As some of the contributors prefer to discuss
the problem of infinity machines, instead
of Zeno’s accounts, and as a number of them relate Zeno’s paradoxes to this
concept, it has been necessary to discuss this subject first.
We can take the right-recursive dichotomy
paradox (cf. Treatise No 8) as a
starting point, and define that the length of the distance is 1 unit of length,
and that the run takes 1 unit of time. The line of thought behind the infinity
machines can then be said to be this. If it is possible 1) to run infinitely many
stages in a finite time, then it must be examined whether it is also possible 2)
to perform some distinct physical act of a given kind at each stage, such that
infinitely many of these acts are performed within a finite space of time, i.e.
such that for each n acts performed in succession, then an n’th act is
performed. Examining this relation, however, does not lead us closer to an
understanding of Zeno’s message, as the discussion of infinity machines takes
place within a common sense frame,
while on the contrary Zeno’s message is a criticism of our common sense view of time.
The physical act can e.g. consist in turning on
and off a lamp that are equipped with a single push-button. The example is put
forward by to James Thomsen in his article Tasks
and Super-Tasks ([Salmon 70] pp. 89-102). We
can imagine that the lamp is on at the start of the run, and that the button is
pushed each time the runner starts a new stretch according the right-recursive
dichotomy paradox. I.e. that when the runner reaches halfway to the goal, the
button is pushed, such that the lamp is switched off, and when he reaches
halfway of the rest, the button is again pushed such that the lamp is on, etc.
It is both a question whether this is
physically possible and whether it is logically possible. The physical problem
is that each time the runner reaches a new stage, the time he has to push on
the button is halved. Therefore, the physical question is whether this thought
experiment is in agreement with the laws of nature, e.g. that the total
physical energy used on the pushes
on the button at least does not go towards infinity for the number of performed
pushes going to infinity. This can be fulfilled if this use of energy is halved
for each traversed stage, such that the sum of these is only the double of the
energy required by the first push.
By this, the length of a push on the button will
soon be less than the diameter of an electron, and this reduction factor will
be gained repeatedly. In
the article Modern science and Zeno’s
Paradoxes of Motion ([Salmon
70] pp. 200-244), Adolf
Grünbaum makes much of showing that this does not make up a problem. However,
devices that can handle such arbitrarily small movements cannot necessarily be
constructed in advance, as each order of magnitude demands an individual device.
At any rate, we must disregard atomic theories and presuppose that the matter
is infinitely divisible, etc. Thus, the remaining question must be whether the
experiment involves a logical inconsistency.
Zeno’s two descriptions of a run make up
paradoxes because we are assured that the run can be finished, though it
according to Zeno’s description will not be finished. According to the
explanations in the subsection “An unfinished account” in Treatise No 8, the reason is that Zeno’s description by virtue of
its recursive form without a stop, i.e. without a non-recursive case, has been worded such that it will never deal
with any termination of the run. However, this is due to facts: For each step
in the recursive description there
is yet a similar step.
Contrary to this, the association from Zeno’s account of the
run, which the concept of infinity machine can be said to be, does not make up a
paradox. The question is instead whether a certain physical fiction is possible,
and whether it implies a logical contradiction. Thus, we are just dealing with
a question that can be answered with a yes
or a no, and a justification. This
question does not doubt our common sense
view.
It must be observed that if a certain infinity
machine is possible, any of its acts must be practicable individually, i.e.
also as the first one. For instance, each push on the button is an individual
act, namely in the sense that it is not conditioned by the preceding acts. This
means that if the act is possible with a “run-up”, i.e. with slower predecessors,
it must also be possible without these. It also means that for each arbitrarily
fast act that is performed at some place in the course, it must also be
possible to carry it through as the first
act. In fact, we can replace all the preceding acts with performances of this
act. As we can always find a yet faster act further on in the course, in fact
an arbitrarily fast one, the whole course can pass off arbitrarily fast. However,
this is only meant as an illustration of what we have to do with, for it hardly
has any decisive importance to the discussion of infinity machines.
We can imagine that the turning on and off the
lamp is operated by a watch, and that a just completed process of turning on
and off has lasted for one time unit. Then we can then start relating the
process in this way (which has no end):
The lamp was switched on at time
0. After ½ unit of time, the button was pushed. After yet ¼ unit of time, the
button was pushed. Etc.
In itself, this is logically unproblematic. Contrary
to this, it is not possible to perform these acts in the opposite order though
all the acts are individual. For this would be the same as carrying the acts through
in the way it is described in the left-recursive dichotomy paradox.
The left-recursive dichotomy paradox could be
solved, or at least clarified, like the right recursive one. The question is
whether a version with an infinity machine can correspond to this paradox. We
can again imagine that the pushing on and off just has taken one time unit. Then
we can relate the process in the following way (which has no end).
Before the time had elapsed, ½ time
unit had to elapse, and the button was pushed. Before the latter time had
elapsed, ¼ time unit had to elapse, and the button was pushed. Etc.
However, the process can be described more
simply:
The button was pushed on repeatedly.
The last push on the button was done at the time ½ and lasted ½ time unit. The push
immediately before this push was done at the time ¼ and lasted ¼ time unit. The
push before this was done at the time ⅛ and lasted ⅛ time unit,
etc.
The question is whether this fiction about the
past is possible at all, especially because there is no first push.
Concerning the right-recursive version, in the
article “Tasks and Super-tasks” James Thomson asserts that either the lamp must
be on or off after the execution of the infinity machine, but that this is
impossible as each switching on was followed by a switching off and vice versa.
([Salmon 70] p. 95) This is countered by Paul Benacerraf in the article “Tasks,
Super-Tasks and the Modern Eleatics” with the allegation that this argument
only concerns the very execution of the machine. ([Salmon 70] p. 108)
This is true, but as a physical object, the lamp must nevertheless be either on or off
after the execution. It is thus reasonable to ask the question which of these
states is present and why.
It is as if at this point the is in a state
that has no a cause.
In fact, it is easy to get these imaginative associations: 1) The intensely frequent
acts somehow take place at a time axis that indefinitely bends away from the
axis we know and, and which after the execution does not hit the place in time
and space where the lamp is. 2) This can be explained with the idea that time
consists of changes, which in these case can take place arbitrarily quickly.
However, this is just only an imaginative association.
In any case, it must be observed that if we
assert that the execution of the machine is impossible because what can be said
of its state after the termination of the run is illogical, we are committed to
explain when it then has to stop. We must thus be able to explain whether it stops
at all because of a future state
that is logically impossible. Besides the fact that this way of presenting the
problem is absurd,
there is also the problem that if n identical acts are physically practicable without
being logically impossible, n+1 acts of the same kind are also practicable
without being logically impossible.
It must be also observed that at any point A’,
at the track from point A to B, a run from A to A’ have been finished. However,
no run of an infinity machine can have been finished in a similar way. For this
reason too, the concept of infinity machine does not contribute to clarifying
Zeno’s paradoxes.
As to the physical practicability of Thomson’s infinity machine, the physical impossibility
of the arbitrarily small pushes on the button can be mentioned. Quite tritely,
it must e.g. be concluded that the duration of the single acts rather quickly becomes
so short that the filament have no time to cool down. The idea of constructing a
device that can handle arbitrarily small movements, that is in advance, is problematic as mentioned.
The physics of today does not make this possible. Thus, this infinity machine
is only possible in a fiction where the laws of nature are quite different than
they really are.
Max Black defines an “act” as “something marked
off from its surroundings by having a definite beginning and end” and he wants
to argue that the expression “infinite
series of acts” is
self-contradicting. According to Black, it is difficult for us to realize this because we confuse the concept of a series of acts
with a mathematical series of numbers. Therefore, Black
will attempt to ‘show some of the absurd consequences of talking about “counting
an infinite number of marbles”’. ([Salmon 70] p. 72), my underlining.)
With his allegation about our
confusion Black commits a psychological digression. This kind of error is commented
on below in the subsection “Classification of the committed errors” in the
section “The paradoxes with runners in Salmon’s
anthology”.
According to Black “Some writers”, however, have the following
viewpoint:
“If only we could count faster and faster, the whole job could be
done in a finite time; there would still never be a time at which we were
ending, but [3] there would be a time at which we already would have ended the
count. It is not necessary to finish counting; it is sufficient that the
counting shall have been finished.” ([Salmon 70] p. 74)
In order to examine this, Black makes
the following thought example:
Black describes an infinity machine
he names “Alpha”. To the left of us there are infinitely many marbles in a
“narrow tray” reaching far out into the distance. To the left there is a
similar, empty tray. Between these trays, there is placed a machine that moves
one marble at a time from the left tray to the right tray. The first move lasts
one minute. Then the machine pauses for one minute (introduced with a view to
examples below). The next move last a half minute. Then the machine pauses for
half a minute. This continues with succeeding halving of the durations. After
four minutes, the machine stops, the left tray is empty and the right tray
contains infinitely many marbles, according to the line of thought. ([Salmon 70] p. 74)
Then, Black imagines yet a machine,
“Beta”, which necessity has infinitely many marbles to count. This works like
Alpha, but contains only one marble in the left tray. On the other hand, the
set up involves a device that moves the marble back from the right tray while
the infinity machine pauses. As Beta works in the same way as Alpha, we can
imagine the two machines Alpha and Beta working in time side by side. Either
both Alpha and Beta succeed or none of them succeeds, Black concludes. ([Salmon
70] pp. 74-75)
As Black emphasizes, we must abstract from the
physical practicability of such a machine, as the case deals with whether we
can imagine such a machine without a
logical contradiction. ([Salmon 70] p.75)
According to Black, the example with
Beta clearly shows that it is impossible to count an infinitely many marbles,
for the moved marble is always returned:
“A man given the task of filling
three holes by means of two pegs can always fill the third hole by transferring
one of the pegs; but this automatically creates another empty place, and it
won't help in the least to run through this futile series of operations faster
and faster.” ([Salmon 70] p.76)
Black concludes that just as this applies
to Beta so it is also applies to Alpha. ([Salmon 70] p. 76)
The above-referred description is
nearly completely convincing. I.e. it is not true without further proof, but demands some scrutiny. It is certainly
true that the statement that there are infinitely many elements means that for
each counted element there is always one more element more than the so far
counted elements. This corresponds quite to the operations of Beta, or rather of
its helping device: These operations bring about that after element number n
there is an element number n+1.
According to common sense, the runner in the right recursive dichotomy-paradox passes
infinitely many stretches, and the moves can be done synchronously. What is it
about them that make them impossible? Has Black just created (yet) a paradox? Yes,
but only according to common sense, according
to which Zeno’s paradox can be rejected. For the possible impossibility of the
counting machine would not contradict Zeno’s paradox.
Besides, the difference between
Zeno’s account and the idea of the infinity machines is that each of the acts
demands use of energy, while the
passage of an object only demands inertia.
This difference between act and passage is diminished or removed in the fiction,
however, because the physical laws necessarily must be reduced in the fiction
about the infinity machines.
