Was reading a bit about history of calculus and its philosophy and stumbled into source of confusion: were infinitesimals in seventeenth century calculus assumed to be actual or potential? Was there any attempt to circumvent problems with infinitesimals the way it was done with the infinite (I mean akin to the distinction between actual and potential infinite)? To begin with, could the problems in Newton and Leibniz calculus be sorted out by assuming potential infinitesimals?...
Actual infinities collected into sets were not officially contemplated by (philosophizing) mathematicians until Cantor (with some anticipation by Bolzano) countered Aristotelian and scholastic arguments that such objects are paradoxical, see How does actual infinity (of numbers or space) work? Ironically, Cantor rejected the infinitesimals themselves, see What was Cantor's philosophical reason for accepting the infinite but rejecting the infinitesimal?
However, already in mid 16th century Stevin advocated admitting infinite decimals as numbers, see When did it become understood that irrational numbers have non-repeating decimal representations?, and by mid 17th century Euclidean strictures about distinguishing magnitudes, numbers and ratios were largely abandoned. Even earlier Cardano started manipulating "impossible" numbers, and others followed. The adventurous attitude of "results are more important than formalities" extended well into 18th century, when Euler did his famous manipulations with divergent series, and Dalambert told his students "keep working, faith will come later". To put it bluntly, most mathematicians did not care for Aristotelian distinction between actual and potential infinities.
Newton and Leibniz were more circumspect, however. Leibniz did contemplate actual infinity of existents, see Leibniz’s Actual Infinite by Arthur, but he was sufficiently impressed by Aristotelian arguments not to collect them into totalities like Cantor. Recent scholarship shows that Leibniz (as Fermat before him) treated infinitesimals as "useful fictions" (his wording), like Cardano's imaginary numbers, but not as purely nominal fictions of today, see Leibniz's Infinitesimals by Katz and Sherry:
"Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof."
Nonetheless, contemporary perceptions were different. Those more concerned with philosophical status of infinitesimals saw Berkeley's criticisms of them as pertinent, and Newton's conception as more coherent (including Kant, his sympathies to Leibniz notwithstanding). One could say that Newton's "kinematic interpretation" of infinitesimals (which he partly owes to Archimedes through Toricelli and Barrow, see Who discovered the power rule for derivatives?) was close to Aristotelian understanding of motion with potential infinities and the classical resolution of Zeno's paradoxes attached. Maclaurin attempted a rigorous exposition of calculus on the basis of the kinematic interpretation in 1742. One could even say that Kant's theory of synthetic construction in mathematics relies on Aristotelian restriction to potential infinities that arguably permeates Euclid's Elements (indeed, Kant explicitly modeled his concept/intuition duality on Aristotle's matter/form). Friedman comments in Kant's Theory of Geometry:
"Moreover, although the kinematic interpretation of the calculus certainly does not meet modern standards of rigor, it is also not afflicted with the obvious problems about consistency and coherence facing an interpretation based on differentials, infinitesimals, and infinitely small quantities. Indeed, when the kinematic interpretation was explicitly criticized by mathematicians like D'Alembert and l'Huilier in the late eighteenth century, it was not on grounds of coherence and consistency but because it was thought to import a "foreign" or "physical" element into pure mathematics. Pure mathematics should be independent of and prior to mathematical physics; therefore, it should be developed in complete independence of the idea of motion. For Kant, on the other hand, this "mixing" of physical and mathematical ideas is not a defect but a virtue."
[...] But why exactly does the kinematic interpretation fail to meet modern standards of rigor? This is a difficult and fascinating question. For now, however, I shall simply hazard the suggestion that the difference between the iterative infinity involved in Euclidean constructions and the stronger use of infinity involved in limit operations plays a central role here. In Euclidean geometry we specify the objects of our investigation — circles, straight lines, and any figures constructible from them-by a well — defined iterative or "inductive" procedure... By contrast, in the fluxional calculus we have no such specification: no step-by-step procedure (nor any other precisely defined method) for constructing all fluents or "fliessende Grissen" has been given. Similarly, our temporal representation of the limit operation does not proceed by repeated application of previously given functions: each new limit has to be constructed "on the spot," as it were. This, in the end, is perhaps the most fundamental advantage of the Cauchy-Bolzano-Weierstrass definition of convergence."
In Newtons calculus, infinitesimals were called fluxions; in a critique, Berkeley called them the 'ghosts of departed things', which suggests potentiality; the normative strategy to give them an ontological basis is through limiting operations - as noted by Aristotle this gave an 'adequate' solution.
A different ontology, which come through Category Theory is to axiomatise the notion of an infinitesimal; algebraically, this is the system of dual numbers; geometrically, synthetic differential geometry; interestingly enough, this does not have any models in the basic category called Set.