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)

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)

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...

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)

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)