A photographic tale

Artef Fierel and the Place of Coincidences

A JOURNEY THROUGH PHILOSOPHY, SCIENCE AND THE ART OF PHOTOGRAPHY

The fact that something happened does not mean it had to happen.

The following morning, Artef Fierel found the notebook still open on the table.

He looked for the blade of grass he had left on the windowsill the evening before. It was no longer there. A moment later he saw it lying on the page, just beneath the sentence he had written before going to sleep: “A proportion can guide me. The responsibility for what I show remains mine.” He picked it up carefully. An almost invisible trace of dust remained on the paper. For a moment, finding it beneath those words seemed like an answer. Then he looked at the half-open window and the table just below the sill. A breath of air could have moved it. That it had come to rest near that sentence was a coincidence; it was he who, on reading it again, had felt a connection. This did not erase his surprise. It allowed him to dwell in it with one more question.

He placed the blade beside the photographs of the garden. In the print of the bee, its wings still held the afternoon light. He remembered choosing a fast shutter speed and moving so that his own shadow would not cover the insect. What he did not remember was the tiny speck suspended above the flower.

He brought the magnifying glass closer. It might have been pollen or dust. In the photograph it had become a bright dot. How many things entered an image without his having invited them? And how many others might have entered if he had pressed the button a moment earlier? He slipped the camera into his bag and went out. He found himself at the gate of the botanical garden without remembering when he had decided to return. On the way he had been thinking about the blade of grass and the bright dot, while his feet followed a route that was now familiar.

The old man’s bench was empty. Near the greenhouse, a few branches used to make the rectangle on the ground were still there. One side had been moved, and the lines inside no longer met.

Artef picked up a small stick and drew a little tree in a patch of fine earth. He traced the trunk, two branches and a third that reached higher than the others. Around them he sketched a crown, leaving it open on one side.

He was going over one of the grooves with the stick when a leaf came spiralling down through the air. It brushed the tip of the stick and settled on the drawing, its stalk pointing towards the highest branch. Artef stood still, looking at it. It seemed to belong to the tree he had traced in the earth: it was too large for the branches, yet that very disproportion gave it a strange vitality.

— Right there… — he murmured.

— Was that the place you had chosen?

Artef turned. The librarian stood a few steps away, holding a bunch of keys in one hand and some folded sheets of paper in the other.

— No, that’s the beauty of it. I drew the tree, and the leaf fell on its own.

The woman came closer, taking care not to step on the grooves, and bent slightly to look.

— It really does look as though you put it there.

— That’s what I was thinking. What were the chances of it falling in that very spot?

The librarian studied the drawing for a few moments.

— Would it have surprised you if it had fallen a little further to the right?

Artef followed her gaze. Indeed, the leaf would have completed the crown there too, as though resting on another branch.

— I think so — he admitted.

— And on the opposite side?

He smiled, beginning to see where she was going.

— I would probably have said the same thing: “Right there.”

— Then there were several places that would have seemed the right one to you.

Artef looked at the tree again. Until a moment earlier, it had seemed to him that the leaf had hit a single target, almost as though it had sought it out as it fell. Now he realised he might have recognised an equally happy encounter at many other points in the drawing.

He remembered the transparent rectangle over the Parthenon: moving it just a little was enough to reveal a new correspondence, as convincing as the one before.

— So it’s as though I decide where the target is after I’ve seen the leaf fall — he said. — I choose the boundaries when I already know the result.

The librarian nodded.

— It can happen. Especially when we like what we see.

A second leaf fell near the bench. Artef followed it with his eyes. It did not seem to have landed anywhere special.

Yet that one, too, had made a journey.

They went into the library. The morning light reached different tables from those of the previous day. Where Artef had photographed the woman with the open volume, there was now a trolley full of books.

The librarian moved the trolley, then opened a drawer and, after rummaging for a moment, took out some cards and a die.

— We use these in our sessions with the children. Sometimes a question becomes clearer if we can try it out — she said with a smile.

She drew six numbered boxes.

— If the die is fair, each face has a one-in-six chance of coming up. It’s the simplest case: we count the outcomes we’re interested in and compare them with all the possible outcomes, provided they are equally likely.

