Page 36 - GALIET PHYSICS BLOSSOMS I+

To Zeno this was a logical challenge, it led to an insuperable dead end, or what Kant called an “antinomy.” It lacked common sense. As a result, Zeno concluded that motion leads to paradoxes or the absurd.
Archimedes’ Infinitesimal
Archimedes explains the infinite series Zeno dislikes. For Archimedes, the idea of an infinitesimal (a finite quantity as small as desired, approaching or getting as close as to “zero” as desired, but never “zero”) was not a paradox, but a practical tool to figure out weighs, volumes, areas, etc. Indeed, one can subdivide things endlessly, infinitesimally, and never end in “zero” given that the intervals are always finite. Consequently, one always ends in 1/2n>0 and reaches 2 only when one subdivides things an infinite number of times.
Archimedes’ infinitesimals were instrumental in calculating areas and volumes and in understanding the differences between rational (fractions) and irrational numbers (can’t be written as fractions) for Archimedes’ series could be used to define an irrational number like pi. Archimedes evaluates the circle’s area by dividing it into successively smaller triangles to approximate pi. In other words, the infinitesimally smaller the segments, the greater the approximation. That is, as the number polygons increases to infinity, the difference in their areas decreases to approximate zero. Archimedes’ idea of dividing areas and volumes into increments, making them successively smaller were instrumental in finding areas and volumes and the center of gravities of things. They were the beginnings of physics and engineering, and they were an enormous aid in predicting the behavior of things. All in all, they anticipated calculus.
How does Archimedes resolve Zeno’s paradox?
Archimedes resolves Zeno’s paradox by finding a limit or summing up an infinite series. That is, one solves things by finding their limit through successive approximation to some quantity simply because a limit is needed before summing up the series. In other words, without limit, one cannot sum! To reiterate, Archimedes believed that even an infinite successive approximation to some quantity could limit things.
Thus, it is possible to construct an infinite series (S=1+1/2+1/4+1/8+....1/2n +....=2) summing up to a finite value as long as the terms decrease fast enough for a finite sum, that is, as long as the infinite series ends at a definite time. In this way, the series is infinite, but the sum is finite; hence the infinite sum is finite, not infinite. That is, Achilles traverses an
infinite series of temporal intervals summing up to a finite value. It will not take
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