Please write on one of the following prompts, explaining the reasoning succinctly and clearly in your own words. Your discussion should be about 300 words. It is due by midnight April 17.

In such a short paper, keeping your discussion focused is important. Quoting the text is not necessary, and will likely take away space you need to fully explain your points. Because the assignment is so short, you won’t have space to explain everything. So concentrate on explaining the most central points.

1. Start by explaining what the learner’s paradox is, and what is paradoxical about it. Then, as clearly as you can, explain Socrates’ solution to it, and his reasoning for this solution. Very briefly indicate a reason to think it works, or a reason to think it fails.

Thursday Q+A Discussion, Session 2

Phil 2, Professor Koslow

The Plan

Admin Questions

Paper 1 topic to be posted later today. Choice of two topics. Due April 17 by midnight.

A paradox is an argument, from apparently acceptable premises by way of apparently acceptable

reasoning to an apparently unacceptable conclusion.

IMPOSSIBLE: Premises false. conclusion true. valid.

Zeno’s Arrow

“At any instant of time, the flying arrow “occupies a space equal to itself.” That is, the arrow at

an instant cannot be moving, for motion takes a period of time, and a temporal instant is

conceived as a point, not itself having duration. It follows that the arrow is at rest at every

instant, and so does not move. What goes for arrows goes for everything: nothing moves.” (20)

1. At each instant, the arrow does not move.

2. A stretch of time is composed of instants.

3. In any stretch of time, the arrow does not move.

Galileo’s Paradox

1. F: The set of natural numbers is bigger than the set of square numbers

2. The set of natural numbers is the same size as the set of square numbers

3. So, the set of natural numbers is bigger than, and the same size as, the set of square

numbers. (CONTRADICTION!)

Hilbert’s Hotel Paradox

Size: odd numbers, even numbers, natural numbers, integers (…-2, -1, 0, 1, 2…), rational

numbers (a/b)

Odds:

1, 3, 5, 7, 9, 10, ….4, 6, 8….

Naturals: 0, 1, 2, 3, 4, 5, …

Odds + 4, 6, 8

Naturals

Different sizes of infinity: proof that there are more real numbers between 0 and 1 than there are

natural numbers (0, 1, 2, 3, …)

Real numbers: rational numbers, irrational numbers

Rational: a/b a and b are integers that are not 0

Irrational: you can’t write a/b where a and b are non-0 integers (square root of 2, pi)

0.

1.

2.

3.

4.

5.

6.

.0101010101010….

.11111111111111…

.212121212…

.3131313131….

.567323493024…a

.2349082349032…

.00000483000003….

….

Make the evil number:

Step 1: .230304….

Step 2: .341415…

Proof that there are more real numbers than natural numbers

Paradox of the Learner

The paradoxical conclusion: we cannot learn the answer to “what is X?” questions

SOCRATES: I know what you want to say, Meno. Do you realize what a debater’s

argument you are bringing up, that a man cannot search either for what he knows or for

what he does not know? He cannot search for what he knows—since he knows it, there is

no need to search—nor for what he does not know, for he does not know what to look for.

[Meno, 80e]

The Argument

1. If you know what you’re looking for, inquiry is unnecessary.

2. If you don’t know what you’re looking for, inquiry is impossible.

3. Either inquiry is unnecessary or impossible.

Priority of Definition: If you don’t know the full nature of X (everything about the nature of X),

then you fail to know anything about X. [e.g., X=toucans, hammers, virtue]

Does the PD seem true to you? Does it only seem true of some sorts of questions?

The Solution: perhaps, but we can remember the answers. Socrates asks a slave a series of

question that lead him to prove the Pythagorean theorem, something he did not know how to do

at the outset of their conversation. Socrates does not seem to supply him with any new

information, yet the slave seems to ‘learn’ how to do the proof. Their whole conversation may as

well have taken place inside the slave’s head. So the knowledge must have been with him all

along.

The Argument:

1. If the slave boy (i) can move from failure to success, without (ii) having been taught,

then the knowledge must have been within him all along.

2. He can move from failure to success (in fact, he did move from failure to success).

3. He was not taught that X is true, but he was guided to consider whether X or not-X is

true.

4. So, the knowledge must have been within him all along.

5. If the knowledge was within him all along, his ‘learning’ is really recollection.

6. So, what people call learning is really recollection. (85d)

The conversation with the slave is meant to show something, but what? What is it that’s meant to be

innate? Whole theorems or theories? Concepts or ways of thinking about the world? or the ability to

reason?

1

Zeno’s paradoxes: space, time, and motion

1.1

Introduction

Zeno the Greek lived in Elea (a town in what is now southern Italy) in the

fifth century BC. The paradox for which he is best known today concerns

the great warrior Achilles and a previously unknown tortoise. For some

reason now lost in the folds of time, a race was arranged between them.

Since Achilles could run much faster than the tortoise, the tortoise was

given a head start. Zeno’s astonishing contribution is a “proof” that

Achilles could never catch up with the tortoise no matter how fast he ran

and no matter how long the race went on.

The supposed proof goes like this. The first thing Achilles has to do is to

get to the place from which the tortoise started. The tortoise, although

slow, is unflagging: while Achilles is occupied in making up his handicap,

the tortoise advances a little bit further. So the next thing Achilles has to

do is to get to the new place the tortoise occupies. While he is doing this,

the tortoise will have gone on a little bit further still. However small the

gap that remains, it will take Achilles some time to cross it, and in that

time the tortoise will have created another gap. So however fast Achilles

runs, all the tortoise need do in order not to be beaten is keep going – to

make some progress in the time it takes Achilles to close the previous gap

between them.

