Infinity is an intriguing topic, with connections to religion, philosophy, metaphysics, logic, and physics as well as mathematics. Its history goes back to ancient times, with especially important contributions from Euclid, Aristotle, Eudoxus, and Archimedes. The infinitely large is intimately related to the infinitely small. Cosmologists consider sweeping questions about whether space and time are infinite. Philosophers and mathematicians ranging from Zeno to Russell have posed numerous paradoxes about infinity and infinitesimals. Many vital areas of mathematics rest upon some version (...) of infinity. The most obvious, and the first context in which major new techniques depended on formulating infinite processes, is calculus. But there are many others, for example Fourier analysis and fractals.In this Very Short Introduction, Ian Stewart discusses infinity in mathematics while also drawing in the various other aspects of infinity and explaining some of the major problems and insights arising from this concept. He argues that working with infinity is not just an abstract, intellectual exercise but that it is instead a concept with important practical everyday applications, and considers how mathematicians use infinity and infinitesimals to answer questions or supply techniques that do not appear to involve the infinite.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. (shrink)

Raymond Aron is widely regarded as the most important figure in the history of twentieth-century French liberalism. Yet his status within the history of liberal thought has been more often proclaimed than explained. Though he is frequently lauded as the inheritor of France's liberal tradition, Aron's formative influences were mostly non-French and often radically anti-liberal thinkers. This book explains how, why, and with what consequences he belatedly defined and aligned himself with a French liberal tradition. It also situates Aron within (...) the larger histories of Cold War liberalism and decolonization, re-evaluating his contribution to debates over totalitarianism, the end of ideology, and the Algerian War. By exposing the enduring importance of Aron's student political engagements for the development of his thought, Iain Stewart challenges the prevailing view of Aron's early intellectual trajectory as a journey from naïve socialist idealism to mature liberal realism, offering a new critical perspective on one of the twentieth century's most influential intellectuals. (shrink)

The purpose of this paper is to describe briefly the institutional arrangements which condition the activities of accountants in the United States; to heighten an awareness of the values which are embodied in the existing structures of accountability; to appraise the consistency with which the established ideals of society have been actualised in financial reporting, and to discern the shape of the emerging history of financial reporting in the light of new values and possibilities. I suggest that the tradition of (...) fair presentation in financial reporting is in danger of being eliminated by a purely political response to previous abuses. Mandatory accounting standards do have a role in preventing these abuses, but they can also be seen as instruments of distributive justice. To promote an awareness and discussion of the issues, I point out some of the goals and values underlying a few of the major structures designed for the distribution of economic costs and benefits. (shrink)

In his testamentary letter to Auguste Chevalier, Évariste Galois states that, in modern terminology, the smallest simple group has order 60. No proof of this statement survives in his papers, and it has been suggested that a proof would have been impossible using the methods available at the time. We argue that this assertion is unduly pessimistic. Moreover, one fragmentary document, dismissed as a triviality and misunderstood, looks suspiciously like cryptic notes related to this result. We give an elementary proof (...) of Galois’s statement, explain why it is likely that he would have been aware of the methods involved, and discuss the potential relevance of the fragment. (shrink)

Taking as its starting point recent claims that Jean-Paul Sartre's Critique de la Raison Dialectique was written as an attempt to overcome the historical relativism of Raymond Aron's Introduction à la philosophie de l'histoire , the present article traces this covert dialogue back to a fundamental disagreement between the two men over the interpretation of Wilhelm Dilthey's anti-positivist theory of Verstehen or 'understanding'. In so doing it counters a longstanding tendency to emphasise the convergence of Aron and Sartre's philosophical interests (...) prior to the break in their friendship occasioned by the onset of the Cold War, suggesting that the causes of their later, radical political divergence were pregnant within this earlier philosophical divergence. (shrink)

Dugald Stewart is usually thought of as the final major figure of the Scottish Enlightenment. But though his name is a recognisable one among intellectual historians, few would probably be able to...

