Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M.; van
Atten, M. "Two-track depictions of Leibniz's fictions." *The
Mathematical Intelligencer* **44** (2022), no. 3,
261-266
https://doi.org/10.1007/s00283-021-10140-3,
https://arxiv.org/abs/2111.00922,
https://mathscinet.ams.org/mathscinet-getitem?mr=4480193
(indexed)

A response by Archibald, Arthur, Ferraro, Gray, Jesseph, Lützen,
Panza, Rabouin, and Schubring in *The Mathematical
Intelligencer* appears at
https://doi.org/10.1007/s00283-022-10217-7,
https://mathscinet.ams.org/mathscinet-getitem?mr=4526076
(indexed),
https://scholar.google.com/scholar?cites=1281376256621624001

Our brief response in *The Mathematical Intelligencer* to
Archibald et al.'s response appears at
23a.
A detailed response in *Antiquitates Mathematicae* appears at
22a.

Our article "Two-track depictions of Leibniz's fictions" in *The
Mathematical Intelligencer*, as the title suggests, presented two
approaches to Leibniz's infinitesimals: (1) an approach that deems his
infinitesimals to be genuine mathematical entities (to be sure,
fictional in the sense of not existing in "nature"); and (2) an
approach that views occurrences of the term "infinitesimal" in Leibniz
as stenography for exhaustion arguments a la Archimedes; accordingly,
the term does not refer to a mathematical entity. We presented
arguments on both sides, drawn from the recent literature. In a
scathing response, Archibald et al. (mostly written by Rabouin)
accused us of using "abusive epithets", and of preferring our
interpretation (1) to the point of suppressing all others (both claims
are false). Ironically, Archibald et al. endorse interpretation (2)
and dismiss ours in derisive terms.

Hidé Ishiguro argued in the second, 1990 edition of her book on Leibniz that the term does not refer (and made a series of embarrassing blunders, such as confusing Nieuwentijt and l'Hopital on page 89). A number of current Leibniz scholars (though graciously not all) follow her teaching, such as Arthur, Levey, Rabouin,... We have argued that the term does refer, namely it refers to a fictional mathematical entity. Rabouin and others claim that "fictional" means something else: namely, that it does not refer.

For Leibniz, there were three realms: (1) the realm of mathematical
entities; (2) the phenomenal world (what we would refer to as physical
reality, or "nature"); and (3) the true (metaphysical) world of
monads. Items in the phenomenal world (2) are merely manifestations
of the true world (3); in this context Leibniz frequently uses the
analogy of a rainbow, which we recognize as a phenomenon, and
similarly a horse (which we don't usually). Mathematical entities can
have fairly direct analogs in the phenomenal realm, in which case they
could be described as "ideal"; and those that don't have fairly direct
analogs, which would then be "fictions". Note however that this is
already an interpretation that does not agree with all of Leibnizian
texts, which on occasion use "fictional" and "ideal" interchangeably.
Up to this point, there are no major disagreements with Rabouin. Also
agreed upon is the fact that Leibniz similarly described negatives,
irrationals, and imaginaries as "fictions". With regard to
infinitesimals, Rabouin wants to lump them together with "infinite
wholes", and declare both to be contradictory. This is a conflation
that we have protested against. Leibniz rejected infinite wholes
(such as an unbounded line) as contradictory (namely contradicting the
part-whole principle, along the lines of the Galilean paradox of n
→ n^{2}). But the contradictory counterparts of
"infinite wholes" (maxima) are minima, namely points (indivisibles),
rather than infinitesimals. This is where the entire Ishiguro-Rabouin
construction collapses as a house of cards.

Infinitesimal topics

Leibniz

Publications on the mathematics, history,
and philosophy of infinitesimals

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