George Berkeley

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, as emphasized by Fermat historian Stromholm. Such a technique was the foundation of Fermat's adequality and Leibniz's transcendental law of homogeneity (see 12e). It is closely related 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 y=x² in his 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 the Leibnizian calculus, Berkeley's The Analyst is itself logically flawed. Berkeley's trolling has been analyzed in Moriarty, Clare. Duelling catechisms: Berkeley trolls Walton on fluxions and faith. Intellectual History Review (2021). https://doi.org/10.1080/17496977.2021.1963933 See Leibniz