1. NicolasArrisola
PHL 363L
Juhl
2-28-15
Zeno’s Dichotomy Paradox and Yanofsky’s Reply to It
In this paper I shall briefly outline Zeno’s dichotomy paradox in Section I, Yanofsky’s
response to it in Section II, an evaluation of Yanofsky’s reply and my proposed argument
invoking the “climax of empirical divisibility” against it in Section III, a colleague’s
counterargument to my proposal in Section IV, and a short conclusion to this paper in Section V.
My main point of contention for this essay is that Yanofsky should endorse the original stance he
entertained regarding the best solution to Zeno’s dichotomy paradox: that space is discrete as
opposed to continuous.
I. The dichotomy paradox as proposed by Zeno and outlined in the text by Yanofsky
follows like so: Suppose that a thoughtful slacker wakes up in the morning and plans to go from
his bed to the door in his room. In order to reach the door, the slacker must reach a halfway point
between the bed and the door. Upon reaching this point, the slacker must travel yet another half
distance to reach the door. According to the paradox, this process will go on ad infinitum given
that numerical values (illustrated here by the distance from any point a to another point b) are
indefinitely divisible; how can the slacker ever reach the door? Why get out of bed at all? It’s as
if one must complete an infinite amount of tasks - traversing a series of halves - within a finite
amount of time.
II. Yanofsky responds to Zeno’s dichotomy paradox by entertaining the notion that Zeno
may be mistaken about the apparent continuity of space itself. Instead of space being like a real-
number line which is infinitely divisible (that is, between any two points there is an infinite
number of other points), Yanofsky states that it seems much more intuitive to view space as

2. NicolasArrisola
PHL 363L
Juhl
2-28-15
discrete (that is, there is a definite distance between any two points that is not infinitely divisible)
since we do in fact reach the door when we get out of bed every morning. He reasons that if one
assumes that space is continuous, then one must also assume that movement is impossible
(infinite tasks in finite time), and since there definitely seems to be movement in the world, it
must be the case that space is not continuous but discrete. He further supports this conclusion for
discrete space by invoking the examples of TV pixels and Planck’s length: If our slacker’s short
venture from the bed to the door were to be featured on a TV screen with dozens of pixels, there
would be a definite x amount of pixels that he would have to cross to get halfway to the door,
and then another x amount of pixels to actually reach the door. The pixels are either crossed or
uncrossed by the slacker; they are discrete. The same can be said for Planck’s length, which is
essentially the smallest length at which classical mechanics concerning gravity and space-time
cease to be a formality and give way to quantum mechanics; to some extent, Yanofsky says,
there’s really nothing smaller that actually exists.
Despite the evidence he cites in support of this discrete space argument however,
Yanofsky seems reluctant in fully subscribing to this theory. His main worry for this is that the
vast majority of mathematical physics is based on calculus which presupposes that reality is
continuous/infinitely divisible; since we build rockets and bridges using math that assumes the
continuity of space, why should we be so hasty to forsake it? I feel that this worry of his,
however, is misgrounded in its undertaking.
III. Considering Yanofsky’s point for why continuous space seems contradictory given
Zeno’s dichotomy paradox and his sound appeals to discrete TV pixels and Planck’s length, his
wariness for completely backing the position of discrete space seems unreasonable. In short, I

