Zeno’s Achilles Has Not Been Solved

Zeno’s paradoxes of motion – particularly “Achilles and the Tortoise” and the “Dichotomy” (aka the “Racetrack”) – are maybe the most well known and historically popular paradoxes in western philosophy. The consensus among philosophers is that they were solved by the insights of calculus. Even those who see a problem beyond what calculus resolves tend to think that set theory takes care of the loose ends. I will argue that neither calculus nor set theory satisfactorily deal with all of the problems involved in Achilles and the Dichotomy, and that the paradoxes have yet to be fully solved. I will further argue that a lesser known of Zeno’s paradoxes – the “Plurality” – also remains unsolved.

I won’t assume prior knowledge of either the paradoxes, calculus, or set theory, and will take care to introduce relevant concepts as they arise. I must also acknowledge that I am myself neither a professional philosopher or mathematician. Despite my attempt to thoroughly research the topic, it is more than likely that I overlook important points (I’ll tackle the paradox of the Preface some other day). Anyone who thinks they see an error, or something that I may be missing, is encouraged to message me.

The paradoxes and the calculus solution
“Achilles and the Tortoise” – they are having a race. Achilles knows he is twice as fast as the Tortoise, and so, wanting to spare the Tortoise’s feelings, decides to give it a head start of 10 meters. It takes Achilles a certain amount of time – let’s say 10 seconds – to catch up to where the Tortoise started; in this time the Tortoise has advanced, albeit, his overall advantage has halved (he’s now at the 15-meter mark). It takes Achilles another 5 seconds to catch up to where the Tortoise has advanced to; in this time the Tortoise has further advanced, albeit, his overall advantage has halved (he’s now at the 17.5-meter mark). It takes Achilles another 2.5 seconds to…. Assuming the Achilles and the Tortoise move continuously, Achilles will advance eternally closer to the Tortoise, each step taking half the time to cover half the distance as the step before. Since this sequence goes on infinitely, he can never actually catch up. Yet we all know that if we hosted this race in the real world, Achilles would pass the Tortoise quickly enough – in 20 seconds, in fact. So, what’s wrong with the reasoning?

The other version of this paradox – the “Dichotomy” – points out that, in order for Achilles to cross that first 10 meters, he’ll need to first cross the first 5 meters. In order to do that, he’ll need to cross the first 2.5 meters. In order to do that, he’ll need to…. So, not only does Achilles have an infinite number of steps to cross in order to pass the Tortoise; there’s also an infinite number of steps he must take before he can even take what we were calling the first step. His journey has no actual first step, and it is impossible for him to even get started!

Because both versions deal with the same issues, I’ll speak from now on about Achilles, unless I say otherwise. Arguments will apply equally well to the Dichotomy.

It’s important, especially if for those not super familiar with paradoxes, to understand what the problem is, and what are and aren’t fair ways to tackle it. The problem is that an apparently valid argument, with apparently true premises, leads to an absurd conclusion. It doesn’t help to host the actual race to empirically prove that Achilles would win – no one questions how the real race would play out. It also doesn’t help to note that, based on the details of the scenario, basic arithmetic tells us that Achilles will catch up to the Tortoise in 20 seconds at the 20-meter mark. This is a different argument, with a different (and not absurd) conclusion. It may be correct, but it tells us nothing about why the argument of the paradox, which seems to also be correct, leads to an absurd conclusion. The task for anyone trying to solve the paradox is to show how and where the apparently sound argument goes wrong. What is the precise logical misstep or incorrect premise that leads to the absurd conclusion?

The most common response to the paradox is to say that calculus took care of it. While the insights of calculus are certainly relevant to the paradox, they by no means solve it.

Here’s what calculus tells us: as the series 1/2 + 1/4 + 1/8 + 1/16 + … approaches an infinite number of terms, the sum of the series approaches 1. This tells us that the infinite number of diminishing distances that Achilles has to run will add up to a finite distance.

Zeno probably thought that any infinite number of distances must add up to an infinite total distance, regardless of whether the distances were diminishing or not. So calculus does solve one aspect of the paradox: it tells us that the total distance Achilles must cover is not infinite, despite having infinitely many parts.

