Open peer commentary on the article “Ethics: A Radical-constructivist Approach” by Andreas Quale. Upshot: The first of my two main goals in this commentary is to establish thinking of ethics as concepts rather than as non-cognitive knowledge. The second is to argue that establishing models of individuals’ ethical concepts is a scientific enterprise that is quite similar to establishing models of individuals’ mathematical concepts. To accomplish these two primary goals, I draw from my experience of working scientifically with von Glasersfeld (...) for 25 years while he was developing radical constructivism as a coherent model of knowing, and appeal to several of his basic insights to establish constructing models of ethical concepts as a scientific enterprise. (shrink)

Open peer commentary on the article “Religion: A Radical-Constructivist Perspective” by Andreas Quale. Upshot: The first of my three main goals in this commentary is to demonstrate that Quale’s radical separation between cognitive and non-cognitive knowledge is not viable. The second is to establish Quale’s assertion that a radical constructivist cannot be genuinely religious is a result of taking radical constructivism and religion as abstracted first-order models and is a result of comparing and contrasting elements of these models. The third (...) goal is to establish how religious radical constructivists establish relations between their religious beliefs and their radical constructivist beliefs. To accomplish the first goal, I appeal to the work of Damasio to establish that what Quale refers to as “non-cognitive knowledge” is inextricably cognitive. I also appeal to the work of both Damasio and von Glasersfeld to demonstrate that the Cartesian duality between mind and body is not viable. To accomplish the last two goals, I make a distinction between first- and second-order knowledge and contrast Quale’s argument that a radical constructivist cannot be religious with the relation between religious beliefs and radical constructivism from the perspective of actual religious radical constructivists. (shrink)

Purpose: One of my goals in the paper is to investigate why realists reject radical constructivism (RC) as well as social constructivism (SC) out of hand. I shall do this by means of commenting on Peter Slezak’s critical paper, Radical Constructivism: Epistemology, Education and Dynamite. My other goal is to explore why realists condemn the use of RC and SC in science and mathematics education for no stated reason, again by means of commenting on Slezak’s paper. Method: I restrict my (...) comments to Slezak’s paper and leave it to the reader to judge which, if any, of the reasons that I advance for these two states of affairs are not specific to Slezak’s paper. Other readers might not agree with my interpretations of Slezak’s paper, including Slezak himself, but I offer them after having worked with von Glasersfeld in interdisciplinary research in mathematics education for over 25 years. Findings: My findings are that Slezak: (1) rejects RC and SC on the basis of unjustified criticisms, (2) does not explore basic tenets of RC nor of SC beyond the unjustified criticisms, (3) rejects how SC and RC have been used in science and mathematics education, based at least in part on the unjustified criticisms, (3) dislikes how SC has been used in science and mathematics education, a dislike that fuels his rejection of any constructivism, and (4) doesn’t explore how RC has been used in scientific investigations in mathematics education. On the basis of these findings, I conclude that how epistemological models of knowing might be used in science or mathematics education would be better left to the educators who use them in interdisciplinary work. (shrink)

Open peer commentary on the target article “Who Conceives of Society?” by Ernst von Glasersfeld. Excerpt: My goal in this commentary is to say enough to suggest that the meanings children impute to the language and actions of other children are based on their current conceptual schemes and that, if the schemes are at different levels of the constructive process, it is no easy feat for children to use their schemes in interactive mathematical communication.

Purpose: In the paper, I discuss how Ernst Glasersfeld worked as a scientist on the project, Interdisciplinary Research on Number (IRON), and explain how his scientific activity fueled his development of radical constructivism. I also present IRON as a progressive research program in radical constructivism and suggest the essential components of such programs. Findings: The basic problem of Glasersfeld's radical constructivism is to explore the operations by means of which we assemble our experiential reality. Conceptual analysis is Glasersfeld's way of (...) doing science and he used it in IRON to analyze the units that young children create and count in the activity of counting. In his work in IRON, Glasersfeld first conducted a first-order conceptual analysis of his own operations that produce units and number, and then participated in a second-order analysis of the language and actions of children and inferred the mental operations that they use to produce units and number. Further, Glasersfeld used Piaget's concept of equilibration in the context of scheme theory in a second-order analysis of children's construction of number sequences and of more advanced ways and means of operating in the traffic of numbers. Research Implications: The scientific method of first- and second-order conceptual analysis transcends our work in IRON and it is applicable in any radical constructivist research program whose problem is to explore the operations by means of which we construct our conceptions. Because of the difficulties involved with introspection, conducting second-order conceptual analyses is essential in exploring these operations and it involves analyzing the language and actions of the observed. But conceptual analysis is only a part of the research process because the researchers are by necessity already involved in creating occasions of observation. The "experimenter" and the "analyst" can be the same person or they can be different people. Either case involves intensive and sustained interdisciplinary thinking and ways of working if the research program is to be maintained over a substantial period of time as a progressive research program. (shrink)