Summary This article approaches post-war debates about the relationship between normative political theory and empirical political science from a French perspective. It does so by examining Raymond Aron's commentaries on a series of articles commissioned by him for a special issue of the Revue française de science politique on this theme as well as through an analysis of his wartime dialogue with the neo-Thomist philosopher, Jacques Maritain. Following a consideration of Aron's critique of contemporary approaches to this issue in France, (...) we discuss his own distinctive attempt to draw normative theory and empirical science into the same orbit by tracing the interaction of these two elements in his work from the late 1930s to the mid-1960s. (shrink)

We introduce a new logical operator CSTC and show that incorporating this operator into first-order logic enables as to capture the complexity class PSPACE. We also show that by varying how the operator is applied we can capture the complexity classes P, NP, the classes of the Polynomial Hierarchy PH, and PSPACE. As such, the operator CSTC can be regarded as a general purpose operator. We also give applications of these characterizations by showing that P and NP coincide with those (...) problems accepted by two new classes of program schemes , by producing some new complete problems for various complexity classes where the reductions are from first principles and do not use known complete problems , and by giving a simple proof that first-order logic incorporated with the least fixed point operator captures P. (shrink)

Jörgen Jörgensen (1938) asks why there should not be a valid deduction even though the premises are imperatives (“Jörgensen’s Dilemma”). Robert Walter (1996; 9 Ratio Juris 168), following Hans Kelsen, thinks that there can be a valid deduction if the premises, although in prescriptive (including imperative) language, are actually descriptions of prescriptions. It is suggested that Walter then has his own dilemma: the more possible it is, for such descriptions to be valid, the less likely it is, that the prescriptions (...) described are illogical; Walter’s Dilemma is a species of Jörgensen’s. (shrink)

This article examines the English scholar James Cowles Prichard's attention to language and comparative philology within his wider project on the natural history of man. It reveals that linguistic evidence was among the most important elements for Prichard in his overarching scientific aim of investigating human physical diversity, and served as the evidential foundation for his ethnology. His work on Celtic comparative philology made him not only one of the earliest British adopters of German comparative grammar, but a comparative philologist (...) of European stature in his own right. More generally, linguistic evidence helped Prichard to keep his magnum opus, Researches into the Physical History of Mankind, as logically ordered as possible, and therefore to turn ethnology into a discipline with analytical aspirations on a global scale. (shrink)

Hans Kelsen’s Pure Theory of Law, presented as a form of legal positivism, is interpreted as establishing metaphysical bases of legal science, on analogy with Kant’s “metaphysical bases of natural science (Metaphysische Anfangsgründe der Naturwissenschaft)”. It is asked, in the light of the Pure Theory’s acknowledged difficulties, how far it succeeds in escaping from the exegetical tradition in Western legal theory.

We investigate why similar extensions of first-order logic using operators corresponding to NP-complete decision problems apparently differ in expressibility: the logics capture either NP or LNP. It had been conjectured that the complexity class captured is NP if and only if the operator is monotone. We show that this conjecture is false. However, we provide evidence supporting a revised conjecture involving finite variations of monotone problems.

We investigate the definability in monadic ∑11 and monadic Π11 of the problems REGk, of whether there is a regular subgraph of degree k in some given graph, and XREGk, of whether, for a given rooted graph, there is a regular subgraph of degree k in which the root has degree k, and their restrictions to graphs in which every vertex has degree at most k, namely REGkk and XREGkk, respectively, for k ≥ 2 . Our motivation partly stems from (...) the fact that REGkk and XREGkk are logspace equivalent to CONN and REACH, respectively, for k ≥ 3, where CONN is the problem of whether a given graph is connected and REACH is the problem of whether a given graph has a path joining two given vertices. We use monadic first - order reductions, monadic ∑11 games and a recent technique due to Fagin, Stockmeyer and Vardi to almost completely classify whether these problems are definable in monadic ∑11 and monadic Π11, and we compare the definability of these problems. (shrink)

The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books.

Scholars of the nineteenth-century race sciences have tended to identify the period from c.1820– c.1850 as a phase of transition from philologically to physically focused study. In France, the physiologist William Frédéric Edwards is normally placed near the center of this transformation. A reconsideration of Edwards’ oeuvre in the context of his larger biography shows that it is impossible to see a clear-cut philological to physical “paradigm shift.” Although he has been remembered almost solely for his principle of the permanency (...) of physical “types,” Edwards was also committed to what he recognized as the new science of “ linguistique” and proposed a new branch of comparative philology based on pronunciation. Bearing Edwards’ attention to linguistics in mind, this article reconstructs his racial theories in their intellectual contexts and suggests that at a time of emergent disciplinary specialization, Edwards tried to hold discrete fields together and mold them into a new “natural history of man.”. (shrink)