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.

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.

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.

Cultural capital and hegemonic masculinity are two concepts that have received intense attention. While both have received serious consideration, critique and analysis, the context or field-specificity of each is sometimes ignored. They have been used in a diversity of ways. Using ethnographic and interview data from a US male bodybuilding community, this study highlights one useful employment. Hegemonic masculinity takes different shapes in different fields of interaction, acting as a form of cultural capital: gender capital. Inherent in this discussion are (...) the cultural contradictions apparent among individuals striving for either physical or ideological embodiments of gender capital. Individuals can attempt to embody hegemonic idealizations, but bodies are not only inscribed with gender, inscriptions are read, and read differently by different social actors and in different settings. The capacity of gender capital to remain elusive is precisely what enables gender practices and projects like bodybuilding to retain passionate participation. (shrink)

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

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.

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.

The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect.

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)

The anti-Specker property, a constructive version of sequential compactness, is used to prove constructively that a pointwise continuous, order-dense preference relation on a compact metric space is uniformly sequentially continuous. It is then shown that Ishihara's principle BD-ℕ implies that a uniformly sequentially continuous, order-dense preference relation on a separable metric space is uniformly continuous. Converses of these two theorems are also proved.

The minimax theorem of matrix game theory is examined from a constructive point of view. It is then shown that the existence of solutions for matrix games cannot be proved constructively, but that a 2-by-2 game with at most one solution has a constructible solution.

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.

It is well known that in Bishop-style constructive mathematics, the closure of the union of two subsets of ℝ is ‘not’ the union of their closures. The dual situation, involving the complement of the closure of the union, is investigated constructively, using completeness of the ambient space in order to avoid any application of Markov's Principle.

Though there is a vast literature on performances of drag, performances of gender and sexual transgressions outside of drag clubs are less studied. This case study of men’s marches protesting violence against women—“Walk a Mile in Her Shoes” marches— examines the politics of such transgressions. Cross-dressing to various degrees is strategically utilized at these events in an attempt to encourage men to become empathetic allies. This article suggests, however, that context is critical to the political potential of performances of drag. (...) The author’s observations of the interactions at the marches suggest that drag at “Walk a Mile” marches often symbolically reproduces gender and sexual inequality despite good intentions. At these marches, feminism is gendered when performances of politics and protest are contextually framed as gender and/or sexual transgressions when “feminism” is understood as “feminine.”. (shrink)

Two new notions of compactness, each classically equivalent to the standard classical one of sequential compactness, for apartness spaces are examined within Bishop-style constructive mathematics.

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).

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 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.

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.

Various types of continuity for preference relations on a metric space are examined constructively. In particular, necessary and sufficient conditions are given for an order-dense, strongly extensional preference relation on a complete metric space to be continuous. It is also shown, in the spirit of constructive reverse mathematics, that the continuity of sequentially continuous, order-dense preference relations on complete, separable metric spaces is connected to Ishihara's principleBD-ℕ, and therefore is not provable within Bishop-style constructive mathematics alone.