# Zeno's Paradoxes

*First published Tue Apr 30, 2002; substantive revision Fri Oct 20, 2017*

Almost everything that we know about Zeno of Elea is to be found in
the opening pages of Plato’s *Parmenides*. There we learn
that Zeno was nearly 40 years old when Socrates was a young man, say
20. Since Socrates was born in 469 BC we can estimate a birth date for
Zeno around 490 BC. Beyond this, really all we know is that he was
close to Parmenides (Plato reports the gossip that they were lovers
when Zeno was young), and that he wrote a book of paradoxes defending
Parmenides’ philosophy. Sadly this book has not survived, and
what we know of his arguments is second-hand, principally through
Aristotle and his commentators (here we draw particularly on
Simplicius, who, though writing a thousand years after Zeno,
apparently possessed at least some of his book). There were apparently
40 ‘paradoxes of plurality’, attempting to show that
ontological pluralism—a belief in the existence of many things
rather than only one—leads to absurd conclusions; of these
paradoxes only two definitely survive, though a third argument can
probably be attributed to Zeno. Aristotle speaks of a further four
arguments against motion (and by extension change generally), all of
which he gives and attempts to refute. In addition Aristotle
attributes two other paradoxes to Zeno. Sadly again, almost none of
these paradoxes are quoted in Zeno’s original words by their
various commentators, but in paraphrase.

- 1. Background
- 2. The Paradoxes of Plurality
- 3. The Paradoxes of Motion
- 4. Two More Paradoxes
- 5. Zeno’s Influence on Philosophy
- Further Readings
- Bibliography
- Academic Tools
- Other Internet Resources
- Related Entries

## 1. Background

Before we look at the paradoxes themselves it will be useful to sketch
some of their historical and logical significance. First, Zeno sought
to defend Parmenides by attacking his critics. Parmenides rejected
pluralism and the reality of any kind of change: for him all was one
indivisible, unchanging reality, and any appearances to the contrary
were illusions, to be dispelled by reason and revelation. Not
surprisingly, this philosophy found many critics, who ridiculed the
suggestion; after all it flies in the face of some of our most basic
beliefs about the world. (Interestingly, general
relativity—particularly quantum general
relativity—arguably provides a novel—if novelty
*is* possible—argument for the Parmenidean denial of
change: Belot and Earman, 2001.) In response to this criticism Zeno
did something that may sound obvious, but which had a profound impact
on Greek philosophy that is felt to this day: he attempted to show
that equal absurdities followed logically from the denial of
Parmenides’ views. You think that there are many things? Then
you must conclude that everything is both infinitely small and
infinitely big! You think that motion is infinitely divisible? Then it
follows that nothing moves! (This is what a ‘paradox’ is:
a demonstration that a contradiction or absurd consequence follows
from apparently reasonable assumptions.)

As we read the arguments it is crucial to keep this method in mind.
They are always directed towards a more-or-less specific target: the
views of some person or school. We must bear in mind that the
arguments are *ad hominem*, not in the ‘bad sense’
that they attack a person rather than his views but in the ‘good
sense’ that they are formulated against a particular
philosopher’s assertions. They work by temporarily supposing
‘for argument’s sake’ that those assertions are
true, and then arguing that if they are then absurd consequences
follow—that nothing moves for example: they are
‘*reductio ad absurdum*’ arguments (or
‘dialectic’ in the sense of the period). Then, if the
argument is logically valid, and the conclusion genuinely
unacceptable, the assertions must be false after all. Thus when we
look at Zeno’s arguments we must ask two related questions: whom
or what position is Zeno attacking, and what exactly is assumed for
argument’s sake? If we find that Zeno makes hidden assumptions
beyond what the position under attack commits one to, then the absurd
conclusion can be avoided by denying one of the hidden assumptions,
while maintaining the position. Indeed commentators at least since
Aristotle have responded to Zeno in this way.

So whose views do Zeno’s arguments attack? There is a huge literature debating Zeno’s exact historical target. As we shall discuss briefly below, some say that the target was a technical doctrine of the Pythagoreans, but most today see Zeno as opposing common-sense notions of plurality and motion. We shall approach the paradoxes in this spirit, and refer the reader to the literature concerning the interpretive debate.

That said, it is also the majority opinion that—with certain qualifications—Zeno’s paradoxes reveal some problems that cannot be resolved without the full resources of mathematics as worked out in the Nineteenth century (and perhaps beyond). This is not (necessarily) to say that modern mathematics is required to answer any of the problems that Zeno explicitly wanted to raise; arguably Aristotle and other ancients had replies that would—or should—have satisfied Zeno. (Nor shall we make any particular claims about Zeno’s influence on the history of mathematics.) However, as mathematics developed, and more thought was given to the paradoxes, new difficulties arose from them; these difficulties require modern mathematics for their resolution. These new difficulties arise partly in response to the evolution in our understanding of what mathematical rigor demands: solutions that would satisfy Zeno’s standards of rigor would not satisfy ours. Thus we shall push several of the paradoxes from their common sense formulations to their resolution in modern mathematics. (Another qualification: we shall offer resolutions in terms of ‘standard’ mathematics, but other modern formulations are also capable of dealing with Zeno, and arguably in ways that better represent his mathematical concepts.)

## 2. The Paradoxes of Plurality

### 2.1 The Argument from Denseness

If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited. (Simplicius(a)On Aristotle’s Physics, 140.29)

This first argument, given in Zeno’s words according to Simplicius, attempts to show that there could not be more than one thing, on pain of contradiction. Assume then that there are many things; he argues that they are both ‘limited’ and ‘unlimited’, a contradiction. First, he says that any collection must contain some definite number of things, neither more nor fewer. But if you have a definite number of things, he concludes, you must have a finite—‘limited’—number of them; he implicitly assumes that to have infinitely many things is to have an ‘indefinite’ number of them. But second, imagine any collection of ‘many’ things arranged in space—imagine them lined up in one dimension for definiteness. Between any two of them, he claims, is a third; and in between these three elements another two; and another four between these five; and so on without end. Therefore the limited collection is also ‘unlimited’, which is a contradiction, and hence our original assumption must be false: there are not many things after all. At least, so Zeno’s reasoning runs.

