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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”
4. Cum Prodiisset
4.4 Status Transitus
Brief Summary:
A state of transition (status transitus) is an infinitely small variation that explains how things can change from one state to a contrary one. For example, parallel lines might converge, so long as the angle of that convergence is infinitely small. That angle would then be the ‘between’ transitional state between its states of being parallel and non-parallel. Such an angle is infinitesimally more than completely superposing the other line, but infinitesimally less than diverging sharply from it, at the vertex of that angle. Looking away from the intersection, we would say it is not yet parallel. Looking toward it, we would say it is not yet superposed. Transition can be explained by means of a law of continuity which says that continuous changes occur by infinitely small bridging variations, allowing us to go from opposing states, so long as that change is so small that we cannot even assign it a magnitude (and thus make an ‘actual’ distinction between the states).
Summary
Previously we noted three of Leibniz’ examples for the Principle of Continuity: (1) lines can both be parallel yet also converge at an infinitely small angle [[because there is so little difference between an infinitely small angle and none at all]] (2) two lines can be equal even if they differ in length by an infinitely small amount, and (3) a parabola is an ellipse with one side extended to infinity. Yet we find him also stating that:
Leibniz introduces his next observation by the clause ‘‘of course it is really true that’’, and notes that ‘‘straight lines which are parallel never meet’’ (Child 1920, p. 148); that ‘‘things which are absolutely equal have a difference which is absolutely nothing’’ (Child 1920, p. 148); and that ‘‘a parabola is not an ellipse at all’’ (Child 1920, p. 149).
[[KS 580]]
To explain then the original three examples we observed, he proposes his notion of status transitus, or ‘state of transition’. [[This concept will introduce the idea of event, change, motion, time, process, and the like, because]] in one such a state of transition, ‘there has not yet arisen exact eqauality’.
a state of transition may be imagined, or one of evanescence, in which indeed there has not yet arisen exact equality … or parallelism, but in which it is passing into such a state, that the difference is less than any assignable quantity; also that in this state there will still remain some difference, … some angle, but in each case one that is infinitely small; and the distance of the point of intersection, or the variable focus, from the fixed focus will be infinitely great, and the parabola may be included under the heading of an ellipse (Child 1920, p. 149).
[[KS 580]]
Recall also Leibniz’s notion of terminus: in a continuous transition, the final ending (the terminus) of the transition may be included with that transition. [[Consider something in motion slowing to a stop. That final ending, rest, can be included with the motion preceding it, as a part of that motion, even if it lies at the end of the motion. The speed while it is moving could be assigned a value, and its speed at rest can be assigned the value of 0. But because of continuity, perhaps we might say that between its states of motion and rest, it is going an inassignable, infinitely slow speed. By ‘between’ we do not mean during some duration of time it is moving some extent of distance. Say we come to the end of the object’s motion. It first comes to complete rest at time point 2 (t2) at location point 2 (p2). Immediately prior it was at p1, t1. There is no p or t between them, if we are dealing with an actual infinity (one that is already divided infinitely and not perpetually divided potentially and thus interminably). But, at p1,t1, its state of motion was immediately in relation with its state of rest at p2,t2. And at p2,t2, it is still in relation to its prior state of motion at p1,t1. This is because motion is always a difference between times and locations. If we divide motion, we do not divide it ultimately into p’s and t’s. Rather, we divide it into the smallest possible differences between p’s and t’s. There is a difference between p1,t1 and p2,t2, because one lies at a state of motion, and the other lies at a state of rest. When we concern ourselves with that tiny moment when the object goes from p1,t1 to p2,t2, then we have the terminus (rest state at p2,t2) included in the motion (as it is part of the interval with the motion state at p1,t1). We also have this state of transition, status transitus.]] “Thus, status transitus is subsumed under terminus, passing into an assignable entity, but is as yet inassignable.” [KS 580] We would not say that the status transitus is a ‘limit’ (as some translators have rendered it), because a limit is an assignable entity, where that status transitus is not. [580]
Yet for Leibniz, the metaphysical reality of the infinitesimal is open to question.
whether such a state of instantaneous transition from inequality to equality, … from convergence [i.e., lines meeting—the authors] to parallelism, or anything of the sort, can be sustained in a rigorous or metaphysical sense, or whether infinite extensions successively greater and greater, or infinitely small ones successively less and less, are legitimate considerations, is a matter that I own to be possibly open to question (Child 1920, p. 149).
[KS 580]
Yet this uncertain to the ontological reality of infinitesimals should not stop mathematicians from using them effectively in their calculations. Leibniz asserts
the possibility of the mathematical infinite: ‘‘it can be done’’, without ontological commitments as to the reality of infinite and infinitesimal objects.
[KS 581]
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.
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