2005-2006 Loemker Conference

The Metaphysical and Mathematical Discussion of the Status of Infinitesimals in Leibniz’s Time

RICHARD ARTHUR (McMaster University, Hamilton)
Leery Bedfellows: Leibniz and Newton on the Status of Infinitesimals

Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to physics, and deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his method of first and ultimate ratios (published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1676) as well as in later writings. I intend to show that both appeal to a version of the Archimedean Axiom already explicated by their predecessors Pascal and Wallis.

O. BRADLEY BASSLER (The University of Georgia, Athens)
Infinitesimals, Differentials, and the Leibnizian Calculus

The metaphysical issues surrounding infinitesimals are for Leibniz first and foremost issues about their fictional status. In the mathematical domain, on the other hand, Leibniz treats infinitesimals largely as differentials, and perhaps the central technical issue surrounding the status of differentials concerns the specification of the “progression of variables.” In this paper, making use of Jan Mycielski’s model of “analysis without actual infinity,” I suggest some ways in which these two sets of issues can be related.

PHILIP BEELEY (Leibniz-Forschungstelle, Münster)
Infinity, Infinitesimals and the Reform of Cavalieri: John Wallis and his Critics

Recent years have seen an increased level of interest in the philosophical background to concepts and techniques employed by mathematicians in the development of calculus in the Seventeenth Century. A good example is provided by Cavalieri's geometry of continua, better known as the method of indivisibles, which was attacked on account of its purported conflict with traditional views on continuity, and which at the same time engendered discussion on questions of rigour and of the foundations of mathematics. As is well known, the reception of Cavalieri saw the transformation of indivisibles into infinitesimals through authors such as Torricelli, Roberval and Pascal. Wallis's response to perceived limitations to their essentially geometrical approaches was arithmetization: in his “Arithmetica infinitorum” he sought for the first time to combine systematically the concepts of infinitesimals and arithmetical limits. The paper will consider Wallis's motives for reforming Cavalieri and give an account of some of the debates which ensued with the likes of Hobbes and Fermat.

HERBERT BREGER (Leibniz-Archiv, Hannover)
Leibniz's Calculation With Compendia

It has often been claimed that Leibniz's verbal statements on infinitesimals seem to be inconsistent. For a proper understanding of these infinitesimals, the discussion should include their mathematical use. According to Leibniz, Pascal and Huygens were of particular importance for his development of infinitesimal calculus. Some of their methods and arguments will be discussed. Nevertheless, not the early Leibniz, but his ideas and statements after his return from Paris will be the main object of investigation. His first publication on the calculus (1684) exposes the rules for the calculation with compendia loquendi. Although Huygens himself had used infinitesimals before 1684, he had difficulties to understand this use of compendia.

MICHEL FICHANT (Université Paris-Sorbonne (Paris IV))
Infini métaphysique et infini mathématique dans Leibniz

Je suis tellement pour l’infini actuel, qu’au lieu d’admette que la nature l’abhorre, comme l’on dit vulgairement, je tiens qu’elle l’affecte partout, pour mieux marquer les perfections de son auteur. Ainsi je crois qu’il n’y a aucune partie de la matière qui ne soit, je ne dis pas divisible, mais actuellement divisée, et par conséquent la moindre particelle doit être considérée comme un monde plein d’une infinité de créatures différentes.
Leibniz à Foucher, 1692, GP I, 416.
Je crois que M. de Fontenelle … en a voulu railler, lorsqu’il a dit qu’il voulait faire des éléments métaphysiques cde notre calcul. Pour dire le vrai, je ne suis pas trop persuadé moi-même qu’il faut considérer nos infinis et infiniment petits autrement que comme des choses idéales ou comme des fictions bien fondées.
Leibniz à Varignon, 20 juin 1702, GM IV, 110.

Le parallèle entre ces deux déclarations de Leibniz pourrait fournir la raison pour laquelle Georg Cantor considérait que la thèse leibnizienne de l’infini était de la plus grande inconséquence : l’infini actuel, attesté dans la nature, est expulsé des mathématiques.
La thèse leibnizienne se subdivise en trois propositions explicites :
1. Il y a une métaphysique de l’infini,
2/ Il y a un calcul de l’infini,
3/ Il n’y a pas de métaphysique du calcul de l’infini.
On pourrait y ajouter une quatrième proposition, non formulée par Leibniz, mais qui a la valeur d’un avertissement pour ses interprètes :
4/ Il n’y a pas de calcul répondant de la métaphysique de l’infini.
Toutefois, la séparation entre infini métaphysique et infini mathématique laisse ouverte la question de leur corrélation, au moins dans la latitude d’un « comme si » : « Le réel ne laisse pas de se gouverner parfaitement par l’idéal et l’abstrait, comme s’il y avait des atomes … quoiqu’il n’y en ait point la matière étant actuellement sousdivisée sans fin ; et que vice versa les règles de l’infini réussissent dans le fini, comme s’il y avait des infiniment petits métaphysiques, quoiqu’on en ait point besoin, et que la division de la matière ne parvienne jamais à des parcelles infiniment petites » (à Varigon, 2 février 1702, GM IV, 93-94.
La communication tentera d’approfondir la situation théorique complexe résultant de ces présuppositions leibniziennes.

DANIEL GARBER (Princeton University)
Infinitesimals and the Physical

In my talk, I will discuss Leibniz's application of infinitesimal mathematics to the physical world. In particular, I will discuss the way the distinction between finite and infinitesimal quantities in mathematics is mirrored in the distinction that Leibniz draws between living and dead force, and the parallel notions of motion with which they correlate. I will try to show how Leibniz's philosophical worries about the status of infinitesimals is reflected in what he says about the corresponding notions of motion and force.

