# Are infinitesimals in the Newton and Leibniz calculus potential or actual?

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?...

• Leibniz wanted to construct calculus with actual infinitesimals but he was never able to do so and when mathematical proofs started to become more rigorous, the epsilon delta definition of limits took precedence over the idea of actual infinitesimal numbers. In the 1960s, after the advent of mathematical logic, Robinson was able to construct nonstandard models of analysis that uses actual infinitesimals, see en.wikipedia.org/wiki/Non-standard_analysis Apr 19 '17 at 2:57
• The point to take away is that all of the founders and major contributors of analysis realized the problem of defining Newton's fluxions and Leibniz's infinitesimals in a rigorous way so they decided to avoid doing so and instead developed the epsilon delta definition of a limit. Robinson's non-standard analysis has helped provide rigor to some specific arguments. Apr 19 '17 at 3:04
• @Not_Here, thank you for your comments. I am aware of Robinson's non-standard analysis via which he formulated the actual infinitesimal through usage of non-standard model of the real numbers. Also am aware of Weierstrass' limit theory which replaced Newton's and Leibniz's infinitesimal-based calculus. Still, what am hoping to know is whether the infinitesimal was problematic due to its actuality or rather both potential and actual infinitesimal caused problems. Apr 19 '17 at 3:05
• @Not_Here, if may further explain part of confusion I arrived at: I ask what I ask since when commentators talk of infinite they accentuate the distinction between potential and actual but they do not do so when they talk of infinitesimals, which made me ponder whether both potential and actual infinitesimals were perceived as problematic. Thank you again for your comments - I think your comments imply that the issue with infinitesimals could not be sort out by assuming potential infinitely small quantities. Apr 19 '17 at 3:09
• I'm assuming what you mean by "potential" is the idea of a limit and something like a hyperreal infinitesimal is "actual." To that extent I would argue, no, Leibniz and Newton and the other founders of analysis did not think potential infinitesimals were problematic, that is why they stuck with defining them in that way. Leibniz thought there was no need for them but he never proved "actual" infinitesimals, Newton eventually considered his fluxions as "approaching zero" similar to the concept of a limit, and the founders of analysis used the epsilon delta definition for their work. Apr 19 '17 at 3:16

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."

• If may ask, @Conifold: what are the main differences between Newton's and Leibniz's calculus? (is it right that Newton's approach was more geometrical whereas Leibniz' arithmetical?) Aug 9 '17 at 20:55
• @L.M.Student You could say that Newton's version was based on geometric (rather kinematic) intuitions, which McLaurin developed, and Leibniz's more on arithmetic/algebraic ones. For conceptual history of calculus look at Boyer, linked Friedman's paper has a more philosophical perspective, but narrower scope. Aug 9 '17 at 21:08
• @L.M.Student It is explained in the linked paper of Katz-Sherry, pp. 7-9, 21-23. "Logical" or "nominal" fictions (syncategoremata) are shorthands that can be fully eliminated by paraphrase into other terms, say into "method of exhaustion". "Pure" fictions, like projective points at infinity or complex numbers, are not seamlessly eliminable by paraphrase, there is tangible loss of meaning when it is attempted. They are instruments but not "mere" instruments, ideal elements that can be said to have "secondary" existence, like centers of mass or equator. Mar 23 '18 at 21:50
• @L.M.Student I am not sure they were ever merged. "Pure mathematics" in the modern sense can be traced to Pythagoreans. Plato already viewed mechanical concepts in mathematics as contamination, in accordance with that Euclid avoided any use of motion and/or congruence in the Elements. Importing mechanical concepts into mathematics was still viewed with suspicion in 17th century, which is one reason why Newton clothed Principia in Euclidean language. It was one of the main philosophical objections to Newton's calculus in 18th, as Friedman mentions. Mar 29 '18 at 20:55
• @L.M.Student Geometers resisted Platonist strictures already in antiquity, see my hsm post When were the concepts of pure and applied Mathematics introduced? But even Archimedes only presented his "mechanical" method in a letter to Eratosthenes, declared it heuristic only, and reproved all the theorems by the Eudoxian double reductio in his "official" works. Dedekind, Cantor, Frege were avowed platonists, but they were decontaminating the pure from the sensible in the form of Kantian intuition rather than of mechanical motion proscribed by Parmenides. Mar 29 '18 at 21:20

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.

• Berkeley hated Newton and other 'free thinkers' of the day for being anti church and this bias was reflected in his critique. I don't think its fair to use his criticisms of Newton's fluxions to express how they were viewed when he had an obvious bias to make them sound impractical and fictitious. Additionally, your comment about Aristotle sounds incredibly anachronistic. I'm not arguing that Aristotle didn't have an understanding of something similar to a limit (although we probably disagree to what extent), but the way you wrote it sounds like he commented on Newton's Fluxions. Apr 24 '17 at 19:11
• "the normative strategy to give them an ontological basis is through limiting operations - as noted by Aristotle this gave an 'adequate' solution." If by 'them' you mean just infinitesimals in general then sure, but from the context of the question it looks like you are talking about Newton's fluxions. Apr 24 '17 at 19:13
• Fluxions were derivatives, not infinitesimals. I believe you're wrong on your history there. Apr 24 '17 at 19:38
• @MoziburUllah Newton didn't talk about infinitesimals. You're thinking of Leibniz. But I hope you're not getting your Newtonian math from Wikipedia. Apr 24 '17 at 20:01
• @MoziburUllah Not a downvoter by the way. I never downvote unless something's really egregious. Apr 25 '17 at 1:27