This
version is preliminary.
Contents.
Summary.
This
treatise contains a slightly revised extracts from the treatise “A criticism of
W.C. Salmon’s Zeno’s Paradoxes” ([Criticism of “Zeno’s),
which addresses a number of the contributions to Wesley C. Salmon’s anthology Zeno’s Paradoxes ([Salmon 70]). Though
this treatise mainly resulted in the pointing out the misleading digressions
and other errors in these contributions, the very work with this criticism has
resulted in further considerations about Zeno’s runner paradoxes than those put
forward in Treatise No 8 ([Treatise No 8]), of which it also must be understood as being
a continuation. These and only these considerations are presented in this
article.
1.
Zeno’s challenge.
1.1
Zeno’s accounts.
This treatise uses the designation, “Zeno’s dichotomy
account” several times. It 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 70], 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” (239b) ([Kirk 70], p. 270)
The paradox about Achilles and the tortoise, i.e. the paradox Aristotle calls “the
so-called ‘Achilles’”, (239b14) ([Kirk 70], p. 272), sounds:
”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)
Zeno’s opponets are common sense and argumentation
errors.
2.1
Common sense.
The concept of common sense is
especially important in the discussion of Zeno’s paradoxes with runners, as
these relate critically to our unreflective common
sense views.
The
expression common sense denotes our
common understanding of the world we live in; it thus concerns the usable meaning or sense of our talk. Our
common sense views are based on our
memories and a model of reality, which in turn is based on our memories. Having
a common sense view consists in
understanding our talk about this model literally. This understanding of the
state of things is not necessarily especially reflected, but ingrained.
A simple,
but relevant part of such a model is the equation s = v * t, where s = covered
stretch, v = velocity, t = elapsed time. According to this, a runner that runs
with a definite velocity will run a stretch of a given length in a computable
time.
It must be observed that since Zeno
asserts a quite different statement, presupposing a literal understanding of this
common sense view would involve an
easily refutable interpretation of the paradoxes, because of the contradiction
that arises in this way. But, we naturally cannot presuppose common sense in the discussion of Zeno’s
paradoxes as this basis is what is criticized. This would thus be to ignore the
displayed problem. Instead, we must address the advanced problem on its own principles.
We cannot just discuss Zeno’s
paradox on the basis of a common sense
view of the paradox of the following kind: “The runner is at A. Zeno asserts
that the runner cannot arrive at B. However, the runner can actually do this
for the evident reason X”. This is exemplified in the section “Argumentation
errors”.
Instead, we must take a step
backwards and considers whether we can derive a message from Zeno’s account
whether this is in contradiction to our common
sense view or not. Thi is done I the section “Zeno’s message”.
It follows from earlier sections in
these treatises that philosophical clarification concerns these questions: What
have we actually to do with, what are we confronted with, what can we at most
put into our existence statements? This also involves a study of our common sense assumptions. It must be
observed that though we regard a philosophical statement as implausible, we
must be able to justify why. It must be observed, that this cannot be done by
means of so-called good assumptions.
Considerations about Zeno’s paradox
results in a view of time, space, and movement that is not in agreement with a common sense view of these concepts.
This means that as long as Zeno’s paradoxes have not been refuted, we cannot
presuppose the common sense view they
criticise, e.g. a view of time that compares the past with the now.
It is thus wrong to presuppose a common sense view in the discussion of Zeno’s account. Thereby, we
do not discuss the latter subject, but a mix of the two subjects, and already
in this way introduce an inconsistence. In this way we fail by presupposing
what Zeno doubts. However, First, we
must understand, then, we can criticize.
2.2
Argumentation errors.
For this reason it is both relevant
and useful to relate to the following argumentation error:
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 of 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.
Moreover, a great number of useless objections
can be refused with this justification.
Some have objected to Zeno’s
paradoxes that the sum of the runner’s partial stretches can be computed as the
limit value of a sum-function and that this has a finite value, wherefore the
runner reaches his goal. Against this, first, it can be advanced that the
definition pictures Zeno’s paradox. Second, the case deals with the runner’s
situation wherever he is.
It has also been objected to the
paradoxes that we are dealing with minimal distances, which imply an infinite
number of leaps, which thus gets over, wherefore the runner reaches the goal.
To this can be said that this objection deals with assertions about a putative
world in itself. The case cannot be about small distances in a literal sense at
all, as we are not confronted with such entities, but only with our
experiences.
Talking about not-observable
physical points that touch each other is scientific realist talk. Of course,
this does not imply that Zeno is a scientific anti-realist, but just that if we
want a usable understanding of Zeno’s paradox it cannot be based on scientific
realism, or other imaginations. However, the contributors’ talk about physical
points concerns exactly these imagined details. Therefore, it is most
advantageous to the philosophical result to choose the most difficultly
refutable interpretation, which means that we must interpret Zeno’s paradox as
an account that takes place in a mathematical model.
