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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
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.
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
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
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
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
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
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,
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
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,
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
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
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
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
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:
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
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
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
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
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.
none of Y, the difficulty of occupying all of Y and none of X – is that which
18
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
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.
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 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