Poincaré Recurrence and Nietzsche's Eternal Return

In Friedrich Nietzsche's first book, The Birth of Tragedy, there is a story in which King Midas compels the satyr Silenus to tell him what is the very best and most excellent thing for mankind. Silenus, with a laugh, says: "Wretched ephemeral one, child of chance and sorrow, why do you compel me to tell you what it would be best for you not to hear? The very best thing is utterly beyond your reach: not to be born, not to be, to be nothing. But the second best thing for you is—to die soon."

Nietzsche's philosophy begins with this story. In The Birth of Tragedy, he raises two questions: Since life is so painful, why should we live at all? And since life is so repetitive, why would we be willing to repeat it?

Before Nietzsche, Arthur Schopenhauer had already described the human predicament: "Life is a mass of desires. When a desire is not satisfied, it is painful; when it is satisfied, it becomes boring. Life, therefore, swings like a pendulum between pain and boredom." In this world, happiness is outweighed by pain and boredom. Moreover, happiness is incredibly brief, whilst the majority of our time is spent in states of suffering or tedium. Boredom is the desire for a desire. Schopenhauer touched upon the most sensitive aspect of human nature: why do people continue to live when they are not happy? Nietzsche's philosophy also proceeds from this question.

"What if, some day or night, a demon were to steal after you into your loneliest loneliness and say to you: ‘This life as you now live it and have lived it, you will have to live once more and innumerable times more; and there will be nothing new in it, but every pain and every joy and every thought and sigh and everything unutterably small or great in your life will have to return to you, all in the same succession and sequence—even this spider and this moonlight between the trees, and even this moment and I myself. The eternal hourglass of existence is turned upside down again and again, and you with it, speck of dust!’"

One day in August 1881, whilst walking by a lake in Switzerland, this idea—Eternal Return (or Eternal Recurrence)—suddenly came to Nietzsche. After introducing it at the end of The Gay Science, he made it one of the fundamental concepts of his next work, Thus Spoke Zarathustra. Zarathustra, the prophet-like figure who proclaims Nietzsche's teachings in the book, is at first reluctant to express the idea, even to himself. Eventually, however, he declares the Eternal Return to be a joyous truth, one that should be embraced by anyone who has lived a full life.

Under the lens of this thought experiment, life would appear to become meaningless. We establish meaning based on the achievement of goals and accomplishments over a period of time. If life were an infinite cycle, it would mean that everything "yet" to happen has already happened, and everything "yet" to be completed is already destined. All actions lose their meaning in an infinite loop, collapsing into nihilism. People are like grains of sand trapped in an hourglass, eternally inverted with the frame of time, unable to make any substantive change to their lives. The nihilism of existence is exposed in this one absurd proposition.

Poincaré Recurrence

In the cryptic aeons, death itself will die.

With a standard 54-card deck of playing cards, it would take approximately 2.3×1072 shuffles on average to return to the initial order. Over a much longer period, the particles in the universe, undergoing a similar "shuffling," will also return to a state infinitesimally close to their initial positions. The atoms that now constitute your body will reassemble you, and the things you have experienced will be experienced again. The time this takes is the Poincaré recurrence time.

Simply put, the Poincaré recurrence theorem states that an isolated and finite system will, over the course of its evolution, return arbitrarily close to its initial state an infinite number of times.

To understand Poincaré recurrence, one must first have a basic grasp of phase space, the Poincaré recurrence theorem, and Liouville's theorem.

Phase Space: An n-dimensional phase space is analogous to rolling n dice simultaneously. The set of all possible outcomes of these n dice is the phase space, where every point in the space represents one possible state of the n dice.

In mechanics, the six-dimensional phase space for a single particle is equivalent to using six dice to describe its state in three-dimensional space. The six parameters required to define the trajectory of a known moving particle are its generalised coordinates (x, y, z) and its generalised momenta (Px, Py, Pz). The space spanned by these parameters is a type of phase space.

Liouville's Theorem: Liouville's theorem can be thought of as the conservation of information in classical mechanics.

