' 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. Friedman comments in Kant's Theory of Geometry: 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). Maclaurin attempted a rigorous exposition of calculus on the basis of the kinematic interpretation in 1742.

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. 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).

Nonetheless, contemporary perceptions were different. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof.' ' 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. 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 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. Newton and Leibniz were more circumspect, however. To put it bluntly, most mathematicians did not care for Aristotelian distinction between actual and potential infinities. 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'. Even earlier Cardano started manipulating 'impossible' numbers, and others followed. 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. The use of infinitesimals can be found in the foundations of calculus independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s.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? The history of non-standard calculus began with the use of infinitely small quantities, called infinitesimals in calculus.

Fermat infinitesimals