Abraham Robinson and nonstandard analysis: history, philosophy, and foundations 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 Pythagoreans, 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:

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