Article ID: CBB001211764

Leibniz's Syncategorematic Infinitesimals (2013)

unapi

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz's mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton's Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.---(Bos 1974--1975, 82--83).

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Authors & Contributors
Katz, Mikhail G.
Knobloch, Eberhard
Blåsjö, Viktor
Bair, Jacques
Ely, Robert
Raffo Quintana, Federico
Concepts
Mathematics
Philosophy of mathematics
Calculus
Infinitesimals
Geometry
Continuity
Time Periods
17th century
18th century
Early modern
Modern
Medieval
Ancient
Places
Germany
Europe
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