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17 Apr 2009

Nietzsche's Eternal Computer in Will to Power §1066



[We build from previous entries (hyperlinked throughout) in order to conclude that Nietzsche's eternal recurrence is a computational engine. The text is reproduced at the end.]


Friedrich Nietzsche

The Will to Power

§ 1066
(March-June 1888)



The world exists.
It does not become.
Nor does it pass away.
Or rather: it becomes, it passes away, but it has never begun to become and never ceased from passing away — it maintains itself in both. — It lives on itself: its excrements are its food. (548b)
It would make no sense to say that the world was created. Some provide a logical account for how the world was created from nothing. But they usually have theological motives. (548bc)

Eugene Dühring, for example, argues that even if time is infinite, it must have a beginning. The only way we can clearly conceive of infinity is if we imagine a continuous process of adding additional numerical units to a series. But this process had to begin before we could start the adding process. So even if the past stretches infinitely to now, it had to begin at some point. Hence, time has a beginning, Dühring concludes.

Nietzsche claims that Dühring makes a mistake. His sense of the future might be correct. It never ends, because there is always something more. But his portrayal of the infinite past is erroneous. For him, the infinite past began at some point, and continued infinitely until now. That of course could not be infinite, because there are two determinate points, the beginning point and the now point. So he wrongly considers a finite progress to be an infinite regress. Indeed, the future is infinite, because we can begin from now, and perpetually add more moments. Nietzsche says we can do the same in reverse for the past. We can begin now and continually subtract moments infinitely into the past.
Only if I made the mistake — I shall guard against it — of equating this correct concept of a regressus in infinitum with an utterly unrealizable concept of a finite progressus up to this present, only if I suppose that the direction (forward or backward) is logically a matter of indifference, would I take the head – this moment – for the tail: I shall leave that to you, my dear Herr Dühring!— (548d)
Nietzsche argues that the universe always must have been in a state of becoming. If it ever did stop, there would be no imbalancing forces to get it going again. Since we still experience a state of flux, there must never have been complete stability in the past.
If the world could in any way become rigid, dry, dead, nothing, or if it could reach a state of equilibrium, or if it had any kind of goal that involved duration, immutability, the once-and-for-all (in short, speaking metaphysically: if becoming could resolve itself into being or into nothingness), then this state must have been reached: from which it follows— (548-549) [again, see this entry for more on Nietzsche's infinite time theory.]
Lord Kelvin worried that entropy would cause all the forces in the universe to someday spend all their energy, causing the world to die [see this entry for more]. Nietzsche says that if this is true, then that means it is possible for there to be equilibrium. But if it were possible, it should have happened already. So if a mechanistic theory concludes that eventually there will be stasis, it is refuted by the very fact that it has not yet occurred and is thus an impossibility. (549b)

Now Nietzsche will describe the eternal return as being both pure chance and yet also an infinite repetition of the same. It's truly a marvel of computational engineering, the ultimate abstract machine of the digital variety. Purely wild, purely unknown, yet it always is the same thing over-and-over again. And we do not just mean it is difference over-and-over, or chance over-and-over. It is not so simple as "the only thing that remains the same is change." Rather, the same combinations of events recur as exactly the same on infinite scales and yet each individual event is decided by chance.

Before continuing, let's first recall Nietzsche's description of the Pythagorean cosmic eternal recurrence. They believed that there were rings of stars in the cosmos that move in circles.
Whenever the stars once more attain the same position, not only the same people but also the same behavior will again occur. (139c)

To better grasp Nietzche's complex theory, we tread carefully through the sixth paragraph of §1066, moving sentence-by-sentence.

If the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force — and every other representation remains indefinite and therefore useless — it follows that, in the great dice game of existence, it must pass through a calculable number of combinations. (549b)
The cosmos always has the same net amount of force. But, it is distributed unevenly. However, the forces are not distributed to an infinite number of places. Rather, the total amount of force is finite, so it must find itself in a "definite number of centers of force."

So, because the quantities and locations are finite and determinate, there are only so many possible arrangements.
In infinite time, every possible combination would at some time or another be realized; more: it would be realized an infinite number of times. (549bc)
So the gods together shake an enormous number of dice, and cast them all together. The outcomes decide a series of events. We are a part of this series. The gods rolled the dice this round in such a way that we were born and came to the present moment as we are now. If just one of those countless dice rolled differently, some detail would have been different. Perhaps everything would be exactly the same, except we all have six fingers instead of five. As you can see, there are so many small ways that things could have been different, even on the atomic level. It's impossible to begin imagining every other possibility. We will have to oversimplify. Let's say that when the gods rolled-out the sequence of events that we live-in now, we call that roll '1.' There are countless other possible rolls, but still a finite number. We will consider just one other possibility and keep in mind that really that are very many others that we are not considering for the moment. In this other roll, everything else is the same as now, except we have six fingers. We call this roll '0.'

The gods have been rolling infinitely long before us. We need to imagine an endless series of rounds of rolls. Every round is in the middle of the series, because the sequence extends infinitely into the past and future. So we need to think of a series like this, but one that really has no beginning or end.


But let's begin with the present roll. They rolled a 1. It's us. We have five fingers.



