Zeno's Dichotomy: Undermining The Modern Response

Some applications of fractional calculus involving summation of infinite series

The Journal of the History of Philosophy: What It All Means Richard A. Watson (bio) The Study of the History of Philosophy as an independent discipline to exhibit and explicate philosophical systems as their originators meant them to be understood is less than one hundred years old. On the other hand, philosophers from Plato and Aristotle through the Middle Ages to Bertrand Russell and Richard Rorty have represented the systems of their predecessors in the light of, and as leading to, their own philosophical positions. It is not surprising then that the study of the long history of the history of philosophy itself discloses a series of often greatly divergent expositions of philosophical systems throughout the ages. The purist conception of the study of the philosophical position of a given philosopher or school of philosophers as meant and understood in the context of the times of that philosopher or school is fairly recent. On this contextual ground, to study any part of the history of philosophy, one must examine and give expositions of philosophers and of the philosophies that influenced them as they were understood in their own times. Such a study of the history of philosophy—call it historicist history of philosophy—further requires knowledge of the history of ideas, of society, and of politics. This historicist stance makes pursuit of the history of philosophy more intricate and more relevant to philosophy per se than is often recognized by philosophers who construct, analyze, and defend contemporary philosophical positions. When contemporary analytic philosophers direct constructive and analytical work on historical philosophers and historical philosophical positions, this work is called analytic history of philosophy, and it is often undertaken with the goal of advancing contemporary philosophy. A recent analytic philosopher who has specialized in this area is Jonathan Bennett.1 An example of a philosopher who provides a history of philosophy in the classic sense to support his own philosophical position is Richard Rorty.2 [End Page 1] In 1983, the History of Philosophy Quarterly was established in some part to publish articles in which historical positions and arguments are analyzed using the techniques of contemporary analytic philosophy. Critics of such analytic history of philosophy point out that the philosophical positions that result are by definition and intent not true to the actual positions of the philosophers under consideration. This is true, but to present it as criticism is to miss the point of analytic history of philosophy, which just is to construct, on the foundations of historical systems, reconstructed systems that are more consistent and more relevant to contemporary philosophical issues and problems than are the unrevised historical systems in themselves. To provide explications of philosophical systems as their creators meant them to be is the work of historicist historians of philosophy. Reconstructions of philosophical systems to rectify and improve them to make them comprehensible to philosophers today and relevant to contemporary philosophical issues is the work of analytic historians of philosophy. Analytic history of philosophy is older than historicist history of philosophy. In fact, it is as old as Western philosophy, and, as I remark above, it has been practiced by philosophers from Plato to Russell and Rorty. The Journal of the History of Philosophy was established fifty years ago to publish articles in which philosophical texts are examined to explicate what their authors meant in historical—as well as in biographical, scientific, religious, and political—context. This requires that a historicist historian of philosophy know the language in which the text was written, the meanings of philosophical words at the time of its composition, and the general historical, social, and political context in which a given text was written. It goes without saying that the historian is presumed also to have enough general philosophical training to be able to understand, analyze, and assess philosophical arguments. If my exposition here seems a bit overdone, it might be because a fine historian of philosophy, Gregor Sebba, took me to task, because while he approved of the historicist history of philosophy in my book The Downfall of Cartesianism,3 he disapproved mightily of my critical analyses of the key notions of Cartesians. He correctly saw the book as a bastard mix...

Those knowledgeable about the subject of Professor Suppes’ paper will appreciate that there are few contemporary philosophers who are equal to debate with him on this topic or to criticize his application of the various probability approaches to specific domains of empirical investigation and practice. In his present paper it is mainly in the appendix, in which he sketches his calculus for determining upper and lower probability approximations, as detailed in his 1974 paper on ‘The Measurement of Belief’ [1], that Suppes exhibits his special technical expertise. There he sets forth his version of Bayes’ well-known theorem as a more plausible approach than other probability techniques for purposes of applying statistical methods to situations of practical belief, decision and practice. As is to be expected, his exposition provides no opportunity for critical comment.

(1907). Summation of Certain Infinite Series. The American Mathematical Monthly: Vol. 14, No. 3, pp. 41-46.

