This paper draws its title from the recent symposium of which it was part; it attempts to respond to the question raised by that title, taking current work in set theory into account. To this end the paper contrasts set theory with number theory, examines a severe brand of set-theoretic realism that is suggested by a passage from Godel, and sketches a first-order way of looking at the results about competing extensions of Zermelo-Fraenkel set theory. A formalistic sentiment may be (...) detectable in some portions of the paper. (shrink)

Peirce writes "It is an indubitable result of the theory of probabilities that every gambler, if he continues long enough, must ultimately be ruined... if he plays long enough he will be sure some time to have such a run against him as to exhaust his entire fortune.... The same thing is true of an insurance company. Let the directors take the utmost pains... according to the doctrine of chances, the time must come, at last, when their losses will bring (...) them to a stop." This note shows that under the most plausible interpretation of the quoted passage Peirce is committed to a mathematical untruth. (shrink)

Peter Abelard (1079–1142 ce) was the most wide‐ranging philosopher of the twelfth century. He quickly established himself as a leading teacher of logic in and near Paris shortly after 1100. After his affair with Heloise, and his subsequent castration, Abelard became a monk, but he returned to teaching in the Paris schools until 1140, when his work was condemned by a Church Council at Sens. His logical writings were based around discussion of the “Old Logic”: Porphyry's Isagoge, aristotle'S Categories and (...) On Interpretation and boethius'S textbook on topical inference. They comprise a freestanding Dialectica (“Logic”; probably c.1116), a set of commentaries (known as the Logica [Ingredientibus], c. 1119) and a later (c. 1125) commentary on the Isagoge (Logica Nostrorum Petititoni Sociorum or Glossulae). In a work Abelard called his Theologia, issued in three main versions (between 1120 and c.1134), he attempted a logical analysis of trinitarian relations and explored the philosophical problems surrounding God's claims to omnipotence and omniscience. The Collationes (“Debates,” also known as “Dialogue between a Christian, a Philosopher and a Jew”; probably c.1130) present a rational investigation into the nature of the highest good, in which the Christian and the Philosopher (who seems to be modeled on a philosopher of pagan antiquity) are remarkably in agreement. The unfinished Scito teipsum (“Know thyself,” also known as the “Ethics”; c.1138) analyses moral action. (shrink)

Yoga has come to be an icon of Indian culture and civilization, and it is widely regarded as being timeless and unchanging. Based on extensive ethnographic research and an analysis of both ancient and modern texts, Yoga in Modern India challenges this popular view by examining the history of yoga, focusing on its emergence in modern India and its dramatically changing form and significance in the twentieth century. Joseph Alter argues that yoga's transformation into a popular activity idolized for (...) its health value is based on modern ideas about science and medicine. Alter centers his analysis on an interpretation of the seminal work of Swami Kuvalayananda, one of the chief architects of the Yoga Renaissance in the early twentieth century. From this point of orientation he explores current interpretations of yoga and considers how practitioners of yogic medicine and fitness combine the ideas of biology, physiology, and anatomy with those of metaphysics, transcendence, and magical power. The first serious ethnographic history of modern yoga in India, this fluently written book is must reading not only for students and scholars but also practitioners who seek a deeper understanding of how yoga developed over time into the exceedingly popular phenomenon it is today. (shrink)

What must the United States do to remain the global superpower and stop alienating the rest of the world? The author of the bestselling "The Paradox of American Power" has one clear answer: soft power.

The main purpose of this paper is to bring out some significant humanistic characteristics of Chinese religious thought. My account is limited to what is originally and typically Chinese. That is to say, it will exclude what has been influenced by Buddhism from India or Christianity from the Western world. Some of the theses of this paper are based on scholarly works, while others are drawn from the author's primary experience.

The author of Governance in a Globalizing World probes the limits of American power, offering a compelling argument for the world's lone superpower to forge cooperative relationships with nations around the world.

In two recent papers Perez Laraudogoitia has described a variety of supertasks involving elastic collisions in Newtonian systems containing a denumerably infinite set of particles. He maintains that these various supertasks give examples of systems in which energy is not conserved, particles at rest begin to move spontaneously, particles disappear from a system, and particles are created ex nihilo. An analysis of these supertasks suggests that they involve systems that do not satisfy the mathematical conditions required of Newtonian systems at (...) the time the supertask is due to be completed, or else they rely on the application of the time-reversal transformation to states which are not well-defined. Consequently, it is unjustified to conclude that the paradoxical results are arising from within the framework of Newtonian mechanics. In the last part of this article, we discuss various aspects of the physics of these supertasks. (shrink)

In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 11–complete. Two of these fragments (...) do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 01–complete. These results go via reduction to problems concerning domino systems. (shrink)

It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely to be hard, (...) we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real. (shrink)

A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in (...) discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time. (shrink)