In this paper, I attempt to describe Mark Burgin’s results in non-Diophantine mathematics, which are important for the foundation of mathematics and its application in quantum field theory. In particular, the elimination of divergences in quantum electrodynamics is described.

In lieu of an abstract, here is a brief excerpt of the content:Book Reviews Zoroaster's Influence on Anaxagoras, the Greek Tragedians, and Socrates. By Ruhi Muhsen Afnan. (New York: Philosophical Library, 1969. Pp. 162. $5.00) Of the author's Zoroaster's Influence on Greek Thought a striking flaw was the misleading rifle. In this earlier volume not one example of Zoroastrian impact was pointed out to corroborate the claim. Now, in the preface to the new work, the author discloses that the original (...) title had merely been "Zoroaster and the Trend of Greek Thought" and that the publishers changed it into "Zoroaster's Influence" etc. "to give the book a more attractive heading" (p. 9). This fiat of the publishers, however, was then gratefully felt by the author a~ a "challenge.... prodding" him to the "venture" (cf. p. 10) of attempting to substantiate (what was before just his belief) "that the illumination vouchsafed by Zoroaster generated the Enlightenment attributed to the Periclean Age of Athens" (p, 12) and to demonstrate, for instance, an ostensible basic similarity between the pfiilosophy of Anaxagoras and the teachings of Zoroaster. This venturous attempt fills six long chapters supposed to show that "Medism," or Zoroastrianism, was the great spiritual power menacing the autochthonous Athenian values, and that Anaxagoras and his circle and Aeschylus, Euripides, Socrates, and Thucydides essentially were propagandists of Zoroastrianism who undermined the pagan culture of Athens. True, the general presentation of Medism and its role in the spiritual struggle of the Periclean Age is sometimes thrillingly interesting reading. Yet, for us the main question is: Has the author succeeded in his attempt? There are prerequisites of such an undertaking. Thorough and first-hand knowledge of both sides of the topic is indispensable, in the first place. To question the historical authenticity of a modern believer's presentation of Zoroastrianism might seem improper. But it should still be permitted sharply to distinguish between the official traditional version and the results of recent serious research. The other side of the topic is concerned with those who, according to the author's claim, were downright prophets of Zoroastrianlsm. Whether this holds true of Aeschylus and Euripides, whether, for instance, really all the dramatis personae of Euripides' Ion were meant as nothing, but symbolizations of the antagonistic spiritual powers of Medism and Athenianism, this to discuss, and perhaps to doubt, is up to literary analysts. But two of those alleged heralds are philosophers: Anaxagoras and Socrates. (Incidentally, what about Heraclitus and Empedocles?) Well, let us forget about Socrates. The "real" Socrates is such an intangible figure anyway that it would not make much sense to argue with any of the author's allegations about him. In the presentation and "analysis" of the philosophy of Anaxagoras, however, the decisive shortcomings come to the fore. The level of the author's general philosophical training is not quite equal to the task. Conceptual haziness, confusion [469] 470 HISTORY OF PHILOSOPHY of fundamental notions (e.g.,"creator" and "demiurge") are omnipresent. Sometimes even a confusion happens of Anaxagoras with Democritus when the "atom" is ascribed to Anaxagoras (p. 48). And the author does not seem to feel the fatal inadequacy of merely second-hand knowledge. While he in longura et latum argues with Aristotelian presentations and misrepresentations of Anaxagorean tenets, there is good reason for the suspicion that he might never have read even one authentic sentence written by Anaxagoras himself. For those interested in general cultural history, this book, the religious diatribe of a modem Zoroastrian believer, is very worthwhile reading. For the historian of philosophy, itis of rather modest value. P-~LIX M. CLEVE New School for Social Research New York City The Esthetics of the Middle Ages. By Edgar De Bruyne. Trans. Eileen B. Hennessy. (New York: Frederick Uugar, 1969. Pp. viii+232. $6.50) This book has a very complex character. It is the English translation of a French work, L'Esth~tique du moyen dge,I which, in turn, is a one-volume abridged version by Edgar De Bruyne of his own original three-volume work, I~tudes d'Esth~tique M~di~vale.2 Thus, one may evaluate this book either as a translation or, presupposing the faithfulness and correctness of the translation itself... (shrink)

The problem of infinities in quantum field theory (QFT) is a longstanding problem in particle physics. To solve this problem, different renormalization techniques have been suggested but the problem persists. Here we suggest another approach to the elimination of infinities in QFT, which is based on non-Diophantine arithmetics – a novel mathematical area that already found useful applications in physics, psychology, and other areas. To achieve this goal, new non-Diophantine arithmetics are constructed and their properties are studied. In addition, non-Diophantine (...) integration is developed in these arithmetics. These constructions allow using constructed non-Diophantine arithmetics for computing integrals associated with Feynman diagrams. Although in the conventional QFT such integrals diverge, their non-Diophantine counterparts are convergent and rigorously defined. As the result, QFT becomes consistent with quantum experiments. (shrink)

Health behaviors such as tobacco use contribute significantly to poor health. It is widely recognized that efforts to prevent poor health outcomes should begin in early childhood. Biomedical enhancements, such as a nicotine vaccine, are now emerging and have potential to be used for primary prevention of common diseases. In anticipation of such enhancements, it is important that we begin to consider the ethical and policy appropriateness of their use with children. The main ethical concerns raised by enhancing children relate (...) to their impact on children’s well-being and autonomy. These concerns are significant, however they do not appear to apply in the case of the nicotine vaccine; indeed the vaccine could even further these goals for children. Nevertheless, concerns about broadly applying this enhancement may be more challenging. The vaccine may be less cost-effective than alternative public efforts to prevent tobacco use, utilizing it could distract from addressing the foundational causes of smoking and it might not be publically acceptable. Empirical research about these concerns is needed to ascertain their likelihood and impact as well as how they could be minimized. This research could help determine whether behavior-related enhancements hold promise for improving children’s health. (shrink)

This book applies Augustine's thought to current questions of teaching and learning. The essays are written in an accessible style and is not intended just for experts on Augustine or church history.

The ninth and last volume of the Adagia of the Amsterdam edition of the Latin texts of Erasmus gives an introduction in German and a critical edition of the Latin text of the Collectanea , an early version of the Adagia that completely differs from the latter collection.