the infinite

Leibniz's Actual Infinite in Relation to his Analysis of Matter”, 137-156 (chapter 7) in G. W. Leibniz on the interrelations between mathematics and philosophy, (Springer: Archimedes Series), ed. Norma Goethe, Philip Beeley and David Rabouin, 2012. [preprint]

The Labyrinth of the Continuum”, in Maria Rosa Antognazza (ed.), Oxford Handbook of Leibniz; published online at (www.oxfordhandbooks.com), Oxford: Oxford University Press, December 2013. [preprint]

Leibniz's Syncategorematic Infinitesimals, Smooth Infinitesimal Analysis, and Second Order Differentials”, Archive for History of Exact Sciences, April 2013 [16,700 words]. [preprint] The final publication is available at link.springer.com.This is a rewrite of a paper submitted to the same journal in July 2010.

Leibniz’s Archimedean infinitesimals,” Proceedings of the Canadian Society for History and Phil. of Mathematics, 21, 1-10, 2008. [preprint]

‘A complete denial of the continuous?’ Leibniz’s philosophy of the continuum,” accepted but never to appear in a special edition of Synthese devoted to the Mathematics and Philosophy of the Continuum. (Written 2005):

  • Abstract: Noting the status of the Law of Continuity as one of Leibniz’s most cherished axioms, Bertrand Russell charged that his philosophy nevertheless amounted to “a complete denial of the continuous”. Georg Cantor made a similar accusation of inconsistency about Leibniz’s philosophy of the actual infinite. But I argue that neither doctrine is inconsistent when the subtleties of Leibniz’s syncategorematic interpretation are properly taken into account. Leibniz rejects the existence of infinite wholes: an infinite aggregate of actual things forms only a fictitious whole. Analogously, infinitesimals are only fictitious parts, this time of ideal wholes. That is, just as an actual infinity of terms can be understood syncategorematically as more terms than can be assigned a number, without there being any infinite numbers, so too the infinitely small can be given a syncategorematic interpretation by means of the Law of Continuity, without there existing any actual infinitesimals. By examining Leibniz’s justification of infinitesimals in his calculus, I argue that the syncategorematic interpretation is also applicable to series of changes, and thus exonerates Leibniz from Russell’s criticism: on this interpretation all naturally occurring transitions are continuous in that the difference between neighbouring states is smaller than any assignable. This means not that there exists a least difference, but that for any assignable finite difference, there exists a smaller one. Thus there is a true continuous transition, even though the states themselves and all assignable differences between them are actually discrete.

Actual Infinitesimals in Leibniz’s Early Thought”, in The Philosophy of the Young Leibniz, Studia Leibnitiana Sonderhefte 35, ed. Mark Kulstad, Mogens Laerke and David Snyder, 2009, pp. 11-28. [preprint]

Here is a longer earlier draft::

"From Actuals to Fictions: Four Phases in Leibniz’s Early Thought On Infinitesimals": [draft]

  • Abstract: In this paper I attempt to trace the development of Gottfried Leibniz’s early thought on the status of the actually infinitely small in relation to the continuum. I argue that before he arrived at his mature interpretation of infinitesimals as fictions, he had advocated their existence as actually existing entities in the continuum. From among his early attempts on the continuum problem I distinguish four distinct phases in his interpretation of infinitesimals: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672-75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus); (iv) (1676 onward) infinitesimals are fictitious entities, which may be used as compendia loquendi to abbreviate mathematical reasonings; they are justifiable in terms of finite quantities taken as arbitrarily small, in such a way that the resulting error is smaller than any pre-assigned margin. Thus according to this analysis Leibniz arrived at his interpretation of infinitesimals as fictions already in 1676, and not in the 1700's in response to the controversy between Nieuwentijt and Varignon, as is often believed.

Leibniz and Cantor on the Actual Infinite,” pp. 41-46 in Nihil sine ratione , Vol. 1, ed. Hans Poser, Berlin: Gottfried-Willhelm-Leibniz-Gesellschaft, 2001. [preprint]

  • Abstract: This constitutes the gist of a dialogue I have written in which Leibniz and Cantor debate the nature of the infinite. Although the paper is rough, the basic argument is discernible: i) Leibniz's syncategorematic actual infinite is a consistent third alternative to the Cantorian actual infinite and the Aristotelian potential infinite; ii) it is appropriate to his conception of the actual infinite division of matter as not involving infinite number; whereas iii) Cantor's actual infinite is not appropriate to such infinite division, since one cannot get to an infinitieth part by recursively dividing.
© Richard Arthur 2012