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IS MOVEMENT AN ILLUSION ?

Zeno’s Paradox From

A Modern Viewpoint.

F. Walter Meyerstein.

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Tavern 45, 08006 Barcelona, Spain.

*E-mail : walter@filnet.es*

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**A B S T R A C T **

The Greek philosopher Zeno presented for the first time in history the problems derived from assuming (or rejecting) the infinite divisibility of space and time. He showed that knowledge of the physical world is dependent on what axioms concerning reality are admitted: either space ant time are atomic or dividablead infinitum. Aristotle, differential calculus, Einstein’s relativity, nonstandard mathematics, and modern philosophers such as Heidegger, all tried to cope with this problem. However, their "solutions" always imply adding controversial new axioms. Thus, a fundamental aspect of how humans understand Nature or, equivalently, the problem of determining which of the possible but indispensable axioms should be given pre-eminence, is reflected in the study of this famous paradox. Finally, a recently characterised subset of the real numbers called "Lexicons" adds a surprising twist to this notorious paradox.

Zeno, the Greek philosopher from Elea in southern Italy, who lived from
*circa* 495 to 445 BC, wanting to prove his teacher’s Parmenides thesis of the
impossibility of all motion, conceived the notorious "paradox" of Achilles and
the tortoise, the solution of which has challenged mathematicians and philosophers
throughout the centuries. He claimed : Not even Achilles, the fastest Greek hero of the *Illiad*,
can ever catch a slow tortoise if that animal is given a head start of, for instance, s1
meters. Obviously, to catch the tortoise, Achilles must first run that distance, say in t1
seconds. But once he gets there the tortoise has already moved further off a distance of
s2 meters, a distance Achilles will cover in t2 seconds. And so on, the procedure must be
repeated, *ad infinitum*. However close Achilles gets to the tortoise, there will
always remain some infinitesimally small distance yet to go. But we do see Achilles in
real life easily catch the ponderous animal. Conclusion : Real life motion is an illusion
!

It is clear that the paradox assumes as true by axiom one of the most
consequential ideas of the Greek philosophers : space and time are a continuum that can be
divided indefinitely, and there is neither and *atom* of space nor an *instant*
of time. And if this is so, Zeno claims, there will always remain some physical distance,
and some duration of time, before Achilles catches the tortoise, no matter how small. In
other words, what Zeno’s paradox asked philosophers to explain, if motion is assumed
to be real, is *how an infinity of acts can be serially completed in finite time*,
even if each act is infinitesimally small.

As stated, the problem raised by Zeno has been the subject of two and a
half millennia of analysis. Plato treated the question *in extenso* in his dialogue *Parmenides*
showing, in truly "Platonic" form, that whichever approach is favoured :
divisible *ad infinitum* or atomic, there will always inescapably result unacceptable
paradoxes. After Plato, Aristotle’s eight books of *Physics* address the problem
. He wrote : "Since Nature is the principle of movement and change, and it is Nature
that we are studying, we must understand what movement is (*Physics* III 200b 12-13) (footnote i). He further claimed that "infinity cannot exist as an
actualised entity, [for then it] must be either altogether indivisible or divisible into
infinities. But for one and the same thing to be many infinities is impossible (*Physics*
III 204a 21-28)". Since "we are engaged in the study of things cognizable by the
senses (*Physics* III 204b 2)", and motion is a fact of the senses, the question
is : How is motion possible ?

For Plato and Aristotle, "motion" (*kinêsis*) means
any kind of *change*, not just how something, Achilles for instance, can pass from
being at rest to being in motion, but also the contrary, how can motion stop, or how
something,* viz*. Achilles, comes into being, or ceases to be, that is, dies. For if
time and space are infinitely divisible Zeno’s paradox applies, and to be born or to die
are then both equally pure illusion! How does Aristotle "solve" the problem ?
Here is what he says : "If we are asked whether it is possible to go through an
unlimited number of points, whether in a period of time or in a length, we must answer
that in one sense it is possible but in another not. If the points are actual, it is
impossible, but if they are potential it is possible. For one who moves continuously
traverses an illimitable number of points [of time and space] only in an accidental, not
unqualified, sense ; it is an accidental characteristic of the line that it is an
illimitable number of half-lengths ; its essential nature is something different (*Physics*
VIII 263 b 4-8)". Otherwise stated, Aristotle is distinguishing between different
kinds of infinities : the actual and the potential. Traversing a region of space (or of
time) does not involve moving across an *actual* infinity, which would be impossible.
However, it is consistent with crossing a *potentially* infinite number of
sub-regions of space (or time intervals), in the sense that there can be no end to the
process of dividing space (or time). Thus Zeno’s paradox is pertinent, but only
potentially, whereas our senses prove that actually Achilles does catch the tortoise. In
summary, the question raised by Zeno is : To which of the two possible means of
acquisition of knowledge about the physical world are we to give priority : to pure
(mathematical) reason or to our (common) senses ? Aristotle’s answer was : Pure
(i.e., independent of all experience) reason shows what is potentially possible, whereas
the senses (i.e., results of performed measurements) teach us the actual world. As a side
note, remark that in our days we see this Aristotelian idea in quantum mechanics : the
quantum wave function is said to represent the linear superposition of all the different
"potentially" possible states of some physical system, whereas nothing but the
"actual" measurement provides the senses with the one and only objective (i.e.,
"eigen-") value. The rest of the maybe infinite possibilities are then said to
have "collapsed" or vanished somehow.

