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Bergson offers his solution to Zeno's paradox in the second chapter of Time and Free Will (Essai sur les données immédiates de la conscience). He then distinguishes his solution from Évellin's, which we describe below.
In Chapter 2 of Infini et quantité, Évellin determines whether or not extension, duration, and movement are infinitely divisible.
Chapitre II: "L'Idole de l'infiniment petit et les quantités analogues a la matière"
I: "Le lieu en soi ou le lieu réel"
A body's real place is the portion of space it occupies. The body cannot be absolutely detached from the real space that contains it. If we could consider its real space as completely distinct from it, then we would be confusing real space with the ideal space that we conceive in our understanding. (65)
Real space is neither a pure abstraction nor a form of a priori sensibility. Real space and the body filling it are divided into a finite number of elementary parts.
A ce point de vue, le lieu, non moins que le corps lui-même, se conçoit comme nécessairement divisible en un nombre fini de parties élémentaires. (66-67)
These elementary parts must be indivisible and unextended (67).
There is a difference between the body's elementary parts and its space's elementary parts: the body's elementary parts need not be touching [they could be like atoms with space in between.] But the points of the space it occupies must be contiguous. (67)
We cannot represent or imagine this contiguity. [Just as Spinoza claimed, space is infinitely divisible in the imagination. So there will always be some space between two points. Hence imagining the contiguity of points in space leads to an infinite regress. However, using our rational faculties, again like Spinoza claims, we may conceive of the indivisibility of space. In this case we might consider it as made up of infinitesimals.] So only with Reason can we conceive the contiguity of space's elementary parts. (67)
We use geometry to measure motion, but the points of geometry are abstractions and are not real. For, points in real space are impenetrable. But geometrical points are empty abstractions through which real things may pass. (68)
Real extension is not the abstract space of geometry, nor is it the extension we form in our imagination. (69)
When bodies move, they cross through a determinate amount of space in a determinate amount of time. This is because bodies move through real space and not through imaginary/geometrical space. (69)
Évellin now describes a situation like the race between Achilles and the Tortoise. He has us consider two moving bodies that begin from different points and move in the same direction [like the Tortoise having the head-start]. As we expect, if the speed of the one trailing is greater than the one ahead, the two bodies will eventually meet. The faster will overtake the slower. Now, we said that ideal space is infinitely divisible, but real space is not. So on account of Zeno's paradox, we know that their encounter cannot occur in this idealized infinitely divisible space. Yet we do know that the two bodies will meet. Hence their encounter can only occur in real space. (70)
When we use geometry to try to understand the race, we are faced with a choice between two impossibilities:
1) motion is continuous. But then it is infinitely divisible. Hence impossible. Or,
2) motion is discontinuous. But then it is not motion. We cannot explain how the body jumps between points with no movement in between. (70c)
Évellin now explicitly describes the race and how real and imaginary space play a role (70-71). He writes that if we presume that the real space of the race-course is the same as the imaginary geometrical space, then we will never be able to explain how the faster Achilles overtakes the head-started Tortoise. We suppose that the Tortoise moves at 1/10th Achille's speed. But, the space the Tortoise advances is always infinitely divisible. So there are continually an infinity of points added between Achilles and the Tortoise. As well, there are an infinity of points of time, and thus an infinity of time-space points that Achilles' motion must traverse before reaching the Tortoise, who is continually creating more infinitely divisible distances for Achilles to cross.
To solve the problem, Évellin returns to his difference between imaginary and real. There is imaginary time, space, and motion. And there is real time, space, and motion. Imaginary time, space, and motion are each made-up of an infinity of geometrical points. For, imaginary extension is infinitely divisible. Between any two points there will always be more. But real space, time, and motion are discrete units. One such unit may be next to another. But that does not mean that there must also be another unit between them, as with the case of geometrical points. Évellin draws up a chart to show us that if the time-units are equal, then Achilles' space-motion units will be larger than the Tortoise's. So after a determinate number of time-units, Achilles will overtake the Tortoise.
The left column shows the standard time-units that are the same for the Tortoise and Achilles, ranging from time-units 1t to 10t. The middle columns show the distances that Achilles and the Tortoise have traveled up through the given time-unit. The right column displays the distance remaining between them at the given time-unit.
[Image from page 75.] We see that slowly the distance between Achilles and the Tortoise narrow until Achilles finally overtakes him.
This is similar to Bergson's solution. Both consider there to be discrete units of motion. But Bergson does not ontologize. He speaks of our own experiences. We do not experience our actions or motions as divisible units. So they are indivisible units. This is all we need to solve the paradox. We do not also need to theorize on the ontological nature of reality and the difference between imaginary and real space.
From:
Évellin, François. Infini et quantité: Étude sur le concept de l'infini en philosophie et dans les sciences. Paris: Librairie Germer Baillière, 1880.
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