BACK Index of Development Graphic Version

## ZENO(ca.450 B.C)

 < Zeno's Paradoxes >     Should we assume that a magnitude is infinitely divisible or that it is made up of a very large number of small indivisible atomic parts?  The first assumption appears the more reasonable to most ofus, but the utility of the second assump tion in the making of discoveries causes it to lose some of its seeming absurdity. There is evidence that, in Greek antiquity, schools of mahtematic reasoning developed that employed each of the above two assumptions.     Some of the logical difficulties encoumtered in either assumption were strikingly brought out in the fifth century B.C. by some paradoxes devised by the Eleatic philosopher Zeno (ca.450 B.C).  These paradoxes, which have had a profound influence on mathimatics, assert that motion is impossible whither we assume a magnitude to be infinitely divisible or to be made up of a large number of atomic parts.  We illustrate the nature of the paradoxes by the follow ing two.     The Dichotomy:If a straight line segment is infinitely divisible, then mo tion is impossible, for in order to traverse the line segment it is necessary first to reach the midpoint, and to do this one must first reach the one-quarter point, and to do this one must first reach the one-eighth point, and so on, ad infinitum.   If follows that the motion can never even begin.     The Arrow:If time is made up of indivisible atomic instants, then a moving arrow is always at rest, for at any instant the arrow is in a fixed position.  Since this is true of every instant, it follows that the arrow never moves.     Many explantions of Zino's paradoxes have been given, and it is not difficult to show that they challenge the common intuitive bilidfs that the sum of an infinite number of positive quantities is infinitely large, even if each quantity is extremely small (¥Ò =©û¥å =¥ö),and that the sum of either a finite or an infinite number of quantities of dimension zero is zero (¥ç¡¿0 = 0 and ¥ö¡¿0 = 0).  Whatever might have been the intended motive of the paradoxes, their dffect was to exclude infinitesimals from Greek demonstrative geometry.