Zeno's
Paradox: A response to Mr. Lynds
by Eric Engle
Paradoxes exist to point out flaws in our reasoning. They are thus heuristic
devices. A paradox occurs when our presumptions are inadequate to solve a
problem. Thus for example, if we believe (erroneously) that all statements
must be either true or false we will quickly run into paradoxes. For example,
the statement "this statement is false" is a classic paradox with no truth
value. The statement is neither true nor false. It is indeterminate. (The
tougher paradox of this art is in fact whether statements about unicorns have
truth value - clearly unicorns do not exist - but does that mean that a statement
about a non-existing entity is false or merely with no truth value?).
Paradoxes such as these exist because people think that all statements must
have a truth value, that is that all statements are either true or false.
In fact Aristotle in Posterior Analytics (1) already recognized
that some statements have no truth value. Thus the solution of the paradox
of the false/true statement is to see that we must exit the category of "true/false"
and enter into the category "indeterminable". It is even clearer since Gödel:
the truth value of some statemens is indeterminable. (2)
Zeno's paradoxes all concern motion. Zeno effectively asks "How can motion
be possible?" This paradox is arguably of little heuristic value today because
we have since Einstein at least recognized that time and matter-energy are
convertible elements, the same thing in fact. Thus rather than seeing a solid
object, an arrow, existing at definite points in its trajectory, the correct
view is to see a wave of energy following the arrows trajectory with much
greater mass/energy presence at certain instances of space time.
That understanding is radically different from the ancients such as Zeno.
For the ancients just as geometric points had no dimension just location so
also material loci were either void (kenon) or contained atoms. It is fair
to say that geometric points and atoms corresponded to each other in the
ancient conception of physics. For some, probably most, ancients matter and
energy were not transmutable: rather the indestructible nature of atoms was
a presumption of at least some ancients.
Given these different assumptions about the nature of time and matter it
is unsurprising that the heuristic value of Zeno's paradox is multi-variate
and historically conditioned. In a world that (generally) presumed that matter
and energy were very different quanta, Zeno's paradox forced one to ask what
is meant by motion and time and to consider the presumptions underlying atomic
theory. For the moderns where both space/time and matter/energy are transmutable
the paradox really only illustrates the erroneous assumptions of pre-relativistic
physics. Zeno's paradox really implodes when one understands that matter/energy
space/time are both elements of a unified field.
It is thus with some surprise that I read Mr. Lynds' article on Zeno. I
would first note that Lynds' methodology is basically correct. When confronted
by paradox, we must somehow "break out of" the flawed presumptions which generate
the paradox. Lynds' solution however is not to re-iterate the relativistic
assumptions of the convertability of matter/energy and space/time. Rather
it is to call into question the concept of time as consisting of discrete
elements which correspond to material loci such that one could say that at
time t1 the arrow is at locus l1. This is an admirable attempt to think "outside
the box". It might eventually prove correct. However it seems that Lynds reaches
the possibly correct conclusion, that time is a continuous entity, by a certainly
incorrect method namely a tautology and reductio. Lynds states:
"We begin by considering the simple and innocuous postulate:
'there is not a precise static instant in time underlying a dynamical [sic]
physical process.' If there were, the relative position of a body in
a relative motion or a specific physical magnitude, although precisely determined
at such a precise static instant, would also by way of logical necessity be
frozen static [sic] at that precise static instant. Furthermore, events and
all physical magnitudes would remain frozen static [sic], as such a precise
static instant in time would remain frozen static [sic] at the same precise
static instant." (3)
Similarly:
"Regardless of how small and accurate the value is made however, it cannot
indicate a precise static instant in time at which a value would theoretically
be precisely determined, because there is not a precise static instant in
time underlying a dynamical physical process. [sic] If there were , the relative
position of a body in relative motion or a specific physical magnitude, although
precisely determined at such a precise static instant, it [sic]
would also by way of logical necessity be frozen static [sic] at that precise
static instant. Furthermore, events and all physical magnitudes would remain
frozen static [sic], as such a precise static instant in time would remain
frozen static a the same precise instant: motion would not be possible."
(4)
Lynds argues that if time were divisible into discrete elements, i.e. if
time were discontinuous, then that would mean that motion would be impossible.
