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.

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.

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.

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.

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.

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)

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.

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).

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.

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)

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.

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.

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.

Our brains are us. But we are neither happy, fulfilled, nor all that we 'should' (or maybe could) be. We have everything previous generations could have dreamed of, but it seems it's never quite enough. What's going on? Has it anything to do with the way those brains have developed, by any chance? Gillian Bridge takes us on a journey through time, history and the mysterious labyrinth that is the brain, investigating strange happenings and unlikely people on the way. The (...) Significance Delusion is a challenging explanation of our species' peculiar vulnerability to disorders of both thinking and social cohesion. It is a power pack of research, ideas, facts and fascinating real life stories which explores our species' obsession with meaning and significance. What does it all mean for me? What do I mean to you? The Significance Delusion is humorous as well as deeply serious, and it is the must read compilation of lifestyle advice for tomorrow's well-adjusted people. It is a book about what it takes to thrive and survive as an individual human and as a society. From it you will learn the meaning of life, and your place in the world. For parents, educators, thinkers, psychologists, psychotherapists and policy makers. (shrink)

One of the chief aims of Donald Davidson's later work was to show that participation in a certain causal nexus involving two creatures and a shared environment–Davidson calls this nexus “triangulation”–is a metaphysically necessary condition for the acquisition of thought. This doctrine, I suggest, is aptly regarded as a form of what I call transcendental externalism. I extract two arguments for the transcendental-externalist doctrine from Davidson's writings, and argue that neither succeeds. A central interpretive claim is that the arguments are (...) primarily funded by a particular conception of the nature of non-human animal life. This conception turns out to be insupportable. The failure of Davidson's arguments presses the question of whether we could ever hope to arrive at far-reaching claims about the conditions for thought if we deny, as does Davidson, the legitimacy of the naturalistic project in the philosophy of mind. (shrink)

One of the chief aims of Donald Davidson's later work was to show that participation in a certain causal nexus involving two creatures and a shared environment–Davidson calls this nexus “triangulation”–is a metaphysically necessary condition for the acquisition of thought. This doctrine, I suggest, is aptly regarded as a form of what I call transcendental externalism. I extract two arguments for the transcendental‐externalist doctrine from Davidson's writings, and argue that neither succeeds. A central interpretive claim is that the arguments are (...) primarily funded by a particular conception of the nature of non‐human animal life. This conception turns out to be insupportable. The failure of Davidson's arguments presses the question of whether we could ever hope to arrive at far‐reaching claims about the conditions for thought if we deny, as does Davidson, the legitimacy of the naturalistic project in the philosophy of mind. (shrink)

It is shown within Bishop's constructive mathematics that, under one extra, classically automatic, hypothesis, a continuous homomorphism from R onto a compact metric abelian group is periodic, but that the existence of the minimum value of the period is not derivable.

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)

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)

This paper offers an interpretation of the later Wittgenstein's handling of the idea of an "essence of human language", and examines in particular his treatment of the 'Augustinean' vision of reference as constituting this "essence". A central theme of the interpretation is the perennial philosophical desire to impose upon linguistic meaning conceptual templates drawn from outside the forms of thought about meaning in which we engage when we exercise our capacity to speak and understand a language. The paper closes with (...) a consideration of ways in which Donald Davidson's generally congenial work on interpretation may diverge from Wittgenstein's thinking in this vicinity. (shrink)

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.

Explores the central role in Wittgenstein's later work of his opposition to a 'mechanistic' conception of understanding. Offers a diagnosis of Kripke's skeptical paradox on this basis.

On the Arbitrary Nature of Things approaches Hegel's Phenomenology of Spirit through a paradigm of agnosticism developed from Hegel's own critique of systems of knowledge. This work traces Hegel's descriptions of the movements of Spirit with equal measures of charity and skepticism. It provokes one to question the level of agnosticism that should be taken toward our various systems of human understanding, both in Hegel's Phenomenology and in our contemporary world. With respect to our contemporary world, Bridges questions whether (...) the nature of things is ultimately arbitrary and finds that phenomena such as the placebo effect and the use of sensoriums in phenomenological anthropology add credence to the position of agnosticism toward the arbitrary nature of things. (shrink)

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.

