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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, though Friedrich Nietzsche resurrected it as a thought experiment to argue for 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. If either time or space is infinite, then mathematics 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 proceeded to show that the eternal recurrence was a mathematical certainty. In the post-Einstein period, there are doubts that time or space is in fact infinite, but many models exist which provide 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. Stephen Hawking'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). 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.
Classical antiquity 
Indian religions 
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.
However it is to be noted that the cycle of life in Buddhism does not involve a soul passing from one body to another, but the karma of the deceased being carrying on to another being born. To get rid of this cycle the person should get rid of its karma through the attainment of enlightenment. (See: Rebirth (Buddhism).)
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, view of time. But cyclical time and return are reflected in Jewish traditions such as: The history of the Jewish people is said to consist of lessons that were needed to be relearned and retaught. By implication these are repeated events that occurred in the lives of its biblical forefathers; people’s lives in the next world, or after death, are to some extent spiritual repetitions of what they did in this world; and some kabbalists wrote  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;
Friedrich Nietzsche 
The concept of "eternal recurrence" 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. It is 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."
Poincaré recurrence theorem 
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).
Modern cosmology 
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).
Modern cosmological proposals have also been made by Peter Lynds, who suggested a model of eternal recurrence in a 2006 paper, and by James Quirk, who approached the question through a hypothesis based on the transactional interpretation of quantum mechanics.
Arguments against eternal return 
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.
See also 
- Jean-Pierre Luminet (2008-03-28). The Wraparound Universe. AK Peters, Ltd. ISBN 978-1-56881-309-7
- August Thalheimer: Introduction to Dialectical Materialism from Google Cache
- citiation needed
- Zev Golan, "God, Man and Nietzsche: A Startling Dialogue between Judaism and Modern Philosophers", iUniverse, 2007, pp4-14
- See Heidegger Nietzsche. Volume II: The Eternal Recurrence of the Same trans. David Farrell Krell. New York: Harper and Row, 1984. 25.
- 1881, (11 )
- Nietzsche, Ecce Homo, "Why I Write Such Good Books", "Thus Spoke Zarathustra", §1
- Walter Benjamin. The Arcades Project. Trans. Howard Eiland and Kevin McLaughlin. Cambridge: Belknap-Harvard, 2002. See chapter D, "Boredom Eternal Return," pp. 101-119.
- See Posthumous fragment, 11  1881; See also Mazzino Montinari, Friedrich Nietzsche, 1974 (German transl. De Gruyter, 1991, French translation PUF, 2001) and also Nietzsche's personal library (see also "?".[dead link] and "revision of previous catalogues".[dead link] on the École Normale Supérieure's website)
- Alfred Fouillée, "Note sur Nietzsche et Lange: le "retour éternel", in Revue philosophique de la France et de l'étranger. An. 34. Paris 1909. T. 67, S. 519-525 (French)
- Kaufmann, Walter. Nietzsche; Philosopher, Psychologist, Antichrist. 1959, page 376.
- Kundera, Milan. The Unbearable Lightness of Being. 1999, page 5.
- Dudley, Will. Hegel, Nietzsche, and Philosophy: Thinking Freedom. 2002, page 201.
- Golan, God, Man and Nietzsche, pp2-3
- Rüdiger Vaas: "Ewig rollt das Rad des Seins": Der 'Ewige-Wiederkunfts-Gedanke' und seine Aktualität in der modernen physikalischen Kosmologie. In: Helmut Heit, Günter Abel, Marco Brusotti (eds.): Nietzsches Wissenschaftsphilosophie. de Gruyter: Berlin, New York 2012, S. 371-390. ISBN 9783110259377 content
- Alex Vilenkin: Many Worlds in One. New York: Hill and Wang 2006
- "Parallel universes. Not just a staple of science fiction, other universes are a direct implication of cosmological observations.", Tegmark M., Sci Am. 2003 May;288(5):40-51.
- Max Tegmark (2003). "Parallel Universes". In "Science and Ultimate Reality: from Quantum to Cosmos", honoring John Wheeler's 90th birthday. J. D. Barrow, P.C.W. Davies, & C.L. Harper eds. Cambridge University Press (2003). arXiv:astro-ph/0302131. Bibcode:2003astro.ph..2131T.
- Tegmark, Max (May 2003). "Parallel Universes". Scientific American: 41.
- Lynds, "On a finite universe with no beginning or end"
- "Circular causality: A physical hypothesis of eternal recurrence"
- Kaufmann, Walter. Nietzsche: Philosopher, Psychologist, Antichrist. (Fourth Edition) Princeton University Press, 1974. p327
- Paolo D'Iorio, "The Eternal Return: Genesis and Interpretation", in The Agonist, vol. III, issue I, spring 2011.
- Hatab, Lawrence J. (2005). Nietzsche's Life Sentence: Coming to Terms with Eternal Recurrence. New York: Routledge. ISBN 0-415-96758-9.
- Lorenzen, Michael (2006). "The Ideal Academic Library as Envisioned through Nietzsche's Vision of the Eternal Return". MLA Forum 5, no. 1.
- Lukacher, Ned (1998). Time-Fetishes: The Secret History of Eternal Recurrence. Durham, N.C.: Duke University Press. ISBN 0-8223-2253-6.
- Magnus, Bernd (1978). ISBN 0-253-34062-4. Bloomington: Indiana University Press.
- Jung, Carl (1988). Nietzsche's Zarathustra: Notes of the Seminar Given in 1934 - 1939 (2 Volume Set). Princeton University Press. ISBN 978-0-691-09953-8.