that led to conclusions contradicting what we all know from our
physical experience–that arrows fly, that runners run, and that there
are many different things in the world. The arguments were paradoxes
for the ancient Greek philosophers. Because the arguments turn
crucially on the notion that space and time are infinitely divisible,
Zeno can be credited with being the first person in history to show
that the concept of infinity is problematical.
In his Achilles Paradox, the fastest runner of antiquity, Achilles,
races to catch a slower runner–for example, a tortoise that is
crawling slowly away from him. The tortoise has a head start, so if
Achilles hopes to overtake it, he must run at least to the place where
the tortoise is when he sees it, but by the time he arrives there, it
will have crawled to a new place, so then Achilles must run to this
new place, but the tortoise meanwhile will have crawled on, and so
forth. Achilles will never catch the tortoise, says Zeno, because the
series of goals has no final member. Therefore, good reasoning shows
that fast runners never can catch slow ones. So much the worse for the
claim that motion really occurs, Zeno says in defense of his mentor
Parmenides who had argued that motion is an illusion.
Although very few scholars today would agree with Zeno's conclusions,
we can not escape them by jumping up from our seat and chasing down a
tortoise. What is required is a principled way out that does not get
us embroiled in new paradoxes nor impoverish our mathematics and
science.
This article explains his nine known paradoxes and considers the
treatments that have been offered. Aristotle accused Zeno of not
paying sufficient attention to the fact that there are no actual
infinities but only potential infinities, and that durations divide
into intervals but never into indivisible instants, and his treatment
became the generally accepted solution until the late 19th century.
The Standard Solution that is favored by most philosophers today
employs the apparatus of calculus which has proved its
indispensability for the development of modern science. Its key ideas
are that space, time and motion are continua; that positions,
distances, times, durations and speeds all should be treated as
continuous variables whose values are real numbers or intervals of
real numbers; that an object can have a positive speed at a
point-place; and that some infinite series of positive terms can have
a finite sum. From this perspective, Zeno was working with false
assumptions. This Standard Solution took hundreds of years to perfect
and was due to the flexibility of intellectuals who were willing to
replace old theories and their concepts with more fruitful ones,
despite the damage done to common sense and our naive intuitions. The
article ends by exploring newer treatments developed since the 1960s.
1. Zeno of Elea
a. His Life
Zeno was born in about 490 B.C.E. in Elea, now Velia, in southern
Italy; and he died in about 430 B.C.E. He was a friend and student of
Parmenides, who was twenty-five years older and also from Elea. There
is little additional, reliable information about Zeno's life. Plato
remarked that Parmenides took Zeno to Athens with him where he
encountered Socrates, who was about twenty years younger than Zeno,
but today's scholars consider this encounter to have been invented by
Plato to improve the story line. Zeno is reported to have been
arrested for taking weapons to rebels opposed to the tyrant who ruled
Elea. When asked about his accomplices, Zeno said he wished to whisper
something privately to the tyrant. But when the tyrant came near, Zeno
bit him, and would not let go until he was stabbed. Diogenes Laërtius
reported this apocryphal story seven hundred years after Zeno's death.
b. His Book
According to Plato's commentary in his Parmenides (127a to 128e), Zeno
brought a treatise with him when he visited Athens. It was said to be
a book of paradoxes defending the philosophy of Parmenides. Plato and
Aristotle probably had access to the book, but Plato did not state any
of the arguments, and Aristotle's presentations of the arguments are
very compressed. A thousand years after Zeno, the Greek philosophers
Proclus and Simplicius commented on the book and its arguments. They
had access to some of the book, perhaps to all of it, but it has not
survived. Proclus is the first person to tell us that the book
contained forty arguments. This number is confirmed by the sixth
century commentator Elias, who is regarded as an independent source
because he does not mention Proclus. Unfortunately, we know of no
specific dates for when Zeno composed any of his paradoxes, and we
know very little of how Zeno stated his own paradoxes. We do have a
direct quotation via Simplicius of the Paradox of Denseness and a
partial quotation via Simplicius of the Large and Small Paradox. In
total we know of less than two hundred words that can be attributed to
Zeno. Our knowledge of these two paradoxes and the other seven comes
to us indirectly through paraphrases of them, and comments on them,
primarily by Aristotle (384-322 B.C.E.), but also by Plato (427-347
B.C.E.), Proclus (410-485 C.E.), and Simplicius (490-560 C.E.). The
paradox names were created by commentators, not by Zeno.
c. His Goals
In the early fifth century B.C.E., Parmenides emphasized the
distinction between appearance and reality. Reality, he said, is a
seamless unity that is unchanging and can not be destroyed, so
appearances of reality are deceptive. Our ordinary observation reports
are false; they do not report what is real. This metaphysical theory
is the opposite of Heraclitus' theory, but evidently it was supported
by Zeno. Although we do not know from Zeno himself what point he was
making with his paradoxes, according to Plato, the paradoxes were
designed to provide detailed, supporting arguments for Parmenides by
demonstrating that our common sense confidence in the reality of
motion, change, and ontological plurality, involve absurdities.
Plato's classical interpretation of Zeno was accepted by Aristotle and
most other commentators throughout the intervening centuries.
Eudemus, a student of Aristotle, offered another interpretation. He
suggested that Zeno was challenging both pluralism and Parmenides'
idea of monism, which would imply that Zeno was a nihilist. Paul
Tannery in 1885 and Wallace Matson in 2001 offer a third
interpretation of Zeno's goals regarding the paradoxes of motion.
Plato and Aristotle did not understand Zeno's arguments nor his
purpose, they say. Zeno was actually challenging the Pythagoreans and
their particular brand of pluralism, not Greek common sense. Zeno was
not trying to directly support Parmenides. Instead, he intended to
show that Parmenides' opponents are committed to denying the very
motion, change, and plurality they believe in, and Zeno's arguments
were completely successful. This controversial issue about
interpreting Zeno's purposes will not be pursued further in this
article, and Plato's classical interpretation will be assumed.
d. His Method
Before Zeno, Greek thinkers favored presenting their philosophical
views by writing poetry. Zeno began the grand shift away from poetry
toward a prose that contained explicit premises and conclusions. And
he employed the method of indirect proof in his paradoxes by
temporarily assuming some thesis that he opposed and then attempting
to deduce an absurd conclusion or a contradiction, thereby undermining
the temporary assumption.
2. The Standard Solution to the Paradoxes
A paradox is an argument that reaches a contradiction or other absurd
conclusion by apparently legitimate steps from apparently reasonable
assumptions, while the experts at the time can not agree on the way
out of the paradox, that is, agree on its resolution. It is this
latter point about disagreement among the experts that distinguishes a
paradox from a mere puzzle in the ordinary sense of that term. Zeno's
paradoxes are now generally considered to be puzzles because of the
wide agreement among today's experts that there is at least one
acceptable resolution of the paradoxes. This Standard Solution
presupposes calculus, the rest of classical real analysis, and
classical mechanics. It implies that motions, durations, distances and
line segments are all linear continua composed of points, then employs
these ideas to challenge various steps made by Zeno. A key background
assumption of the Standard Solution is that this resolution is not
simply employing some concepts that will undermine Zeno's
reasoning–Aristotle does that, too, at least for many of the
paradoxes–but that it is employing concepts which not only do that but
also are needed in the development of a coherent and fruitful system
of mathematics and physical science.
What are continua? Intuitively, a continuum is a continuous entity; it
is a whole thing that is smooth, having no gaps. Two abstract examples
are the path of a runner's center of mass and the time during this
motion. Two concrete examples of continua are oceans and metal rods
because treating them as continua is very useful for many calculations
in physics even though we know the objects are lumpy with atoms, and
thus discontinuous at the microscopic level. The ocean has the
continuous property of degree of salinity, and the rod has the
continuous property of temperature, so these properties are assigned
real numbers as values rather than merely fractions or integers.
The distinction between "a" continuum and "the" continuum is that
"the" continuum is the paradigm of "a" continuum. The continuum is the
mathematical line, which has the same structure as the real numbers in
their natural order. Real numbers are assigned one-to-one to its
points; there are not enough rational numbers for this assignment. For
Zeno's paradoxes, the most important features of any continuum are
that (a) it is undivided yet infinitely divisible, (b) it is composed
of points, (c) the measure (such as length) of a continuum is not a
matter of adding up the measures of its points nor adding up the
number of its points, (d) any connected part of a continuum is also a
continuum, (e) it is so dense that no point has any point next to it
since the distance between distinct points is always positive and
finite, and (f) the total distance traveled when crossing a convergent
series of point places is defined by an infinite sum.
Knowing a continuous object is infinitely divisible does not tell you
how many elements or points or ultimate parts it has, other than that
there are an infinite number. The Standard Solution says there are in
fact an aleph-one number of elements between any two elements in a
continuum.