— So a four has a one-in-six chance.

— And an even number has three chances out of six: two, four or six.

— Half.

The woman handed him the die.

Artef chose four and rolled. It came up two. On the second roll, five; on the third, two again. Four appeared on the fifth.

— It kept us waiting — he said.

— Yes, but the die didn’t know we were waiting for it.

Artef turned it between his fingers.

— If it hadn’t come up for twenty rolls, would it have been more likely on the twenty-first?

— Still a one-in-six chance, if the rolls are independent. The die doesn’t keep count, and it doesn’t feel the need to make amends.

— But we do.

— We remember absences. And sometimes we mistake them for promises.

Artef looked at the numbers on the card. He remembered times when, while waiting, he had thought: it has to happen now; it’s been too long since it last happened. The die remained on the table, utterly indifferent.

· · ·

They kept rolling, noting down every result.

For a while, two came up more often than the other numbers. Then several sixes appeared. The boxes filled unevenly.

— These probabilities don’t seem very orderly — Artef observed.

— With only a few trials, chance can make quite a lot of noise. The law of large numbers tells us that, when we repeat the same experiment a great many times under the same conditions, the relative frequencies tend to approach the probabilities.

Artef looked up.

The woman pointed to the boxes.

— It means that, with many rolls, we expect each number to account for close to one sixth of the total. Not that all six faces must appear in every six rolls.

— Nor that the totals must end up perfectly even.

— Exactly.

Artef rolled again. Another two.

— This one doesn’t seem to have been listening.

The librarian laughed again.

— The law describes a tendency. It doesn’t give orders to the next roll.

In his notebook, Artef wrote:

“A regularity can emerge from a multitude without telling me what will happen a moment from now.”

He read the sentence again. He seemed to recognise the garden in it: the return of the seasons, the leaves falling every autumn, and the impossibility of saying which would touch the ground first.

— Sooner or later, though, four will come up — he said.

— With more rolls, the probability of seeing it at least once increases. But no finite number of attempts guarantees it.

The woman took a fresh card.

— In one roll, you have about a seventeen per cent chance of getting a four. In six rolls, the probability of getting at least one rises to about sixty-seven per cent.

— Six opportunities, and still no certainty.

— To work that out, it helps to look at the opposite possibility: that four never comes up. We calculate that and subtract it from the total.

Artef smiled.

— To understand whether something will arrive, you also look at all the times it might fail to appear.

— It’s the complement rule. Sometimes the simplest path leads through what doesn’t happen.

He thought of the pine cones from the day before. He had kept the photographs of those in which he could trace the spirals. Anyone who saw only those images would not know how many other specimens he had left on the path.

— And my leaf? If I drew enough trees, would a leaf eventually complete one of them?

— You could give it more opportunities. But the garden isn’t a die: the wind, the season and the ground change. And we would have to decide what “complete” means.

Artef wrote:

“To understand an encounter, I must also remember the occasions when it did not happen.”

The old man’s book was still on the desk.

Artef opened it at the sequence of numbers. The pencilled sentence was there:

“Count what you see. Then look at what has been left out of the count.”

The librarian began sorting some cards. Each showed a date and the number of entries into the library.

Artef brought together two with the same total.

— Did the same people come in on these days?

— There were the same number of visits. One person can come in more than once, and two days with the same total can be very different.

Within those figures there might be a school group, someone returning a novel, a man who had come in to shelter from the rain.

— Statistics begins here too — the woman said. — Deciding what we want to know and what we are really counting. Then looking for regularities, without forgetting how incomplete the data are.

— And probability?

— It helps us reason about what might happen. In statistics, we often start from what we have observed to learn something we do not yet know.

Artef looked towards the garden.

— Like wondering how many people will come in tomorrow.

— Yes. And knowing that a school visit is scheduled for tomorrow would change our prediction.

— Because we have an extra piece of information.

— That’s the idea of conditional probability: assessing a possibility in the light of a condition. The probability that the path is wet changes if we know it has rained.

Just then, the sprinklers came on.

Artef pointed to them.

