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 → n2). 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.
For further analysis, see this answer: https://math.stackexchange.com/a/4725050/72694.
Arthur and Rabouin published a follow-up "unviability" paper; see https://mathscinet.ams.org/mathscinet/article?mr=4721554. Stay tuned.
Infinitesimal topics
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