Open peer commentary on the article “Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom” by Philip Borg, Dave Hewitt & Ian Jones. Upshot: My commentary has two general goals. First, I investigate how basic principles of radical constructivism might be used in constructing models of mathematics teaching. Toward that end, I found that I was not in complete intersubjective agreement with Borg et al.’s use of some basic terms. Second, I (...) explore what mathematics teaching might look like in what is construed as constructivist mathematics teaching. Toward that end, I comment on Borg et al.’s use of “negotiation” and explain how a constructivist teacher can establish experiential models of students’ mathematics. (shrink)

Context: Ernst von Glasersfeld introduced radical constructivism in 1974 as a new interpretation of Jean Piaget’s constructivism to give new meanings to the notions of knowledge, communication, and reality. He also claimed that RC would affect traditional theories of education. Problem: After 40 years it has become necessary to review and evaluate von Glasersfeld’s claim. Also, has RC been successful in taking the “social turn” in educational research, or is it unable to go beyond “private worlds? Method: We provide an (...) overview of contributed articles that were written with the aim of showing whether RC has an impact on educational research, and we discuss three core issues: Can RC account for inter-individual aspects? Is RC a theory of learning? And should Piaget be regarded as a radical constructivist? Results: We argue that the contributed papers demonstrate the efficiency of the application of RC to educational research and practice. Our argumentation also shows that in RC it would be misleading to claim a dichotomy between cognition and social interaction (rather, social constructivism is a radical constructivism), that RC does not contain a theory of mathematics learning any more or less than it contains a theory of mathematics teaching, and that Piaget should not be considered a mere trivial constructivist. Implications: Still one of the most challenging influences on educational research and practice, RC is ready to embark on many further questions, including its relationship with other constructivist paradigms, and to make progress in the social dimension. (shrink)

The gauge compensation fields induced by the differential operators of the Stueckelberg-Schrödinger equation are discussed, as well as the relation between these fields and the standard Maxwell fields; An action is constructed and the second quantization of the fields carried out using a constraint procedure. The properties of the second quantized matter fields are discussed.

It is well known that when resolution occurs in the stichic iambics and trochaics of tragedy word-end is not found between the two shorts so produced: w or, more accurately, that the first short of resolution must not be the last syllable of a polysyllabic word. Moreover, the syllables in resolution most often form part of the same word as the following short or anceps, e.g.: Ion 1143.

Some of the problems associated with the construction of a manifestly covariant relativistic quantum theory are discussed. A resolution of this problem is given in terms of the off mass shell classical and quantum mechanics of Stueckelberg, Horwitz and Piron. This theory contains many questions of interpretation, reaching deeply into the notions of time, localizability and causality. A proper generalization of the Maxwell theory of electromagnetic interaction, required for the well-posed formulation of dynamical problems of systems with electromagnetic interaction is (...) discussed, and some of the significance of recently found (classical) relativistic chaotic behavior is pointed out. Many results of a technical nature have been achieved in this framework; in this paper, some of these are reviewed, but I shall concentrate on a discussion of the basic ideas and foundations of the theory. (shrink)