Physical space is not a linear continuum because it is a
three-dimensional continuum. But it has one-dimensional subspaces such
as paths of runners and orbits of planets; and these are linear
continua if we use the line created by only one point on the runner
and only one point on the planet. Regarding time, each (point) instant
is assigned a real number as its time, and the duration of an instant
is zero. Well-defined events that are not instantaneous are assigned
an interval of real numbers rather than a single real number. For
example, the time taken by Achilles to catch the tortoise is an
interval, a linear continuum of instants, according to the Standard
Solution (but not according to Zeno or Aristotle).
Of the nine known paradoxes, The Achilles attracted the most attention
over the centuries. Aristotle's treatment of these involved accusing
Zeno of using the concept of an actual or completed infinity instead
of the concept of a potential infinity, and accusing Zeno of failing
to appreciate that a line cannot be composed of points. Aristotle's
treatment is described in detail below. It was generally accepted
until the 19th century, but slowly lost ground to the Standard
Solution. Some historians say he had no solution but only a verbal
quibble. This article takes no side on this dispute and speaks of
Aristotle's "treatment."
Why the slow acceptance of the Standard Solution? There are four
reasons. (1) It took time for the relative shallowness of Aristotle's
treatment to be recognized. (2) It took time for philosophers of
science to appreciate that each theoretical concept used in a physical
theory need not have its own correlate in our experience. (3) It took
time for the calculus of Newton and Leibniz and the rest of real
analysis to prove its applicability and fruitfulness in physics. (4)
It took time for certain problems in the foundations of mathematics to
be resolved.
Point (1) is discussed in section 4 below.
Point (2) is about the time it took for philosophers of science to
reject the demand, favored by many Logical Positivists, that
meaningful terms in science must have "empirical meaning." This was
the demand that each physical concept must be separately definable
with observation terms. It was thought that, because our experience is
finite, the term "actual infinite" could not have empirical meaning,
but "potential infinity" could. Today, most philosophers would not
restrict meaning to empirical meaning, but for an interesting
exception see Dummett (2000) which contains a theory in which time is
composed of overlapping intervals rather than of instantaneous or
durationless instants, and in which the endpoints of those intervals
are the initiation and termination of actual physical processes. This
idea of treating time without instants develops a proposal of Russell
and Whitehead from 1936.
The time in Point (3) is the time it took for classical mechanics, as
opposed to quantum mechanics, to develop to the point where it was
accepted as giving correct solutions to problems involving motion.
Point (3) was challenged in the metaphysical literature on the grounds
that the abstract account of continuity in real analysis does not
truly describe either time, space or concrete physical reality. This
challenge is discussed in later sections.
Regarding (4), the standard of rigorous proof and rigorous definition
of concepts has increased over the years. As a consequence, the
difficulties in the foundations of real analysis, which began with
George Berkeley's criticism of inconsistencies in the use of
infinitesimals in the calculus of Newton and Leibniz, were not
satisfactorily resolved until the early 20th century with the
development of Zermelo-Fraenkel set theory. The key idea was to work
out the necessary and sufficient conditions for being a continuum. To
achieve the goal, the conditions for being a mathematical continuum
had to be strictly arithmetical and not dependent on our intuitions
about space, time and motion. The idea was to revise or "tweak" the
definition until it would not create new paradoxes and would still
give useful theorems. When this revision was completed, it could be
declared that the set of real numbers is an actual infinity, not a
potential infinity, and that not only is any interval of real numbers
a linear continuum, but so are the spatial paths, the temporal
durations, and the motions that are mentioned in Zeno's paradoxes. In
addition, it was important to clarify how to compute the sum of an
infinite series and how to define motion in terms of the derivative.
This new mathematical system required new or better-defined
mathematical concepts of compact set, connected set, continuity,
continuous function, convergence of an infinite series, curvature at a
point, cut, derivative, dimension, function, integral, limit, measure,
reference frame, set, and size of a set. Similarly, rigor was added to
the definitions of the physical concepts of place, instant, duration,
distance, and instantaneous speed. The relevant revisions were made by
Euler in the 18th century and by Bolzano, Cantor, Cauchy, Dedekind,
Frege, Hilbert, Lebesque, Peano, Russell, Weierstrass, and Whitehead,
among others, during the 19th and early 20th centuries.
What about infinitesimals? In 1734, Berkeley had properly criticized
the use of infinitesimals in calculus as inconsistent. Earlier Newton
had defined instantaneous speed as the ratio of an infinitesimally
small distance and an infinitesimally small duration, and he and
Leibniz produced a system of calculating variable speeds that was very
fruitful. But nobody in that century or the next could adequately
explain what an infinitesimal was. Newton had called them "evanescent
divisible quantities," whatever that meant. Leibniz called them
"vanishingly small," but that was just as vague. The practical use of
infinitesimals was unsystematic. For example, they were sometimes
treated as being equal to zero, but at other times they were not
treated as zero because division by zero would produce trouble. In
addition, consider the seemingly obvious Archimedean property of pairs
of positive numbers: given any two positive numbers A and B, if you
add enough copies of A, then you can produce a sum greater than B.
This property fails if A is an infinitesimal. Finally, mathematicians
gave up on answering Berkeley's charges because, in 1821, Cauchy
showed how to achieve the same useful theorems of calculus by using
the idea of a limit instead of an infinitesimal. Later in the 19th
century, Weierstrass resolved some of the inconsistencies in Cauchy's
account and satisfactorily showed how to define continuity in terms of
limits (his epsilon-delta method). As J. O. Wisdom points out (1953,
p. 23), "At the same time it became clear that Newton's theory, with
suitable amendments and additions, could be soundly based." In an
effort to provide this sound basis according to the latest, heightened
standard of what counts as "sound," Peano, Frege, Hilbert, and Russell
attempted to properly axiomatize real analysis. This led in 1901 to
Russell's paradox and the fruitful controversy about how to provide a
foundation to all of mathematics. That controversy still exists, but
the majority view is that axiomatic Zermelo-Fraenkel set theory with
the axiom of choice blocks all the paradoxes and legitimizes Cantor's
theory of transfinite sets, and provides the proper foundation. This
is the mathematics that the Standard Solution applies to Zeno's
Paradoxes.
That solution recommends using very different concepts and theories
than those used by Zeno. The argument that this is the correct
solution was presented by many people, but it was especially
influenced by the work of Bertrand Russell (1914, lecture 6) and the
more detailed work of Adolf Grünbaum (1967). In brief, the argument
for the Standard Solution is that we have solid grounds for believing
our best scientific theories, but theories of mathematics such as
calculus and Zermelo-Fraenkel set theory are indispensable to these
theories, so we have reason to use and believe in the mathematical
concepts and objects of those mathematical theories, such as continua
composed of actual infinities of points. The theories require a
resolution of Zeno's paradoxes and the other paradoxes, and the
Standard Solution is indispensable to this resolution. Therefore, the
Standard Solution is correct. In the next section, this solution will
be applied to each of Zeno's paradoxes.
To be optimistic, the Standard Solution represents a counterexample to
the claim that philosophical problems never get solved. However, for
some commentators the Solution has its drawbacks and its alternatives,
and these have generated new and interesting philosophical
controversies beginning in the last half of the 20th century, as will
be seen in later sections. The primary alternatives contain different
treatments of calculus from that developed at the end of the 19th
century. Whether this implies that Zeno's paradoxes have multiple
solutions or only one is still an open question.
Did Zeno make mistakes? And was he superficial or profound? These
questions are a matter of dispute in the philosophical literature. The
majority position is as follows. If we give his paradoxes a
sympathetic reconstruction, he correctly demonstrated that some
important, classical Greek concepts are logically inconsistent, and he
did not make a mistake in doing this, except in the Moving Rows
Paradox and the Grain of Millet Paradox, two of his weakest paradoxes.
Zeno did assume that the classical Greek concepts were the correct
concepts to use in reasoning about his paradoxes, and now we prefer
revised concepts, though it would be unfair to say he blundered for
not foreseeing later developments in mathematics and physics. He did
make a mistake in concluding that motion and plurality are impossible,
and in concluding that the opponents of Parmenides have been refuted.
3. The Nine Paradoxes
Zeno probably created forty paradoxes, of which only the following
nine are known. Only the first four have standard names. They are of
uneven quality. Zeno and his ancient interpreters usually stated his
paradoxes badly, so it has taken some clever reconstruction over the
years to reveal their full force. Below, the paradoxes are
reconstructed sympathetically. This reconstruction is just one of
several reasonable schemes for presenting the paradoxes, but the
present article does not explore the historical research about the
variety of interpretive schemes and their relative plausibility.
a. The Achilles
The fleet-of-foot Achilles is racing to catch the tortoise that is
slowly crawling away from him. Both are moving along a linear path at
constant speeds. In order to catch the tortoise, Achilles will have to
reach the place where he presently sees the tortoise to be. However,
by the time Achilles gets there, the tortoise will have crawled to a
new location. Achilles will then have to reach this new location.
However, by the time Achilles reaches that location, the tortoise will
have moved on to yet another location, and so on forever. Achilles
will never catch the tortoise because the series of goals has no final
member, Zeno claims. However, if we do believe that Achilles succeeds
and that motion is possible, then we are victims of illusion, as
Parmenides says we are. The source is Aristotle (Physics 239b14-16).