But why are there ‘always others between the things that are’? (In modern terminology, why must objects always be ‘densely’ ordered?) Suppose that we had imagined a collection of ten apples lined up; then there is indeed another apple between the sixth and eighth, but there is none between the seventh and eighth! On the assumption that Zeno is not simply confused, what does he have in mind? There are two possibilities: first, one might hold that for any pair of physical objects (two apples say) to be two distinct objects and not just one (a ‘double-apple’) there must be a third between them, physically separating them, even if it is just air. And one might think that for these three to be distinct, there must be two more objects separating them, and so on (this view presupposes that their being made of different substances is not sufficient to render them distinct). Second, one might hold that any body has parts that can be densely ordered. Of course 1/2s, 1/4s, 1/8s and so on of apples are not dense—some such parts are adjacent—but there may be sufficiently small parts—call them ‘point—parts’—that are. Indeed, if between any two point-parts there lies a finite distance, and if point-parts can be arbitrarily close, then they are dense; a third lies at the half-way point of any two. In particular, familiar geometric points are like this, and hence are dense.

And thus we should read the argument as follows: if you suppose that the world contains many things, then you are faced with a contradiction, for the collection must be both finite and infinite—finite because it contains a definite number of things, and infinite because they are dense. The assumption that any definite number is finite seems intuitive, but we now know, thanks to the work of Cantor in the Nineteenth century, how to understand infinite numbers in a way that makes them just as definite as finite numbers. One central element of this theory of the ‘transfinite numbers’ is a precise definition of when two infinite collections are the same size, and when one is bigger than the other—with such a definition in hand it is then possible to order the infinite numbers just as the finite numbers are ordered. For example, both the fractions and geometric points in a line are dense, but there are different, definite infinite numbers of them. (See Further Reading below for references to introductions to these mathematical ideas, and their history.) Of course, settling the mathematical question of whether infinite numbers can be definite doesn’t show that real physical objects actually have geometric point parts, all it shows is that it is a logical possibility.

### 2.2 The Argument from Finite Size

… if it should be added to something else that exists, it would not make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing. (Simplicius(a)On Aristotle’s Physics,139.9)But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited. (Simplicius(a)

On Aristotle’s Physics, 141.2)

Once again we have Zeno’s own words. According to his conclusion, there are three parts to this argument, but only two survive. The first—missing—argument purports to show that if many things exist then they must have no size at all. Second, from this Zeno argues that it follows that they do not exist at all; since the result of joining (or removing) a sizeless object to anything is no change at all, he concludes that the thing added (or removed) is literally nothing. The argument to this point is a self-contained refutation of pluralism, but Zeno goes on to generate a further problem for someone who continues to urge the existence of a plurality. This third part of the argument is rather badly put but it seems to run something like this: suppose there is a plurality, so some spatially extended object exists (after all, he’s just argued that inextended things do not exist). Since it is extended, it has two spatially distinct parts (one ‘in front’ of the other). And the parts exist, so they have extension, and so they also each have two spatially distinct parts; and so on without end. And hence, the final line of argument seems to conclude, the object, if it is extended at all, is infinite in extent.

But what could justify this final step? It doesn’t seem that because an object has two parts it must be infinitely big! And neither does it follow from any other of the divisions that Zeno describes here; four, eight, sixteen, or whatever finite parts make a finite whole. Again, surely Zeno is aware of these facts, and so must have something else in mind, presumably the following: he assumes that if the infinite series of divisions he describes were repeated infinitely many times then a definite collection of parts would result. And notice that he doesn’t have to assume that anyone could actually carry out the divisions—there’s not enough time and knives aren’t sharp enough—just that an object can be geometrically decomposed into such parts (neither does he assume that these parts are what we would naturally categorize as distinct physical objects like apples, cells, molecules, electrons or so on, but only that they are geometric parts of these objects). Now, if—as a pluralist might well accept—such parts exist, it follows from the second part of his argument that they are extended, and, he apparently assumes, an infinite sum of finite parts is infinite.

Here we should note that there are two ways he may be envisioning the result of the infinite division.

First, one could read him as first dividing the object into 1/2s, then one of the 1/2s—say the second—into two 1/4s, then one of the 1/4s—say the second again—into two 1/8s and so on. In this case the result of the infinite division results in an endless sequence of pieces of size 1/2 the total length, 1/4 the length, 1/8 the length … . And then so the total length is (1/2 + 1/4 + 1/8 + …) of the length, which Zeno concludes is an infinite distance, so that the pluralist is committed to the absurdity that finite bodies are ‘so large as to be unlimited’.

What is often pointed out in response is that Zeno gives us no reason
to think that the sum is infinite rather than finite. He might have
had the intuition that any infinite sum of finite quantities, since it
grows endlessly with each new term must be infinite, but one might
also take this kind of example as showing that some infinite sums are
after all finite. Thus, contrary to what he thought, Zeno has not
proven that the absurd conclusion follows. However, what is not always
appreciated is that the pluralist is not off the hook so easily, for
it is not enough just to say that the sum *might* be finite,
she must also show that it *is* finite—otherwise we
remain uncertain about the tenability of her position. As an
illustration of the difficulty faced here consider the following: many
commentators speak as if it is simply obvious that the infinite sum of
the fractions is 1, that there is nothing to infinite summation. But
what about the following sum: 1 − 1 + 1 − 1 + 1 −
… . Obviously, it seems, the sum can be rewritten (1
− 1) + (1 − 1) + … = 0 + 0 + … = 0. Surely
this answer seems as intuitive as the sum of fractions. But this sum
can also be rewritten 1 − (1 − 1 + 1 − 1 + …)
= 1 − 0—since we’ve just shown that the term in
parentheses vanishes—= 1. Relying on intuitions about how to
perform infinite sums leads to the conclusion that 1 = 0. Until one
can give a theory of infinite sums that can give a satisfactory answer
to any problem, one cannot say that Zeno’s infinite sum is
obviously finite. Such a theory was not fully worked out until the
Nineteenth century by Cauchy. (In Cauchy’s system 1/2 + 1/4 +
… = 1 but 1 − 1 + 1 − … is undefined.)

Second, it could be that Zeno means that the object is divided in
half, then both the 1/2s are both divided in half, then the 1/4s are
all divided in half and so on. In this case the pieces at any
particular stage are all the same finite size, and so one could
conclude that the result of carrying on the procedure infinitely would
be pieces the same size, which if they exist—according to
Zeno—is greater than zero; but an infinity of *equal*
extended parts is indeed infinitely big.

