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[The following is summary; my commentary is in brackets.]
Bergson, Time and Free Will
Chapter II, "The Multiplicity of Conscious States," "The Idea of Duration"
Part XXIII: The Eleatic Paradox
§70 "The Common Confusion between Motion and the Space Traversed Gives Rise to the Paradoxes of the Eleatics"
Previously we saw how we confuse motion with the space that a body traverses. We know that space is infinitely divisible. So if we confuse motion with space, we conclude that motion is divisible as well. But, if space is infinitely divisible, then there are an infinite number of intervals to cross in a finite area. So motion across a finite extent could never be completed. Hence when we confuse space and motion, we obtain Zeno's paradoxes. (112-113)
This confusion forms the basis of Zeno's paradox of Achilles and the Tortoise. The two race. The Tortoise starts further down the track. But they begin at the same time. Before Achilles can overtake the Tortoise, he must first reach the spot where the Tortoise began. Quickly Achilles arrives there. But, the Tortoise has been moving continuously since then. So by the time Achilles arrives at the Tortoise's starting spot, the Tortoise has already advanced a bit down the track. So now for Achilles to overtake the Tortoise, he must first cross the Tortoise's more advanced position. But again, as soon as Achilles arrives, the Tortoise has advance and created a new spot that Achilles must cross. So long as the Tortoise continues moving, there will aways be points in advance for Achilles to cross. Hence Achilles can never overtake the Tortoise.
Bergson explains that indeed space is infinitely divisible. But the motion through it is an act. And acts are indivisible. So "each of Achilles' steps is a simple indivisible act." (113a) After so many of these acts, Achilles will pass the tortoise. Zeno's mistake is that he believes motion to be as homogeneous and divisible as space is. Rather, in motion there is a heterogeneous series of indivisible acts.
Zeno and the Eleatics homogenize the movement of Achilles and the Tortoise. So in the end, each step of Achilles is equal to each of the Tortoise's step. In a way, Zeno really is comparing the movement of two tortoises who "agree to make the same kind of steps or simultaneous acts, so as never to catch one another." (113bc)
To help us better understand the problem, Bergson has us make a distinction.
1) As motions, the steps of both Achilles and the Tortoise are indivisible acts.
2) As spatial motions, the steps of both racers have extensive magnitudes. So their movements are made up of discrete simple motions. Now, the extensive magnitude of each of Achilles' steps is greater than the lengths of the Tortoise's steps. So even if the Tortoise has a head start, eventually Achilles' longer strides will allow him to pass the Tortoise.
To visualize, consider each discrete movement-act of Achilles and the Tortoise as having disproportionate extensive magnitudes.
The Tortoise begins the race further down the track. They take their first step.
Second:
And then the following:
So Bergson sees movement-actions as descrete units. These may be of different spacial magnitudes. Thus we can see why there really is no paradox.
Bergson then addresses Évellin's similar solution to the Achilles paradox. Both Évellin and Bergson solve the problem by considering Achilles' motion as made up of discrete units. Achille's units are greater than the Tortoise's, so Achilles will overtake him. For Bergson, we experience discrete acts of motion in a non-spatial pure duration. Motions, as durational acts, then, do not extend in time. However, any motion may correspond to a different spatial extent. So this way Achilles' steps are larger. For Bergson, the problem is that we fail to distinguish inextensive duration from extensive time and space. So it does not matter if space is infinitely divisible, for Bergson, because motion-acts are indivisible units corresponding to different extensions in space.
Yet, for Évellin, real space is not infinitely divisible. There are basic discrete units of space and time. But we use our imagination to think geometrically about space. We can imagine any two neighboring points. But no matter how close we picture them to be, we can always imagine another point between them. However, this leads to Zeno's paradoxes. So Évellin argues that there is a real space that is not infinitely divisible. Then, in each common discrete time moment, Achilles will travel more distance-units than the Tortoise. This way, no matter how far ahead the Tortoise begins, Achilles will overtake him, given enough time. So we see that the same diagram that we used for Bergson applies to Évellin as well.
Except now the numbers represent discrete time moments, and not discrete motion-acts.
Bergson thinks it is unnecessary to resort to metaphysical abstractions about the nature of reality. All we need to do is presuppose that motion is qualitative and not quantitative.
But if motion is not quantitative, then what do scientists measure when they calculate velocity? At the start of motion, the object is in one spatial place, and its destination is somewhere else. Eventually, there will be another simulteneity when the object and its destination coincide. Bergson claims that when physicists measure velocity, they are really determining when they can expect the simultaneity during when the object and its destination coincide. So mathematics is perfectly capable of making calculations regarding the starting and ending simulteneities. However, math and physics go beyond their "province" when they try to explain what happens between those simulteneities. (114-115)
So when we looked for a homogeneous medium in duration, we found time, which was really space. But there is no duration in space. Hence what is homogeneous in duration is not durational. And we see now that when we seek something homogeneous in motion, all we find is the line of space traversed. But geometrical lines are motionless. Hence likewise, what is homogeneous in motion is really motionless. (115b)
[Directory of other entries in this series.]
Images from the pages summarized above, in the English Translation [click on the image for an enlargement]:
Images from the pages summarized above, in the original French [click on the image for an enlargement]:
Bergson, Henri. Time and Free Will: An Essay on the Immediate Data of Consciousness, Transl. F. L. Pogson, (New York: Dover Publications, Inc., 2001).
Available online at:
http://www.archive.org/details/timeandfreewill00pogsgoog
French text from:
Bergson, Henri. Essai sur les données immédiates de la conscience. Originally published Paris: Les Presses universitaires de France, 1888.
Available online at:
http://www.archive.org/details/essaisurlesdonn00berguoft
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