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. See Fermat

*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 transcendental law of homogeneity (see 12e). 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
x^{2} 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. See Leibniz

*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. See Bishop

*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 - allegedly 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 theBeitraege, 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 convergence. 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 convergence. We argued in 2011 that the received approach
to Cauchy's ideas about continuity contains an outright contradiction.
See Cauchy

Connes See Connes

Delta function

Diophantus

Eudoxus hyperreals

*Fermat, Pierre* (1601-1665) See
Fermat

Goedel

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

Leibniz See Leibniz

Platonism

Russell

Solovay model

Syncategorematic interpretation

Stevin

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 the significance and coherence of
Berkeley's critique 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

More on Infinitesimals

Fermat

Leibniz

Euler

Cauchy

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