Summary.
The
subject of this treatise is to discuss Zeno’s paradoxes with runners on the
basis of immediate considerations.
Some
philosophers understand the concept of limit
of a sum-function as being identical with the concept of a sum of infinitely many numbers, as far
as such a sum exists. It proves, however, that the solution of these paradoxes is
found in (1) the way the story of the race between Achilles and the tortoise is
told, which can be rendered recursively without a stop. Moreover, the right
understanding of the concept of infinity proves to dissolve an alleged paradox concerning
(2) how it is possible to have passed
infinitely many points (in a finite time). This corresponds to the difference
between respectively 1) being in the present and 2) considering a
past-construction.
A
later treatise deals with the discussion in Wesley C. Salmon’s anthology Zeno’s Paradoxes ([Salmon 70]).
1.
Achilles and the tortoise.
The paradoxes with runners of Zeno of Elea can
obviously be solved by mathematical formalism, e.g. that of the race between Achilles
and the tortoise. However, the question is whether they deal with the
mathematical world or with the physical world, i.e. the phenomenal, in which
not all theoretical concepts have a counterpart. The fact that the theories of
natural science can be used to predict the phenomena is a different matter, as
this does not demand an analogy between the mathematical and the physical world,
but just their usability or being empirically adequate, such as Bas van
Fraassen argues in The Scientific Image ([Fraassen 80], p. 12, 18).
1.1
A description of the paradox.
In the popularized version of one of Zeno’s
paradoxes, Achilles and the tortoise, Achilles has to catch up with a tortoise
that has a lead. It must be observed that the paradox consists of two parts: 1)
our common sense view of the race, which is based on memory impressions of
finished processes, and 2) Zeno’s account of it.
According to the implicit reasoning in (2)
Zeno’s account of the run, Achilles will never catch up the tortoise. Nevertheless,
we will assert (1) that we know that Achilles will catch up with the tortoise. This
is a paradox concerning our conception of time. Of course, Zeno has a point
here.
Re 1: Zeno’s reasoning can be recounted in this
way: Let us assume that Achilles runs G times as fast as the tortoise, and that
the leap is F units of length. According to principles that are in accordance
to our experience, it can be figured out that that Achilles has cached up with
the tortoise when he has run the distance F G/(G - 1).
Re 2: When Achilles has run F units of length,
the tortoise has run F/G. When Achilles has cached up with this lead, the
tortoise has run F/G2. When Achilles has run F/G2, the tortoise
has run F/G3. Etc. After each of these parts of the run, the
tortoise still has a lead (that certainly gets smaller and smaller, but never
zero).
This means two things: 1) Each time one of
these stages has been finished, Achilles has yet to traverse a stage. I.e. that
Achilles is always in the same situation. 2) In a situation where Achilles has
caught up with the tortoise, he must have run through infinitely many stages.
This is another aspect of the paradox above. The first one can be comprised in:
1.1.1
A recursive description of the race.
After each of the described parts of the race,
the same situation is present, except that the tortoise has a lead that is 1/G
times what it was at the start of the run.
We can call the length of the stretch to the
starting point of the tortoise d, and relation between Achilles’ velocity and
that of the tortoise g. Then, the race between Achilles and the tortoise can be
described by an account, B(d).
B(d): “Achilles catches up with the tortoise
that has a leap d”.
This account has the following recursive form
where the expression R(s) means, “Achilles runs the distance d”:
B(d)
-> R(d) + B(d/g).
If the initial leap was F and the relation
between the velocities g = G, the whole account can be expressed as
B(F).
If this account is regarded as a pure mathematical
description, the situation after each step is quite the same as at the start of
the race, for there are no absolute lengths on the line in geometry. Thus, according
to this mathematic description, no changes are present that imply that the race
will stop.
Thus, from the description above we
get:
B(F) -> R(F) + B(F/G).
B(F/G) -> R(F) + R(F/G) + B(F/G2).
B(F/G2) -> R(F) + R(F/G) + R(F/G2) + B(F/G3).
B(F/G3) -> R(F) + R(F/G) + R(F/G2) + R(F/G3) + B(F/G4).
