Summary.
This article addresses the following paradoxes of Zeno: 1) The dichotomy
paradox, 2) The paradox about Achilles, 2) The flying arrow, and 4) The moving
rows.
Introduction.
The dichotomy paradox and that about Achilles have been earlier treated in
two related articles (Cf. [Treatise 8] and [Treatise 8F]) and some previous
articles, that is mentioned in the list of literature at “Essays and treatises”.
The dichotomy paradox has two interpretations, a right recursive and a
left recursive. The relation between these two paradoxes is shown. (Cf. the
subsection “The right recursive
dichotomy-paradox” in Treatise No 8.)
The paradox about the flying arrow, on the other hand, immediately
reminds of the left recursive dichotomy paradox, according to which it is
impossible to point at any first place the object reaches.
The paradox about the moving rows, which is explained rather
circumstantially in Aristotle’s rendition, however, only deals with the consistency
of a scientific fiction, but not with our experienced world.
The dichotomy
paradox.
The dichotomy paradox sounds
like this:
“... 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
70], p. 270, my square brackets.)
With this it is suggested that this
can take place arbitrarily many times, as the same also applies to the remaining
stretch.
Some put forwards a counter argument
based on making an expression of the sum of the length of the first thus
described stretches. Next, they advance that this sum will go towards the
length of the whole stretch for n going towards infinity. This limit value is
then called a “sum”.
0------------------------------1/2-------------3/4-----7/8------1
Cf. The section “Summation of
infinitely many values” in Treatise No 8,
in which this is put forward:
“A limit value 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 later (for any number higher than
n).”
Instead, it can be realized that
after having traversed n such stretches the remaining stretch has the length (½)n.
as its s halved after each of these arrivals.
This illustrates that it is possible
to get arbitrarily close at the computed endpoint of the stretch without
reaching it. Thus, no moment is found before the goal in which the goal is
reached. And in the very goal the goal is not reached for the moving object is
already at the goal.
Achilles.
The paradox that is known by the
name as “Achilles and the tortoise” is the paradox that 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
70], p. 272)
This paradox is apparently just a
more complicated version of the dichotomy paradox with a moving goal. With this
moving goal as a reference system, it is not essentially different from the
dichotomy paradox.
On common sense premises, we can
compute when the quickest runner catches up the slowest thus:
Let us assume that the velocity of
the fastest runner is V, that it is G times the velocity of the slowest runner,
and that the leap was F at the beginning of the run.
It can then be calculated that the
whole stretch the length:
F’ = F * G / (G - 1).
This means that when the fastest runner
reaches the starting point of the slowest runner, he has run F/F’ = (G - 1)/G of the whole stretch F’ and lacks 1/G of F’.
Thus, the stretch F’ is divided in
the ratio (G – 1) to 1.
After n cases of this, (1/G)n of the whole stretch remains.
Therefore, this paradox illustrates
the same as the dichotomy paradox:
When an object gets closer to a
slower object in the same direction as this, it can come arbitrarily close to
the foremost without reaching it. I.e. there is no point on the line where the
foremost object is reached, but only a point at which it is already reached.
Let us just accept the common-sense
view that an object that moves at a uniform velocity, moves in accordance with
this equation:
s = v * t, where s = elapsed stretch, v = the
velocity of the object, t = elapsed time.
Moreover, let us accept that the
object therefore will reach the goal at the time T = S / V.
This means that a runner that runs at
a certain velocity will run a stretch of certain length during a certain time.
Exactly therefore there is a paradox. For we cannot
point any point before the goal is reached.
The flying arrow.
This paradox sounds thus:
“Zeno argues fallaciously; for if,
he says, everything always rests when it is against what is equal, and what is
in locomotion is always in the now, the arrow in locomotion is motionless. But
this is false, for time is not composed of indivisible ‘nows’, no more than is
any other magnitude.” (239b30-33) ([Kirk 70], p. 273)
“The
third [argument concerning motion] is the one just mentioned, that the arrow in
locomotion is at rest. This follows from assuming that time is composed of ‘nows’;
for if that is not granted, the conclusion will not follow.” (239b30-33) ([Kirk 70], p. 273)
‘Zeno abolishes motion, saying “What
is in motion moves neither in the place it is nor in one in which it is not.”’ ([Kirk 70], p. 273)
We can here assume that the
expression “what is equal” refers to the space
the object fills up.
It is obvious to think that
naturally the space that an object fills up follows the movement of the object,
and that form this it follows that the first premise is wrong.
