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.

David Bridges; Education, Women and Human Nature, Journal of Philosophy of Education, Volume 11, Issue 1, 30 May 2006, Pages 136–143, https://doi.org/10.1111/j.

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)

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.

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.

David Bridges; Non-paternalistic Arguments in Support of Parents’ Rights, Journal of Philosophy of Education, Volume 18, Issue 1, 30 May 2006, Pages 55–61, http.

Constructive mathematics – mathematics in which ‘there exists’ always means ‘we can construct’ – is enjoying a renaissance. Fifty years on from Bishop’s groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject’s myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, (...) constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches. (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)

Epistemic Contextualism: A Defense By BaumannPeterOxford University Press, 2017. x + 266 pp.

Relying on more expansive criteria for defining “mass shootings” than much existing research, we examine a subset of a unique dataset incorporating 7,048 news documents covering 2,170 shootings in the United States between 2013 and 2019. We analyze the descriptive language used to describe incidents and perpetrators and discover significant racial disparities in representation. This research enables a critical examination of the explanatory frames utilized by news media to tell the public who mass shooters are and journalistic attempts to explain (...) why they occur. Data were analyzed utilizing a mixed methods approach, relying on content analysis to inductively code emergent categories of descriptions of shooters and binary logistic regressions to analyze the preponderance of descriptive categories when comparing news articles reporting on shootings committed by differently racialized shooters. Our results confirm some recent research showing that mass shooters racialized as white are more likely to be described with kind and compassionate language. With our larger sample, however, we also find that mass shooters racialized as white are additionally more likely to be described with negative language as “bad” or “evil” in comparison to shooters of color. We discuss how these data demonstrate that media reports present a more complex picture of white mass shooters for the public than shooters of color. (shrink)

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.

Ethical mysticism, by S. Coit.--The ethical import of history, by D. S. Muzzey.--The tragic and heroic in life, by W. M. Salter.--Distinctive features of the ethical movement, by A. W. Martin.--Ethical experience as the basis of religious education, by H. Neumann.--"All men are created equal," by G. E. O'Dell.--How far is art an aid to religion? by P. Chubb.--Evolution and the uniqueness of man, by H. J. Bridges.--The spiritual outlook on life, by H. J. Golding.--The ethics of Abu'l Ala al (...) Ma'arri, by N. Schmidt.--Life's unused moral force, by H. Snell.--Is the ideal real? by G. A. Smith.--Some ethical tendencies in the professions, by R. D. Kohn.--On the art of living, by W. Boerner.--The relation of the ethical ideal to social reform, by J. L. Elliott.--Concerning tolerance, by R. F. Dewey.--Ethical culture in Germany after the war, by R. Penzig.--A confession of faith, by S. B. Weston.--"Hearing the witnesses," by J. Gutmann. (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)

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.

© 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)

This international collection forms a response from 22 educators to our changing political environment and to the reassessment they provoke of the principles shaping educational thought and practice. The philosophical discussion, however, remains clearly rooted in the world of educational practice and its political content.

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.

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 article is centrally concerned with the sort of knowledge that can and should inform educational policy—and it treats this as an epistemological question. It distinguishes this question from the more extensively explored question of what sort of knowledge in what form policy-makers do in fact commonly take into account. The article examines the logical and rhetorical character of policy and the components of policy decisions and argues that policy demands a much wider range of information than research typically provides. (...) Either the research task or commission has to be substantially extended or the gap will be filled by information or thinking that is not derived from research. One of the gaps between research of an empirical kind and policy is the normative gap. In the final section the article points to the inescapably normative character of educational policy. Of course the values that inform policy can be investigated empirically, but this kind of enquiry cannot tell us what we should do. There is a role for research/scholarship and more rather than less intelligent and critical argumentation in addressing these normative questions as well as the empirical questions that underpin policy. (shrink)

Some philosophers hold that rational explanations­—explanations of people’s attitudes and actions that cite their reasons for forming these attitudes or performing these actions—are dispositional. The hold that rational explanations do their explanatory work by representing these attitudes and actions as the product of dispositions on the part of the subject. I challenge arguments to this effect by Barry Stroud and Michael Smith. And I argue that human beings do not possess, and could not possess, the dispositions required for the dispositionalist (...) account. I propose an alternative account of rational explanation, one that exploits the connection between rational explanations and rational deliberation to show how such explanations can be simultaneously normative and causal. (shrink)

This chapter contains sections titled: Introduction I II.

Fred Dretske’s teleofunctional theory of content aims to simultaneously solve two ground-floor philosophical puzzles about mental content: the problem of naturalism and the problem of epiphenomenalism. It is argued here that his theory fails on the latter score. Indeed, the theory insures that content can have no place in the causal explanation of action at all. The argument for this conclusion depends upon only very weak premises about the nature of causal explanation. The difficulties Dretske’s theory encounters indicate the severe (...) challenges involved in arriving at a robust naturalistic understanding of content. (shrink)

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 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.

In A General View of Positivism French philosopher Auguste Comte gives an overview of his social philosophy known as Positivism. Comte, credited with coining the term 'sociology' and one of the first to argue for it as a science, is concerned with reform, progress and the problem of social order in society. In this English edition of the work, published in 1865, he addresses the practical problems of implementing his philosophy or doctrine, as he also refers to Positivism, into society. (...) He believes that society evolves through a series of stages that are ruled by social laws and culminate in a superior form of social life. During this reorganisation of society, which will find its greatest supporters among women and the working class, a 'new moral power' will emerge. Under the motto 'love, order and progress' Comte wishes humanism to replace organised religion as the object of spiritual worship. (shrink)

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

The desire to use established corporate law skill sets in the pro bono context has lead some lawyers to extend pro bono services to charitable and non-profit organisations. But does the provision of free legal services to well-funded organisations constitute pro bono work, and how can providers of pro bono legal services best prioritise among competing organisations? The author surveys various sources of formal and informal regulation in Singapore and selected Asian and other common law jurisdictions and suggests that when (...) distinguishing between non-profit organisations with different levels of funding, a coherent guideline would evaluate whether the payment of legal fees would deplete economic resources to the extent of significantly impairing organisation programmes. (shrink)

Recent initiatives from the Employment Department in the UK have promoted ‘enterprise education’. This paper discusses the relationship of enterprise education to the more established notion of a liberal education. It is argued that enterprise education should be understood not as replacing the aspirations of a liberal education, but rather as supporting or extending them. It does this (i) by helping pupils to understand what is arguably a significant form of life; (ii) by developing understanding of the economic conditions of (...) our social functioning; and (iii) by developing personal qualities which equip us practically to pursue our chosen paths in a competitive and sometimes hostile world. (shrink)