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.

As in literature poetry precedes prose, so in poetry a special and ‘heightened’ diction seems to precede everyday language. Mr.T.S.Eliot has put it thus: ‘Every revolution in poetry is apt to be, and sometimes to announce itself as, a return to common speech.’ How does this apply to Greek and Latin ? There are objections to considering words in isolation from this point of view, since neutral ones are apt to go now grey, now purple, according to their company; but (...) if we do not do so, we deny ourselves the only considerable method of investigation that is still open to us. Again, we must recognize that most poems are composed largely of ordinary words, though these are often used in a way that is not ordinary. (shrink)

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)

The elements which every schoolboy learns on beginning Latin Verse Composition include a number of rules which seem arbitrarily designed to make the game harder. In hexameters, he is told, he must have a masculine caesura either in the third foot or in the second and fourth, and end normally with a disyllabic or a trisyllable; in pentameters he must end with a disyllabic; and in neither line may a single monosyllable stand at the end. Rarely, in my experience, is (...) any reason given him by way of redress, and he will search for one in vain in most of the school text-books, in introductions like Postgate's to Tibullus and Propertius, and in histories of Latin literature like Wight Duff's and Mackail's. This reticence may be due to the dissensions of experts on this subject and on the subject of Latin accentuation in general, but the theory that predominates in England, among those who hold a theory at all, explains so many of the phenomena that it deserves to be more widely and precisely known. The most detailed exposition of it is by E. H. Sturtevant, who summed up his analyses in a pair of articles in the Transactions of the American Philological Association in 1923–4. While his careful work is invaluable as marshalling the statistics and evidence, it errs on the side of excessive minuteness, and leaves room not merely for some additions, but for a different kind of treatment concerned less with bare statistics and more with poetic principles and historical development. Since Latin Verse Composition plays such an important part in English higher classical education, it seems desirable that a less technical and more accessible account should be available for English readers. Such an account I attempt to give here, keeping separate as far as possible the exposition and the consideration of the criticisms and rival theories that have been advanced. (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.

In C.Q. xliii , p. 39, Mr. J. H. Quincey quotes the opening lines of Catalepton 5 as, Ite hinc,-inanes, ite, rhetorum ampullae, inflata rhoso* non Achaico verba, and adds, ‘the second line is corrupt and no satisfactory emendation has been proposed’. The MS. readings are: rhorso B, roso Mu, om. in lacuna Ar. In face of these voces nihili many have fallen back on the rore of the Aldine edition of 1517. But this does not really help, for one (...) does not inflate with dew: orators are not dew-bags, but wind-bags. It occurred to me some years ago that what is needed is some word meaning breath or wind to go with inflata, and that in view of the rh in rhorso it was probably a Greek word which a scribe had failed to recognize. I conjectured οζ, and found subsequently that this had been proposed by K. Münscher in Hermes, xlvii , pp. 153–4. οȋζος used of any rushing sound, is applied to speech by Philostratus , and by Pollux . It is easy to see how rhoezo could degenerate into rhoeso-rhoso-roso. (shrink)

It has been pointed out to me that in my article on ‘The Augustan Rules for Dactylic Verse’ I misrepresented an observation of Maas as reported by Wilamowitz in his Griechische Verskunst, p. 53. Wilamowitz' words are: ‘Wenn Tibull und Ovid den Pentameter so bauen, dass die vorletzte Silbe betont wird, tun sie das nach dem Vorgange gleichzeitiger griechischer Epigrammatiker.’ This means, of course, that the Greek writers mentioned ended with a paroxytone word, not necessarily with a disyllable, as I (...) assumed it to mean, and I apologize for the oversight. (shrink)

The fragmentary columns of the Fourth Book of Philodemus' περ μονσικς were the first-fruits of Herculaneum, published in 1793, with venturesome reconstructions and learned notes, by the Academici of Naples. Fragments of the other books occur in four volumes of the Collectio Altera of 1862–5. A. Teubner Text by J. Kemke, a pupil of Bücheler, appeared in 1884, upon which Gomperz made a number of improvements in a pamphlet published in the following year. Otherwise, save for a few pages in (...) H. Abert's Die Lehre vom Ethos in der griechischen Musik published in 1899, the work has attracted very little attention; which is scarcely surprising in view of the state of the text and the peculiar awkwardness of Philodemus' Greek. (shrink)