This is an argument by reductio ad absurdam. However as we see the
reductio appears to be founded on a tautology. It seems that Lynds
believes merely be reaffirming the static nature of objects that they somehow
become necessarily frozen forever at one point in space time. Further, even
if Lynds' statement were grammatically correct and not a tautological statement,
Lynds' reductio does not compel the conclusion he believs it does.
Lynds fails to consider other possibilities than that motion be continuous
and time discontinuous. What if motion were also discontinuous? Then Zeno's
arrow could occupy locus l1 at time t1 and locus l3 at t2 without ever transiting
locus l2. If motion were a series of very tiny (even infinitely tiny?) "jumps"
(teleportations if you will). This in fact does appear to reflect sub-atomic
physics where, as I understand, particles mysteriously appear and disappear
as if teleported. If we presume motion is in fact discontinuous then we are
in no way compelled to admit Lynds' arguement by reductio, that time
cannot be divided into discrete elements.
Methodologically, an argument by reductio ad absurdam is actually
quite weak. A reductio does not actually prove its affirmation. Rather
it disproves the negation and from that "dis-proof" attempts to infer the
opposite conclusion. "If A then B. A is not so. Thus B is not so." Logically
however reductio seriously risks confusing the possible with the necessary.
Arguments by reductio can be summarized as "Let us presume the opposite
of what we wish to prove. That supposition however would lead to an impossibility,
i.e. an absurdity. That cannot be the case. Thus the opposite must be true."
As was said, reductios risk confusing the possible with the necessary.
If we prove that presumption X leads to an impossible consequence that does
not mean that presumption not-X leads to necessary consequence. By showing
a statement to be asurd it remains possible that the opposite statement is
true - but that is not a necessary consequence. It is only where all other
possibilities have been eliminated that the reductio necessarily holds
- and even here, if a new presumption can be developed, say due to new evidence
(for example, that matter/energy are one and the same and are convertiblee)
then the reductio fails.
Those then are the weak points of Lynds' argument: he appears to attempt
- using an impermissible tautology - to prove by reductio a possible
but not necessary proposition: that time is not divisible into discrete elements
but rather is a continuous entity. That conclusion may well be true. However
Lynds has not reached it in a methodologically sound fashion because he is
relying on a weak reductio. Further he asserts the reductio
using a tautology. Finally and most critically an alternative hypothesis to
the reductio can be readily found: whether time is continuous (Lynds'
position) or discontinuous (the position he attributes to classical physics)
if we presume motion to be itself discontinuous we are in no wise compelled
to accept his argument by reductio.
Lynds' articles also feature minor grammatical flaws. He occasionally omits
the apostrophe on posessive pronouns.(5) He even states
"It is doubtful with his paradoxes, Zeno was attempting to argue that motion
was impossible as is sometimes claimed. Zeno would of [SIC] known full well
that..."(6) Of course Lynds means "Would have". These basic
grammatical flaws mar Lynds' work and basically open it to a criticism of
being slipshod and ill thought out. Naturally these critiques are much less
important than the philosophical critique. However taken together, while
Lynds may be right - time and motion may well be continuous entities, other
possibilities doe exist, namely that time and motion are both discontinuous
entities. Thus Lynds has simply not proven that which he claims to have proven.
His positions may be provable. But he has not proven them.
We can also criticize Lynds for appearing to combine modern and ancient
conceptions of physics at will. However while I am a logician I am no physicist.
I will simply content myself with noting that to the physicist mass/energy
and space/time have all been exchangable and part of a unified field since
at least Einstein. I would also note that, as I understand it, we may gain
some insight if we remember that we can only predict location or velocity
of sub-atomic particles. Thus Lynds is actually not saying much new. Locus
of objects since Einstein at least is no longer seen as perfectly determinate
since objects are in fact energy and since at the sub-atomic level particles
behave in startling ways, and are literally capable of teleportation (which
is in keeping with a model of motion as a discontinuous process).