In the fiction, we are dealing with
an infinitude of removals that is just asymptotic compressed. There would nevertheless
be an ongoing removal arbitrarily close to the time 1 and nevertheless be
infinitely many removals left. At the time 1, no removals would be ongoing, but
there would be no time at which they stopped. Where the marble is, the fiction
simply says nothing about, as it (the fiction) is just a mathematical
construction.
We can alternatively imagine that
the length of the removal of the marble was halved for each move to prevent
that the energy becomes infinite. According to a common sense view, the mass midpoint of the marble will be within this
interval when the time is 1, irrespective of how small an interval we state. In
this way, there is no decisive difference between 1) the runner’s or the moving
body’s situation in Zeno’s paradox and 2) the fictions about infinity machines.
To prove that it is impossible for
Beta to finish with the marble in the right track, Black introduces a new
variant of the example: In this, the marbles are moved from the right track to the
left by machine called “Gamma” that functions like Beta, but works when this machine
pauses. If
Gamma works, the marble shall end up in the left tray. However, Beta does not
work. Only one of the machines can function; however, as they are alike both
must function if the one function. This is a logical contradiction. ([Salmon
70] pp. 76-77)
Again, should this really mean that
the machines stop for logical reasons in order to avoid a contradiction?
It must be observed that the mere idea
of the infinity machine Beta implies a self-contradiction. For if we do not take
into account the first move, we have a machine that starts with a marble in the
right tray, and which according to the line of thought must finish with the
marble in the left tray. This means that it is a human decision that decides
where the marble will end, though physically the same happens in both cases.
Black advances that the infinite
number of oscillations with which the machine must finish must contain ‘a part
that oscillated “infinitely fast”, as it were´, but that this is impossible.
([Salmon 70] pp. 77-78)
Black also justifies that the
concept of an infinite numbers of acts is self-contradicting with the fact that
the performance of them involves a discontinuous movement, and that this
is impossible. ([Salmon 70] p. 81)
However, it does not appear what
Black means with “infinitely fast”. We can at most be dealing with arbitrarily fast oscillations. With
this, we now have both an unclear logical and an unclear physical argument for
the impracticability of Black’s infinity machines. Black has explained in what
the discontinuous movement consists either, and eo ipso not what makes the infinity machine be interrupted.
Black concludes that the same must
be true of not just machines that can count an infinite set, but of all
machines that can perform infinitely many acts, as these acts have that in
common that they have a beginning and an end. ([Salmon 70] p. 79)
However, has Max Black really
justified the postulated likeness? As far as I can see, Blacks argumentation is
based on the run of a specific infinity machine that consists of two coupled infinite
machines that count objects, and not just on the fact that each of it acts has
a beginning and an end, such as Black suggests.
The problem remains how a logical
contradiction can make a physical process stop. (Cf. the subsection “An overview”
above.) It cannot be so that the process itself finds out that it involves or
leads to a logical contradiction after which it decides to stop in time before
it so to say “comes into a mess”, for a physical process can only stop because
it is physical impossible for it to continue.
As to the inconsistency in Blacks
examples with counting of marbles, it disappears according to Zeno’s argument because
it presupposes that the process comes to an end, which does not happen,
according to this argument. At any moment before the time 1, the process is not
finished; at any moment, infinitely many partial stretches or tasks are
remaining.
Contrary to this, we can ask where
the marble from the left tray is, if we imagine that a run of Black’s counting
machine has just finished. It seems reasonable to assert that the ball must be
at a definite place, just as it is at a definite, computable place at any time during
the run. However, there is no answer, as there is not more in this
not-practicable fiction than what has been put into it and what can be
concluded from that. Moreover, as the physical laws are reduced, a concept of
cause and effect is lacking. Alternatively, it can be put forward that the
problem so to say has shrunk together with the distances.
In short, as the practicability
demands that (inter alia) the mass of the ball or the length of the stretch it
is moved goes towards 0 for the time going towards 1, the question of existence
or place irrelevant or not existing.
That the concept of infinity machines
does not contribute to the understanding of Zeno’s account follows from this
too:
Zeno’s right-recursive dichotomy
paradox also applies within the
individual passages in the course, especially the first passage. This
results in the left-recursive paradox. The paradox also applies to the
down-going move of the bottom in James Thomson’s lamp example, and to the single
moves of the marble from left to right in Blacks example with counting.
However, an infinity machine cannot be running for each of these passages.
Regarded as a comment to Zeno’s
paradox, the concept of infinity machines presupposes what is discussed, namely
that common sense view of time and space Zeno criticizes.
Benacerraf argues that Thomson’s
example with switching on and off a lamp does not imply a contradiction, such
as Thomson says, as the example does not say anything about the state of the
lamp after the switchings on and off are finished. ([Salmon 70] pp. 107-8)
Exactly therefore it can be advanced
against this that the lamp is in an unknown state at this moment; in other
words, it is in a state that does not have a cause. This problem misses a
solution.
Benacerraf certainly admits that the
lamp must be either on or off when the process is finished. ([Salmon 70] p.
108) However, However, Benacerraf does not concentrate on solving this problem,
only on proving that Thomson’s lamp example does not imply the mentioned
logical contradiction:
According to Benacerraf, the
solution of his problem is that though the lamp example says nothing about the
state of the lamp after the completion, it is not self-contradicting, for “there
need not be an ωth act of the relevant kind!” Benacerraf argument is that we can do anything else. ([Salmon 70] p. 115)
The latter statement is certainly true,
but the case is not about a specific subsequent act, but about the fact that a working lamp must be either on or
off, and that its state, whatever it is, must have a cause.
Benacerraf wants to examine whether
the concept of super-task is self-contradicting,
and advances that to prove that it is self-contradictory, it must be shown “that
there is something self-contradictory in the concept of a completed infinite
series of tasks”. Benacerraf thinks that it may help to examine whether the concept
of infinity together with the
concept of a finished number of tasks
is self-contradicting.
([Salmon 70] p. 125)
To this, it can be answered that
there is a difference between 1) the very completion of infinitely many tasks,
and 2) the pointing out of infinitely many completed tasks. We cannot talk
about the completion (1) as such, whereas the mentioned pointing out (2) has
semantic meaning in a mathematical model. This applies whether we only
understand performing an act with distinct termination and start as a task, or
whether we also understand passing a partial stretch as a task.
Moreover, it must be remarked that
the case just involves the question of completing infinitely many tasks. We can
go to the point itself: We are dealing with an ordered row of physical tasks
that have to be carried out one by one. This means that if the tasks up to and
including number n have just been carried through, there is still a task number
n+1 that has not been carried out yet. Black has clearly illustrated this. (Cf.
[Salmon 70] p.76.) Thus, the task can certainly be started, but that we cannot
point at a moment when it is being finished.
Benacerraf asserts that this
question is complicated by our not being able to confirm whether an infinite series
of tasks has been accomplished. ([Salmon 70] pp. 125-6)
This is true enough, for as
mentioned it is difficult to demonstrate signs of an accomplishment of a
finished run of an infinity machine we can look back at. The reason is that
even if the task should leave physical traces, it would be just as difficult to
ascertain the result of the accomplishment as to perform the left-recursive
procedure.
It must also be observed that if the
physical tasks are physically possible irrespective of how fast they are
performed, they can be compared to passing each of the infinitely many
intervals in which the track can be divided. Under this condition, performing
the tasks is therefore just as possible or impossible as traversing the track
is.
As Thomson refers to his concept of
an ω-task in both this section and the subsequent one, the definition of it is
put forward here:
“To complete an ω-task is to
complete each of an ω-sequence of tasks. Thus an ω-task is the kind of task I
previously called a super-task, although in fact only ω-tasks were there in
question.” ([Salmon 70] p. 130)
James Thomson wants to scrutinize
whether the very idea of the
performance on infinitely many tasks is self-contradicting, just as the
lamp-example is. In order to clarify this, Thomson describes a system S, in
which changes occur from a state A to a state B at the beginning of each
time-period, just as in the case with the lamp. By this, a new state is not
attributed to S at the final moment. However, though the system necessarily must
be in a definite new state, e.g. A, we cannot say that the system changes into the state A. Thus, we must either say that the system C can “change (or even go)” or “undergo a
transition” to a state without doing it from another state. ([Salmon 70] pp.
131-2)
It must be observed that though Thomson’s
system S deals with physical systems, it can only be a fiction. For it has
above been justified that infinity machines with such traits are not realizable
in the physical reality, as they demand reduced physical laws. However, they
can exist as a mathematical model added time, as nothing stands in the way of
defining a mathematical function with those characteristics. This means that
Thomson’s system S just represents what we already and necessarily are dealing
with in connection to these infinity machines. However, in these fictions, it
has no sense to talk about verifying executions of them, as they are defined.
Thomson likewise asserts that we
must examine the problem of “a last uncaused transition” ([Salmon 70] p. 134), but
without advancing any idea of a solution.
It must be observed that the concept
of cause only belongs to the physical world with all the acknowledged physical
laws, and thus cannot concern the presented fictions in which we disregard
essential parts of the physical theories to get the machines work in a fiction,
and in which a lot more can therefore happen. We can for instance unproblematically
decide that such a system is in a state of infinite changes in a period and
then in a definite state in a new period, as we have defined it in that way.
However, Thomson also touches the
question whether it is legitimate to solve the problem by e.g. letting the lamp
disappear at the decisive moment. Thomson rejects this for the reason that
though something like that could happen, it is not a necessary condition of an
ω-task. ([Salmon 70] p. 134)
To get infinity machines work in a
fiction without contradictions, we must presuppose more or less reduced laws of
nature, e.g. except conservation of energy, or simply by ignoring them. This
will demand either that the sizes by degrees approaches zero or that we have a
fiction without causes and effects. In this way, the problem about a final
state disappears. However according to Zeno’s paradox, the problem is how we
reach the final moment at all.
It must be observed that tasks with
a beginning and a termination, such as Black defines them, do not exist in
isolation, i.e. such that a task stops, where after a new one begins again.
They can only be concomitant phenomena to certain forms of continuous use of
energy. This will pass off just as steadily as an object that moves along by
means of its energy...
Grünbaum asks whether it is useful
to examine whether infinity machines are possible, if this presupposes that we
disregard all other problems than kinematic ones. Nevertheless, Grünbaum hopes
to show that this can illuminate the kinematical component of the physical theories
that assert that space and time are mathematically continuous. [Salmon 70] p. 202)
As earlier mentioned in these
treatises, science does not have to regard the theoretic entities and their qualities
as having a real counterpart in the physical world. This also applies to the
question of whether space and time have a mathematically continuous structure.
Moreover, Zeon’s paradox with runners must be regarded as a fiction about a
mathematical world added a concept of time if it is to be must be taken seriously
and thus not a literally understood model.
Grünbaum calls the runner in Zeno’s paradox a
“legato runner” and invents what he call a “staccato runner” who makes a pause
for each stage, but runs with the same average velocity as the legato runner. ([Salmon
70] p. 204)
As this infinity machine resembles James
Thomson’s example with the on an off button and Max Black’s counting machine, and
thus does not contribute with anything more to the clarification of the concept
of infinity machines than what has have already been looked at, it will not be
went through here.
