The **sequence**:

1/2, 1/4, 1/8, 1/16, . . . .

has infinitely many terms, each of which is of finite size greater than zero.
The sequence has no *last* term, but it does have a **limit**, namely
zero. (Roughly: no matter how close to zero you want to get, there is a term
in the sequence that gets you at least that close.)

If we add together all the terms in the sequence above, we get an infinite
**series**:

1/2 + 1/4 + 1/8 + 1/16 + . . . .

To calculate the sum of this series, one must consider the infinite sequence
<**S**_{1}, **S**_{2}, **S**_{3}, . .
. .>, where:

S_{1}= 1/2

S_{2}= 1/2 + 1/4

S_{3}= 1/2 + 1/4 + 1/8etc.

<**S**_{1}, **S**_{2}, **S**_{3}, . .
. .> is called the **sequence of partial
sums** of the infinite series 1/2 + 1/4 + 1/8 + . . . .

Our sequence of partial sums is thus:

1/2, 3/4, 7/8, 15/16, . . . .

This sequence also has a limit, namely 1.

The sum of an
infinite series is defined as the limit (where one exists) of the sequence
of partial sums. |

Therefore, our infinite series has a finite sum. That is:

1/2 + 1/4 + 1/8 + . . . . = 1.

Zeno seems, in some sense, to recognize that this is so, since he concedes
that a race-course of length **d** can be divided into infinitely many
parts, 1/2**d** + 1/4**d** + 1/8**d** + . . . . That is, he concedes
that

d/2 +d/4 +d/8 + . . . =d.

But there is evidence that he *also* thought that anything divisible
into infinitely many parts must be infinitely large. (Cf. the second limb
of the argument against plurality [as recorded by Simplicius, in DK1].) Hence
he might have felt entitled to the conclusion that to traverse a finite
race-course one must travel an infinitely great distance.

Is Zeno simply confused? Not necessarily. For he may think that this bizarre
conclusion is one that his opponent (the pluralist), but not he, must accept.
If, as Zeno thinks, the very notion of plurality is incoherent, it would
not be surprising for him to think that a plurality (if there were one) would
have to be *both* finite and infinite.

Return to lecture
on Zeno against plurality

Go to next
lecture on Zeno’s race course, part 2

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Copyright © 2002, S. Marc Cohen