There is no evidence that Zeno used a tortoise rather than a human
competitor for Achilles; the tortoise is a commentator's addition.
Aristotle spoke simply of "the runner" who competes with Achilles.
It won't do to react and say the solution to the paradox is that there
are biological limitations on how small a step Achilles can take.
Achilles' feet aren't obligated to stop and start again at each of the
locations described above, so there is no limit to how close one of
those locations can be to another. It is best to think of the change
from one location to another as a movement rather than as incremental
steps requiring halting and starting again.
The Standard Solution to the Achilles Paradox employs calculus and
other parts of real analysis to describe the situation. For example,
the path Achilles travels is a linear continuum and is composed of an
actual infinity of points. Achilles travels a distance d1 in reaching
the point x1 where the tortoise starts, but by the time Achilles
reaches x1, the tortoise has moved on to a new point x2. When Achilles
reaches x2, having gone an additional distance d2, the tortoise has
moved on to point x3, and so forth. This sequence of distances or
sub-paths is an actual infinity, a completed infinity, but happily the
sum of its terms d1 + d2 + d3 +… is a finite distance that Achilles
can readily cover while moving at a constant speed, because the series
of terms converges due to the fact that Achilles travels faster than
the tortoise.
b. The Dichotomy (Racetrack)
In his Progressive Dichotomy Paradox, Zeno argued that a runner will
never reach a fixed goal along the racetrack. The reason is that the
runner must first reach half the distance to the goal, but when there
he must then cross half the remaining distance, then half of the new
remainder, and so on. If the goal is one meter away, the runner must
cover a distance of 1/2 meter, then 1/4 meter, then 1/8 meter, and so
on ad infinitum. Because there is no final member of this sequence,
the runner will never reach the goal. Or, one might say, because there
are so many tasks to complete, the runner will never reach the goal.
This manner of speaking assumes that tasks can be arbitrarily short,
but some scholars believe this is an improper use of the term "task,"
and the discussion of this interesting point continues in Section 5c.
The problem of the runner getting to the goal can be viewed from a
different perspective. According to the Regressive version of the
Dichotomy Paradox, the runner cannot even take a first step. Here is
why. Any step may be divided conceptually into a first half and a
second half. Before taking a full step, the runner must take a 1/2
step, but before that he must take a 1/4 step, but before that a 1/8
step, and so forth ad infinitum, so Achilles will never get going. The
original distance between the runner and the goal is not relevant.
Like the Achilles Paradox, this paradox also concludes that any motion
is impossible. The original source is Aristotle (Physics, 239b11-13).
The Dichotomy paradox, in either its Progressive version or its
Regressive version, assumes for the sake of simplicity that the
runner's positions are point places. Actual runners take up some
space. But this is not a controversial assumption because Zeno could
have reconstructed his paradox by speaking of the point places
occupied by the tip of the runner's nose.
In the Dichotomy Paradox, the runner reaches the points 1/2 and 3/4
and 7/8 and so forth on the way to his goal, but under the influence
of Bolzano and Cantor, who developed the first theory of sets, the set
of those points is no longer considered to be potentially infinite. It
is an actually infinite set of points abstracted from a continuum of
points–in the contemporary sense of "continuum" at the heart of
calculus. And the ancient idea that the actually infinite sequence
1/2, 3/4, 7/8, … never converges had to be rejected in favor of the
new theory that it converges to 1. This is key to solving the
Dichotomy Paradox, according to the Standard Solution. It is basically
the same treatment as that given to the Achilles. The Dichotomy
Paradox has been called "The Stadium" by some commentators, but that
name is also commonly used for the Paradox of the Moving Rows.
c. The Arrow
Zeno's Arrow Paradox takes a different approach to challenging the
coherence of our common sense concepts of time and motion. As
Aristotle explains, from Zeno's "assumption that time is composed of
moments," a moving arrow must occupy a space equal to itself at any
moment. That is, at any moment it is at the place where it is. But
places do not move. So, if at each moment, the arrow is occupying a
space equal to itself, then the arrow is not moving at that moment
because it has no time in which to move; it is simply there at the
place. The same holds for any other moment during the so-called
"flight" of the arrow. So, the arrow is never moving. Similarly,
nothing else moves. The source for Zeno's argument is Aristotle
(Physics, 239b5-32).
The Standard Solution to the Arrow Paradox uses the "at-at" theory of
motion, which says that being at rest involves being motionless at a
particular point at a particular time, and that being in motion does,
too. The difference between rest and motion has to do with what is
happening at nearby moments. An object cannot be in motion in an
instant, but it can be motion at an instant in the sense of having a
speed at that instant, provided the object occupies different
positions at times before or after that instant so that the instant is
part of a period in which the arrow is continuously in motion. Zeno
would have balked at the idea of motion at an instant, and Aristotle
explicitly denied it, believing that all motion occurs only over a
duration of time, and that durations divide into intervals but never
into indivisible instants. However, in calculus, the derivative of
position x with respect to time t, namely dx/dt, is the arrow's speed,
and it has non-zero values at specific places at specific instants
during the flight. The speed during an instant or inan instant would
be 0/0 and so is undefined. Using these modern concepts, Zeno cannot
successfully argue that the speed of the arrow is zero at every
instant. Advocates of the Standard Solution conclude that Zeno's Arrow
Paradox has false assumptions and so is unsound.
d. The Moving Rows
It takes a body moving at a given speed a certain amount of time to
traverse a body of a fixed length. Passing the body again at that
speed will take the same amount of time, provided the body's length
stays fixed. Zeno challenged this common reasoning. According to
Aristotle (Physics 239b33-240a18), Zeno considered bodies of equal
length aligned along three parallel racetracks within a stadium. One
track contains A bodies (three A bodies are shown below); another
contains B bodies; and a third contains C bodies. Each body is the
same distance from its neighbors along its track. The A bodies are
stationary, but the Bs are moving to the right, and the Cs are moving
with the same speed to the left. Here are two snapshots of the
situation, before and after.
Diagram of Zeno's Moving Rows
Zeno points out that, in the time between the before-snapshot and the
after-snapshot, the leftmost C passes two Bs but only one A,
contradicting the common sense assumption that the C should take
longer to pass two Bs than one A.
Aristotle argues that the common sense assumption is fallacious.
Reading between the lines, his point is that the assumption fails to
pay attention to the fact that how long it takes to pass a body
depends on the speed of the body; for example, if the body is coming
towards you, then you can pass it in less time than if it is
stationary. Today's analysts agree with Aristotle's diagnosis, and
historically this paradox of motion has seemed weaker than the
previous three. This paradox is also called "The Stadium," but so is
the Dichotomy Paradox.
Some analysts, such as Tannery (1887), believe Zeno may have had in
mind that the paradox was supposed to have assumed that space and time
are discrete (quantized) as opposed to continuous, and Zeno intended
his argument to challenge the coherence of this assumption about space
and time. Well, the paradox could be interpreted this way. Then, if
the Cs were moving at a speed of, say, one atom of space in one atom
of time, the leftmost C would pass two atoms of B-space in the time it
passed one atom of A-space, which is a contradiction. Or else we'd
have to say that in that atom of time, the leftmost C somehow got
beyond two Bs by passing only one of them, which is also absurd.
Interpreted this way, Zeno's argument produces an interesting
challenge to the idea that space and time are discrete. However, what
is controversial is whether Zeno himself interpreted his paradox this
way, and most commentators believe he did not.
e. Limited and Unlimited
The previous four paradoxes of motion can all be seen as indirect
criticisms of plurality. This is because, if motion is considered to
be covering a plurality of places in a plurality of times, Zeno's
criticisms of motion are then criticisms of this particular sort of
plurality. Strictly interpreted, though, these criticisms of motion,
if successful, would not rule out other sorts of plurality. However,
Zeno also provided more direct criticisms of plurality. The first is
the Limited and Unlimited Paradox, also called the Paradox of
Denseness.
Suppose there exist many things rather than, as Parmenides would say,
just one thing. Then there will be a definite or fixed number of those
many things, and so they will be "limited." But if there are many
things, say two things, then they must be distinct, and to keep them
distinct there must be a third thing separating them. So, there are
three things. But between these, …. In other words, things are dense
and there is no definite or fixed number of them, so they will be
"unlimited." This is a contradiction, because the plurality would be
both limited and unlimited. Therefore, there are no pluralities; there
exists only one thing, not many things. This argument is reconstructed
from Zeno's own words, as quoted by Simplicius in his commentary of
book 1 of Aristotle's Physics.