Though the concept of act or task does not enter in Zeno’s paradox, it is naturally acceptable
to take inspiration from it and invent a different problem, such as one in which
an “infinity machine” enters, and then seek to solve this problem. This problem
has just nothing to do with Zeno’s paradoxes, as it takes place exactly within
that common sense frame Zeno questions
with his paradoxes.
All the contributors agree, however,
that infinity machines cannot work according to the laws of physics. That they
are in fact right in this can be illustrated by this example:
We can imagine a vehicle that colours the
partial sections of the trace alternately red and green for each partial
stretch. However, soon the lengths of the line pieces become shorter than the
pigments. To this, we can add that at some moment in the course of the halving
of the times, these become so short that a gamma-wave cannot manage to pass. Soon
we cannot measure the length of the time intervals. All measuring equipment
that demands definite energy quantities, lengths or time periods become too
clumsy after a number of tasks. We must therefore ignore the details of the
physical laws in these fantasies about infinity machines of they have to be
coherent.
Therefore, infinity machines only belong to a
scientific realist fiction in which the scientific rules for the behaviour of
the phenomena have been removed or reduced. In this fantasy, the concepts of
cause and effect do thus not exist. In this way, the idea of an infinity
machine approaches, when it is extricated from all physical problems concerning
its fulfilment, a mathematical abstraction added the concept of time.
In this sense, it is realizable, as
the accomplishment just corresponds to the description of a mathematical function.
We can thus imagine a wavelike function (a compressed sinus function) that is
alternating over and under the x-axis in these intervals. A different, simpler
example of such a function may be a function whose value is alternating 1 and 0
in intervals of the length ½, ¼, ⅛, etc. The concept of absolute magnitude even
becomes subordinate here. In fact, this description is nearly identical to the
description of the division of runners track in Zeno’s right-recursive
dichotomy paradox.
We would neither be able to verify what takes place in this imagined infinity
machine, even if we could decide whether the machine is possible, i.e. whether
it has finished the run. However,
an infinity machine cannot be called an
infinity machine until it can be verified that its actions have finished
correctly, i.e. not until the necessary time has elapsed. In order to find out
whether the machine in the above example has run correctly, we must examine the
colouring of the track still closer to the goal. However, there will always be
infinitely many not-examined succeeding partial stretches. Perhaps we can
decide the question indirectly by measuring the used amount of dye, which
however demands unlimitedly exact measurements. Moreover, the result will only
be a so-called good assumption, and
such a one cannot be literally confirmed.
We cannot even regard the acts as theoretical
assumptions that belong to a scientific model that can be used to predict the phenomena or be useful in any way, just as
theories about elementary particles and gravitational fields can be.
There is thus a contradiction
between 1) scientific realism and 2) disregarding exactly those laws of nature
that make the infinity machine impossible. For the rules abandoned in this scientific realist fiction are exactly the rules
that are the very basis of the
phenomena according to scientific realism. In this way, the question about the
state of the lamp after 1 time unit loses its relevance in these fictions.
Finally, e running infinity machines
cannot stop for logical reasons, not either to avoid a logical self-contradiction that threatens, but only for
physical reasons. These may certainly be described logically in certain cases, but
the reasons are still physical. (It can e.g. be explained logically why three
toothed wheels that gear into each other cannot move, but the real reasons are
physical.)
If an infinity machine cannot finish
correctly for logical reasons, such as Max Black asserts, it would be a
question when it would “discover it”, for the single act is not yet an act can
always be performed.
We cannot follow or examine the
process in detail either: it is as if
the time has been stretched out but not to us. Inside the computed period, the
process is always running such as Thomson and Black has made it clear. However,
in our fantasy, we cannot just let the necessary time elapse, and then examine
what has happened. For this examination would be just as an attempt to manage
the left recursive dichotomy paradox.
We can either simply ignore the
physical laws in our fictions about infinity machines or just reduce them. In
both cases, the concept of cause and effect has been annulled in the fiction.
In the latter case, the very basis of contradiction has been removed as it is
downgraded.
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” or the more clear statement “There is no stretch such
that the runner reaches the goal after his passage of it”. We have then to
refute the most difficultly refutable version of the latter statement.
This problem is not elucidated by
refuting the statement U about the infinity machine: “There is no task so that
the infinity machine finishes after the completion of this task”; for if the
infinity machine is physical, the statement is easily refutable, and the more
abstract the fiction about the machine becomes, the more it will resemble
Zeno’s account. Moreover, the fiction about the infinity machine remains within
a common sense frame whereas Zeno’s account escapes from this.
The only
thing the concept of “tasks” and infinity machine do is to add an extra
complexity to the existing paradox. Moreover, it is easier to refute that the
infinity machine succeeds than to refute that the runner reaches his goal. For
that reason, the discussion of infinity machines says nothing about Zeno’s
paradox. Finally, this added complexity moves the focus from the purpose Zeno’s
account as it takes place within a common sense view.
As the concept of infinity machines
as such has already been treated above, below, passages will be looked at in
preference to acts, also in those cases where these are discussed by the
contributors. To this it may be objected
“By this you change the subject, which you have criticized yourself”. However,
as the subject has just been discussed and rejected as being irrelevant to the
discussion of Zeno’s paradoxes, we just have to scrutinize whether anything
interesting remains.
The contributors to the anthology discuss
Zeno’s paradox, as if we can just presuppose 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”. None of them takes 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:
The right-recursive dichotomy
paradox says, “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, irrespectively of how close A’ was to B’”. The left-recursive
dichotomy paradox says that the runner cannot either arrive at any place, B’,
to which he is said to arrive, however close B’ is to the starting point A. It
must be observed that what can be said about distances and places can just as
well be said about time intervals and moments.
The right-recursive dichotomy paradox can be
interpreted thus:
“The runner is at a point A’,
wherever he is, but he cannot arrive at a definite place B, irrespective of how
close A’ is to B”.
The left-recursive dichotomy paradox says
“The runner is at definite point
A. but he cannot arrive cannot arrive at any place, B’, irrespective of how
close B’ is to the starting point A.
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.
Despite of this, most of the
contributors presuppose their common sense view in their discussion of
Zeno’s account. Thereby, they do not discuss the latter subject, but a mix of
the two subjects, and already in this way introduce an inconsistence. They fail
by presupposing what Zeno doubts. For, first,
we must understand, then, we can criticize.
Some of the contributors 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. One of them involves the runner’s steps and
breathing in his rejection of the paradox. This comes under the same error with
the same consequences.
However, a number of the
contributors point out, at least, 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. Some of
the contributors even approach the clarification, presented in Treatise No 8, according to which Zeno’s
account is recursive without a stop-clause. However, they do not do this
explicitly.
? where?
In order to throw into relief the
contributors’ misinterpretations, argumentation errors and wrong attitude, an
attempt at interpreting Zeno’s account is put forward here:
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.
Thus, the following applies:
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 arrives at the now.
We can 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 statement that 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 also
agrees with the fact that our experiences, i.e. our memory impressions make up the basis of our past-fictions and
future-fictions.
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. 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, 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.
We can take inspiration from the
quote from ([Salmon 70] p. 74) that is referred in the subsection “Max Black’s infinity machine”, and look at a similar statement
about the race, that deals with passages of intervals instead of counting. The
original statement about Black’s counting machine sounded:
“[1] If only we could count faster
and faster, the whole job could be done in a finite time; [2] there would still
never be a time at which we were ending, but [3] there would be a time at which
we already would have ended the count. [4] It is not necessary to finish counting;
it is sufficient that the counting shall have been finished.” ([Salmon 70] p. 74,
my square brackets.)
Obviously, Black dissociates himself
from the quoted view of his counting machine. However, in fact, the view
touches the core of the matter. This is more directly seen in a similar version
about passing of intervals:
[1] If only the runner could pass
the intervals faster and faster, the whole passing of intervals could be done
in a finite time; [2] there would still never be a time at which he would be finishing
the run, but [3] there would be a time at which he already would have ended the
passing of intervals. [4] It is not necessary to finish the passage of
intervals; it is sufficient that the passing of intervals shall have been
finished.
The core of the matter is touched in
this way:
According to Common
sense, [1] the runner can in fact pass the intervals faster and faster,
even so that the whole passing of intervals can be done in a finite time. [3] There
will be a time at which the runner would already have ended the passing of
intervals.
[2] According to Zeno, despite the runner getting closer
to the goal, there will never be a time at which the runner will be finishing
the run.
However, it is neither in accordance to Zeno’s
account to assert that in the now the runner has finished a passage of intervals,
not in a past-fiction.
[4] According to common sense, the passage of intervals
will be finished. According to Zeno, it does not. However, we can look back at
a finished passage of intervals.
In fact, the quote mirrors the message of Zeno’s account: In our
past-fiction, we can come closer and closer to the present time, but we can
never reach it, in the sense that there is never a moment at which we reach it. In this way, there is an unbridgeable
border between past and present irrespective of how close we reach the present
time, in our consideration over our past fiction. This is in accordance to the
fact that we cannot get from a fiction to a reality outside this, or from a
past-fiction to the now. According to this, there is thus a dualism between past and present.
It must be observed, that wherever
the runner is, he has just finished infinitely many passages. In the now, thus,
we are at an arrival at which we can look back at a finished passage of
intervals that has started at some earlier moment in our past fiction. However,
we cannot explain how the passage was finished.
According to the two versions of Zeno’s
dichotomy paradox, we can explain neither how the run is finished nor how it
begins. I.e. we can explain neither how we come from an ongoing progress of
events to a finished one, nor how we come from a standstill to an ongoing
progress. According to common sense,
there is literally a time before the now and a time after the now, and the now
is just a point in a progress of events. However, we cannot justify this view
of time. Zeno’s paradox clarifies this fact. In this sense, it is not a paradox,
but the disclosure of a fact.
It must also be observed that the just
mentioned fact that wherever the runner is, he has just finished infinitely
many passages, also applies to the first half of any of the passages. By this,
the left recursive dichotomy paradox arises.
This paradox especially clearly
illustrates that it has no sense to say that there exists a future understood
as new, not-experienced nows: We cannot explain how we reach a new now; we can only be
in a now.
At any moment during the run, the
runner is a definite place. At an earlier moment, he was at the distance d from
this place. To reach it, he had to run ½ d, ¼ d, etc. We cannot point at
anything in that past that can lead the runner to the place where he is. “Yes,
his run”, somebody will answer. It is true that according to our experiences the
runner’s run brings him still closer to the place where he is now. However, the
question is when he comes into the now. We only have to do with the now and a
past-fiction. Something similar applies to the future. The left-recursive
dichotomy-paradox shows this. Here the question is how the runner comes from
the now to the future.
The going through of the
contributors’ arguments has unfortunately become a study of erroneous
argumentation and not least of the contributors’ unfruitful attitude to a philosophy that they
obviously dislike. However, some of their misunderstandings have contributed to
making the paradox plain.
It is evident that a statement must
be understood, in order that it can be criticized on the right basis. Nevertheless,
none of the contributors has presented an understanding of what can be put into
Zeno’s account. It must here be observed that understanding a statement does of
course not imply acceptance of it.
