One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets.

One of the greatest revolutions in mathematics occurred when Georg Cantor promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula.Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's (...) own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory. (shrink)

The term "fuzzy logic," as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic---the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediate truth values between true and false are allowed, which are interpreted as degrees of truth. This opens a (...) new way of thinking---thinking in terms of degrees rather than absolutes. For example, it leads to the definition of a new kind of sets, referred to as fuzzy sets, in which membership is a matter of degree. The book examines the genesis and development of fuzzy logic. It surveys the prehistory of fuzzy logic and inspects circumstances that eventually lead to the emergence of fuzzy logic. The book explores in detail the development of propositional, predicate, and other calculi that admit degrees of truth, which are known as fuzzy logic in the narrow sense. Fuzzy logic in the broad sense, whose primary aim is to utilize degrees of truth for emulating common-sense human reasoning in natural language, is scrutinized as well. The book also examines principles for developing mathematics based on fuzzy logic and provides overviews of areas in which this has been done most effectively. It also presents a detailed survey of established and prospective applications of fuzzy logic in various areas of human affairs, and provides an assessment of the significance of fuzzy logic as a new paradigm. (shrink)

I first met Ivor Grattan-Guinness and his wife Enid in the late summer of 1970. I was in England following an intensive course in German at the Goethe Institute in Prien am Chiemsee, and had arrang...

In December and January of 1983–1984, archaeologists excavating the tomb of an ancient Chinese provincial bureaucrat at a Western Han Dynasty site near Zhangjiashan, in Jiangling county, Hubei Province, discovered a number of books on bamboo strips, including inter alia works on legal statutes, military practice, and medicine. Among these was a previously unknown mathematical work on some 200 bamboo strips, the Suan shu shu, or Book of Numbers and Computations. Based upon other works found in the tomb, especially a (...) copy of the Er nian lü ling (Laws and Decrees of the Second Year (of the reign of empress Lü, i.e. Lü Hou)), archaeologists have dated the tomb to ca. 186 BCE (Lü Hou’s regency lasted from 188 to 180 BCE). The Suan shu shu, as the earliest yet discovered work devoted specifically to mathematics from ancient China, has stirred considerable interest among Chinese historians of science. The translation and commentary offered here draw extensively on the works cited in Sect. 3 below. Several appendixes devoted to specific issues related to translating the Suan shu shu, including its title and the problem of determining English equivalents for various commodities that arise in the text, may be found in Appendix II. (shrink)

Shaughan La Vine, Understanding the Infinite.Cambridge, Massachussets :Harvard University Press, 1994, ix + 372 pp.£31.95/$47.95 B.Russell, Foundations of logic 1903‐05, The Collected Papers of Bertrand Russell, Volume 4, Edited by Urquhart, A.with the assistance of Lewis, A.C.London and New York:Routledge, 1994, Hi+ 743 pp.£100 Ray Monk and Anthony Palmer, Bertrand Russell and the Origins of Analytical Philosophy.Introduction by Ray Monk and Anthony Palmer.Bristol, U.K.:Thoemmes Press, 1996. xvi + 383 pp.£48.00/$78.00 ; £16.95/$29.95 T.J.Holopainen, Dialectic & Theology in the Eleventh Century.Leiden:E.J.Brill, 1996. (...) vii+171pp.$78. (shrink)