According to the Standard Solution to this paradox, the weakness of
Zeno's argument can be said to lie in the assumption that "to keep
them distinct, there must a third thing separating them." Zeno would
have been correct to say that between any two physical objects that
are separated in space, there is a place between them, because space
is dense, but he is mistaken to claim that there must be a third
physical object there between them. Two objects can be distinct at a
time simply by one having a property the other does not have.
f. Large and Small
Suppose there exist many things rather than, as Parmenides says, just
one thing. Then every part of any plurality is both so small as to
have no size but also so large as to be infinite, says Zeno. His
reasoning for why they have no size has been lost, but many
commentators suggest that he'd reason as follows. If there is a
plurality, then it must be composed of parts which are not themselves
pluralities. Yet things that are not pluralities cannot have a size or
else they'd be divisible into parts and thus be pluralities
themselves.
Now, why are the parts of pluralities so large as to be infinite?
Well, the parts cannot be so small as to have no size since adding
such things together would never contribute anything to the whole so
far as size is concerned. So, the parts have some non-zero size. If
so, then each of these parts will have two spatially distinct
sub-parts, one in front of the other. Each of these sub-parts also
will have a size. The front part, being a thing, will have its own two
spatially distinct sub-parts, one in front of the other; and these two
sub-parts will have sizes. Ditto for the back part. And so on without
end. A sum of all these sub-parts would be infinite. Therefore, each
part of a plurality will be so large as to be infinite.
This sympathetic reconstruction of the argument is based on
Simplicius' On Aristotle's Physics, where Simplicius quotes Zeno's own
words for part of the paradox, although he does not say what he is
quotingfrom.
There are many errors here in Zeno's reasoning, according to the
Standard Solution. He is mistaken at the beginning when he says, "If
there is a plurality, then it must be composed of parts which are not
themselves pluralities." A university is an illustrative
counterexample. A university is a plurality of students, but we need
not rule out the possibility that a student is a plurality. What's a
whole and what's a plurality depends on our purposes. When we consider
a university to be a plurality of students, we consider the students
to be wholes without parts. But for another purpose we might want to
say that a student is a plurality of biological cells. Zeno is
confused about this notion of relativity, and about part-whole
reasoning; and as commentators began to appreciate this they lost
interest in Zeno as a player in the great metaphysical debate between
pluralism and monism.
A second error occurs in arguing that the each part of a plurality
must have a non-zero size. In 1901, Henri Lebesgue showed how to
properly define the measure function so that a line segment has
nonzero measure even though (the singleton set of) any point has a
zero measure. Lebesgue's theory is our current civilization's theory
of measure, and thus of length, volume, duration, mass, voltage,
brightness, and other continuous magnitudes.
Thanks to Aristotle's support, Zeno's Paradoxes of Large and Small and
of Infinite Divisibility (to be discussed below) were generally
considered to have shown that a continuous magnitude cannot be
composed of points. Interest was rekindled in this topic in the 18th
century. The physical objects in Newton's classical mechanics of 1726
were interpreted by R. J. Boscovich in 1763 as being collections of
point masses. Each point mass is a movable point carrying a fixed
mass. This idealization of continuous bodies as if they were
compositions of point particles was very fruitful; it could be used to
easily solve otherwise very difficult problems in physics. This
success led scientists, mathematicians, and philosophers to recognize
that the strength of Zeno's Paradoxes of Large and Small and of
Infinite Divisibility had been overestimated; they did not prevent a
continuous magnitude from being composed of points.
g. Infinite Divisibility
Shakespeare's Hamlet says, "I could be bounded in a nutshell and count
myself a king of infinite space." Are nutshells and other objects
infinitely divisible, theoretically? Consider the difficulties that
arise if we assume that an object theoretically always can be divided
into parts. There is a reassembly problem. Take a normal object that
is of finite non-zero size. Imagine cutting the object into two
non-overlapping parts, then similarly cutting these parts into parts,
and so on until the process of repeated division is complete. Assuming
the hypothetical division is "exhaustive" or does comes to an end,
then at the end we reach what Zeno calls "the elements." Here there is
a problem. There are three possibilities. (1) The elements are
nothing. In that case the original objects will be a composite of
nothing, and so the whole object will be a mere appearance, which is
absurd. (2) The elements are something, but they have zero size. So,
the original object is composed of elements of zero size. Adding an
infinity of zeros yields a zero sum, so the original object had no
size, which is absurd. (3) The elements are something, but they do not
have zero size. If so, these can be further divided, and the process
of division was not complete after all, which contradicts our
assumption that the process was already complete. In summary, there
were three possibilities, but all three possibilities lead to
absurdity. So, objects are not infinitely divisible.
Simplicius says this argument is due to Zeno even though it is in
Aristotle (On Generation and Corruption, 316a15-34, 316b34 and
325a8-12) and is not attributed there to Zeno, which is odd. Aristotle
says the argument convinced the atomists to reject infinite
divisibility. The argument has also been called the Paradox of Parts
and Wholes, but it has no traditional name.
The Standard Solution says we should ask Zeno to be clearer about what
he is dividing. Is it concrete or abstract? When dividing a concrete,
material stick into its components, we reach ultimate constituents of
matter such as quarks and electrons that cannot be further divided.
These have a size, a zero size (according to quantum electrodynamics),
but it is incorrect to conclude that the whole stick has no size if
its constituents have zero size. [Due to the forces involved, point
particles have finite "cross sections," and configurations of those
particles, such as atoms, do have finite size even if composed of
zero-size quarks and electrons.] So, Zeno is wrong here. On the other
hand, is Zeno dividing an abstract path or trajectory? If so, this
should be considered to be a continuum, and he is mistaken in his
discussion of "the elements" of the abstract objects. The abstract
object, say a line segment, is infinitely divisible, but the Standard
Solution rejects Zeno's assumption that a length of a line segment is
the sum of the lengths of its point-elements. Instead, the size
(length, measure) of any line segment is determined by a certain
distance function on the uncountably many ordered points of that
segment. This distance function retains the important idea behind
ordinary intuitions about distance, namely that the sum of the lengths
of non-overlapping line segments is the mathematical sum of the
lengths of the segments. You can break a line segment into separate
sub-segments whose lengths add up to the original length. Also, you
can break a line segment at a point, but even if you were to break it
at all its points, its length still would not be a simple sum of the
lengths of its points.
h. The Grain of Millet
There are two common interpretations of this paradox. According to the
first, when a bushel of millet (or corn) grains crashes to the floor,
it makes a sound. Since the bushel is composed of individual grains,
each individual grain also makes a sound, as does each thousandth part
of the grain. But we hear no sound for portions like a thousandth part
of a grain, and so we would hear no sound for an ultimate part of a
grain. So, how can the bushel make a sound if none of its ultimate
parts make a sound? The original source is Aristotle Physics
(250a.19-21).
We do not have Zeno's words on what conclusion we are supposed to draw
from this. Perhaps, he would conclude that we are mistakenly supposing
that whole bushels of millet have millet parts. This is an attack on
plurality. The Standard Solution to the paradox says Zeno mistakenly
assumes that there is no lower bound on the size of something that can
make a sound. There is no problem, we now say, with parts having very
different properties from the wholes that they constitute. Zeno is
committing both the fallacy of division and the fallacy of
composition.
Some analysts interpret Zeno's paradox a second way, as challenging
our trust in our sense of hearing. When a bushel of millet grains
crashes to the floor, it makes a sound. The bushel is composed of
individual grains, so they, too, make an audible sound. But if you
drop an individual millet grain or a small part of one, your hearing
detects no sound, even though there is one. Therefore, you cannot
trust your sense of hearing.
This reasoning about our not detecting low amplitude sounds is similar
to making the mistake of arguing that you cannot trust your
thermometer because there are some ranges of temperature that it is
not sensitive to. One reason given in the literature for believing
that this second interpretation of the fallacy is not the one that
Zeno had in mind is that Aristotle's criticism given below applies to
the first interpretation and not the second, and it's unlikely that
Aristotle would have misinterpreted this paradox.
i. Against Place
Given an object, we may assume that there is a single, correct answer
to the question, "What is its place?" Because everything that exists
has a place, and because place itself exists, so it also must have a
place, and so on forever. That's too many places, so there is a
contradiction. The original source is Aristotle'sPhysics (209a23-25
and 210b22-24).
The standard response to Zeno's Paradox Against Place is to deny that
places have places, and to point out that the notion of place should
be relative to reference frame. But Zeno's assumption that places have
places was common in ancient Greece at the time, and Zeno is to be
praised for showing that it is a faulty assumption.
4. Aristotle's Treatment of the Paradoxes
Aristotle's views about Zeno's paradoxes can be found in Physics, book
4, chapter 2, and book 6, chapters 2 and 9. About the Achilles
Paradox, Aristotle said, "Zeno's argument makes a false assumption in
asserting that it is impossible for a thing to pass over…infinite
things in a finite time." Aristotle believed that if one added the
lengths of all the potentially infinite steps taken by Achilles in
catching the tortoise, this should be a finite sum and thus not
prevent Achilles from achieving his goal in a finite time. Not having
the Standard Solution's concept of a convergent infinite series having
a finite sum, Aristotle achieved finiteness of the sum by saying
Zeno's key mistake in the Achilles and the Dichotomy paradoxes was to
use the notion of an actual infinity of sub-paths and sub-goals. If
Zeno had properly confined himself to a "potential infinity," then the
argument of the paradoxes of motion would not have produced any
contradiction because potential infinities always have a finite number
of terms at any time. Here is how Aristotle expressed the point:
For motion…, although what is continuous contains an infinite
number of halves, they are not actual but potential halves. (Physics
263a25-27). …Therefore to the question whether it is possible to pass
through an infinite number of units either of time or of distance we
must reply that in a sense it is and in a sense it is not. If the
units are actual, it is not possible: if they are potential, it is
possible. (Physics 263b2-5).
