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A century ago, Georg Cantor demonstrated the possibility of a series of transfinite infinite numbers. His methods, unorthodox for the time, enabled him to derive theorems that established a mathematical reality for a hierarchy of infinities. Cantor's innovation was opposed, and ignored, by the establishment; years later, the value of his work was recognized and appreciated as a landmark in mathematical thought, forming the beginning of set theory and the foundation for most of contemporary mathematics. As Cantor's sometime collaborator, David Hilbert, remarked, "e;No one will drive us from the paradise that Cantor has created."e; This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory; logical objects and logical types; and independence results and the universe of sets. She concludes with views of the constructs and reality of mathematical structure. Philosophers with only a basic grounding in mathematics, as well as mathematicians who have taken only an introductory course in philosophy, will find an abundance of intriguing topics in this text, which is appropriate for undergraduate-and graduate-level courses.
A thorough account of the philosophy of mathematics. In a cogent account the author argues against the view that mathematics is solely logic.
This is the first critically evaluative study of Gaston Bachelard's philosophy of science to be written in English. Bachelard's professional reputation was based on his philosophy of science, though that aspect of his thought has tended to be neglected by his English-speaking readers. Dr Tiles concentrates here on Bachelard's critique of scientific knowledge. Bachelard emphasised discontinuities in the history of science; in particular he stressed the ways of thinking about and investigating the world to be found in modern science. This, as the author shows, is paralleled by those debates among English-speaking philosophers about the rationality of science and the 'incommensurability' of different theories. To these problems Bachelard might be taken as offering an original solution: rather than see discontinuities as a threat to the objectivity of science, see them as products of the rational advancement of scientific knowledge. Dr Tiles sets out Bachelard's views and critically assesses them, reflecting also on the wider question of how one might assess potentially incommensurable positions in the philosophy of science as well as in science itself.