Zeno’s paradox, solved again Page 1 of 1
Stephen Berer
[email protected] 9/22/12, 6 Tishray 5773
Zeno’s Paradox, another solution Meditating the other day, maybe 9/20, I solved, again, Zeno’s paradox. But this time the solution is mathematical, not logical, as in some of my other solutions. The paradox is more accurately just a mathematical equation with a limit solution, and that limit solution, by the equation’s own definition, assures that Hercules cannot surpass the turtle. Why? Because the turtle’s position is the limit solution, arbitrarily set at the outset! The paradox appears to address the problem of infinite regression, and the infinite divisibility of time and space, but these matters are only secondary details in an equation that postulates the turtle’s position as the endpoint of the series. At first, as I thought about this, I wanted to quantize space and time, in a quantum mechanical sense, but for space/time measured in meters and seconds rather than billionths of a meter/second. Interesting, but unnecessary for Zeno’s problem, and probably untrue or at least unnecessary, in general. Thus, imagine a football field, which is how I’ve always imagined it, with Hercules on the field’s one endzone, and the turtle on the 10 yard line at the other end of the field, 90 yards ahead of Hercules. For every 10 yards Hercules runs, the turtle runs 1 yard. If Hercules is allowed to run for as long as he wants, in 10 seconds he passes the turtle. But NO! Our equation doesn’t allow him to run as long or as far as he wants. The equation continually postulates that the distance Hercules must run is the distance from him to the turtle, and each of these distances become terms in an infinite series, the limit being the integral of the series. So, first Hercules must run 90 yards, but the turtle, in that time, has run 9 yards. So now Hercules must run 9 yards, but in that time the turtle runs .9 yards, and so on. The turtle is ever the end point, and our equation is represented by a graph that looks like a log function, rising sharply at first, then leveling off to nearly horizontal. The convergence point, which in an infinite series as this is, is only theoretical, and which in this problem is when Hercules catches up to the turtle (where the momentary velocity is 0) is easily computed, and just as easily intuited. It is the goal line that started out being 100 yards from Hercules and 10 yards from the turtle. Hercules can’t pass the turtle because the equation doesn’t allow him to! Had Zeno made a small adjustment in his logic and his math, setting the divisible distances not at the turtle’s nose but say, 10 yards past the far goal line, that is, the far end of the endzone, Hercules would have run his 110 yards, the turtle would have run her 11 yards, race over, Hercules wins with 9 yards to spare. Or suppose there is some other, moving object against which Hercules’s progress is measured (rather than the static far end of the endzone, which doesn’t require the sum of an infinite series), that, then becomes the equation’s limit-solution. If the turtle also runs that race, and depending on the turtle’s relative position to Hercules at the race’s start, if the race is long enough, Hercules will surpass the turtle, but of course, by definition, he will only reach, but never surpass, whatever it is Hercules is racing to reach. QED (altho, I suppose, after all, this is yet only another logical solution to the paradox).
