It has been observed that whereas painters and musicians are likely to be embarrassed by references to the beauty in their work, mathematicians instead like to engage in discussions of the beauty of mathematics. Professional artists are more likely to stress the technical rather than the aesthetic aspects of their work. Mathematicians, instead, are fond of passing judgment on the beauty of their favored pieces of mathematics. Even a cursory observation shows that the characteristics of mathematical beauty are at variance (...) with those of artistic beauty. For example, courses in art appreciation are fairly common; it is however unthinkable to find any mathematical beauty appreciation courses taught anywhere. The purpose of the present paper is to try to uncover the sense of the term beauty as it is currently used by mathematicians. (shrink)
Offers a glimpse into the world of science and technology between 1950 and 1990 as seen through the eyes of a mathematician, and debunks various myths of scientific philosophy. Portrays some of the great scientific personalities of the period, including Stanislav Ulam, who patented the hydrogen bomb, and Jack Schwartz, one of the founders of computer science. Also discusses phenomenology of mathematics, and philosophy and computer science. Includes book reviews. For students and academics. Annotation copyright by Book News, Inc., Portland, (...) OR. (shrink)
We shall argue that the attempt carried out by certain philosophers in this century to parrot the language, the method, and the results of mathematics has harmed philosophy. Such an attempt results from a misunderstanding of both mathematics and philosophy, and has harmed both subjects.
Husserl’s Third Logical Investigation, ostensibly dealing with the phenomenology of whole and parts, is actually meant to introduce the notion of Fundierung. This term is frequently used in the phenomenological literature, although little has been written about Fundierung itself since Husserl introduced it. Husserl himself, although he used it extensively, never again felt the need to reopen the discussion.
a Mathematicians, like Proust and everyone else, are at their best when writing about their first lovea (TM) a ] They are among the very best we have; and their best is very good indeed. a ] One approaches this book with high hopes. Happily, one is not disappointed. a ]In paperback it might well have become a best seller. a ]read it. From The Mathematical Intelligencer Mathematics is shaped by the consistent concerns and styles of powerful minds a three (...) of which are represented here. Kaca (TM)s work is marked by deep commitment and breadth of inquiry. Rota is the easiest of these authors to reada ]a delight: witty and urbane, with a clear and interesting agenda and an astonishing intellectual range. To read him is to be a part of a pleasant and rewarding conversation. Jacob T. Schwartz attacks problems ina ]computer science, mathematical economics, computer and instructional a ] No matter how slippery the problem, Schwartz manages to capture a substantial chunk of it in his mathematical net. a Mathematics Magazine This is a volume of essays and reviews that delightfully explore mathematics in all its moods a from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, Heidegger among others. (shrink)
A plurality of axiomatic systems can be interpreted as referring to one and the same mathematical object. In this paper we examine the relationship between axiomatic systems and their models, the relationships among the various axiomatic systems that refer to the same model, and the role of an intelligent user of an axiomatic system. We ask whether these relationships and this role can themselves be formalized.
LIKE ARTISTS WHO FAIL TO GIVE an accurate description of how they work, like scientists who believe in unrealistic philosophies of science, mathematicians subscribe to a concept of mathematical truth that runs contrary to the truth.
ARE MATHEMATICAL IDEAS INVENTED OR DISCOVERED? This question has been repeatedly posed by philosophers through the ages, and will probably be with us forever. We shall not be concerned with the answer. What matters is that by asking the question, we acknowledge the fact that mathematics has been leading a double life.