The present essay describes Leibniz's foundational studies on continuity in geometry. In particular, the paper addresses the long-debated problem of grounding a theory of intersections in elementary geometry. In the early modern age, in fact, several mathematicians had claimed that Euclid's Elements needed to be complemented with additional axioms in order to ground the existence of the intersection points between straight lines and circles. Leibniz was sensible to similar foundational issues in the Euclidean tradition, and dedicated several studies to investigate a good definition of the continuity of space, in order to ground a general theory of intersections of curves and surfaces. While Leibniz's researches on continuity in relation with the Calculus have been extensively studied in the past, the present essay deals with the less-known Leibnizian notion of a continuous space, as it is to be found in several unpublished writings preserved in Hannover. The subject widens as to encompass the relation between mereology and analysis situs, Leibniz's studies on a geometrical characteristics, and Leibniz's theory of space at large. Acknowledgements. I thank the Max Planck Institute for Mathematics in the Sciences and his director Jürgen Jost for providing me with the possibility of writing a good part of this paper while I was based in Leipzig, and more in general for having fostered important occasions of discussion with other Leibniz scholars. I also thank Richard Arthur, Eberhard Knobloch, Julien Narboux and Erich Reck, who have discussed with me this paper in several occasions; and Andrea Costa, Siegmund Probst and Javier Echeverría for sharing with me some unpublished papers by Leibniz.
The paper lists several editions of Euclid’s Elements in the Early Modern Age, giving for each of them the axioms and postulates employed to ground elementary mathematics.
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The "common notions" prefacing the Elements of Euclid are a very peculiar set of axioms, and their authenticity, as well as their actual role in the demonstrations, have been object of debate. In the first part of this essay, I offer a survey of the evidence for the authenticity of the common notions, and conclude that only three of them are likely to have been in place at the times of Euclid, whereas others were added in Late Antiquity. In the second part of the essay, I consider the meaning and uses of the common notions in Greek mathematics, and argue that they were originally conceived in order to axiomatize a theory of equivalence in geometry. I also claim that two interpolated common notions responded to different epistemic needs and regulated diagrammatic inferences. §1. IntroductionThe Elements of Euclid are introduced by three sets of principles: definitions, postulates and common notions. The number of common notions (κοιναὶ ἔννοιαι) varies in different Greek, Arabic and Latin manuscripts. Most of these manuscripts comprise either nine or ten of them. Ancient commentators inform us that still further principles had been suggested in late antiquity but never found their way into the text of the Elements. 1 Following the textual indications of these commentaries and other philological considerations, Johan Ludvig Heiberg produced a critical edition of the Elements containing five of these common notions: CN1: Things which are equal to the same thing are also equal to one another. CN2: If equals be added to equals, the wholes are equal. CN3: If equals be subtracted from equals, the remainders are equal. CN4: Things which coincide with one another are equal to one another. CN5: The whole is greater than the part.This list is accepted nowadays by most historians of mathematics, all of whom agree with Heiberg's excision of many other principles from the text. Nonetheless, a few authoritative Aknowlwdgments. I thank Gregg de Young, Eduardo Giovannini, Marco Panza, and an anonymous referee, for their insightful remarks. I am especially grateful to Mattia Mantovani, whose many suggestions and relentless requests of better evidence substantially improved the final version of this essay. 1 For a first appraisal of the number of common notions in various manuscripts and Euclidean editions, see my V. DE RISI, The Development of Euclidean Axiomatics. The systems of principles and the foundations of mathematics in editions of the Elements from Antiquity to the Eighteenth Century, "Archive for History of Exact Sciences", 70 (2016), pp. 591-676.
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