In QED, an external electromagnetic field can be accounted for non-perturbatively by replacing the causal propagators used in Feynman diagram calculations with Green’s functions for the Dirac equation under the external field. If the external field destabilises the vacuum, then it is a difficult problem to determine which Green’s function is appropriate, and multiple approaches have been developed in the literature whose equivalence, in many cases, is not clear. In this paper, we demonstrate for a broad class of external fields (...) that includes all that act for a finite time, that four Green’s functions used in the literature are equivalent: Schwinger’s “proper-time” propagator; the “causal propagator” used in the “Bogoliubov transformation” method based on the canonical quantization of the field operator; and two defined using analytic continuation from complexified parameters. To do so, we formulate Schwinger’s “proper-time quantum mechanics" as a Schrödinger wave mechanics, and use this to re-derive Schwinger’s expression for the propagator as a statement relating solutions of the inhomogeneous Dirac equation to those of the inhomogeneous “proper-time Dirac equation”. This is done by constructing direct integral spectral decompositions of the Hamiltonians of both equations, and deriving a form for solutions of the inhomogeneous Dirac equation in terms of these decompositions. We then show that all four propagators return solutions of the inhomogeneous Dirac equation that satisfy the same boundary condition, which under a physically reasonable assumption is sufficient to specify the solution uniquely. (shrink)

This book, based on lectures, examines ways that the tradition of 'Middle' Platonism can point us to aspects of Plato's thinking which contemporary discussions often overlook, and which give us a better-rounded account of Plato.

The definition of acedia presents unique conceptual problems among the eight Evagrian logismoi. Its descriptions are so complex and varied as to render the concept seemingly incoherent. This article argues that the conceptual unity of acedia has been obscured by the translation of Evagrian logismoi into the ‘deadly sins’ tradition, resulting in a category error. Acedia is more properly understood, not as a psychological state or a sin, but rather as an array of demonic temptations with the unifying end-goal of (...) interrupting the anchoritic ideal of sustained contemplation. (shrink)

Although the theory of definability had many important antecedents—such as the descriptive set theory initiated by the French semi-intuitionists in the early 1900s—the main ideas were first laid out in precise mathematical terms by Alfred Tarski beginning in 1929. We review here the basic notions of languages, explicit definability, and grammatical complexity, and emphasize common themes in the theories of definability for four important languages underlying, respectively, descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic. We review (...) the history of previous studies of the similarities and differences in the theories of definability of the first three of these four languages. A seminal role leading toward unification of the theories has been played by the separation principles introduced by Nikolai Luzin in 1927. Emphasizing analogies and driving toward further unification embracing finite-universe logic we concentrate on a simple example—the first and second separation principles for existential-universal first-order sentences . Using this as a test case for the fundamental problem of how to “finitize” arguments in classical pure logic to the finite-universe case, we are led to the analogous negative solution by using the theory of certain special graphs: a graph is - special for any positive integers m , n , p , q iff it is bipartite with m red points and n blue points and for every p -tuple of red points there is a blue point to which they are all connected . As an aside we introduce for further study a natural “Ramseyesque” increasing sequence A of positive integers, where A is the least positive integer n for which an -special graph exists. (shrink)

Introduction, by J. L. Hevesi.--Days of reading, by M. Proust.--Poetry and abstract thought, by P. Valèry.--Jacob Cow the pirate; or, Whether words are signs, by J. Paulhan.--Concerning the pebble, by F. Ponge.--The journey and the return, by J. P. Sartre.--The power of words, by B. Parain.