This recursive description can be compared to a
recursive description that contains a not-recursive case that will even be
satisfied and thus finishes the description.
Let “Run(N)” mean: The runner runs the
remaining N/M parts of the stretch, and let R(M) mean: ”The runner runs 1/M of the
whole track”. Then, the
run can then be described by these recursive definitions:
Run(0).
Run(N) ->
R(M) + Run(N-1).
Then the run
can be described in this way:
Run(M).
Example.
M=3:
Run(3) ->
R(M) + Run(3-1).
Run(2) ->
R(M) + Run(2-1).
Run(1) ->
R(M) + Run(1-1).
Run(0).
1.2
Mathematical proposals for solution.
This
subsection looks at two mathematical proposals for solution based on
mathematical formalism for so-called summation of infinite many values. It
proves that none of these attempts at solving the paradox succeeds. The first
attempt, which consists in a simple computation, is circular, while the other cannot
make good that which takes place in the description of the race between
Achilles and the tortoise will lead to a termination.
1.2.1
Summation of infinite many values.
We
can attempt to let the paradox deal with a summation S∞ of all the
steps until Achilles has caught up with the tortoise according to the common
sense view[1]:
S∞
= F ∑∞i=1 G-i+1
From this, we get:
S∞ = F +F ∑∞i=2
G-i+1
S∞ G = F G + F ∑∞i=2
G-i+2
S∞
G = F G + F ∑∞i=1 G-i+1
By this, we get the following computation of
the summation:
S∞
G = F G + S∞
S∞ = F G/(G – 1)
However, this calculus with a thought summation
of infinitely many numbers presupposes that the summation can be
carried out, i.e. that the paradox is solved. Thus, understood as a solution of
the paradox, this attempt is circular. The circularity is committed in the line
in which the summation is substituted by S∞, as this substitution presupposes
that the summations understood as a summation of infinitely many numbers can be
carried out, and that it has semantic meaning.
1.2.2
Mathematical limit.
In mathematics, the summation symbol including
its parameters denotes the limit of Sn for n going towards ∞, i.e. for n increasing without
limit. Thus, it does not denote a summation of infinitely many numbers in any
literal way.
More specifically, a function f(x) is said to
go towards the limit b for x going towards ∞, if
"εÎR+: $hÎN: "x:
x > h => 0 < |f(x) - b| ≤ ε.
This definition can be applied to
Sn = F ∑ni=0 G-i.
This means that the following must be proved:
"εÎR+: $hÎN: "n:
n > h => 0 < |Sn - S∞| ≤ ε, where Sn and S∞ can be substituted by the expressions in the above subsection, after
which the correctness can be confirmed.
However,
this mathematical formalism does not solve the problem, since as just mentioned
it does not deal with any sum, but with something else, for the definition
above just means this:
A
limit of a sum of n numbers for n going towards infinity is not a sum of infinitely
many numbers in some literal sense, but is a value to which this sum of n
numbers can get arbitrarily close without coming farther away for any higher n.
Our talk about infinitely many elements in a
certain set does not mean that the set contains a number of elements we can call “infinite” just as when we talk
about there being 1017 elements in a set, but that the set of these elements has no end, i.e. that we can continue
pointing out new elements in the set. Here we have an ordered set of infinitely
many intervals on a piece of the line, which means that we can continue
pointing out new, still smaller intervals, even succeeding ones in the ordered
set. The proof above is not in contradiction to Zeno’s paradox, but rather copies
it.
Finally, it must be observed that the above
definition only means that Sn comes relatively close to S∞, but not absolutely closer, as there are no absolute distances in geometry.
It is even so that the division of the line from Sn+1 to S∞ has
the same shape as the division from Sn to
S∞. So much the more the runner’s situation is
not changed.
1.3
An unfinished account.
The following attempt at a solution is inspired
by the fact that if Achilles velocity is v, the sum of the first n intervals of
time is Sn/v. Number
n of these time intervals has the duration F/(Gn-1 v).