No one nowadays would dream of accepting the conclusion that

Achilles cannot catch the tortoise. (I will not vouch for Zeno’s reaction

to his paradox: sometimes he is reported as having taken his paradoxical

conclusions quite seriously and literally, showing that motion was impossible.) Therefore, there must be something wrong with the argument.

Saying exactly what is wrong is not easy, and there is no uncontroversial

diagnosis. Some have seen the paradox as produced by the assumption

that space or time is infinitely divisible, and thus as genuinely proving that

space or time is not infinitely divisible. Others have seen in the argument

nothing more than a display of ignorance of elementary mathematics – an

ignorance perhaps excusable in Zeno’s time but inexcusable today.

4

Zeno’s paradoxes: space, time, and motion

5

The paradox of Achilles and the tortoise is Zeno’s most famous, but

there were several others. The Achilles paradox takes for granted that

Achilles can start running, and purports to prove that he cannot get as far

as we all know he can. This paradox dovetails nicely with one known as the

Racetrack, or Dichotomy, which purports to show that nothing can begin

to move. In order to get anywhere, say to a point one foot ahead of you,

you must first get halfway there. To get to the halfway point, you must first

get halfway to that point. In short, in order to get anywhere, even to begin

to move, you must first perform an infinity of other movements. Since this

seems impossible, it seems impossible that anything should move at all.

Almost none of Zeno’s work survives as such. For the most part, our

knowledge of what his arguments were is derived from reports by other

philosophers, notably Aristotle. He presents Zeno’s arguments very

briefly, no doubt in the expectation that they would be familiar to his

audience from the oral tradition that was perhaps his own only source.

Aristotle’s accounts are so compressed that only by guesswork can one

reconstruct a detailed argument. The upshot is that there is no universal

agreement about what should count as “Zeno’s paradoxes,” or about

exactly what his arguments were. I shall select arguments that I believe

to be interesting and important, and which are commonly attributed to

Zeno, but I make no claim to be expounding what the real, historical Zeno

actually said or thought.

Aristotle is an example of a great thinker who believed that Zeno was to

be taken seriously and not dismissed as a mere propounder of childish

riddles. By contrast, Charles Peirce wrote of the Achilles paradox: “this

ridiculous little catch presents no difficulty at all to a mind adequately

trained in mathematics and in logic, but is one of those which is very apt to

excite minds of a certain class to an obstinate determination to believe a

given proposition” (1935, vol. VI, §177, p. 122). On balance, history has

sided with Aristotle, whose view on this point has been shared by thinkers

as dissimilar as Hegel and Russell.

I shall discuss three Zenonian paradoxes concerning motion: the

Racetrack, the Achilles, and a paradox known as the Arrow. Before doing

so, however, it will be useful to consider yet another of Zeno’s paradoxes, one

that concerns space. Sorting out this paradox provides the groundwork for

tackling the paradoxes of motion.

1.2

Space

In ancient times, a frequently discussed perplexity was how something (“one

and the same thing”) could be both one and many. For example, a book is one

but also many (words or pages); likewise, a tree is one but also many (leaves,

6

Paradoxes

branches, molecules, or whatever). Nowadays, this is unlikely to strike anyone

as very problematic. When we say that the book or the tree is many things, we

do not mean that it is identical with many things (which would be absurd), but

rather that it is made up of many parts. Furthermore, at least on the face of it,

there is nothing especially problematic about this relationship between a whole

and the parts which compose it (see question 1.1).

1.1

Appearances may deceive. Let us call some particular tree T, and the

collection of its parts at a particular moment P. Since trees can survive the

loss of some of their parts (e.g. their leaves in the fall), T can exist when P no

longer does. Does this mean that T is something other than P or, more

generally, that each thing is distinct from the sum of its parts? Can P exist

when T does not (e.g. if the parts of the tree are dispersed by timberfelling operations)?

Zeno, like his teacher Parmenides, wished to argue that in such cases

there are not many things but only one thing. I shall examine one ingredient of this argument. Consider any region of space, for example the

region occupied by this book. The region can be thought of as having parts

which are themselves spatial, that is, they have some size. This holds

however small we make the parts. Hence, the argument runs, no region

of space is “infinitely divisible” in the sense of containing an infinite

number of spatial parts. For each part has a size, and a region composed

of an infinite number of parts of this size must be infinite in size.

This argument played the following role in Zeno’s attempt to show that

it is not the case that there are “many things.” He was talking only of

objects in space, and he assumed that an object has a part corresponding

to every part of the space it fills. He claimed to show that, if you allow that

objects have parts at all, you must say that each object is infinitely large,

which is absurd. You must therefore deny that objects have parts. From

this Zeno went on to argue that plurality – the existence of many things –

was impossible. I shall not consider this further development, but will

instead return to the argument against infinite divisibility upon which it

draws (see question 1.2).

1.2

* Given as a premise that no object has parts, how could one attempt to

argue that there is no more than one object?

Zeno’s paradoxes: space, time, and motion

7

The conclusion may seem surprising. Surely one could convince oneself that any space has infinitely many spatial parts. Suppose we take a

rectangle and bisect it vertically to give two further rectangles. Taking the

right-hand one, bisect it vertically to give two more new rectangles.

Cannot this process of bisection go on indefinitely, at least in theory? If

so, any spatial area is made up of infinitely many others.