It is naturally possible that a
statement can be so evidently wrong that there is not much to comment on, but
that does not apply to Zeno’s account. Contrary to this, some of the
contributors’ statements can only be discussed with some indulgence.
The same types of argumentation
errors occur in a number of the contributions. Some of the errors have an immediately
general character; others are specifically related to Zeno’s paradoxes, but are
of course subordinated some general type of error. The errors can be divided into
these types:
G1: 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, the most easily refutable interpretation of it ought
not to be chosen, 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.
G2: Unwillingness
consists in lacking interest in understanding the opponent’s message. This may
be due to lacking knowledge of the need of understanding before criticising.
Often unwilling persons effectively defend themselves against understanding
their opponent’s message by presupposing
their own view in their criticism.
G3: Lacking
understanding of the problem often consists in not understanding what the
core of the matter is. The solution of a problem does not consist in not
acknowledging it. Lacking understanding may sometimes be due to unwillingness (G2).
G4: Lacking
justification consists in unjustified assertions and lacking references, and
can e.g. be pure attributions.
G5: The
psychological digression consists in referring to what other people possibly
imagine. This is irrelevant to the debate, and disturbing.
G6: The circular argument consists in presupposing what has
to be proved, e.g. by returning to the starting point.
G7: The common sense error consists in presupposing a common sense view of the problem or in
regarding a common sense explanation
or -view as a solution of the problem.
G8: Change of subject simply consists in discussing a different subject
than what is at issue. This error can be caused by lacking understanding of the
core of the matter and by the common
sense error.
G9: The limit value error is the view that
the runners distance is identical to a sum of infinitely many numbers, understood
as a limit value of a sum function. This view ignores that we are dealing with
a definition, and not literally a sum.
The committed errors that are here classified
under general type of errors are of course also problem related errors of some
kind. However, there is no reason to invent new terms for that reason alone.
This type of error occurs in different
versions:
This
error consists in including the runner’s steps
and their details in an argumentation against Zeno’s paradox, which makes the
paradox easy to refute.
Thus, Henry Bergson asserts that we can regard the race between Achilles
and the tortoise as consisting of Achilles’ steps and those of the tortoise,
and that it is therefore evident that Achilles will have reached the tortoise
after a number of steps. ([Salmon 70] p. 65)
This
error consists in involving the physical surroundings
in an argumentation against Zeno’s paradox.
According to Max Black, it is not
necessary to Achilles to ‘to perform an
infinite number of acts, as Achilles only have to do with things that
are limited in number, such as steps, heartbreaks, pebbles, etc. Black advances
that we can of course divide the lengths of things in our thoughts, and that it
would then be a priori true that there are infinitely many distances. We would thus
just dealing with pairs of numbers, which has nothing to do with a concept of
infinitely many things that Achilles must touch. ([Salmon 70] pp. 79-80)
With the former of these
commentaries, like Bergson, Max Black chooses the easiest refutable reading of
his counterpart: First, touching infinitely many things do not necessarily involve
a pointing out of points individually. Second, the paradox exists irrespective
of circumstances such as grains of sand. Third, as mentioned, this talk about
particles and touching becomes superfluous by choosing the most difficultly
refutable interpretation.
In fact, Zeno does not just talk
about a runner but also about “that which is in locomotion” ([Kirk 70], p. 270).
However, even if Zeno had only talked about a runner, we should still choose a
more difficultly refutable interpretation. Thus, Zeno’s account is more
abstract than suggested here. (G1)
Therefore, these details are
irrelevant to the problem. To understand a problem as dealing with a statement
that can be refuted so easily indicates either a wrong interpretation or an
avoidance of the problem. (G3)
It must be observed, that these
remarks are not intended as psychological attributions, but deals with the
contributors’ expressed dislike.
For instance, Max Black wants to solve Zeno’s paradox by finding “exactly what mistake
is committed in this argument”, if it is possible. ([Salmon 70] P. 68, my
underlining.)
Quite fundamentally, we cannot talk
about an error in an argumentation until we have understood the reasoning. What
is argued for in Zeno’s account is the fact that our common sense view of time is wrong. Zeno uncovers a paradox that
makes clear that we have not understood the concept of time. By not seeing such
possibilities, Black expresses a kind of unwillingness (G2).
In fact, there is no error in Zeno’s
argument, such as it is presupposed by Max Black and further contributors. This
has been justified in Treatise No 8. Therefore,
Black also expresses lacking understanding (G3).
Some contributors advance as a solution of the paradox that the runner
just runs from point A to point B. This is of course not a solution but a
description of the subject that Zeno analyses. Cf. Benacerraf ([Salmon]
p 103), Thomson ([Salmon 70] pp.
98, 99, 130).
Such an objection is expressive of lacking
understanding of Zeno’s paradox, as it is this run Zeno analyses. It is therefore
this analysis the critics must address - in order first to understand it and
then to consider it.
According to Russell, it is obviously a refutation of
Zeno’s paradox to assert that there is a point 1 after the series 1, ½, ¼,... ([Salmon
70] p. 49) On the contrary, this
fact is essential to the understanding of the paradox, which contains e.g. the
fact that Zeno’s account deals with the runner’s situation, which can be
opposed to a completed whole.
To Black’s above-referred statement
that of course we can divide the length of things in the thought, it can be
said that this is exactly what Zeno does and thus invites his reader to do in
order to understand his message. We must also here understand the paradox
before we can conclude anything about it. (G3)
Black draws the conclusion that what
can then be called “distances” will be a series of pairs of numbers and not an
infinite series of spatiotemporal objects. Though these can be used to describe the physical reality, this does
not mean that Achilles must do infinitely many things, Black concludes.
([Salmon 70] p. 80)
Certainly, Zeno’s description of the
race implies the use of an algorithm for dividing a line in infinitely many
pieces. However, as mentioned, the paradox does not consist in this as
mentioned but in the fact that after each catching up with the lead of the
tortoise in principle Achilles has not come further with his catching up with
the tortoise. (G3)
It appears from the above remarks
that Black’s rejection of the paradox is just an easy rejection of an easily
refutable interpretation. In fact, it is not Zeno who talks about pars of
figures, but those who write about his paradoxes. Zeno just describes the race
in a recursive way. This at most involves some divisions, on which the debaters
elaborate. Left is the interpretation, that the paradox deals with a fiction that
takes place in a mathematical universe added time. (G1, G3)
Lack of justification can consist in
lacking reference or mere postulations.
Thus, Bertrand Russell says about Achilles and the tortoise that it is
not true that an infinite number of
instants make up an infinitely long time wherefore it does not follow that Achilles
will never overtake the tortoise. ([Salmon 70] p. 50)
Perhaps some has drawn this conclusion
in the literature. However, from Zeno’s paradoxes with runners it does not
follow that an infinite numbers of moments make up an infinite long time. Thus,
Russell groundlessly attributes a view to Zeno. (G4)
The psychological digression consists
in asserting that some people may be inveigled into believing that something is
in a certain way and then discus this view. Such a digression makes the debate
obscure and does not lead it any further. What has to be done is to address the
case itself and find out how it is in fact.
Black thinks that we erroneously
think that Achilles must do what is logically impossible, because we mix
the finite number of acts the runner has to do with the infinite number of
figures by means of which we describe what he really does. ([Salmon 70] p. 81)
From the above commentaries to
Salmons anthology, it must appear that at least the author of this treatise
does not make the postulated mix. Moreover, the paradox has been clarified
without such confusion in the subsections “A description of the paradox” and
“An unfinished account” in Treatise No 8.
Thus, Black makes a psychological digression here. (G5)
Wisdom refers Black for the view
that the illusion that Achilles must do an infinite number of acts
originates in the mathematics we use for describing time, space and movement. ([Salmon
70] p. 84)
Discussing whether anybody should
have such an illusion is not to discuss the very matter. Therefore, Wisdom’s
remark makes up an irrelevant psychological digression (G5).
Let us suppose that we are not
dealing with acts but with continuous passages. Then, it is certainly true, that
there are infinitely many partial stretches to be passed in the sense that for
each passed partial stretch, a new has to be passed; however, this very status
quo is the interesting thing to the understanding of Zeno’s message. In this
way, both Wisdom’s and Black’s interpretation makes up an unfounded distortion
of Zeno’s philosophy. (G3)
The circular argument consists in
presupposing what has to be proved, or simply in returning to the starting
point.
Presupposing a common sense view of time and space is of course to make a circular
argument in a criticism of Zeno’s account, as it is this view that are under
debate.
As mentioned, some of the
contributors reach the solution that Achilles just runs, e.g. Benacerraf ([Salmon] p 103), Thomson ([Salmon 70] pp. 98, 99, 130), the latter even three times. This
is naturally circular because it is exactly this run Zeno analyses.
We may also here be dealing with a
denial of the problem or with:
This
error consists in presupposing that common
sense view on time and space Zeno criticizes, and even presupposing that it
is true during the whole discussion. It is true that also Zeno presupposes it,
but only as a point of departure in order to find a contradiction in it.
According to the nature of the case,
especially in a discussion of philosophically subjects such as Zeno’s
paradoxes, this error will often be a case of the circular argument.
This
error consists in discussing a quite different subject than what is to be
discussed. It may certainly be related but also less philosophical that the real
subject. This applies to:
By not letting Zeno’s paradoxes
about runners just deal with passing of intervals, but also with tasks with a beginning and an end, an
interpretation that is not at all suggested in the original text is chosen. For
in the thus derived account, not only the same occurs in each interval, but
something more, e.g. a change of some state between the intervals, which makes
the course more difficult. It is easier to prove the impossibility of carrying
this through than to disprove Zeno’s paradoxes, such as these are worded. This
is only apparently to Zeno’s advantage, as it would be a wrong justification.
Moreover, Zeno’s left-recursive dichotomy paradox cannot
exist in a version with an infinity machine, or rather the runner or the object
cannot be replaced by an infinity machine in Zeno’s left-recursive dichotomy
paradox in the way it was possible in Zeno’s right-recursive dichotomy paradox.
In addition, both the right-recursive and the left-recursive dichotomy paradox
apply to any period that has preceded the object’s arrival at any point on the track;
however, there cannot be infinitely many infinity machines running. Likewise, these
paradoxes also apply to the single movements of a marble and to the pushes on
the switch bottom.
This makes clear that the concept of
infinity machines does not contribute to clarifying Zeno’s runner paradoxes.
Overall, Zeno’s paradox must be discussed and solved on the basis of the
content it in fact has.
This
error consists in talking about so-called “sum” of infinitely many numbers, as
if we are dealing with a value that is in fact reached and not just a limit
value. (Cf. the
subsection “Mathematical limit” in Treatise
Part 8.)
Bertrand Russell certainly refutes proposal made by C.D. Broad that Zeno may have
appealed to the view that the successive
halving of distances lead to infinitely many distances, which each has to be
reached in a finite time, and that this will never be fulfilled as the sum of
infinitely many finite times is infinite. According
to Russell, this is wrong as the sum of the required times, ½ minute, ¼ minute
etc., is 1 minute. ([Salmon 70] p. 49.)
