Paradoxes are an excellent way of introducing someone to philosophy if that someone shows at least some interest in questioning their most closely held assumptions. The first and most famous of these come from Zeno of Elea (not to be confused with the other Zeno, the founder of Stoic philosophy). Continuing and probably modifying the central points made by Parmenides, Zeno is said to have written a book of paradoxes to defend Eleatic philosophy (Huggett, 2019). This book has not survived, but we know some of these through second-hand sources like Aristotle. Believe it or not, some 2500-year-old arguments that purport to show how motion is impossible continue to influence our understanding of space, time, change, motion, and infinity. With that said, I’d like to invite you to consider one of these arguments in more detail.

Here I’m concerned with the relationship between Zeno’s first paradox of motion (“the dichotomy”), Aristotle’s response to it, modern mathematics, infinity machines, and supertasks. I will attempt to show, as clearly as I can, how advances made in 19th-century mathematics by Augustin-Louis Cauchy can supplement Aristotle’s response to the paradox and thereby take us a step further toward resolving it. I will also show how it is possible to complete an infinite number of tasks in a finite amount of time. I say “take us a step further” because a “complete” resolution that would satisfy even the Eleatics themselves seems near-impossible. As Wesley C. Salmon wrote:

It would, of course, be rash to conclude that we had actually arrived at a complete resolution of all problems that come out of Zeno’s paradoxes. Each age, from Aristotle on down, seems to find in the paradoxes difficulties that are roughly commensurate with the mathematical, logical, and philosophical resources then available. When more powerful tools emerge, philosophers seem willing to acknowledge deeper difficulties that would have proved insurmountable for more primitive methods. We may have resolutions which are appropriate to our present level of understanding, but they may appear quite inadequate when we have advanced further. (Salmon, 1970/2001, pp. 43-44)

- The Dichotomy
- Aristotle’s Response
- Convergence and Divergence
- Infinity Machines
- Atalanta’s Supertask
- Conclusion
- References

## The Dichotomy

I won’t attempt to extend the discussion to other paradoxes of motion or plurality, even if the paradox of Achilles and the tortoise expresses the same idea as the dichotomy. I chose the latter because it is simpler.

Here’s the dichotomy paradox, as conveyed by Aristotle:

The first is the one which declares movement to be impossible because, however near the mobile is to any given point, it will always have to cover the half, and then the half of that, and so on without limit before it gets there. (Aristotle, Physics, 239b)

In other words, Atalanta (a heroine in Greek mythology known for her swiftness) has to run a 100-meter track. Before she gets to the finish line, she must get 1/2 of the way there. Before she can get halfway there, she must get 1/4 of the way there. Before that, she must get 1/8 of the way there, and so on, *ad infinitum.* So Atalanta seems to be facing an impossible “supertask”: she must complete an infinite series of smaller and smaller runs. Is completing this task possible? Zeno seems to have taken it as self-evident that the answer is “no.”

The obvious response to this is that of Diogenes of Sinope. According to one story, he stood up and walked out after hearing Zeno’s argument (Huggett, 2019). But to dismiss the argument in such a way is to miss out on its value. Sure, you won’t start believing that motion is impossible because you can’t resolve Zeno’s paradoxes, but they force us to rethink our assumptions about space, time, change, motion, and infinity.

## Aristotle’s Response

Aristotle, unlike Diogenes, tried to combat the Eleatics on their own terms. So here is Aristotle’s response: suppose that with constant speed, we believe that Atalanta would need 10 seconds to complete the run. If this is so, the time needed to complete the infinite series of tasks can be divided just the same way as the distance. So, Atalanta would need 5 seconds to cover the first half of the whole distance, 2.5 seconds to complete another 1/4th of the distance, and so on.

Some think that this is enough to resolve the paradox (For example). But even by Aristotle’s own admition (Physics, 263a), this isn’t enough: what is needed is an account of how a sum of an infinite number of finite things can be equal to anything other than infinity, and this is exactly what 19th-century advances in mathematics can provide.

