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Eternal return (also known as "eternal recurrence") is a concept that the universe has been recurring, and will continue to recur, in a self-similar form an infinite number of times across infinite time or space. The concept is found in Indian philosophy and in ancient Egypt and was subsequently taken up by the Pythagoreans and Stoics. With the decline of antiquity and the spread of Christianity, the concept fell into disuse in the Western world, with the exception of Friedrich Nietzsche, who connected the thought to many of his other concepts, including amor fati.
In addition, the philosophical concept of eternal recurrence was addressed by Arthur Schopenhauer. It is a purely physical concept, involving no supernatural reincarnation, but the return of beings in the same bodies. Time is viewed as being not linear but cyclical.
The basic premise proceeds from the assumption that the probability of a world coming into existence exactly like our own is greater than zero (we know this because our world exists). If space is infinite, then cosmology tells us that our existence will recur an infinite number of times.
In 1871, Louis Auguste Blanqui, assuming a Newtonian cosmology, where time and space are infinite, claimed to have shown that the eternal recurrence was a mathematical certainty. In the post-Einstein period, there were doubts that time or space was in fact infinite, but many models existed which provided the notion of spatial or temporal infinity required by the eternal return hypothesis.
The oscillatory universe model in physics could be provided as an example of how the universe cycles through the same events infinitely. Arthur Eddington's concept "arrow of time", for example, discusses cosmology as proceeding up to a certain point, after which it undergoes a time reversal (which, as a consequence of T-symmetry, is thought to bring about a chaotic state due to entropy). Stephen Hawking and J. Richard Gott have also proposed models by which a universe could undergo time travel, provided the balance between mass and energy created the appropriate cosmological geometry.
The concept of cyclical patterns is very prominent in Indian religions, such as Jainism, Hinduism, Sikhism and Buddhism among others. The wheel of life represents an endless cycle of birth, life, and death from which one seeks liberation. In Tantric Buddhism, a wheel of time concept known as the Kalachakra expresses the idea of an endless cycle of existence and knowledge.
Judaism posits a creation narrative "In the beginning" and a redeemed "Olam Haba" at the end, which means Judaism has a linear, not a cyclical, at least regarding the physical world. However, as the Creator is eternal and without beginning or end, so is the Jewish concept of time. Additionally, according to the Kabbalah, time is circular and as such, historical events in the history of the Jewish people are said to repeat themselves spiritually, and the influence is transmitted year after year, no matter the "distance" from the original event. As a result, the actions in a person's life directly affect their life in the next world, or after death, and are direct manifestations of the spiritual influences of what they did in this world. (see the Kabbalah of the Ariza'l).  There is also a perspective that time is composed of seven cycles, which repeat every seven thousand years (a view rejected by Isaac Luria). These concepts give human choices to do good deeds in Olam HaZeh — "this world" - some of what Nietzsche called the eternal recurrence's "infinite weight".
The symbol of the Ouroboros, the snake or dragon devouring its own tail, is the alchemical symbol par excellence of eternal recurrence. The alchemist-physicians of the Renaissance and Reformation were aware of the idea of eternal recurrence; the physician-philosopher Sir Thomas Browne in his A Letter to a Friend c. 1657 linked the Uroboros symbol with the idea of eternal return thus -
that the first day should make the last, that the Tail of the Snake should return into its Mouth precisely at that time, and they should wind up upon the day of their Nativity, is indeed a remarkable Coincidence, which tho Astrology hath taken witty pains to salve, yet hath it been very wary in making Predictions of it.