Some may assert that the runner will
reach his goal when his distance to the goal is smaller than the diameter of a
physical point. Exactly therefore, however, the paradox must interpreted more
abstractly.
It must be pointed out, that a limit
value as a concept is not a sum but
just a value, to which the summing up can come arbitrarily close without coming
farther away from it later.
3.
The derivation of Zeno’s message.
3.1The
abstract version of Zeno’s accounts.
Of course, Zeno’s accounts does not
have to be literally understood: Instead of being about two runners in the
examples, the paradox could be about e.g. two rolling vehicles with each
individual velocity, or it could be about two objects that in virtue of their
inertia rush along out in the space with individual velocities relative to the
fix stars. This means that a number of physical objections are annulled. We can
take one more step and look at a version that is not based on the immediate
phenomena, but lets the process take place in the imagination. Such an
interpretation is more difficult to refute and can thus be more profitable.
Thus, we only need to talk about
what takes place in a mathematical model, e.g. how far an object hat reached at
certain moments.
If we want to refute Zeno’s paradox
we have refute the statement Z: “The runner does not reach the goal after the
passage of any stretch” in the account or the more clear statement “There is no
stretch such that the runner reaches the goal after his passage of it”. Of
course, we have to refute the most difficultly refutable version of the latter
statement.
The
right-recursive dichotomy paradox can be interpreted thus: “The runner is at a point B’, wherever he is, but
he cannot have arrived at it from any place, A’, where he was previously,
irrespective of how close A’ was to B’”.
The left-recursive
dichotomy paradox says “The runner cannot arrive at any place, B’, to which he is
said to arrive, irrespective of how close B’ is to the starting point A”.
It must be observed that what can be
said about traversing distances and arriving at places can just as well be said
about time intervals and moments.
3.2
Zeno’s message.
Zeno’s dichotomy paradox and that
about Achilles, which are both quoted in the sub section “Introduction”, make
clear that we do not fully understand what movement is and thus what time is
neither. The fact that science can describe movement by means of equations is
something else. For these can just be used for computing the results of certain
acts.
Above, it has been argued that we do
not need to understand the investment that Zeno gives his account literally,
but instead can use an abstract version.
The following can even be said to apply:
If we want
to benefit from Zeno’s account, in the last end we must regard it as a mathematical allegory that takes
place in a mathematical world added time, in which there is a one-to-one
relation between time and the place that the runner is passing.
It is true that Zeno’s account starts from a common sense view of time and space, but only to show that this
view is insufficient:
With
the common sense view of time as a
starting point, Zeno challenges us to explain how on this basis we can reach a
future moment, and thus how we have arrived at the now, too. We could come
closer and closer to the now, but there remains an insurmountable barrier in
front of us.
Some
may assert that we naturally reach the now in
the now. However, in the now we are
simply in the now. The question is how the last leap happened, as there is no
fundamental difference between this problem for 1 minute ago, for ½ minute ago,
for ¼ minute ago, etc.
The
result agrees with this view:
The question
about how the discussed points (or the track’s partial stretches) are reached one
by one deals with our present time in which this is experienced.
Contrary to
this, the fact that we can look back at a finished period concerns a past-fiction.
This agrees with the fact that our memory
impressions make up the basis of our past-fictions and future-fictions.
In these treatises, a similar
conclusion has already been reached in two different ways, viz. in the
subsection “Time” in Treatise no 1
and in the section “The velocity of time” in Treatise no 5. However, philosophy does not deal with gathering
points like the natural sciences, e.g. geology, which seeks for answers to the
question about the age of the earth, or about where there can be water in the
underground. Philosophy deals with inter alia the semantic meaning of our
existence-statements and what we can at most be said to be confronted with,
versus what are just unreflected common sense imaginations.
3.4
Dualism.
Thus, we are dealing with a dualism
between past and present time. We cannot come from past time to present, as the
present is real, whereas the past cannot even be said to be former present, but
just a construction. This resembles the realists’ so-called soul-body dualism,
i.e. the dualism between experience and matter:
With his
paradox, Zeno seeks to illustrate that the concept of time contains nothing
that can lead us forward to a new now. Thus, it has no sense to talk about
anything we can call the flow of time.
Thus, it has no sense either to talk
about a past as something literally existing, but only about a past-fiction. Inspired
by Zeno’s paradox and by Aristotle[1], the following can be put forward:
One unit of
time ago, the time in our past-fiction was one unit of time less than now. When
half of the time had elapsed, the time was half a unit of time more and there
was half a time to now. When half of the resting time had passed, the time was
a quarter of a unit more and there was a quarter of a time left to now. Etc.