Wait one moment! Suppose that I am drawing the bisections with a

ruler and pencil. However thin the pencil, the time will fairly soon come

when, instead of producing fresh rectangles, the new lines will fuse into a

smudge. Alternatively, suppose that I am cutting the rectangles from

paper with scissors. Again, the time will fairly soon come when my strip

of paper will be too small to cut. More scientifically, such a process of

physical division must presumably come to an end sometime: at the very

latest, when the remainder of the object is no wider than an atom (proton,

hadron, quark, or whatever).

The proponent of infinite divisibility must claim to have no such physical process in mind, but rather to be presenting a purely intellectual

process: for every rectangle we can consider, we can also consider a

smaller one having half the width. This is how we conceive any space,

regardless of its shape. What we have to discuss, therefore, is whether the

earlier argument demonstrates that space cannot be as we tend to conceive it; whether, that is, the earlier argument succeeded in showing that

no region could have infinitely many parts.

We all know that there are finite spaces which have spatial parts, but

the argument supposedly shows that there are not. Therefore we must

reject one of the premises that leads to this absurd conclusion, and the

most suitable for rejection, because it is the most controversial, is that

space is infinitely divisible. This premise supposedly forces us to say

that either the parts of a supposedly infinitely divisible space are finite

in size, or they are not. If the latter holds, then they are nothing, and no

number of them could together compose a finite space. If the former

holds, infinitely many of them together will compose an infinitely large

space. Either way, on the supposition that space is infinitely divisible,

there are no finite spaces. Since there obviously are finite spaces, the

supposition must be rejected.

The notion of infinite divisibility remains ambiguous. On the one hand,

to say that any space is infinitely divisible could mean that there is no

upper limit to the number of imaginary operations of dividing we could

effect. On the other hand, it could mean that the space contains an infinite

number of parts. It is not obvious that the latter follows from the former.

The latter claim might seem to rely on the idea that the process of

imaginary dividings could somehow be “completed.” For the moment

8

Paradoxes

let us assume that the thesis of infinite divisibility at stake is the thesis that

space contains infinitely many non-overlapping parts, and that each part

has some finite size.

The most doubtful part of the argument against the thesis is the claim

that a space composed of an infinity of parts, each finite in size, must be

infinite. This claim is incorrect, and one way to show it is to appeal to

mathematics. Let us represent the imagined successive bisections by the

following series:

1 1 1

; ;

2 4 8

where the first term 12 represents the fact that, after the first bisection,

the right-hand rectangle is only half the area of the original rectangle;

and similarly for the other terms. Every member of this series is a

finite number, just as each of the spatial parts is of finite size. This does

not mean that the sum of the series is infinite. On the contrary, mathematics texts have it that this series sums to 1. If we find nothing problematic in the idea that an infinite collection of finite numbers has a finite

sum, then by analogy we should be happy with the idea that an infinite

collection of finite spatial parts can compose a finite spatial region (see

question 1.3).

This argument from mathematics establishes the analogous point

about space (namely, that infinitely many parts of finite size may together

form a finite whole) only upon the assumption that the analogy is good: that

space, in the respect in question, has the properties that numbers have. This

is controversial. For example, we have already said that some people take

Zeno’s paradoxes to show that space is not continuous, although the series

of numbers is. Hence we would do well to approach the issue again. We do

not have to rely on any mathematical argument to show that a finite whole

can be composed of an infinite number of finite parts.

There are two rather similar propositions, one true and one false, and

we must be careful not to confuse them.

(1) If, for some finite size, a whole contains infinitely many parts none

smaller than this size, then the whole is infinitely large.

(2) If a whole contains infinitely many parts, each of some finite size, then

the whole is infinitely large.

Statement (1) is true. To see this, let the minimum size of the parts be δ

(say linear or square or cubic inches). Then the size of the whole is ∞ × δ,

which is clearly an infinite number. However, (1) does not bear on the

case we are considering. To see this, let us revert to our imagined bisections. The idea was that however small the remaining area was, we could

Zeno’s paradoxes: space, time, and motion

9

1.3

Someone might object: is it not just a convention in mathematics to treat this

series as summing to 1? More generally, is it not just a convention to treat the

sum of an infinite series as the limit of the partial sums? If this is a mere

mathematical convention, how can it tell us anything about space? Readers

with mathematical backgrounds might like to comment on the following

argument, which purports to show that the fact that the series sums to 1 can

be derived from ordinary arithmetical notions, without appeal to any special

convention. (Warning: mathematicians tell me that what follows is highly

suspect!)

The series can be represented as

x þ x2 þ x3 þ

where x = ½. Multiplying this expression by x has the effect of lopping off the

first term:

xðx þ x2 þ x3 þ Þ ¼ x2 þ x3 þ x4 þ

Here we apply a generalization of the principle of distribution:

a:ðb þ cÞ ¼ ða:bÞ þ ða:cÞ:

Using this together with a similar generalization of the principle that

ð1 aÞ:ðb þ cÞ ¼ ðb þ cÞ a:ðb þ cÞ

we get:

ð1 xÞ:ðx þ x2 þ x3 þ Þ ¼ ðx þ x2 þ x3 þ Þ ðx2 þ x3 þ x4 þ Þ

Thus

ð1 xÞ:ðx þ x2 þ x3 þ x . . .Þ ¼ x

So, dividing both sides by (1 – x):

x þ x2 þ x3 þ ¼

x

ð1 xÞ

So where x = ½, the sum of the series is equal to 1.

always imagine it being divided into two. This means that there can be no

lower limit on how small the parts are. There can be no size δ such that all

the parts are at least this big. For any such size, we can always imagine it

being divided into two.

