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I am reading Everything and more: A brief history of infinity by David Foster Wallace and came across this quote:

Broadly stated, Cauchy’s project involves trying to rescue calculus from its metaphysical difficulties by defining infinitesimals rigorously in terms of limits; but much of Cauchy’s analysis is still beholden to geometry in ways that end up causing problems.

The context is what D. F. Wallace calls the arithmeticization of Analysis, which is basically divorcing proofs of analysis from any reference to geometry for the sake of mathematical rigor. The prominent figures of arithmeticization would be Fr. Bernard Bolzano and Karl Weierstrass.

Where can I read more about Cauchy's efforts to avoid such "metaphysical difficulties"? Was it something that he set out to do on purpose? Did he ever articulate the difficulties he was addressing?

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    $\begingroup$ Jourdain, P. E. (1913). The origin of Cauchy's conceptions of a definite integral and of the continuity of a function. Isis, 1(4), 661-703 and Smithies, F. (1986). Cauchy's conception of rigour in analysis. Archive for history of exact sciences, 41-61. I am not aware that Cauchy used the notion of metaphysics in this context. For general discussion of Cauchy vis-a-vis metaphysics see: Belhoste, B. (1991). Augustin-Louis Cauchy: A Biography. Springer. $\endgroup$ Commented 15 hours ago
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    $\begingroup$ There are plenty of answers on this site that had already talked about this issue. It is clear that Cauchy did not originally recognise that he had arrived at the answer to the thousand-year-old quarrel. He was using both the $\varepsilon$-$\delta$ arguments next to limits (as in limit definition of calculus) for years and in many papers, before it is finally realised that the $\varepsilon$-$\delta$ method can replace all cases and solve the metaphysical argument once and for all. $\endgroup$ Commented 11 hours ago

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Wallace is merely using "metaphysical difficulties" as shorthand for "justifying the use of infinitesimals". Authors ranging from Leibniz to Carnot have written texts entitled "the metaphysics of the calculus" or something of that order, and one of the issues had traditionally been the justification of infinitesimals.

Contrary to Wallace's claim, Cauchy did not "define infinitesimals in terms of limits"; such a claim would be incomprehensible even today. What Cauchy did do is define both infinitesimals and limits in terms of a notion of a variable quantity, taken as primitive. If such a quantity gets arbitrarily small, it "becomes" an infinitesimal; the limit of such a quantity is $0$ (by definition).

Wallace's claim that Cauchy's analysis is "beholden to geometry" is similarly dubious. There is not a single picture in Cauchy's foundational texts on analysis. Cauchy put much effort into avoiding purely geometrical arguments. His proof of the intermediate value theorem, given in an appendix, is a convincing example.

On the other hand, Cauchy's book on differential geometry does use some geometric arguments, including some which exploit infinitesimals. You can read more about this in

Katz, M. "Episodes from the history of infinitesimals." British Journal for the History of Mathematics 40 (2025), no. 2, 123-135. https://doi.org/10.1080/26375451.2025.2474811, https://arxiv.org/abs/2503.04313

The best 20th century Cauchy historian is Detlef Laugwitz. You can consult some of his work linked at https://u.cs.biu.ac.il/~katzmik/laugwitz.html.

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