Imagine taking a small, fixed volume element around a point in phase space. Since we cannot determine the initial state of all points in a system with perfect certainty, we consider a degree of uncertainty, measured by this volume element. As the system evolves, the points within this volume move. After a unit of time, the volume of the element containing these points remains unchanged. This demonstrates the conservation of information; the initial uncertainty, according to the laws of classical mechanics, does not change as the system evolves.

For example, if at time t=0 we select a circle in phase space (where every point in the circle is a state of the system), and we then track the motion of every point in that circle, we would find that as time progresses, the circle moves through the phase space. It might be stretched into an ellipse or a long, thin line, but its total area remains constant.

An isolated and finite system will, over the course of its evolution, return arbitrarily close to its initial state an infinite number of times. In simple terms, this is the recurrence of a finite system.

According to the law of increasing entropy, a closed system will always tend towards disorder. Counter-intuitively, however, the Poincaré recurrence theorem suggests that there will always come a moment when something miraculous happens in a gas chamber: all the air molecules will spontaneously return to one half of the chamber.

To use a more common example, if you place a sugar cube in a cup of coffee and stir, the sugar will dissolve. Yet, if you could continue to stir for an astronomically long time, the sugar cube could spontaneously re-form. An event with a probability that approaches zero will, given an infinite number of trials, necessarily occur. (This does not violate the second law of thermodynamics, which is based on statistics. The second law states that a system will, with very high probability, move towards greater entropy, but Poincaré recurrence deals with the probability of rare fluctuations).

It is just that the entire history of humanity, and even the lifespan of the universe, is infinitesimally short compared to the time required for such a spontaneous decrease in entropy to occur.

M-theory, however, allows for universes to be born spontaneously from nothing, a process that can occur any number of times without a creator.

According to the holographic principle, the amount of information contained within a specified region of spacetime is finite, proportional to its surface area. This means that the possible arrangements of matter, energy, and information within a given volume of spacetime are finite. Across any number of universes and over any length of time, any arrangement of matter, energy, and information can be reproduced an arbitrary number of times.

Therefore, supported by this concept of recurrence, even if we, the things around us, and the universe we inhabit are all destined for destruction, all of it will be faithfully reproduced in other universes, a process that can happen repeatedly. This is equivalent to saying that the universe and everything it contains are eternal in their flux. The universe is eternal, humanity is eternal, you and I are eternal. Existence is absolute and eternal; death is relative and transient.

Unfortunately, we as human beings cannot carry our consciousness from this universe's evolution across the Poincaré recurrence to the universe that follows. (The only possibility for crossing the divide would be for humanity to use an extremal black hole—one with balanced charge, mass, and angular momentum that does not decay via Hawking radiation—as a carrier of information).

Yet, this is perhaps a stroke of luck. As carbon-based life forms, humans could not endure the endless solitude of an empty vacuum. Even when the beauty of the cycle has no end, boundless rest would eventually transform into eternal suffering.

However, since we cannot change this despairing eternal return, let us, like Zarathustra—the seeker who, after a long struggle in the abyss of reason, appeals to existence itself—appeal to our own being!

After a fierce internal battle, Zarathustra makes his final choice: to affirm and embrace this highest truth of existence. He defeats the "dwarf"—a symbol of the pity for human existence deep in his soul—who had appeared to him in a vision. Relying on a courage or will he feels "in his body," he vanquishes all his despair, dispels the "dizziness at the abyss," and eradicates compassion and pity. He finally vanquishes death itself, "for it said: 'Was that life? Well then! Once more!'"

In his autobiography, Nietzsche called Thus Spoke Zarathustra a book "for none and for all," and specified that "the fundamental conception of this work is the idea of the eternal recurrence." The idea of "Eternal Return" can be seen as the zenith of Nietzsche's lifelong thought experiments and philosophical quests. It is the spiritual ground that this profoundly religious spirit found after sensing that "God is dead," illuminating modern experience with the lightning of his thought, exhausting the "circumference of the modern soul" with his acute deliberations, and being guided by an extreme reverence for life.

