Conclusion
As Keisler showed us, the infinitesimal, that good old heuristic tool, can be used in teaching calculus with a very slight departure from the original spirit of Leibniz. The main difference is in the explicit distinction between ≈ and = and the use of notions such as “standard part” which were not explicitly clarified before. At the classroom level, the main importance of Robinson’s contribution is that it reassures us, the teachers, that when we say “infinitesimal”, we can finally claim that we know what we are talking about.
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References
H. J. Keisler,Elementary Calculus, Prindle, Weber & Schmidt, Boston, 1976. (Review: M. Davis and M. Hausner, “The Joy of Infinitesimals,”The Mathematical Intelligencer 1 (1978), pp. 168-170).
H. J. Keisler,Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston, 1976.
I. Lakatos, “Cauchy and the Continuum”.The Mathematical Intelligencer 1 (1978), pp. 151–161.
A. Robinson,Non-standard Analysis, North Holland Publishing Co., Amsterdam, 1974.
K. Sullivan, “The Teaching of Elementary Calculus Using the Nonstandard Approach”,The American Mathematical Monthly 83 (1976), pp. 371–375.
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It is my pleasure to express my gratitude to Prof. A. Shenitzer who persuaded me to publish this note. His assistance went beyond the warm words of encouragement. He introduced improvements in a first draft and had it typed at York University. Needless to say, the remaining imperfections are all mine. The text expands a talk delivered to an audience of high school teachers from the Haifa area.
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Harnik, V. Infinitesimals from Leibniz to Robinson time to bring them back to school. The Mathematical Intelligencer 8, 41–47 (1986). https://doi.org/10.1007/BF03026834
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DOI: https://doi.org/10.1007/BF03026834