But this line of thought can be resisted. First, suppose that the
procedure just described completely divides the object into
non-overlapping parts. (There is a problem with this supposition that
we will see just below.) It involves doubling the number of pieces
after every division and so after *N* divisions there are
2^{N} pieces. But it turns out that for any natural
or infinite number, *N*, 2^{N} >
*N*, and so the number of (supposed) parts obtained by the
infinity of divisions described is an even larger infinity. This
result poses no immediate difficulty since, as we mentioned above,
infinities come in different sizes. The number of times everything is
divided in two is said to be ‘countably infinite’: there
is a countable infinity of things in a collection if they can be
labeled by the numbers 1, 2, 3, … without remainder on either
side. But the number of pieces the infinite division produces is
‘uncountably infinite’, which means that there is no way
to label them 1, 2, 3, … without missing some of them—in
fact infinitely many of them. However, Cauchy’s definition of an
infinite sum only applies to countably infinite series of numbers, and
so does not apply to the pieces we are considering. However, we could
consider just countably many of them, whose lengths according to
Zeno—since he claims they are all equal and non-zero—will
sum to an infinite length; the length of *all* of the pieces
could not be less than this.

At this point the pluralist who believes that Zeno’s division completely divides objects into non-overlapping parts (see the next paragraph) could respond that the parts in fact have no extension, even though they exist. That would block the conclusion that finite objects are infinite, but it seems to push her back to the other horn of Zeno’s argument, for how can all these zero length pieces make up a non-zero sized whole? (Note that according to Cauchy 0 + 0 + 0 + … = 0 but this result shows nothing here, for as we saw there are uncountably many pieces to add up—more than are added in this sum.) We shall postpone this question for the discussion of the next paradox, where it comes up explicitly.

The second problem with interpreting the infinite division as a
repeated division of all parts is that it does not divide an object
into distinct parts, if objects are composed in the natural way. To
see this, let’s ask the question of what parts are obtained by
this division into 1/2s, 1/4s, 1/8s, …. Since the division is
repeated without end there is no last piece we can give as an answer,
and so we need to think about the question in a different way. If we
suppose that an object can be represented by a line segment of unit
length, then the division produces collections of segments, where the
first is either the first or second half of the whole segment, the
second is the first or second quarter, or third or fourth quarter, and
in general the segment produced by *N* divisions is either the
first or second half of the previous segment. For instance, writing
the segment with endpoints *a* and *b* as
[*a*,*b*], some of these collections (technically known
as ‘chains’ since the elements of the collection are
ordered by size) would start {[0,1], [0,1/2], [1/4,1/2], [1/4,3/8],
…}. (When we argued before that Zeno’s division produced
uncountably many pieces of the object, what we should have said more
carefully is that it produces uncountably many chains like this.)

The question of which parts the division picks out is then the
question of which part any given chain picks out; it’s natural
to say that a chain picks out the part of the line which is contained
in *every one* of its elements. Consider for instance the chain
{[0,1/2], [1/4,1/2], [3/8,1/2], …}, in other words the chain
that starts with the left half of the line and for which every other
element is the right half of the previous one. The half-way point is
in every one of the segments in this chain; it’s the right-hand
endpoint of each one. But no other point is in all its elements:
clearly no point beyond half-way is; and pick any point *p*
before half-way, if you take right halves of [0,1/2] enough times, the
left-hand end of the segment will be to the right of *p*. Thus
the only part of the line that is in all the elements of this chain is
the half-way point, and so that is the part of the line picked out by
the chain. (In fact, it follows from a postulate of number theory that
there is exactly one point that all the members of *any* such a
chain have in common.) The problem is that by parallel reasoning, the
half-way point is also picked out by the distinct chain {[1/2,1],
[1/2,3/4], [1/2,5/8], …}, where each segment after the first is
the left half of the preceding one. And so both chains pick out the
same piece of the line: the half-way point. And so on for many other
pairs of chains. Thus Zeno’s argument, interpreted in terms of a
repeated division of *all* parts into half, doesn’t
divide the line into distinct parts. Hence, if we think that objects
are composed in the same way as the line, it follows that despite
appearances, this version of the argument does not cut objects into
parts whose total size we can properly discuss.

(You might think that this problem could be fixed by taking the elements of the chains to be segments with no endpoint to the right. Then the first of the two chains we considered no longer has the half-way point in any of its segments, and so does not pick out that point. The problem now is that it fails to pick out any part of the line: the previous reasoning showed that it doesn’t pick out any point greater than or less than the half-way point, and now it doesn’t pick out that point either!)

### 2.3 The Argument from Complete Divisibility

… whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible will have resulted if it has actually been divided … though perhaps nobody in fact could so divide it.What then will remain? A magnitude? No: that is impossible, since then there will be something not divided, whereas

ex hypothesithe body was divisiblethrough and through. But if it be admitted that neither a body nor a magnitude will remain … the body willeitherconsist of points (and its constituents will be without magnitude)orit will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus presumably the whole body will be nothing but an appearance. But if it consists of points, it will not possess any magnitude. (AristotleOn Generation and Corruption, 316a19)

These words are Aristotle’s not Zeno’s, and indeed the
argument is not even attributed to Zeno by Aristotle. However we have
Simplicius’ opinion ((a) *On Aristotle’s Physics*,
139.24) that it originates with Zeno, which is why it is included
here. Aristotle begins by hypothesizing that some body is completely
divisible, ‘through and through’; the second step of the
argument makes clear that he means by this that it is divisible into
parts that themselves have no size—parts with any magnitude
remain incompletely divided. (Once again what matters is that the body
is genuinely composed of such parts, not that anyone has the time and
tools to make the division; and remembering from the previous section
that one does not obtain such parts by repeatedly dividing all parts
in half.) So suppose the body is divided into its dimensionless parts.
These parts could either be nothing at all—as Zeno argued
above—or ‘point-parts’. If the parts are nothing
then so is the body: it’s just an illusion. And, the argument
concludes, even if they are points, since these are unextended the
body itself will be unextended: surely any sum—even an infinite
one—of zeroes is zero.

Could that final assumption be questioned? It is (as noted above) a
consequence of the Cauchy definition of an infinite sum; however
Grünbaum (1967) pointed out that that definition only applies to
countable sums, and Cantor gave a beautiful, astounding and extremely
influential ‘diagonal’ proof that the number of points in
the segment is uncountably infinite. There is no way to label
*all* the points in the line with the infinity of numbers 1, 2,
3, … , and so there are more points in a line segment than
summands in a Cauchy sum. In short, the analysis employed for
countably infinite division does not apply here.