Nietzsche's "God is dead" negates the eternal hereafter constructed by traditional religion and Greek philosophy, causing the worldview and view of life built upon them to collapse. But after God's death, how can life be sustained? Through the mouth of Zarathustra, Nietzsche on the one hand preaches his idea of "Eternal Return," and on the other hand, fervently calls on humanity to bravely face this fact of existence, to abandon all nihilistic thoughts and attitudes, and to embrace and immerse itself in the experience of life itself.

Upon the ruins of this nihilism, Nietzsche reconstructs the structure of temporality to grant the "Übermensch" (Overman) the existential mode of the "eternal return of the same."

The traditional structure of time is linear, where time is a "river of nows": the "past" is a "now" that has vanished, the "present" is currently passing, and the "future" is a "now" that has not yet arrived. Linear time is hopeless, because in this view, everyone is a helpless "waiter for death," watching as life mercilessly slips away, waiting for its cruel and relentless passage. To escape the constant loss of "linear time," various peoples created religions of eternity, telling people of a timeless, eternal shore—a "heaven" or an "afterlife."

Nietzsche opposed traditional linear time—that is the time of the "Last Man." (Nietzsche constructed a sequence: "ape—man—Übermensch." The Übermensch is to man as man is to the ape. Man, caught in the middle, is not only a "transition" but also a "decadent," the "last man"). Nietzsche inaugurates a circular time centred on the moment-opportunity (the Augenblick)—a creative time, the time of the Übermensch.

According to Martin Heidegger's interpretation, "The heaviest and most authentic thing in the doctrine of eternal recurrence is this: eternity is in the moment. The moment is not the fleeting now, not a mere instant that flashes past for an observer, but the collision of the future and the past. In this collision, the moment comes to itself. The moment determines how everything returns."

In this three-dimensional, cyclically emergent structure of "circular time," how does the moment become eternal? For the acting individual Dasein (the human being who is always becoming, always transcending itself), the decisive opportunity of the present moment is paramount. Only for one situated in the "moment" can their actions penetrate the "future" whilst simultaneously accepting and affirming the "past." The seemingly lofty and abstract doctrine of "Eternal Return" in fact points to the present existence of the individual Dasein.

When Nietzsche proposed the idea of Eternal Return, he asked us not to treat it as a truth, but to ask ourselves what we would do if it were true. He assumed our first reaction would be utter despair—and indeed it would be. The human condition is tragic; life contains much suffering; the thought of having to relive it all countless times is desperate.

But then he imagined a different reaction. What if we could welcome the Eternal Return, embracing it as something we desire?

In every moment of the Dionysian experience of life, the existing being exists ecstatically, selflessly, overflowing with an absolute self-affirmation. In terms of the meaning of existence itself, this is not temporal; it is itself a form of eternity. If every moment of life embodies the true meaning of existence, or is lived ecstatically, then it has already overcome temporality within its own sphere. In this, it also represents the fundamental characteristic of existence itself and is homogenous with eternity.

At this point, Nietzsche's thought experiment has become an "attitude": You should live in every moment in such a way that you believe each moment is eternal, part of the Eternal Return.

This is the ultimate expression of Nietzsche's love for life and his desire to be "loyal to the earth": to want this life again and again, with all its pain, boredom, and frustration—moving from a love for this life, to a craving for this life, and finally to a resolution to choose this life. The philosophy of the Übermensch points towards the meaning of the earth and of life itself. It rejects any otherworldly ideals and calls for a state of vital self-affirmation and self-intoxication. Here there are no formulas, no concepts, no alienation—only the truest form of life itself. This idea is related to the main theme of Book Four of The Gay Science: the importance of becoming a "yea-sayer," an affirmer of life, and of embracing amor fati (love of one's fate).

The core of Nietzsche's faith in life shines through in Zarathustra's prayer-like words, echoing again and again:

The world is deep,

And deeper than day had ever been aware.

Deep is its pain—

Joy—deeper still than agony:

Pain says: "Pass away!"

But all joy wants eternity—

Wants deep, wants deep eternity!

The question in each and every thing, 'Do you desire this once more and innumerable times more?' would lie upon your actions as the greatest weight. Or how well disposed would you have to become to yourself and to life to crave nothing more fervently than this ultimate eternal confirmation and seal?