First, it must be observed that Russell
here expresses himself as if this sum is 1 in the same way as e.g. the sum of ½
minute, ¼ minute and ¼ minute is 1.
Second, Russell’s rejection is
misleading irrespective of how mathematically correct it is. The reason is that
the quoted reasoning does not address the philosophical core of the matter, which
consists in the status quo of the runner’s situation, Contrary to this, this
talk about a sum is to regard the run as a whole in accordance to the common sense view on time Zeno
criticizes, and is thus circular. (G6)
Not least, Berkeley is unjustified
in attributing to Zeno this idea, which is clearly wrong and thus superfluous
to be occupied with.
Finally, the proof that enters into
the computation or rather the determination of the sum of the infinitely many
numbers is based on the definition of convergence of series. Thus, it cannot
refute Zeno’s argument. On the contrary, the following comparison can even be
made:
The definition of convergence has
that in common with Zeno’s account of runners that it deals with coming arbitrarily
close to a value without reaching it.
Bertrand Russell puts forward that according
to Aristotle’s interpretation, Zeno asserts that it is impossible to touch
infinitely many points “one by one” in a finite time. Russell emphasizes that this
expression may concern either [1] all the points at the stretch or [2] only the successively passed
middle points. [3] In case one, the points are certainly passed continuously, but not “one
by one”, as there is always a point between two points. ([Salmon 70] pp. 48-9)
Russell is clearly right in this, as
it is contained in the concept of continuity. However, obviously, Russell does
not want to focus on this case, as he continues:
In case two, however, all the the successive middle points are reached one by one, and within
a finite time, though they are infinite in number. ([Salmon 70] p. 49)
This is discussed below ... ?
Instead of addressing the text
itself, Bertrand Russell tries to find a few suggestions for what we may call “the
historical background of Zenons unknown argumentation” and then to criticize
them:
According to Russell, a possible
argument may be directed against the view “that
a finite time must consist of a finite number of instants”. Zeno’s
argumentation is then supposed to be that as there are infinite many divisions,
there are also infinitely many instances, wherefore the elapsed time cannot be
finite. ([Salmon 70] p. 49.)
First, with its talk of what “time must consist of” this line of thought
only concerns a realist’s scientific model of the world, which even cannot be
used in science. Second, it addresses neither Zeno’s account the way it is
presented nor its message.
Russell thinks that an alternative argument may be directed against “the partisans of infinite
divisibility”. According to Russell, the line of thought can have been that the
infinite divisibility leads to infinitely many stretches and durations, that
each have a positive length, wherefore the total duration must be infinite. Russell asserts that this is wrong, for “If half the course takes half a minute, and the
next quarter takes a quarter of a minute, and so on, the whole course will take
a minute.” ([Salmon 70] p. 49.)
This argumentation has already been
refuted in Treatise No 8, the
subsection “Mathematical limit”.
Primarily, however, none of Russell’s
arguments deals with Zeno’s argument such as it is worded, and its possible
message. For when Russell talks about the successive
middle points being
reached within a finite time, he talks about a course of events that are gone
through, i.e. a course regarded as a
whole that can be embraced whereas
Zeno’s account can be understood as being about the runner’s actual situation that in principle does
not change, stretch after stretch. The paradox consists in these two facts. (G3)
Moreover, Russell’s idea of Zeno’s
line of thought is a line of thought that is clearly wrong and thus
philosophically uninteresting.
Russell adds that [1] the apparent force
of the Zeno’s argument lies in the wrong supposition that there can be nothing
after an infinite series. This is wrong, Russell asserts, because [2] the
number 1 follows the infinite series ½, ¾, 7/8, 15/16,... ([Salmon 70] p. 49)
Russell’s unfounded attribution (1) is
elaborated on in the appendix. However, with his objection (2) concerning the
point 1, Russell is in fact near an understanding of Zeno’s paradox that there
is no change from the past to the
present time. With his statement, Russell touches the core of Zeno’s paradox, just
as others of the contributors, but without using it for a clarification of the
paradox. (G3)
Moreover, it is crucial to the clarification
of the paradox that no time at which Achilles reaches the tortoise will arrive in the account. Cf. the argumentation
above in the subsection “A description of the paradox” in Treatise No 8.
It must be concluded that Bertrand
Russell’s rejection of Zeno’s paradox is based on unreasonable and ill-founded
attributions. Moreover, they occur within a common
sense view of time as a progress. In fact, however, the paradox leads us outside
this common sense frame.
Nevertheless, Russell presupposes
a common sense view of time during his criticism, which is of course clearly
circular and leads to a contradiction. The message of Zeno’s account is not
that everything is in agreement with common sense except that the runner does
not reach his goal. It is a misunderstanding to argue against Zeno’s account on
the basis of a common sense
understanding of it. Certainly, this account takes its point of departure
a common sense view of time, but implies
that this view is wrong.
Bergson asserts that we cannot
divide a movement in smaller movements as that would correspond to introduce a
stop. This means that a movement cannot be divided into arbitrary parts just as
the track can. ([Salmon 70] p. 64)
It seems here that Bergson takes the
concept of division of a movement too literally by regarding it as a stop, though
the movement is not physically broken by Zeno’s ideas. Possibly Bergson has a point,
but it is difficult to get to the bottom solution of this, as Bergson is more
appealing that arguing such as it follows from this:
Bergson asserts that the division of
the movement is absurd and only something we can imagine when we observe
a movement from the outside. “The absurdity
vanishes”, as Bergson puts it, if we conceive a movement as continuous, which
we do when we move an arm. ([Salmon 70] p. 65)
Here, Bergson talks about our
situation as sensing beings that experience themselves. However, when we
describe the runner’s situation, we do this by observation of it, i.e. exactly from the outside. By this,
Bergson’s objection disappears. Thus, Bergson has chosen an interpretation that
is more easily refutable than necessarily. We can even choose a more difficultly refutable interpretation
than the version with the runner. For as mentioned, Zeno’s account can be
understood in a more abstract way than Bergson does, viz. as dealing with an
object driving along out among the fix stars. According to this view, it is
naturally more difficult to use Bergson’s idea of regarding a movement from
within. (G8)
In addition, the movement of an
object naturally remains continuous, irrespective of how we divide it in our
thoughts. We can compute its place coordinates (x1, x2, x3)
in space as a function of the time t, and photograph it where it is. In this
way, it has sense to talk about the division of a move. From this, we can even take
a step further and regard Zeno’s account as being about something that occurs
in a mathematical model.
Henry Bergson’s conclusion is (1)
that we cannot regard the race between Achilles and the tortoise as consisting
of an accidental division of a racing track into distances, such as Zeno does,
but (2) as consisting of Achilles’ steps and those of the tortoise. According
to the latter view, it is clear that Achilles will have reached the tortoise
after a number of steps, Bergson concludes. ([Salmon 70] p. 65)
However, Zeno’s description does not
contradict Bergson’s description. Bergson just understands Zeno’s description
in a very literal way. First, Zeno’s description of the dichotomy paradox could
also be applied to the single details in the runner’s steps and to the movement
of his gravity point. Second, as mentioned, instead, the paradox could e.g. be
about two objects that rush along, as just described in which case Bergson’s
objections cannot be applied. (G8)
With his description, Zeno attempts
to express a criticism of our view of time. Contrary to this, Bergson just presupposes
the common sense view that Zeno
criticizes. Bergson’s interpretation is therefore not fruitful, but represents the
error of choosing an easily refutable interpretation. (G1, G6)
Moreover, Bergson’s arguments deal
with an analysis of the phenomena. Such analyses can certainly lead to detailed
scientific theories, but not to an understanding of Zeno’s abstract account. In
fact, Bergson does not even approach the above-mentioned idea of finding abstract
interpretations, but argues as if a superficial description of the paradox is the
same as a solution of it, as he asserts:
First Achilles reaches the place
from which the tortoise stared, the place to which the tortoise in the meantime
has reached, etc. “But obviously, to overtake the tortoise, he goes about it in
quite another way.” ([Salmon 70] pp. 65-6, my underlining.)
Bergson does not explain in what
this “quite another way” consists. We can only guess that it just consists in
running; and in that case, we are back at the starting point. For the first
description is exactly a special description of this other way. Thus, Bergson
does not address Zeno’s paradox, but argues circularly. With his account, Zeno
seeks to express a message, which Bergson does not take pains to understand. (G3,
G6)
Max Black asserts [1] that the
paradox about Achilles and the tortoise cannot be solved [1.1] by adducing an “independent
argument” that Achilles will catch up with the tortoise, as all already know this,
and [1.2] that “the puzzle arises because the conclusion of Zeno's argument is
known to be absurd”. Instead it must be solved by [2] finding “exactly what mistake
is committed in this argument”, if it is possible. ([Salmon 70] pp. 68)
Re 1.1: It is certainly true that
searching for such an “independent argument” is a wrong approach to the paradox,
but Black’s argument is rather unclear. The reason is that if a statement is known to be absurd it must be a fact
that it is absurd, i.e. the statement must have been proved, and then there is no
“puzzle” to be solved.
Re 1.2: Moreover, the mere fact that
it is a widespread view that Zeno’s paradox is absurd is
no counterargument. With the referred statements, Max Black expresses a preconceived
view that does not bring him close to an understanding of what Zeno’s account
deals with. Instead, we are dealing with a kind of unwillingness. (G2)
Re 2: It must finally be observed
that we cannot assert that there is an error in Zeno’s argument without having
understood the line of thought in the argument and its possible conclusion.
Therefore, we do not primarily have to find “what mistake is committed” in
Zeno’s argument, but first to understand it.
Black puts forward that the paradox
cannot be explained by the fact that the stages of the run can be described as
a convergent series. ([Salmon 70] pp. 68-69)
In this Black is right, in turn.
This has been clearly justified in the subsection “Mathematical limit” in Treatise No 8. Instead, as mentioned, the
paradox must be clarified by analyzing Zeno’s arguments without minimizing
them.
Black adduces that the mathematical
summation only tells where and when Achilles will catch up with the tortoise
provided this happen. ([Salmon 70] p.70)
To this must be added the
above-mentioned comparison between the definition of convergence and Zeno’s
account of runners, according to which both is about coming arbitrarily close
to a value without reaching it.
Black asserts about Zeno’s account
[1] that “after each step there are infinitely many steps still to be
taken”, and concludes [2]: “The logical difficulty is that Achilles seems called upon to
perform an infinite series of tasks”
([Salmon 70] p.72, my underlinings and square brackets.)
True enough the first statement (1)
expresses a certain understanding of Zeno’s paradox. (2) However, it is misleading to say that
Achilles “seems called upon
to perform an infinite series of tasks”,
even if these tasks are passages, for Achilles just runs in fact. The problem is that this run can be
divided in the passages Zeno talks about, and which we have
to discuss. (G3)
Black points out that Achilles’
logical difficulties with catching up the tortoise exist regardless of how
small the lead of the tortoise becomes, and he compares the task with drawing a
small square circle rather than a big one. ([Salmon 70] p. 71)
This evaluation is in agreement with
the justifications in the just mentioned subsection. Moreover, it has already been justified in the
subsection “The right recursive dichotomy-paradox” in Treatise No 8 that Zeno’s
paradoxes with runners, the way they are presented, are the fact that the
runner’s situation is unchanged after each passage of a stretch.