So suppose that you are just given the number of points in a line and
that their lengths are all zero; how would you determine the length?
Do we need a new definition, one that extends Cauchy’s to
uncountably infinite sums? It turns out that that would not help,
because Cauchy further showed that any segment, of any length
whatsoever (and indeed an entire infinite line) *have exactly the
same number of points as our unit segment*. So knowing the number
of points won’t determine the length of the line, and so nothing
like familiar addition—in which the whole is determined by the
parts—is possible. Instead we must think of the distance
properties of a line as logically posterior to its point composition:
*first* we have a set of points (ordered in a certain way, so
that there is some fact, for example, about which of any three is
between the others) *then* we define a function of pairs of
points which specifies how far apart they are (satisfying such
conditions as that the distance between *A* and *B* plus
the distance between *B* and *C* equals the distance
between *A* and *C*—if *B* is between
*A* and *C*). Thus we answer Zeno as follows: the
argument assumed that the size of the body was a sum of the sizes of
point parts, but that is not the case; according to modern
mathematics, a geometric line segment is an uncountable infinity of
points plus a distance function. (Note that Grünbaum used the
fact that the point composition fails to determine a length to support
his ‘conventionalist’ view that a line has no determinate
length at all, independent of a standard of measurement.)

As Ehrlich (2014) emphasizes, we could even stipulate that an
‘uncountable sum’ of zeroes is zero, because the length of
a line is not equal to the sum of the lengths of the points it
contains (addressing Sherry’s, 1988, concern that refusing to
extend the definition would be ad hoc). Hence, if one stipulates that
the length of a line is the sum of any complete collection of proper
parts, then it follows that points are not properly speaking
*parts* of a line (unlike halves, quarters, and so on of a line).
In a strict sense in modern measure theory (which generalizes
Grünbaum’s framework), the points in a line are
incommensurable with it, and the very set-up given by Aristotle in
which the length of the whole is analyzed in terms of its points is
illegitimate. (This discussion has been revised from earlier versions
of the entry, in light of Ehrlich’s paper.)

## 3. The Paradoxes of Motion

### 3.1 The Dichotomy

The first asserts the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal. (AristotlePhysics, 239b11)

This paradox is known as the ‘dichotomy’ because it involves repeated division into two (like the second paradox of plurality). Like the other paradoxes of motion we have it from Aristotle, who sought to refute it.

Suppose a very fast runner—such as mythical Atalanta—needs
to run for the bus. Clearly before she reaches the bus stop she must
run half-way, as Aristotle says. There’s no problem there;
supposing a constant motion it will take her 1/2 the time to run
half-way there and 1/2 the time to run the rest of the way. Now she
must also run half-way to the half-way point—i.e., a 1/4 of the
total distance—before she reaches the half-way point, but again
she is left with a finite number of finite lengths to run, and plenty
of time to do it. And before she reaches 1/4 of the way she must reach
1/2 of 1/4 = 1/8 of the way; and before that a 1/16; and so on. There
is no problem at any finite point in this series, but what if the
halving is carried out infinitely many times? The resulting series
contains no first distance to run, for any possible first distance
could be divided in half, and hence would not be first after all.
However it does contain a final distance, namely 1/2 of the way; and a
penultimate distance, 1/4 of the way; and a third to last distance,
1/8 of the way; and so on. Thus the series of distances that Atalanta
is required to run is: …, then 1/16 of the way, then 1/8 of the
way, then 1/4 of the way, and finally 1/2 of the way (for now we are
not suggesting that she *stops* at the end of each segment and
then starts running at the beginning of the next—we are thinking
of her continuous run being composed of such parts). And now there is
a problem, for this description of her run has her travelling an
*infinite* number of *finite* distances, which, Zeno
would have us conclude, must take an infinite time, which is to say it
is never completed. And since the argument does not depend on the
distance or who or what the mover is, it follows that no finite
distance can ever be traveled, which is to say that all motion is
impossible. (Note that the paradox could easily be generated in the
other direction so that Atalanta must first run half way, then half
the remaining way, then half of that and so on, so that she must run
the following endless sequence of fractions of the total distance:
1/2, then 1/4, then 1/8, then ….)

A couple of common responses are not adequate. One might—as
Simplicius ((a) *On Aristotle’s Physics*, 1012.22) tells
us Diogenes the Cynic did by silently standing and walking—point
out that it is a matter of the most common experience that things in
fact do move, and that we know very well that Atalanta would have no
trouble reaching her bus stop. But this would not impress Zeno, who,
as a paid up Parmenidean, held that many things are not as they
appear: it may appear that Diogenes is walking or that Atalanta is
running, but appearances can be deceptive and surely we have a logical
proof that they are in fact not moving at all. Alternatively if one
doesn’t accept that Zeno has given a proof that motion is
illusory—as we hopefully do not—one then owes an account
of what is wrong with his argument: he has given reasons why motion is
impossible, and so an adequate response must show why those reasons
are not sufficient. And it won’t do simply to point out that
there are some ways of cutting up Atalanta’s run—into just
two halves, say—in which there is no problem. For if you accept
all of the steps in Zeno’s argument then you must accept his
conclusion (assuming that he has reasoned in a logically deductive
way): it’s not enough to show an unproblematic division, you
must also show why the *given* division is unproblematic.

Another response—given by Aristotle himself—is to point
out that as we divide the distances run, we should also divide the
total time taken: there is 1/2 the time for the final 1/2, a 1/4 of
the time for the previous 1/4, an 1/8 of the time for the 1/8 of the
run and so on. Thus each fractional distance has just the right
fraction of the finite total time for Atalanta to complete it, and
thus the distance can be completed in a finite time. Aristotle felt
that this reply should satisfy Zeno, however he also realized
(*Physics*, 263a15) that it could not be the end of the matter.
For now we are saying that the *time* Atalanta takes to reach
the bus stop is composed of an infinite number of finite
pieces—…, 1/8, 1/4, and 1/2 of the total time—and
isn’t that an infinite time?

Of course, one could again claim that some infinite sums have finite
totals, and in particular that the sum of these pieces is 1 ×
the total time, which is of course finite (and again a complete
solution would demand a rigorous account of infinite summation, like
Cauchy’s). However, Aristotle did not make such a move. Instead
he drew a sharp distinction between what he termed a
‘continuous’ line and a line divided into parts. Consider
a simple division of a line into two: on the one hand there is the
undivided line, and on the other the line with a mid-point selected as
the boundary of the two halves. Aristotle claims that these are two
distinct things: and that the latter is only ‘potentially’
derivable from the former. Next, Aristotle takes the common-sense view
that time is like a geometric line, and considers the time it takes to
complete the run. We can again distinguish the two cases: there is the
continuous interval from start to finish, and there is the interval
divided into Zeno’s infinity of half-runs. The former is
‘potentially infinite’ in the sense that it could be
divided into the latter ‘actual infinity’. Here’s
the crucial step: Aristotle thinks that since these intervals are
*geometrically* distinct they must be *physically*
distinct. But how could that be? He claims that the runner must do
something at the end of each half-run to make it distinct from the
next: she must stop, making the run itself discontinuous. (It’s
not clear why some other action wouldn’t suffice to divide the
interval.) Then Aristotle’s full answer to the paradox is that
the question of whether the infinite series of runs is possible or not
is ambiguous: the potentially infinite series of halves in a
continuous run is possible, while an actual infinity of discontinuous
half runs is not—Zeno does identify an impossibility, but it
does not describe the usual way of running down tracks!

