# How can Zeno's dichotomy paradox be disproved using mathematics?

A brief description of the paradox taken from Wikipedia:

Suppose Sam wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin.

The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion.

How can this be disproved using math, as obviously we can all move a walk from one place to another?

• I can't believe this question wasn't previously asked! – PA6OTA May 29 '14 at 16:17
• See Henning's answer to this post – Mauro ALLEGRANZA May 29 '14 at 16:19
• @PA6OTA: It actually was, see the above link. – Moishe Kohan May 29 '14 at 16:32
• In case anyone is interested, I posted several references in my answer to Achilles and the tortoise paradox?. Regarding these references, an enormous amount has been written about Zeno's paradoxes, and in my opinion much of it is overly verbose and mathematically naive for someone with a background in mathematics. I chose those references carefully. – Dave L. Renfro May 29 '14 at 20:24
• math.stackexchange.com/questions/161756/… – Alex Gordon May 29 '14 at 20:24

It can't. It's not a mathematical statement, it's a statement about the nature of physical space.

At least for the first problem, the obvious mathematical answer is that the "total distance" is finite, because it's the infinite sum $\sum 2^{-n}$, which converges. But the whole point of the paradox is that it's making a statement about the physical world. It's philosophically difficult to say whether or not the above infinite series argument can really be applied to physical space. In particular, is it even meaningful to subdivide a physical length indefinitely? Are physical lines fundamentally continuous or discrete? Do any of these questions really mean anything?

No matter how far you postpone it, at some point you're going to have to cross the bridge from the mathematical model into the real world, and that will always be a philosophical problem, not a mathematical one.

• You are right ! My personal not scientific answer to this riddle is "If we model the race with the mathematical continuum we must not make the mistake of describing the progress of the runner as made of successive move from on point to 'the next one': in the real number line, a point has no 'next one'. We may say that Achilles will win because he is not counting the point in space; he is 'traversing' intervals in time." – Mauro ALLEGRANZA May 29 '14 at 16:23
• This isn't the "it takes forever for Achilles to get there" paradox, it's the "Achilles can't start" paradox. – user14972 May 29 '14 at 16:40
• I've downvoted this answer, because I feel like it's reinforcing the the paradox: the (erroneous) conclusion that infinite divisibility implies motion is impossible and thus we have to resort to "but the real world may not be a continuum" type arguments to avoid contradicting our experience that motion is possible. – user14972 May 29 '14 at 16:47
• @PA6OTA: I'm not downvoting for POV reasons; this post doesn't even differ from my POV. I'm downvoting because the difference between the real world and the physical theory is emphasized in a way that reinforces the misconception that the physical theory is self-contradictory. – user14972 May 29 '14 at 17:28
• Moreover, mathematics resolves the paradox in both discrete and non-discrete models of the world, so it's unnecessary to make a philosophical choice of which model is more "correct" in order to address the issue. – Kyle Strand May 30 '14 at 18:50

The implicit assumption here is that 1. cutting distances into infinitely many pieces is different than cutting times into infinitely many pieces, and/or 2. an infinite sum cannot converge. Neither of which are true.

The sum of distances $1/2+1/4+1/8+1/16...$ equals $1$ as expected. We must also split time up into correspondingly small steps: adding intervals of $1/2+1/4+1/8...=1$ (possibly scaled for the appropriate speed), which also sums to $1$. The times add to a finite time in the exact same way as the distances add to a finite distance. The claim that one cannot complete infinitely many tasks implicitly assumes that infinitely many smaller and smaller tasks cannot add together into one well-defined task that takes finite time, which is not true.

One could of course instead reject the idea that distance and time can be split infinitely in this way at all, claiming that actual motion cannot be split in this way and that the difference between this thought experiment and reality rests crucially on that.

• This isn't the "it takes forever for Achilles to get there" paradox, it's the "Achilles can't start" paradox. – user14972 May 29 '14 at 16:40
• @Hurkyl my understanding was that there isn't a significant difference to them. One says "he must always do something else first", the other says "he must always do something else before he passes". I know I wrote my sums in the backward order for this problem, but I only meant that for notational convenience. – Robert Mastragostino May 29 '14 at 19:01
• @Hurkyl They are functionally identical, one is a perfect mirror of the other. And the logic error (not cutting time in the same manner as distance) is also the same, which is the part described in this answer. – Izkata May 29 '14 at 21:28
• @Izkata: There is a very significant difference: which side of the problem "now" is on. The arrow of time introduces an additional conceptual difference. Incidentally, there are actually a number of logical errors one can make that lead to the paradox. – user14972 May 30 '14 at 4:31
• @Izkata I don't see how cutting the time in the same manner as the distance helps; Zeno can just say "ok, so $\dfrac 12$ of second must pass. But before that, $\dfrac 14$ of a secod must pass. And before that, $\dfrac 18$ of a second must pass... Which one passes first?" Or something like that. – Ovi Sep 26 '18 at 4:04

Zeno may want us to infer that the time necessary to complete these infinite number of tasks is infinite. However, he omits any mention of the speed at which the traveler is moving. There's nothing in this paradox that says the traveler can't move at a constant speed, which simply means that the time taken to move a given distance is proportional to the distance.

Whether Zeno understood infinite sums and convergence would be interesting background to how he arrived at his conclusion, but it's irrelevant to the mathematics known today.

So what's obvious mathematically is that the infinite sum of the distances from these infinite number of tasks is still a finite distance and (for a traveler moving a constant speed) the time it takes to travel that distance is proportional and therefore finite.

The same conclusion can be reached even if the speed is not constant, and may be answered using calculus, which Zeno wasn't familiar with.