This paper is mainly intended for postgraduate orthodontic students especially for those who will learn and practice the Fixed Appliance Technique. With the advancement of research, introduction of new biomaterials and techniques, the orthodontic treatment has been scientifically affordable for the last few decades. In Bangladesh the orthodontic treatment and education is also becoming very popular in the private and institutional level day by day. In fact, post-graduate trainee doctors in the department of orthodontics & Dentofacial Orthopedics, Dhaka Dental College and Hospital were the once who had the idea of writing this paper. While supervising & conducting FCPS Part II examination as a convener, I felt this sort of clinical and technical review would be very helpful for the trainees as well as practitioners. Keeping that thought in my mind, I also intend to write a series of papers that will contain the basic as well as contemporary orthodontic techniques and philosophy. Post-graduate trainee doctors, faculty members and private practitioners will all find this paper as a guideline during their training as well as in professional practicing period. I am very much delighted to present this paper and series of papers in the next subsequent issues. The present paper describes training techniques, the design of standard edge-wise-technique in sequential stages for treating Angle Class II maxillary protrusion with excessive over jet, overbite (upper and lower 1st premolar extraction case), and illustrated case reports1,2 of class II malocclusion. The author acknowledges that the paper is summarized from lectures, handouts during his postgraduate studies in Kyushu University and Hiroshima University, Japan and also clinical experience acheived from the Department of Orthodontics and Dentofacial Orthopedics, Dhaka dental College and Hospital and private practice in Bangladesh. DOI: http://dx.doi.org/10.3329/bjodfo.v1i1.15976 Ban J Orthod & Dentofac Orthop, October 2010; Vol-1, No.1, 27-37

I. In the Paper, which the Royal Society did me the honour to print, on Summation of Series, is given a method of finding the sum of series

In Sect. 1.9 we considered summation of finite series. In practice it is often necessary to deal with infinite series which contain an infinite number of terms a n . Here a n is a general (we say “the n-th”) term of the series which is constructed via a some kind of formula depending on an integer n = 1, 2, 3, …. For instance, the rule \(a_{n} = a_{0}q^{n}\) with n = 0, 1, 2, 3, … corresponds to an infinite geometric progression $$\displaystyle{a_{0} + a_{0}q + a_{0}q^{2} + \cdots = a_{ 0}\left (1 + q + q^{2} + \cdots \,\right ).}$$

The present paper is an abridged version of another paper of mine titled “The Ethical Turn of Contemporary Western Philosophy, Semiotics, and Linguistics” delivered at the 11th World Congress of Semiotics in response to related essays by Augusto Ponzio, Susan Petrilli, John Deely, and others. A review of contemporary Western philosophy reveals a confluence and integration of scientism and humanism, a linguistic turn, and a formation of postmodernism (poststructuralism), all of which marks an ethical turn (or focus) in contemporary Western philosophy. As semiotics is based on philosophy, and linguistics is a branch of semiotics, this ethical turn in philosophy concurrently occurs in contemporary Western semiotics and linguistics.

The theory of distributions has a flavor similar to the theory of summation of divergent series and integrals and, as we have seen in Chapter 6, is closely related to the theory of singular integrals. A generic alternating series $$1 - 1 + 1 - 1 + \ldots = \sum { \pm 1} $$ is a good example here. It seems to make no sense to assign a specific value to this infinite sum. Nevertheless, mathematicians have produced certain reasonable rules of summation that assign to it value 1/2. Such an assignment is in complete agreement with the intuition of physicists who encounter similar series. In this chapter, we will see how one can sum this, or even more strange, divergent series and integrals. To gain a better insight into the essence of this problem, let us begin with elementary examples and recall basic notions and theorems of the ordinary theory of convergent infinite series.

By a series we mean a set of numbers a1, a2, a3… such that we have a rule for calculating a2, a3 etc. from the first number a1.Series occur in many problems in chemistry such as specific heats of solids, the theory of black-body radiation, solution of the Schrödinger equation, statistical thermodynamics and Fourier series in X-ray crystallography. In this chapter we consider the summation of series and the convergence of infinite series. We shall develop methods for the expansion of functions such as transcendental functions in terms of infinite series.

