Abstract
Once the continuum had been provided with a set-theoretic foundation, the use of the infinitesimal in mathematical analysis was largely abandoned. And so the situation remained for a number of years. The first signs of a revival of the infinitesimal approach to analysis surfaced in 1958 with a paper by A. H. Laugwitz and C. Schmieden. But the major breakthrough came in 1960 when it occurred to the mathematical logician Abraham Robinson (1918–1974) that “the concepts and methods of contemporary Mathematical Logic are capable of providing a suitable framework for the development of the Differential and Integral Calculus by means of infinitely small and infinitely large numbers.” This insight led to the creation of nonstandard analysis (NSA), which Robinson regarded as realizing Leibniz’s conception of infinitesimals and infinities as ideal numbers possessing the same properties as ordinary real numbers. In the introduction to his book on the subject he writes:
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Notes
- 1.
However, nonarchimedean ordered structures, containing infinite and infinitesimal elements, continued to be studied, chiefly by algebraists, notably Hölder, Hahn, Baer, and Birkhoff. See Fuchs (1963) and Ehrlich (2006).
- 2.
Laugwitz, A.H. and Schmieden, C., Eine Erweiterung der Infinitesimalrechnung, Mathematische Zeitschrift 69, pp. 1–39.
- 3.
Robinson [1996], p. xiii.
- 4.
So-called, Robinson says, because his theory “involves and was, in part, inspired by the so-called Non-standard models of Arithmetic whose existence was first pointed out by T. Skolem.” (Ibid.)
- 5.
Robinson, op. cit., p. 2.
- 6.
- 7.
It follows that ℝ★ is a nonarchimedean ordered field. One might question whether this is compatible with the facts that ℝ★ and ℝ share the same first-order properties, but the latter is archimedean. These data are consistent because the archimedean property is not first-order. However, while ℝ★ is nonarchimedean, it is∗-archimedean in the sense that, for any a ∈ ℝ★ there is n ∈ ℝ★ for which a < n.
- 8.
Robinson (1996), Ch. 3. A number of “nonstandard” proofs of classical theorems may also be found there.
- 9.
Bell and Machover (1977), p. 560.
- 10.
Bibliography
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Bell, J.L. (2019). Nonstandard Analysis. In: The Continuous, the Discrete and the Infinitesimal in Philosophy and Mathematics. The Western Ontario Series in Philosophy of Science, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-030-18707-1_8
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