10

Paradoxes

To see that (2) is false, we need to remember that it is essential to the

idea of infinite divisibility that the parts get smaller, without limit, as the

imagined process of division proceeds. This gives us an almost visual way

of understanding how the endless series of rectangles can fit into the

original rectangle: by getting progressively smaller.

It would be as wrong to infer “There is a finite size which every part

possesses” from “Every part has some finite size or other” as it would be to

infer “There is a woman who is loved by every man” from “Every man

loves some woman or other.” (Readers trained in formal logic will recognize a quantifier-shift fallacy here: one cannot infer an 98 conclusion from

the corresponding 89 premise.)

The explanation for any tendency to believe that (2) is true lies in a

tendency to confuse it with (1). We perhaps tend to think: at the end of

the series the last pair of rectangles formed have some finite size, and all the

other infinitely many rectangles are larger. Therefore, taken together they

must make up an infinite area. However, there is no such thing as the last

pair of rectangles to be formed: our infinite series of divisions has no last

member. Once we hold clearly in mind that there can be no lower limit on

the size of the parts induced by the infinite series of envisaged divisions,

there is no inclination to suppose that having infinitely many parts entails

being infinitely large.

The upshot is that there is no contradiction in the idea that space is

infinitely divisible, in the sense of being composed of infinitely many

non-overlapping spatial parts, each of some finite (non-zero) size. This

does not establish that space is infinitely divisible. Perhaps it is granular, in

the way in which, according to quantum theory, energy is. Perhaps there

are small spatial regions that have no distinct subregions. The present

point, however, is that the Zenonian argument we have discussed gives us

no reason at all to believe this granular hypothesis.

This supposed paradox about space may well not strike us as very deep,

especially if we have some familiarity with the currently orthodox mathematical treatment of infinity. Still, we must not forget that current orthodoxy was not developed without a struggle, and was achieved several

centuries after Zeno had pondered these questions. Zeno and his contemporaries might with good reason have had more trouble with it than we

do. The position of a paradox on the ten-point scale mentioned in the

introduction can change over time: as we become more sophisticated

detectors of mere appearance, a paradox can slide down toward the

Barber end of the scale.

Clearing this paradox out of the way will prove to have been an essential

preliminary to discussing Zeno’s deeper paradoxes, which concern

motion.

Zeno’s paradoxes: space, time, and motion

1.3

11

The Racetrack

If a runner is to reach the end of the track, he must first complete an

infinite number of different journeys: getting to the midpoint, then to the

point midway between the midpoint and the end, then to the point midway between this one and the end, and so on. Since it is logically impossible for someone to complete an infinite series of journeys, the runner

cannot reach the end of the track. It is irrelevant how far away the end of

the track is – it could be just a few inches away – so this argument, if sound,

will show that all motion is impossible. Moving to any point will involve an

infinite number of journeys, and an infinite number of journeys cannot be

completed in a finite time.

Let us call the starting point Z (for Zeno), and the endpoint Z*.

The argument can be analyzed into two premises and a conclusion, as

follows:

(1) Going from Z to Z* would require one to complete an infinite number

of journeys: from Z to the point midway to Z*, call it Z1; from Z1 to the

point midway between it and Z*, call it Z2; and so on.

(2) It is logically impossible for anyone (or anything) to complete an

infinite number of journeys.

Conclusion: It is logically impossible for anyone to go from Z to Z*. Since

these points are arbitrary, all motion is impossible.

Apparently acceptable premises, (1) and (2), lead by apparently acceptable reasoning to an apparently unacceptable conclusion.

No one nowadays would for a moment entertain the idea that the

conclusion is, despite appearances, acceptable. (I refrain from vouching

for Zeno’s own response.) Moreover, the reasoning appears impeccable.

So for us the question is this: which premise is incorrect, and why?

Let us begin by considering premise (1). The idea is that we can

generate an infinite series, let us call it the Z-series, whose terms are

Z; Z1 ; Z2 ; :::

These terms, it is proposed, can be used to analyze the journey from Z to

Z*, for they are among the points that a runner from Z to Z* must pass

through en route. However, Z* itself is not a term in the series; that is, it is

not generated by the operation that generates new terms in the series –

halving the distance that remains between the previous term and Z*.

The word “journey” has, in the context, some misleading implications.

Perhaps “journey” connotes an event done with certain intentions, but it

is obvious that a runner could form no intention with respect to most of

the members of the Z-series, for he would have neither the time, nor the

12

Paradoxes

memory, nor the conceptual apparatus to think about most of them.

Furthermore, he may well form no intention with respect to those he

can think about. Still, if we explicitly set these connotations aside, then

(1) seems hard to deny, once the infinite divisibility of space is granted; for

then all (1) means is the apparent platitude that motion from Z to Z*

involves traversing the distances Z to Z1, Z1 to Z2, and so on.

Suspicion focuses on (2). Why should one not be able to complete an

infinite number of journeys in a finite time? Is that not precisely what does

happen when anything moves? Furthermore, is it not something that could

happen even in other cases? For example, consider a view that Bertrand

Russell once affirmed: he argued that we could imagine someone getting

more and more skillful in performing a given task, and so completing it

more and more quickly. On the first occasion, it might take one minute to

do the job, on the second, only a half a minute, and so on, so that,

performing the tasks consecutively, the whole series of infinitely many

could be performed in the space of two minutes. Russell said, indeed, that

this was “medically impossible” but he held that it was logically possible:

no contradiction was involved. If Russell is right about this, then (2) is the

premise we should reject.