In the late middle ages, chemistry was the science and technology closest to philosophy, the material realization of the method of analysis and synthesis. No longer. Contemporary philosophy is concerned with many sciences—physics, psychology, biology, linguistics, economics—but chemistry is not among them. Why not? Every discipline has particular problems with some philosophical coloring. Those in quantum theory are famous; those in psychology seem endless; those in biology and economics seem more sparse and esoteric. If, for whatever reason, one’s concern is the conceptual or theoretical problems of a particular science, there is no substitute for that science, and chemistry is just one among others. Certain sciences naturally touch on substantive areas of traditional philosophical concern: quantum theory on metaphysics, for example, psychology on the philosophy of mind, and economics and statistics on theories of rationality. In these cases, there is a special interest in particular sciences because they may reform prior philosophical theories or recast philosophical issues or, conversely, because philosophy may inform these subjects in fundamental ways. That is not true, in any obvious way, of chemistry. So what good, then, what special value, does chemistry offer contemporary philosophy of science? Typically philosophical problems, even problems in philosophy of science, are not confined to a particular science. For general problems—problems about representation, inference, discovery, explanation, realism, intertheoretic and interdisciplinary relations, and so on—what is needed are scientific illustrations that go to the heart of the matter without requiring specialized technical knowledge of the reader. The science needed for most philosophy is familiar, not esoteric, right in the middle of things, mature and diverse enough to illustrate a variety of fundamental issues. Almost uniquely, chemistry fits the description. In philosophy of science, too often an effort gains in weight and seriousness merely because it requires mastery of an intricate and arcane subject, regardless of the philosophical interest of what it says. Yet, surely, there is something contrived, even phony, in illustrating a philosophical point with a discussion of the top quark if the point could be shown as well with a discussion of the ideal gas law.

In this book, Mackie critically examines various philosophical problems raised in John Locke's An Essay Concerning Human Understanding. Mackie carefully considers Locke's treatment of these problems, but also proposes his own resolution of the related issues in contemporary philosophy. In the course of these discussions, Mackie also proposes his theory of a realism combined with a moderate empiricism. In Ch. 1, Mackie considers the distinction between primary and secondary qualities and argues that they might still have some legitimate application. In Ch. 2, he also defends a version of Locke's representative theory of perception. In Ch. 3, Locke's theory of substance is examined and compared with the theories of more recent philosophers, such as Saul Kripke. In Ch. 4, Mackie defends Locke's theory of abstraction against Berkeley's criticisms and suggests his own revision of the theory considered in relation to the problem of universals. In Chs. 5 and 6, Mackie examines and revises Locke's consideration of identity over time and personal identity. Finally, in Ch. 7, Mackie critically discusses Locke's rejection of innate ideas and considers the possibility of a reconciliation of innate ideas and empiricism.

Bertrand Russell’s work in philosophy of science has been identified as a progenitor of structuralism in contemporary philosophy. It is often unclear, however, how the philosophical problems facing contemporary structuralist programmes relate to the problems of philosophy as Russell saw them. We contend that Russell has been mistakenly identified as an epistemic structural realist. The goal of this essay is to clarify the relationship between Russell’s programme and contemporary structuralist projects. In doing so, we hope to display the motivation for a broad, truly Russellian structuralist project in the philosophy of science.

A familiar view among Nietzsche's admirers whose training is in Anglo-American philosophy is that he would, up to a point, have reciprocated and admired the enterprise of analytical philosophy—as far it goes. Richard Schacht, for instance, writes that Nietzsche "would find no fault with the most exacting projects of linguistic-analytical inquiry—as far as they go," but that he would reject the idea that such inquiries comprise "the sole and entire business of philosophy" (1983, 42). I agree with the second point: philosophers, says Nietzsche, should not simply "accept concepts as gifts," but "should make and create them" (WP, 409). 1 Philosophy should be revisionary, not just descriptive. I agree, too, that Nietzsche would applaud several virtues often associated with analytical philosophy—objectivity and tough-mindedness, say, and the close attention to language, not only to particular words but to the grammatical structures that, for Nietzsche as for many twentieth-century philosophers, constrain or reflect our thinking.

Summation of Certain Infinite Series