What it doesn’t do is solve the logical problems involved in completing an infinite number of tasks, regardless of whether they’re of diminishing size or not. It is important to remember that the sum of the infinite series is a tendency – it tells us that as the series approaches having an infinite amount of members, their sum approaches 1. In other words, calculus treats infinity as a limit – a point endlessly being approached by certain functions – and not necessarily as a something completed or achieved in reality. Ultimately, calculus tells us nothing about whether a realized infinity can exist. And it certainly has nothing to say about whether it is possible for an infinite series of tasks to be achieved, or about the problem of completing an ordered series of tasks that has no first or last step.

At this point, it is useful to state in precise terms what remains tough to swallow about the paradox after calculus solves part of the problem. Consider the following:

Proposition 1: It is possible to complete an infinite series of tasks.

Proposition 2: It is possible to complete an ordered series of tasks that has no first task, no last task, and in which no task has a distinct next task following it.

If we can accept both propositions, Zeno’s argument no longer flies, and the absurd conclusion doesn’t follow. However, these conclusions seem at least very counterintuitive. Are there good reasons, besides avoiding the paradox, to accept or deny them?

Proposition 1: Completing an infinite series of tasks
Regarding the first proposition – that an infinite series of tasks is achievable – the mathematically inclined might note that it’s commonplace in set theory to discuss and operate on sets that have an infinite number of members.

This is true, but is it relevant? Nothing in set theory requires that sets be mappable onto series of achievable tasks. There is therefore no reason to think that simply because a series could be a well-defined set, it must therefore be achievable when interpreted as a series of tasks.

This is obvious if we consider the set of all natural numbers. It is perfectly well-defined. For any possible thing, you could say of it whether it belongs in the set or not. There is even an operation (+1) that, if allowed to repeat endlessly, would enumerate each possible member of the set. However, if we interpret the set as a series of tasks that must all be achieved – for instance, the task of writing down every member of the set – that series of tasks is not achievable.

So, we must be careful when using set theory’s insights, and ask whether they are applicable to the question at hand. Clearly, in this case, to assume that the possibility of infinite sets implies the achievability of infinite series of tasks is to beg the question about whether infinite series of tasks can be achievable.

The infinity of tasks that Achilles must achieve is an infinity that must be allowed to come to an end-point, or point of completion. And yet, the infinite is defined as precisely that which has no end (ie, the endless, the boundless). So a completed infinity isn’t just difficult to accept – it is, simply, a contradiction. To say that Achilles successfully completes an infinite series of tasks is equivalent to saying that the series is both infinite and not infinite (in the same sense).

So, what then? Given the Dichotomy formulation, denying that a completed infinity is possible means denying that all motion is possible. That’s not a realistic option. Should we just accept that the contradiction is true – that Achilles’s series is both infinite and not? Sometimes accepting a contradiction isn’t so problematic – in the case of the Liar, for example, the repercussions aren’t all that rough. In this case, however, accepting the contradiction is incredibly counterintuitive. Unlike the Liar, where the proposition in question is abstract and lacks real-world manifestation, here it seems utterly impossible to deny that the series either is or is not made up of an infinite number of steps (in the same sense). It can’t be both.

One could try to avoid the infinity issue by questioning whether Achilles’s journey should actually be described as being composed of an infinite number of distinct tasks. Achilles’s journey is, practically speaking, one continuous run. Why are we justified in splitting it up into an infinite number of components?

Rather than directly answer that question, I find it more useful to consider a modified version of Achilles: staccato Achilles, proposed by Adolf Grünbaum. Staccato Achilles catches up to the Tortoise after 20 seconds, just as our usual (legato) Achilles does. Staccato Achilles differs in that he takes a finite (non-zero duration) rest between each of the infinite series of steps. This means, of course, that this Achilles reaches much higher velocities than legato Achilles. Following Grünbaum, physicist Richard Friedberg developed a version of staccato Achilles that does not violate any physical laws (such as, eg, against infinite acceleration). Since staccato Achilles takes a full stop in between each of the infinite series of runs, his series of runs can be described as infinite in number without controversy. Dealing with proposition 1 remains unavoidable, and, so far, impossible to accept.

Proposition 2: Thomson’s Lamp
Even if, somehow, it were possible to get around proposition 1, it is independently problematic to accept proposition 2. Consider Thomson’s Lamp. It has a toggle switch – a flick turns the lamp on, another flick turns it off. Start with the lamp off. After 1/2 min., flick the switch. Flick it again after 1/4 min. Again after 1/8 min…. As we’ve seen, this sequence will last 1 minute. After the minute is done, will Thomson’s Lamp be on or off? Since the series has no last step, there can be no answer: the lamp can be neither on nor off, or must be both on and off, depending on how you look at it. Both ways of phrasing it are absurd – in the real world, the lamp must be either on or off, and not both.

