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.

The truly great ones, the giants, the really original thinkers, the pure philosopher types, these are to be found in the time before Plato.

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)

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)