Actual infinities, if they were to exist, would exist all at once.
Potential infinities exist over time, as processes that can always be
continued at a later time. Zeno made the mistake, said Aristotle, of
conceiving of the continuous path taken by Achilles as being composed
of an actual infinite aggregate of sub-paths, and Zeno envisioned the
whole as dependent on these parts. That's the mistake, says Aristotle.
Instead, the whole path is there, and then the analyst envisions a
process of potentially dividing the whole into its parts. In reality,
the path is given first, and it is continuous, whole, and finite. The
potential infinity of sub-paths are created over time by the analyst,
and at no time is there an actual infinity of sub-paths marked out in
reality beyond the analyst's mind, yet Zeno needs this actual infinity
in order to complete his argument. If we reject actual infinities, we
have a way out of these paradoxes, claims Aristotle. Notice that
Aristotle is using the word "potential" in a special sense because a
potential president can later become an actual president, but a
potential infinity cannot become an actual infinity.
From what Aristotle actually says, one can infer between the lines
that he believes there is another reason to reject actual infinities:
doing so is the only way out of the paradoxes. Today we know better.
There is another way out–the Standard Solution that uses actual
infinities.
Aristotle's treatment by disallowing actual infinity while allowing
potential infinity was clever, and it satisfied nearly all scholars
for 1,500 years. George Berkeley, Immanuel Kant, Karl Friedrich Gauss,
and Henri Poincaré were influential defenders of potential infinity.
Leibniz accepted actual infinity, but other mathematicians and
physicists in European universities were careful to distinguish
between actual and potential infinities and to avoid using actual
infinities.
Given 1,500 years of opposition to actual infinities, the burden of
proof was on anyone advocating them. Georg Cantor accepted this burden
in the last quarter of the 19th century. He argued that any potential
infinity must be interpreted as varying over a predefined set of
possible values, a set that is actually infinite. He put it this way:
In order for there to be a variable quantity in some mathematical
study, the "domain" of its variability must strictly speaking be known
beforehand through a definition. However, this domain cannot itself be
something variable…. Thus this "domain" is a definite, actually
infinite set of values. Thus each potential infinite…presupposes an
actual infinite. [Bihand Till Koniglen Svenska Vetenskaps Akademiens
Handigar 11 (19), (1886)]
Dedekind's 1872 axiom of continuity and his definition of real numbers
as certain infinite subsets of rational numbers suggested to Cantor
and then to many other mathematicians that arbitrarily large sets of
rational numbers are most naturally seen to be subsets of an actually
infinite set of rational numbers. The same can be said for sets of
real numbers.
Cantor's primary contribution to our topic was to give the first
rigorous definition of actual infinity, showing that the notion is
useful and not self-contradictory. Cantor provided the missing
ingredient–that we need a dense linear ordering of uncountably many
points.
These ideas now form the basis of modern real analysis. The
implication for the Achilles and Dichotomy paradoxes is that, once the
rigorous definition of a linear continuum is in place, and once we
have Cauchy's rigorous theory of how to assess the sums of infinite
series, then we can be confident that the sequence of intervals or
paths described by Zeno is most properly treated as a sequence of
subsets of an actually infinite set, and we can be confident that
Aristotle's treatment of the paradoxes is inferior to the Standard
Solution's.
To summarize the above discussion, Zeno said Achilles cannot achieve
his goal in a finite time because there are too many goals to achieve
along the way, that is, too many sub-paths to cover. Aristotle
objected that Achilles achieves his goal in a finite time by covering
only a potential infinity of paths, but an advocate of the Standard
Solution disagrees and says Achilles achieves his goal by covering an
actual infinity of paths. (The discussion of whether Achilles can
properly be described as completing an actual infinity of tasks will
be considered in Section 5c.)
Let's turn to the other paradoxes. In proposing his treatment of the
Paradox of the Large and Small and of the Paradox of Infinite
Divisibility, Aristotle said that
…a line cannot be composed of points, the line being continuous
and the point indivisible. (Physics, 231a 25)
In modern real analysis, a continuum is composed of points, but
Aristotle claimed that a continuum cannot be composed of points.
Aristotle believed a line can be composed only of smaller,
indefinitely divisible lines and not of points without magnitude.
Similarly a distance cannot be composed of point places and a duration
cannot be composed of instants. This is one of Aristotle's key errors,
according to advocates of the Standard Solution. In addition to
complaining about points, Aristotelians object to the idea of an
actual infinite number of them.
In his analysis of the Arrow Paradox, Aristotle said Zeno mistakenly
assumes time is composed of indivisible moments, but "This is false,
for time is not composed of indivisible moments any more than any
other magnitude is composed of indivisibles." (Physics, 239b8-9) Zeno
needs those instantaneous moments; that way Zeno can say the arrow
doesn't move during the moment. Aristotle recommends restricting Zeno
to saying motion be divided only into a potential infinity of
intervals. That restriction implies the arrow's path can be divided
only into finitely many intervals at any time. So, at any time, there
is a finite interval during which the arrow can exhibit motion by
changing location. Therefore, Aristotle declares Zeno's argument is
based on false assumptions without which there is no problem with the
arrow's motion. However, the Standard Solution agrees with Zeno that
time can be composed of indivisible moments or instants, and it
implies that Aristotle has mis-diagnosed where the error lies in the
Arrow Paradox.
Aristotle's treatment of The Paradox of the Moving Rows is basically
in agreement with the Standard Solution to that paradox–that Zeno
didn't appreciate the difference between speed and relative speed.
Regarding the Paradox of the Grain of Millet, Aristotle said that
parts need not have all the properties of the whole, and so grains
need not make sounds just because bushels of grains do. (Physics,
250a, 22) And if the parts make no sounds, we should not conclude that
the whole can make no sound. It would have been helpful for Aristotle
to have said more about what are today called the Fallacies of
Division and Composition that Zeno is committing. However, Aristotle's
response to the Grain of Millet is brief but accurate by today's
standards.
In conclusion, are there two adequate but different solutions to
Zeno's paradoxes, Aristotle's Solution and the Standard Solution? No.
Aristotle's treatment does not stand up to criticism in a manner that
most scholars deem adequate. The Standard Solution uses contemporary
concepts that have proved to be more valuable for solving and
resolving so many other problems in mathematics and physics. Replacing
Aristotle's common sense concepts with the new concepts from real
analysis and classical mechanics has been a key ingredient in the
successful development of mathematics and science in recent centuries.
5. Other Issues Involving the Paradoxes
a. Consequences of Accepting the Standard Solution
There is a price to pay for accepting the Standard Solution to Zeno's
Paradoxes. The following–once presumably safe–intuitions or
assumptions must be rejected:
1. A continuum is not divisible into point elements.
2. The sum of an infinite series of positive terms is always infinite.
3. For each instant there is a next instant and for each place
along a line there is a next place.
4. A finite distance along a line cannot contain an actually
infinite number of points.
5. The more points there are on a line, the longer the line is.
6. It is absurd for there to be numbers that are bigger than every integer.
7. A one-dimensional curve can not fill a two-dimensional space.
8. A whole is always greater than any of its parts.
The loss of intuition (1) has caused the greatest stir, because so
many philosophers object to a continuum being constructed from points.
The Austrian philosopher Franz Brentano believed with Aristotle that
scientific theories should be literal descriptions of reality, as
opposed to today's more popular view that theories are idealizations
of reality. Continuity is something given in perception, said
Brentano, and not in a mathematical construction; therefore,
mathematics misrepresents. In a 1905 letter to Husserl, he said, "I
regard it as absurd to interpret a continuum as a set of points."
But the Standard Solution's point-set analysis of continua has
withstood the objections and demonstrated its value in mathematics and
mathematical physics. As a consequence, advocates of the Standard
Solution say we must live with rejecting the eight intuitions listed
above, and accept the counterintuitive implications such as there
being divisible continua, infinite sets of different sizes, and
space-filling curves. They agree with the philosopher W. V .O. Quine
who demands that we be conservative when revising the system of claims
that we believe and who recommends "minimum mutilation," but advocates
of the Standard Solution say no less mutilation will work.
b. Criticisms of the Standard Solution
Balking at having to reject so many of our intuitions, the 20th
century philosophers Henri-Louis Bergson, Max Black, Franz Brentano,
L. E. J. Brouwer, Solomon Feferman, William James, James Thomson, and
Alfred North Whitehead argued in very different ways that the standard
mathematical account of continuity does not apply to physical
processes, or is improper for describing those processes. Here are
their main reasons: (1) the physical world comes into existence in
chunks, (2) the actual infinite cannot be encountered in experience
and thus is unreal, (3) human intelligence is not capable of
understanding motion, (4) the series of tasks that Achilles performs
is finite and the illusion that it is infinite is due to
mathematicians who confuse their mathematical representations with
what is represented. (5) motion is unitary even though its spatial
trajectory is infinitely divisible, (6) treating time as being made of
instants is to treat time as static rather than as the dynamic aspect
of consciousness that it truly is, (7) actual infinities and the
contemporary continuum are not indispensable to solving the paradoxes,
and (8) the Standard Solution's implicit assumption of the primacy of
the coherence of the sciences is unjustified because coherence with a
priori knowledge and common sense is primary.