Aristotle extended the paradox in a way that lets the Parmenidean claim that there can be no finite time. He even conceded that if an infinite amount of intervals existed in actuality (instead of existing merely in potentiality), then the sum would indeed equal infinity.

This answer is, of course, unsatisfactory: a 100-meter track isn’t only ‘potentially’ made up of smaller intervals. Yet there is an argument to be made about whether physical space is a continuum:

The track on which he runs has a finite number of pebbles, grains of earth, and blades of grass!’ each of which in turn has a finite, though enormous number of atoms. For all of these are things that have a beginning and end in space or time. But if anybody says we must imagine that the atoms themselves occupy space and so are divisible “in thought,” he is no longer talking about spatia-temporal things. To divide a thing “in thought” is merely to halve the numerical interval which we have assigned to it. Or else it is to suppose what is in fact physically impossible beyond a certain point, the actual separation of the physical thing into discrete parts. We can of course choose to say that we shall represent a distance by a numerical interval, and that every part of that numerical interval shall also count as representing a distance; then it will be true a priori that there are infinitely many “distances.” But the class of what will then be called “distances” will be a series of pairs of numbers, not an infinite series of spatia-temporal things. The infinity of this series is then a feature of one way in which we find it useful to describe the physical reality. (Black, 1950, p. 80)

On this interpretation, Zeno’s paradoxes arise from our failure to separate how we conceive of space, time, and motion in mathematics from how space, time, and motion operate in physical reality. Considering that nothing guarantees, a priori, that space is a continuum, this resolution seems plausible:

… the premiss really is “Achilles’ physical distance is 1+1/2+1/4+etc.” It is this that is self-contradictory. In other words “Achilles’ distance is 1+1/2+1/4+etc.” and “Achilles’ distance is physical” are contradictory. (Wisdom, 1951, p. 86)

For some, including A. N. Whitehead, Zeno’s arguments don’t prove that motion isn’t real, but they do show that changes occur discretely rather than continuously. But we don’t necessarily have to buy into the idea that space is not a continuum. Even if we agree with Zeno and other Eleatics that there is an infinity of smaller and smaller intervals, we can, with Cauchy’s help, still show that the sum is finite.

## Convergence and Divergence

Before we prove that a sum of an infinite number of finite things can be equal to a finite number, we must understand that even simple addition doesn’t work conventionally when we’re working with infinite sums. Consider Nick Huggett’s (2010, p. 20) example: x=1-1+1-1+1-1+… What is the value of x? Suppose that we rewrite it in the following way: x=(1-1)+(1-1)+(1-1)+… Then, x=0+0+0+…=0. Now suppose that we rewrite the original equation in the following way: x=1-(1-1+1-1+…) but 1-1+1-1+…=0 so x=1-0=1. Now we have a contradiction whereby x=0 and x=1 are both true (some even think, mistakenly, that 1-1+1-1+…=1/2). So our assumption that we can apply the same reasoning to infinite and finite sums is somehow false.

How so? Modern mathematics can resolve this by dividing infinite sums into two different categories: convergent and divergent sums. If an infinite sum approaches something (such as infinity, minus infinity, or some finite number), then the sum is **convergent**. If not, then it is **divergent** or **undefined**. For example, the sum 1+1+1+… is convergent and approaches infinity, but our example above (1-1+1-1+…) does not approach anything, so it is divergent or undefined.

So is the geometric sequence in Zeno’s dichotomy paradox (1/2+1/4+1/8+…) divergent or convergent? There are many ways to prove that the sequence is convergent and approaches 1. First, we know that the series will never exceed 1 because of Zeno’s own reasoning (the series has no last member or last half-run). Second, we know that an infinite geometric series is convergent if the absolute value of the common ratio is less than 1:

- S=a/(1-r) (where
**S**is the sum,**a**is the first term of the series, and**r**is the common ratio); - In our case, S=(1/2)/(1-(1/2))=(1/2)/(1/2)=1.