It is hard—from our modern perspective perhaps—to see how this answer could be completely satisfactory. In the first place it assumes that a clear distinction can be drawn between potential and actual infinities, something that was never fully achieved. Second, suppose that Zeno’s problem turns on the claim that infinite sums of finite quantities are invariably infinite. Then Aristotle’s distinction will only help if he can explain why potentially infinite sums are in fact finite (couldn’t we potentially add 1 + 1 + 1 + …, which does not have a finite total); or if he can give a reason why potentially infinite sums just don’t exist. Or perhaps Aristotle did not see infinite sums as the problem, but rather whether completing an infinity of finite actions is metaphysically and conceptually and physically possible. We will briefly discuss this issue—of ‘Supertasks’—below, but note that there is a well-defined run in which the stages of Atalanta’s run are punctuated by finite rests, arguably showing the possibility of completing an infinite series of finite tasks in a finite time (Huggett 2010, 21–2). Finally, the distinction between potential and actual infinities has played no role in mathematics since Cantor tamed the transfinite numbers—certainly the potential infinite has played no role in the modern mathematical solutions discussed here.

One last point: Zeno’s argument seeks most obviously to establish the impossibility of motion, but he also intended it (and the following arguments) as further refutations of plurality—certainly, Plato interprets Zeno’s intentions in this way. How might the argument seek to establish this conclusion? Presumably Zeno has in mind the view that spatial (and perhaps temporal) distances have a plurality of parts; parts which are infinitely divisible into two. Given that assumption, supposedly finite distances (or times) can be decomposed into an infinity of finite parts with no first (or alternatively, last) one. And how can such distances be finite after all? And if the pluralist also believes in motion, how can such a distance be traversed? It seems it cannot be.

### 3.2 Achilles and the Tortoise

The [second] argument was called “Achilles,” accordingly, from the fact that Achilles was taken [as a character] in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … . And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount. (Simplicius(b)On Aristotle’s Physics, 1014.10)

This paradox turns on much the same considerations as the last.
Imagine Achilles chasing a tortoise, and suppose that Achilles is
running at 1 *m/s*, that the tortoise is crawling at 0.1
*m/s* and that the tortoise starts out 0.9 *m* ahead of
Achilles. On the face of it Achilles should catch the tortoise after
1*s*, at a distance of 1*m* from where he starts (and so
0.1*m* from where the Tortoise starts). We could break
Achilles’ motion up as we did Atalanta’s, into halves, or
we could do it as follows: before Achilles can catch the tortoise he
must reach the point where the tortoise started. But in the time he
takes to do this the tortoise crawls a little further forward. So next
Achilles must reach this new point. But in the time it takes Achilles
to achieve this the tortoise crawls forward a tiny bit further. And so
on to infinity: every time that Achilles reaches the place where the
tortoise was, the tortoise has had enough time to get a little bit
further, and so Achilles has another run to make, and so Achilles has
an infinite number of finite catch-ups to do before he can catch the
tortoise, and so, Zeno concludes, he never catches the tortoise.

One aspect of the paradox is thus that Achilles must traverse the
following infinite series of distances before he catches the tortoise:
first 0.9*m*, then an additional 0.09*m*, then
0.009*m*, … . These are the series of distances
ahead that the tortoise reaches at the start of each of
Achilles’ catch-ups. Looked at this way the puzzle is identical
to the Dichotomy, for it is just to say that ‘that which is in
locomotion must arrive [nine tenths of the way] before it arrives at
the goal’. And so everything we said above applies here too.

But what the paradox in this form brings out most vividly is the
problem of completing a series of actions that has no final
member—in this case the infinite series of catch-ups before
Achilles reaches the tortoise. But just what is the problem? Perhaps
the following: Achilles’ run to the point at which he should
reach the tortoise can, it seems, be completely decomposed into the
series of catch-ups, none of which take him to the tortoise.
Therefore, nowhere in his run does he reach the tortoise after all.
But if this is what Zeno had in mind it won’t do. Of course
Achilles doesn’t reach the tortoise at any point of the
sequence, for every run in the sequence occurs *before* we
expect Achilles to reach it! Thinking in terms of the points that
Achilles must reach in his run, 1m does not occur in the sequence
0.9*m*, 0.99*m*, 0.999*m*, … , so of
course he never catches the tortoise during that sequence of runs! The
series of catch-ups does not after all completely decompose the run:
the final point—at which Achilles does catch the
tortoise—must be added to it. So is there any puzzle? Arguably
yes.

Achilles’ run passes through the sequence of points
0.9*m*, 0.99*m*, 0.999*m*, … ,
1*m*. But does such a strange sequence—comprised of an
infinity of members followed by one more—make sense
mathematically? If not then our mathematical description of the run
cannot be correct, but then what is? Fortunately the theory of
transfinites pioneered by Cantor assures us that such a series is
perfectly respectable. It was realized that the order properties of
infinite series are much more elaborate than those of finite series.
Any way of arranging the numbers 1, 2 and 3 gives a series in the same
pattern, for instance, but there are many distinct ways to order the
natural numbers: 1, 2, 3, … for instance. Or … , 3,
2, 1. Or … , 4, 2, 1, 3, 5, … . Or 2, 3, 4,
… , 1, which is just the same kind of series as the
positions Achilles must run through. Thus the theory of the
transfinites treats not just ‘cardinal’
numbers—which depend only on how many things there are—but
also ‘ordinal’ numbers which depend further on how the
things are arranged. Since the ordinals are standardly taken to be
mathematically legitimate numbers, and since the series of points
Achilles must pass has an ordinal number, we shall take it that the
series is mathematically legitimate. (Again, see
‘Supertasks’ below for another kind of problem that might
arise for Achilles’.)

### 3.3 The Arrow

The third is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (AristotlePhysics, 239b30)Zeno abolishes motion, saying “What is in motion moves neither in the place it is nor in one in which it is not”. (Diogenes Laertius

Lives of Famous Philosophers, ix.72)

This argument against motion explicitly turns on a particular kind of
assumption of plurality: that time is composed of moments (or
‘nows’) *and nothing else*. Consider an arrow,
apparently in motion, at any instant. First, Zeno assumes that it
travels no distance during that moment—‘it occupies an
equal space’ for the whole instant. But the entire period of its
motion contains only instants, all of which contain an arrow at rest,
and so, Zeno concludes, the arrow cannot be moving.