See Salmon (1970, Introduction) and Feferman (1998) for a discussion
of the controversy about the quality of Zeno's arguments, and an
introduction to its vast literature. This controversy is much less
actively pursued in today's mathematical literature, and hardly at all
in today's scientific literature. A minority of philosophers are
actively involved in opposing the rejection of one or more of the
eight intuitions listed in the previous section. The central
philosophical issue is whether the paradoxes should be solved by
assuming that a line is not composed of points but of intervals, and
whether use of infinitesimals is essential to a proper understanding
of the paradoxes. See below for more on this ongoing issue.
c. Supertasks and Infinity Machines
Zeno's Paradox of Achilles was presented as implying that he will
never catch the tortoise because the series of goals to be achieved
has no final member. In that presentation, use of the terms "task" and
"act" was intentionally avoided, but there are interesting questions
that do use those terms. In reaching the tortoise, doesn't Achilles
need to complete an infinite series of actions? In other words,
assuming Achilles does complete the task of reaching the tortoise,
does he thereby complete a supertask, that is, an actual infinity of
tasks in a finite time?
Bertrand Russell said "yes." Using Zeno's form of reasoning, he argued
that in principle you could perform a task in one-half minute, then
perform another task in the next quarter-minute, and so on, keeping up
this rate for a full minute. At the end of the minute, an infinite
number of tasks would have been performed, said Russell. In the
mid-twentieth century, Hermann Weyl, Max Black, and others objected,
and thus began an ongoing controversy about the number of tasks that
can be completed in a finite time.
That controversy has sparked a related discussion about whether there
could be a machine that can perform an infinite number of tasks. A
machine that can is called an infinity machine. In 1954, the
philosopher James Thomson described a lamp that is intended to be an
infinity machine. Let a machine switch it on for a half-minute; then
switch it off for a quarter-minute; then on for an eighth-minute; off
for a sixteenth-minute; and so on. Would the lamp be lit or dark at
the end of minute? Thomson argued that it must be one or the other,
but it cannot be either because every period in which it is off is
followed by a period in which it is on, and vice versa, so there can
be no such lamp, and the specific mistake in the reasoning was to
suppose that it is logically possible to perform a supertask. The
implication for Zeno's paradoxes is that, although Thomson is not
denying Achilles catches the tortoise, he is denying Russell's
description of Achilles' task as being the completion of an infinite
number of sub-tasks.
Paul Benacerraf (1962) complains that Thomson's reasoning is faulty
because it fails to notice that the initial description of the lamp
determines the state of the lamp at each period in the sequence of
switching, but it determines nothing about the state of the lamp at
the limit of the sequence. So, Thomson has not established the
impossibility of completing supertasks. Could some other argument
establish this impossibility? Benacerraf suggests that an answer
depends on what we ordinarily mean by the term "completing a task." If
the meaning does not require that tasks have minimum times for their
completion, then maybe Russell is right that supertasks can be
completed, he says; but if a minimum time is always required, then
Russell is wrong. Grünbaum objects to Benacerraf's reliance on
ordinary meaning. "We need to heed the commitments of ordinary
language," says Grünbaum, "only to the extent of guarding against
being victimized or stultified by them."
The Thomson lamp has generated a great literature in recent
philosophy. Here are some of the issues. What is the proper definition
of "task"? For example, does it require a minimum amount of work, in
the physicists' technical sense of that term? Even if it is physically
impossible to flip the switch in Thomson's lamp, suppose physics were
different and there were no limit on speed; what then? Is the lamp
logically impossible? Is the lamp metaphysically impossible, even if
it is logically possible? Was it proper of Thomson to suppose that the
question of whether the lamp is lit or dark at the end of the minute
must have a determinate answer? Does Thomson's question have no
answer, given the initial description of the situation, or does it
have an answer which we are unable to compute? Should we conclude that
it makes no sense to divide a finite task into an infinite number of
ever shorter sub-tasks? Even if completing a countable infinity of
tasks in a finite time is possible, is completing an uncountable
infinity also possible? Interesting issues arise when we bring in
Einstein's theory of relativity and consider a bifurcatedsupertask.
This is an infinite sequence of tasks in a finite interval of an
external observer's proper time, but not in the machine's own proper
time. See Earman and Norton (1996) for an introduction to the
extensive literature on these topics. Unfortunately, there is no
agreement in the philosophical community on the main question of
whether Achilles performs a supertask in reaching the tortoise.
d. Constructivism
The spirit of Aristotle's opposition to actual infinities persists
today in the philosophy of mathematics called constructivism.
Constructivism is not a precisely defined position, but it implies
that acceptable mathematical objects and procedures have to be founded
on constructions. Some constructivists would say these constructions
must be performable ideally by humans independently of practical
limitations of time or money. Some constructivists would say recursive
functions, mathematical induction, and Cantor's diagonal argument are
constructive, but the following are not: the axiom of choice, the law
of excluded middle, the law of double negation, completed infinities,
and the classical continuum of the Standard Solution. The implication
is that Zeno's Paradoxes were not solved correctly by using the
methods of the Standard Solution.
L. E. J. Brouwer's intuitionism was the leading constructivist theory
of the early 20th century. In response to suspicions raised by the
discovery of Russell's Paradox and the introduction into set theory of
the controversial axiom of choice, Brouwer attempted to place
mathematics on what he believed to be a firmer epistemological
foundation by arguing that mathematical concepts are admissible only
if they can be constructed from, and thus grounded in, an ideal
mathematician's vivid temporal intuitions, the a priori intuitions of
time. Brouwer's intuitionistic continuum has the Aristotelian property
of unsplitability. What this means is that, unlike the Standard
Solution's set-theoretic composition of the continuum which allows,
say, the closed interval of real numbers from zero to one to be split
into (that is, be the union of sets of) those numbers in the interval
that are less than one-half and those numbers that are greater than or
equal to one-half, the corresponding closed interval of the
intuitionistic continuum cannot be split this way into two disjoint
sets. This unsplitability or inseparability agrees in spirit with
Aristotle's idea of the smoothness of a real continuum, but disagrees
in spirit with Aristotle by allowing the continuum to be composed of
points. [Posy (2005) 346-7]
Although everyone agrees that any legitimate mathematical proof must
use a finite number of steps and be constructive, the majority of
mathematicians in the first half of the twentieth century claimed that
constructive mathematics could not produce an adequate theory of the
continuum because, they believed, essential theorems will no longer be
theorems, and its principles and procedures are too awkward to use
successfully. In 1927, David Hilbert exemplified this attitude when he
objected that Brouwer's restrictions on allowable mathematics–such as
rejecting the law of excluded middle–were like taking the telescope
away from the astronomer. But thanks in large part to the further
development of constructive mathematics by Errett Bishop and Douglas
Bridges in the second half of the 20th century, most contemporary
philosophers of mathematics believe the question of whether
constructivism is successful is still open [see Wolf (2005) p. 346,
and McCarty (2005) p. 382], and to that extent so is the question of
whether the Standard Solution to Zeno's Paradoxes needs to be rejected
or perhaps revised to embrace constructivism. Frank Arntzenius (2000),
Michael Dummett (2000), and Solomon Feferman (1998) have done
important philosophical work to promote the constructivist tradition.
Nevertheless, the vast majority of today's practicing mathematicians
routinely use nonconstructive mathematics.
e. Nonstandard Analysis
Although Zeno and Aristotle had the concept of small, they did not
have the concept of infinitesimally small, which is the informal
concept that was used by Leibniz and Newton in their development of
calculus. In the 19th century, infinitesimals were eliminated from the
standard development of the calculus due to the work of Cauchy and
Weierstrass on defining a derivative in terms of limits using the
epsilon-delta method, but in 1881, C. S. Peirce advocated restoring
infinitesimals. Unfortunately, he was unable to work out the details,
as were all mathematicians until 1960 when Abraham Robinson produced
his nonstandard analysis. Robinson extended the standard real numbers
to include infinitesimals, using this definition: h is infinitesimal
if and only if its absolute value is less than 1/n, for every positive
standard number n. Robinson went on to create a nonstandard model of
analysis using hyperreal numbers. The class of hyperreal numbers
contains counterparts of the reals, but in addition it contains any
number that is the sum, or difference, of both a standard real number
and an infinitesimal number, such as 3 + h and 3 – 4h2. The reciprocal
of an infinitesimal is an infinite hyperreal number. These hyperreals
obey the usual rules of real numbers except for the Archimedean axiom.
