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The fundamental theorem of algebra: a constructive development without choice
 Pacific Journal of Mathematics
, 2000
"... Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice come ..."
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Cited by 12 (3 self)
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Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice comes into play andhow to proceedin its absence. Along the way, a notion of a complete metric space, suitable for a choiceless environment, is developed. By constructive mathematics I mean, essentially, mathematics that is developed along the lines proposed by Errett Bishop [1]. More precisely, I mean mathematics that is done in the context of intuitionistic logic — without the lawof excluded middle. My reasons for identifying these notions are discussed in [9] and [10], the basic contention being that constructive mathematics has the same subject matter as classical mathematics. Ruitenburg [11] treated the fundamental theorem of algebra in a choiceless
Constructive Mathematics, in Theory and Programming Practice
, 1997
"... The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part ..."
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Cited by 7 (2 self)
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The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on MartinLof's theory of types as a formal system for BISH.
The Constructive Implicit Function Theorem and Applications in Mechanics
 Chaos, Solitons and Fractals
, 1997
"... We examine some ways of proving the Implicit Function Theorem and the Inverse Function Theorem within Bishop's constructive mathematics. Section 2 contains a new, entirely constructive proof of the Implicit Function Theorem. The paper ends with some comments on the application of the Implicit F ..."
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Cited by 4 (0 self)
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We examine some ways of proving the Implicit Function Theorem and the Inverse Function Theorem within Bishop's constructive mathematics. Section 2 contains a new, entirely constructive proof of the Implicit Function Theorem. The paper ends with some comments on the application of the Implicit Function Theorem in classical mechanics. 1 Introduction In this paper, which is written entirely within the framework of constructive mathematics #BISH# erected by the late Errett Bishop #2#, we examine a standard proof of the Implicit Function Theorem and give a completely new proof. As far as understanding constructive mathematics goes, the reader need only be aware that when working constructively,weinterpret #existence" strictly as #computability". To do so, we need to be careful about our logic. For example, when we prove a disjunction P Q; we need to either produce a proof of P or produce a proof of Q; it is not enough, constructively, to show that : #:P :Q#:To understand this better, con...
DSGE AND BEYOND  COMPUTABLE AND CONSTRUCTIVE CHALLENGES
, 2011
"... The genesis and the path towards what has come to be called the DSGE model is traced, from its origins in the ArrowDebreu General Equilibrium model (ADGE), via Scarf’s Computable General Equilibrium model (CGE) and its applied version as Applied Computable General Equilibrium model (ACGE), to its ..."
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The genesis and the path towards what has come to be called the DSGE model is traced, from its origins in the ArrowDebreu General Equilibrium model (ADGE), via Scarf’s Computable General Equilibrium model (CGE) and its applied version as Applied Computable General Equilibrium model (ACGE), to its ostensible dynamization as a Recursive Competitive Equilibrium (RCE). An outline of a similar nature, albeit very brie‡y, of the development and structure of AgentBased Economics (ABE) is also included. It is shown that these transformations of the ADGE model are computably and constructively untenable. Suggestions for going ’beyond DSGE and ABE’ are, then, outlined on the basis of a framework that is underpinned –from the outset –by computability and constructivity considerations.
Constructive Aspects of the Dirichlet Problem 1
"... Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we ..."
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Abstract: We examine, within the framework of Bishop's constructive mathematics, various classical methods for proving the existence of weak solutions of the Dirichlet Problem, with a view to showing why those methods do not immediately translate into viable constructive ones. In particular, we discuss the equivalence of the existence of weak solutions of the Dirichlet Problem and the existence of minimizers for certain associated integral functionals. Our analysis pinpoints exactly what is needed to nd weak solutions of the Dirichlet Problem: namely, the computation of either the norm of a linear functional on a certain Hilbert space or, equivalently, the in mum of an associated integral functional.
Do Noetherian Modules Have Noetherian Basis Functions?
"... Abstract. In Bishopstyle 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 Bishopstyle 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
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.