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 121.2 (2000) 279-289 [Access article in PDF] Dice and Facie: Quintilian, Institutio Oratoria 1.7.23 And 9.4.39 J. Bradford Churchill In his discussion of orthography in book 1 of his Institutio Oratoria Quintilian mentions several examples of archaic spelling conventions, among them a practice of the Elder Cato, which I present here with readings I shall propose and defend during the course of this essay: Non (...) Cato Censorius dicam et faciam dice et facie scripsit eundemque in ceteris quae similiter cadunt modum tenuit? (Inst. 1.7.23 5 Cato, Inc. Lib. fr. 55 Jordan, p. 90) Dice et facie is the reading of one manuscript (N); two other manuscripts (BH) have dicae et faciae; the other major manuscripts (APMS) have the similar but obviously corrupt dice et face. The vast discrepancies between the readings accepted by various editors demonstrate the need for a clarification of the evidence: dice et facie; 1 dicem et faciem; 2 dicae et faciae; 3. 4 The reading dicae et faciae found its way into the most recent edition, the Oxford Classical Text of Michael Winterbottom ([1970] 1989). A close examination of the evidence will confirm that this is not the correct reading. It is time to put to rest a good deal of needless controversy by showing that Cato and his contemporaries used a distinct first-person form in the future indicative of the third conjugation and that they usually or always spelled the active form without the final -m. The reading dice et facie accepted by Radermacher (1959), Fierville (1890), and Halm (1869, 367) should stand. [End Page 279]A century ago (1898) Moore argued, in a brief note influential enough to be cited in Winterbottom's apparatus, that behind Quintilian's citation as preserved lies an innovative orthographical practice not otherwise attested. Moore constructed his hypothesis on the assumption that the variants adduced were simply different ways of spelling the forms dicam and faciam. 5 Cato was not satisfied, Moore proposed, simply to follow the usual practice of the time and omit spelling the final -m, which was in many environments only partly articulated. Since the -m did, however, have some impact on the sound, Cato invented a new custom, turning M on its side at the end of the word to indicate the nasal quality of the final vowel. This anomalous symbol was preserved by Quintilian but later misconstrued as E. In this way Moore attempted to render the otherwise nonsensical forms dicae and faciae comprehensible as textual corruptions of Cato's innovation.Klotz (1931) dealt in greater depth with the problem, and although his solution is almost certainly inadequate, he touched on most of the evidence needed to solve it. Because Quintilian quotes Cato's archaic spelling convention again in book 9 (Inst. 9.4.39) in a discussion of sounds which are not spelled, Klotz conjectured (142) that Verrius Flaccus had quoted Cato as a precedent for his own practice of indicating the attenuated pronunciation of an elided final -m by writing only the first half of the letter. So, Klotz's argument continues, it was from Verrius Flaccus (GL 7.80) that Messala drew the example, and from him Quintilian derived the extant version. The "half-M" was misconstrued by Quintilian as E, and so the textual tradition preserves what he wrote: dicae et faciae.Both Moore and Klotz have read Quintilian too narrowly. Each of their arguments is predicated on the assumption (in Moore's case) or argument (in Klotz's) that Cato's forms were morphologically identical to Quintilian's dicam and faciam. They dismissed the possibility that Cato was spelling forms distinct from the classical counterparts listed by Quintilian. This is not a surprising dismissal: among the examples cited by Quintilian, these are the only ones which turn out (if the current argument is accepted) to have been morphologically distinct from the [End Page 280] classical "equivalents." Nevertheless, a careful examination of all the evidence, and in particular a close reading of the cognate passage... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:714 BOOK REVIEWS learning; the Jesuits lean to the voluntarist. Possessed of a unitary academic model, he is really arguing for Aristotle's analogy of attribution. Apparently, no one of his 200 plus Jesuit contacts told him that Nastri prefer St. Thomas and his analogy of proper proportionality. The historian Daniel Boorstin spent 25 yeari!l writing his trilogy on The Americans. What emerges from this analysis? Boorstin points out that (...) Americans like ambiguity, that they function best amid flux; ambivalence becomes their proper medium. For this reason, I have accepted Nathaniel Hawthorne as our most central writer. He situates American questions within a landscape of chiaroscuro. The Jesuit experience in the United States belongs essentially to the same landscape. But today, as formerly, it remains a landscape of questions. McDonough, on the other hand, has too many certitudes, too many answers, too much high definition-what Henry Adams regards as "mere facts." Instead of illuminating the landscape from within, his study inevitably recedes into the larger ambiguity reserved for what Wallace Stevens has called, " Prologues to what is possible." Saint Paul University Ottawa, Ontario JOHN P. McINTYRE, S.J. The Dilemma of Freedom and Foreknowledge. By LINDA TRINKAUS ZAGZEBSKI. New York: Oxford University Press, 1991. viii + 215 pp. $29.95. As a Christian and a philosopher, Linda Trinkaus Zagzebski wants to understand why it is reasonable to affirm that we remain free despite God's infallible knowledge of our future acts. Moreover, she is not content simply to demonstrate that this affirmation involves no logical con· tradiction, for "reasonable people expect more" (p. 33) than a merely negative result. Hence the author sets herself the more ambitious task of working toward a positive solution that would explain why the compatibility of divine foreknowledge and human freedom is plausible and not a bare logical possibility. The hook sorts out recent opinions on the subject and sketches an original position that she hopes will advance the discussion. Zagzehski begins by constructing the most formidable version of the dilemma. Assuming that God is infallible and essentially omniscient, that the past has a kind of necessity that the future lacks ("accidental necessity"), and that this necessity can he transferred (in some sense) from the antecedent to the consequent of a conditional proposition, the dilemma may he stated as follows: BOOK REVIEWS (1) Necessarily, God believes at time t 1 that I will do S at t 3 if and only if I do S at t 3. (2) It is accidentally necessary at t 2 that God believes at t 1 that I will do S at t 3. (3) If (1) and (2) are true, then it is accidentally necessary at t 2 that I do Sat t 3. (4) It follows that it is accidentally necessary at t 2 that I do S at t 3. (5) But if it is accidentally necessary at t 2 that I do S at t 3, I do not do S at t 3 freely. (6) Therefore, I do not do S at t 3 freely. 715 Much of the first chapter is devoted to establishing that this form of the dilemma is more resistant to solution than any other, especially the so-called problem of future truth. Zagzebski goes on to evaluate three broad approaches to solving the dilemma. In Chapter 2 she takes up the " timelessness " solution. Its proponents, among whom the author numbers Boethius, Thomas Aquinas, Eleonore Stump, and Norman Kretzmann, negate premise (2) of the foreknowledge dilemma by contending that God is not in time. Zagzebski considers this an acceptable negative solution to the dilemma and sees in it the potential to ground a more comprehensive, positive solution. Nonetheless, she refrains from subscribing wholeheartedly to the timelessness solution because it confronts one with a dilemma at least as perplexing as the one it solves. She puts the problem this way: like the past, eternity is fixed, for it is constituted by a state of affairs about which we cannot do anything; therefore my future acts are determined no less by God's timeless knowledge than by God's foreknowing them in a strictly temporal sense. In Chapter 3, Zagzebski considers... (shrink)

