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)

Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics illustrates the much greater variety of the actual developments in the foundations during the period covered. The viewpoints that serve this purpose included the foundational ideas of working mathematicians, such as Kronecker, Dedekind, Borel and the early Hilbert, and the development (...) of notions like model and modelling, arbitrary function, completeness, and non-Archimedean structures. The philosophers discussed include not only the household names in logic, but also Husserl, Wittgenstein and Ramsey. Needless to say, such logically-oriented thinkers as Frege, Russell and Gödel are not entirely neglected, either. Audience: Everybody interested in the philosophy and/or history of mathematics will find this book interesting, giving frequently novel insights. (shrink)

David Cahan has assembled an outstanding group of European and North American historians of science and philosophy for this intellectual biography of Helmholtz, ...

The contributions of the great physicist Ludwig Boltzmann to philosophy and biology are not known sufficiently. In philosophy, he was a realist, and much opposed to his colleague's, Mach's, positivism, but also to Berkeley's, Kant's, Hegel's and Schopenhauer's idealisms. In biology, Boltzmann was a passionate Darwinist and tried to explain on the basis of evolution the meaning of photosynthesis as well as the origin of life and of the mind. Boltzmann argued for evolutionary epistemology. Opposing Kant, he derived the fundamental (...) ideas of space, time and causality from the experience of mankind and its ancestors. The laws of thought were acquired in evolution. While they must be broadly true they need not be faultless. Antinomies arise when thought overshoots the mark, i. e. exceeds the limits of the area for which it evolved. As for Boltzmann the experience of the organism in the course of evolution is the only and exclusive source of knowledge, he may be called an absolute Darwinist. A short account of Boltzmann's Darwinian ideas on the origin of morality, on the sense of beauty and on happiness is also given. (shrink)