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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 statements 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 arrow's 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. On this point we should note the structural similarity between Euclid's elements and Newton's principia. For some ancients (probably most in fact) 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 collapeses 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' heuristic is basically correct. When confronted by paradox we must somehow "break out of" the flawed presumptions which generate it. That is exactly what Lynds attempts to do. 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)
"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 at 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 absurdum. 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) then Lynds' reductio fails. This "teleportation" model does in fact 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 absurdum is actually quite weak. A reductio does not in fact prove its affirmation. Rather it disproves the negation and from that "dis-proof" attempts to infer the opposite conclusion. Commonly reductio is mis-applied. "If P then Q. P is not so. Thus Q is not so." That mis-application risks confusing the possible with the necessary. In fact
"If P then Q. P is not so. Thus Q is not so." is not necessarily true: only "If and only if A then B. A is not so. Thus B is not so." Lynds appears to presume that motion is possible only if time is discontinuous. Since that presumption is not necessarily true his reductio fails.

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." Reductio is only a valid inference where we are dealing with statements in the form of "Only if P then Q" which explains why it is easy to make an erroneous reductio. "If P then Q; Not P; Therefore Not Q" is not a valid inference in the sense that while possibly true it is not necessarily true. In contrast "Only if P then Q; Not P; Therefore not Q" is a valid inferential mode.

As can be seen, 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 absurd 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 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 then we are in no way 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, [SIC - a "that" is obviously missing] 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 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 (or negative depending on sign) infinite limit with a positive denominator (but at different rates depending on the numerator or denominator). 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 infinities 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, distance (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 that which cannot be solved within its own terms.

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: paradoxes 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

Notes on infinity:

Four cases may illustrate how we should think about division by zero:

1. Positive series approaching positive infinity (the numberless)
10/5= 2; 10/2=5; 10/1=1; 10/.1=100... 10/0 approaches positive infinity;
100/5=20; 100/2=50; 100/1=100; 100/.1=1000... 100/0 approches positive infinity faster than 10/0;
Thus the greater the numerator the more rapid the limit approaches positive infinity.

2. Negative series approaching positive infinity (the numberless)

a. Denominator - This is the problematic case:
10/-5= -2; 10/-2=-5; 10/-1=-1; 10/-.1=-100... 10/-0 approaches negative infinity;

This is perhaps the only troublesome case. Clearly we are approaching negative infinity but to reach it requires a concept of "negative zero" which I would call "approaching zero from the negative".

b. Numerato - this case is not at all problematic:
-10/5= -2; -10/2=-5; -10/1=-1; -10/.1=-100... -10/0 approaches negative infinity;
-100/5=-20; -100/2=-50; -100/1=-100; -100/.1=-1000... -100/0 approches negative infinity faster than -10/0;

3. Positive series approaching zero
(let x = 2; 1/2, 1/4, 1/16, 1/256...) approaches zero faster than
1/x*2 (let x = 2; 1/2, 1/4, 1/8, 1/16...) which also approaches zero

4. Negative series approaching zero from the negative
(let x = -2; 1/-2, 1/-4, 1/-16, 1/-256...) approaches zero from the negative faster than
1/x*-2 (let x = 2; 1/-4, 1/-8, 1/-16...) which also approches zero from the negative

Though division by zero is considered undefined by contemporary mathematics I believe it is more accurate to speak of division by zero as approaching infinity or negative infinity. Similarly infinite series of division approaching zero should be seen as approaching zero either from the negative or positive.

Division by zero also raises the issue of paradox: 0 * a rational number = 0. However since infinity is a limit it is not a rational number. More problematic however any positive integer divided by zero yields infinity. However if we multiply positive infinity by zero any positive integer could result! That does not mean however that division by zero is undefined, only that multiplication by positive infinity yields a positive rational number


(1)Aristotle, Posterior Analytics (ca. 350 B.C.)  Translated by G. R. G. Mure, Book I, Part 1. Available at: and at:

(2) Kurt Gödel, On formally undecidable propositions of Principia Mathematica and related systems, (1931) available at:

3) Peter Lynds, "Time and Classical and Quantum Mechanics" 2.

Peter Lynds, "Zeno's Paradoxes: a Timely Solution"

p. 6.

5) Peter Lynds, "Time and Classical and Quantum Mechanics"  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: