THE AMBIGUITY OF MATHEMATICS

May 19th, 2015
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Abstract: This paper deals with mathematical rigor and the notion of ambiguity in mathematics. It takes the counter-intuitive position that ambiguity is of central importance to the mathematical endeavor—that it is essential and cannot be avoided. In our view, rigor and ambiguity form two complementary dimensions of mathematics—what we characterize as the surface versus the depth dimensions of the subject.

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KARL JASPERS FORUMTARGET ARTICLE 59THE AMBIGUITY OF MATHEMATICSby Bill Byers[1]Abstract: This paper deals with mathematical rigor and the notion of ambiguity in mathematics. It takes the counter-intuitive position that ambiguity is of central importance to the mathematical endeavorthat it is essential and cannot be avoided. In our view, rigor and ambiguity form two complementary dimensions of mathematicswhat we characterize as the surface versus the depth dimensions of the subject.[2]IntroductionIn this paper I shall attempt to get the reader to think about mathematics in a different and slightly unusual wayi, one that is closer to the way mathematics is seen by mathematicians when they are doing mathematics. I hope that this new point of view will suggest a new approach to some of the key questions that any philosophy of mathematics must inevitably deal with. Is mathematics discovered or invented? Is it objectively true or is it constructed by human beings? Why has mathematics had such a profound influence on human thought both in the humanities and in the sciences? Finally, the question of whether one can imagine a computer doing mathematics.[3] Now mathematics is usual approached in one of two ways. The first way is instrumentally as a body of useful results, techniques, formulas, etc. that are assumed to be valid and can be applied to solve various kinds of problems. This is how engineers use mathematics, for example, or psychologists

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