An allusion to eternal recurrence also occurs at the conclusion of Browne's The Garden of Cyrus-
All things began in order, so shall they end, and so shall they begin again;
The concept of "eternal recurrence", the idea that with infinite time and a finite number of events, events will recur again and again infinitely, is central to the writings of Friedrich Nietzsche. As Heidegger points out in his lectures on Nietzsche, Nietzsche's first mention of eternal recurrence, in aphorism 341 of The Gay Science (cited below), presents this concept as a hypothetical question rather than postulating it as a fact. According to Heidegger, it is the burden imposed by the question of eternal recurrence—whether or not such a thing could possibly be true—that is so significant in modern thought: "The way Nietzsche here patterns the first communication of the thought of the 'greatest burden' [of eternal recurrence] makes it clear that this 'thought of thoughts' is at the same time 'the most burdensome thought.' "
The thought of eternal recurrence appears in a few of his works, in particular §285 and §341 of The Gay Science and then in Thus Spoke Zarathustra. The most complete treatment of the subject appears in the work entitled Notes on the Eternal Recurrence, a work which was recently published alongside Søren Kierkegaard's own version of eternal return, which he calls 'repetition'. Nietzsche sums up his thought most succinctly when he addresses the reader with: "Everything has returned. Sirius, and the spider, and thy thoughts at this moment, and this last thought of thine that all things will return". However, he also expresses his thought at greater length, and perhaps in its finest form, when he says to his reader:
"Whoever thou mayest be, beloved stranger, whom I meet here for the first time, avail thyself of this happy hour and of the stillness around us, and above us, and let me tell thee something of the thought which has suddenly risen before me like a star which would fain shed down its rays upon thee and every one, as befits the nature of light. - Fellow man! Your whole life, like a sandglass, will always be reversed and will ever run out again, - a long minute of time will elapse until all those conditions out of which you were evolved return in the wheel of the cosmic process. And then you will find every pain and every pleasure, every friend and every enemy, every hope and every error, every blade of grass and every ray of sunshine once more, and the whole fabric of things which make up your life. This ring in which you are but a grain will glitter afresh forever. And in every one of these cycles of human life there will be one hour where, for the first time one man, and then many, will perceive the mighty thought of the eternal recurrence of all things:- and for mankind this is always the hour of Noon".
This thought is indeed also noted in a posthumous fragment. The origin of this thought is dated by Nietzsche himself, via posthumous fragments, to August 1881, at Sils-Maria. In Ecce Homo (1888), he wrote that he thought of the eternal return as the "fundamental conception" of Thus Spoke Zarathustra.
Several authors have pointed out other occurrences of this hypothesis in contemporary thought. Rudolf Steiner, who revised the first catalogue of Nietzsche's personal library in January 1896, pointed out that Nietzsche would have read something similar in Eugen Dühring's Courses on philosophy (1875), which Nietzsche readily criticized. Lou Andreas-Salomé pointed out that Nietzsche referred to ancient cyclical conceptions of time, in particular by the Pythagoreans, in the Untimely Meditations. Henri Lichtenberger and Charles Andler have pinpointed three works contemporary to Nietzsche which carried on the same hypothesis: J.G. Vogt, Die Kraft. Eine real-monistische Weltanschauung (1878), Auguste Blanqui, L'éternité par les astres (1872) and Gustave Le Bon, L'homme et les sociétés (1881). Walter Benjamin juxtaposes Blanqui and Nietzsche's discussion of eternal recurrence in his unfinished, monumental work The Arcades Project. However, Gustave Le Bon is not quoted anywhere in Nietzsche's manuscripts; and Auguste Blanqui was named only in 1883. Vogt's work, on the other hand, was read by Nietzsche during this summer of 1881 in Sils-Maria. Blanqui is mentioned by Albert Lange in his Geschichte des Materialismus (History of Materialism), a book closely read by Nietzsche. The eternal recurrence is also mentioned in passing by the Devil in Part Four, Book XI, Chapter 9 of Dostoevsky's The Brothers Karamazov, which is another possible source that Nietzsche may have been drawing upon.
[T]ime is infinite, but the things in time, the concrete bodies, are finite. They may indeed disperse into the smallest particles; but these particles, the atoms, have their determinate numbers, and the numbers of the configurations which, all of themselves, are formed out of them is also determinate. Now, however long a time may pass, according to the eternal laws governing the combinations of this eternal play of repetition, all configurations which have previously existed on this earth must yet meet, attract, repulse, kiss, and corrupt each other again...
Nietzsche calls the idea "horrifying and paralyzing", referring to it as a burden of the "heaviest weight" ("das schwerste Gewicht") imaginable. He professes that the wish for the eternal return of all events would mark the ultimate affirmation of life:
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' ... Would you not throw yourself down and gnash your teeth and curse the demon who spoke thus? Or have you once experienced a tremendous moment when you would have answered him: 'You are a god and never have I heard anything more divine.' [The Gay Science, §341]
My formula for human greatness is amor fati: that one wants to have nothing different, not forward, not backward, not in all eternity. Not merely to bear the necessary, still less to conceal it—all idealism is mendaciousness before the necessary—but to love it.