It does not work to object that the 1-minute mark lies outside the series, and that therefore the infinite alternating series does not apply to the moment after the series is complete. This may be the case, but it doesn’t change the fact that once the series is complete – that is, after 1 minute passes – the lamp must be either on or off.

A reasonable objection is that Thomson’s Lamp has a physical limitation: flicking the toggle switch requires that a finger cross some finite distance. This distance, unlike Achilles’s distances, does not diminish with every iteration. An infinite number of toggle flicks does require that the finger traverse an infinite distance in a finite time.

We could remove this problem by designing the switch such that the distance it must travel to move from the on-state to the off-state shrinks by half every time the switch is flicked. This creates an infinitely diminishing sequence of distances that the finger must cross in order to flick the switch, and thus makes the total distance the finger must move finite. However, this also makes the difference in distance between the on-state and off-state of the toggle switch converge to zero. At the end of this minute-long sequence, the on-state and off-state of the toggle switch are in identical positions. The question of the whether the lamp is on or off then depends on some arbitrary quirk of how the lamp reacts to the toggle switch being stuck exactly halfway between its two states. Whichever way that quirk goes can be arbitrarily stipulated. The takeaway is that the lamp being either on or off at the end of the sequence is no longer absurd.

A question then arises: should the physical impossibility of the lamp matter? The issue addressed by the thought experiment is a logical one, not a mechanical one. So long as Thomson’s Lamp is logically coherent, isn’t its physical impossibility irrelevant to the question? This attitude is tempting, though it welcomes the following objection: if the physical impossibility of the lamp is merely a mechanical issue, and not a logical one, why is it that attempts to alter the hypothetical design to avoid impossibility lead to the dissolution of the absurd conclusion? Does this not suggest that the impossibility of the lamp is in fact relevant to the thought experiment?

With this question in mind, let us return to staccato Achilles. Imagine he wears a blinking sensor during the race. The sensor is programmed to alternate abruptly from green to red every time Achilles comes to a full stop. Achilles begins the race with the sensor at green. Will the sensor be green or red at the exact moment Achilles catches up to the Tortoise? As with Thomson’s Lamp, there is no last step, so there can be no answer. However, it also true that there can physically be no such sensor – mechanisms within the sensor must travel finite non-diminishing distances in order for the sensor to work. These mechanisms would therefore have to travel an infinite total distance in order for the sensor to continue alternating throughout Achilles’s series of runs.

We could make one final modification – forget the physical sensor. We can map out exactly when, along Achilles’s route, the sensor would have switched colors if it didn’t have the physical limitations. We could simply write that series down. It would look something like: GRGRGRG…. The series does not terminate, so, as we knew all along, there is no value, “G” or “R,” applicable to the moment at which the series terminates.

To the best of my knowledge, no machine has been conceived that:

a) can successfully be mapped to the Achilles GRG… sequence;
b) doesn’t converge toward an undefined intermediate state that can be arbitrarily stipulated (as the diminishing toggle switch version of Thomson’s Lamp does); and
c) doesn’t face a problem of inherent (even if only physical) impossibility.

It has therefore not been shown that it is impossible to accept proposition 2. However, the foregoing discussion goes a long way to spelling out just how very counterintuitive it is to do so.

The Plurality and the applicability of set theory
Given the difficulties so far faced in accepting propositions 1 and 2, it becomes reasonable to consider a more radical possibility: rejecting the premise that space and time are composed of dimensionless points/instants. Some important philosophers, notably Aristotle, Henri Bergson, and William James, have rejected this notion.

There is no question that this move dissolves the paradoxes. Without the instant in time, we can no longer ask about the precise moment Achilles catches up to the Tortoise. Without the the point in space, we cannot talk about the precise points that demarcate the various steps of Achilles’s journey.

Zeno left us another paradox, perhaps his most fundamental – the “Plurality” – that hints toward this solution. Consider a line in space – say, the line that marks Achilles’s first 10-meter journey. As we’ve seen, that line can be halved, and the halves can be halved, and so on. So, the line can be shown to be composed of infinitely many parts. What is the length of each of those infinitely many individual parts?