However, consider the following argument, in which the word “task” is

used in quite a general way, so as to subsume what we have been calling

“journeys.”

There are certain reading-lamps that have a button in the base. If the lamp is off

and you press the button the lamp goes on, and if the lamp is on and you press the

button the lamp goes off.

Suppose now that the lamp is off, and I succeed in pressing the button an infinite

number of times, perhaps making one jab in one minute, another jab in the next

half-minute, and so on, according to Russell’s recipe. After I have completed the

whole infinite sequence of jabs, i.e., at the end of two minutes, is the lamp on or

off? It seems impossible to answer this question. It cannot be on, because I did not

ever turn it on without at once turning it off. It cannot be off, because I did in the

first place turn it on, and thereafter I never turned it off without at once turning it

on. But the lamp must be either on or off. This is a contradiction. (Thomson 1954;

cited in Gale 1968, p. 411)

Let us call the envisaged setup consisting of me, the switch, the lamp, and

so on, “Thomson’s lamp.” The argument purports to show that

Thomson’s lamp cannot complete an infinite series of switchings in a

finite time. It proceeds by reductio ad absurdum: we suppose that it

can complete such a series, and show that this supposition leads to an

absurdity – that the lamp is neither on nor off at the supposed end of the

series of tasks.

Zeno’s paradoxes: space, time, and motion

13

The argument is not valid. The supposition that the infinite series has

been completed does not lead to the absurdity that the lamp is neither on

nor off. Nothing follows from this supposition about the state of the lamp

after the infinite series of switchings.

Consider the series of moments T1, T2, …, each corresponding to a

switching. According to the story, the gaps between the members of this

T-series get smaller and smaller, and the rate of switching increases. At T1

a switching on occurs, at T2 a switching off occurs, and so on. Call the first

moment after the (supposed) completion of the series T*. It follows from the

specification of the infinite series that, for any moment in the T-series, if

the lamp is on at that moment there is a later moment in the series at which

the lamp is off; and vice versa. However, nothing follows from this about

whether the lamp is on or off at T*, for T* does not belong to the T-series.

T* is not generated by the operation that generates new members of the

T-series from old, being a time half as remote from the old member as its

predecessor was from it. The specification of the task speaks only to

members of the T-series, and this has no consequences, let alone contradictory consequences, for how things are at T*, which lies outside the

series (see question 1.4).

1.4

Are we entitled to speak of “the first moment after the (supposed) completion of the task”?

The preceding paragraph is not designed to prove that it is logically

possible for an infinite series of tasks to be completed. It is designed to

show only that Thomson’s argument against this possibility fails. In fact,

someone might suggest a reason of a different kind for thinking that there

is a logical absurdity in the idea of Thomson’s lamp.

Consider the lamp’s button. We imagine it to move the same distance

for each switching. If it has moved infinitely many times, then an infinite

distance has been traversed at a finite speed in a finite time. There is a

case for saying that this is logically impossible, for there is a case for saying

that what we mean by average speed is simply distance divided by total

time. It follows that if speed and total time are finite, so is distance. If this is

allowed, then Thomson was right to say that Thomson’s lamp as he

described it is a logical impossibility, even though the argument he gave

for this conclusion was unsatisfactory.

This objection might be countered by varying the design of the

machine. There are at least two possibilities. One is that the machine’s

14

Paradoxes

switch be so constructed that if on its first switching it travels through a

distance δ, then on the second switching it travels δ/2, on the third δ/4, and

so on. Another is that the switch be so constructed that it travels faster and

faster on each switching, without limit (see questions 1.5, 1.6).

1.5

Does this mean that it would have to travel infinitely fast in the end?

1.6

* Does this mean that the switch would have to travel faster than the speed of

light? If so, does this mean that the machine is logically impossible?

It is hard to find positive arguments for the conclusion that this machine is

logically possible; but this machine is open neither to Thomson’s objection, which was invalid, nor to the objection that it involves an infinite

distance being traveled in a finite time. Therefore, until some other

objection is forthcoming, we can (provisionally, and with due caution)

accept this revised Thomson’s lamp as a logical possibility. What is more,

if it is a possibility, then there is nothing logically impossible about a

runner completing an infinite series of journeys (see question 1.7).

1.7

Evaluate the following argument:

We can all agree that the series of numbers 12 ; 14 ; 18 ; sums to 1. What is

controversial is whether this fact has any bearing on whether the runner

can reach Z*. We know that it would be absurd to say that energy is

infinitely divisible merely because for any number that is used to measure

some quantity of energy there is a smaller one. Likewise, Zeno’s paradox

of the runner shows that motion through space should not be thought of as

an endless progression through an infinite series. It is as clear that there is a

smallest motion a runner can make as it is that there is a smallest spatial

distance that we are capable of measuring.

One does not need to establish outré possibilities, such as that of a

Thomson’s lamp that can complete an infinite number of tasks, in order to

establish that the runner can reach Z*. The argument is supposed to work

the other way: if even the infinite Thomson’s lamp is possible, then there

can be no problem about the runner.

Zeno’s paradoxes: space, time, and motion

15

In the next section, I discuss a rather sophisticated variant of the

Racetrack. The discussion may help resolve some of the worries that

remain with this paradox.

1.4

The Racetrack again

Premise (1) of the previous section asserted that a necessary condition for

moving from Z to Z* is moving through the infinite series of intermediate

Z-points. In this rerun, I want to consider a different problem. It is that

there appear to be persuasive arguments for the following inconsistent

conclusions:

(a) Passing through all the Z-points is sufficient for reaching Z*.

