Two-Track Depictions of Leibniz’s Fictions

Abstract

Some prominent twentieth-century scholars were still opposed to, or uncomfortable with, both irrationals and infinitesimals. Thus, Errett Bishop opposed both the classical development of the real numbers and the use of infinitesimals in teaching calculus [5]. For a discussion, see [9][10][11]31]. 2 Galileo (1564-1642) observed that the natural numbers admit a one-to-one correspondence with their squares. 3 The part-whole principle, which goes back to Euclid, asserts that a (proper) part is smaller than the whole. 4 ''fraction infiniment petite, ou dont le denominateur soit un nombre infini' ' [19, p. 93].

Three case studies in current Leibniz scholarship

“…Such difficulties do not arise for interpretations of Leibnizian infinitesimals that accord them the status of mathematical entities, that have recently appeared in the scholarly literature; see e.g., [8], [23, pp. 620, 641], [30], [31].…”

“…Leibniz's references to infinity and infinitesimals have been a source of continued debate in current Leibniz scholarship; see e.g., [31] for a comparison of rival viewpoints. Ishiguro developed a so-called syncategorematic interpretation of Leibnizian infinities and infinitesimals in her [28,Chapter 5].…”

Leibniz on bodies and infinities: rerum natura and mathematical fictions

“…. yet it is plain from what I have said that, at least in our minds, 31 the unassignables [inassignabiles in the original Latin] dx and dy may be substituted for them by a method of supposition even in the case when they are evanescent . .…”

“…33 See [9], Section 4 for a formalisation of the dichotomy in modern mathematics. 31 For Leibniz on minds, see Section 5.1 and the main text at note 38. 32 "Although we may be content with the assignable quantities (d)y, (d)v, (d)z, and (d)x, since in this way we can perceive the whole fruit of our calculus, namely a construction using assignable quantities, still it is clear from this that we may, at least by feigning, substitute for them the unassignables dx, dy by way of fiction even in the case where they vanish, since dy : dx can always be reduced to (d)y : (d)x, a ratio between assignable or undoubtedly real quantities" (Leibniz as translated by RA in [71, p. 439]).…”

“…Arthur [4, p. 557] and Rabouin-Arthur [71, p. 405] quote this passage but fail to account for the fact that Leibniz never refers to such entities as well-founded fictions. For more details see [31]. 46 In the original Latin: "Utrasque enim per modum loquendi compendiosum pro mentis fictionibus habeo, ad calculum aptis, quales etiam sunt radices imaginariae in Algebra" [53, p. 32].…”