Alternatively:

We can also prove this by using partial sums:

- x=1/2+1/4+1/8+…
- 2x=2(1/2+1/4+1/8+…)=1+1/2+1/4+1/8+…
- 2x=1+x
- x=1

Notice that we needed a new rule of addition for distinguishing different infinite sums: the rules of finite addition gave us 1=0 when we used them for infinite sums, so we had to introduce the concepts of convergence and divergence. With this, we have shown that mathematics can explain how an infinite sum of a finite number of things can be equal to something other than infinity.

## Infinity Machines

Is the paradox completely resolved? Of course not, because one could still question whether these definitions of infinity correspond to physical reality. We still haven’t proved if Cauchy’s definitions can be correctly applied to physical actions (like Atalanta’s half-runs). This involves a premise about the applicability of mathematical language to physical phenomena in general. For example, simple laws of arithmetic break down if we mix an x amount of water and a y amount of alcohol, because the volume of the mixture won’t be x+y. So how do we know that infinite sequences of physical actions are, in all relevant respects, analogous to Cauchy sums? The points about the supposed infinite divisibility of physical space raised by Max Black and J. O. Wisdom (as quoted above) are still relevant. Furthermore, the relevance of infinity machines becomes apparent.

Max Black, James Thomson, and their followers attempted to show that Zeno’s paradoxes of motion (the dichotomy and Achilles and the tortoise in particular) are based on contradictory premises. Thomson brings out this self-contradictory nature of “supertasks” by use of thought experiments that involve infinity machines, the most famous of which is the lamp:

There are certain reading lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off. So if the lamp was originally off, and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half minute, and so on (…) After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction. (Thomson, 1954, pp. 94-95)

But this apparent contradiction can be resolved if we realize that Thomson’s description of how the infinity machine operates applies only to any point in time **before **the two minutes are up (Benacerraf, 1962). It tells us nothing about the lamp after the two minutes have passed. Thomson’s infinite series has no last element. In mathematical terms, Thomson describes a set exhibiting order type ** ω**, which means that:

- The set has a first element;
- Each element except the first has a unique immediate predecessor;
- Every element has a unique immediate successor.

Because of the last characteristic, the set has no last element. Atalanta’s run in Zeno’s paradox has the same character: there is no last half of the remaining track, yet she completes the course. In essence, Thomson’s argument gives information about the state of an object at any point in time (t), where t₀≤t<t₁, and asks us to deduce from this the state of the object at t₁. The reason it “seems impossible to answer this question” is that we are given absolutely no information about the state of the lamp at t₁.

There are, of course, other infinity machines, but I won’t discuss each. What I want to end this essay with is a demonstration of how Atalanta could complete her run even if we conceive of it as an infinite series of runs that are separated with periods of rest.

## Atalanta’s Supertask

So we know that ‘infinity machines’ might not show an inherent contradiction in the idea of a supertask, but can we show how Atalanta can complete her supertask? Adolf Grünbaum (1970) offered an interesting proof of this which has been updated by Huggett (2010). Here’s the solution: imagine that Atalanta has to run a 100 meter course in 10 seconds. Instead of a continuous run, she will have to complete ℵ₀ runs that are separated by ℵ₀ periods of rest (ℵ₀ being the cardinal number of the set of natural numbers). The purpose of separating each run with a period of rest is to show that an infinite number of runs can be completed in a finite amount of time **even if **space and time are infinitely divisible.

Each run involves covering 3/4 of the remaining distance at 1/2 the speed of the previous run and then resting for the same amount of time as she ran. For example, imagine that the initial speed is 30m/s. She would need 2.5 s to cover 75 m; she would then rest for 2.5 s; then run the next 18.75 m at 15 m/s; rest for 1.25 s; and so on (Huggett, 2010, p.21).