This lead to that
solution of the paradox about Achilles and the tortoise that after each of its
steps the above account of it has only
concerned a certain space of time
and not the whole run.[2] Thus,
it only deals with what takes place within a sum of time intervals each of
which has become still smaller until some moment before and arbitrarily close
to the point where we would expect that Achilles would catch up with the tortoise:
It
is thus not the race that never
finishes but the very account of the
race that is put in such a way that
it never finishes, and therefore it self-evidently does not deal with the whole
fiction about the race. What is the case is therefore that Achilles does not catch
up with the tortoise within the space of
time that the account deal is
supposed to deal with. The reason to this is that the account has the
above-described recursive form.
We can imagine that that we have just attended
the race between Achilles and the tortoise and seen Achilles catch up with the
tortoise. Nevertheless, we can subsequently start describing the race in the
following way that can never end:
The
tortoise started with a lead of the length F. Achilles ran G times as fast as
the tortoise. At first, Achilles reached the place from which the tortoise
started. At that time, the tortoise had moved a stretch of the length F/G.
Thereafter Achilles reached the place where the tortoise was after the first
stage, etc.
As
mentioned, this account can never end for the reason that it is told in such a way that the sum of
these intervals of time at no point exceeds the duration of the race.
However, it is also told in way that makes
clear that Achilles’ situation is in principle is unchanged for each stage.
Does this mean that the paradox is solved? It
can be put forward that the account just points out a number of points one after
one on the time axis to which this fiction about past can be attached. This
takes place according to a definite algorithm defined by the account. It is not
paradoxical that, just as we cannot finish many other algorithms, we cannot
finish this algorithm either. In fact, we can both put forward and combine many
algorithms to division of a piece of line or a track into intervals, even with
more than one point of accumulation. These can of course be passed. It can also
be put forward that they just make up an intellectual construction that as such
has nothing with the race to do. However, this way of looking at the paradox is
to ignore Zeno’s message.
1.4
Scientific limitations.
After this going through, scientific considerations
concerning the paradox are strictly speaking irrelevant. However, this shall
not prevent us from looking at a scientific based proposal for a solution of
the problem the account obviously implies, viz. that Achilles only gets
arbitrarily close to the tortoise, but never catches up with it. The proposal
is based on the idea that the mathematical distances can be insignificant
according to scientific theories. Thus, it could be asserted that when, for the
one or the other reason, the difference in position cannot be measured,
Achilles and the tortoise have reached the same position on the racing track.
According to this, the race would simply be finished when the distance between
the two competitors and less than a certain magnitude. On its own premises,
however, this does not explain the fact that Achilles not only catches up with
the tortoise, but also runs past the tortoise, i.e. also with a length of half
a point.
Below it is explained that this proposal for a
solution does not fall in with Zeno’s premises, but just rejects the paradox.
This will also be discussed in a later treatise.
Zeno’s account cannot fully deal with the
physical world, for it is a question, which two physical points on the tortoise
and Achilles have to be compared during the race, especially as they are
moving. If we also involve Achilles’ steps and movements, when we talk about
minimal distances, the discussion becomes boundlessly complex. Thus, we must
regard the race as dealing with two points A and B that are moving along a line.
Mixing physical contingencies into the paradox is not to look at its
fundamental character. On the other hand, the mathematical definitions have to
be correctly understood and their terms not to be literally understood.
1.5.
Zeno’s dichotomy paradox.
In The Presocratic Philosophers, it is
advanced about “Zeno’s arguments about motion ...”:
“The
first asserts the non-existence of motion on the ground that that which is in
locomotion must arrive at the half-way stage before it arrives at the goal…” ([Kirk
1983] p. 270)
Because of its short form, this argument it can be interpreted in two ways.
(Cf. [Kirk 1983] p. 270.)
1.5.1
The right recursive dichotomy-paradox.
One of the interpretations reminds of the
paradox about Achilles and the tortoise, and is right recursive like this paradox:
To
reach his goal a runner must first run halfway to the goal, then the half of
the rest, etc.
Let us presuppose that the goal is the point 1
on the x-axis, and define the statement R(s).
R(s):
“The runner runs a distance of the length s”.
We can then define a recursive account C(d)
about how the runner runs from the point d to the point 1:
C(d)
-> R((1-d)/2) + C((1+d)/2).
The
whole account can thus expressed by:
C(0).
