It all began with Zeno of Elea.

In the fifth century BC, the philosopher imagined an impossible race between Achilles, the swiftest of heroes, and a slow Tortoise. Achilles gave her a small head start. To catch her, however, he first had to reach the point from which she had set off. Meanwhile, the Tortoise moved forward. By the time Achilles reached that new position, she was already further ahead. The distance kept shrinking, yet it could still be divided into an infinite number of intervals. According to the paradox, Achilles seemed condemned to pursue her for ever. This famous argument has come down to us through Aristotle, among others.

Many centuries later, Lewis Carroll brought the two characters together again. In his short dialogue What the Tortoise Said to Achilles, the physical race became a logical one: the Tortoise forced Achilles to add ever more premises in an attempt to prove a conclusion that kept receding. The finish line was no longer a point in space, but certainty in reasoning.

Douglas R. Hofstadter later returned to Achilles and the Tortoise in Gödel, Escher, Bach: an Eternal Golden Braid. In his dialogues, the pair explore paradoxes, musical structures, images, levels of reality and stories contained within other stories.

The story that follows imagines a new race. This time, Achilles and the Tortoise will compete in more than space or logic. They will enter a camera and confront each other on the terrain of time: Achilles will choose the speed of a quick exposure, while the Tortoise will defend the patience of a long one. Both will try to prove that their way of photographing is the truest to reality. But, as always when Achilles and the Tortoise meet, the finish line may not be where it seems.

The track had been prepared by Zeno himself.

It was no ordinary track: it was a long strip of photosensitive paper laid out in a completely dark room. At the starting line stood Achilles, the Tortoise and a camera mounted on a tripod. At the finish line there was no trophy, only an empty frame.

ACHILLES — This time I will not let a sophism stop me from winning.

TORTOISE — Are you certain that winning means arriving first?

ACHILLES — In a race, that seems a reasonable definition.

TORTOISE — But this is not merely a race.

The Tortoise pointed to the camera.

TORTOISE — It is also a photograph.

Then Zeno threw open the walls of the room as though they were the doors of a wardrobe. Beyond the track appeared a meadow, the sky and a small crowd of spectators. Finally, he explained the rules: Achilles would run at full speed; the Tortoise would set off with a modest head start. The winner, however, would not be decided by the stopwatch. Whoever appeared most clearly in the final image would win.

ACHILLES — In that case, we shall use one thousandth of a second. I shall be so sharp that you will even see the dust kicked up by my sandals.

TORTOISE — I would prefer four minutes.

ACHILLES — Four minutes? In four minutes I shall have left the track, crossed the city and had the photograph developed.

TORTOISE — That is precisely what worries me.

The camera was set to 1/1000 of a second.

Achilles set off. When he reached the point from which the Tortoise had moved, the shutter fired.

Click.

In the photograph, Achilles appeared suspended in mid-air. Every muscle was perfectly defined. A drop of sweat, detached from his forehead, looked like a tiny motionless planet. A little further ahead, the Tortoise had one foot raised. Neither of them was moving in the image.

ACHILLES — Behold my victory! The photograph proves my speed.

TORTOISE — On the contrary. It proves that you were motionless.

ACHILLES — But I was motionless only in the photograph.

TORTOISE — Then the photograph does not show what you were doing.

ACHILLES — Of course it does. You can tell from the position of my body that I was running.

TORTOISE — Then the movement is not in the image. It is supplied by the mind of whoever looks at it.

Achilles studied the photograph again. It was true: speed was not visible as speed. It had become posture, tension, balance. The fast exposure had stripped movement of its duration, preserving only its anatomical imprint.

ACHILLES — My image has isolated the decisive moment.

TORTOISE — Decisive for whom?

ACHILLES — For the photographer.

TORTOISE — And if the photographer had pressed the shutter one thousandth of a second later?

ACHILLES — He would have captured another decisive moment.

TORTOISE — How many decisive moments does a second contain?

ACHILLES — A thousand, if we shoot at one thousandth.

TORTOISE — And how many of those thousand must we look at to truly understand the race?

Achilles sighed. He recognised the Tortoise’s tone: it usually heralded an infinite regress.

The fast exposure had performed an act much like Zeno’s: it had divided movement, extracted a slice and called that slice reality. Yet the image possessed an undeniable power. Time, precisely interrupted, became form. Achilles’ body looked like a statue that light had sculpted in the space of a flash.

