This discussion begins from the dilemma, posed in some earlier writing by Alasdair MacIntyre, that education is essential but also, in current economic and cultural conditions, impossible. The potential for resolving this dilemma through appeal to ‘practice’, ‘narrative unity’, and ‘tradition’(three core concepts in After Virtue and later writings) is then examined. The discussion also explores the relationship of education to the modern state and the power of a liberal education to create an ‘educated public’ very different in character (...) from the electorates of contemporary democratic regimes. It concludes with some remarks on the role of education in combating prejudice against certain kinds of human difference. (shrink)

It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected to Ax's theory of pseudofinite fields, and analogues are obtained for most of Ax's classical results. Some indication is given of the connection to nonstandard Frobenius maps.

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. (...) To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable, this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly. (shrink)

A nontrivial ring with unit eliminates imaginaries just in case its complete theory has the following property: every definable m-ary equivalence relation E may be defined by a formula f = f, where f is an m-ary definable function. We show that for certain natural expansions of the field of p-adic numbers, elimination of imaginaries fails or is independent of ZPC. Similar results hold for certain fields of formal power series.

We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” , which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define (...) composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered. (shrink)

Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

We give foundational results for the model theory of AfinK, the ring of finite adeles over a number field, construed as a restricted product of local fields. In contrast to Weispfenning we work in the language of ring theory, and various sortings interpretable therein. In particular we give a systematic treatment of the product valuation and the valuation monoid. Deeper results are given for the adelic version of Krasner's hyperfields, relating them to the Basarab–Kuhlmann formalism.

In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.

We construct, by model-theoretic methods, several exponentiations on the universal enveloping algebra U of the Lie algebra.

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform (...) definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)

This essay takes issue with Alasdair MacIntyre’s denial that teaching is a practice. It does so less by appeal to MacIntyre’s concept of practice than by criticism of his conception of teaching. It argues that this conception, as reconstructed from adversions to teaching in a range of his writings, does less than justice to what good teachers accomplish; and that, if this inadequacy is rectified—as much else in his writings suggests that it ought to be—there are clearer grounds (...) for acknowledging teaching as a practice. (shrink)

Background:The debate over the ethical implications of care robots has raised a range of concerns, including the possibility that such technologies could disrupt caregiving as a core human moral activity. At the same time, academics in information ethics have argued that we should extend our ideas of moral agency and rights to include intelligent machines.Research objectives:This article explores issues of the moral status and limitations of machines in the context of care.Design:A conceptual argument is developed, through a four-part scheme derived (...) from the work of Alasdair MacIntyre. No empirical data are used.Ethical consideration:No primary data were gathered for this study. Secondary sources and authorship have been acknowledged throughout.Findings / discussion:Certain kinds of social experience, including the narrative unity of a life, and the giving and receiving of care, are essential for moral development. Machines, no matter how advanced, cannot participate in such experiences in key respects, and thus cannot develop as practical reasoners.Conclusion:It follows that they cannot be moral agents and that they cannot care. There are, it seems, no such things as care robots. In view of the institutional power of tech companies and commissioning bodies, care practitioners need to take more of a lead in developing new assistive technologies which are appropriate to their practice. (shrink)

Over the past quarter of a century the work of few philosophers has exerted such powerful influence, or been the centre of such vigorous debate, as that of Alasdair MacIntyre. And although MacIntyre has not often formally addressed educational issues, the thrust of his writing has seemed to bear more clearly on education than that of most philosophers. His assault on central tenets of the Enlightenment in After Virtue already contained an implicit critique of public education in the (...) modern era. That critique was extended and made more explicit in later writings such as his Richard Peters Lecture ‘The Idea of an Educated Public’ (1987), the concluding chapters of Whose Justice? Which Rationality? (1988), the final chapter of Three Rival Versions of Moral Enquiry (1990), and the essay ‘Aquinas’s Critique of Education: Against his Age, Against Ours’ (1998). With less polemical intensity he furnished some further arguments on education and upbringing in his book Dependent Rational Animals (1999a). And his most recent comments on the subject are presented in the dialogue published in the Journal of Philosophy of Education (MacIntyre and Dunne, 2002). Moreover, even in his most critical writing MacIntyre has also developed an affirmative vision that has seemed to offer resources for a fruitful reconceptualisation of education: his steady concern with human flourishing, strongly linked to his own distinctive conceptions of virtue and of practice, as well as to his ideas of the biographical unity of a life and the encompassing power of traditions, flows easily into discourse about the aims of education and the kind of undertaking engaged in by teachers. (shrink)

This essay takes issue with Alasdair MacIntyre's denial that teaching is a practice. It does so less by appeal to MacIntyre's concept of practice than by criticism of his conception of teaching. It argues that this conception, as reconstructed from adversions to teaching in a range of his writings, does less than justice to what good teachers accomplish; and that, if this inadequacy is rectified—as much else in his writings suggests that it ought to be—there are clearer grounds (...) for acknowledging teaching as a practice. (shrink)

Journal of Philosophy of Education, EarlyView.

This volume explores the distinctiveness of teaching and learning as a human undertaking and the nature and scope of the philosophy of education. An investigation of the distinctiveness of teaching and learning as a human undertaking. Provides fresh thinking on the nature and scope of the philosophy of education. Draws on the original insights of an international group of experts in philosophy and education. Includes an interview on education with Alasdair MacIntyre, together with searching investigations of his views by (...) other contributors. (shrink)