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Mikhail G. Katz and David Sherry
“Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond”
2. Preliminary Developments
Brief Summary:
Indivisibles are not infinitesimals, although they are often confused. Indivisibles have one dimension less than what they divide. Infinitesimals have the same dimensions as what they are a part of.
Summary
We should first distinguish indivisibles from infinitesimals.
Leibniz first uses the term ‘infinitesimal’ in 1673, but he credits the coinage to Mercator.
There is more than one conception of the infinitely small. Many commentators do not differentiate them. For example, Boyer seems to imply that Archimedes infinitesimal and kinematic methods provided the basis for Leibniz’ differential calculus. (Boyer, The concepts of the calculus, p.59.) (KS 573d). However, Archimedes’ infinitesimal method uses indivisibles and not infinitesimals. His indivisibles are the limits of division, and thus they have one dimension less than the areas they are dividing (they are one dimensional while what they divide is two dimensional).
Archimedes’ infinitesimal method employs indivisibles. For example, in his heuristic proof that the area of a parabolic segment is 4/3 the area of the inscribed triangle with the same base and vertex, he imagines both figures to consist of perpendiculars of various heights erected on the base (ibid., 49–50). The perpendiculars are indivisibles in the sense that they are limits of division and so one dimension less than the area. Qua areas, they are not divisible, even if, qua lines they are divisible. In the same sense, the indivisibles of which a line consists are points, and the indivisibles of which a solid consists are planes. We will discuss the term ‘‘consist of’’ shortly.
(KS 574a, boldface and underlining mine)
[See Boyer’s treatment of Archimedes here, and see the original Archimedes’ text] However, Leibniz’ infinitesimals have the same dimension as the figures they make-up, and thus they are not like Archimedes’ indivisibles.
Leibniz’s infinitesimals are not indivisibles, for they have the same dimension as the figures that consist of them. Thus, he treats curves as composed of infinitesimal lines rather than indivisible points. Likewise, the infinitesimal parts of a plane figure are parallelograms. The strategy of treating infinitesimals as dimensionally homogeneous with the objects they compose seems to have originated with Roberval or Torricelli, Cavalieri’s student, and to have been explicitly arithmetized by Wallis (Beeley 2008, [Infinity, Infinitesimals, and the Reform of Cavalieri: John Wallis and his Critics. In Goldenbaum and Jesseph Infinitesimal Differences: Controversies between Leibniz and his Contemporaries] p. 36ff).
(KS 574)
Democritus used both indivisibles and infinitesimals. (KS 574)
Plutarch notes a puzzle of Democritus’ that would arise for indivisibles but not for infinitesimals. The puzzle is this. A cone can be thought as being made of many surfaces parallel to the base. If they are all equally, it would be a cylinder. But if they were all different, it would look like a staircase. However, we do not encounter this problem when we use the concept of the infinitesimal. “This puzzle need not arise for infinitesimals of the same dimension, with an infinitesimal viewed as a frustum of a cone rather than a plane section.” (KS 574) So we are to think of each slice as being infinitesimally thin, but the outer part is at an angle. Thus one slice picks up where the prior leaves off, and we have an infinity of slices all making a smoothly tapering cone. (see diagrams of frustums below)
(thanks wikimedia commons)
(thanks wikimedia commons)
[We give a more detailed treatment of the Plutarch text here. From that entry, here is a moving diagram for the puzzle, including the distinction between the indivisibles and infinitesimals:]
[Moving diagram by Corry Shores, made with Open Office Draw and Unfreez]
Zeno’s “metrical paradox proposes a dilemma: If the indivisibles have no magnitude, then a figure which consists of them has no magnitude; but if the indivisibles have some (finite) magnitude, then a figure which consists of them will be infinite.” (574) There is a further problem for indivisibles. They are boundaries of what they limit. But this means they are not immediately up against one another. So we seem unable to concatenate them in order to increase a magnitude.
If a magnitude consists of indivisibles, then we ought to be able to add or concatenate | them in order to produce or increase a magnitude. But indivisibles are not next to one another; as limits or boundaries, any pair of indivisibles is separated by what they limit. Thus, the concepts of addition or concatenation seem not to apply to indivisibles.
(574-575)
These problems do not apply to Leibniz’ infinitesimals. They do not have a zero magnitude, so they do not have the problem of being unable to add up to a larger magnitude. However, their magnitude is not finite, so an infinity of them is not infinitely large. Infinitely many infinitely small magnitudes make up finite magnitudes. This also allows us to perform arithmetic operations on them, which distinguishes them from Archimedes’ methods.
The paradox may not apply to infinitesimals in Leibniz’s sense, however. For, having neither zero nor finite magnitude, infinitely many of them may be just what is needed to produce a finite magnitude. And in any case, the addition or concatenation of infinitesimals (of the same dimension) is no more difficult to conceive of than adding or concatenating finite magnitudes. This is especially important, because it allows one to represent infinitesimals by means of numbers and so apply arithmetic operations to them. This is the fundamental difference between the infinitary methods of Archimedes (and later Cavalieri) and the infinitary methods of Leibniz and his followers.
(575)
Not rigorously making this distinction has led to misleading claims being made about 17th century calculus. In the following, the authors will “say that a magnitude consists of infinitesimals just in case the infinitesimals and the original magnitude have the same dimension. Otherwise, we shall use the term indivisible.” (575)
Bibliography:
Katz, M.; Sherry, D. Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. Erkenntnis 78 (2013), no. 3, 571-625. See http://dx.doi.org/10.1007/s10670-012-9370-y, http://www.ams.org/mathscinet-getitem?mr=3053644, and http://arxiv.org/abs/1205.0174
The above bibliography material taken from the following source, a page by Mikhail Katz, which links to many other recent publications on infinitesimals.
http://u.cs.biu.ac.il/~katzmik/infinitesimals.html
Image credits:
Frustrum:
http://commons.wikimedia.org/wiki/File:Frustum_of_a_cone.jpg
http://commons.wikimedia.org/wiki/File:Frustum_of_a_Decagonal_Pyramid.svg
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