Abstract
This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in turn, to three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status.
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We will refer to Leibniz (1923) following the standard abbreviation: A, followed by series (in Roman numerals), volume (in Arabic numerals) and page number. Ex.: A VII 6, 600.
For the moment, we consider both concepts as equivalent, but later it will be necessary to distinguish them.
This phenomenon of vague or “global” understanding occurs mainly in verbal language, although it does not necessarily have to accompany every use of symbolic notions. See Esquisabel (2012a, pp. 10–18), where the distinction between two kinds of symbolic thought is proposed.
Leibniz is not always terminologically consistent in relation to the distinction between notion and idea. Here “idea” must be understood in the sense of “notion” or “concept.”
Its full title is Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus, GM 5 220–233, originally published in Acta Eruditorum, 1684. French translation: Leibniz (1995, pp. 104–117).
Regarding this question, it is worth mentioning that Levey (2008, pp. 123–128) exhibited three senses in which fictions can be conceived, without trying to unravel Leibniz’s own notion, but anachronistically, based on three ways in which scientific theories can be interpreted. Thus, he distinguished (1) “reductionism,” according to which Leibniz’s infinitesimal language can be reduced to a language that includes only finite terms (that is, the syncategorematic interpretation to which the author subscribes); (2) “pragmatism,” according to which the infinitesimal language is an adequate way, scientifically speaking, to describe the data that the theory attempts to organize, explain and predict; and (3) “ideal-theory instrumentalism,” according to which an infinitesimal is a device “for inferring meaningful results from meaningful premises” (p. 124).
Leibniz seems to have developed this conception of mathematics as something “ideal” at the end of his Parisian period and especially in the 1680s. Cf. A II 2 75; A VI 4 991; GP 4 490, 561; GP 2 225/OFC 16B 1164, inter alia. For the question of the origins of the “ideal” conception of mathematical entities, see Esquisabel and Raffo Quintana (2020).
It is outside the scope of this paper to explain the difficult connections between our thoughts and those of the divine mind.
It may be surprising to say that infinitely small quantities could be incompatible with the principle of continuity. But we take that principle here in the sense of a principle of order, that is, nature should be orderly constructed (see, for example, A VI 3, 564–565, and GP 2 193; 282). Thus, to suppose the existence of infinitely small real quantities could entail the thought that motion would really be composed of infinitely small jumps in infinitely small parts of space and time, and this goes against the order of nature.
In accordance with this and in an illustrative way, in the Parisian period Leibniz noted: “It is not admirable that the number of all numbers, all possibilities, all relations, that is, reflections, are not distinctly intelligible; in effect, they are imaginary and do not have anything that corresponds in reality [a parte rei]” (A VI 3, 399). However, there is also a difference between what is manifested in this passage and what he does later. For, in this passage epistemic elements prevail for possibility and impossibility, expressed in the fact that they “are not distinctly intelligible,” while the “logical” criterion based on consistency, which already appears in the Parisian period, is however much more clearly formulated after this period.
Actually, Leibniz appeals to various ways of referring to infinitesimal or infinitely small quantities: “quantity smaller than any assignable quantity” is one of them, but there is a plurality of characterizations that are only apparently equivalent. Moreover, it can be shown that there is an evolution in the way that Leibniz characterizes infinitely small quantities. Although we cannot develop it here and we will do so in a later study, we maintain that the different ways of designating or characterizing infinitely small quantities denote an evolution in the way that Leibniz conceived of the mathematical function of such fictions.
Apart from that, the notion of the infinitesimal that follows from the case of Numeri infiniti previously mentioned does not seem to coincide with the one we pointed out before: it is not a quantity smaller than any given one, but of a quantity smaller than any than can be given, or, as Leibniz literally says, a “last number.”
That is, they can be applied in mathematics as fictions without problems and can be substituted by other methods, but they do not exist in the actual world. An anonymous referee has objected that, in the question of Leibniz’s treatment of infinitely small quantities, methodological and of existence questions must be distinguished, since Leibniz himself dealt with them separately. As an answer to this objection, we fully agree with this approach, as can be seen, for example, in a forthcoming paper of ours (Esquisabel and Raffo Quintana 2021). In the same way, our final brief reference to Leibniz’s solution of the continuum problem, namely, the distinction between an ideal continuum (or “syncategorematic,” in the sense of potential), and a real continuum, in which there is an infinite actual division, refers to the problem of existence, and not to methodological questions, regarding which Leibniz just appeals to infinitary fictions. On the other hand, it seems clear enough to us, as it is to Rabouin and Arthur (2020), that in his maturity Leibniz deals with the question of the justification for the introduction of infinitely small quantities by appealing to the principle of continuity.
This role is often characterized in terms of introducing “ideal” concepts, as Sherry and Katz (2012) do. However, we think that the concept of “ideal,” which corresponds to the concept introduced in the geometry of the nineteenth century, should be applied, in the case of Leibniz, cum grano salis. As we could show, towards the last stage of his thought, Leibniz conceives that all mathematical entities, and not only infinitary ones, are “ideal.”
One might wonder if other mathematical truths, different from those that rule our world, could be possible for Leibniz (as, for example, non-Euclidean geometries). The answer should apparently be negative, since mathematical truths are true in all possible worlds. But there are nuanced opinions on this topic. See, for example, Rescher 1981 and Debuiche and Rabouin 2019.
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Acknowledgements
We would like to express our thanks to the anonymous referees of Archive for History of Exact Sciences for their observations and suggestions, which have contributed to considerably improve this paper.
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This paper is supported by the projects “La Ciencia General de Leibniz como fundamentación de las ciencias: lógica, ontología y filosofía natural” (ANPCyT, Argentina, PICT-2017-0506) and “Resultados de imposibilidad en geometría: perspectivas históricas y semánticas” (ANPCyT, Argentina, PICT-2017-0443).
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Esquisabel, O.M., Raffo Quintana, F. Fiction, possibility and impossibility: three kinds of mathematical fictions in Leibniz’s work. Arch. Hist. Exact Sci. 75, 613–647 (2021). https://doi.org/10.1007/s00407-021-00277-0
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DOI: https://doi.org/10.1007/s00407-021-00277-0