(b) Passing through all the Z-points is not sufficient for reaching Z*.

We cannot accept both (a) and (b). The contradiction might be used to

disprove the view that the runner’s journey can be analyzed in terms of an

infinite series, and this would throw doubt on our earlier premise (1) (p. 11).

Let us look more closely at an argument for (a):

Suppose someone could have occupied every point in the Z-series without having

occupied any point outside it, in particular without having occupied Z*. Where

would he be? Not at any Z-point, for then there would be an unoccupied Z-point to

the right. Not, for the same reason, between Z-points. And, ex hypothesi, not at any

point external to the Z-series. But these possibilities are exhaustive. (Cf. Thomson

1954; cited in Gale 1968, p. 418)

In other words, if you pass through all the Z-points, you must get to Z*.

Contrasted with this is a simple argument against sufficiency – an argument for (b):

Z* lies outside the Z-series. It is further to the right than any member of

the Z-series. So going through all the members of the Z-series cannot take

you as far to the right as Z*. So reaching Z* is not logically entailed by

passing through every Z-point.

The new twist to the Racetrack is that we have plausible arguments for

both (a) and (b), but these are inconsistent.

The following objection to the argument for (a) has been proposed by Paul

Benacerraf (1962, p. 774). A possible answer to the question “Where would

the runner be after passing through all the Z-points?” is “Nowhere!” Passing

through all the Z-points is not sufficient for arriving at Z* because one might

cease to exist after reaching every Z-point but without reaching Z*. To lend

color to this suggestion, Benacerraf invites us to imagine a genie who “shrinks

from the thought” of reaching Z* to such an extent that he gets progressively

16

Paradoxes

smaller as his journey progresses. By Z1 he is half his original size, by Z2 a

quarter of it, and so on. Thus by the time he has passed through every Z-point

his size is zero, and “there is not enough left of him” to occupy Z*.

Even if this is accepted (see question 1.8), it will not resolve our problem. The most that it could achieve is a qualification of (a). What would

have to be said to be sufficient for reaching Z* is not merely passing

through every Z-point, but doing that and also (!) continuing to exist.

However, the argument against sufficiency, if it is good at all, seems just as

good against a correspondingly modified version of (b). Since Z* lies

outside the Z-series, even passing through every Z-point and continuing

to exist cannot logically guarantee arriving at Z*.

1.8

* Can the following objection be met?

Where is the runner when he goes out of existence? He cannot be at any

Z-point since, by hypothesis, there is always a Z-point beyond it, which

means that he would not have gone through all the Z-points; but if he goes

out of existence at or beyond Z*, then he reached Z*, and so the sufficiency claim has not been refuted.

Part of the puzzle here lies, I think, in the exact nature of the correspondence that we are setting up between mathematical series and physical space.

We have two different things: on the one hand, a series of mathematical

points, the Z-series, and on the other hand, a series of physical points

composing the physical racetrack. A mathematical series, like the Z-series,

may have no last member. In this case, it is not clear how we are to answer

the question “To what physical length does this series of mathematical

points correspond?” That this is a genuine question is obscured by the

fact that we can properly apply the word “point” both to a mathematical

abstraction and to a position in physical space. However, lengths as ordinarily thought of have two ends. If a length can be correlated with a

mathematical series with only one end, like the Z-series, this can only be

by stipulation. So if we are to think of part of the racetrack as a length, a

two-ended length, corresponding to the mathematically defined Z-series, a

one-ended length, we can but stipulate that what corresponds to the physical length is the series from Z to Z*. Given this, it is obvious that traversing

the length corresponding to the Z-series is enough to get the runner to Z*.

On this view, the paradox is resolved by rejecting the argument for (b), and

accepting that for (a) – modified, perhaps, by the quibble about the runner

continuing to exist.

Zeno’s paradoxes: space, time, and motion

X

17

Y

B

Figure 1.1

This suggestion can be strengthened by the following consideration.

Suppose we divide a line into two discrete parts, X and Y, by drawing

a perpendicular that cuts it at a point B, as in figure 1.1. The notions

of line, division, and so on are to be just our ordinary ones, whatever

they are, and not some mathematical specification of them. Since B is a

spatial point, it must be somewhere. So is it in X or in Y or both? We

cannot say that it is in both X and Y, since by hypothesis these are

discrete lines; that is, they have no point in common. However, it would

seem that any reason we could have for saying that B is in X is as good a

reason for saying that it is in Y. So, if it is in either, then it is in both,

which is impossible.

If we try to represent the intuitive idea in the diagram in mathematically

precise terms, we have to make a choice. Let us think of lengths in terms of

sets of (mathematical) points. If X and Y are to be discrete (have no points

in common), we must choose between assigning B to X (as its last

member, in a left-to-right ordering) and assigning B to Y (as its first

member). If we make the first choice, then Y has no first member; if we

make the second choice, then X has no last member. So far as having an

adequate model for physical space goes, there seems to be nothing to

determine this choice – it seems that we are free to stipulate.

Suppose we make the first choice, according to which B is in X.

Imagine an archer being asked to shoot an arrow that traverses the

whole of a physical space corresponding to X, without entering into any

of the space corresponding to Y. There is no conceptual problem about

this instruction: the arrow must be shot from the leftmost point of X and

land at B. Now imagine an archer being asked to shoot an arrow that

traverses the whole of a physical space corresponding to Y, without

entering into any of the space corresponding to X. This time there

appears to be a conceptual problem. The arrow cannot land at the point

in space corresponding to B because, by stipulation, B has been allocated

to X and so lies outside Y; but nor can the arrow land anywhere in Y,

since for any point in Y there is one between it and B. There is no point

that is the first point to the right of B.