The fast photograph did not narrate movement: it transformed it into gesture.

The Tortoise set the camera to Bulb mode. She stopped down the aperture, lowered the ISO and fitted a neutral-density filter in front of the lens.

ACHILLES — You have put dark glasses on the camera.

TORTOISE — When you want to look for a long time, you must protect yourself from too much light.

The shutter opened.

Achilles darted forward, crossed the field of view and disappeared. The Tortoise began her journey. The clouds moved. The wind bent the grass. Some spectators changed position; others left the track. The sun traced a small arc across the sky.

After four minutes, the shutter closed.

When they looked at the result, Achilles turned pale. The track seemed empty. All that remained of him was a faint, translucent trail, like a sentence spoken too quickly. The Tortoise, by contrast, had left a continuous dark trace. She did not look like a stationary animal, but like a brushstroke slowly crossing the scene.

ACHILLES — I have disappeared.

TORTOISE — You did not stay anywhere long enough.

ACHILLES — And you have become a streak.

TORTOISE — I stayed long enough in every place.

ACHILLES — But this photograph does not show how we looked.

TORTOISE — It shows how we endured through time.

The long exposure had done the opposite of the fast one. It had not isolated one instant from the others: it had gathered them together. The light from each fragment of time had entered the lens and been added to that of the fragment before it.

The clouds had become veils. The grass, waves. The moving people were ghosts. What had remained still appeared solid; what had passed quickly had become transparent.

ACHILLES — So, for photography, to exist means to stand still?

TORTOISE — For this photograph, to exist means to persist.

The Tortoise took out a notebook.

TORTOISE — Let us consider three propositions: A, a photograph represents what has been in front of the lens. B, you have been in front of the lens. Z, therefore the photograph represents you.

ACHILLES — The reasoning is impeccable.

TORTOISE — Yet in the long exposure you are almost invisible.

ACHILLES — Then we must add a new premise: whatever passes in front of the lens is recorded recognisably.

TORTOISE — But that premise is false. You passed in front of it, and you are not recognisable.

ACHILLES — We shall replace it: whatever remains in front of the lens long enough is recorded recognisably.

TORTOISE — Not always. If I stand still, I shall be sharp; if I move slowly, I shall become a trace.

ACHILLES — Then let us add that being recorded does not necessarily mean being recognisable.

TORTOISE — I accept that premise.

ACHILLES — So you also accept the conclusion?

TORTOISE — Which conclusion?

Achilles leafed through the notebook. The premises now filled several pages, but the conclusion seemed to keep moving a little further ahead, just like the Tortoise in Zeno’s race.

In Carroll’s dialogue, the Tortoise had forced Achilles to keep adding new propositions. Now an echo of that mechanism seemed to have slipped inside the camera: to establish what an image showed, they first had to establish what it meant to show, to endure, to appear and to be.

Every definition needed another definition. Every photograph seemed to require a photograph of the photograph.

ACHILLES — I propose an experiment. Let us photograph the photograph.

They placed the long-exposure image on the track and photographed it at 1/1000 of a second. In the new shot, the trace of four minutes was frozen in a single thousandth of a second. Accumulated time became an instantaneous object.

ACHILLES — Now your duration is contained within my instant.

TORTOISE — Or your instant contains a representation of my duration.

ACHILLES — Is that not the same thing?

TORTOISE — If you photograph a fire, does the photograph burn?

ACHILLES — Of course not.

TORTOISE — Then a photograph of a duration does not last as long as what it represents.

The Tortoise then took the instantaneous photograph of Achilles and placed it in front of the lens for four minutes. In the new image, the frozen moment remained unchanged throughout the long exposure.

TORTOISE — Now your thousandth of a second has lasted four minutes.

Achilles began to feel slightly dizzy. They had produced an instant that showed a duration, a duration that showed an instant, a fast photograph of the slow photograph and a slow photograph of the fast photograph.

ACHILLES — What if we photographed this latest image?

TORTOISE — We would obtain a fast photograph of a slow photograph of a fast photograph.

ACHILLES — And after that?

TORTOISE — A slow photograph of a fast photograph of a slow photograph of a fast photograph.

ACHILLES — How long would this go on?

