A new look at E.G. Björling and the Cauchy sum theorem

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

“…Such a turn of phrase occurs both in his letter to Coriolis of 1837, and in his 1853 text [36, p. 35]. 44 Cauchy transforms perceptual continuity into a mathematical notion by exploiting his conception of an infinitesimal as being generated by a null sequence (see [26]). Both in 1821 and in 1853, Cauchy defines continuity of y = f (x) in terms of an infinitesimal x-increment resulting in an infinitesimal change in y.…”

“…60 Namely, an infinitesimal being generated by a null sequence, we evaluate f at it by applying f to each term in the sequence. Bråting [26] analyzes Cauchy's use of the particular sequence x = 1 n in [36].…”

“…A majority of historians have followed suit, though some truly original thinkers differed. These include C. S. Peirce, Felix Klein, N. N. Luzin [107], Hans Freudenthal, Robinson, Lakatos [108], Laugwitz, Teixeira [136], and Bråting [26].…”

“…26 The law of excluded middle (see Section 2) is the mathematical counterpart of geocentric cosmology (alternatively, of phlogiston, see [125, p. 299]), slated for the dustbin of history. 27 The anti-realist narrative dismisses the Quine-Putnam indispensability thesis (see Feferman [57, Section IIB]) on the grounds that a philosophyfirst examination of first principles is the unique authority empowered to determine the correct way of doing mathematics.…”

“…The most detailed statement of Boyer's position may be found in Grabiner [69]. Robinson's perspective was developed most successfully by D. Laugwitz [102] in 1989, and by K. Bråting [26] in 2007.…”