This handbook presents a comprehensive introduction to the core areas of philosophy of education combined with an up-to-date selection of the central themes. It includes 95 newly commissioned articles that focus on and advance key arguments; each essay incorporates essential background material serving to clarify the history and logic of the relevant topic, examining the status quo of the discipline with respect to the topic, and discussing the possible futures of the field. The book provides a state-of-the-art overview of philosophy (...) of education, covering a range of topics: Voices from the present and the past deals with 36 major figures that philosophers of education rely on; Schools of thought addresses 14 stances including Eastern, Indigenous, and African philosophies of education as well as religiously inspired philosophies of education such as Jewish and Islamic; Revisiting enduring educational debates scrutinizes 25 issues heavily debated in the past and the present, for example care and justice, democracy, and the curriculum; New areas and developments addresses 17 emerging issues that have garnered considerable attention like neuroscience, videogames, and radicalization. The collection is relevant for lecturers teaching undergraduate and graduate courses in philosophy of education as well as for colleagues in teacher training. Moreover, it helps junior researchers in philosophy of education to situate the problems they are addressing within the wider field of philosophy of education and offers a valuable update for experienced scholars dealing with issues in the sub-discipline. Combined with different conceptions of the purpose of philosophy, it discusses various aspects, using diverse perspectives to do so. Contributing Editors: Section 1: Voices from the Present and the Past: Nuraan Davids Section 2: Schools of Thought: Christiane Thompson and Joris Vlieghe Section 3: Revisiting Enduring Debates: Ann Chinnery, Naomi Hodgson, and Viktor Johansson Section 4: New Areas and Developments: Kai Horsthemke, Dirk Willem Postma, and Claudia Ruitenberg. (shrink)

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.

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.

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.

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)

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 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.

This chapter contains sections titled: Introduction I II.

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.

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.

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.

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)

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.

This book evaluates the increasingly wide variety of intellectual resources for research methods and methodologies and investigates what constitutes good educational research. Written by a distinguished international group of philosophers of education Questions what sorts of research can usefully inform policy and practice, and what inferences can be drawn from different kinds of research Demonstrates the critical engagement of philosophers of education with the wider educational research community and illustrates the benefits that can accrue from such engagement.

This special issue of the Journal of Philosophy of Education is addressed both to philosophers of education and to the many colleagues in the wider educational.

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

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)

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.

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.

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.

This book evaluates the increasingly wide variety of intellectual resources for research methods and methodologies and investigates what constitutes good educational research. Written by a distinguished international group of philosophers of education Questions what sorts of research can usefully inform policy and practice, and what inferences can be drawn from different kinds of research Demonstrates the critical engagement of philosophers of education with the wider educational research community and illustrates the benefits that can accrue from such engagement.

The focus of this paper is on a variety of practices associated with the transfer of educational policy and practice from one national education system to another – practices sometimes referred to as ‘policy borrowing’. Its concern is with the ethical and political issues raised by these practices. In particular, it discusses concerns that these practices might be practically inappropriate, that they might be culturally insensitive or inappropriate, and that they might be impositional, exploitative perhaps or even oppressive. Such concerns (...) arise particularly in contexts in which the transfer is from relatively rich and powerful countries to relatively poor and less powerful countries. But policy transfer is a feature of relations between the rich and powerful too, and to some extent the issues are then turned on their head. Those engaged in the business of transfer become more clearly identified as service providers and they have to ask questions about to whom and under what conditions they might provide this service. Finally, the paper considers policy transfer as a form of pedagogy and asks whether the kind of ethical considerations which underpin any properly educational transaction might not provide a guide to behaviour by the agents of policy transfer. (shrink)

This chapter examines criticisms made by or on behalf of ‘disempowered’ groups against outsider research into their experience: that outsiders cannot properly understand and represent their experience and are exploitative and disrespectful, and that having outsiders articulate your views for you is intrinsically disempowering. I argue that ‘outsider research’ can contribute to the better understanding of the researcher, of the community engaged in the research and of the wider community. Nevertheless the claim ‘nothing about us without us’ expresses an ethical (...) and epistemological truth in educational research: as a statement about the kind of relationship which should obtain between researcher and participants. (shrink)

This publication arises in a context in which policy makers and educational researchers are increasingly vocal in their demands that educational policy and prac.

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 shown, with intuitionistic logic, that if every locally constant function from to has a property akin to constancy, then the fan theorem for -bars holds, and conversely.

This paper starts from the point in the early 1970s at which educational theory and research was temporarily structured under the ‘foundation’ disciplines of psychology, sociology, philosophy and history of education. It observes the way the intellectual resources of educational research have become enlarged and enriched and these disciplines themselves fragmented and hybridised to a degree that prompts talk not just of interdisciplinarity but of ‘postdisciplinarity’. The paper argues, however, that without discipline, in the sense of a shared language, a (...) rule governed structure of enquiry—something ‘systematic’—we lose the conditions that make a community of arguers possible. Further, we lose the basis for the special claim which research might otherwise make on our attention and on our belief. (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.

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 notes the critical response that the ‘competence movement’ has received from writers in philosophy of education and argues for a more positive assessment of what it offers in relation to: (i) the place of practical competence in a liberal education, (ii) the meritocratic principles underlying the competence movement, (iii) the ‘transparency’ of expectations in assessment, and even (iv) the element of practical competence in moral performance. It emphasises, however, that not all versions of ‘competence’ can be defended in (...) these terms and that this requires a more generic and cognitively laden concept of personal and professional competence. (shrink)

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.

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 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.

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.

In education issues to do with insider and outsider understanding arise in debates about religious education and about certain areas of research, and in argument about education for international understanding. Here I challenge the dichotomy between insider and outsider, arguing that a more collectivist view of human identity combined with elements of 'the self which we share with our fellows' means that we always stand in part as an insider and in part as an outsider in relation to others. I (...) argue further that 'understanding' needs always to be thought of in the plural, as a matter of 'understanding s ', since members of any community, and even any one individual, have different understandings at different times. Against what may seem the strong case in favour of the superiority or exclusivity of insider understanding, I note the force of claims in a variety of theoretical traditions (expressed notably in the Marxist tradition in the notion of 'false consciousness' and in psychoanalytic attention to the unconscious) concerning the limitations of the exclusively insider perspective and in contrast the particular authority of the outsider perspective. Finally, I acknowledge the discomfort with which people respond to outsiders' claims to understand them, especially where such understanding fails to support their own self-understanding, and identify ethical considerations, which might shape sensitive negotiation between insider and outsider perspectives. (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)

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)