In 1952, Cambridge mathematician Ian Haldane posed a question in a letter to a colleague: If there existed a computer that could run indefinitely, would it be able to calculate all the digits of π before the end of the universe?

The question is obviously absurd. π is irrational—there are no “all digits.” But what Haldane really wanted to ask was something else: Suppose the universe has an end, and the computer’s processing speed can increase without limit—at that terminus, would the final digit the machine outputs carry some kind of meaning?

His colleague never wrote back.

Seventy years later, my advisor recounted this question to me in his office. It was four in the afternoon, winter, and the light outside the window had already begun to turn gray. On his bookshelf stood an entire row of Borges, Spanish originals, their spines worn white from use.

“What did Haldane say afterward?” I asked.

“There was no afterward,” he said. “He died the following year. Heart attack.”

He took a piece of paper from his drawer, handwritten, the script very small. I recognized it as his own.

“But I came up with an answer for him.”

My advisor’s answer required some premises.

First, the universe has no “end,” but it has a “horizon.” Accelerating expansion means that sufficiently distant galaxies are receding faster than the speed of light—the light they emit can never reach us. This is a kind of ending: not time stopping, but a boundary of information.

Second, any computer—no matter how fast—is constrained by Landauer’s principle: erasing one bit of information requires energy expenditure and releases heat. This means computation has a cost.

Finally, in the late universe, when all stars have extinguished, when protons begin to decay, time itself will become thin. Not “slow”—thin. The interval between two events will be stretched to an indefinable degree.

“So,” he said, “if we place a computer on a spacecraft falling into a black hole, from an external observer’s perspective, it will asymptotically approach the event horizon but never arrive. From the computer’s own perspective, it will cross the horizon in finite time.”

“And then?”

“Then we have two versions of π,” he said. “One is what the external observer sees—infinite digits, forever growing, forever without end. The other is what the computer itself experiences—finite time, finite digits, a final number that actually exists.”

He pushed the paper toward me.

“The question is: Are these two π the same π?”

I didn’t answer his question. Not because I didn’t know the answer, but because I knew the answer, and that answer made me uncomfortable.

The answer is: They are not the same.

If the observer determines the state of the observed, then π is not a fixed number but a function that depends on who is asking, and from where. This means mathematics is not discovery but construction. This means that before the Big Bang, before any observer existed, π was nothing.

I told this idea to a friend who studied physics. He was another student of my advisor’s, researching cosmology, his office at the other end of the corridor.

“You’ve got the question backwards,” he said. “It’s not that π depends on the observer—it’s that the observer depends on π.”

“What do you mean?”

“The fine-structure constant of the universe,” he said. “If its value deviated by one percent, stars wouldn’t form. If π weren’t π, you wouldn’t be here asking this question.”

He smiled slightly.

“π doesn’t need to be calculated. π is the precondition for your existence and your ability to ask questions.”

My advisor died three years later. Not a heart attack—something else.

In our last few meetings, he didn’t talk much about mathematics anymore. He talked about coincidences he’d encountered when he was young—meeting someone on a train who later became his collaborator; finding a line in the margin of an old book that happened to be the answer to a problem he was contemplating.

“Do you believe in synchronicity?” he asked me.

I said I didn’t know.

“Jung didn’t mean mysticism,” he said. “He meant: causality is not the only order. Some things are meaningful not because A caused B, but because A and B occurred simultaneously.”

He pointed out the window. It had grown dark.

“Every digit of π and every instant of the universe,” he said, “they’re not causally related—they occur simultaneously.”

There’s a variant of Haldane’s question that my advisor never mentioned.

Suppose we don’t care about “all” the digits of π, only one of them—say, the 10^10^10th digit. This number is determined. It’s either 0, or 1, or 2… or 9. Whether or not anyone calculates it, it’s there.

But what does “there” mean?

If no physical process can ever reach that digit—if all the computers in the universe, all the time, all the energy, are insufficient to calculate to that position—then what does it mean to say it “exists”?

I remembered the paper my advisor had pushed toward me. I had never examined it carefully. Now I found it in my drawer and unfolded it.

There was only one line, written in his small script:

“Some truths are not unknowable—it’s that ‘being known’ is simply not part of their definition.”

Below it, he had drawn a circle. No notation for π, just a circle.

I don’t know how to end this report.

Last year I started a research project, using computers to simulate complex systems—not π, something else. Large numbers of individuals, simple rules, let them run, see what emerges.

One night I let the program run overnight. The next morning when I woke, there was a pattern on the screen I had never seen before. I hadn’t designed it, no single rule had directly caused it, but there it was.

I stared at it for a long time.

I don’t know what this has to do with π. Maybe nothing. Maybe I just want to say: some things really can emerge from infinite computation, but after they emerge, we don’t know what to do with them either.

My advisor loved Borges. Borges wrote a story about an old man who meets his younger self on a park bench. They talk, but nothing important is said. Because what’s important isn’t in the conversation—it’s in the fact that the conversation happens at all.

Two versions of the same person, sitting on the same bench, existing simultaneously.

Maybe this is the last digit of π. It exists, but not to be read. It exists only to give all the other digits a direction to go.

Footnotes

[1] Ian Haldane’s (1901-1953) letter was never publicly published. This account of its contents is based on secondhand sources and may contain inaccuracies.

[2] I haven’t written my advisor’s name. Not for privacy, but because I found that when I tried to write it down, I couldn’t remember his face. I only remember his handwriting, and that row of Borges on his shelf.