TORTOISE — Until the camera begins to photograph itself.

At that very moment, they noticed a small camera mounted on a tripod in the corner of the image. Achilles and the Tortoise were reflected in the little camera’s lens. Achilles picked up a magnifying glass.

In the reflection, a very tiny Tortoise was handing a very tiny notebook to a very tiny Achilles. On the page was written the first premise: a photograph represents what has been in front of the lens.

Achilles dropped the magnifying glass.

ACHILLES — Fast photography honours precision. It separates, chooses, makes visible what the eye cannot hold on to.

TORTOISE — But by selecting one instant, it pretends that instant is independent of what comes before and after it.

ACHILLES — Every art must choose.

TORTOISE — Long exposure chooses too. It simply chooses a duration instead of a point.

Achilles looked at the two images side by side. In the first, line, geometry and tension prevailed. His body formed a sharp diagonal. The Tortoise served as a small visual counterweight. Movement had been converted into composition.

In the second, transformation, atmosphere and continuity prevailed. The landscape seemed to remember what had happened within it. The photograph was no longer the recording of an appearance, but the condensation of an interval.

ACHILLES — My image is an exclamation mark.

TORTOISE — Mine is a subordinate clause.

ACHILLES — Mine says: “There!”

TORTOISE — Mine says: “Meanwhile…”

ACHILLES — My photograph asserts.

TORTOISE — Mine remembers.

The fast exposure offered the culminating moment: form reached an unrepeatable balance, and the photographer intervened to wrest it from the flow. It came close to the idea of the decisive moment associated with Henri Cartier-Bresson’s work: the fraction of a second in which gesture, geometry and meaning seem to converge.

Long exposure, by contrast, created a synthesis. It transformed becoming into surface, a path into a mark, a crowd into mist and light into calligraphy. It did not show one privileged moment, but the relationship among many moments.

One photograph said: “This happened.” The other replied: “This passed.”

TORTOISE — Tell me, how long does the present last?

ACHILLES — An instant.

TORTOISE — How long is an instant?

ACHILLES — It has no duration.

TORTOISE — Then how can anything happen in it?

Achilles did not answer.

In Zeno’s old race, before catching the Tortoise, Achilles had to reach the point from which she had set off, but meanwhile the Tortoise moved forward. Fast photography resembled Zeno’s arrow: considered in isolation, the body occupied a precise position and seemed motionless.

Long exposure did not isolate a single position. It showed many superimposed positions, paying for continuity with the loss of contours.

TORTOISE — Your fast exposure knows the where perfectly, but risks losing the during.

ACHILLES — Your exposure knows the during, but makes the where uncertain.

TORTOISE — Perhaps movement arises precisely from the impossibility of possessing both completely.

ACHILLES — You speak as though the camera were a philosopher.

TORTOISE — Every camera proposes a theory of time. The photographer adopts it when choosing the exposure time.

Zeno announced a new race.

This time, Achilles and the Tortoise would choose the shutter speed together.

ACHILLES — One thousandth?

TORTOISE — Four minutes?

They negotiated at length and settled on a quarter of a second.

Achilles slowed until he was alongside the Tortoise, who sped up as much as she could. The photographer followed them with a slight movement of the camera.

Click.

In the final image, Achilles was neither completely sharp nor entirely dissolved. His face was recognisable, while his legs became trails. The Tortoise’s shell remained fairly well defined, but the ground behind her stretched into a horizontal blur.

For the first time, the photograph did not set stillness against duration. It made them work together.

ZENO — Who won?

Achilles looked at the Tortoise. The Tortoise looked at Achilles.

ACHILLES AND TORTOISE — The relationship won.

The camera, which until then had remained silent, made a small sound.

Click.

ACHILLES — Who took that?

No one answered.

A newly taken photograph appeared on the back of the camera. It showed Achilles and the Tortoise looking at a photograph that showed Achilles and the Tortoise looking at a photograph.

In the smallest image, almost invisible, the Tortoise seemed to be speaking. Achilles picked up the magnifying glass and deciphered the movement of her lips.

TORTOISE IN THE IMAGE — The race has ended only within the image.

ACHILLES — And outside the image?

The Tortoise took another step.

⁂

A tale in homage to Zeno’s paradox, Lewis Carroll’s dialogue and the dialogic inventions of Douglas R. Hofstadter.