High quality images sell products. Here's how you do it. From cereal boxes to billboards to photos on Amazon, product photos have a strong impact on viewers. Now you can master the secrets of effective product photography with this essential guide. Author J. Dennis Thomas guides you through the basics, from selecting the right equipment and practicing different lighting techniques to controlling exposure, using backgrounds and props, and much more. Whether it's jewelry, food, fashion, or other products, learn how to (...) photograph for effective selling, while building the skills and tools you need for a career. Explains how to produce quality photos for product or commercial photography, including fashion, food, jewelry, technology, and more The author is a professional photographer whose work has been published in major U.S. magazines including Rolling Stone, Elle, W Magazine, and US Weekly Covers choosing the right equipment, practicing different lighting techniques, controlling exposure, using backgrounds and props, and more Gives new and even experienced photographers the tools they need to build careers in product photography Take photographs that impress, intrigue, dazzle, and sell with The Art and Style of Product Photography. (shrink)

It would be a mistake to attempt to identify in modern terms the disease of Galerius described so graphically by Lactantius, Mort. 33. Consumption by lice or worms, if not genital ‘gangrene’, was a typical end for a tyrant or the impious, and there must be an element of literary exaggeration in Lactantius' account. But whatever one makes of the nature of the illness, Lactantius did set out to give the passage a scientific plausibility by his use of technical medical (...) phraseology, and by an allusion to a medical theory at 33.7. Recognition of this theory allows one to settle the text at one point, where editors have failed to agree. There is also a second place in the chapter where familiarity with medical Latin points one towards the solution of a textual problem. (shrink)