3. NicolasArrisola
PHL 363L
Juhl
2-28-15
believe that he should cast this worry about mathematical physics aside and embrace his original
view that space is indeed discrete and not continuous.
The reason for this is actually quite simple: continuous real number mathematics/calculus
is merely idealistic1 and only seems to be a good model for the physical world because calculus
can account for exceedingly small or large numbers that are attributed to physical properties of
objects in reality; this doesn’t entail that reality has continuous, infinitely divisible space. It may
certainly be the case (or it’s at least rational to consider the possibility) that there is an
unimaginably small object - in fact, the smallest object - that exists out in space somewhere that
is even smaller than Planck’s length. This object, spectacular in its conception, would have quite
a prolific series of digits attributed to its physical properties of length, mass, etc. (perhaps
something like 1.6*10^-309, or even smaller). This purported value is undeniably discrete. Any
formulation of divisibility would have to be conceptual as opposed to physical given that this is
the smallest thing that can ever exist.
Theoretically, if we were to base our real number calculus on this notion that this is the
definite point where the divisibility of matter ceases, the construction of buildings, cars, boats,
and airplanes using this new calculus would still be viable given that this value is so
unimaginably small that it can be accounted for in all physical phenomena (it is, after all, the
fundamental physical length) and could even be considered mathematical physics’ successor to
infinite divisibility which I would like to call the “climax of empirical divisibility.” In other
words, our conception of the climax of empirical divisibility would be on par with our
1 When I say “idealistic,” I mean it in thesense that the infinitely divisible nature of real numbers is only a mathematical concept;
as far as contemporary science and human intellect/understanding is concerned, there is no feasible way of proving that physical
space is infinitely divisible (see Planck’s length).

4. NicolasArrisola
PHL 363L
Juhl
2-28-15
conception of infinite divisibility given its obscure, practically unfathomable nature of
minuteness; the critical difference distinguishing the two, however, is the fact that the climax of
empirical divisibility would be proven and discrete.
***As a small side note, I do contend that space may be infinite as a continuing set or potential infinite2
,
but not as an actual infinite3
as examined in Zeno’s dichotomy paradox and Yanofsky’s appeal to practical
calculus.
IV. Garrett Stanton, a philosophy colleague of mine in the same class, brought up an
interesting point against my “climax of empirical divisibility” argument that appeals to the
uncertainty principle in quantum mechanics. Stanton asserts that anything smaller than Planck’s
length will stray from our classical understanding of physics and adhere to the probabilistic
world of the quantum. He invokes Heisenberg’s uncertainty principle which states that certain
pairs of physical properties (such as the location and momentum) of a subatomic particle have a
limit for how precisely they can be known simultaneously. Basically, the more one knows about
where subatomic particle a is (position x), the less they know about particle a’s momentum p,
and vice versa.
Stanton contends that this is troublesome for my proposed argument for continuous space
because the tiny object known as the “climax of empirical divisibility” wouldn’t actually be
discrete but probabilistic since it would be within the realm of quantum mechanics and therefore,
adhere to the Heisenberg uncertainty principle; there wouldn’t be a perfectly accurate way to
locate such an object for discrete measurement, only a probability.
2 Think of thegrouping of natural numbers and their function n+1.
3 Think of theset of points between any two natural numbers a and b which have a finite beginning, the first natural number a,
and a finite end, the second natural number b, but have an infinite number of members between the two.

5. NicolasArrisola
PHL 363L
Juhl
2-28-15
I have taken into consideration the concerns Stanton has outlined in his counterargument
and I must say I don’t have much of a reply other than that if we were to concede that space is
indeed continuous, even on the quantum level, then movement would have to be impossible
since a series of infinite tasks would have to be completed in a finite time and we’re right back to
where we started in Zeno’s dichotomy paradox. While this may seem unsatisfactory as a reply to
Stanton, I stand by the logic in this reductio ad absurdum.
V. As I have argued thus far, Yanofsky has no reason to not champion the ideal he set out
for answering Zeno’s dichotomy paradox, that discrete space is the best approach for resolving
this metaphysical dilemma given that continuous space leads to the contradiction of being able to
complete an infinite number of tasks within a finite amount of time. Yanofsky’s worry about
practical calculus isn’t of much concern since replacing the mathematical physics assumption of
space’s infinite divisibility with a “climax of empirical divisibility” would yield an arguably
more “complete” mathematical physics which utilizes a discrete, proven fundamental length. The
only real problem for the discrete space argument, as pointed out by Garrett Stanton, is the ever-
looming presence of quantum mechanics whose probabilistic operations founded in uncertainty
seem almost mystical when compared to classic Newtonian physics. Until we have a
comprehensive understanding of quantum mechanics, however, we shouldn’t rule out the
possibility of space existing as a finite set of discrete points.