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In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to infinitesimal calculus. It provides a rigorous justification that were previously considered merely heuristic. Nonrigourous calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless. Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."
References
- Scott, J.F. 1981. "The Mathematical Work of John Wallis, D.D., F.R.S. (1616–1703)". Chelsea Publishing Co. New York, NY. p. 18.
- Katz, Mikhail; Tall, David (2011), Tension between Intuitive Infinitesimals and Formal Mathematical Analysis, Bharath Sriraman, Editor. Crossroads in the History of Mathematics and Mathematics Education. The Montana Mathematics Enthusiast Monographs in Mathematics Education 12, Information Age Publishing, Inc., Charlotte, NC, Bibcode: 2011arXiv1110.5747K http://adsabs.harvard.edu/abs/2011arXiv1110.5747K
- Kevin Houston, How to Think Like a Mathematician, ISBN:978-0-521-71978-0
- Blass, Andreas (1978), "Review: Martin Davis, Applied nonstandard analysis, and K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, and H. Jerome Keisler, Foundations of infinitesimal calculus", Bull. Amer. Math. Soc. 84 (1): 34–41, doi:10.1090/S0002-9904-1978-14401-2, http://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/home.html , p. 37.