An immediate concern is why Zeno is justified in assuming that the arrow is at rest during any instant. It follows immediately if one assumes that an instant lasts 0s: whatever speed the arrow has, it will get nowhere if it has no time at all. But what if one held that the smallest parts of time are finite—if tiny—so that a moving arrow might actually move some distance during an instant? One way of supporting the assumption—which requires reading quite a lot into the text—starts by assuming that instants are indivisible. Then suppose that an arrow actually moved during an instant. It would be at different locations at the start and end of the instant, which implies that the instant has a ‘start’ and an ‘end’, which in turn implies that it has at least two parts, and so is divisible, contrary to our assumption. (Note that this argument only establishes that nothing can move during an instant, not that instants cannot be finite.)

So then, nothing moves during any instant, but time is entirely
composed of instants, so nothing ever moves. A first response is to
point out that determining the velocity of the arrow means dividing
the distance traveled in some time by the length of that time.
But—assuming from now on that instants have zero
duration—this formula makes no sense in the case of an instant:
the arrow travels 0*m* in the 0*s* the instant lasts,
but 0/0 *m/s* is not any number at all. Thus it is fallacious
to conclude from the fact that the arrow doesn’t travel any
distance in an instant that it is at rest; whether it is in motion at
an instant or not depends on whether it travels any distance in a
*finite* interval that includes the instant in question.

The answer is correct, but it carries the counter-intuitive
implication that motion is not something that happens at any instant,
but rather only over finite periods of time. Think about it this way:
time, as we said, is composed only of instants. No distance is
traveled during any instant. So when does the arrow actually move? How
does it get from one place to another at a later moment? There’s
only one answer: the arrow gets from point *X* at time 1 to
point *Y* at time 2 simply in virtue of being at successive
intermediate points at successive intermediate times—the arrow
never changes its position during an instant but only over intervals
composed of instants, by the occupation of different positions at
different times. In Bergson’s memorable words—which he
thought expressed an absurdity—‘movement is composed of
immobilities’ (1911, 308): getting from *X* to *Y*
is a matter of occupying exactly one place in between at each instant
(in the right order of course). For a discussion of this issue see
Arntzenius (2000).

### 3.4 The Stadium

The fourth argument is that concerning equal bodies which move alongside equal bodies in the stadium from opposite directions—the ones from the end of the stadium, the others from the middle—at equal speeds, in which he thinks it follows that half the time is equal to its double…. (AristotlePhysics, 239b33)

Aristotle goes on to elaborate and refute an argument for Zeno’s
final paradox of motion. The text is rather cryptic, but is usually
interpreted along the following lines: picture three sets of touching
cubes—all exactly the same—in relative motion. One
set—the *A*s—are at rest, and the others—the
*B*s and *C*s—move to the right and left
respectively, at a constant equal speed. And suppose that at some
moment the rightmost *B* and the leftmost *C* are
aligned with the middle *A*, as shown (three of each are
pictured for simplicity).

A |
A |
A |
|||

B |
B |
B |
|||

C |
C |
C |

Since the *B*s and *C*s move at same speeds, they will
be aligned with the *A*s simultaneously.

A |
A |
A |
|||

B |
B |
B |
|||

C |
C |
C |

At this moment, the rightmost *B* has traveled past all the
*C*s, but only half the *A*s; since they are of equal
size, it has traveled both some distance *and* half that
distance. The putative contradiction is not drawn here however,
presumably because it is clear that these contrary distances are
*relative* to the *C*s and *A*s respectively;
there’s generally no contradiction in standing in different
relations to different things. Instead, the distances are converted to
times by dividing the distances by the speed of the *B*s; half
the distance at a given speed takes half the time. Then a
contradiction threatens because the time between the states is
unequivocal, not relative—the process takes some (non-zero) time
and half that time.

The general verdict is that Zeno was hopelessly confused about
relative velocities in this paradox. If the *B*s are moving
with speed *S m/s* to the right with respect to the
*A*s, and if the *C*s are moving with speed *S
m/s* to the left with respect to the *A*s, then the
*C*s are moving with speed *S*+*S* = 2*S
m/s* to the left with respect to the *B*s. And so, of
course, while the *B*s travel twice as far relative to the
*C*s as the *A*s, they do so at twice the relative
speed, and so the times are the same either way. But could Zeno have
been this confused? (Sattler, 2015, argues against this and other
common readings of the stadium.)

Perhaps (Davey, 2007) he had the following in mind instead (while Zeno
is smarter according to this reading, it doesn’t quite fit
Aristotle’s words so well): suppose the *A*s, *B*s
and *C*s are of the smallest spatial extent,
‘point-sized’, where ‘points’ are of zero size
if space is continuous, or finite if space is ‘atomic’.
Suppose further that there are no spaces between the *A*s, or
between the *B*s, or between the *C*s. During the motion
above the leading *B* passes all of the *C*s, and half
of the *A*s, so half as many *A*s as *C*s. Now,
as a point moves continuously along a line with no gaps, there is a
1:1 correspondence between the instants of time and the points on the
line—to each instant a point, and to each point an instant.
Therefore, the number of ‘*A*-instants’ of time the
leading *B* takes to pass the *A*s is half the number of
‘*C*-instants’ takes to pass the
*C*s—even though these processes take the same amount of
time. If we then, crucially, assume that half the instants means half
the time, we conclude that half the time equals the whole time, a
contradiction.

We saw above, in our discussion of complete divisibility, the problem
with such reasoning applied to continuous lines: any line segment has
the same number of points, so nothing can be inferred from the number
of points in this way—certainly not that half the points (here,
instants) means half the length (or time). The paradox fails as
stated. But doesn’t the very claim that the intervals contain
the same number of instants conflict with the step of the argument
that concludes that there are half as many *A*-instants as
*C*-instants? This issue is subtle for infinite sets: to give a
different example, 1, 2, 3, … is in 1:1 correspondence with 2,
4, 6, …, and so there are the same number of each. It is in
this sense of 1:1 correspondence—the precise sense of
‘same number’ used in mathematics—that any finite
line has the same number of points as any other. However, informally
speaking, there are also ‘half as many’ even numbers as
whole numbers: the pairs (1, 2), (3, 4), (5, 6), … can also be
put into 1:1 correspondence with 2, 4, 6, …. Similarly, there
are—informally speaking—half as many *A*-instants
as *C*-instants: *A*-instants are in 1:1 correspondence
with pairs of *C*-instants. So there is no contradiction in the
number of points: the informal half equals the strict whole (a
different solution is required for an atomic theory, along the lines
presented in the final paragraph of this section).

