08.10.2026 Diary XXXVIII Zeno: no reaching the half before its half

08.10.2026 Diary Page XXXVIII

Every road has a half. So does its half.

Thursday. I had to write an email. Three lines. Before starting it there was something I had to prepare: the final version of the file I would send. Before looking at the final version of the file I had to check another file. And for that I had to find a folder.

When I found the folder, an hour later I found myself at a desk where the email had never been written but seven files had been corrected.

The name of this is something a man from Elea described two thousand five hundred years ago.

Bisection

One of Zeno of Elea's paradoxes goes like this. Someone who wants to go from one place to another first has to reach the half of the road. Before reaching the half, the half of the half, that is, the quarter. Before that, the eighth. This goes on like this forever.

The conclusion: to be able to move, you have to finish an infinite number of things, and an infinite number of things cannot be finished in a finite time. So there is no motion.

Zeno built this to defend the claim of his teacher Parmenides that "change is an illusion." Other paradoxes like Achilles and the tortoise and the flying arrow do the same job.

Everyone knows this argument is absurd: people walk. But showing exactly where the absurdity lies took two thousand years.

The two layers of the answer

Aristotle's answer is in the sixth book of the Physics. According to him a road has the possibility of being divisible forever but is not divided into an infinite number of parts. Divisibility is a potential; not an act.

I described capacity and act in the Entelekheia piece. Here the same distinction helps: the halves carry the potential of existing but the person walking does not make them one by one.

The modern answer comes from mathematics. A half, plus a quarter, plus an eighth and so on, the sum that goes on being added forever comes to exactly one. What is added infinitely many times can approach a finite value, because the steps shrink.

These two answers rest on different logics but for me they arrive at the same place: something that can be divided forever does not have to last forever.

The regress of starting

I go back to my Thursday morning. What I lived through was bisection itself.

To write the email I first have to prepare the file. To prepare the file I first have to find the folder. To find the folder I first have to... Every precondition looks like half of the previous job, and every half has a half too.

This is one of the most common forms of starting in ADHD. In the task paralysis article I had said "knowing what to do and not being able to start." Actually it is not only not knowing: it feels as if there is always something that has to be started before starting.

Zeno drew this picture correctly. What he drew wrong was the conclusion.

The steps are shrinking

Here I learn something from mathematics. The sum is finite because every step is smaller than the one before.

Looking at Thursday morning, I see that the preconditions really did shrink. The email is three lines. Preparing the file twenty minutes. Finding the folder two minutes. Every precondition is smaller beside the job it serves.

But my mind does not feel it that way. The mind shows every precondition at equal weight: a precondition is a precondition. It does not see the sizes.

The solution is to write down the size of the steps. Email: three minutes. File: twenty minutes. Folder: two minutes. Total twenty-five minutes. When I write the numbers, the chain that looked infinite turns into a finite sum.

The how long will it take tool exists for exactly this: seeing the size of a job beforehand.

Starting from which half

Zeno's paradox has one more subtle side. As the road is being halved, the question of the "first step" disappears. From which half do you begin? You cannot begin from the smallest, because there is no smallest.

In a real walk, though, the first step is simple: you lift your foot. Lifting your foot is not an infinitely small piece of the road; it is walking itself.

Starting is like this too. Looking for the first step as "the smallest precondition" is wrong. The first step is the shortest form of the job itself. For the email, that is writing the first sentence.

The idea in the stopping halfway article comes from here too: even writing the first sentence and leaving it starts the movement.

Misreading Zeno

It is very easy to use this paradox as an excuse. The sentence "starting is impossible because there are infinite preconditions" sounds logical.

But you have to look at Zeno's own aim. He wanted to show that motion is impossible, and motion is not impossible. Starting is not impossible either. The paradox is not a proof, it is a puzzle: knowing it is wrong, we look for the place of the wrongness.

For me the place of the wrongness is in not seeing the size of the steps. The mind shows small steps as big.

The arrow and the moment

Another of Zeno's paradoxes is also about starting but from another angle: the flying arrow.

An arrow flying in the air is at a single place at any moment. Something at a place at a single moment is not moving, it is still. If every moment is still, the sum of all moments is still too. So the arrow does not move.

The error of this argument is also that an instant cannot capture motion: motion is not an instant, it is a process. But the effect it leaves on me is different. Looking at a snapshot of a day, progress does not show. At twelve noon I look at the desk, it is the same as twenty minutes ago. At six in the evening I look, still the same.

Progress stands not in snapshots but in the difference between them. That is why at the end of the day I need to count not what I did but what changed since morning. The end-of-day closing is a way of doing this: four items, fifteen minutes, a comparison with the previous day.

Achilles and the tortoise

This is Zeno's best-known paradox: the fast runner can never catch the slow tortoise, because each time he first has to reach the place where the tortoise was, and in the meantime the tortoise has moved a little ahead.

It is a relative of the bisection paradox and goes with the same solution: the sum of infinitely many shrinking steps is finite. Achilles catches the tortoise, because the elapsed times shrink too.

But when writing I give this another name. The tortoise is what the work cannot catch up with, always standing a step ahead; and I am Achilles. Each time I reach where it was and in the meantime the scope of the work has widened a little. As the work grows, catching up gets harder.

Here the mathematical solution of the paradox helps: I cannot catch up only if the growth speed of the work is greater than mine. If the growth slows, that is, if I stop adding new requests, I catch up. Freezing the scope is therefore a speed setting.

The unfinished ones

This paradox has a reverse side too: reaching the half and being left half done are not the same thing.

When someone setting out on a road reaches the half, they have completed half of the road. An unfinished job, on the other hand, is a job that has stopped halfway. One is progress, the other abandonment.

What makes the difference is the movement going on. If yesterday's first half-done job is the beginning of a job that will be completed today, it is not left half done, it continues. I described the mechanism of this in the stopping halfway article: when a job is left open, the mind returns to it.

The difference between leaving halfway and being left half done is the next step being written down. If it is written, it was left halfway; if it is not written, it was left half done.

What I listened to today

Arvo Pärt: Cantus in Memoriam Benjamin Britten.

For strings and a bell, written after Britten's death. In the piece the same descending A minor scale is played at a different speed in each group of instruments; each group twice as slow as the one before. The bell tolls at intervals.

This structure is called a mensuration canon and it makes Zeno's halves audible. The same line at half speed, at quarter speed, at eighth speed. The slow voices come later but all of them play the same line.

The piece announces that something that can be divided forever still arrives somewhere.

Tomorrow

Tomorrow is Friday. In the morning I will write the email first. I will look at the file afterward. This morning I saw that there is such a thing as starting without knowing half of the precondition.