What is odd about this contrast – the ease of occupying all of X and

none of Y, the difficulty of occupying all of Y and none of X – is that which

18

Paradoxes

task is problematic depends upon a stipulation. If we had made the other

choice, stipulating that B is to belong to Y, the difficulties would have been

transposed.

Two real physical tasks, involving physical space, cannot vary in their

difficulty according to some stipulation about how B is to be allocated.

There is some discrepancy here between the abstract mathematical

space-like notions, and our notions of physical space.

If we think of X and Y as genuine lengths, as stretches of physical space,

the difficulty we face can be traced to the source already mentioned:

lengths – for example, the lengths of racetracks – have two ends. If B

belongs to X and not Y, then Y lacks a left-hand end: it cannot have B as its

end, since B belongs to X and not Y (by hypothesis); but it cannot have

any point to the right of B as its left end, for there will always be a Y-point

to the left of any point that is to the right of B.

The difficulty comes from the assumption that the point B has partially

to compose a line to which it belongs, so that to say it belongs to X and Y

would be inconsistent with these being non-overlapping lines. For an

adequate description of physical space, we need a different notion: one

that allows, for example, that two distinct physical lengths, arranged like X

and Y, should touch without overlapping. We need the notion of a

boundary that does not itself occupy space.

If we ask what region of space – thought of in the way we think of

racetracks, as having two ends – corresponds to the points on the

Z-series, the only possible answer would appear to be the region from

Z to Z*. This explains why the argument for sufficiency is correct,

despite the point noted in the argument against it. Z* does not belong

to the Z-series, but it does belong to the region of space that corresponds

to the Z-series.

In these remarks, I have assumed that we have coherent spatial notions,

for example, that of (two-ended) length, and that if some mathematical

structure does not fit with these notions, then so much the worse for the

view that the structure gives a correct account of our spatial notions. In

the circumstances, this pattern of argument is suspect, for it is open to the

following Zeno-like response: the only way we could hope to arrive at

coherent spatial notions is through these mathematical structures. If this

way fails – if the mathematical structures do not yield all we want – then we

are forced to admit that we were after the impossible, and that there is no

way of making sense of our spatial concepts.

The upshot is that a full response to Zeno’s Racetrack paradox would

require a detailed elaboration and justification of our spatial concepts.

This is the task Zeno set us – a task that each generation of philosophers of

space and time rightly feels it must undertake anew.

Zeno’s paradoxes: space, time, and motion

1.5

19

Achilles and the Tortoise

We can restate this most famous of paradoxes using some Racetrack

terminology. The Z-series can be redefined as follows: Z is Achilles’

starting point; Z1 is the tortoise’s starting point; Z2 is the point that the

tortoise reaches while Achilles is getting to Z1; and so on. Z* becomes

the point at which, we all believe, Achilles will catch the tortoise, and the

“proof” is that Achilles, like the runner before him, will never reach Z*.

We can see this as nothing more, in essentials, than the Racetrack, but

with a receding finishing line. The paradoxical claim is this: Achilles can

never get to Z* because however many points in the Z-series he has

occupied, there are still more Z-points ahead before he gets to Z*.

Furthermore, we cannot expect him to complete an infinity of “tasks”

(moving through Z-points) in a finite time. An adequate response to the

Racetrack will be easily converted into an adequate response to this version

of the Achilles.

In such an interpretation of the paradox, the tortoise has barely a

walk-on part to play. Let us see if we can do him more justice. One attempt

is this:

The tortoise is always ahead of Achilles if Achilles is at a point in the Z-series. But

how is this consistent with the supposition that they reach Z* at the same time? If

the tortoise is always ahead in the Z-series, must he not emerge from it before

Achilles?

This makes for a rather superficial paradox. It is trivial that the tortoise is

ahead of Achilles all the time until Achilles has drawn level: he is ahead

until Z*. Given that both of them can travel through all the Z-points,

which was disputed in the Racetrack but which is not now challenged,

there is no reason why they should not complete this task at the same point

in space and time. So I have to report that I can find nothing of substantial

interest in this paradox that has not already been discussed in connection

with the Racetrack.

1.6

The Arrow

At any instant of time, the flying arrow “occupies a space equal to itself.”

That is, the arrow at an instant cannot be moving, for motion takes a

period of time, and a temporal instant is conceived as a point, not itself

having duration. It follows that the arrow is at rest at every instant, and so

does not move. What goes for arrows goes for everything: nothing moves.

Aristotle gives a very brief report of this paradoxical argument, and

concludes that it shows that “time is not composed of indivisible instants”

(Aristotle 1970, Z9. 239b 5). This is one possible response, though one

20

Paradoxes

that would nowadays lack appeal. Classical mechanics purports to make

sense not only of velocity at an instant but also of various more sophisticated notions: rate of change of velocity at an instant (i.e. instantaneous

acceleration or deceleration), rate of change of acceleration at an instant,

and so on.

Another response is to accept that the arrow is at rest at every instant,

but deny that it follows that it does not move. What is required for the

arrow to move, it may be said, is not that it move-at-an-instant, which

is clearly an impossibility (given the semi-technical notion of instant in

question), but rather that it be at different places at different instants.

An instant is not long enough for motion to occur, for motion is a

relation between an object, places, and various instants. If a response

along these lines can be justified, there is no need to accept Aristotle’s

conclusion.