If we say that they are of any finite length, then the total length of the line must be infinite. We’re no longer in a diminishing series, but dealing with an infinite number of equally sized constituent parts, so the convergence of the infinite diminishing series provides no help. The sum of an infinite number of finite distances of equal length, no matter what that length is, must be infinite.

But if we say that they have no length – that they are dimensionless, as we typically say about points – then what we’re left with is the sum of an infinite number of zeroes. And that sum is simply zero – the line is shown to have no dimension. Since all extension in space consists of the sum of its infinite number of dimensionless components, and since that sum must always be zero, extension is shown to be impossible – there can be no positive space at all. So, whether or not the infinitely many constituent parts of space have length, Zeno’s “Plurality” demonstrates finite extension to be impossible.

Set theory has a solution to this problem. To begin, it is important to understand the term “cardinality”: the number of members in a given set. So, say we have a set that includes Achilles, the Tortoise, and Zeno. That set has a cardinality of 3.

The set of all natural numbers, the set of all integers, and the set of all rational numbers all have an infinite cardinality. However, set theory distinguishes between different magnitudes of infinite cardinality, so more needs to be said. The three sets listed above are all considered to have a countably infinite cardinality. It’s called “countable” infinity because, though you can’t complete the enumeration of every member of the set (as discussed above), you can enumerate each individual member of the set by letting some function run endlessly. The countably infinite cardinality is called “aleph null”: \aleph 0.

Now let’s look at the set of all real numbers. This set includes every number in the three sets listed above, and also includes all irrational numbers (real numbers, π like or e, that cannot be expressed by a fraction). This set also has an infinite cardinality. However, it is a different kind of infinite cardinality. The set of irrational numbers (and therefore the set of all real numbers as well) is uncountably infinite: its members cannot be enumerated with a function. This is also true of any set of real numbers within a given range (eg, between 1 and 10). The uncountability of these sets means that their cardinality is infinitely greater than that of the countably infinite sets. This infinitely greater “kind” of infinity is “aleph one”: \aleph 1. According to set theory, \aleph 0 and \aleph 1 are only the first two in an infinite series of infinite cardinalities: \aleph 0, \aleph 1, \aleph 2, \aleph 3….

It is standard to consider the sum of an uncountably infinite series of numbers to be undefined. So, while the sum of a countably infinite (\aleph 0) number of zeroes is zero, the sum of an uncountably infinite (\aleph 1 or greater) number of zeroes is simply undefined. If we assume that the infinite divisibility and continuity of a line in space are analogous to the infinite divisibility and continuity of the real number line (or some range within the number line), then any line in space contains \aleph 1 quantity of dimensionless points. Therefore, we cannot say that the \aleph 1 number of zeroes that constitute a line in space add up to zero; they don’t “add up” to anything.

Return to the 10-meter line that marked Achilles’s distance from the Tortoise’s starting point. Map that line to the number line from 1 to 10. As we saw, the \aleph 0 number of dimensionless points on that line that correspond to rational numbers will add up to zero. Since the length of the line is 10 meters, it follows that the total measure of the \aleph 1 number of dimensionless points that correspond to irrational numbers is 10 meters. This is so despite the fact that this total measure cannot be viewed as a sum because the sum is undefined. The same reasoning holds for any length of line, even though all lines – regardless of their length – have the same number of points that correspond to irrational numbers: \aleph 1. In other words, the same \aleph 1 number of dimensionless points can create a total measure (though not sum) of any possible length of line.

The counterintuitiveness of these conclusions poses no problem for mathematics, so long as there is no inconsistency. I trust that there isn’t, as that is the general consensus among mathematicians. But we must remember that mathematical systems cohere around stipulated axioms and definitions. These do not necessarily correspond to the nature of reality, to our generally agreed-upon linguistic conventions, or even to our logic.

One might object that mathematical axioms and definitions are chosen precisely because they do correspond to reality. It is easy enough to say that the Pythagorean theorem is nothing more than a mathematical construct. But find any shape that approximates a right triangle, and you will inevitably also find that the sum of the squares of its sides is equal to the square of its hypotenuse. The applicability of the vast majority of mathematics to real-world questions has been thoroughly verified by experience.

Maybe so, but this cannot be said of set theory. There are no observable infinities with which to corroborate its findings. Set theory’s function – to create a formal “foundation” for mathematics – has always been too abstract for empirical verification.