Notice that each distance is 1/4 times as long as the previous one. The total distance is, therefore, 75 m ×(1+1/4+[1/4×1/4]+[1/4×1/4×1/4]+…). And 1+1/4+[1/4×1/4]+[1/4×1/4×1/4]+… is convergent, so we can calculate the sum:

- x=1+1/4+[1/4×1/4]+[1/4×1/4×1/4]+…
- 4x=4+1+1/4+[1/4×1/4]+[1/4×1/4×1/4]+…
- 4x=4+x
- 3x=4
- x=4/3

Therefore, 75×(1+1/4+[1/4×1/4]+[1/4×1/4×1/4]+…)=100, which means that the infinite sequence of runs does cover the whole distance. Now we need to check whether she can do this in a finite amount of time.

Each distance is 1/4 as large as the previous, the speed is 1/2 of the previous, and the time required to complete the run must be doubled because of the rest time. Taking all of this into account, we see that each consecutive run takes 1/2 as long as the previous one. If she needed 2.5 seconds to cover the first run and 2.5 more for rest, the whole run would take 2×2.5×(1+1/2+1/4+1/8+…) seconds. Since we already established that 1/2+1/4+1/8+…=1, we can calculate the time needed:

- 2×2.5×(1+1/2+1/4+1/8+…)=2×2.5×(1+1)
- 2×2.5×(1+1)=10

Therefore, Atalanta will have traveled 100 m in exactly 10 s. In other words, she will have completed an infinite number of **discrete** tasks in a finite amount of time. All this shows that there can be a supertask-like conception of Atalanta’s run that doesn’t involve any logical contradictions.

## Conclusion

So where does all this leave us? We saw that Aristotle’s response wasn’t sufficient. We also saw that Cauchy’s contributions could supplement Aristotle’s response. We also saw that even if we think that a run is infinitely divisible, Thomson’s arguments about the contradictory nature of supertasks don’t necessarily hold up. So we know that there is no reason to think that it is impossible to perform an infinite series of tasks in a finite amount of time. We even saw a proof of how Atalanta could complete her run even if it involved ℵ₀ rest periods.

What we haven’t done is give a conclusive answer to whether using mathematical language to describe physical phenomena is appropriate in this case. Whatever theory one subscribes to or finds useful, the preceding discussion clearly shows that for any theory of space, time, motion, change, or infinity, Zeno’s paradoxes can be a sharpening stone.

## References

Aristotle. (1934). *Physics* (P. H. Wicksteed & F. M. Cornford, Trans.). Harvard University Press. https://doi.org/10.4159/DLCL.aristotle-physics.1957

Benacerraf, P. (1962). Tasks, Super-Tasks, and the Modern Eleatics. In W. C. Salmon (Ed.), *Zeno’s Paradoxes* (pp. 103-129). Bobbs-Merrill.

Black, M. (1950). Achilles and the Tortoise. In W. C. Salmon (Ed.), *Zeno’s Paradoxes* (pp. 67–81). Bobbs-Merrill.

Grünbaum, A. (1970). Modern Science and Zeno’s Paradoxes of Motion. In W. C. Salmon (Ed.), *Zeno’s Paradoxes* (pp. 200–250). Bobbs-Merrill.

Huggett, N. (2019). Zeno’s Paradoxes. In E. N. Zalta (Ed.), *The Stanford Encyclopedia of Philosophy* (Winter 2019). Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/win2019/entries/paradox-zeno/

Huggett, N. (2010). *Everywhere and Everywhen: Adventures in Physics and Philosophy.* Oxford University Press.

Salmon, W. C. (2001). *Zeno’s Paradoxes*. Bobbs-Merrill. (Original work published 1970)

Thomson, J. (1954). Tasks and Supertasks. In W. C. Salmon (Ed.), *Zeno’s Paradoxes* (pp. 89-102). Bobbs-Merrill.

Wisdom, J. O. (1951). Achilles on a Physical Racecourse. In W. C. Salmon (Ed.), *Zeno’s Paradoxes* (pp. 82-88). Bobbs-Merrill.