The demonstration by E. Wölfflin that between theHistoriesandAnnalsTacitus progressed towards a more archaic and artificial style is well known. From the outset Tacitus adhered to the traditional Roman view that history should be composed in an archaic language remote from everyday usage ; but he was apparently at first not fully aware of the possibilities of the archaizing style. New archaisms and artificial usages suggested themselves as he advanced ; and others, which he had used sporadically even early in theHistories, (...) were allowed to oust ordinary alternatives completely. (shrink)

Takeuti introduced an infinitary proof system for determinate logic and showed that for transitive models of Zermelo-Fraenkel set theory with the Axiom of Dependent Choice that contain all reals, the cut-elimination theorem is equivalent to the Axiom of Determinacy, and in particular contradicts the Axiom of Choice. We consider variants of Takeuti's theorem without assuming the failure of the Axiom of Choice. For instance, we show that if one removes atomic formulae of infinite arity from the language of Takeuti's proof (...) system, then cut elimination is equivalent to a determinacy hypothesis provable, e.g., in ZFC + “there are infinitely many Woodin cardinals.” A slight extension of the proof system admits cut elimination for countable sequents under the same assumptions. A simple modification of the proof system yields analogs of the results above for the Axiom of Real Determinacy and uncountably many Woodin cardinals. (shrink)

This essay answers the question: Which are the themes often considered as “Philosophy of Law” that are of concern to legal scholars? My answer seeks to sort out how the “thematic content” has become into what today is known as “Philosophy of Law” to legal scholars (against “Philosophy of Law” to philosophers). To deal with this question I will consider Norberto Bobbio’s point of view.Resumen:En este texto trato de responder a la pregunta: ¿cuáles son los temas calificados generalmente como “de (...) filosofía del derecho” que nos interesan a los juristas? Mi respuesta se dirige a desgranar cómo se ha ido construyendo el “contenido temático” más habitual que hoy identifica a la filosofía del derecho de los juristas (frente a la filosofía del derecho de los filósofos). Para tratar esta cuestión examino el punto de vista de Norberto Bobbio. En nuestro tiempo, dicha perspectiva constituye el punto de partida en la discusión de cuáles son los ámbitos problemáticos más habituales de la reflexión teórico-jurídica. (shrink)

Judges have a duty to decide judicial cases, however, judicial decision will vary and will have certain characteristics depending on the kind of case at hand: this article address three kinds of judicial cases: clear cases, borderline cases and pivotal cases. The purpose: discuss if relying on these classifications is useful or not in light of the unity in judicial adjudication principle.Resumen:Los jueces tienen el deber general de resolver los casos judiciales; sin embargo, la decisión judicial variará y tendrá ciertas (...) características dependiendo del tipo de caso; son tres los tipos de casos judiciales que este artículo estudia: clear cases, borderline cases y pivotal cases. El objetivo: intentar descubrir si adoptar una de las tipologías señaladas vulnera o reafirma el principio de unidad propio de la aplicación judicial. (shrink)

This essay subjects to criticism the historical and philosophical attitudes to political thought found in the writings of John Dunn and Michael Oakeshott. The essay does not limit itself to criticism but attempts to elaborate what is indicated by criticism for the sake of the modern understanding of political thought. The argument is that history and philosophy as they have recently been practised suffer from limitations that can only be addressed by a recognition of something which is here called theology. (...) The significance of such a recognition is considered with regard to the history of political thought, political philosophy and finally to theology itself. The emphasis on theology makes it necessary to say that the essay is written for the benefit of historians and philosophers, not theologians. Although it is an argument for theology it is in no sense an argument from theology. Whether this is a contradictory position may require consideration elsewhere; but since the essay is a first step rather than a final step in an argument, it may gain more than it loses by such a contradiction, if there is one. (shrink)