— But finding the path wet isn’t enough to say it has rained.

The woman nodded.

— We also have to consider the other ways it might have got wet.

Artef watched the water beyond the glass. A clue could guide him without giving him the whole story.

Beside the cards lay a copy of Alice in Wonderland. The White Rabbit’s ears peeped out from the cover.

Artef picked it up.

— Today I feel as though I’ve ended up inside this book.

— Because of the die?

— Because of all the things that can happen. A leaf falls and seems to know my drawing. A blade of grass comes to rest beneath a sentence. I open a door, and behind it there are ten more.

He leafed through the book.

He remembered Alice in the hall of doors, the little key, the garden glimpsed but out of reach. Each passage called for a different size, an attempt, something she did not yet know.

— There could be a Land of Probabilities — he said. — A place where all the things that haven’t happened yet are waiting.

The librarian leaned against the desk.

— What would it be like?

Artef looked at the illustrations.

— There would be the leaf that falls on my tree, but also the one that ends up in the fountain. The gardener who walks past as I take a photograph, and the one who stops before he reaches me. The speck of pollen that enters the photograph, and the one that passes outside the frame.

— And would all the doors be equally easy to open?

Artef thought about it.

— No. Under a tree, you’re more likely to come across a leaf than an umbrella.

— Then in your land, the possibilities carry different weights. And some doors, under certain conditions, are closed.

He smiled.

— Alice would have to read the labels carefully.

— And find out whether she can trust them.

Artef brushed the cover with his fingers.

— The world we observe could be like this: a collection of possibilities. Some become facts, and others remain in the Land of Probabilities.

The librarian let a moment pass.

— It’s an evocative image. As long as we remember that we’re using a metaphor. Probability helps us describe uncertainty; on its own, it doesn’t prove that there is a place where everything that did not happen remains.

Artef nodded. He liked that land even without having to look for it on a map.

It was a way of remembering how much wider what he imagined was than what he could see at any given moment.

· · ·

They returned to the garden.

The leaf was still on the drawing. An insect had crossed the crown, leaving a mark in the earth finer than the one made by the stick.

Artef crouched down.

— So this possibility came true. The others stayed behind the doors.

He followed the outline of the leaf with a finger, without touching it.

— But something brought it here.

— Certainly. It came away from a branch, it had a weight and a shape, it met the air. Talking about probability doesn’t erase causes. Sometimes we need it precisely because we don’t know all the conditions well enough.

Artef looked up towards the branches.

— My drawing didn’t attract it.

— We have no reason to think so.

— I’m the one who recognised an encounter.

A little further away, the second leaf had come to rest in a crack in the ground. He tried to imagine that he had drawn that crack. Perhaps he would have bent down with the same sense of wonder.

— Then can anything seem like a coincidence to me?

— If you look for enough similarities, you will find many. To understand how surprising they are, you also have to count how many you looked for.

Artef thought of clouds, faces glimpsed in walls, spirals sought among pine cones, and the shadows and reflections he had photographed so often.

In his Land of Probabilities there were also doors that resembled the one he wanted. If he were free to choose afterwards, he could convince himself that he had arrived exactly where he had expected to.

The leaf did not become any less beautiful.

Its beauty was simply not a measure of its improbability.

He raised the camera.

The first frame included the drawing, the leaf and the tip of his shoe. He moved. The shoe disappeared, but the camera’s shadow covered the highest branch. He tried the other side. The light revealed the grooves clearly. Seen at an angle, however, the leaf seemed far from the point where it had landed.

Artef returned almost to his original position, keeping his feet further back.

He took the photograph.

On the screen appeared a little tree with a single leaf.

— Someone seeing this photograph might think I put it there.

— They might.

— The image doesn’t show that it fell.

The photograph showed a relationship in space. The moment that had produced it remained outside the frame. He could describe it in a caption. Anyone reading it would also have to trust his words.

— And if I moved it to make the stalk line up better?

— You would have made a new choice.

He looked at the small gap between the stalk and the groove.

He left the leaf where it was.

That distance belonged to the encounter he wanted to preserve.

The librarian went back inside to open the reading room. Artef stayed in the garden.