(Let me mention a similar paradox of motion—the ‘millstone’—attributed to Maimonides. Imagine two wheels, one twice the radius and circumference of the other, fixed to a single axle. Let them run down a track, with one rail raised to keep the axle horizontal, for one turn of both wheels [they turn at the same rate because of the axle]: each point of each wheel makes contact with exactly one point of its rail, and every point of each rail with exactly one point of its wheel. Does the assembly travel a distance equal to the circumference of the big wheel? Of the small? Both? Something else? How? This problem too requires understanding of the continuum; but it is not a paradox of Zeno’s so we shall leave it to the ingenuity of the reader.)

A final possible reconstruction of Zeno’s Stadium takes it as an
argument against an atomic theory of space and time, which is
interesting because contemporary physics explores such a view when it
attempts to ‘quantize’ spacetime. Suppose then the sides
of each cube equal the ‘quantum’ of length and that the
two moments considered are separated by a single quantum of time. Then
something strange must happen, for the rightmost *B* and the
middle *C* pass each other during the motion, and yet there is
no moment at which they are level: since the two moments are separated
by the smallest possible time, there can be no instant between
them—it would be a time smaller than the smallest time from the
two moments we considered. Conversely, if one insisted that if they
pass then there must be a moment when they are level, then it shows
that cannot be a shortest finite interval—whatever it is, just
run this argument against it. However, why should one insist on this
assumption? The problem is that one naturally imagines quantized space
as being like a chess board, on which the chess pieces are frozen
during each quantum of time. Then one wonders when the red queen, say,
gets from one square to the next, or how she gets past the white queen
without being level with her. But the analogy is misleading. It is
better to think of quantized space as a giant matrix of lights that
holds some pattern of illuminated lights for each quantum of time. In
this analogy a lit bulb represents the presence of an object: for
instance a series of bulbs in a line lighting up in sequence represent
a body moving in a straight line. In this case there is no temptation
to ask when the light ‘gets’ from one bulb to the
next—or in analogy how the body moves from one location to the
next.

## 4. Two More Paradoxes

### 4.1 The Paradox of Place

Zeno’s difficulty demands an explanation; for if everything that exists has a place, place too will have a place, and so onad infinitum. (AristotlePhysics, 209a23)

When he sets up his theory of place—the crucial spatial notion in his theory of motion—Aristotle lists various theories and problems that his predecessors, including Zeno, have formulated on the subject. One can again see here a problem for pluralism, for the second step of the argument concludes that there are many places. It is, however, a little hard to feel the full force of the conclusion, for why should there not be an infinite series of places of places of places of …? Presumably the worry would be greater for someone who (like Aristotle) believed that there could not be an actual infinity of things, for the argument seems to show that there are. But certainly today we need have no such qualms; there seems nothing problematic with an actual infinity of places; indeed, it seems very natural to think that every point of space is a distinct place, even if there are an infinity of points.

The only other way one might find the regress troubling is if one had reason to suppose that objects must have ‘absolute’ places, in the sense that there is always a unique answer to the question ‘where is it’? For example, where am I as I write? If the paradox is right then I’m in my place, and I’m also in my place’s place, and my place’s place’s place, and my … . Since I’m in all these places any might seem an appropriate answer to the question. But why think that there must be a unique answer to the question? Why shouldn’t I have many locations? At my desk, in my apartment, in Chicago, Illinois, USA, North America, the Earth, Solar System … . (In fact there is a reason that Aristotle might have had this concern about Zeno’s argument, for in his theory of motion, the natural motion of a body is determined by the relation of its place to the center of the universe: an account that requires place to be determinate.)

### 4.2 The Grain of Millet

… Zeno’s reasoning is false when he argues that there is no part of the millet that does not make a sound; for there is no reason why any part should not in any length of time fail to move the air that the whole bushel moves in falling. (AristotlePhysics, 250a19)

This argument is a Parmenidean one against the reliability of the senses. It goes like this: if you drop a sack of millet on the floor then you hear a loud thud; but this noise is the result of the noise made by every grain of millet in the sack; and the result of the noise made by every part of every grain; therefore every part of every grain makes a noise as it hits the ground. But now consider dropping a tiny part of a grain; we know that we won’t hear it. Therefore our sense of hearing is deceptive—there are noises it cannot hear—and so we should not trust it. Aristotle’s response seems to be that the part would not move as much air as the sack, but the paradox is not that the part should make as much noise as the sack, but that it should make some noise. A better reply is surely that not every disturbance in the air is audible by us: that a measuring instrument is unreliable over some range is no argument that it is unreliable over every range.

## 5. Zeno’s Influence on Philosophy

In this final section we should consider briefly the impact that Zeno has had on various philosophers; a search of the literature will reveal that these debates continue.

The Pythagoreans: For the first half of the Twentieth century, the majority reading—following Tannery (1885)—of Zeno held that his arguments were directed against a technical doctrine of the Pythagoreans. According to this reading they held that all things were composed of elements that had the properties of a unit number, a geometric point and a physical atom: this kind of position would fit with their doctrine that reality is fundamentally mathematical. However, in the middle of the century a series of commentators (Vlastos, 1967, summarizes the argument and contains references) forcefully argued that Zeno’s target was instead a common sense understanding of plurality and motion—one grounded in familiar geometrical notions—and indeed that the doctrine was not a major part of Pythagorean thought. We have implicitly assumed that these arguments are correct in our readings of the paradoxes. That said, Tannery’s interpretation still has its defenders (see e.g., Matson 2001).

The Atomists: Aristotle (

*On Generation and Corruption*316b34) claims that our third argument—the one concerning complete divisibility—was what convinced the atomists that there must be smallest, indivisible parts of matter. See Abraham (1972) for a further discussion of Zeno’s connection to the atomists.Temporal Becoming: In the early part of the Twentieth century several influential philosophers attempted to put Zeno’s arguments to work in the service of a metaphysics of ‘temporal becoming’, the (supposed) process by which the present comes into being. Such thinkers as Bergson (1911), James (1911, Ch 10–11) and Whitehead (1929) argued that Zeno’s paradoxes show that space and time are not structured as a mathematical continuum: they argued that the way to preserve the reality of motion was to deny that space and time are composed of points and instants. However, we have clearly seen that the tools of standard modern mathematics are up to the job of resolving the paradoxes, so no such conclusion seems warranted: if the present indeed ‘becomes’, there is no reason to think that the process is not captured by the continuum.