Almost since its inception, this foundational project has had to reject correspondence with intuitive logic in order to maintain consistency. Most famously, modern set theory fixes restrictions on what counts as a “set,” exclusively in order to avoid Russell’s paradox. That the sum of an uncountably infinite quantity of numbers is necessarily undefined is arguably another such ad-hoc stipulation.

While the foundational work that set theory provides serves its limited purpose well, it’s not necessary to accept it in order to accept applicable mathematics. The “foundation” in question can be as easily stipulated as the axioms of set theory. There is therefore nothing terribly compelling to stop us from disregarding the axioms and definitions of set theory when practical or philosophical intuitions suggest that we do so.

The relevant question, then, is: does the set theoretical position on the sum of uncountably infinite zeroes apply to real space and time? I have to answer no. The zero – as an intuitively graspable concept – is simply composed of nothing. Whether an infinity of them is countable or not makes no difference – it must combine to zero. Zeno’s Plurality therefore remains unsolved, and serves as an added reason to consider views of space and time that reject the point and instant as their respective building blocks.

Rejecting the point
There are three obvious ways to reject the point/instant. One is to consider space/time as altogether indivisible. This rejects not only the point, but also the possibility of the universe having separate parts. If it had separate parts, these parts would occupy separate spaces, which would mean that space wasn’t indivisible after all. This option therefore leads to unqualified monism. This was Zeno’s preferred solution to his own paradoxes. To say the least, it runs contrary to experience and logic.

Another way to reject the point is to describe space as divisible in such a way that its finite constituent regions overlap and are not divisible at distinct points. This description runs into a logical problem. If regions A and B overlap, the area where they overlap would constitute a region of its own: the region of their overlap. This region – call it AB – would have a distinct point of separation from the region of region A that does not overlap with B – call it region A1. While it is possible to propose that the separation between A1 and AB is not a distinct point but another, smaller overlap region, this approach quickly falls into an infinite regress that converges back at the point.

The final method for rejecting the point is also the most common way of dealing with Zeno’s paradoxes (besides outright dismissal): to consider space/time discrete. This would mean that there is a smallest possible amount of space, and all extension occurs in packets of this elementary unit size. This makes it senseless to talk of anything – even an empty region of space – being smaller than one of these chunks. It is an open debate in theoretical physics whether space and time are continuous or discrete.

It’s easy to miss, however, just how much discrete space/time would upend our usual view of the world. Space and time, in themselves, are empty. They consist of nothing at all. So how can they come in some minimum size? Pick any size you like for the minimum chunk. If space and time are insubstantial media, then why couldn’t you just halve that chunk into two equally empty chunks? Accepting that you can’t isn’t quite a contradiction, but it’s damn counterintuitive.

Then there’s the geometric issue posed by mathematician Hermann Weyl: imagine a square plane, composed of squares tiles, each the size of the minimum quantum of space (this thought experiment occurs in two-dimensional space, but it’s translatable to three). In this plane, construct a right triangle that is three tiles high and three tiles wide. This triangle’s hypotenuse moves diagonally across three tiles. Since the minimum measure of distance is one tile, the hypotenuse of this right triangle is equal in length to its sides. You might be tempted to object that square tiles are “longer” diagonally than across, but this isn’t true in this scenario. One tile is the smallest possible anything, in any direction. Weyl’s thought experiment seems to require that we throw away our entire geometry – a geometry that, as noted above, has empirically tested very well throughout human history.

Solutions to the Weyl’s tiles problem have been proposed. Their level of mathematical complexity lies outside the scope of both this essay and my understanding. They seem to point to a redesign of everyday geometry. So, while discrete space/time has not been shown to be contradictory, it remains extremely counterintuitive. Even so, it comes out the most plausible of the possibilities I’ve considered.

The counterintuitive conclusions for geometry are perhaps not as tough to swallow, however, as the thought that Zeno’s scenario is sufficient reason to accept as profound a metaphysical fact as discrete space/time would be. Epistemically, it seems more likely to think that we misunderstand the problem than to think that such a seemingly innocuous thought experiment could tell us something so fundamental. I will someday return to this issue of whether it’s possible that we simply misunderstand the problem.

Until then, given all the above considerations, and to the best of my knowledge, no satisfactory solution to Zeno’s Achilles has been found.

Other posts about Zeno’s paradoxes:

Zeno’s Arrow

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