This paper explores the political potential of digital failure as a refusal to work in service of today’s dataveillance society. Moving beyond criticisms of flawed digital systems, this paper traces the moments of digital failure that seek to break, rather than fix, existing systems. Instead, digital failure is characterized by pesky data that sneaks through the cracks of digital capitalism and dissipates into the unproductive; it supports run-away data prone to misidentifications by digital marketers, coders, and content moderators; and it (...) celebrates data predisposed to “back-talk” with playful irreverence toward those that seek to bring order through normative categorization and moderation. I call these data entropic, fugitive, and queer and explore their mischievous practices through three case studies: the unaccountable data in identity resolution, public shaming of the ImageNet training data, and reading practices of sex worker and influencer, @Charlieshe. Together these case studies articulate the political potential of digital failure as a process of unbecoming the good data subject by pushing past the margins of legibility, knowability, and thinkability, to reveal what is made illegible, unknowable, and unthinkable to data’s seeing eye. As predictive analytics, automated decision-systems, and artificial intelligence take on increasingly central roles in public governance, digital failure reveals how data itself is a flawed concept prone to political abuse and social engineering to protect the interests of the powerful, while keeping those marginalized over-surveilled and underrepresented. (shrink)

© The Authors 2017. Published by Oxford University Press on behalf of The Analysis Trust. All rights reserved. For Permissions, please email: [email protected] Blome-Tillmann’s Knowledge and Presuppositions proposes and defends a novel form of epistemological contextualism. As the title would suggest, the view’s novelty lies in its deployment of the pragmatic-theoretic concept of a conversational presupposition to delineate a role for context in shaping the meaning of our knowledge claims. Over the course of six dense, argument-filled chapters, Blome-Tillmann brings his (...) approach into focus and claims for it several advantages. While existing varieties of contextualism can account for widely shared intuitions about the truth values of knowledge ascriptions in various familiar thought experiments, the distinctive flexibility of Presuppositional Epistemological Contextualism, Blome-Tillmann argues, enables it to account as well for cases in which our intuitions diverge. This flexibility also allows for especially satisfying analyses of scepticism... (shrink)

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)

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.

A critique of attempts by Charles Travis and others to read contextualism back into Philosophical Investigations. The central interpretive claim is that this reading is not only unsupported; it gets Wittgenstein's intent, in the parts of the text at issue, precisely backwards. The focus of the chapter is on Wittgenstein's treatment of explanation, understanding, proper names, and family-resemblance concepts.

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.

On the horizon between metaphysics and philosophy of religion stands the question of God’s relation to various abstracta. Like other contemporary philosophical debates, this one has resulted in a broadly dichotomous stalemate between Platonic realists on the one hand and varieties of nominalism/antirealism on the other. In this paper, I offer Aquinas’s moderaterealism as a true middle ground between realist or nominalist solutions. What Platonists take to be abstracta are actually the result of intellect’s abstractive work on sensible objects. Further, (...) the Christian philosopher should be concerned as much, if not more so, by nominalism than by Platonism. Given the problems associated with either Platonist or nominalist solutions, one should be open to a Thomistic moderate-realist solution to the problem of God and abstracta. (shrink)

One might presume that elite educational environments, including colleges and universities, have been the exclusive venues for rhetorical theory. Rhetorical theory typically takes the form of published treatises, monographs, and essays. Thus, the status of theorist would have been denied to, among others, African Americans during the nineteenth century because they were not afforded opportunities to become literate or, even if literate, not admitted into realms of elite literacy. But there are the seeds of a competing story, particularly regarding the (...) narratives written by Frederick Douglass. Henry Louis Gates Jr., for example, has stated that Douglass used... (shrink)