That an object moves, entails that
its distance to this place becomes positive. This involves the same problem as the left-recursive dichotomy paradox points
out. In this way, the paradox about the flying arrow clearly resembles the left
recursive dichotomy paradox.
It must be observed that irrespective
of how short we can ascertain that an object has moved from the mentioned place,
this is not to observe a movement. For we must distinguish between our very
sensation of the move of an object as such and the realized change of the
position of an object.
Movement as such is thus something
we sense (in the now), including its various aspects, just as we have sensation
of colour. Our very impression of the movement of an object is elusive and
cannot be kept. We can certainly follow the movement of an object with our
sight, but then the object in question does not move relative to our focus, of course.
It has earlier, in the Treatises no 1-6, been maintained that the
only thing we deal with or is confronted is the now with all its content. Thus,
we do not experience the whole passage of an object from a point A to a point B
as such, except when we actively follow it, but of course not at once.
We can have 1) memory or
documentation of an object having been at a certain place at a certain earlier
moment. Likewise, we can have 2) memories about an object having been at a
certain place at a certain moment before the moment that is remembered or noted
down, and memory or documentation of ur subsequently having seen it turning up.
For that reason, we can have certain expectations concerning point 1.
Our picture of this can be
illustrated by the above-mentioned equation of the motion that involves the
described paradox. WE can see the bus moving when it turns up at the corner,
however, the picture that the equation of movement give us cannot be perceived
as a literal reality only as a useful imagination or theoretical construction.
The
moving rows.
This paradox sounds thus:
“The fourth is the
one about equal bodies which move in opposite directions past equal bodies in a
stadium at equal speed, the one row from the end of the stadium [towards us]
and the other from the middle [away from us] – in which he thinks it follows
that half the time is equal to [its] double. The fallacy consists in requiring
that things which move at equal speed past a moving body and past a body at
rest of equal magnitude take an equal time. But this is false. For example, let
the stationary equal bodies be Α, Α …; let Β, Β … be those starting from the
middle, equal in number and magnitude to them; let Γ, Γ … be those starting
from the end, equal in number and magnitude to them [sc. the As], and equal in
speed to the Βs. Now it follows that the first Β and the first Γ are at the end
at the same time, as they [sc. the Βs and Γ s] move past each other. And it
follows that the Γ [sc. the first Γ] has gone right past all of them [sc. the
Βs], but the Β [sc. the first Β] past only half [what it passes, sc. the Αs]:
so the time is half, for each is alongside each for an equal time. And
at the same time it follows that the first Β has gone past all the Γs; for the
first Γ and the first Β will be at opposite ends at the same time, because both
are an equal time alongside the Αs. This then is his argument, and it depends
on the falsehood we have mentioned.” ([Kirk 70], p. 275, my underlining.)
326 “Α = stationary bodies.
Β = bodies moving from Δ towards E.
Γ = bodies moving from E towards Δ.
Δ = starting-place.
E = goal.” ([Kirk 70], p. 275)
We only need to assume that the
purpose of this paradox is to prove that movement is impossible, by proving
that the idea of it involves a paradox.
It must thus be observed that also a
discussion of an old text as this deals with deducing a possible message from
it, a message that has relevance here and now. Statements about that at its
time the text was so and so profound, is not relevant philosophically, but only
historically.
Thus, it has no sense to involve a
concept of the author’s possible intentions with the text, but instead to
discuss what it implicates or can be used to.[i]
Most of the text deals with
technicalities that, when they are uncovered, reveals the substance, that each
time a B has passed an A, then it has passed two Γs. This means, according to
common sense, that if a B has moved one length unit per time unit relative to
the row of As, then it has moved two length units per time unit relative to the
row of Γs that are moving with the same velocity in the opposite direction.
However, according to the quote, it also
takes the same time for a B to pass an Γ as to pass an A, because it is said
that each passage of another object takes the same time.
This is certainly quite wrong
according to common sense and science. However, kit must be remembered that
many mathematicians thinks that Zeno’s account of the two runners just involves
that the hindmost runner catches up with the foremost runner, and nothing else.
Perhaps I commit a similar error, not by ascertaining this error, but by
overlooking something else. And therefore, I must reserve my point of view.
