Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a continuum taken more or less as a primitive notion. Modern foundational theories starting around 1870 are based on a continuum made of points and therefore cannot serve as a basis for interpreting the thinking of the earlier mathematicians as far as the foundations are concerned.
What one can however seek to interpret are the techniques and procedures (rather than foundations) of the earlier authors, using techniques and procedures available in modern frameworks. In the case of analysis, the modern frameworks available are those developed by Weierstrass and his followers around 1870 and based on an Archimedean continuum, as well as more recently those developed by Robinson and his followers, and based on a continuum containing infinitesimals, and other frameworks such as the one developed by Lawvere, Kock, and others.
I was therefore a bit puzzled by the following comment by a historian expressed here:
Recently there have been attempts to argue that Leibniz, Euler, and even Cauchy could have been thinking in some informal version of rigorous modern non-standard analysis, in which infinite and infinitesimal quantities do exist. However, a historical interpretation such as the one sketched above that aims to understand Leibniz on his own terms, and that confers upon him both insight and consistency, has a lot to recommend it over an interpretation that has only been possible to defend in the last few decades. It is parsimonious and requires no expert defence for which modern concepts seem essential and therefore create more problems than they solve (e.g. with infinite series). The same can be said of non-standard readings of Euler; etc.
Question 1. Is this historian choosing one foundational framework over another in interpreting the techniques and procedures of the historical authors?
Question 2. What exactly is a Parsimonious History?
Question 3. Gray and Bottazzini reportedly make a rather poetic proposal in the following terms: "The best policy is to read on in a spirit of dialogue with the earlier authors." The proposal of such a conversation with, say, Euler sounds intriguing. I am only wondering about Gray's comment here that "Euler’s attempts at explaining the foundations of calculus in terms of differentials, which are and are not zero, are dreadfully weak." Isn't such an opening line in a conversation likely to be a conversation-stopper?
Question 4. In connection with the work of Laugwitz mentioned by one of the responders, one could ask why Gray does not cite explicitly the work of any of the authors that he wishes explicitly to criticize for using modern infinitesimals? Laugwitz's article (Laugwitz, Detlef. Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39 (1989), no. 3, 195–245) appeared in Archive for History of Exact Sciences, clearly a reputable journal since Jeremy Gray happens to be its Editor-in-Chief. Similarly, the article McKinzie, Mark; Tuckey, Curtis. Hidden lemmas in Euler's summation of the reciprocals of the squares. Arch. Hist. Exact Sci. 51 (1997), no. 1, 29–57 appeared in the same journal and exploited Robinson's framework to clarify some of Euler's procedures; it, too, is being stonewalled by the Editor-in-Chief.