It can be summarized that we have to do with two different things: 1)
Zeno’s account deals with the runner’s his-et-nunc
ongoing passing of a line segment, i.e. the
present time. 2) The division of the line, however, deals with either an embraced past-fiction or an embraced
future-fiction.
Black asserts that it would be logically
impossible for Achilles to
pass the tortoise, if he had
‘to perform an infinite number of acts or, as Aristotle says “to
pass over or severally to come in contact with infinite things” (Physics, 233a),’ ([Salmon 70] pp. 79-80,
my underlinings.)
Here, Black compares passages with acts. It is certainly true that the
runner must perform infinitely many passages of the defined kind to reach the
goal, but only in the sense that for each performed passage a new is necessary.
This is in agreement with Zeno’s account.
However, Black rejects Zeno’s paradox with the argument that Achilles
only have to do with finitely many physical thing and circumstances and not
with infinitely many sections of the track. By this, he commits the physical
explanation, which is an example of the easily refutable interpretation. This
has been discussed above in the subsection “The physical error”. (G1)
What is left, then, is a paradox that as already explained consists in
the fact that each time Achilles reaches the point where the tortoise was at
the beginning of the stage, Achilles’ situation concerning his catching up with the tortoise can be described
in the fundamentally same was as at the beginning of the stage. In other words,
nothing new has happened, and Black has not solved the problem but only looked
at a simplification of it. Black’s termination of his article is also postulating:
“We create the illusion of the infinite tasks by the kind of mathematics
that we use to describe space, time, and motion.” ([Salmon 70] p. 81)
By this remark Black clearly makes a psychological digression, which
shortly put is irrelevant, if we address the case itself (G5):
As it has already appeared, the very core of Zeno’s paradox does not
deal with mathematics though fractions and factors inter into them. Thus, the
solution attempts based on mathematical conclusions that use these entities do
not approach an understanding of Zeno’s account. Moreover, as just mentioned, the
core of Zeno’s paradox is not that the runner must do infinitely many acts
before he reaches the goal, but that in a certain sense he makes no progress. (G3)
J.O. Wisdom wants to clarify some points in Max Black’s article, and starts
with describing the paradox as consisting of premise, inference, and
conclusion. First Wisdom clarifies what he calls Zeno’s premise in this way:
[P1] “For [P1.1] Achilles to catch the tortoise, [P1.2] he must run
to the tortoise's initial place; then the tortoise will have advanced to a
second place, so that Achilles must run on to the tortoise's second place; and
so on.” ([Salmon 70] p. 82, my square brackets.)
Wisdom asserts that under the precondition that Achilles’ runs double as
fast as the tortoise, and that this has a head start of one unit distance, the premise can be expounded as:
[P2] “Achilles’ distance is 1 + ½ + ¼ + etc.”
According to Wisdom, the conclusion of both versions is
[C]: “Achilles never catches the tortoise”.
According to Wisdom, either the premise must be false or the inference must
be invalid. ([Salmon 70] pp. 82-3)
It must be observed that the premise P1 contains the necessary condition P1.2, whereas P.2 expresses a
limit value.
Wisdom’s account is also made unclear by the fact that the word “inference”
can mean both the act of concluding
and conclusion (the result). The
least restrictive of the choices the reader is left with is that the conclusion
is a statement, viz. C, and that the inference is some reasoning leading from
premise [P1]/[P2] to conclusion[K]. However, the text says nothing about what
the inference should consist in. Moreover, it must be observed that Wisdom
presupposes that the paradox has to be rejected and that the question is just
how. Thus, Wisdom expresses a preconceived unwillingness towards Zeno’s
paradox. (G2)
Wisdom agrees with Black that the case is not 1) about proving that
Achilles will catch up with the tortoise, but 2) about working out where Zeno’s
argument fails. According to Wisdom, in order to solve the paradox we must
prove either that the inference (“Achilles never catches the tortoise”) is invalid
or that the premise (P1/P2) is false. Wisdom himself wants to prove that the premise
is false. [Salmon 70] p. 83)
It is of course true that the case is not about the mentioned proof (1),
not only because this kind of occurrences is a part of our experiences, but
also because the case is about understanding Zeno’s message. As to the proposed
alternative (2), Zeno’s argument does not
fail, however. First, the premise P1
is in fact logically correct, and second, it is circular to interpret the
conclusion from the discussed common
sense view. For as mentioned, Zeno’s argument involves that Achilles does
not catch up with the tortoise within Zeno’s
account, as in principle the situation remains the same after each step of
the recursive account. (G2, G6)
Wisdom asserts like Black that Zeno’s description of the run presupposes
that matter is infinitely divisible, and that this is impossible from a certain
point. ([Salmon 70] p. 84)
Here, it must be noted that J.O. Wisdom and Max Black, just like Bergson,
chooses the most easily refutable interpretation of the philosophic statements
they want to criticize. First, the paradox also applies to objects that move
along out in the universe and this objection cannot concern them. Second, as
mentioned too, the paradox can be understood even more abstractly. Wisdom’s
objection just expresses a superficial attitude to Zeno’s paradox. (G1)
Wisdom thinks that by regarding Zeno’s premise as equivalent to
performing an infinite number of tasks Black has been able to prove that the
premise is self-contradictory. ([Salmon 70] p. 84)
It is possible that Black’s example with the counting of marbles shows
self-contradiction, which Black’s argument is based on, but the example has
nothing to do with Zeno’s paradox, though it can be interesting enough as such.
For it does not clarify this paradox, but is simply yet a problem we can invent
and then seek to solve. Moreover, it must be possible to point out in what the
error consists, for if a statement implies a self-contradiction, this error
must arise from something, i.e. have logical or conceptual reasons. (G8)
First of all, what Wisdom calls “Zeno’s premise”, but which in fact is a
description, is not self-contradicting, but logically true. What Black has
probed is that his own infinity machine cannot succeed its run. However, that
is something different. Here Thomson commits an error that is more basic than
lack of understanding of the problem (G3).
Wisdom advances that the premise being self-contradictory does not imply
that the conclusion is false, as everything can be concluded from false
statements. ([Salmon 70] p. 84)
It is certainly true that from false statements everything can be
concluded – in the propositional calculus. This deals with logical consistence,
however, not with what is asserted. Thus, from the false statement, “Acropolis
is a river” we can conclude neither “2 + 2 = 5” nor “2 + 2 = 4”.
Moreover, it must be observed that Wisdom’s premise represents Zeno’s account
about Achilles, which is about what must necessarily apply, whereas Black talks
about the result of already performed actions. What is self-contradicting according
to Wisdom’s rendering is the premise, that to
catch up with the tortoise, Achilles must successively reach the points mentioned,
while what is self-contradicting according to Black is the idea that a removal
of infinitely many marbles can been completed.
According to Zeno, 1) for each traversed stage, yet a stage is to be
traversed in order that the run can be finished. Thus, there are potentially
infinitely many stages to traverse. This corresponds to the fact that for each
marble to be traversed in Black’s example, yet a marble has to be moved.
However, Black talks about 2) the situation after infinitely many marbles having been counted, which
corresponds to infinitely many stages having been traversed. As these two
discussions have individual subjects, Black’s argumentation cannot be a proof that
the discussed premise is self-contradicting.
Wisdom puts forward that in order to get the paradox disappear,
it is not sufficient to know, [1] that an infinite geometric series cannot
describe a physical run for logical reasons, or [2] that Achilles cannot
be demanded to perform an infinite series of tasks. For to say
that the demand is impossible implies “that Achilles cannot catch the tortoise
if he runs as prescribed by the premiss”, and this supports Zeno’s conclusion: that Achilles cannot catch the
tortoise. ([Salmon 70] p. 85)
As to point one, it can at most be due to practical reasons that a physical run cannot be described by an
infinite series of physical points, e.g. concerning measurability. Logic does
not enter here. Concerning point 2, it must again be remarked that it is
misleading to say that Achilles “runs as prescribed by the premiss”, for
Achilles simply runs. The fact that Zeno analyses this run the way he does s another
thing, as mentioned. Thus, we are dealing with a misunderstanding of the
problem. (G3)
The decisive point is naturally, that it is true that the discussion of
task apparently supports Zeno’s paradox, as the discussion of infinity machines
has a certain similarity to the discussion of passages of infinitely many
intervals. However, as mentioned, this “support” is misleading as Black’s
argumentation takes place within a common
sense frame, whereas Zeno’s conclusion is true within Zeno’s account, whose single passages can be said to take
place in the now, while the goal of the run is a moment in the future.
According to Wisdom, the premise discussed seems plausible because “it
seems reasonable to ask Achilles to carry out the infinite number of tasks
stated by the premiss.” The first step of Wisdom’s justification is that the premise
“Achilles’ distance is 1 + ½ + ¼ + etc.” contains no contradiction if
“Achilles’ distance” is understood as a mathematical distance. ([Salmon 70] pp.
85-6)
This is a too easy rejection, for as
mentioned we can imagine an abstract
world added time, perhaps a mathematical model of reality, and in this way we
can reach an interpretation of Zeno’s account. (G1)
Wisdom puts forward that the premise
is instead “Achilles physical
distance is 1 + ½ + ¼ + etc.”, and that this is self-contradictory as the
distance cannot be both “1 + ½ + ¼ + etc.” and physical. Wisdom wants to use this to
“dispel the plausibility of the paradox”. ([Salmon 70] p. 86)
Here again not the most difficultly refutable interpretation has
been chosen, but an immediately refutable interpretation. The erroneousness of
this has already been justified (G1, G8).
According to Wisdom, we cannot mark
this distance with physical points because they have a size, wherefore there
cannot be infinitely many of them on a finite distance. ([Salmon 70] p. 86)
Wisdom concludes that at some moment
two neighbouring points will touch each other such that the distance between
them cannot be subdivided any longer. Wisdom imagines that this can be “what
Black had in mind”. ([Salmon 70] p. 87)
The suggested difficulties with our
sensation of very small distances is immediately evident to anybody, and
therefore the pointing out of it can hardly be made to a result of a deep
philosophical scrutiny. This has been touched on in connection with the
commentaries to Max Black.
It can again be objected that Zeno’s
argument does not involve any marking of the partial intervals with physical
points. Moreover, the concept of physical point is not immediately evident.
Thus, it must also be considered what really happens when two points start
overlapping each other, as they must be supposed to have a size whatever they
may consist in. Thus, we are still dealing with a distance. We can ask these
two questions: Has Achilles reached the tortoise when two points on the
division of the track overlap each other? Does Achilles not move past the
point? The answers to this only involve a secondary discussion on the
contributors’ premises.
