The ABCs of the history of infinitesimal mathematics

Some topics from the history of infinitesimals appear below in alphabetical order.

Adequality is a technique used by Fermat to solve problems of tangents and maxima and minima. Adequality derives from Diophantus' parisotes, and involves an element of approximation and "smallness", represented by a small variation E, as in the familiar difference f(A+E)-f(A). Fermat used adequality in particular to find the tangents of transcendental curves such as the cycloid, that were considered to be "mechanical" curves off-limits to geometry, by Descartes. Fermat also used it to solve the variational problem of the refraction of light so as to obtain Snell's law. Adequality incorporated a procedure of discarding higher-order terms in E (without setting them equal to zero). Such a heuristic procedure was ultimately formalized mathematically in terms of the standard part function in Robinson's theory of infinitesimals dating from 1961.

Berkeley, George (1685-1753) was an English cleric whose empiricist (i.e., based on sensations, or sensationalist) metaphysics tolerated no conceptual innovations, like infinitesimals, without an empirical counterpart or referent. Berkeley was similarly opposed, on metaphysical grounds, to infinite divisibility of the continuum (which he referred to as extension), an idea widely taken for granted today. In addition to his metaphysical criticism of the infinitesimal calculus of Newton and Leibniz, Berkeley also formulated a logical criticism, claiming to have detected a logical fallacy at the foundation of the method. In terms of Fermat's E (see entry Adequality), his objection can be formulated as follows: the increment E is assumed to be nonzero at the beginning of the calculation, but zero at its conclusion, an apparent logical fallacy. In reality, Berkeley's criticism in his book The Analyst was a misunderstanding on his part. Namely, E is not assumed to be zero at the end of the calculation, but rather is discarded at the end of the calculation. Such a technique was the content of Fermat's adequality and Leibniz's law of homogeneity. It is equivalent to taking the limit (of a typical expression such as (f(A+E)-f(A))/E) in the Weierstrassian approach, and to taking the standard part in Robinson's approach. Meanwhile, Berkeley's own attempt to explain the calculation of the derivative of x2 in The Analyst contains a logical circularity. Namely, Berkeley's argument relies on the determination of the tangents of a parabola by Apollonius (which is eqivalent to the calculation of the derivative). This circularity in Berkeley's argument is analyzed in the 2011 article by Kirsti Andersen in Historia Mathematica. Far from exposing logical flaws in Leibnizian calculus, Berkeley's The Analyst is itself logically flawed.

Bernoulli, Johann (1667-1748) was a disciple of Leibniz's who, having learned an infinitesimal methodology for the calculus from the master, never wavered from it. This is in contrast to Leibniz himself, who, throughout his career, used both (A) an Archimedean methodology (proof by exhaustion), and (B) an infinitesimal methodology, in a symbiotic fashion. Thus, Leibniz relied on the A-methodology to underwrite and justify the B-methodology, and he exploited the B-methodology to shorten the path to discovery (Ars Inveniendi). Historians often name Bernoulli as the first mathematician to have adhered systematically to the infinitesimal approach as the foundation for the calculus. We will therefore refer to an infinitesimal-enriched number system as a B-continuum, as opposed to an Archimedean A-continuum.

Bishop, Errett (1928-1983) was a mathematical constructivist who, unlike the intuitionist Arend Heyting (see entry below), held a dim view of classical mathematics in general and Robinson's infinitesimals in particular. Discouraged by the apparent non-constructivity of his early work in functional analysis under Halmos, he believed to have found the culprit in the law of excluded middle (LEM). He spent the remaining 18 years of his life in an effort to expunge the reliance on LEM from analysis, and sought to define meaning itself in mathematics in terms of such LEM-extirpation. Accordingly, he described classical mathematics as both a debasement of meaning and sawdust, and did not hesitate to speak of both crisis and schizophrenia in contemporary mathematics, predicting an imminent demise of classical mathematics. His criticism of calculus pedagogy based on Robinson's infinitesimals was a natural outgrowth of his general opposition to the logical underpinnings of classical mathematics.