To travel any distance, a traveler must not take the path that Zeno took. There are several responses to your question that begin with Zeno's original perceptions as if they are somehow entrenched canon in philosophy (in understanding physical nature) and that one must start there to begin to answer the OP's question. But to start there is just as fruitless as traversing the distance in an infinite number of individual tasks, where even the first task (of allowing the traveller to traverse that first infinitesimal distance) is hobbled by awkward concepts on motion.

• One can see Zeno's argument as disproving infinite divisibility: that it doesn't make sense to split a problem into infinitely many parts. If Zeno didn't mean it that way, there are people today certainly do. And people today still make mistake like your last sentence. – user14972 Jun 4 '14 at 17:18

This one's easy; sequences don't have to have a "first" element, nor does any particular term in a sequence have to have a "next" element.

This "paradox" is not really any different from being confused about the fact that the integers do not have a smallest element, nor the fact that in the extended integers, the element $-\infty$ does not have a successor; the confusion is just disguised better.

We often label points in a sequence with natural numbers, as this is the most common use case for the notion of a sequence, and thus are in the habit of thinking any sequence must have a first element, and every other point has a predecessor, and conversely every point is either last or it has a successor.

However, if we work with sequences that cannot be labeled in such a way -- e.g. marking the midpoint, the quarter point, the one-eighth point and so forth of our journey, along with marking the two endpoints, and observe that we have to transverse them in order -- we can make grave errors if we treat them as if they can be.

• What is your definition of a sequence? I cannot understand your post since I do not know what you mean by a sequence, if not a function from $\mathbb{N}$. – Ovi Sep 26 '18 at 4:11
• @Ovi: A "function from $\mathbb{N}$" is what I allude to by "label points ... with natural numbers". But for any set $I$, one can talk about "$I$-indexed sequences", which are precisely functions from $I$. And if we give $I$ an ordering, we can talk about places in an $I$-indexed sequence coming before or after one another. – user14972 Sep 26 '18 at 4:24
• @Ovi: Ordinal numbers (i.e. "well-order types"), in particular, are popular choices for index sets. Ordinal-indexed sequences are often called "transfinite sequences". – user14972 Sep 26 '18 at 4:27

There is nothing "easy" about this paradox. It can be overcome using integral calculus, which assigns meaning to infinite sums described here. But it is, for me, too close to the foundations of mathematics to be "disproved" by any conventional argument.

My reasoning is as follows. Suppose it takes a total of one minute to get to his destination. So to get half way there, it takes half a minute. Then to go the extra quarter of a distance, it takes him a quarter of a minute. And etc, etc. So after $n$ of these steps, he gets a distance $1-2^{-n}$ of the way to where he is going. But this whole thing only took him $1-2^{-n}$ minutes. So the reason we think he never gets to his destination is that we only consider how far he has travelled before the first minute is finished. And we correctly conclude that he does not arrive before the allotted minute is completed.

We know that if Sam runs fast enough and long enough, he will eventually catch up to the bus. If both are moving at a constant speed, there is no need to decompose their motion into infinitely many, ever decreasing intervals. A simple application of the speed-distance-time formula will tell us that Sam will catch up to the bus in $\frac d{s_2 - s_1}$ seconds where $d$ = the head start by the bus (m), $s_1$ = the speed of the bus (m/s), and $s_2$ = Sam's speed (m/s).

In any finite time interval, we know that Sam and the bus with pass through infinitely many points in space, with an event being associated with their arrival at each point. To the modern mind, there is nothing "paradoxical" or even counter-intuitive about this.

Historical note: It wasn't until Galileo's pioneering efforts in physics and the introduction of the scientific method several centuries after Zeno and Aristotle that we were able to actually measure the speed of an object.

Even if we allow for successively smaller intervals of time, and ignore the infinite sum having a finite answer, we can introduce some physics and bring it into the real world.

I'll introduce a concept called Planck's Length : 1.61619926 × 10^-35 meters.

This is the smallest measure that exists in reality. It is the smallest distance you can travel. If you like, it is the Pixel Size of reality.

The Corollary is Planck Time : 5.39106(32) × 10−44 s

Which is the smallest time in which anything can happen (It's the time it takes light to travel one Planck length). You could consider it the clock rate of reality.

So while mathematics is happy to allow our two running sums (time and distance) to get successively closer but never touching the end, Physics dictates that eventually, you can't divide by two, because distance and or time can't be measured that finely, and eventually, you run into the bus.

The impact of this is several paradoxical concepts, such as Zeno's (from the OP,) or Gabriel's horn (finite volume, infinite surface area) suddenly collapse in the face of reality. Or, if you like, dx and dt cease to exist as infinitesimals, and just become the smallest possible Delta-x and Delta-t.

An interesting philosophical upshot of Planck Length and Planck Time is that reality as we know it, could actually be a simulation running on a real computer somewhere.

• Even without the Planck Length, reality could still be computed by a Turing machine as long as all the real numbers we can observe are computable numbers, and we can't even write a rigorous description of any of the non-computable numbers, let alone measure one on a ruler. – Keen May 30 '14 at 18:14
• This is plain wrong; the Planck length is not the smallest possible distance and Planck time is not the smallest unit of time. – user21820 Jun 10 '14 at 16:05
• @user21820 Here's a simple discussion on it. fromquarkstoquasars.com/the-smallest-possible-length – Chris Cudmore Jun 10 '14 at 16:18
• @ChrisCudmore: I know quantum physics. Do you? If you don't, keep in mind that there are many fancy unfalsifiable theories floating around. What is reasonably certain is that the Planck length is the smallest resolution at which a measurement can possibly be accurate. Any other theories concerning what is below that scale is almost purely speculation, and hence does not prove or disprove Zeno's paradox. – user21820 Jun 11 '14 at 2:14