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.

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.

Robertson's earlier work, The New Renaissance projected the likely future impact of computers in changing our culture. Phase Change builds on and deepens his assessment of the role of the computer as a tool driving profound change by examining the role of computers in changing the face of the sciences and mathematics. He shows that paradigm shifts in understanding in science have generally been triggered by the availability of new tools, allowing the investigator a new way of seeing into questions (...) that had not earlier been amenable to scientific probing. (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).

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.

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.

The best interest standard is the threshold most frequently employed by physicians and ethics consultants in challenging a parent’s refusal to provide consent for a child’s medical care. In this article, I will argue that the best interest standard has evolved to serve two different functions, and that these functions differ sufficiently that they require separate standards. While the best interest standard is appropriate for choosing among alternative treatment options for children, making recommendations to parents, and making decisions on behalf (...) of a child when the legal decision makers are either unable to make a decision or are in dispute, a different standard is required for deciding when to seek state interference with parental decision-making authority. I will suggest that the harm principle provides a more appropriate threshold for determining when to seek state intervention than the best interest standard. (shrink)

Tibetan Buddhist Philosophy of Mind and Nature is a philosophical overview of Tibetan Buddhist thought. Charting the different ways Buddhist traditions in Tibet configure the relationship between Madhyamaka and Mind-Only, Duckworth shows how these configurations inform the shape of distinct contemplative practices.

The case of Ashley X involved a young girl with profound and permanent developmental disability who underwent growth attenuation using high-dose estrogen, a hysterectomy, and surgical removal of her breast buds. Many individuals and groups have been critical of the decisions made by Ashley's parents, physicians, and the hospital ethics committee that supported the decision. While some of the opposition has been grounded in distorted facts and misunderstandings, others have raised important concerns. The purpose of this paper is to provide (...) a brief review of the case and the issues it raised, then address 25 distinct substantive arguments that have been proposed as reasons that Ashley's treatment might be unethical. We conclude that while some important concerns have been raised, the weight of these concerns is not sufficient to consider the interventions used in Ashley's case to be contrary to her best interests, nor are they sufficient to preclude similar use of these interventions in the future for carefully selected patients who might also benefit from them. (shrink)

Clinical ethics training programs are responsible for preparing their trainees to be competent ethics consultants worthy of the trust of patients, families, surrogates, and healthcare professionals. While the American Society for Bioethics and Humanities (ASBH) offers a certification examination for healthcare ethics consultants, no tools exist for the formal evaluation of ethics trainees to assess their progress toward competency. Medical specialties accredited by the Accreditation Council for Graduate Medical Education (ACGME) use milestones to report trainees’ progress along a continuum of (...) professional development as a means of “operationalizing and implementing” medical competencies. Utilizing the Core Competencies for Healthcare Ethics Consultation and the ACGME and American Board of Pediatrics’ (ABP) Pediatric Milestones Project, we developed milestones for 17 subcompetencies in clinical ethics consultation and academic bioethics. As the field of clinical ethics becomes more standardized, such tools will be needed to promote the development of robust training programs and to certify that their graduates are competent practitioners. (shrink)