However, if we imagine that we are
dealing with a fiction about the regularity of nature, in which these objects
can only move a minimal length unit at time without intermediate positions per
minimal time unit, then what has been said about the B’s passage past the A’s
is correct, isolated seen. Likewise, what has been said about the Bs and the Γs
passage past each other is correct isolated seen, but together the two facts
contain the inconsistencies mentioned. However, this does just mean that this
freely invented construction is self-contradicting, but says nothing
philosophical about the reality.
So for this reason alone, it is
unjustified when the author of the article asserts that this paradox have
occasioned the following question.:
“(…) if the distance a body moves is simply a function
of its positions relative to other bodies, is there any absolute basis for
ascribing movement to it at all?” ([Kirk 70], p. 276)
Moreover, to this it must of course
be answered negatively, as the word “relative” denotes the opposite of absolute. In fact, the linear movement
of an object can only be described
relative to another body. So there is no absolute velocities.
We can compare the situations by use
of the understood reference system and of an alternative reference system.
Reference system: The A-row.
Movements in length units per
2 time unites.
B: +2
Γ: -2
Here, the B’s move 2 length units to the right. The Γ’s move 2 length
units to the left in the reference system A. This is two independent movements
irrespective the moment of the start: the total change of distance becomes 4
length units
Reference system: The Γ-row.
Movements in length units per 2 time unites.
A: +2
B: +4
Γ: 0
In the reference system Γ, it is
clearly seen that the B’s are moving double so quickly as the As. But it is the
same that happens, physically.
[Salmon
70] Wesley C. Salmon, ed.: Zeno’s
Paradoxes
Bobbs-Merrill (Indianapolis & New York, 1970)
Bobbs-Merrill (Indianapolis & New York, 1970)
[Treatise 8] Achilles and the tortoise. Fra Treatise No 8.
http://ovemk2.blogspot.dk/2015/06/achilles-and-tortoise.html
http://ovemk2.blogspot.dk/2015/06/achilles-and-tortoise.html
[Treatise 8F] Zeno’s runner-paradoxes.
http://ovemk2.blogspot.dk/2015/05/zenos-runner-paradoxes.html
http://ovemk2.blogspot.dk/2015/05/zenos-runner-paradoxes.html
Zeno’s paradoxes about movement, a
short version: http://ovemk2.blogspot.dk/2015/10/9102015-zenosparadoxes-about-movement.html
Endnotes.
[i] Thus, it can be considered whether this paradox’s description
of the movement of the referred uniform objects and the placing of them can be
used to discuss the consistency of the idea about minimal length and time
units. It has been put forward that such an assumption can be used to refute
the paradox about the two runners.
This
is mentioned in the Appendix “A.2 Wisdom’s counter arguments based on physics”
in the treatise A criticism of some
points in Salmon’s Zeno’s Paradoxes.
Thus,
Wisdom advances as an argument for the falsity of Zeno’s account that Achilles
certainly can reach the place where and that this can be repeated a number of
times but that at a point the description does not longer apply to a physical
race. ([Salmon 70] p. 85.)
The carrying through of this, however, presupposes that this idea is
consistent. Zenons argumentation can then be regarded as an attempt to prove
its inconsistence.
On
the other side, this paradox cannot be used to refute the two first paradoxes,
as the idea of minimal lengths and time units would involve a more easily
refutable interpretation than the omission of it, e.g. the mentioned mathematical
interpretation.
On
the own premises of the ides, it must likewise be observed that movement per
definition is relative to other objects. It is this misleading to say that the
As are stationary. Instead, however, we can choose the As as a reference
system.
If
we nevertheless accept the premises, we can (tentatively) assume that such
minimal units exist, and that the length of each object is equal to the minimal
length unit, and that the time it takes for a B to pass an A is identical to
the minimal time unit. However, it must be observed that such an idea, however,
is only a fiction belonging to scientific realism, just as Democritus’ ideas
does.
It
must naturally be observed that there is a difference between talking about
what happens during a specific time unit and talking about velocity, i.e. what
happens per time unit. The latter subject is just an abstraction in the form of
the result of a division. Besides, how should we talk about a velocity that is
different from one length unit per time unit in this universe?
Thus,
we can certainly talk about a velocity that is two length units per time unit,
or ½ length unit per time unit, if it is interpreted such that it just means
that it takes two time units to move an object one length unit in this
universe.
According
to the referred line if thought, we can only talk about velocities that is one
length unit per time unit. However, the case is problematic:
If
a minimal object e.g. passes three objects per time unit, it cannot be at the
side of the middle one, bit only at the side of the last one.
It
can then be asserted to be justification that we cannot use our daily day
experiences in this (thought) micro-world.