Suppose we set out Zeno’s argument like this:

(1) At each instant, the arrow does not move.

(2) A stretch of time is composed of instants.

Conclusion: In any stretch of time, the arrow does not move.

Then the response under discussion is that this argument is not valid: the

premises are true, but they do not entail the conclusion.

If the first premise is to be acceptable, it must be understood in a rather

special way, which provides the key to the paradox. It must be understood

as making a claim which does not immediately entail that the arrow is at

rest. The question of whether something is moving or at rest “at an

instant” is one that essentially involves other instants. An object is at rest

at an instant just on condition that it is at the same place at all nearby

instants; it is in motion at an instant just on condition that it is in different

places at nearby instants. Nothing about the arrow and a single instant

alone can fix either that it is moving then or at rest then. In short, the

first premise, if acceptable, cannot be understood as saying that at each

instant the arrow is at rest.

Once the first premise is properly understood, it is easy to see why the

argument is fallacious. The conclusion that the arrow is always at rest says

of each instant that the arrow is in the same place at neighboring instants.

No such information is contained in the premises. If we think it is implicit

in the premises, this is probably because we are failing to distinguish

between the claim – interpretable as true – that at each instant the arrow

does not move, and the false claim that it is at rest at each instant.

If this is correct, then the Arrow paradox is an example of one in which the

unacceptable conclusion (nothing moves) comes from an acceptable premise (no motion occurs “during” an instant) by unacceptable reasoning.

Hilbert’s Hotel (A Paradox)

We begin our journey with Hilbert’s Hotel, an imaginary hotel named after the great German mathematician

David Hilbert. Hilbert’s Hotel has infinitely many rooms, one for each natural number (0, 1, 2, 3, …), and every

room is occupied: guest 0 is staying in room 0, guest 1 is staying in room 1, and so forth:

When a finite hotel is completely full, there is no way of accommodating additional guests. In

an infinite hotel, however, the fact that the hotel is full is not an impediment to accommodating extra guests.

Suppose, for example, that Natalia shows up without a reservation. If the hotel manager wants to accommodate

her, all she needs to do is make the following announcement: “Would guest n kindly relocate to room n + 1?”

Assuming everyone abides by this request, guest 0 will end up in room 1, guest 1 will end up in room 2, and so

forth. Under the new arrangement each of the original guests has her own room, and room 0 is available for

But what about the guest in the last room?” you might wonder, “What about the guest in the room with the

biggest number?” The answer is that there is no such thing as the last room, because there is no such thing as

the biggest natural number. Every room in Hilbert’s Hotel is followed by another.

Now suppose 5 billion new guests arrive at our completely full but infinite hotel. We can accommodate

them too! This time, the manager’s announcement is this: “Would guest n kindly relocate to room n + 5

billion?” In a similar way, we can accommodate any finite number of new guests! We can even accommodate

infinitely many new guests. For suppose the manager announced, “If you are in room n, please relocate to room

2n.” Assuming everyone complies, this will free infinitely many rooms while accommodating all of the hotel’s

original guests, as shown:

Hilbert’s Hotel might seem to give rise to paradox. Say that the “old guests” are the guests who were occupying

the hotel prior to any new arrivals and that the “new guests” are the old guests plus Natalia. We seem to be

trapped between conflicting considerations. On the one hand, we want to say that there are more new guests

than old guests. (After all, the new guests include every old guest plus Natalia.) On the other hand, we want to

say that there are just as many new guests as there are old guests. (After all, the new guests and the old guests

can be accommodated using the exact same rooms, without any multiple occupancies or empty rooms.).

What’s paradoxical about Hilbert’s Hotel is very like something that perplexed Galileo.

Galileo’s Paradox

Galileo Galilei’s 1638 masterpiece, Dialogues Concerning Two New Sciences, has a wonderful discussion of

infinity, which includes an argument for the conclusion that “the attributes ‘larger,’ ‘smaller,’ and ‘equal’ have

no place […] in considering infinite quantities.”

Galileo asks us to consider the question of whether there are more squares (1, 4, 9, 16, …) or more roots

(1, 2, 3, 4, …). One might think that there are more roots on the grounds that, whereas every square is a root,

not every root is a square. But one might also think that there are just as many roots as squares, since, as Galileo

puts it, “Every square has its own root and every root has its own square, while no square has more than one

root and no root has more than one square.”

We are left with a paradox. And it is a paradox of the most interesting sort. A boring paradox is a

paradox that leads nowhere. It is due to a superficial mistake and is no more than a nuisance. An interesting

paradox, on the other hand, is a paradox that reveals a genuine problem in our understanding of its subject

matter. The most interesting paradoxes of all are those that reveal a problem interesting enough to lead to the

development of an improved theory.

Such is the case of Galileo’s infinitary paradox. In 1874, almost two and a half centuries after Two New

Sciences, Georg Cantor published an article that describes a rigorous methodology for comparing the sizes of

infinite sets. Cantor’s methodology yields an answer to Galileo’s paradox: it entails that the roots are just as

many as the squares. It also delivers the arresting conclusion that there are different sizes of infinity. Cantor’s

work was the turning point in our understanding of infinity. By treading on the brink of paradox, he replaced a

muddled pre-theoretic notion of infinite size with a rigorous and fruitful notion, which has become one of the

essential tools of contemporary mathematics.

We’ll talk more about Cantor’s insights in lecture. For now, can you set out as Galileo’s paradox as an

argument? What about the paradox of Hilbert’s Hotel?

Purchase answer to see full

attachment