The
first four stages have the following form:
1:
C(0) -> R(1/2) + C(1/2).
2:
C(1/2) -> R(1/4) + C(3/4).
3:
C(3/4) -> R(1/8) + C(7/8).
4:
C(7/8) -> R(1/16) + C(15/16).
This
gives the following composite result:
1-4:
C(0) -> R(1/2) + R(1/4) + R(1/8) + R(1/16) + C(15/16).
The account is analogous to that about Achilles
and the tortoise. Thus, after each stage, the situation in this account is the
same as before the stage. For the remaining account still deals with the fact
that the runner is running a stretch that can described in the same way as before
the stage except that its length is here halved for each stage. Thus, we are
again dealing with an account that due
to its form never finishes. Moreover, it must be observed that just as it
was the case with the paradox about Achilles and the tortoise, after each stage
no real change has occurred, as there are no absolute distances in the mathematical
universe.
We can imagine that that we have just attended
the runner run a distance of 1 length unit. Nevertheless, we can start describing
the run in the following retrospective way that never reaches an end:
The
runner started from point 0. At first, he reached the middle of the distance.
Then he reached the middle of the remaining distance, etc.
As just mentioned, this account will never end
for the reason that it is told in such a
way that the sum of these intervals of time at no point exceeds the
duration of the run and especially that the runner’s situation remains
unchanged in principle.
1.5.2
The left recursive dichotomy paradox.
The other interpretation of the paradox is left
recursive and can be rendered in this way:
To
reach a point P halfway before his goal, a runner must first run to a point
halfway before the point P, etc.
Let us presuppose that the goal is the point 1
at the x-axis, and define the statement R(s):
R(s):
“The runner runs a distance of the length s”.
We can then define a recursive account C(d)
about how the runner runs from the point 0 to the point d:
D(d)
-> D(d/2) + R(d/2).
By
means of this, the whole account can be expressed by:
D(1).
The
first four stages have the following form:
1:
D(1) -> D(1/2) + R(1/2).
2:
D(1/2) -> D(1/4) + R(1/4).
3:
D(1/4) -> D(1/8) + R(1/8).
4:
D(1/8) -> D(1/16) + R(1/16).
This
gives the following composite result:
1-4:
D(1) -> D(1/16) + R(1/16) + R(1/8)+ R(1/4)+ R(1/2).
By this, it has been illustrated that the
account D about the whole race has to be finished before the account of runs of
the single stages can begin. This has already been established by the left recursivity.
Therefore, the account D will never explicitly mention or concern the start of the run in any of its stages. This
version is retrospective, and regarded
in this way, it is analogous to the first version.
We can again imagine that that we have just
attended the runner run a distance of 1 length unit. Nevertheless, we can start
describing the run in the following way that never will end:
The
runner started from point 0. Before he reached the middle of the distance, he
had to run half way to this point. Before he reached the latter point, he had to
run half way to this point, etc.
1.7
Aristotle’s arguments.
Aristotle
asserts that length and time can be called “infinite” as regards both divisibility and their
extremities, and concludes from this:
“So while a thing in a finite time cannot come in contact
with things quantitatively infinite, it can come in contact with things
infinite in respect of divisibility: for in this sense the time itself is also
infinite: and so we find that the time occupied by the passage over the infinite
is not a finite but an infinite time, and the contact with the infinites is
made by means of moments not finite but infinite in number. (After Gaye)” ([Kirk
70], p. 270)
Aristotle
naturally has a point in saying that a piece of line can be both 1) infinitely
long and 2) infinitely divisible, i.e. also if its length is finite, such as
Zeno demonstrates. Moreover, it is clear that about a runner’s passage of a
line piece we can say the same about the
division of the time the run lasts as about the division of the line piece
traversed. For the passage of the line piece and the passage of time goes
together. This means, however, that with his above pointing out Aristotle does
not solve the paradox, but only shows two sides of it.
Thus,
we can again look at the situation in which a runner has reached his goal. This
is a paradox, as it is impossible to him according to Zeno’s account. For in
order to arrive at it, the runner first must run half-way, and to arrive at
that point he must run half-way of this distance, etc. However, if the runner’s
velocity was h, he reached the halfway after the time (1/2)/h, he reached 1/4
of the distance at the time (1/4)/h, etc.
