A reply to Richard Burke on Bernardette’s Paradox1
Alejandro Martín Maldonado
martinmaldonado@yahoo.com
Richard Burke exposes a new zenonian paradox designed by José Bernardette2 in the 60’s, that was resucited a little time ago by Priest3 I would like to pose a particular case that makes it look even more paradoxical, in a way that I wish it would help us to find another way out different of the one slightly proposed by Burke. The only thig he finds there to doubt about is the continuity of space, which for me, as I will shortly argue, is out of question.
Burke, before formalizig it (something that I think we should not need here), explains the paradox in the following way:
Point -1 is one meter west of point 0, which is one meter west of point 1. The ground between -1 and 1 is smooth and level. A ball at -1 is rolling eastward with sufficient momentum to reach 1 and beyond, if nothing (other than friction) impedes it. But rising from the ground between 0 and 1 (as they have from all eternity) are infinitely many barriers. Specifically, there are barriers at points ½, ¼, _, and so on. (The barriers are equal in height and width, but they differ in thickness. The barrier at ½ is one centimeter thick. Each of the other barriers is half as thick as the first barrier to its east.) Each barrier is strong enough to stop the rolling ball. (This does not seem problematic logically. But if it were, we could replace the ball with a massless particle, such as a photon.) Now here is the problem: It seems obvious that the ball cannot progress beyond point 0, since to do so it would have to get past an infinitude of barriers, none of which it is able to get past. But since there is no first barrier, the ball does not reach any barrier (since it can't get past the preceding barriers) and thus is not stopped by any barrier. But there's nothing to stop the ball other than a barrier. And it may be assumed, in accordance with Newton's first law, that the ball will not stop unless something stops it. Thus we arrive at a contradiction — and a paradox.
The version I will propose goes like this: let us supose that the first barrier begins in 1/2 and ends in 1/4 (its isn’t 1 cm thick, but 1/4 m thick), the second one goes from 1/4 to 1/8, the third from 1/8 to 1/16 and so on … . Here we have the same conditions of the paradox: a series of infinite barriers without a first one, so that when the ball comes from –1, it doesn’t find a first barrier to reach and to get stopped by. This is the way it looks when we get too close, but if we look it with a little more of distance we will find that, instead of the series of infinite barriers, there is just one barrier made of contiguos barriers. We get that looking it in one way there is no fist barrier for the ball to reach, and in the other way we get a solid wall that would stop the ball just when it gets to 0.
The thing that called my attention was the note where Burke states that "Benardete's paradox will stand as a substantial challenge to a presupposition of (1): the continuity of the spatial continuum". I have been allways challenged by the posibility of doubting about the continuty of space, if it wasn’t continuos then: where would then be the "discontinous" pieces? in "another" space, this time continuous? or we should go this way ad infinitum? But solely the first step, asking for "another" space is absurd, because space is the "totality" of spaces, and we won’t find any space out of it.
I will try to deviate the point of view: the problem is not the continuity, but of our comprehension of continuity and the problems that come with it. For me, the concepts we are challenged to think about by the paradox are those of "unity" and "contiguity" (contact). ¿What does it mean to be "one"? ¿What does mean to be "in contact"?
Any object that is "one", with no need of going to far, any concret wall, is composed by a series of other objets (other unities) of minor size, in the specific case of the wall this "components" would be grains of sand and cement, wich themselves are also composed of other things, and so on ad infinitum. What does it mean that a wall goes to the border? That for any given x > 0 , the distance between the wall and the border is less than x. So we get that contiguity is nothing different that a distance less that any finite distance given. In my version of the paradox it is evident that the ball gets to be in contact with the wall, the challenge is to think that in Burke’s version it does the same way, and that the series of barries constitute "one" barrier reached by the ball in 0.
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