Infinitesimal distances between distinct points are allowed, unlike
with standard real analysis. The derivative is defined in terms of the
ratio of infinitesimals, rather than in terms of a limit as in
standard real analysis.
Nonstandard analysis is called "nonstandard" because it was inspired
by Thoralf Skolem's demonstration in 1933 of the existence of models
of first-order arithmetic that are not isomorphic to the standard
model of arithmetic. What makes them nonstandard is especially that
they contain infinitely large (hyper)integers. For nonstandard
calculus one needs nonstandard models of real analysis rather than
just of arithmetic. An important feature demonstrating the usefulness
of nonstandard analysis is that it achieves essentially the same
theorems as those in classical calculus. The treatment of Zeno's
paradoxes is interesting from this perspective. See McLaughlin (1994)
for how Zeno's paradoxes may be treated using infinitesimals.
McLaughlin believes this approach to the paradoxes is the only
successful one, but commentators generally do not agree with that
conclusion, and consider it merely to be an alternative solution.
f. Smooth Infinitesimal Analysis
Abraham Robinson in the 1960s resurrected the infinitesimal as an
infinitesimal number, but F. W. Lawvere in the 1970s resurrected the
infinitesimal as an infinitesimal magnitude. His work is called
"smooth infinitesimal analysis" and is part of "synthetic differential
geometry." In smooth infinitesimal analysis, a curved line is composed
of infinitesimal tangent vectors. One significant difference from a
nonstandard analysis, such as Robinson's above, is that all smooth
curves are straight over infinitesimal distances, whereas Robinson's
can curve over infinitesimal distances. In smooth infinitesimal
analysis, Zeno's arrow does not have time to change its speed during
an infinitesimal interval. Smooth infinitesimal analysis retains the
intuition that a continuum should be smoother than the continuum of
the Standard Solution. Unlike both standard analysis and nonstandard
analysis whose real number systems are set-theoretical entities and
are based on classical logic, the real number system of smooth
infinitesimal analysis is not a set-theoretic entity but rather an
object in a topos of category theory, and its logic is intuitionist.
(Harrison, 1996, p. 283) Like Robinson's nonstandard analysis,
Lawvere's smooth infinitesimal analysis may also be a promising
approach to a foundation for real analysis and thus to solving Zeno's
paradoxes, but there is no consensus that Zeno's Paradoxes need to be
solved this way.
6. The Legacy and Current Significance of the Paradoxes
What influence has Zeno had? He had none in the East, but in the West
there has been continued influence and interest up to today.
Let's begin with his influence on the ancient Greeks. Before Zeno,
philosophers expressed their philosophy in poetry, and he was the
first philosopher to use prose arguments. This new method of
presentation was destined to shape almost all later philosophy,
mathematics, and science. Zeno probably also influenced the Greek
atomists to accept atoms. Aristotle was influenced by Zeno to use the
distinction between actual and potential infinity as a way out of the
paradoxes, and careful attention to this distinction has influenced
mathematicians ever since. The proofs in Euclid's Elements, for
example, used only potentially infinite procedures. Awareness of
Zeno's paradoxes made Greek and all later Western intellectuals more
aware that mistakes can be made when thinking about infinity,
continuity, and the structure of space and time. "Zeno's arguments, in
some form, have afforded grounds for almost all theories of space and
time and infinity which have been constructed from his time to our
own," said Bertrand Russell. There is controversy in the recent
literature about whether Zeno developed any specific, new mathematical
techniques. Aristotle claimed Zeno was the first person to use
"dialectical" argumentation (indirect argumentation or reductio ad
absurdum), but today's scholars do not agree on whether Zeno was
actually the first. He may have discovered the technique, which was
then adopted by the Greek mathematicians; or it may have been the
other way around. G. E. L. Owen (Owen 1958, p. 222) argued that Zeno
influenced Aristotle's concept of motion as not existing at an
instant, which implies there is no moment when a body begins to move,
nor a moment when a body changes its speed. So, his conception is an
obstacle to a Newton-style concept of acceleration, and this hindrance
is "Zeno's major influence on the mathematics of science." Other
commentators consider Owen's remark to be slightly harsh because, if
Zeno had not been born, would Aristotle have been likely to develop
any other concept of motion? No.
Zeno's paradoxes have received some explicit attention from scholars
throughout the centuries. Pierre Gassendi in the early 17th century
mentioned Zeno's paradoxes as the reason to claim that the world's
atoms must not be infinitely divisible. Pierre Bayle's 1696 article on
Zeno drew the skeptical conclusion that, for the reasons given by
Zeno, the concept of space is contradictory. In the early 19th
century, Hegel suggested that Zeno's paradoxes supported his view that
reality is inherently contradictory.
Zeno's paradoxes cause mistrust in infinites, and this mistrust has
influenced the contemporary movements of constructivism, finitism, and
nonstandard analysis, all of which affect the treatment of Zeno's
paradoxes. Dialetheism, the acceptance of true contradictions via a
paraconsistent formal logic, provides a newer, but unpopular, response
to Zeno's paradoxes, but dialetheism was not created specifically in
response to worries about Zeno's paradoxes. With the introduction in
the 20th century of thought experiments about supertasks, interesting
philosophical research has been directed towards understanding what it
means to complete a task.
So, Zeno's paradoxes have had a wide variety of impacts upon
subsequent research. Little research today is involved directly in how
to solve the paradoxes themselves, especially in the fields of
mathematics and science, although discussion continues in philosophy
primarily on whether a line should be composed of points. This raises
the issue of whether there is a single solution to the paradoxes or
several solutions. Also, the answer to whether the Standard Solution
is the correct solution to Zeno's paradoxes may depend on whether the
best science of the future that reconciles the theories of quantum
mechanics and general relativity will require us to assume spacetime
is composed at its most basic level of points, or regions, or loops,
or something else.
From the perspective of the Standard Solution, the two most
significant lessons learned by researchers who have tried to solve
Zeno's paradoxes of applied mathematics are that the way out requires
revising many of our old theories and their concepts, and that math
and science are not as certain as they were believed to be centuries
ago. To find a way out of the paradoxes, we have to be willing to rank
the virtues of preserving logical consistency and promoting scientific
fruitfulness above the virtue of preserving our intuitions. Zeno
played a significant role in causing this progressive trend in
philosophy, physics and mathematics.
7. References and Further Reading
* Arntzenius, Frank. (2000) "Are there Really Instantaneous
Velocities?", The Monist 83, pp 187-208.
o Examines the possibility that a duration does not consist
of points, that every part of time has a non-zero size, that real
numbers cannot be used as coordinates of times, and that there are no
instantaneous velocities at a point.
* Barnes, J. (1982). The Presocratic Philosophers, Routledge &
Kegan Paul: Boston.
o A well respected survey of the philosophical contributions
of the Pre-Socratics.
* Barrow, John D (2005). The Infinite Book: A Short Guide to the
Boundless, Timeless and Endless, Pantheon Books, New York.
o A popular book in science and mathematics introducing
Zeno's Paradoxes and other paradoxes regarding infinity.
* Benacerraf, Paul (1962). "Tasks, Super-Tasks, and the Modern
Eleatics," The Journal of Philosophy, 59, pp. 765-784.
o An original analysis of Thomson's Lamp and supertasks.
* Bergson, Henri (1946). Creative Mind, translated by M. L.
Andison. Philosophical Library: New York.
o Bergson demands the primacy of intuition in place of the
objects of mathematical physics.
* Black, Max (1950-1951). "Achilles and the Tortoise," Analysis
11, pp. 91-101.
o A challenge to the Standard Solution to Zeno's paradoxes.
Blacks agrees that Achilles did not need to complete an infinite
number of sub-tasks in order to catch the tortoise.
* Cajori, Florian (1920). "The Purpose of Zeno's Arguments on
Motion," Isis, vol. 3, no. 1, pp. 7-20.
o An analysis of the debate regarding the point Zeno is
making with his paradoxes of motion.
* Chihara, Charles S. (1965). "On the Possibility of Completing an
Infinite Process," Philosophical Review 74, no. 1, p. 74-87.
o An analysis of what we mean by "task."
* Copleston, Frederick, S.J. (1962). "The Dialectic of Zeno,"
chapter 7 of A History of Philosophy, Volume I, Greece and Rome, Part
I, Image Books: Garden City.
o Copleston says Zeno's goal is to challenge the
Pythagoreans who denied empty space and accepted pluralism.
* Dauben, J. (1990). Georg Cantor, Princeton University Press: Princeton.
o Contains Kronecker's threat to write an article showing
that Cantor's set theory has "no real significance."
* De Boer, Jesse (1953). "A Critique of Continuity, Infinity, and
Allied Concepts in the Natural Philosophy of Bergson and Russell," in
Return to Reason: Essays in Realistic Philosophy, John Wild, ed.,
Henry Regnery Company: Chicago, pp. 92-124.
o A philosophical defense of Aristotle's treatment of Zeno's
paradoxes.
