Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics

Mathematics is the subject in which we don't know* what we are talking about. —Bertrand Russell * Don't care would be more to the point. —Martin Davis I never understood why logic should be reliable everywhere else, but not in mathematics. —A. Heyting

1.                 Infinitesimals and the History of Mathematics Historically, the dual concepts of infinitesimals and infinities have always been at the center of crises and foundations in mathematics, from the first "foundational crisis" that some, at least, have associated with discovery of irrational numbers (more properly speaking, incommensurable magnitudes) by the pre-socartic Pythagoreans1, to the debates that are currently waged between intuitionists and formalist—between the descendants of Kronecker and Brouwer on the one hand, and of Cantor and Hilbert on the other. Recently, a new "crisis" has been identified by the constructivist Erret Bishop: There is a crisis in contemporary mathematics, and anybody who has

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