With implicit reference to the right-recursive dichotomy paradox, James
Thomson presents this argumentation, which we can call “A”:
“[P1] To complete any journey you must complete an infinite number
of journeys. [P1.1] For to arrive from A to B you must first go from A to A’,
the mid-point of A and B, and thence to A”, the mid-point of A’ and B, and so
on. [P2] But it is logically absurd that someone should have completed
all of an infinite number of journeys, just as it is logically absurd that
someone should have completed all of an infinite number of tasks.
Therefore [C:] it is absurd that to suppose that [C’:] anyone has ever
completed any journey.” ([Salmon 70] p. 89, my square brackets and underlinings.)
James Thomson assets about this:
“It may seem that this argument is valid; and then, since the conclusion
is absurd, we must deny one of the premises.” ([Salmon 70] p. 89, my underlining.)
To this, it must first be said that as a start of a discussion Thomson’s
reasoning is unfortunately rather intricately worded. For Thomson does not
conclude that the conclusion C is
absurd, but that C’ is absurd. So
the premise Thomson wants to put
forward in the latter quote cannot be “the conclusion is absurd”, but must be “C’
is absurd”.
Secondly, later in his article, Thomson refers to what is here called premise
P1 and premise P2; but obviously, the reader must seek to delimit these
himself. However, their boundary is made unclear because the first premise P1
contains a premise itself (P1.1).
Thirdly, it must be observed that premise P1 expresses a necessary condition and not a sufficient
condition. However, later Thomson
renders P1 as a sufficient condition.
Fourth, the quotation deals with journeys,
which in premise P2 are compared with tasks
without the purpose of this being obvious. Moreover, as mentioned, the
following must be observed:
Zeno’s paradoxes do not deal with tasks. They do neither deal with
journeys, unless passing a stretch is regarded as a journey. However, a journey
is usually a task. So we can temporarily conclude that very little is clear
her.
However, it must be observed that in spite of this criticism it proves
that with his considerations Thomson gets close to the essential in Zeno’s message
just as Black did.
As to the premise P1, the sub-premise
P1.1 is evident as it is possible to
arrive from A to B without passing
the intervening points. In addition, if we call the passage of the segment
between one of these points and the next “a journey”, it follows from this that
infinitely many journeys must be performed in order to reach B - according to our
common sense view.
As to the premise P2, it must
be observed that the fact that a statement is illogic follows from the choice
of words or the relation between statements, which cannot the case concerning a
passage of stretches.
In fact, it is neither illogical nor absurd at all that “someone should have completed all of an infinite number of journeys”. Certainly, we cannot point out a moment at which the run is being
finished, but notwithstanding where the runner or the object is we can refer to
infinitely many stretches that are passed after an earlier moment. Here, it
must again be observed that the discussed premise P2 deals with a surveyed period in the past, while Zeno’s account
deals with the runner’s actual situation.
Moreover, it must be observed that because of the law of inertia, passing
a stretch does not have to be “a task”.
To Thomson’s conclusion about the existence of an absurdity, it can be remarked
that Zeno just clarifies that there is something we cannot understand. As to
what it means to have performed
infinitely many tasks, as just touched on, we can explain this but not what it
means to perform infinitely many
tasks. This may be a source of the clarification of Zeno’s paradox.
According to Thomson, Max Black has argued that the expression “an
infinite numbers of acts” is self-contradicting, and thus affirmed premise P2. ([Salmon 70] p. 90)
As mentioned, Blacks argumentation is based on the run of a specific
infinity machine consisting of two coupled infinite machines that move objects.
As a matter of form, it must be remarked that this is different from Zeno’s
account. Black has therefore just argued
that it leads to a self-contradiction to assert that this machine has finished its run.
Thomson asserts that the quoted argument commits “the fallacy of equivocation”
and promises to explain this point. ([Salmon 70] p. 90, my underlining.)
I have fruitlessly attempted to find out where Thomson presents his explanation.
It can be ascertained, however, that the argument simply deals with two
different subjects, namely an ongoing
process versus a state. It is in
this connection, it proves that Thomson’s analysis approaches a clarification of
Zeno’s point.
First Thomson looks at premise P2
according to which it is absurd that anyone should have done infinitely many journeys.
Thomson calls the act of having performed infinitely many tasks a “super-task”
and asks whether this is a concept at all. ([Salmon 70]
p. 90)
As this section deal with Zeno’s paradox, we only need to talk about
passing of stretches, and as mentioned, in our hic-et-nunc situation we can
look back at infinitely many passages. So having performed infinitely many
tasks is obviously a reasonable concept in this way.
Thomson asserts that there are good reasons to assume that super-tasks
are impossible. As an example, Thomson puts forward his lamp example, which is
described above (in the section “Infinity-machines”). Thomson concludes that
such a task cannot be carried through as the idea of it leads to a
self-contradiction. This consist in the fact that the lamp must be either on or
off when the super-task is finished but that it cannot be either. ([Salmon 70] pp. 94-95)
According to this line, it must obviously be concluded that premise P2 is true. However, as already substantiated, the concept of infinity
machine does not contribute to the clarification of Zeno’s paradox, but makes
up an erroneous approach to this paradox, or rather a digression from it. For Thomson’s argumentation certainly seems to
support Zeno’s argumentation, but only within a common sense frame, and thus without being able to clarify Zeno’s
message. (G8)
The argumentation is based on the idea that we can presume that infinity
machines are physically possible.
However, as infinity machines can only be possible in a fiction with reduced
physical laws or in mathematically defined fiction, the idea of them cannot
lead to any self-contradiction. Their job will not differ decisively from
passing stretches.
Next, Thomson looks at premise P1,
(according to which completing a journey demands infinitely many of journeys) and
concludes [1] that it is in fact possible
to perform infinitely many journeys. For if one travels from point 0 to point 1,
one first travels from point 0 to point ½, then from point ½ to point ¾, etc. Thomson
says [2] that this is done when one
arrives at point 1 and that it corresponds to premise 1, but also [3] that this fulfilment does
nevertheless not contradict the
impossibility of super-tasks. ([Salmon 70] p. 97)
Re 1: As mentioned, premise P1
deals with logical necessity, which makes Thomson’s talk about possibility
irrelevant and misleading to this discussion.
Re 2: To be relevant, premise P1 must
refer to Zeno’s account the way as it is put forward. In this account, in a
certain sense the runner is in a standstill. With his talk about the runner’s
arrival at the point 1, Thomson ignores this decisive part of Zeno’s account. Zeno
exactly argues against the possibility of this arrival, so this cannot be
presupposed in an argument against Zeno. Moreover, Thomson talks about what has
been accomplished, and this has
nothing to do with premise P1, which
is about what is necessary. Our task
is to clarify the importance of this demand. Thomson’s reasoning expresses lack
of understanding of Zeno’s account. In addition, with its talk about an
arrival, the argument is circular. (G3, G6)
Re 3: Thomson does not substantiate this point here. However, this talk
about super-tasks is luckily irrelevant to this section, as it discusses the
passage of partial stretches referred to in Zeno’s paradox.
In what follows, Thomson commits further errors concerning sufficient
versus necessary condition. These are described in the appendix.
Perhaps Thomson’s justification concerning point 3 follows here. Thomson
focuses on the dividing points on the track, i.e. {0, ½, ¾,...}. Thomson calls
these points “Z-points” and asserts that it is impossible to have been at all
the Z-points without having passed the point 1, as there is always a Z-point to
the right before this point. According to Thomson, the question about where one
is if one has been at all these points without having passed point 1 is like
the question about the state of the lamp after its having been switched infinitely
often. ([Salmon 70] pp. 97-98)
This comparison is not evident, however, as the latter question about
the state of the lamp is a question about cause and effect, whereas the former
question about placing deals with a placing in which for logical reasons the
runner cannot be.
Thomson’s solution of the presented problem again consists in ignoring
it:
“To arrive at 1 you do not have to occupy all the Z-points and then
do something else. If you have completed all the journeys that have end-points
in Z, there is no further distance to run before arriving at 1. Arriving is not
running a last distance.” ([Salmon 70] p. 98)
The arrival to the point 1 is the only possible completion of all the
infinitely many passages, Thomson concludes. ([Salmon
70] p. 98)
It is completely absurd that Thomson (again) talks about what happens in
a situation that cannot take place
in Zeno’s account, viz. the arrival. For in order to reject the paradox, Thomson
exactly has to explain how this arrival can take place. Therefore, it is
erroneous to presuppose this arrival in a counter argument. As Thomson’s argumentation
is thus circular, we still lack a justification (G6).
Thomson asserts that this talk about occupying Z-points is something
strange:
‘[1] For when the order ”Run an infinite number of journeys!” is so
explained as to be intelligible, it is seen to be the order “Run!” [2] And
indeed how could one run any distance without being at some point midway between
point of departure and destination? [3] If running to catch a bus is performing
a super-task, then this super-task is, for some people at some times, medically
possible. But this super-task is just a task.’ ([Salmon 70] p. 99, my square
brackets.)
We can ignore the misleading talk about the runner in Zeno’s paradoxes
being ordered to run infinitely many
stretches. Contrary to this, it must be observed, that Thomson does not address
what the case is about. It is true that the
runner just runs in Zeno’s account, such as Thomson suggests, however, it
is exactly this run Zeno describes in a recursive way in order to express a
message that is in contradiction to our common
sense view. Asserting that the runner just runs is therefore not a solution,
but a description of what Zeno analyses, i.e. the discussed problem. Thus,
Thomson leads us nowhere. The argumentation is again circular. (G6)
In fact, there is nothing untrue in Zeno’s right-recursive dichotomy
paradox of which Thomson certainly expresses understanding in point 2, but as if
he is presenting a counter-argument. In point 3, however, Thomson refers to the
run as “a super-task”, i.e. as an embraced whole composed of the divisions of
the run, Zeno suggests, where after Thomson repeats his conclusion that we are
just dealing with a run. In this way Thomson turns Zeno’s description up and
down twice, but without coming nearer to an understanding of Zeno’s account
With this, Thomson’s solution is yet an ignoring of the difference between
Zeno’s account of the run and the common sense
view of the run. Apparently, Thomson does not quite accept that philosophically
we must address Zeno’s paradox on the basis of how it is described. Instead,
Thomson only addresses our common sense
ideas. Thus we are dealing with lacking understanding of the problem. (G7)
Thus Thomson puts forward the following conclusion, which contains a
relatively great complex of errors:
“when [1] we explain in what sense a man who completes a
journey completes an infinite number of journeys, and thus [2] explain in
what sense the first premiss is true, [3] we thereby explain that
what is said to be possible by the first premiss is not what is said to
be impossible by the second.” ([Salmon 70] p. 100, my underlinings and square
brackets.)
First, Thomson has not put forward any explanation of the first
mentioned kind, as we cannot point out any moment at which the run is finished.
Second, this would not have been able to explain why the first premise (P1) is
true, for this premise is true simply because it expresses a logically
necessary demand. Third, therefore and as mentioned, premise P1 does not deal
with what is possible, but with what is necessary. Moreover, we can in fact explain
what it means to have passed a
stretch, i.e. in which way the second premise (P2) is false. This has been
explained above.