Therefore, nothing has decisively happened concerning the finish of this unit
of time.
It is remarkable, that the dualism
between the present time and the future can be compared to the dualism between
matter and our sense impressions. The latter dualism arises to the scientific
realists because they presuppose that our sense impressions have a material
basis. According to the realists, this basis starts a chain of cause and effect
originating from the sensed matter, first outside our body, then in a sense
organ, next in our brain where a number of material
processes, i.e. changes in mater, finally turn into impressions in the
consciousness of the person in question. According to scientific realism,
science can explain all the first steps; but its adherents cannot explain the
last step from matter to sense impression in any similar way, only postulate.
The last material explanations can be refined more and more, but this does not
lead us to the goal. The realists’ material explanation can even be followed as
a line that possibly is divided in a number of lines in the brain of the
sensing person for lastly to end neither here nor there without approaching any
sense impression.
4.
Conclusion.
In the subsection “Zeno’s message”, a
justified interpretation of Zeno’s account is put forward. It has above been
emphasized that we must understand before we can criticize. So now, when we
have a justified interpretation, the question is not only how good this
interpretation is, but also whether it expresses something true about the
world. In other words, can we substantiate
that there is a dualism between the now and the future (or between the past and
the now) as described? A result of this interpretation was that though we can
come closer and closer to the now, an insurmountable barrier remains in front
of us.
What we have to do with, are first
the phenomena and next our own constructed mathematical models understood as
such and not a race in all its imagined details as if these had literal
existence. The question is how the difference between present tine and the
future is exceeded. The concept of really existing arbitrarily small entities does
not even belong to a scientific model. Thus, we are dealing with the fact that
irrespectively of how close we imagine that we come to a certain moment there
remains a distance to the moment.
In Zeno’s paradoxes, we do not have to
do with such entities at all, but only with a literary fiction:
In
order to be interesting at all and not trite, Zeno’s account must be
interpreted as a fiction that as
such is not about the phenomena, but takes space in a mathematical model added
a concept of time. This is supported by the fact that according to the nature
of the case, the small entities that are discussed cannot be phenomena but only
our imaginations.
Zeno’s
dichotomy paradox consists of a single statement. However, just as it applies
to literary fictions, this account
must be understood under the headline “Imagine what follows and only what can
be inferred from it in the world in
which it takes place”. Thus, talking about “physical points” is a digression
from the text. This only deals with our imaginations of being at a definite place
now and arriving at a different place at a later moment.
Here follows an everyday example of
the dichotomy paradox: In Albert Camus’ novel, L’etranger Mersault has gone to his mother’s funeral
and is waiting to meet the principal of the old people's home. Without saying
that it is intended, Camus’ description of the waiting time can illustrate
Zeno’s paradox:
”Comme il était
occupé, j’ai attendu un peu. Pendant tout ce temps, le concierge a parlé et ensuite,
j’ai vu le directeur: il m‘a reçu dans son bureau.” ([Camus 02] p. 11, my underlining.)
When do Mersault meet the principal?
“Thereafter”! That is when the waiting time is over. However, it is solely over
by Mersault’s meeting the principal. Here nothing leads to the cease of the
waiting time: the situation remains the same and the same... The cease of the waiting
time is a paradox. In the next part of the text, Mersault has so to say arrived
at the goal. The impossible has here come, but only by virtue of our dealing
with a text. In our real life, instead, we can be dealing with the now versus our
memory impressions. Some may object that a number of actions concurrently take
place behind the principal’s closed door, which thus makes up an ongoing
process. However, these have nothing to do with Mersault’s experiences, and besides
they would subject to the dichotomy paradox, too.
It can be objected that we live and
act in accordance with our common sense
view of the world, and that this goes on without problems. Against this, it can
be put forward that this just means that our common sense view is consistent,
at least within the frame in which we act: our horizon. In other words, our common sense view of the world exactly only
concerns our common sense view of the
world. It never reaches outside itself.
We can have a fiction about the
future, but we never experience any future. Contrary to this, we can have
memory impressions of past fictions about the now. This does not occasion any
contradiction. Only when we begin more closely to consider how we reach this
future, do we come up against a contradiction. This is derivable from Zeno’s runner
paradoxes.
Other
treatises.
[Treatise
No 8] https://antiintroduktionisme.wordpress.com/2015/09/23/anti-introduktionisme-afhandling-nr-8/
[Criticsm of “Zeno’s”] http://filosofisk-debat.blogspot.dk/2014/06/en-kritik-af-wc-salmons-zenos-paradoxes.html
Literature.
[Camus 78] Albert
Camus: L’etranger
Gallimard (France 2002).
Gallimard (France 2002).
[Dummett 78] Michael Dummett: Truth and Other Enigmas
Duckworth (London 1978).
Duckworth (London 1978).
[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)
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