URSULA GOLDENBAUM (Emory University, Atlanta)
“Indivisibilia vera” in Leibniz’s Early Philosophy of Mind

There is no longer any doubt that Hobbes had a powerful impact on Leibniz, especially in the fields of mechanics (conatus) and mathematics (point). His impact is seen most clearly in the TMA. This influence is usually traced back to Leibniz's reading of De Corpore, though I plan to show that Leibniz already knew of the controversy between Hobbes and Wallis in 1669/70 (at least from the perspective of Hobbes). In my paper I will discuss (1) how this knowledge deeply influenced his approach to the questions of indivisibles and infinitesimals in the TMA and (2) to what extent his solution (and indeed the entire TMA) was determined by his theological purposes. I conclude by showing the significance of these points for the interpretation of his later positions as well.

EMILY GROSHOLZ (PennState University, University Park)
Leibniz's Calculus: Ambiguous Notation and Metaphysics

The bridging between levels of analysis that ambiguous notation accomplished in chemistry (like Berzelian formulas) led directly to a new realism about atoms in the late nineteenth century. But what can we say about the bridging that Leibniz's notation accomplished in the seventeenth century to bring the infinitesimal, finitary, and infinitary into rational relation? Does it give evidence for a metaphysics of infinity, for mathematical realms beyond the finite or "constructible"? To answer this question, we can look to Leibniz's own opinions, and we can look to the mathematical practice his notation begat; the two kinds of testimony may not agree.

HIDÉ ISHIGURO (University of Tokyo)
Infinitesimals and Points: Philosophical Reflections on Some Recent Theories

Leibniz, with his "infinitesimals", is often grouped among thinkers who believe that continua are irreducible to points, e. g. Aristotle, Pierce, Weyl, Brouwer. If this is true, it is a position which is incompatible with the Cantorian continua constituted by points, clarified by A. Robinson in the mid 20th century. The logician John Bell has recently obtained non-punctiform continua through actual-linear infinitesimals, which the philosopher Richard Arthur finds, to be close to those of Leibniz. But if infinitesimals are point determining limit processes, then, must we not reconsider in what way Leibniz can stand aloof from the view that lines are constituted by points?

DOUG JESSEPH (North Carolina State University, Raleigh)
Truth in Fiction: Origins and Consequences of Leibniz's Doctrine of Infinitesimal Magnitudes.

This paper investigates the background to Leibniz's doctrine of the fictionality of infinitesimal magnitudes and the consequences the doctrine has for his account of the foundations of the calculus. It first traces the connection between Leibniz's doctrine of "incomparably small" magnitudes and Hobbes's doctrine of conatus, particularly as it is applied to the study of geometric figures. The concluding sections consider the application of this doctrine to disputes about the reality of infinitesimal magnitudes.

EBERHARD KNOBLOCH (Technische Universität Berlin)
Generality and Infinitely Small Quantities in Leibniz's Mathematics: The Case of His "Arithmetical Quadrature of Conic Sections and Related Curves"

In 1675/76 Leibniz wrote his treatise Arithmetical quadrature of the circle, the ellipse and the hyperbola etc. (published in 1993, French/Latin ed. in 2004). Therein Leibniz used infinitely small and infinite quantities in order to give a rigorous foundation of
integration theory and to deduce more than a dozen general theorems on the quadrature of hyperboloids and paraboloids. He again and again underlines the generality of his method. The paper will discuss Leibniz's notion of generality in mathematics with special reference to this treatise.

MARK KULSTAD (Rice University, Houston)
Leibniz's Discussion of Spinoza's Notion of Infinity

An intriguing point of entry into Leibniz's thought on the infinitely small and large is his discussion of Spinoza's notion of infinity. A key text involved here is Spinoza's letter on the infinite in conjunction with Leibniz's comments on this. In these comments Leibniz develops a three-fold distinction of kinds of infinity, omnia, maximum, infinitum. In this paper I examine this and other important texts of Leibniz on Spinoza's notion of infinity, with the goal of gaining insights into Leibniz's own emerging conception of infinity.

SAMUEL LEVEY (Dartmouth College, Hanover)
Leibniz on Indivisibles in Motion: Early Foundations

Some of Leibniz's early treatments of the structure of motion involve the hypothesis that motion contains indivisible elements, the beginnings and ends of individual motions. In two documents in particular -- On Minimum and Maximum (1672-3) and Pacidius to Philalethes (1676) -- he argues expressly for the existence of indivisibles in motion by appeal to classical paradoxes concerning change. Here I examine Leibniz's use of the paradoxes and consider how it suggests a metaphysical conception of motion as something "complete in itself."

FRITZ NAGEL (Bernoulli-Edition, Zürich)
Johann I Bernoulli and Jacob Hermann on Infinitesimals

The notion of the infinitesimal small has been discussed in the Bernoulli circle several times. It was Jacob Hermann (1678-1733), a disciple of Jacob Bernoulli, who first has published some arguments for the legitimate use of this quantities. The first part of my paper therefore will deal with Hermann's "Responsion ad Clarissimi viri Bernh. Nieuwentijt Considerationes secundas circa calculi differentialis principia, editas", Basel 1700. In the second part of my paper I will focus on some arguments of Johann Bernoulli (1667-1748) for the use of the infinitesimal quantities discussed for example in his correspondence with Bernard de Fontenelle.

DONALD RUTHERFORD (University of California, San Diego)
Leibniz on Infinitesimals and the Reality of Physical Force

Leibniz is committed to three prima facie inconsistent propositions: 1) force is the only "real and absolute" property of body; 2) forces are infinitesimal quantities or compounds of such quantities; 3) infinitesimal quantities are not real, but only "fictions of the mind" which allow for the “abbreviation of thought.” My paper will explore the support Leibniz offers for these three propositions and the means he has for resolving the apparent conflict among them.