Near the fountain, he found a blade of light crossing the path. People coming from the gate stepped into it for a moment, then returned to the shade. A gardener was taking some plants towards the greenhouse. Artef asked whether he could photograph him as he passed. The man agreed and continued with his trolley.

Artef chose the framing and waited. When the gardener reached the light, he lowered his head to check a wheel. Artef took the photograph, but the face remained hidden.

On the next pass, a plant obscured part of his figure.

The third time, the man stopped in the patch of light to straighten a pot. A long leaf brushed his sleeve; he guided it with his hand before moving on. Artef took another photograph.

He immediately preferred the last image. The gesture was clear, the light separated the hand from the jacket, and the wheel remained just inside the frame.

That photograph had needed his waiting. It had also needed a tilted pot.

Being prepared had allowed him to catch the moment. It had not guaranteed that the moment would come. He thought of Alice before the little door: to pass through it, she also had to be the right size.

He adjusted the shoulder strap and wrote:

“Time. Event. Possibility.”

Beneath it he added:

“I was there, but being there was not enough. I also had to notice.”

Sitting on the bench, he began scrolling through the photographs.

Between two successful images, he found an empty view of the path. He had pressed the button too soon. He was about to delete it, then stopped.

That image preserved one of the opportunities he would otherwise have forgotten.

If he showed only the gardener’s gesture, the meeting of hand, leaf and light would seem immediate. The sequence also contained the waiting, the hidden face, the plant in front of the figure.

He thought back to the library cards. To understand a collection of data, you had to know how it had been put together. The successful photographs alone told a different story from the entire sequence. Then he noticed something else.

At first he had been waiting for a face in the light. In the end, he had chosen a hand. What he wanted had changed as he watched.

He wrote: “I must distinguish what I was waiting for from what I learnt to see afterwards.” He did not need to assign a percentage to the gardener’s hand. But if he wanted to call it an almost impossible event, the single photograph would not be enough. He looked again at the empty path. That had happened too. It was just that, until a moment earlier, it had not seemed worth keeping.

Before leaving, he returned to the little tree.

The leaf had moved. Now it covered part of the trunk, and its stalk pointed towards the path.

Artef felt a slight pang of disappointment. He looked at the photograph to recover its earlier position, then lowered the screen.

Things had continued to happen on the ground even after his image.

He took a second, wider photograph. He included the other leaves, the branches left from the previous day’s lesson and part of the bench.

The drawing occupied very little space.

In the first photograph, the encounter seemed to belong to a world made expressly to contain it. In the second, you could see how much ground surrounded it.

He kept both.

They seemed to him like two pages of the same story: in one, he had followed the White Rabbit; in the other, he had raised his eyes and looked at the meadow.

That evening he arranged the new prints beside those from the garden of proportions. The leaf, the gardener’s hand, the bee, the visitor on the stairs. In each he recognised a decision and something that had come to meet him.

He had chosen the edges. Within those edges, the world had kept moving.

He opened the afternoon’s entire sequence on the screen. The less successful photographs made visible the work that the chosen print left behind. One he had overlooked showed the gardener already in the shade, with the trolley almost out of the frame.

He looked at it longer than he had expected.

Then he picked up the notebook.

“The world I observe appears to me as an interweaving of possibilities. Some come true and become encounters, gestures, images. Others, which I had imagined and which did not happen, remain in my Land of Probabilities.”

He paused, then added:

“They were not all equally likely. And the fact that something happened does not mean it had to happen.”

He thought again of the die, the unevenly filled boxes, the records of visits. The numbers had not taken the garden away from him. They had taught him to ask more careful questions.

Beneath the print of the leaf, he wrote:

“I had just drawn the tree. The leaf fell while I was still bent over it.”

Those words told what he remembered, without asking the photograph to prove it on its own.

Beside them, on a blank page, he sketched a little door. He left it ajar, like the door to a land he could return to in his imagination.

On the table, the blade of grass had remained beside the portrait of the philosophers. Artef picked it up and placed it between the pages of the notebook. This time he had put it there himself. Before closing the notebook, he wrote that down too.