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DIR Return to: Greek Philosophy: The Pre-Socratics and Plato
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#Post#: 112--------------------------------------------------
Sample Answer: Zeno
DIR By: CaseyEnos
Date: August 8, 2012, 8:18 pm
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Another essay I wrote while rehearsing for last years exams; and
actually used on the test. Every year Zeno comes up, with a
slightly different twist on the question: for instance, can Zeno
teach us anything positive about motion; does it dissolve Zeno's
paradoxes to hold that space can be divided infinitely, and a
few others.
"Can we we learn anything positive about the nature of motion
from Zeno's paradoxes?"
In treating Zeno's paradoxes, possibly the single most
satisfying answer was that of Doignes the Cynic, who refuted
Zeno by walking around the room. In declaring movement to be
impossible, Zeno is clearly running afoul of empirical evidence,
and it is therefore unlikely that his paradoxes have much to say
on the nature of movement. However, such hasty answers miss the
opportunity to examine exactly where the fault in his reasoning
lies, and therefore to learn a great deal about the tools and
models with which we examine movement.
Poor treatment of dynamic processes is symptomatic o f classical
Greek thought, with its focus on understanding static states. To
Zeno, a finite section, be it of time, motion of space, can only
be divided into finite sized chunks, although he held, oddly
that the division can be performed an infinite number of times,
leading directly to a paradox. Using a system of mathematics
without the concept of zero and with an undeveloped idea of
infinity, Zeno could not do other than generate a paradox in his
treatment of both physical extension and motion. The infinitely
small points of modern calculus were as unavailable to him as
to any of his contemporaries.
Modern treatment of Zeno, as most famously presented by Russell,
hinges on the idea of an infinite number of points which can be
fitted, in a gap-free manner, on a line finite extension so that
the addition or subtraction of points cannot cause an alteration
in the size of the line. Importantly the modern calculus based
solution is primarily advocated over Zeno's presentation or
classical attempts at solutions, largely because of the success
at calculus in building scientific models describing motion. No
other set of solutions has provided such support to theories
describing empirical reality, surely an indication of which
direction practical handling of the nature of motion may lie.
Zeno's most discussed paradoxes are "The Dichotomy" and "The
Achilles". In the Dichotomy, a runner must complete a course
from point A to B. In order to reach B, she first must traverse
1/2 the distance, then 3/4, then 7/8...Each diminishing fraction
must take a finite amount of time to complete; presumably as
there is no lower limit to the fractions remaining yet each
requires a finite time to traverse, the result is an ever
growing time spiraling to infinity to traverse a finite space.
The same treatment can actually be applied to the runner's first
step, so that the course can not only be completed, it will not
even be started. The Achilles, involving the efforts of a swift
runner to overtake a tortoise who enjoys a head start, is
essentially the same paradox with a more developed story line.
Each time the runner arrives at the animal's previous position,
it will have crawled ahead, resulting in an ever-diminishing gap
that cannot be closed.
A standard solution is point out that the continuously
decreasing fractions will sum to 1;g with the along with the
diminishing amount of time required to traverse each. Such a
treatment is not entirely satisfactory, although it does point
in the right direction. Zeno is not dealing with a completed
division which can be summed to one, but with an ongoing and
never completed one that never reaches the sum of one.
Aristotle famously refuted to even consider the division into
points, addressing the fact that lines may be potentially, or
mathematically, divisible, yet real distances in nature do not
consist of points or fractions. In this assertion he was clearly
correct an established an extremely salient point about the
limitations of human ability to conduct accurate analysis of
nature; however in taking such a route he also closes off any
access to the practical application of mathematics to analyze
movement. Building the modern edifice of physics on Aristotle's
foundations would be simply impossible...
Another potential solution, similar to the calculus based one,
to regard the division of the line and the time required to
traverse it as already complete; this time into infinitely small
units, or points. As before, the runner is faced with an
infinity of tasks to complete, however for each task she has a
corresponding infinitely small fragment of time in which to
complete it. Both series sum to one, in the same way a finite
line can be divided into an infinity of points and a decimal
number between zero and be assigned to each point. The key is to
regard the division as already completed, not as one that occurs
continuously as the runner moves and thus never arrives at its
sum.
The completion of "super-tasks" of course remains controversial
and has spawned a great deal of philosophical literature, most
famously "Thompson's Lamp". This lamp is equipped with a toggle
switch which is flicked from "on" to "off" an infinite number of
times in a minute, and the question, probably unanswerable, is
posed of which position it comes to rest at after the minute.
However, for our purposes a better analysis is not of the end
position of the switch, but the motion of the switch as it
traverses an infinite number of points back forth in its
movement. Under such an analysis, more relevant to consideration
of Zeno, a super-task is performed every time a light switch is
turned on, indeed every time this pen draws a line on the paper
(*note: the actual exam paper was, of course, hand written).
While not exactly resolving the problem of super-tasks, it does
shed light on the irrelevance of the question to that of motion.
The third paradox of motion reported in the doxology is that of
"The Arrow", which when shot from the bow will occupy a space
equal to itself at any "instant". Since time consists only of
instants, motion is impossible as the arrow never moves from the
space it is occupying.
Again, Aristotle makes a correct answer, that the division into
instants is artificial, yet again it is an answer that does not
allow anything further to be learned. A more enlightening answer
is that at every instant the arrow possesses "speed", that is
the quality of movement, a characteristic which is carried over
from instant to instant. Commentators have pointed out that in
modern terms, by assigning a value of 0 to the arrow's speed at
any instant, Zeno is effectively dividing by zero and thus
rendering the analysis of the arrow's movement meaningless.
Another solution, similar to that of the runner, is to look at
the arrow's flight as a whole, divided into an infinity of
points, each one of which is traversed at each instantly, of
which there are also an infinity. Since the points in both time
and space are completely gap-less, the result would be a smooth
flight.
The final paradox of motion, and perhaps the easiest to resolve,
is that of "the stadium", or "moving bands". As reconstructed by
Aristotle, three bands of equally sized objects are presented,
two moving parallel to one another in opposite directions, and
both passing an unmoving one. The paradox is generated by the
fact that the two moving bands pass each other more quickly than
the third:
(*here there was a hand drawn diagram difficult to reproduce in
type, but similar to
T1: AAA
BBB
CCC
T2:
AAA
BBB
CCC
with A stationary, B moving to the right and C to the left)
In the diagram, the leading edge of the two moving bands have
passed three units of one another, and only two of the
stationary one.
This paradox is usually regarded as simply a serious mishandling
on Zeno's part of the notion of relative speed. More charitable
reconstructions involve the motion of point-particles or quantum
style "jumping", unfortunately there is very little in the
doxology to support such reconstructions and Zeno lacked the
technical equipment to postulate points of zero size. Therefore
the moving bands, at least as Zeno presented them, is probably
the least edifying if the four paradoxes of motion.
As a whole, we can learn a great deal more about motion really
consists of from Aristotle and his successors, such as Aquinas,
who denied the actual divisibility of time and space, than from
Zeno. In maintaining the use of calculus to dispel the
paradoxes, we must be careful to remember that we are operating
with an approximate tool. Mathematics is internally consistent
only because of the definitions of the units and operators
involved, just as Zeno's text works only because of the
definitions in the terms used. The most important lesson may be
simply that nature contains motion and it does not contain
points, fractions or calculus.
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