Western philosophy spent almost two-thousand years trying to shake
loose from Aristotelian physics and recover the Platonic approach put forward in the *Timaeus*
(reference as in note iii below), claiming to give absolute pre-eminence to the
mathematical (pure reason) *model* over the (common) sense apperception of Nature.
Consistent with this paradigm shift, after Newton and Leibniz sanctioned differential
calculus as the pre-eminent tool to "explain" the world, a new solution to
Zeno’s paradox was proposed. Clearly, before catching the tortoise Achilles must
traverse ever smaller segments of space in ever smaller intervals of time. Does the
infinite (as admitted by the pure-reason hypothesis) sum of these ever smaller terms
converge to a finite value, or is it itself infinite ? What is the *limit* of such a
summation when its infinite terms tend to zero ? At the limit, the ratio of the space
segments divided by the time intervals in which Achilles traverses them approaches the
ratio of 0/0, which is indeterminate. But this ratio of the space segments over the time
intervals is equal to Achilles’ instantaneous speed : how can that be indeterminate
precisely at the point where he finally catches the tortoise ? Thus, Achilles can only
catch the tortoise if the infinite terms of the above mentioned sum converge to some
finite value. It follows that in the real world Achilles’ continuous velocity
function is at least one-time differentiable everywhere.

But, initially, Achilles is at rest. How can he pass from rest to
motion? According Zeno this is impossible since, if time is illimitable divisible, at the
immediate next "instant" in time after rest he must already possess some finite
velocity ; otherwise, he remains at rest, and so never could start moving. Whatever that
initial (finite) velocity, when divided by the very first interval of time, and if time is
infinitely divisible then that "first" interval may be taken as close to zero as
you wish, it results that the quotient of that initial (finite) speed divided by the very
first interval of time, that is, Achilles *acceleration*, now tends to infinity. And
since the reverse argument also applies, motion can neither start nor stop ! Unless, of
course, the summation of the ratio of the ever smaller velocities over the ever smaller
time intervals converges. That is, throughout the interval, the acceleration of our hero
must furthermore conform to a *continuous (footnote ii) function
everywhere two-times differentiable*. But the subset of the continuous everywhere
two-times differentiable functions is a vanishing small fraction of the set of all
possible continuous functions ; and so, in spite of all the effort, we are back where
Aristotle left us ! Since we do observe Achilles catching the tortoise, we must conclude
that the "potential" set of all possible continuous functions that are *not*
everywhere two-times differentiable will simply not obtain in the "actual" case.
For, unless everything our senses transmit is pure illusion, Achilles can be born, start
running, catch the tortoise, stop, and someday die, only if he always manages to traverse
the space-time continuum in this particular everywhere two-times differentiable manner.
Potentially anything can happen, or everything can be an illusion ; actually Achilles has
a speed and an acceleration conforming to differential calculus. The physical world may be
"explained" by our (pure-reason) mathematical model, but only as far as it
conforms to observed phenomena ; all potentially possible but unobserved outcomes
"collapse" and vanish.

More recent approaches to Zeno’s paradox, while still adhering to
the Platonic point of view, come however to the contrary solution : all movement *is *illusion.
This can be seen in what may be called "Einstein’s solution". In
Einstein’s Theory of Relativity the continuous (i.e., non-atomic) space and the
continuous (i.e., non-atomic) time are fused into one continuous (i.e., non-atomic) entity
: space-time. This four-dimensional *topological space* (footnote iii)
possesses this characteristic : it is entirely frozen, in it there is simply no change, no
movement, no *kinêsis*. This follows from the condition that time is already
incorporated as the fourth dimension of space-time ; so, how could anything *change *?
Consequently, in four-dimensional space-time all movement, all change is an illusion, as
in a reel of film, where "time" appears as a number, a dimension : the number of
each frame. The illusion of *kinêsis* (from which "cinema") is achieved in
the usual manner, and the illusion of future and past are the result of running the film
in one or the other direction. Zeno and Parmenides would have enthusiastically endorsed
this way of understanding reality.

