bishop’s constructive mathematics both requests can be fulfilled simultaneously, but then the reciprocal funct ..."

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Abstract: The concept of continuity for mappings between metric spaces should coincide with that of uniform continuity in the case of a compact domain, and still give rise to a category. In Bishop’s constructive mathematics both requests can be fulfilled simultaneously, but then the reciprocal function has to be abandoned as a continuous function unless one adopts the fan theorem. This perhaps little satisfying situation could be avoided by moving to a point–free setting, such as formal topology, in which infinite coverings are defined mainly inductively. The purpose of this paper is to discuss the earlier situation and some recent developments.