The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility innature, rather than inmathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in (...) their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, arementis fictiones, i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything inrerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

El objetivo de este trabajo es examinar el concepto leibniziano de ficción matemática, con especial énfasis en la tesis de Leibniz acerca del carácter ficcional de las nociones infinitarias. Se propone en primer lugar, como marco general de la investigación, un conjunto de cinco condiciones que una ficción tiene que cumplir para ser matemáticamente admisible. Sobre la base de las concepciones de Leibniz acerca del conocimiento simbólico, se propone la ficción matemática como la clase de nociones confusas que carecen de (...) denotación a raíz de la imposibilidad de su objeto. Tomando como punto de partida el análisis de la imposibilidad en términos de inconsis tencia, se muestra que Leibniz admite otras formas de imposibilidad, que afectan especialmente a las nociones infinitarias. Proponemos así la imposibilidad como irrepresentabilidad geométrica y la imposibilidad por incompatibilidad con principios arquitectónicos. Así, el resultado de nuestro examen fundamenta la admisión de tres tipos de ficcionalidad matemática, la ficción 1, que se corresponde con la noción inconsistente, la ficción 2 que incluye las nociones geométricamente irrepresentables y la ficción 3, que se aplica a las nociones “arquitectónicamente” imposibles. En conclusión, los conceptos infinitarios, sin ser por sí mismos inconsistentes, corresponden al tipo de ficcion 2 y ficción 3. Finalmente, se concluye que Leibniz está más preocupado por la imposibilidad por incompatibilidad con principios arquitectónicos que por la cuestión de la irrepresentabilidad geométrica. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...) interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (shrink)

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. (shrink)

In Book 1 of the Principia, Newton presented two different descriptions of orbital motion under the action of a central force. In Prop. 1, he described this motion as a limit of the action of a sequence of periodic force impulses, while in Prop. 6, he described it by the deviation from inertial motion due to a continuous force. From the start, however, the equivalence of these two descriptions has been the subject of controversies. Perhaps the earliest one was the (...) famous discussion from December 1704 to 1706 between Leibniz and the French mathematician Pierre Varignon. But confusion about this subject has remained up to the present time. Recently, Pourciau has rekindled these controversies in an article in this journal, by arguing that “Newton never tested the validity of the equivalency of his two descriptions because he does not see that his assumption could be questioned. And yet the validity of this unseen and untested equivalence assumption is crucial to Newton’s most basic conclusions concerning one-body motion” (Pourciau in Arch Hist Exact Sci 58:283–321, 2004, 295). But several revisions of Props. 1 and 6 that Newton made after the publication in 1687 of the first edition of the Principia reveal that he did become concerned to provide mathematical proof for the equivalence of his seemingly different descriptions of orbital motion in these two propositions. In this article, we present the evidence that in the second and third edition of the Principia, Newton gave valid demonstrations of this equivalence that are encapsulated in a novel diagram discussed in Sect. 4. (shrink)