A different approach originated in the 1960s with the work of A.
Robinson, followed by that of E. Nelson (footnote iv): nonstandard
mathematics. Once again, the question is : Is a line segment divisible without limit ?
Take a segment of finite length, call the first "point" on it *zero *and
the last "point" *one*. The *distance* from the origin (zero) to any
point on the segment is given by a (standard) real number of the form : zero-point (0.),
followed by an *infinite* expansion (sequence) of digits (for instance,
0.2854618326580009276492651648206517848.....). This is the mathematical version of
Zeno’s axiom, inaccessible to the Greek who did not know the zero. Accepting the
"existence" of the real numbers is a strong hypothesis and, consequently, their
study has become a key element in the search for a set of axioms or fundamental
assumptions on which elementary number theory, and by extension, the whole of mathematics,
might be firmly based. It is generally accepted that the set of 10 or so statements
supporting most mathematical systems is the Zermelo-Fraenkel set theory. To these
statements the nonstandard approach adds three additional axioms. They are based on the
definition of a new nonstandard number : the infinitesimal. An infinitesimal nonstandard
number is a new type of number : by definition, it is greater than zero but always less
than any standard real number, however small.

These infinitesimals possess a thoroughly elusive character because they can never be captured through any possible measurement. The reason : measurements have always as result a standard real number. Furthermore, the difference between two standard real numbers can never be a nonstandard number, which is by definition always less than any standard number. Thus the interval between two nonstandard points on the line, or two nonstandard intervals of time, can never be measured, and so these intervals are forever beyond the range of observation. They exist only by axiomatic (Platonic) definition but can never become actual in Aristotle’s sense.

The nonstandard theory adds two more nonstandard numbers as axioms. The nonstandard unlimited number, which is the inverse of an infinitesimal number, is greater than any standard number but nevertheless smaller than infinity. The nonstandard unlimited numbers are thus very large, larger than any standard number, but always finite, that is, always less than the truly infinite numbers. The nonstandard mixed numbers are so to speak in between : around each standard number, on both sides of it on the line segment, a particular set of nonstandard numbers is, on the left, greater than any other standard number but less than this particular standard number ; on the right side, it is smaller than any standard number greater than this particular number, but it is still greater than this number. In summary, between zero and infinity, a new infinity of nonstandard numbers has been added by axiom.

How does this nonstandard mathematics "solve" Zeno’s paradox ? Achilles, as he gets closer and closer to the tortoise, will be traversing an infinite series of ever smaller space segments until eventually he will be at nonstandard infinitesimal distance from the tortoise. From this point on, his progress until he catches his prey escapes all possibility of measurement : all the final segments being nonstandard infinitesimal distances. In other words, what "really" happens when Achilles catches the tortoise can never be known, by definition, and so the case rests.

In the early twentieth century some philosophers have been particularly
intrigued by the time continuum : if the instant is zero, when do we *exist* ? The
past is already gone, the future not yet here, and the present instant zero : when can we
claim that we *are *? Martin Heidegger’s attempt to crack this problem in *Being
and Time*, first published in 1927, may in a certain way be considered to be one more
"solution" to Zeno’s paradox. This philosopher suggests that humans are never
authentically "being" ; instead, from the very moment one is born, one is
already dying, i.e., not-being. "The moment you are born you are old enough to
die". He furthermore claims that the only "time" that has a sense is the
unknown period a human still has before he dies, that is, only the yet non-existent future
is real. If one is asked : Will you die ? the answer is : Yes, of course,* but not yet*.
Like Achilles : will he ever catch the tortoise ? Yes, of course, *but not yet*. So,
Heidegger states, there are two manner of being : the inauthentic and the authentic. In
the former, which is where most of us choose to *be*, our allotted time-span is this *not
yet*, this unknown future which allows us to escape from, to conceal, the unbearably
displeasing fact : we are mortal. In this manner of being, we never *actually* die,
we are *always* alive ; death is "only" a *potentiality*. And we can
say such strange things as : I* have* no time, don’t *waste* your time,
etc. Whereas *being*** **authentically is equivalent, in a sense, to *dying *!
That is, to fully accept human mortality. Put differently, inauthentic *being* is
equivalent to live, to *be*, in the standard part of the time scale : there we can
"measure" time with watches in standard numbers. Whereas authentic being is
traversing our life-span in the nonstandard numeration, which is beyond measure, and
where, in a sense, as soon as we are born we are already dying (footnote v).

