Therefore,
Is the argument valid? Is it sound (i. e., are all the premises true)?
Aristotle rejects premise (2) on the grounds that Zeno has failed to distinguish between infinite divisibility and infinite extension. How might one justify (2)? Aristotle suggests the following argument:
Therefore,
Aristotle rejects this argument in terms of the distinction between infinite divisibility and infinite extension. What then, he asks, is the justification for (2)?
But, Aristotle's argument is not quite right, since the infinite divisibility of the motion is reduced to the infinite divisibility of the temporal interval of the motion. If this interval is infinitely divisible then it has an infinite number of parts and to get from one moment to the next requires an infinite series of steps. Aristotle introduces a new distinction between potential and actual infinities. Infinitely divisible intervals are only potentially infinite.
But, the fact remains that the interval (either of motion or time or space)
contains an infinite sequence of non-zero intervals.
The following Zenoesque argument for (2) can be constructed:
(Z1) Every interval can be subdivided into an infinite number of non-zero parts.
(Z2) To reconstruct the interval we would have to "stitch together" all the parts.
(Z3) The sum of an infinite number of non-zero parts is infinite, i. e., can never be completed.
Therefore,
(Z4) No intervals can be constructed.
Mathematical techniques developed in the 19th century (2500 years or so after Zeno first put forth the paradoxes) suggest that (Z3) is false. This argument for (2) fails. But, we have no argument that (2) is false. What, then, if anything is wrong with the argument?
(1) For every i, if Achilles is at T(i), then the tortoise is at T(i+1).
Therefore,
(C) For every point P on the race course, if Achilles is at P, the tortoise
is at some point P" such that P" is farther from A(0) than P.
In order to derive (C) from (1), Zeno needs another premise
Therefore,
Therefore,
This is a modern restatement of the paradox as it has been passed down to us from Aristotle.
Aristotle challenges premise (3) and what he takes to be the atomistic assumption underlying the argument. For Aristotle, an object, to be at rest, must remain in the same position for a certain non -zero interval of time. For Aristotle, there is no "rest at an instant." Aristotle's objection to premise 3 then, is that, at an instant, no object is either at rest or moving. Both these predicates are only applicable to objects considered over finite intervals of elapsed time.
Even if you do not accept Aristotle's argument (from "ordinary Greek," as it were) the argument, as formulated is invalid. The conclusion (5) does not follow from (4). This inferential step commits what is called the fallacy of composition (whatever is true of all the parts of a thing is true of the whole).
But Zeno is not down and out yet. The fact that one way of formulating a line of reasoning is invalid does not mean that all formulations of the line of reasoning are invalid. Zeno, were he alive today, might well respond by offering the following reformulation of his view.
A Valid Reconstruction of the Arrow.
Therefore,
This reconstruction of the argument IS VALID. Since the conclusion, presumably, is unacceptable either premise 6 or premise 7 must be false. But, which one and why?
Note that this formulation implicitly rejects Aristotle's argument that one cannot meaningfully talk of rest (or motion) at an instant.
SOLUTION (Uncover only after you have given up or think you have the answer)
From Aristotle:
The fourth is an argument concerning two rows with an equal number of bodies
all of equal length, the rows extending from the opposite ends of the stadium
to the midpoint and moving in opposite directions with the same speed; and
the conclusion in this argument, so Zeno thinks, is that the half of an
interval of time is equal to its double . . . For example, let {A1,A2,A3,A4}
be a set of stationary bodies all of equal length, {B1,B2,B3,B4} another equal
set of moving bodies starting on the right from the middle of the A's and
having lengths equal to the A's, and {C1,C2,C3,C4} a third equal set with speed
equal to and contrary to that of the B's, also of lengths equal to those
of the A's and ending on the right with the end of the stadium
This is illustrated in the Figure.
Now as the B's and the C's pass over one another, B1 will be over C4 at
the same time that C1 will be over B4, shown in the following Figure.
From (6), (7) and (8), we conclude
But, this is impossible. Therefore,
Aristotle rightly focusses on (8) as suspect and he argues that it is false. But, he does not pick up on the significance of relative motion. No one does until the 17th century AD.
Nonetheless, the argument can be reconstrued as an argument against the
view that spatial and temporal magnitudes can be decomposed into atoms.
The first seven premises are the same. The argument proceeds
Therefore,
But, (10') contradicts the assumption that T is an atomic (i. e., indivisible)
unit of time. Therefore,
Does this argument rule out indivisible units of space and time? If so,
then it seems to conflict with modern quantum theory which suggests that
space and time might be quantized - i. e., be composed of minimal indivisible
units.
1)The magnitude of any finite extended interval is 0.
2)The interval can be infinitely subdivided.
3)The ultimate parts produced by this division, once completed, will
4)If 3a, then the sum of the parts will be an interval of infinite magnitude.
5)If 3b, then the sum of the parts will be an interval of zero magnitude.
