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.
 |
