The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics. 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 Martin-L6f 's theory of types as a formal system for BISH.

This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.

In this essay David Bridges argues that since most families choose to realize their responsibility for the major part of their children's education through state schools, then the way in which the state constructs parents' relation with these schools is one of its primary levers on parenting itself. Bridges then examines the way in which parent‐school relations have been defined in England through government and quasi‐government interventions over the last forty‐five years, tracing these through an awakening interest in (...) the relation between social class and unequal school success in the 1960s, passing through the discourse of accountability in the 1970s, marketization in the 1980s and 1990s, performativity extending from this period into the first decade of the twenty‐first century, and, most recently, more direct interventions into parenting itself and the regulation of school relations with parents in the interests of safeguarding children. These have not, however, been entirely discrete policy themes, and the positive and pragmatic employment of the discourse of partnership has run throughout this period, albeit with different points of emphasis on the precise terms of such partnership. (shrink)

The theory of apartness spaces, and their relation to topological spaces (in the point–set case) and uniform spaces (in the set–set case), is sketched. New notions of local decomposability and regularity are investigated, and the latter is used to produce an example of a classically metrisable apartness on R that cannot be induced constructively by a uniform structure.

It is argued that Hellman's arguments purporting to demonstrate that constructive mathematics cannot cope with unbounded operators on a Hilbert space are seriously flawed, and that there is no evidence that his thesis is correct.

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 Martin-L6f 's theory of types as a formal system for BISH.

Cultures of low aspirations, and more particularly young people's adaptation to them, are often presented as the major obstacle to an economic development agenda which requires more higher-level skills and a social agenda which is about enabling people from ‘non-traditional’ backgrounds to go to university. The article analyses and discusses some of the different sorts of constraints on the choices which we make and which may become unconsciously internalised and so constitute our adaptive preference. It argues, however, that all choice (...) is significantly adaptive and has its roots in a self which neither in its development nor in its current agency is detached from the social context in which it has been constructed, with which it identifies and from which that identity itself derives many of its features. Finally the article discusses briefly the grounds on which intervention in the life of such a chooser might be justified and some implications for interventionist strategies which are sensitive to such a socially embedded view of the self. (shrink)

We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into R is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.

An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure is defined and analysed.

An axiomatic development of the theory of apartness and nearness of a point and a set is introduced as a framework for constructive topology. Various notions of continuity of mappings between apartness spaces are compared; the constructive independence of one of the axioms from the others is demonstrated; and the product apartness structure is defined and analysed.

The nature of modern constructive mathematics, and its applications, actual and potential, to classical and quantum physics, are discussed.

The constructive functional calculus for a sequence of commuting selfadjoint operators on a separable Hilbert space is shown to be independent of the orthonormal basis used in its construction. The proof requires a constructive criterion for the absolute continuity of two positive measures in terms of test functions.

The main result of this paper is a weak constructive version of the uniform continuity theorem for pointwise continuous, real-valued functions on a convex subset of a normed linear space. Recursive examples are given to show that the hypotheses of this theorem are necessary. The remainder of the paper discusses conditions which ensure that a sequentially continuous function is continuous. MSC: 03F60, 26E40, 46S30.

In this chapter, which has evolved over the last ten years to what I hope will be its perfect Platonic form, I shall first discuss those features of constructive mathematics that distinguish it from its traditional, or classical, counterpart, and then illustrate the practice of that distinction in aspects of complex analysis whose classical treatment ought to be familiar to a beginning graduate student of pure mathematics.

A weak constructive sequential compactness property of metric spaces is introduced. It is proved that for complete, totally bounded metric spaces this property is equivalent to Brouwer's fan theorem for detachable bars. Our results form a part of constructive reverse mathematics.

This paper is dedicated to Prof. Dr. Günter Asser, whose work in founding this journal and maintaining it over many difficult years has been a major contribution to the activities of the mathematical logic community.At first sight it appears highly unlikely that Urysohn's Lemma has any significant constructive content. However, working in the context of an apartness space and using functions whose values are a generalisation of the reals, rather than real numbers, enables us to produce a significant constructive version (...) of that lemma. (shrink)

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.

We examine, from a constructive perspective, the relation between the complements of S, T, and S ∩ T in X, where X is either a metric space or a normed linear space. The fundamental question addressed is: If x is distinct from each element of S ∩ T, if s ϵ S, and if t ϵ T, is x distinct from s or from t? Although the classical answer to this question is trivially affirmative, constructive answers involve Markov's principle and (...) the completeness of metric spaces. (shrink)

Continuing the study of apartness in lattices, begun in [8], this paper deals with axioms for a product a-frame and with their consequences. This leads to a reasonable notion of proximity in an a-frame, abstracted from its counterpart in the theory of set-set apartness.

The existence of either a maximum or a minimum for a uniformly continuous mapping f of a compact interval into ${\mathbb{R}}$ is established constructively under the hypotheses that f′ is sequentially continuous and f has at most one critical point.

Working within Bishop-style constructive mathematics, we derive a number of results relating the nonconstructive LPO and sequential compactness property on the one hand, and the intuitionistically reasonable anti-Specker property on the other.

Two classically equivalent, but constructively inequivalent, strict convexity properties of a preference relation are discussed, and conditions given under which the stronger notion is a consequence of the weaker. The last part of the paper introduces uniformly rotund preferences, and shows that uniform rotundity implies strict convexity. The paper is written from a strictly constructive point of view, in which all proofs embody algorithms. MSC: 03F60, 90A06.