* Diels, Hermann and W. Kranz (1951). Die Fragmente der
Vorsokratiker, sixth ed., Weidmannsche Buchhandlung: Berlin.
o A standard edition of the pre-Socratic texts.
* Dummett, Michael (2000). "Is Time a Continuum of Instants?,"
Philosophy, 2000, Cambridge University Press: Cambridge, pp. 497-515.
o Promoting a constructive foundation for mathematics,
Dummett's formalism implies there are no instantaneous instants, so
times must have rational values rather than real values. Times have
only the values that they can in principle be measured to have; and
all measurements produce rational numbers within a margin of error.
* Earman J. and J. D. Norton (1996). "Infinite Pains: The Trouble
with Supertasks," in Paul Benacerraf: the Philosopher and His Critics,
A. Morton and S. Stich (eds.), Blackwell: Cambridge, MA, pp. 231-261.
o A criticism of Thomson's interpretation of his infinity
machines and the supertasks involved, plus an introduction to the
literature on the topic.
* Feferman, Solomon (1998). In the Light of Logic, Oxford
University Press, New York.
o A discussion of the foundations of mathematics and an
argument for semi-constructivism in the tradition of Kronecker and
Weyl, that the mathematics used in physical science needs only the
lowest level of infinity, the infinity that characterizes the whole
numbers. Presupposes significant knowledge of mathematical logic.
* Freeman, Kathleen (1948). Ancilla to the Pre-Socratic
Philosophers, Harvard University Press: Cambridge, MA. Reprinted in
paperback in 1983.
o One of the best sources in English of primary material on
the Pre-Socratics.
* Grünbaum, Adolf (1967). Modern Science and Zeno's Paradoxes,
Wesleyan University Press: Middletown, Connecticut.
o A detailed defense of the Standard Solution to the paradoxes.
* Grünbaum, Adolf (1970). "Modern Science and Zeno's Paradoxes of
Motion," in (Salmon, 1970), pp. 200-250.
o An analysis of arguments by Thomson, Chihara, Benacerraf
and others regarding the Thomson Lamp and other infinity machines.
* Harrison, Craig (1996). "The Three Arrows of Zeno: Cantorian and
Non-Cantorian Concepts of the Continuum and of Motion," Synthese,
Volume 107, Number 2, pp. 271-292.
o Considers smooth infinitesimal analysis as an alternative
to the classical Cantorian real analysis of the Standard Solution.
* Heath, T. L. (1921). A History of Greek Mathematics, Vol. I,
Clarendon Press: Oxford. Reprinted 1981.
o Promotes the minority viewpoint that Zeno had a direct
influence on Greek mathematics, for example by eliminating the use of
infinitesimals.
* Kirk, G. S., J. E. Raven, and M. Schofield, eds. (1983). The
Presocratic Philosophers: A Critical History with a Selection of
Texts, Second Edition, Cambridge University Press: Cambridge.
o A good source in English of primary material on the
Pre-Socratics with detailed commentary on the controversies about how
to interpret various passages.
* Maddy, Penelope (1992) "Indispensability and Practice," Journal
of Philosophy 59, pp. 275-289.
o Explores the implication of arguing that theories of
mathematics are indispensable to good science, and that we are
justified in believing in the mathematical entities used in those
theories.
* Matson, Wallace I (2001). "Zeno Moves!" pp. 87-108 in Essays in
Ancient Greek Philosophy VI: Before Plato, ed. by Anthony Preus, State
University of New York Press: Albany.
o Matson supports Tannery's non-classical interpretation
that Zeno's purpose was to show only that the opponents of Parmenides
are committed to denying motion, and that Zeno himself never denied
motion, nor did Parmenides.
* McCarty, D.C. (2005). "Intuitionism in Mathematics," in The
Oxford Handbook of Philosophy of Mathematics and Logic, edited by
Stewart Shapiro, Oxford University Press, Oxford, pp. 356-86.
o Argues that a declaration of death of the program of
founding mathematics on an intuitionistic basis is premature.
* McLaughlin, William I. (1994). "Resolving Zeno's Paradoxes,"
Scientific American, vol. 271, no. 5, Nov., pp. 84-90.
o How Zeno's paradoxes may be explained using a contemporary
theory of Leibniz's infinitesimals.
* Owen, G.E.L. (1958). "Zeno and the Mathematicians," Proceedings
of the Aristotelian Society, New Series, vol. LVIII, pp. 199-222.
o Argues that Zeno and Aristotle negatively influenced the
development of the Renaissance concept of acceleration that was used
so fruitfully in calculus.
* Posy, Carl. (2005). "Intuitionism and Philosophy," in The Oxford
Handbook of Philosophy of Mathematics and Logic, edited by Stewart
Shapiro, Oxford University Press, Oxford, pp. 318-54.
o Contains a discussion of how the unsplitability of
Brouwer's intuitionistic continuum makes precise Aristotle's notion
that "you can't cut a continuous medium without some of it clinging to
the knife," on pages 345-7.
* Proclus (1987). Proclus' Commentary on Plato's Parmenides,
translated by Glenn R. Morrow and John M. Dillon, Princeton University
Press: Princeton.
o A detailed list of every comment made by Proclus about
Zeno is available with discussion starting on p. xxxix of the
Introduction by John M. Dillon. Dillon focuses on Proclus' comments
which are not clearly derivable from Plato's Parmenides, and concludes
that Proclus had access to other sources for Zeno's comments, most
probably Zeno's original book or some derivative of it. William
Moerbeke's overly literal translation in 1285 from Greek to Latin of
Proclus' earlier, but now lost, translation of Plato's Parmenides is
the key to figuring out the original Greek. (see p. xliv)
* Russell, Bertrand (1914). Our Knowledge of the External World as
a Field for Scientific Method in Philosophy, Open Court Publishing
Co.: Chicago.
o Russell champions the use of contemporary real analysis
and physics in resolving Zeno's paradoxes.
* Salmon, Wesley C., ed. (1970). Zeno's Paradoxes, The
Bobbs-Merrill Company, Inc.: Indianapolis and New York. Reprinted in
paperback in 2001.
o A collection of the most influential articles about Zeno's
Paradoxes from 1911 to 1965. Salmon provides an excellent annotated
bibliography of further readings.
* Tannery, Paul (1885). "'Le Concept Scientifique du continu:
Zenon d'Elee et Georg Cantor," pp. 385-410 of Revue Philosophique de
la France et de l'Etranger, vol. 20, Les Presses Universitaires de
France: Paris.
o This mathematician gives the first argument that Zeno's
purpose was not to deny motion but rather to show only that the
opponents of Parmenides are committed to denying motion.
* Tannery, Paul (1887). Pour l'Histoire de la Science Hellène: de
Thalès à Empédocle, Alcan: Paris. 2nd ed. 1930.
o More development of the challenge to the classical
interpretation of what Zeno's purposes were in creating his paradoxes.
* Thomson, James (1954-1955). "Tasks and Super-Tasks," Analysis,
XV, pp. 1-13.
o A criticism of supertasks. The Thomson Lamp
thought-experiment is used to challenge Russell's characterization of
Achilles as being able to complete an infinite number of tasks in a
finite time.
* Tiles, Mary (1989). The Philosophy of Set Theory: An
Introduction to Cantor's Paradise, Basil Blackwell: Oxford.
o A philosophically oriented introduction to the foundations
of real analysis and its impact on Zeno's paradoxes.
* Vlastos, Gregory (1967). "Zeno of Elea," in The Encyclopedia of
Philosophy, Paul Edwards (ed.), The Macmillan Company and The Free
Press: New York.
o A clear, detailed presentation of the paradoxes. Vlastos
comments that Aristotle does not consider any other treatment of
Zeno's paradoxes than by recommending replacing Zeno's actual
infinities with potential infinites, so we are entitled to assert that
Aristotle probably believed denying actual infinities is the only
route to a coherent treatment of infinity.
* White, M. J. (1992). The Continuous and the Discrete: Ancient
Physical Theories from a Contemporary Perspective, Clarendon Press:
Oxford.
o A presentation of various attempts to defend finitism,
neo-Aristotelian potential infinities, and the replacement of the
infinite real number field with a finite field.
* Wisdom, J. O. (1953). "Berkeley's Criticism of the
Infinitesimal," The British Journal for the Philosophy of Science,
Vol. 4, No. 13, pp. 22-25.
o Wisdom clarifies the issue behind George Berkeley's
criticism (in 1734 in The Analyst) of the use of the infinitesimal
(fluxion) by Newton and Leibniz. See also the references there to
Wisdom's other three articles on this topic in the journal Hermathena
in 1939, 1941 and 1942.
* Wolf, Robert S. (2005). A Tour Through Mathematical Logic, The
Mathematical Association of America: Washington, DC.
o Chapter 7 surveys nonstandard analysis, and Chapter 8
surveys constructive mathematics, including the contributions by
Errett Bishop and Douglas Bridges.
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