23/10 2015
Zeno’s so-called paradoxes about movement we know
from Aristotle’s rendition:
The paradox about Achilles and the tortoise is the paradox Aristotle calls “the
so-called ‘Achilles’”:
”In
a race the quickest runner can never overtake the slowest, since the pursuer
must first reach the point whence the pursued started, so that the slower must
always hold a lead.” (239b14) ([Kirk 83], p. 272)
The dichotomy paradox.
This treatise uses the designation, “Zeno’s dichotomy
account” several times. This account enters as a part of the dichotomy paradox:
“...
The first [paradox] 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...” (239b11) ([Kirk 83], p. 270, my square bracket.)
Zeno’s dichotomy
account is simply
the latter statement:
”that
which is in locomotion must arrive at the half-way stage before it arrives at
the goal” (239b11) ([Kirk 83], p. 270)
Introduction.
These paradoxes have been discussed
ever since they were first put forward. One may ask himself, or at least
someone will ask why they have been that really, as we manage splendidly without
thinking about them. To this, it may be said that if a person thinks that it is
unworthy to let questions lie without seeking to understand and answer them, he
will have to attempt to make a contribution to this. Moreover, if one does not
in advance know whether these efforts will lead to a profitable result it is
extra unwise to leave them undone.
Here they are scrutinized and
interpreted.
General remarks about
interpretation and criticism.
When criticising a statement and
interpreting it not only necessary, but also profitable to avoid the serious
argumentation error that consists in “The easily refutable interpretation explained”.
This is explained here:
The easily refutable
interpretation
consists in not choosing the most difficultly refutable interpretation of a
statement one wants to refute:
When criticizing an argument, in the interest
of the truth, we ought not to choose the most easily refutable interpretation
thereof it, but the most difficultly refutable. For when the most easily
refutable interpretation has been refuted, the most difficultly refutable
interpretation has still not been refuted.
A more difficultly refutable
interpretation can be gained by removing criticisable subordinate details of
the text. This idea of concentrating on the matter itself leads to the most
fruitful interpretation in the easiest way. For by this, we can concentrate on
the matter itself and avoid futile speculations about what the author might
have thought.
Thus, it is not profitable to refute
the above-mentioned paradoxes by reference to ideas of a least measurable
distance or ideas about undividable particles or points or something similar.
The final version we will have to look at thus becomes an account that takes
place in a mathematical universe added a parameter we can call “time”
The understanding and the interpretation of
Zeno’s two accounts deals with both their mathematical content and the
interpretation of them. On the one side, it is not sufficient to understand
them as mathematical or physical statements. On the other side, their
mathematical and logical content must be included in the interpreting of them.
The mathematical content.
As to the mathematical and logical considerations,
we can focus on the dichotomy paragraph, as this corresponds to Achilleus run
relatively to the position of the tortoise.
As there is no absolute distances on
a line in geometry, no change has happened concerning the situation of the
object on the line after each traversing of the first half of the rest. This
can be illustrated with these two graphs:
0------------------------------1/2-------------3/4-----7/8------1
1/2-----------------------------3/4-------------7/8----15/16-----1
Therefore, task remains completely
the same after each passage of a partial stretch. Thus, the moving object in
Zeno’s analysis remain to the left of the goal, i.e. the open interval from and
including the starting point t0 unto t0 + 1/v, where v is
the velocity of the moving object. The division of the remaining part of the
line is even isomorph to division of the line before the last passage.
As Aristotle suggests, a similar
division of the time into time intervals can be made, wherefore there is time enough
to traverse all the intervals in the stretch.
However, all what can be said
about the stretch can be said about
the whole time interval, just about times instead of partial stretches. Thus, time
courses are just as paradoxical as passing stretches.
A
number of mathematicians have got the idea that of summing up all the partial
stretches (½)1, (½)2,
(½)3,...
The intention of the
mathematicians in question is obviously to prove that at the sum of all these
distances is equal to the whole distance, and that the traverse of them thus
lead to the goal.
Let Sn denote ∑ni=0(½)i, and let S∞ denote the limit value of Sn for
n going towards ∞, i.e. for n increasing unlimited. This expression does thus not denote
a summation of infinitely many numbers in any literal way.
In general a function f(x) is said to go towards
the limit value b for x going to wards ∞, if
"εÎR+: $hÎN: "x:
x > h => 0 < |f(x) - b| ≤ ε.
This
definition can be applied at Sn, which means that the following must be
proved:
"εÎR+: $hÎN: "n: n > h => 0 < |Sn - S∞|
≤ ε, where Sn can be substituted by the above definition, and S∞ with a proposal for a limit value, e.g. the value 1, where after the
correctness of this can be verified.
This
mathematical formalism however does not solve the paradox, as it, as just
mentioned, does not deal with any, but about a limit value, which is something
else, for the above definition 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 number
higher than n.
Thus, the above proof does not contradict
Zeno’s paradox, but rather copies it. In both the proof and the paradox, we are
dealing with something that comes arbitrarily close to a certain entity, but
without reaching it, namely respectively what is converged towards and the
goal.
There is nothing paradoxical in this. The same applies
to other function, for example the function f(x): 1 – 1/x.
The interpretation.
The interpretation is important and
necessary, for just as all statements is supposed to have semantic meaning,
they ae also presupposed to have a content in the form of a message.
It can be objected, quite true, that
the dichotomy account is simply told in such a way that it does not get to an
end. However, it is exactly the message of the account that it is possible to
describe the run in this way. For this account aims at showing that we can come
closer and closer to a definite future moment without reaching it, just as an
object can come still closer to a point at a line without reaching it.
It can also be objected that according
to common sense and our structured experiences, we can reach a point farer away
at a line according to the equation:
s = v * t, where s denotes the
length of the traversed stretch of an object, v the velocity of the object, and
t the elapsed time.
This is certainly based on memorized
experiences about an object at a certain moment having been at a certain place
and later having been at another place, i.e. that it has reached this, i.e. the
goal. However, the question is how this movement happens.
We cannot describe this passage point
after point and neither as partial interval after partial interval. We can only
imagine that we have been going from the past and arbitrarily close to the now,
but not how the transfer to this happens, as the last part of this transition
can be divided just as the whole passage can. Thus, the past is only something
we have some diffuse ideas about. However, in the now we are already in the
now.
Cf. the links below.
Conclusion.
It has appeared that the function f(n)
= ∑ni=0(½)i, which expresses the length of the first
n partial stretches, has the limit value 1 for n going towards ∞,
but does not assume the value 1 for any n. This very fact is not paradoxical.
It
can be objected that in practise this have no sense used on the division of a
stretch in partial stretches of the length (½)i. This is not just to choose the most easily
refutable interpretation, though, but also to omit an understanding of Zeno’s
account. Alternatively, one could just as well assert that the function s(t) =
v * t gradually assumes the value 1. However what Zeno analyses is just the
gradually gaining any value of this function. The state that s(t) < 1 we can
call being in the past. The state that s(t) = 1, we can call being in the now.
The transition to this state does not happen to any value of t in the past.
Treatise No 8:
Aristoteles advances
that distance and time can be called “infinite” both as to divisibility and
their utmost end, and concludes from this that the goal can be reached in a
finite time:
“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 83], p. 270, my
underlinings.)