Recently (footnote vi) a new, fascinating twist to
this old conundrum appeared : Cris Calude’s *Lexicon*. Here is succinctly how it
applies. Zeno’s paradox, and its possible "solutions", must be somehow
explicitly stated in a communicable language. This implies some linear sequence of
symbols, the set of allowed symbols constituting the pertinent alphabet. Any *finite *sequence
can be unambiguously coded in binary (or decimal) and thus corresponds exactly to some
rational number. This paper, for example, corresponds to the rational number "**w**".
On the other hand, real numbers are infinite sequences of digits (in whatever chosen code
or *base*). Question : Is there a real number that with certainty contains the *word*
**w** (i.e. : *exactly* this paper) ? Answer, Yes, and furthermore, there exists a
real number that contains *every possible "word"*. That is, that contains *everything
that can be explicitly stated, coded, communicated.* Here is how that number is
constructed, in binary : simply add one after the other every possible binary sequence of
1,2,3,4,.....bits :

0,1,00,01,10,11,000,001,011,111,110,100, 010,101, 0000,0001,............

all the way to infinity. By construction, absolutely everything that can be explicitly stated is represented, at least once, in this sequence.

Now it can be shown that this special real number not only contains, by
construction, every possible finite linear sequence, say William Shakespeare’s
complete works, but also that it contains every possible linear sequence *infinitely
many times !* This is easily proved. Again, call some sequence, say this paper, **w**.
Now construct these sequences :

**w**0

**w**00

**w**000

**w**0000

.................

all the way to infinity. Since by construction, each of them is already
on our specially constructed binary real number, all of these "words" or binary
sequences must also appear, at least once. But in each of them **w** appears, hence **w
**appears infinitely many times. And this is the case for every possible **w**, QED.

It has been shown in 1998 by Calude and Zamfirescuvi (footnote
vi) that there exist real numbers that present this remarkable property *independent
of the employed code or alphabet *(binary, decimal, or, for instance, all the symbols
on a computer keyboard). These are the Lexicons. Thus a Lexicon contains infinitely many
times anything imaginable and not imaginable, everything ever written, or that will ever
be written, any description of anything, of any phenomenon, real or imaginary, etc., etc.
But where are these monsters to be found ? The amazing result is : almost every real
number is a Lexicon ! That means, if you put all the reals in an urn, and blindly pick out
one, with almost certainty it will be a Lexicon.

So what is the relation of this surprising result with Zeno’s
paradox ? We humans are limited and mortal. We can only name, we can only put our fingers
on, we can only *actually* exhibit, the rational (finite) numbers. But underlying all
our mathematics, and Zeno’s paradox, are the real numbers. These, at least the
immense majority of them, contain *potentially* everything, and infinitely many
times. What we can *actually* exhibit is only a vanishing small subset of the
underlying *potentially* possible number set. Now, we may call some
"fact"corresponding to some rational number a *novelty*. But this is just a
Zeno-type illusion : everything is already contained (or expressed) infinitely many times
in the infinite set of the reals almost all being Lexicons.

But what is a fact ? Something that may have happened and that is
amenable to a communicable description (measurement). For instance : the result of the
sense-perception by which we observe Achilles catching the tortoise. Thus, something that
corresponds to some finite sequence, to some rational number. But this "fact",
this sequence, this change, movement, *kinêsis*, is infinitely many times recorded
in infinitely many real numbers.....if they exist, of course. Then it follows from the
admittance of the existence of the real numbers, of Zeno’s infinite divisibility
axiom of space and time, that there can never be any novelty. Everything *is*,
always, and nothing ever *becomes* : Parmenides was right after all !

"The thing that has been, it is that which shall be ; and that which is done is that which shall be done : and there is no new thing under the sun" ; so says the Ecclesiastes 1.9 (composed after 250 BC (footnote vii)). Everything, the theory tells us, is right there under our nose, so to speak, potentially, but inaccessible. Whereas anything that actually happens, happened, or will happen, is an illusion, since it has been "there", already, always. A final question remains, however : has anybody anytime laid his fingers on such a Lexicon ? Surprisingly, yes, Greg Chaitin’s marvelous and mysterious real number is a Lexicon. But that is another story.

In conclusion, after twenty-five centuries Zeno’s paradox is still
with us: if we admit the existence of the real numbers we run into trouble; we deny it,
and we find a different set of equally intractable problems since now mathematics, and
thus physics describing "change", become problematic. We are left with
admiration for those early Greek philosophers, who unveiled the fundamental *limits*
of human reason (and of mathematics (footnote viii)).

*Copenhagen 1996, Auckland 1998.*

**ACKNOWLEDGEMENT**. I wish to thank Prof. Cristian Calude for
encouragement and very helpful discussion and comments.

**NOTES and REFERENCES**

**viii** See Gregory J,
Chaitin, *The Limits of Mathematics*, Springer-Verlag Singapore, 1998.