Well, if we are not compelled to accept Lynds' position that still leaves
us with Zeno to contend with. I've already pointed out that the presumptions
of modern physics are radically different from that of the ancients: namely,
that mass and energy are convertible as are time and space. What do we see
in Zeno? First, it is worth noting that Zeno is playing with the concept of
the infinite. However contemporary mathematics still seems troubled by the
infinite, considering division by zero to be undefined, even though such division
in fact clearly approaches a positive infinite limit with a positive denominator
(albeit at different rates depending on the numerator). Modern thought does
distinguish between positive and negative infinity. However it does not appear
to distinguish between the infinitely small and the infinitely large. We
must always of course remember that infinity is not a number nor even, really,
a limit, but rather an ever receding horizon. In all of Zeno's paradoxes
however the horizon is not ever farther from us, rather it is ever closer.
That is Zeno's infinities are all aproaching zero, and not the numberless.
It would be useful if modern mathematical theory clarifies the concept of
rates at which numbers approach infinity and infities which approach zero
and infinties which aproach the numberless.
That being said what else can we make of Zeno? When we consider Zeno we
are really looking at comparing two incommensuarete ratios. Namely, distanc
(d) over time (t) with distance (d1) over distance (d2). Thus when we look
at the arrow traveling say 10 meters per second we are looking at d/t. In
contrast when we look at the arrow which travels half the distance and the
arrow which traels again half that distance we are comparing d1/d2. These
are not commensurate. It is like Zeno is asking us to compare apples (distance
over time) with oranges (distance over distance). They do have something
in common however which is why we are taken in to try to solve the insoluble.
Lynds notes that we can almost solve Zeno's paradox using calculus. That
however raises the problem that calculus, like Catholicism, assumes the impossible,
namely that we can actually complete an infinite series of additions thus
reaching (instead of merely approaching) a limit. It's not really a satisfying
answer philosophically. More importantly, Lynds seems to miss the point: paradoxed
exist not to be solved but rather to teach problem solving! It is axiomatic
that a paradox presents a "red herring" - that it present a problem other
than the real problem that it presents - in order to force the student to
discover a solution by questioning their ordinarily unquestioned assumptions.
Lynds definitely succeeds in doing what the paradox is intended to compel,
"thinking outside of the box". But Lynds does not compel a solution to the
paradox first due to flawed method, and second because the paradox is unsolvable
as it is comparing incommensurates, namely distance/time with distance/distance.
The paradox however is not incommensurate in the sense of the trisection
of an angle. Rather it is incommensurate because it is comparing two different
entities (at least for classical physics...). Thus Zeno posed what, in his
time, was a false question. In our time however since we see that space/time
are one convertible thing the question may not in fact be paradoxical and
may be able to be solved: though Lynds has not compelled the solution he
proposes.
Eric Engle
http://www.geocities.com/engleerica/ZENO.html
REFERENCES:
(1) Aristotle, Posterior Analytics
(ca. 350 B.C.) Translated by G. R. G. Mure, Book I, Part 1. Available
at: http://classics.mit.edu/Aristotle/posterior.1.i.html
and at: http://www.rbjones.com/rbjpub/philos/classics/aristotl/o4219c.htm
(2) Kurt Gödel, On formally undecidable
propositions of Principia Mathematica and related systems, (1931) available
at:
http://nago.cs.colourado.edu/~hirzel/papers/canon00-goedel.pdf
(3) Peter Lynds,
"Time and Classical and Quantum Mechanics"
href="http://doc.cern.ch//archive/electronic/other/ext/ext-2003-045.pdf">http://doc.cern.ch//archive/electronic/other/ext/ext-2003-045.pdf
p. 2.
(4) Peter Lynds, "Zeno's
Paradoxes: a Timely Solution"
http://philsci-archive.pitt.edu/archive/00001197/02/Zeno_s_Paradoxes_-_A_Timely_Solution.pdf
p. 6.
(5) Peter Lynds,
"Time and Classical and Quantum Mechanics" http://doc.cern.ch//archive/electronic/other/ext/ext-2003-045.pdf
e.g. p.4 " (trains). This is not the only example of omitted apostrophes which
occur in both articles.
(6) "Zeno's Paradoxes: a
Timely Solution", supra note 4 at 2.
ERIC ENGLE'S, ARTICLE Zeno's Paradox:
A response to Mr. Lynds, IS
FOUND AT:
http://www.lexnet.bravepages.com/ZENO.html
http://www.geocities.com/engleerica/ZENO.html