Gilbert Ryle in The Concept of Mind pronounced the “official doctrine” regarding the nature of the mind and the body as “hailing chiefly from Descartes.” That doctrine, anathematized by Ryle as “the dogma of the ghost in the machine,” is said to hold that every human being is composed of a body and a mind, that the body is physical whereas the mind is not, and that the mind may continue to exist when the body is destroyed. Ryle’s famous attack (...) on the official doctrine has undoubtedly stimulated Cartesian studies, both in those who wish to understand Descartes better in order to make the defense of the doctrine more effective and in those who wish to attack it, sometimes even with arguments which Ryle himself would have rejected. Occasionally, one reads a reference to “Platonic dualism,” not so much as a clearly defined alternative to Cartesian dualism but rather as an inchoate or loose version of Descartes’ doctrine. A recent exception is Richard Rorty who in the first chapter of his book Philosophy and the Mirror of Nature has insisted on the difference between the Platonic and the Cartesian positions, although it should perhaps be added that his hostility to the former is if anything greater than his hostility to the latter. Without suggesting that there are not obvious similarities in the Platonic and Cartesian versions of dualism, it will perhaps be useful to try to penetrate beneath the slogans in order to get a clearer picture of the arguments which impelled Plato to hold a position the strangeness which was evident even to his contemporaries. (shrink)

In this paper we show that the prime ideal space of an MV-algebra is the disjoint union of prime ideal spaces of suitable local MV-algebras. Some special classes of algebras are defined and their spaces are investigated. The space of minimal prime ideals is studied as well.

The paper is aimed at analyzing the contribution that the Global Reporting Initiative makes to the field of sustainability reporting. It provides an overview of the multitude of initiatives aimed at standardizing corporate social responsibility efforts on a global scale and highlights the ways in which the GRI can be distinguished from other international initiatives. By evaluating GRI's goals and its claims, the paper provides an overview of the strengths and weaknesses of this critical initiative. It includes a discussion of (...) changes and new strategies that GRI proposes as part of its recently introduced G3 Guidelines. The authors contend that, despite certain remaining challenges, GRI has much to offer a stakeholder community that has for many decades been starved of quality, measurable and accountable corporate social information presented in an accessible and understandable format. (shrink)

1. Bog i tvarnyi mīr -- 2-3. Bog i mīr, v Bogochelovi︠e︡ki︠e︡ vi︠e︡chnyĭ ad pobi︠e︡zhdai︠u︡shchīĭ.

---- t. 6. O nachalakh : opyt khristīanskoĭ metafiziki. [no.] 1. Bog i tvarnyi mīr. [no.] 2-3. Bog i mīr, v Bogochelovi︠e︡ki︠e︡ vi︠e︡chnyĭ ad pobi︠e︡zhdai︠u︡shchīĭ.

As described by the ancient metricians, catalexis is a matter of arithmetic rather than rhythm. They develop the idea in their usual way, mechanically and mathematically, adding and subtracting elements, so as to produce ‘brachycatalexis’ and ‘hypercatalexis’. These are now mere metrical-glossary terms, but in catalexismodern metricians have seen a genuine relationship between cola and a rhythmic effect more or less comprehensible even to us. Wilamowitz, T.D. Goodell, and A.M. Dale explore the concept to some extent, but current hand books (...) and general treatises give it only perfunctory and superficial treatment, while casual appearances of the term offer the reader glimpses of a theoretical substructure which hasnot been explicitly and coherently explained. (shrink)

We discuss some of the experimental motivation for the need for semigroup decay laws and the quantum Lax-Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax-Phillips S-matrix. In the case of discrete (complex) spectrum of the generator of the semigroup, associated with resonances, the decay law is exactly exponential. The states corresponding to these (...) resonances (eigenfunctions of the generator of the semigroup) lie in the Lax-Phillips Hilbert space, and, therefore, all physical properties of the resonant states can be computed. We show that the parametrized relativistic quantum theory is a natural setting for the realization of the Lax-Phillips theory. (shrink)

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a (...) more general type. In particular, detailed discussion is given of the simplest generalization of the complex Hilbert space, that of the quaternion Hilbert module. (shrink)

A fundamental problem in the construction of local electromagnetic interactions in the framework of relativistic wave equations of Klein-Gordon or Dirac type is discussed, and shown to be resolved in a relativistic quantum theory of events described by functions in a Hilbert space on the manifold of space-time. The relation, abstracted from the structure of the electromagnetic current, between sequences of events, parametrized by an evolution parameter τ (“historical time”), and the commonly accepted notion of particles is reviewed. As an (...) illustration of these ideas, a perturbative calculation is made for photon emission from a charged two-body system in which the electromagnetic field is quantized in the usual way. The result is in essential agreement with calculations in which the charged particles are treated in the framework of nonrelativistic quantum mechanics, and provides them with a relativistic interpretation. In particular, we obtain a relativistically invariant form of the Bohr radiation condition. (shrink)