In Carl Jung's seminar on Thus Spoke Zarathustra, Jung claims that the dwarf states the idea of the Eternal Return before Zarathustra finishes his argument of the Eternal Return when the dwarf says, "'Everything straight lies,' murmured the dwarf disdainfully. 'All truth is crooked, time itself is a circle.'" However, Zarathustra rebuffs the dwarf in the following paragraph, warning him against over-simplifications.
A late 1880s comment by Nietzsche, "In an infinite period of time, every possible combination would at some time be attained," has been cited to argue that Nietzsche dropped his plans to try to scientifically prove the theory because he realized that if he would have to eventually repeat life as it is, his presumption of infinite time means "he" would also have to "repeat" life differently, since every configuration of atoms and events will occur. Instead, according to this interpretation of Nietzsche, he continued to propound the doctrine for its psychological and philosophical import. Though section 1063 of his posthumous notebooks "The Will To Power" states, “The law of conservation of energy demands eternal recurrence."
Related to the concept of eternal return is the Poincaré recurrence theorem in mathematics. It states that a system whose dynamics are volume-preserving and which is confined to a finite spatial volume will, after a sufficiently long time, return to an arbitrarily small neighborhood of its initial state. "A sufficiently long time" could be much longer than the predicted lifetime of the observable universe (see 1 E19 s and more).
The philosopher and writer Albert Camus explores the notion of "eternal return" in his essay on "The Myth of Sisyphus," in which the repetitive nature of existence comes to represent life's absurdity, something the hero seeks to withstand through manifesting what Paul Tillich called, "The Courage to Be." Though the task of rolling the stone repeatedly up the hill without end is inherently meaningless, the challenge faced by Sisyphus is to refrain from despair. Hence Camus famously concludes that, "one must imagine Sisyphus happy."
While the big bang theory in the framework of relativistic cosmology seems to be at odds with eternal return, there are now many different speculative big bang scenarios in quantum cosmology which actually imply eternal return - although based on other assumptions than Nietzsche's. So there are competing models and hypotheses with a temporal, spatial or spatio-temporal eternal return of everything in all variations as Nietzsche has envisaged.
The oscillating universe theory - that the universe will end in a collapse or 'big crunch' followed by another big bang, and so on - dates from 1930. Cosmologists such as professor Alexander Vilenkin from Tufts University and Massachusetts Institute of Technology professor Max Tegmark suggest that if space is sufficiently large and uniform, or infinite as some theories suggest, and if quantum theory is true such that there is only a finite number of configurations within a finite volume possible, due to Heisenberg's uncertainty principle, then identical instances of the history of Earth's entire Hubble volume occur every so often, simply by chance. Tegmark calculates that our nearest so-called doppelgänger, is 1010115 meters away from us (a double exponential function larger than a googolplex). In principle, it would be impossible to scientifically verify an identical Hubble volume. However, it does follow as a fairly straightforward consequence from otherwise unrelated scientific observations and theories. Tegmark suggests that statistical analyses exploiting the anthropic principle provide an opportunity to test multiverse theories in some cases. Generally, science would consider a multiverse theory that posits neither a common point of causation, nor the possibility of interaction between universes, to be an ideal speculation. However, it is a fundamental assumption of cosmology that the universe continues to exist beyond the scope of the observable universe, and that the distribution of matter is everywhere the same at such a large scale (see cosmological principle).
Nietzsche scholar Walter Kaufmann has described an argument originally put forward by Georg Simmel, which rebuts the claim that a finite number of states must repeat within an infinite amount of time:
Even if there were exceedingly few things in a finite space in an infinite time, they would not have to repeat in the same configurations. Suppose there were three wheels of equal size, rotating on the same axis, one point marked on the circumference of each wheel, and these three points lined up in one straight line. If the second wheel rotated twice as fast as the first, and if the speed of the third wheel was 1/π of the speed of the first, the initial line-up would never recur.
Thus a system could have an infinite number of distinct physical configurations that never recur. However the example presupposes the possibility of perfect continuity: for instance, if the universe proves to have a quantum foam nature, then the exact quantity of an irrational number cannot be expressed by any physical object.