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In §4 of the Stanford Encyclopedia of Philosophy article on continuity and infinitesimals, the author (John L. Bell) mentions that:

... Johann Bernoulli (1667–1748) [in a] letter of his to Leibniz written in 1698 contains the forthright assertion that “inasmuch as the number of terms in nature is infinite, the infinitesimal exists ipso facto” (Boyer 1939 [1959: 239], quoting Leibniz, Mathematische Schriften, III (Part 2), 555). One of his arguments for the existence of actual infinitesimals begins with the positing of the infinite sequence 1/2, 1/3, 1/4,…. If there are ten terms, one tenth exists; if a hundred, then a hundredth exists, etc.; and so if, as postulated, the number of terms is infinite, then the infinitesimal exists.

Shortly below that he notes:

... Nieuwentijdt’s infinitesimals have the property that the product of any pair of them vanishes; in particular each infinitesimal is nilsquare, i.e., its square and all higher powers are zero (see Mancosu 1996: 159). ... It is of interest to note that Leibnizian infinitesimals (differentials) are realized in nonstandard analysis, and nilsquare infinitesimals in smooth infinitesimal analysis (for both types of analysis see below). In fact it has been shown to be possible to combine the two approaches, so creating an analytic framework realizing both Leibniz’s and Nieuwentijdt’s conceptions of infinitesimal.

Following that is a short discussion of Euler's quasi-theory that (hypothetical) evaluations of $0/0$ are the background sense we attach to infinitesimal semantics (so to speak), or more precisely, "Euler treated infinitesimals as formal zeros..."

I get the impression that Nieuwentijdt was inspired, to some extent (maybe subconsciously), by the (at the time) sort-of-recent emergence of imaginary numbers as (socially) acceptable mathematical objects. At least, it would not be hard to see how the algebraism involved in the bare acceptance of the imaginary unit can be reflected in the assertion of nilpotent infinitesimals (they, like i, are arbitrary nonreal-number solutions to a specific equation template, and both involve, initially anyway, squaring). However, did any mathematician try out a definition like:

$$\forall n\exists\epsilon(\frac{\epsilon}{n} = \epsilon, \epsilon \neq 0, n \neq 0)$$

... albeit without, of course, the modern quantifiers in the historical notation?

I ask this in part because it was discerned much later (this is from over on the MathSE) that $\frac{\aleph_1}{\aleph_0} = \aleph_1$, and generally when a transfinite cardinal numerator $\lambda$ exceeds the denominator $\kappa$, the lower cardinal is "absorbed" (like with algebraic absorbing elements, even) into the upper one. Moreover, $\frac{\lambda}{\lambda}$ is indeterminate "just like" $0/0$ is (with the exception of $0 \times \lambda = 0$). So transfinite numbers would have this idquotient(?) feature $\frac{\lambda}{n} = \lambda, \lambda \neq 0$, and their conceptual inverses (not their operational inverses, at least not "at first") would then be the kind of infinitesimals triangulated between Bernoulli, Nieuwentijdt, and Euler's definitions/characterizations.

It seems like a natural enough idea to have tried out, and on my end of things, I look at nilsquare infinitesimals as implying that adding $\epsilon + \epsilon$ together $\epsilon$-many times zeroes the infinitesimal out, which would make it both positive and negative somehow (like zero, granted), especially in the act of its own addition, or then adding it to itself is equivalent to self-subtraction in some manner, and that hardly seems less difficult to imagine and make sense of than idquotient(?) infinitesimals would be. I don't know that a system of such things would have worked out, of course, even had it been tried, but again, in smooth infinitesimal analysis (SIA), they introduce nilpotent infinitesimals as suspensions of not only general commensurability but outright transgressions of the Law of the Excluded Middle—so if there had been any mathematician, in those days, who tried out the idquotient algebraism instead, maybe I should expect that an attempt would have been made nowadays to recapitulate such a definition.

Or, again, was it tried out and never made workable in the way reciprocal infinitesimals have been provided for via hyperreals and surreals, or nilpotent ones modulo SIA?

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    $\begingroup$ I am struggling to see motivation for this outside of cardinal arithmetic, which even in 19th century Cantor was selling as unrelated to infinitesimals. $\frac{\epsilon}{n} = \epsilon$ means $(n-1)\epsilon=0$, which suggests modular arithmetic considered since Bachet de Méziriac, rather than infinitesimals. Nilpotent (of any degree) 'infinitesimals' are implicit in the "lost calculus" of Hudde, but he presented it as an algebraic trick for simplifying formulas modulo polynomial relations. $\endgroup$
    – Conifold
    Aug 8, 2023 at 6:02
  • $\begingroup$ @Conifold what I tried out at first was taking $1/{\infty}$ but replacing the lemniscate with $\aleph_0$ (I read somewhere that Fraenkel was unhappy with such notation, but beyond that I didn't see any a priori reason to reject the possibility). Then I thought $\frac{\frac{1}{\aleph_0}}{n} = \frac{1}{\aleph_0} \times 1/n = 1/(\aleph_0 \times n) = 1/{\aleph_0}$, which satisfied the $\epsilon$-definition I wanted to test. Unfortunately, I ended up with something incoherent a little ways down the road, so IDK... "Intuitively," $\epsilon$-infinitesimals here would be indivisible, which fit one $\endgroup$ Aug 9, 2023 at 15:58
  • $\begingroup$ of the themes Bell kept bringing up in the SEP article (the thing about infinitely small straight lines). $\endgroup$ Aug 9, 2023 at 15:59
  • $\begingroup$ @KristianBerry, I don't follow your remark about "indivisible" infinitesimals. In Nieuwentijt's theory, in Leibniz's theory, and in all modern theories, infinitesimals are divisible: half an infinitesimal is again an infinitesimal. As far as Cantorian infinities are concerned, there does exist a way of turning them into a number system with infinitesimals, called the surreals, but it's not at all straightforward. There is a number of posts at MSE and MO about this. The main shortcoming of the surreals is that they don't possess a transfer principle, which is what makes... $\endgroup$ Aug 9, 2023 at 16:04
  • $\begingroup$ ... Robinson's infinitesimals a powerful tool in analysis and other fields, including measure theory, statistics, physics, you name it. $\endgroup$ Aug 9, 2023 at 16:05

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These are assorted comments too long for a comment.

  1. Boyer is unreliable and should not be used as a source.

  2. Bernoulli's "proof" of the existence of infinitesimals was rejected by Leibniz, who always (in his mature period starting around 1676) viewed infinitesimals as fictional.

  3. It may not be appropriate to describe Euler's contribution as "quasi-theory [of] (hypothetical) evaluations". Euler's approach was more coherent than this description tends to imply; see this publication in Journal for General Philosophy of Science.

  4. Cardinal arithmetic does not have good algebraic properties and therefore is not suitable for doing infinitesimal analysis, for the very reason that you point out.

  5. Cardinal arithmetic was not developed until infinitesimal analysis was already a full-fledged field. In particular, it was not available as a tool to the classical authors such as Leibniz and Cauchy. One of the reasons is that they rejected "infinite wholes", i.e., actual infinity.

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