Obviously,
there is no end to the related problems we can think over in connection with
the three discussed paradoxes. According to The
Presocratic Philosophers, from Aristotle’s’ discussion the following left-recursive
reasoning can be deduced:
“(1) To reach his goal a runner must touch
infinitely many points ordered in the sequence 1/2, 1/4, 1/8, ...
(2) It is impossible to touch infinitely many
points in a finite time.
So
(3)
the runner cannot reach his goal.” ([Kirk 70], p. 270)
According
to Aristotle’s argumentation above n, it is possible to pass infinitely many
points in a finite time, wherefore (2) is false. As this argument, however,
deals with a space of time regarded as a
whole, it does not deal with the runner’s situation, which is in the now.
Therefore, the demonstration that (2) s false does not solve the paradox, but
only illustrates one aspect of it.
Aristotle
puts forward that this argument presupposes that the single intervals only
exist potentially. However, Aristotle wants to discuss this condition:
”...
when someone asks the question whether it is possible to traverse infinite
things - either in time or in distance - we must reply that in a way it is but
in a way it is not. For if they exist actually it is not possible, but if
potentially, it is; for someone in continuous movement has traversed infinite
things incidentally,...” ([Kirk 70], p. 271)
It
is true that this applies to e.g. the retrospective description. The question
is what it would mean that the intervals should exist actually, i.e. as intervals. The answer is that this is the case
for the single interval when it is
pointed out, which it is in the now. It is exactly intervals pointed out we
have to do with in the recursive interpretation of Zeno’s description. It is
true that according to this recursive account the run has no beginning, just as
it has no ending according to the right recursive account.
It must
be observed, that it cannot be the case that the intervals exist all of them
actually as something different from existing potentially; for we cannot point
them all out at one time.
1.7
Summarizing considerations.
As
it is simpler to comment on the right recursive account C above, a concluding
remark shall deal with this account. We can imagine that we are in a situation
in which a runner has reached the goal, and we can then ask how the runner has
been able to pass infinitely many distances of the lengths (½)n. The fact that there are infinitely many
distances of this kind before the point 1 just means that for each such distance
we can point at, we can point at one distance more. The fact that the runner has
run infinity many of this kind of distances before the point 1, therefore just means
that for each distance of this kind the runner has run, we can point at yet a
distance the runner has run further along the running track. To each n
corresponds the interval from 1-(½)n-1 to 1-(½)n. In this way, the runner passes
infinitely many points within a finite time: there is no end of the set of
points he touches in a finite time.
It
can be noted that Zeno’s account of the run deal with the present time, i.e. the runner’s
situation, while the retrospective explanations deal with past fictions. We
cannot explain the flight of time. We can only explain how it is to be in the
present time and look at a past-construction, as here, and at a future-construction.
As
already touched on we can certainly imagine and that a person objects: “A
little while ago you were talking to me. That was both real and present at that
time. Now it is past time. This proves that time is passing.”[3]
However, this only proves the fact that we have memory impressions, and that
the person in question is placing them in a past-construction. Naturally, an
unsolved philosophic problem cannot be solved by insufficient explanations. It
must rather remain unsolved for the moment, only illuminated.
Literature.
[Fraassen
80] Bas C. van Fraassen: The
Scientific Image
Oxford University Press (Oxford 1980)
Oxford University Press (Oxford 1980)
[Kirk
83] G.S. Kirk,
J.E. Raven),
M. Schofield:
The Presocratic Philosophers, 2. ed.,
(Cambridge 1983)
(Cambridge 1983)
[Salmon 70] Wesley C. Salmon, ed.: Zeno’s Paradoxes
Bobbs-Merrill (Indianapolis & New York, 1970)
Bobbs-Merrill (Indianapolis & New York, 1970)
[1] I have seen this argumentation somewhere, but have not been able to
find it again.
[2] Here I
owe to refer to Aristotle who discusses this subject in ([Kirk 70], p. 270).The
subject id treated in the subsection “Summarizing considerations”.
[3] Cf. the subsection, “The present
and past time and the future of the past” in Treatise No 1.