The relation between near continuity and sequential continuity for mappings between metric spaces is explored constructively. It is also shown that the classical implications “near continuity implies sequential continuity” and “near continuity implies apart continuity” are essentially nonconstructive.

It is shown how, under certain circumstances and within Bishop‐style constructive mathematics, one can construct a product of two a‐frames (the structures underlying the constructive theory of apartness on frames).

Uniform sequential continuity, a property classically equivalent to sequential continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, uniform sequential continuity implies strong continuity if and only if one adopts a certain boundedness principle that, although valid in the classical, recursive and intuitionistic setting, is independent of Heyting arithmetic.

It is proved, within Bishop's constructive mathematics , that, in the context of a Hilbert space, the Open Mapping Theorem is equivalent to a principle that holds in intuitionistic mathematics and recursive constructive mathematics but is unlikely to be provable within BISH.

Within Bishop's constructive mathematics we provide conditions that ensure weak continuity properties of mappings between metric and normed spaces.

It is shown that in any model of constructive mathematics in which a certain omniscience principle is false, for strongly extensional functions on an interval the distinction between sequentially continuous and regulated disappears. It follows, without the use of Markov's Principle, that any recursive function of bounded variation on a bounded closed interval is recursively sequentially continuous.

Working within Bishop-style constructive mathematics, we examine some of the consequences of the anti-Specker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics.

It is shown, within constructive mathematics, that the unit ball B1 of the set of bounded operators on a Hilbert space H is weak-operator totally bounded. This result is then used to prove that the weak-operator continuity of the mapping T → AT on B1 is equivalent to the existence of the adjoint of A.

This article addresses a paradoxical stance taken by young straight men in three groups who identify aspects of themselves as “gay” to construct heterosexual masculine identities. By subjectively recognizing aspects of their identities as “gay,” these men discursively distance themselves from stereotypes of masculinity and privilege and/or frame themselves as politically progressive. Yet, both of these practices obscure the ways they benefit from and participate in gender and sexual inequality. I develop a theory of “sexual aesthetics” to account for their (...) behavior and its consequences, contributing to a growing body of theory regarding the hybridization of contemporary masculinities and complicating theories of sexual practice. (shrink)

The two assumptions of this article are that the mainstream ecumenical paradigm of the 20th century is no longer viable, and that the gifts of global Christianity are adequate for the cause of mission and unity. The Christian landscape has vastly changed. Its centre of gravity has shifted to the South. A new form of ecumenism is needed. The vision of unity ‘made visible as all in each place who are baptized into Jesus Christ’, which involves death and rebirth, is (...) now a calling for a new generation. Constructing a new ecumenical table requires re-conception and re-visioning. North and South are henceforth equal partners.The gifts of Christianity in North and South must be re-conceived.The journey toward Christian unity must be re-visioned. For a viable future there will need to be conversion, openness and humility. (shrink)

It is shown constructively that, on a metric space that is dense in itself, if every pointwise continuous, real-valued function is uniformly sequentially continuous, then the space has the anti-Specker property. The converse is also discussed. Finally, we show that the anti-Specker property implies a restricted form of countable compactness.

In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof-technique is extracted and then applied in several different situations.

In this paper, I argue that informational semantics, the most well-known and worked-out naturalistic account of intentional content, conflicts with a fundamental psychological principle about the conditions of belief-formation. Since this principle is an important premise in the argument for informational semantics, the upshot is that the view is self-contradictory??indeed, it turns out to be guilty of a sophisticated version of the fallacy famously committed by Euthyphro in the eponymous Platonic dialogue. Criticisms of naturalistic accounts of content typically proceed piecemeal (...) by narrowly constructed counterexamples, but I argue that the current result is more robust. It affects a broad family of accounts, and provokes a wider doubt about the possibility of successful execution of the naturalistic project. (shrink)

The notions of permutable and weak-permutable convergence of a series|$\sum _{n=1}^{\infty }a_{n}$|of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD-|$\mathbb {N}$|implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for (...) series holds but BD-|$\mathbb{N}$|does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD-|$\mathbb {N}$|. We show that this is the case when the property is weak-permutable convergence. (shrink)

In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof—technique is extracted and then applied in several different situations.

This paper provides a Bishop-style constructive analysis of the contrapositive of the statement that a continuous homomorphism of R onto a compact abelian group is periodic. It is shown that, subject to a weak locatedness hypothesis, if G is a complete (metric) abelian group that is the range of a continuous isomorphism from R, then G is noncompact. A special case occurs when G satisfies a certain local path-connectedness condition at 0. A number of results about one-one and injective mappings (...) are proved en route to the main theorem. A Brouwerian example shows that some of our results are the best possible in a constructive framework. (shrink)

Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property, a property antithetical to that in Specker’s theorem in recursive analysis, and the uniform continuity theorem for integer-valued functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics.

A characterisation of a type of weak-operator continuous linear functional on certain linear subsets of B, where H is a Hilbert space, is derived within Bishop-style constructive mathematics.

The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.

It is shown that, relative to Bishop-style constructive mathematics, the boundedness principle BD-N is equivalent both to a general result about the convergence of double sequences and to a particular one about Cauchyness in a semi-metric space.