Applying the Mathematical Continuum to Physical Space and Time: Following a lead given by Russell (1929, 182–198), a number of philosophers—most notably Grünbaum (1967)—took up the task of showing how modern mathematics could solve all of Zeno’s paradoxes; their work has thoroughly influenced our discussion of the arguments. What they realized was that a purely mathematical solution was not sufficient: the paradoxes not only question abstract mathematics, but also the nature of physical reality. So what they sought was an argument not only that Zeno posed no threat to the mathematics of infinity but also that that mathematics correctly describes objects, time and space. It would not answer Zeno’s paradoxes if the mathematical framework we invoked was not a good description of actual space, time, and motion! The idea that a mathematical law—say Newton’s law of universal gravity—may or may not correctly describe things is familiar, but some aspects of the mathematics of infinity—the nature of the continuum, definition of infinite sums and so on—seem so basic that it may be hard to see at first that they too apply contingently. But surely they do: nothing guarantees

*a priori*that space has the structure of the continuum, or even that parts of space add up according to Cauchy’s definition. (Salmon offers a nice example to help make the point: since alcohol dissolves in water, if you mix the two you end up with less than the sum of their volumes, showing that even ordinary addition is not applicable to every kind of system.) Our belief that the mathematical theory of infinity describes space and time is justified to the extent that the laws of physics assume that it does, and to the extent that those laws are themselves confirmed by experience. While it is true that almost all physical theories assume that space and time do indeed have the structure of the continuum, it is also the case that quantum theories of gravity likely imply that they do not. While no one really knows where this research will ultimately lead, it is quite possible that space and time will turn out, at the most fundamental level, to be quite unlike the mathematical continuum that we have assumed here.One should also note that Grünbaum took the job of showing that modern mathematics describes space and time to involve something rather different from arguing that it is confirmed by experience. The dominant view at the time (though not at present) was that scientific terms had meaning insofar as they referred directly to objects of experience—such as ‘1 m ruler’—or, if they referred to ‘theoretical’ rather than ‘observable’ entities—such as ‘a point of space’ or ‘1/2 of 1/2 of … 1/2 a racetrack’—then they obtained meaning by their logical relations—via definitions and theoretical laws—to such observation terms. Thus Grünbaum undertook an impressive program to give meaning to all terms involved in the modern theory of infinity, interpreted as an account of space and time.

Supertasks: A further strand of thought concerns what Black (1950–51) dubbed ‘infinity machines’. Black and his followers wished to show that although Zeno’s paradoxes offered no problem to mathematics, they showed that after all mathematics was not applicable to space, time and motion. Most starkly, our resolution to the Dichotomy and Achilles assumed that the complete run could be broken down into an infinite series of half runs, which could be summed. But is it really possible to complete any infinite series of actions: to complete what is known as a ‘supertask’? If not, and assuming that Atalanta and Achilles can complete their tasks, their complete runs cannot be correctly described as an infinite series of half-runs, although modern mathematics would so describe them. What infinity machines are supposed to establish is that an infinite series of tasks cannot be completed—so any completable task cannot be broken down into an infinity of smaller tasks, whatever mathematics suggests.

Infinitesimals: Finally, we have seen how to tackle the paradoxes using the resources of mathematics as developed in the Nineteenth century. For a long time it was considered one of the great virtues of this system that it finally showed that infinitesimal quantities, smaller than any finite number but larger than zero, are unnecessary. (Newton’s calculus for instance effectively made use of such numbers, treating them sometimes as zero and sometimes as finite; the problem with such an approach is that how to treat the numbers is a matter of intuition not rigor.) However, in the Twentieth century Robinson showed how to introduce infinitesimal numbers into mathematics: this is the system of ‘non-standard analysis’ (the familiar system of real numbers, given a rigorous foundation by Dedekind, is by contrast just ‘analysis’). Analogously, Bell (1988) explains how infinitesimal line segments can be introduced into geometry, and comments on their relation to Zeno. Moreover, McLaughlin (1992, 1994) shows how Zeno’s paradoxes can be resolved in non-standard analysis; they are no more argument against non-standard analysis than against the standard mathematics we have assumed here. It should be emphasized however that—contrary to McLaughlin’s suggestions—there is no need for non-standard analysis to solve the paradoxes: either system is equally successful. (Reeder, 2015, argues that non-standard analysis is unsatisfactory regarding the arrow, and offers an alternative account using a different conception of infinitesimals.) The construction of non-standard analysis does however raise a further question about the applicability of analysis to physical space and time: it seems plausible that all physical theories can be formulated in either terms, and so as far as our experience extends both seem equally confirmed. But they cannot both be true of space and time: either space has infinitesimal parts or it doesn’t.

## Further Readings

After the relevant entries in this encyclopedia, the place to begin any further investigation is Salmon (2001), which contains some of the most important articles on Zeno up to 1970, and an impressively comprehensive bibliography of works in English in the Twentieth Century.

One might also take a look at Huggett (1999, Ch. 3) and Huggett (2010,
Ch. 2–3) for further source passages and discussion. For
introductions to the mathematical ideas behind the modern resolutions,
the Appendix to Salmon (2001) or Stewart (2017) are good starts;
Russell (1919) and Courant *et al*. (1996, Chs. 2 and 9) are
also both wonderful sources. Finally, three collections of original
sources for Zeno’s paradoxes: Lee (1936 [2015]) contains
everything known, Kirk *et al* (1983, Ch. 9) contains a great
deal of material (in English and Greek) with useful commentaries, and
Cohen *et al*. (1995) also has the main passages.

## Bibliography

- Abraham, W. E., 1972, ‘The Nature of Zeno’s Argument
Against Plurality in DK 29 B I’,
*Phronesis*, 17: 40–52. - Aristotle, ‘On Generation and Corruption’, A. A.
Joachim (trans), in
*The Complete Works of Aristotle*, J. Barnes (ed.), Princeton: Princeton University Press, 1984. - Aristotle, ‘Physics’, W. D. Ross(trans), in
*The Complete Works of Aristotle*, J. Barnes (ed.), Princeton: Princeton University Press, 1984. - Arntzenius, F., 2000, ‘Are There Really Instantaneous
Velocities?’,
*The Monist*, 83: 187–208. - Belot, G. and Earman, J., 2001, ‘Pre-Socratic Quantum
Gravity’, in
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### Acknowledgments

This entry is dedicated to the late Wesley Salmon, who did so much to educate philosophers about the significance of Zeno’s paradoxes. Those familiar with his work will see that this discussion owes a great deal to him; I hope that he would find it satisfactory. This material is based upon work supported by National Science Foundation Grant SES-0004375. I would also like to thank Eliezer Dorr for bringing to my attention some problems with my original formulation of the argument from finite size, and Georgette Sinkler for catching errors in earlier versions. I also revised the discussion of complete divisibility in response to Philip Ehrlich’s (2014) enlightening paper.