Thus, the former premise, P1, can be said to deal with an ongoing
passage, while the latter premise, P2, deals with a now, in which we are
looking back at passages. In this way, Thomson in fact touches Zeno’s message the way it has been interpreted above.
As mentioned above, in his article “Tasks, Super-Tasks and the Modern
Eleatics”, Paul Benacerraf advances that Zeno’s idea with the paradoxes was to
prove that movement is impossible; and as to day the opposite is evident even
to philosophers there is no reason to disproof this. ([Salmon 70] p. 130)
First, as mentioned, our task is to clarify the argumentation Zeno puts
forward. Second, there is in fact an
important kernel of truth in Zeno’s paradoxes, namely that the concepts of
movement and time are problematic, as we cannot argue that they are in
accordance with our common sense view.
Therefore, Poul Benacerraf’s irony is misplaced and expressive of both unwillingness
and lacking understanding (G2, G3).
Benacerraf puts forward the following two commentaries to Thomson’s
rendering of Zeno’s argument:
1) If a journey from A to B is understood as passing the described
number of stretches then the first premise (“To complete any journey you must
complete an infinite number of journeys”) is true and not absurd; and therefore
the second premise (“it is logically absurd that someone should have completed
all of an infinite number of journeys”) false. For it is just as easy to
perform infinitely many travels that are just defined by a peculiar division of
the track, as it is to perform one. ([Salmon 70]
p. 104)
2) If, on the contrary, we understand this journey as performing a
number of separate acts then this is absurd, and therefore the second premise (“it
is logically absurd that someone should have completed all of an infinite
number of journeys”) is true. However, then the journey from A to B is just a
single journey, and thus the first premise (“To complete any journey you must
complete an infinite number of journeys”) is false. For:
“one need not complete an infinite numbers of journeys of this kind
in order to complete a single one.” ([Salmon 70] pp. 104-105)
Re 1: It is true that Zeno’s account defines a peculiar division of the
track just as in the now we can look back at potentially infinite many traversed
stages. However, this does not change Zeno’s account. According to Zeno’s
right-recursive dichotomy-paradox, after each stage, the runner is in principle
in the same situation as before the stage. Benacerraf’s conclusion leads us
nowhere, as it is exactly this “journey” from A to B Zeno analyses. It is also
this analysis Benacerraf must address in order first to understand Zeno’s
message and then to criticize it. Instead, Benacerraf expresses both unwillingness
and lacking understanding. (G2, G3)
Re 2: The first part of this point only concerns the concept of infinity
machines and not Zeno’s paradox, as Zeno’s account does not contain any talk of
separate actions at all. As mentioned, involving them is discussing another
problem. (G1)
The last part of the comment is simply just a rendition of the common sense view of the run. In other
words, also this comment is futile, as it is this single “journey” Zeno analyses.
This view expresses unwillingness by presupposing common sense in the criticism of Zeno’s criticism of common sense. In this way, Benacerraf
simply ignores Zeno’s paradox. (G7, G2)
The following concerns the concept
of ω-task versus the passages in Zeno’s paradoxes.
As a reaction on Benacerraf’s criticism
James Thomson advances that he (Thomson) meant that the idea of a practicable
ω-task was suspect, as the result of such a task could not be decided. Therefore,
it was only meant as a joke to refer to the passing the points 0, ½, ¾,... on a
track as an ω-task, since carrying through the run was the same as running
from 0 to 1. ([Salmon 70] p. 130)
Again, it must be emphasized that it
is not a solution of Zeno’s paradox to assert that the runner just runs through
the track. Thomson has to address Zeno’s analysis of this run and the common sense view Zeno questions with
his accounts about runners. Therefore, with the last part of the quote, James
Thomson expresses lacking understanding of the idea of Zeno’ paradox. Thomson’s
reasoning is futile in the same way as Benacerraf’s, as it leads us nowhere. Thus,
Thomson expresses the same lack of understanding as a number of the other
contributors. (G6, G3)
It is Owen’s intention to explain why some proposed solutions of Zeno’s
paradoxes are beside the point. ([Salmon 70] p. 139) Here, however,
we shall only look at the paradoxes with runners.
According to Owen, we can consider Aristotle’s solution that the less
the distance is, the less also the necessary time to pass the distance is.
([Salmon 70] p. 144)
To this, it can be added that just as the goal is not reached within Zeno’s account, then the moment, when the goal would be reached
according to the formula s = v * t, where s denotes the traversed distance, t
denotes the elapsed time, and v denotes the runner’s velocity, is not reached in
Zeno’s account. The reason is both that 1) the traversed distance and the time go
together, so to say, and that 2) the runner first reaches the temporal halfway,
then the half of the rest of lapse of time, etc. Thus, Zeno’s paradox applies
to both distance and to time.
Owen asserts that he understands Zeno such that if Achilles has caught
up with the tortoise, we can ask about the position of the two last
divisions of the line. Whether there is a distance or not, there are
problems: either there is no stretch between them or the total stretch is
infinitely long. Therefore Achilles cannot have caught up with the tortoise. ([Salmon 70]
p. 145)
This cannot be inferred from Zeno’s account, which exactly never will
end, for which reason we cannot be dealing with any last division.
On the basis the referred text, Owen describes Zeno’s argumentation thus:
1) There cannot be any last movement in Achilles’ series of movements. 2) If
this series of movements can be carried through, it must be possible to
describe the result without absurdities. From this, Zeno concludes that Achilles
can never catch up with the tortoise. ([Salmon 70]
p. 145)
This argumentation appears reasonable. It must even be observed, that if
point 1 is justified, there is no reason to discuss point 2, which is certainly
correct. The core of the matter is that we cannot point out any particular moment
at which the run is being finished rather than at any other moment, and thus no
last divisions. The point is simply point 1.
Owen solemnly concludes, “Any hope of salvation lies in looking in this
inference”. ([Salmon 70] p. 145)
However, from what are we to be salvaged? May there not be a paradox? The
existence of this paradox just means that there is something we have not quite
understood or clarified concerning the elapse of time, which we than have to study.
With his remark, Owen expresses an unwilling attitude. (G2)
Owen refers to Black’s description of Hercules’ cutting off Hydra’s heads,
which grow up just as fast as Hercules cuts them off, for which reason there
never arises a situation in which Hercules has finished his cutting off.
According to Owen it is otherwise with Achilles:
“There are plenty of states of affairs compatible with Achilles’
having achieved his task of overtaking the tortoise: plenty of positions beside
or beyond the tortoise that Achilles can have reached.” ([Salmon 70] p. 146)
It does neither help here to put forward this common sense description, as Achilles’ run does not finish within
Zeno’s account. The account, or the runner, does simply not reach any following
state. Thus, Owen’s description presupposes that common sense assumption Zeno
criticizes, viz. that Achilles can reach the tortoise. (G6)
Owen, and other contributors, asserts altogether just “Zeno puts
forward a paradox, but we have experienced that the case is otherwise”.
However, Zeno’s account cannot be disposed off in this way. Our philosophical
task is to clarify this account and clarify what we can put into the statement
that the case is otherwise, if we think so.
It must be emphasized that Owen slightly touches the above mentioned clarification
of Zeno’s paradox about Achilles and the tortoise according to which we are
dealing with a recursively told account (Cv. the subsection “An unfinished
account” in Treatise No 8), as Owen
asserts:
“It is just the case here [1] that Achilles’ movements have been so
described that they have no last term, but not so [2] that no subsequent
state of affairs is compatible with his having completed the series.” ([Salmon 70] p. 146, my square brackets and underlining.)
However, Owen does not focus on the clarification in question expressed
in the first part of the quote (1), and does not even mention it as a
clarification. The second part (2) ignores Zeno’s account whose point is that
its single stages cannot lead us from the past to the now including its
“subsequent states of affairs”. However, these stages are present in the now in
which we can consider a past-fiction about a passage of stretches.
Grünbaum asserts that what he calls
Zeno’s claim that the run lasts infinitely long has been made plausible by our
idea that there is lower bound of the duration of each individual partial
stretch. Grünbaum declares that he will demonstrate this error of Zeno. Moreover,
Grünbaum asserts that we let us inveigle into believing this, because we
imagine the each period must have a minimal length. ([Salmon 70] pp. 205-6)
It can be objected to this that Zeno
does not assert that the run lasts infinitely long. Unfortunately, Grünbaum
does not tell the reader to what he refers with his talk about our ideas of the
minimal length of periods. However, at least the paradox itself is not based on
this. This has been clarified above. It must be observed that now when we have
understood this alleged delusion, and are thus not inveigled any longer, Grünbaum’s
talk about it is irrelevant, wherefore we can concentrate our attention on the
real content of the paradox. Grünbaum’s line of thought is a clear case of
psychological digression. (G5)
Some of Grünbaum’s further attributions
are gone through in the appendix below.
Despite his reluctance, however, Grünbaum
is in fact close to a solution when he asserts that
“Zeno illicitly exploits the fact
that [1] it is logically impossible for the terminal
instant of the motion to belong to any of the subintervals of the unending
progression, since it is later than all of them. Zeno appeals to this fact to
infer wrongly [2] that there cannot
be any terminal instant at which the runner reaches his destination.” ([Salmon
70] p. 207, my bracketed parentheses.)
It is correct that (1) Zeno’s
account never will comprise anything else than the stretch arbitrarily close to
the goal, however it contains nothing wrong. For (2) we can in fact not point
at any moment in the account at which this passage of intervals finishes, no
matter how late in the account. (Cf. the subsection “An unfinished account” in Treatise No 8.)
Zeno’s message is described above
and can be summarized thus: A) the runner, or the moving body out in the space
among the fixed stars, is at any moment at a definite place. B) As just
described, Zeno’s account deals with the problem of how the runner has arrived
at the place where he is in time and space. We are not dealing with an exploitation of any fact, but with a
message. Grünbaum is close to an interpretation of it, but without reaching it.
(G3)
Grünbaum asserts that as “the
terminal instant” does not belong to the single subintervals it does not belong
to the complete stretch. Then Grünbaum concludes somewhat clumsy, that its
half-openness does not mean that it must be of infinite duration, just because
it has no terminal instant, and just because the sub-intervals have no last
element. ([Salmon 70] p. 207)
Here too, Grünbaum needs to state a reference,
viz. to his allegation about infinite duration. As mentioned, Zeno’s paradoxes with
runners do not deal with infinite
duration, but with a lack of transition from the past to the now. We are
again dealing with lacking justification. (G4)
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 they do 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.
In the subsection “Zeno’s message”, a
justified interpretation of Zeno’s account is put forward. The contributors,
however, have not expressed any interest in understanding Zeno’s account, but have
only regarded it as an idea we must reject.
As a criticism of this attitude, 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.
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.
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 just 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.
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. 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. The question is how the difference between present tine and the future
is exceeded.
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. Generally, the most advantageous
to the philosophical result is 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.
The concept of really existing arbitrarily
small entities does not even belong to a scientific model. 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 two statements. 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”. Contrary to
this, 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 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, viz. 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 here be objected that we live
and act in accordance with our common
sense view of the world, and that this takes place 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 the content of Zeno’s
paradoxes with runners.
