Zeno's Regressive Dichotomy and the 'Calculus Solution'

The Regressive Dichotomy is one of Zeno's paradoxes of motion. How can I get from point A, where I am, to point B, where I want to be? It seems I can't get started.

A_______1/8_______1/4_______________1/2_________________________________ B

To get from A to B, I must go halfway. But to travel halfway, I must first traverse half of the halfway distance, and thus 1/4 of the total distance. But to do this I must move 1/8 of the total distance. And so on. The sequence of runs I must complete in order to reach my goal has the form of an infinite regress with no first term:

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

Since there is no first term, I can't get started.

Let us consider the 'calculus solution' to the paradox. Consider the infinite sequence

1/2, 1/4, 1/8, . . . ,1/2ⁿ, . . . .

A sequence is said to be convergent if it has a limit. Roughly, the limit of a sequence is a number L such that the terms of the sequence become and remain arbitrarily close to L as we consider the terms in succession. Thus the limit of the sequence displayed above is 0, since as we run through the sequence, each term gets closer and closer to 0. The larger n becomes, the smaller is the difference between 1/2ⁿ and 0. The sequence converges to 0 as its limit.

We can also speak of the convergence of an infinite series, where an infinite series is the result of adding all the terms in an infinite sequence. But how do we add the terms in an infinite sequence? What meaning can we attach to 'sum of an infinite sequence of terms'? In standard analysis (calculus), the sum of an infinite sequence is the limit of the sequence of partial sums:

1/2, 1/2 + 1/4 = 3/4, 1/2 + 1/4 + 1/8 = 7/8, . . . .

This sequence of partial sums converges to the limit 1. Thus the sum of a convergent infinite series is a number that can be approximated arbitrarily closely by adding up sufficiently many finite terms.

Now how is this supposed to solve the Regressive Dichotomy paradox? Since the sum of an infinite series has been defined as the limit of the sequence of partial sums, and since this sequence converges to a finite limit (1 in our example), it is intelligible how infinitely many terms in a convergent series can have a finite sum. Note, however, that this intelligibility is a matter of stipulative definition: it is just stipulated that 'sum' in the case of infinite series shall have the meaning 'limit of a convergent sequence.' And since this stipulation is intelligible, it is supposed also to be intelligible how a runner can traverse the infinity of points lying between A and B.

The rub, however, is in applying the mathematical formalism, which I cannot fault, to the physical world. Our convergent infinite series has a sum, but only in the sense that the sequence of partial sums converges to the limit 1. The series does not have a sum in the sense of some number that is actually computed by someone or some computer. But to get from A to B I must actually complete an infinite sequence of runs. There is an infinite number of acts that I must perform assuming that space is composed of points. It is not enough that I converge upon point B; I must actually arrive there. In the mathematical case, however, it is enough that the infinite series converges upon the limit 1. There is no need that one actually compute up to 1.

Although I haven't expressed this as clearly as I ought to, my sense is that the 'calculus solution,' although fine as mathematics, is no solution to the Dichotomy paradox. One can render intelligible how an infinite series of terms can have a finite sum. But this does not render intelligible how an infinite series of acts can be performed in a finite time.

REFERENCE: Wesley C. Salmon, Space, Time, and Motion, Chapter 2.