Though Nietzsche briefly touches on ‘eternal recurrence’ a few more times in other works, the quotes given above are the most fundamental, and stand out like two supporting pillars of the doctrine. Right off the bat one can see from these quotes, ‘eternal recurrence’ is a thread woven out of thin air, a simple fantasy presented without any argument. It is a tale, a story told as part of the narrative of that old oracle, Zarathustra, along with his eagle, serpent, spider and other animals. This narrative, which at its very best, is only a fabrication of the writer’s imagination, does not amount to a reasoned argument in support of a theory or doctrine. In the narrative, ‘eternal recurrence’ is just a hypothesis put forward by the author. None of the exhaustive arguments, axioms, theorems, syllogisms, etc., required to prove or support a ‘philosophical’ theory, are ever given either here, or any other works of Nietzsche. Eternal Recurrence is just an idea, a concept, thrown at us much like a ghost story. There are those, who, like little children, believe any ghost story at face value. But there are also some of us who demand a more substantive and objective proof before we can accept an idea.

The strongest of the ‘pseudo scientific’ arguments, put forward to support this doctrine by the followers of Neitzsche, is a mixture of statistical mathematics, physics and astrophysics. It goes something like this:- ‘…Time is infinite, an endless eternity, but since space and matter in the universe are finite, limited, all the matter in the universe, therefore, can be combined, arranged and rearranged in a finite number of permutations. Given the eternity of time, these permutations must therefore, repeat themselves over and over again, and must already have repeated themselves many, many times in the eternal past. And they will also continue to repeat themselves going in circles in the eternity of the future. Bingo! Therefore, they say: eternal recurrence is a scientific fact! Q.E.D.

For those who don’t have a strong background in math and statistics, and are unfamiliar with terminology like permutation, here is a simplified version of what permutation means and a brief introduction to the concept. If we toss a coin, the most probable outcome of heads or tails coming up is fifty/fifty, (fifty percent.) That means, since the coin has only two sides, a throw of one hundred times will probably result in heads and tails coming fifty times each. This is only ‘probability’, though. There is no absolute way of knowing for sure. To simplify the example, let us examine only ten throws. According to the statistical probability we are very likely to get five heads and five tails. But from experience we know it is also possible to get all ten heads and no tails. Or nine heads and one tails, or eight heads and two tails, or all the other combinations of heads and tails up to one heads and nine tails, and no heads and ten tails. That is why mathematics can only give us the statistical ‘probability.’ Taking each of these combination of heads and tails and performing some mathematical operation would then give us the famous ‘bell curve’ which shows the percentage probability of each combination of heads and tails. The fifty/fifty heads to tails ratio would have the highest probability at the top of the dome, right on the top of the bell. The nine heads to one tail and nine tails to one heads would be at each wing of the bottom of the bell curve, having the lowest probability of occurrence.

Having bored the reader with all these mathematical concepts, we are still left without any way of knowing with ‘certainty’ what the next coin throw will produce. It can be a head or tail. Not only that, but the next ten throws can be all heads and no tails, or all tails and no heads. Put bluntly: we have no way of knowing what the future holds. That precisely, is the reason why fortune-tellers, Tarot card readers, the entire horoscope industry, the Las Vegas slot machines, the horse races, the dog races, and all other forms of gambling will always be in business.

Regardless of the odds, the gambler keeps hoping, and betting that the next few coin tosses or dice throws will bring a series of results favorable to him. Occasionally, this turns out to be true, and the gambler wins, proving all the statistical probability wrong! In the case of the dice, since it has six sides, each face has a one sixth probability of coming up on top. That is a smaller percentage than the coin. That is only 16.66% as compared to 50%. But when we apply the mathematical operation on the combinations of the dice, we still end up with the bell curve. The 16.66% for every face on the dice coming up will be on the top of the bell, but all other combinations will be spread on the bell curve from the top all the way to the bottom fringes.

To simplify the theory of eternal recurrence further and give one more concrete example, suppose we have a necklace with an arrangement of a variety of colored beads on a string. These colored beads can be arranged in a sequence: red, blue, yellow, orange, green, purple… etc. Suppose that we keep a record, and change the sequence of colors after wearing the necklace every day. Since there are a limited number of beads on the necklace, we soon find that we will have exhausted all possible sequence of colors, and would need to repeat a sequence previously used. If we continue this for a long time, we will have pattern after pattern repeating again and again. That, in short, is the sum total of the doctrine of ‘eternal recurrence’, and its proof. The colored beads would represent ‘matter’ while the string in which the beads are threaded represents ‘space.’

From this simplistic ‘pseudo scientific’ explanation, one is expected to arrive at that horrendous conclusion that we must have lived this same life many times in the past, and everything goes on and on in an ever repeating ‘single’ circle. So we have no alternative or power to do otherwise, but to come again and again and relive this very same life over and over again through all eternity. That is the basis of the doctrine or philosophy, if you will, of ‘Eternal Recurrence.’ God can then be seen as a child with a single film, audio or videocassette, and who replays that single tune or film again, and again, for eternity.

http://www.meskot.com/recurrence.htm

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 universe (see 1 E19 s and more).

Controversial theoretical physicist Peter Lynds suggested a model of eternal recurrence in a 2006 paper.[11] Lynds hypothesizes that if the universe undergoes a big crunch, the arrow of time may reverse. Others have approached the question of eternal recurrence from a physics perspective in different ways, including a hypothesis based on the Transactional interpretation of quantum mechanics.[12]

http://en.wikipedia.org/wiki/Eternal...dern_cosmology

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