Cantor, Georg (1845-1918) is intimately familiar to the modern reader as the underappreciated creator of the "Cantorian paradise" which David Hilbert would not be expelled out of, as well as the tragic hero (persecuted by Kronecker) who ended his days in a lunatic asylum. Cantor historian J. Dauben notes, however, an underappreciated aspect of Cantor's scientific activity, namely his principled persecution of anything and anyone related to infinitesimals:

Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beitraege, to attacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics', which had spread from germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz and Veronese was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible (Dauben 1980, pp. 216-217).

Cauchy, Augustin-Louis: Augustin-Louis Cauchy (1789-1857) is often viewed in the historical literature as a precursor of Weierstrass. Note, however, that contrary to a common misconception Cauchy never gave an epsilon, delta definition of either limit or continuity. In a series of recent articles Borovik & Katz; Katz & Katz; Katz & Tall), we have argued that a proto-Weierstrassian view of Cauchy is one-sided and obscures Cauchy's important contributions, such as his infinitesimally defined (``Dirac'') delta function with applications in Fourier analysis and evaluation of singular integrals, his infinitesimal definition of continuity, and his study of orders of growth of infinitesimals that anticipated the work of du Bois-Reymond. A recent article in Perspectives on Science (Katz & Katz 2011) examined Cauchy's 1853 paper on a notion closely related to uniform continuity. Cauchy handles the said notion using infinitesimals, including one generated by the null sequence (1/n). The received approach to this work of Cauchy's cannot account for its treatment of Cauchy's notion related to uniform continuity. We argued in 2011 that the received approach to Cauchy's ideas about continuity contains an outright contradiction.

Delta function
Eudoxus hyperreals

Fermat, Pierre (1601-1665)

Great triumvirate

Heyting, Arend (1898-1980) was a mathematical Intuitionist whose lasting contribution was the formalisation of the Intuitionstic logic underpinning the semi-mystical Intuitionism of his teacher Brouwer. While Heyting never worked on any theory of infinitesimals, he had several opportunities to present an expert opinion on Robinson's theory. Thus, in 1961, Robinson made public his new idea of non-standard models for analysis, and "communicated this almost immediately to ... Heyting" (see Dauben 2003, 259). Robinson's first paper on the subject was subsequently published in Proceedings of the Netherlands Royal Academy of Sciences (Robinson 1961). Heyting praised non-standard analysis as "a standard model of important mathematical research" (Heyting 1973, 136). Addressing Robinson, he declared:

you connected this extremely abstract part of model theory with a theory apparently so far apart as the elementary calculus. In doing so you threw new light on the history of the calculus by giving a clear sense to Leibniz's notion of infinitesimals (ibid).
Intuitionist Heyting's admiration for the application of Robinson's infinitesimals to calculus pedagogy is in stark contrast with the views of constructivist E. Bishop (see entry above).

Incompleteness theorem
Law of continuity
Solovay model
Syncategorematic interpretation
Stevin number
Transcendental law of homogeneity
Triumvirate history

The general outline of what I am interested in is to understand to what extent the famous dictum that "history is always written by the victors" applies to the history of mathematics, as well. A convenient starting point for this is a remark made by Felix Klein in his book Elementary mathematics from an advanced standpoint, to the effect that there are not one but two separate tracks for the development of analysis: (A) the Weierstrassian approach (in the context of an Archimedean continuum); and (B) the approach with indivisibles and/or infinitesimals (in the context of a Bernoullian continuum). In the historical literature, there is a common assumption, sometimes explicit and sometimes implicit, that the A-approach is the "true" one, and the infinitesimal approach was a kind of neanderthal dead-end. Such an assumption can influence our appreciation of historical mathematics, and make us blind to certain significant developments due their automatic placement in a "neanderthal" track. One example is the visionary work of Enriques exploiting infinitesimals, recently analyzed in an article by David Mumford. Another example is important work by Cauchy on singular integrals and Fourier series using infinitesimals and infinitesimally defined Dirac delta functions (these precede Dirac by a century), which was forgotten for several decades because of shifting foundational biases. We have recently published two papers on Leibniz arguing that his system for infinitesimal calculus was consistent, contrary to widespead perceptions. Here Berkeley's significance and coherence have been exaggerated because Berkeley fits well with the A-approach favored by many historians. Our text on Fermat shows how the nature of his contribution to the calculus has been systematically distorted, also due to the "neanderthal" bias. There is a lot of work to be done here to bring this all together. An expanded version of these remarks may be found here

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