Why do we long for someone? Perhaps because time, in our intuition, is a road that runs in only one direction. Someone leaves, a relationship ends, an afternoon disappears many years into the past. All are left behind, farther and farther from the present.

On that straight line, farewell feels permanent by nature.

Yet mathematics once discovered that time may do more than carry everything away. In certain worlds, if we wait long enough, a system returns to somewhere it has been before. It may not coincide perfectly with the past, but it can come arbitrarily close.

This is Poincaré recurrence.

Time does more than say goodbye

Imagine a completely closed room. Its particles move and collide without end, forming an extremely complex dynamical system.

We tend to think that once the particles disperse, their original arrangement is gone forever. The Poincaré recurrence theorem says otherwise. If the system is closed, its available state space has finite measure, and the evolution neither compresses nor loses the ‘volume’ of states, then after enough time the system will again come arbitrarily close to its initial state.

Not because time has reversed, and not because the particles remember the past.

A system that never loses states but must keep moving through a finite space cannot visit only new places forever.

Eventually, it returns.

Why the return is mathematically inevitable

Let a discrete-time dynamical system be

\[ (X,\Sigma,\mu,T), \]

where \(X\) is the space of every possible state of the system, \(\mu\) is a measure of the size of sets of states, and \(T:X\to X\) is the evolution of the system after one unit of time.

The system preserves measure. For every measurable set \(E\),

\[ \mu(T^{-1}E)=\mu(E). \]

The total state space also has finite measure:

\[ \mu(X)<\infty. \]

Now choose a region \(A\subset X\) with positive measure. It may represent a family of states once experienced by the system, such as every possible state near ‘an encounter.’

The Poincaré recurrence theorem states that for almost every state \(x\) in \(A\), there are infinitely many positive integers \(n\) for which \(T^n(x)\in A\).

In other words, the system does not return only once. It returns infinitely many times.

The proof is not complicated.

Let \(B\subset A\) be the set of points that never return to \(A\) after leaving it:

\[ B=\{x\in A:T^n(x)\notin A,\ \forall n\geq1\}. \]

Consider the following sets:

\[ B,\quad T^{-1}B,\quad T^{-2}B,\quad\ldots \]

They must be pairwise disjoint. Suppose a point belonged to both \(T^{-m}B\) and \(T^{-n}B\), with \(n>m\). After \(m\) steps it would enter \(B\), then after another \(n-m\) steps it would enter \(B\subset A\) again. That contradicts the definition of points in \(B\), which never return to \(A\).

Because the system preserves measure,

\[ \mu(T^{-n}B)=\mu(B). \]

If \(\mu(B)>0\), infinitely many disjoint sets of exactly the same positive measure would all have to fit inside the finite state space \(X\). Thus

\[ \mu(X)\geq\sum_{n=0}^{\infty}\mu(B)=\infty, \]

which contradicts \(\mu(X)<\infty\). The only possibility is therefore

\[ \mu(B)=0. \]

The points that return only finitely many times also form a set of measure zero. If a point has a final return, then from that return onwards it belongs to some \(T^{-m}B\). A countable union of measure-zero sets still has measure zero.

Therefore, almost every state in \(A\) returns infinitely often.

That is why recurrence is inevitable.

Writing ‘reunion’ as a mathematical proposition

Suppose the complete state of the world at one encounter is \(x_0\).

That state includes more than the positions of two people. It includes the light, the air, the words spoken, and the people each of them had become.

Take a neighbourhood of radius \(\varepsilon\) centred on \(x_0\):

\[ A_\varepsilon=\{x\in X:d(x,x_0)<\varepsilon\}. \]

In an ordinary separable metric state space, Poincaré recurrence can be written more strongly: for almost every recurrent point \(x_0\), there is a sequence of times \(n_k\) tending to infinity such that

\[ d(T^{n_k}x_0,x_0)\longrightarrow0. \]

No matter how finely we define ‘meeting again,’ the system will re-enter an arbitrarily small neighbourhood of that state at some future time.

It may not repeat the past word for word, but it can come arbitrarily close.

In this sense, reunion need not mean that two people recognise each other at the same railway station. Years later, in another city, another piece of music or another person’s eyes, we may suddenly meet the selves we once were.

Poincaré recurrence guarantees no repetition of plot. It guarantees that a state approaches itself again.

Longing does not summon the return. It gives the return a name

Mathematics, of course, has not proved that every pair of lovers in the real world will meet again.

Poincaré recurrence has strict assumptions. The system must have finite measure and its evolution must preserve that measure. The result holds ‘almost everywhere,’ not without exception. It guarantees an arbitrarily close return, not that every particle will occupy exactly the same position. More importantly, the recurrence time may exceed the present age of the universe.

‘Every longing must meet again’ is therefore not the literal conclusion of the Poincaré recurrence theorem. It is a human translation of it.

What mathematics actually allows us to say is this:

In a finite, closed, measure-preserving world, almost every encounter that once occurred will, after enough time, come arbitrarily close to itself again.

Longing does not force the world to return.

It merely keeps the name of one region of state space alive in human consciousness. When the world passes through that region again, we can recognise it: this was the place I had been waiting for.

Perhaps reunion does not mean that time returns what was lost in its original form.

Perhaps, after a long evolution, the world finally comes close to those coordinates again. We have changed, but still recognise them in the smallest resemblance.

So if time is long enough, if that world continues to run as a closed system, and if we are willing to understand ‘reunion’ as an arbitrarily close return rather than an exact repetition, then this follows:

No encounter that truly happened has disappeared completely.

Every longing waits, somewhere deep in time, for its recurrence.