[Each of the rolls we take from a random number generator.] Events in the cosmos go on-and-on according to all the many dice-throws that make-up this roll. There is a finite number of those rolls, which takes the cosmos so far in its becoming. Then, they roll again. This time, a zero.



Nietzsche writes:
And since between every combination and its next recurrence all other possible combinations would have to take place, and each of these combinations conditions the entire sequence of combinations in the same series, a circular movement of absolutely identical series is thus demonstrated: the world as a circular movement that has already repeated itself infinitely often and plays its game in infinitum. (549c)
We have not yet encountered a recurrent combination. So let's see the next roll.



The gods roll a one again. Everything is exactly the same as now. We live every detail of our lives again the same way, with our five fingers, and every other detail of the cosmos is the same as well. So here we have "a combination and its next recurrence."



As we saw, Nietzsche writes that there will be a "circular movement of absolutely identical series." This could be so if the gods rolled 1010101010101.... from now on, and have rolled that way up to now. But that's extraordinarily improbable. Each time it is left to chance. So let's see what they actually roll the next time.



We see already that the pattern of 101010101... was broken. However, perhaps there is a higher-order repetition. Given this next roll, it could be that 101 repeats, to be 101101101101101... This would fulfill Nietzsche's criteria that the repetition be "absolutely identical." In fact, he writes that each of the combinations "conditions the entire sequence of combinations in the same series." So when we roll 101, this imposes the condition that the next set of repetitions will need to have a 101, at least somewhere in the series. So we look now for this block to recur.



Next they roll a zero.



That is promising. Then they roll a one.



Perfect. So now it will be absolutely identical if they continue rolling 10110101101...
Let's see if they do. They roll a one. That's good.



Then a zero, also good.



All we need now is a one and then we can say we have an absolutely identical series. But, chance decided otherwise. They roll another zero.



We might ask when the next recurrence of 101101 is. That will give us another longer series, and we can see if that is the series that repeats eternally. Here are some more rolls.



Still not what we need. It is not until much later that 101101 repeats. But sure enough, it does.



Let's now consider this whole sequence from end-to-end. Then let's see if it as a whole series repeats again after more dice throws.


But given its complexity, the entire sequence does not repeat again any time soon. It is not until much later that we see this whole block again.



So then we might ask, when will that whole block repeat?



It will repeat, given that it has an infinite number of chances to do so. However, it will not be for very many more rolls, so many that we cannot show it here. But it will repeat. And that will create another much longer sequence. And that even longer sequence will repeat too. Over-and-over again, there will show to be higher orders of identical series that repeat themselves. And because the series is infinite, there is no end to this repetition of same series, each one found in an even longer such repetition.

Now consider that we arbitrarily started with the roll for the present time. But the series has already been going on infinitely. So even the present roll is right now instantiating the infinite repetition of absolutely and precisely the exact same sequences. The gods rolled a one this time. That one is part of an infinite number of series that repeat exactly. Not because the gods roll the same sequences over and over like A B A B A B A.... No, it's because, as Nietzsche writes, "each of these combinations conditions the entire sequence of combinations in the same series." In other words, each new roll is purely produced by chance. But that chance-roll influences the individuality of the higher-and-higher orders of absolutely same sequences that infinitely repeat. So the gods rolled a one now. Then they rolled a zero. So we know that "1-0" will repeat again. Future repetitions must have that sequence "10". But the next rolls were decided by chance, not by algorithms. So they gave us "1011" instead. Now this longer series conditions the rest of the sequence. It must have "1011" again. Each new roll is by chance, and conditions what will be the same. But note also that this has been going on infinitely long. That means, past rolls conditioned that the gods would roll a 1 sometime in the future, with that time happening to be the present.

We can see then that Nietzsche's circular movement of eternal recurrence is not circular like A B A B. It is circular in a fractal way. Each repetition is at a higher order. So the eternal recurrence is a recurrence happening each time at a higher power, order, exponent, level, etc. In each instance is implied the infinity of its recurrences. [In this way, each time is a first time that is carried to infinitely higher levels, that is, to the nth power.]

Nietzsche ends by writing that the eternal recurrence is not a "mechanistic conception." (549c) There are no laws that determine what the next roll will be. For, that would "not condition an infinite recurrence of identical cases, but a final state." Instead, we see that because each roll is pure chance, there is no final state. Yet, although there is no final state, there still is an absolutely-identical perfect repetition of the same. And it all makes logical sense, when we combine these principles:
a) finite possibilities,
b) pure chance,
c) infinite time.

That is the recipe for eternal recurrence.

And we see also that it is a computational process. He says in his Pythagoreans lecture that they believed becoming is a calcating. However, they were not able to say what was doing that calculating. Here we have Nietzsche's answer. The eternal recurrence performs this computation. Each roll is another computational operation. This produces longer-and-longer series of combinations, each having infinitely many more recurrences, every time at a higher order. Out of the infinity emerges pure becoming as nothing more than wild chance computation calculated eternally.

We see then why we must affirm chance. Nothing is determined. But whatever happens, occurs infnitely more times. Because the chain of rolls is infinitely long, there is no pattern to allow us to predict the rest of the rolls based on a finite sample. However, we can be sure that whatever happens in any finite sample will repeat infinitely, and not in a reduntant way. It is repetition without redudency. It produces novelty despite absolutely identical cycles. And it is only on account of higher-order pattern-emergences ascending to infinite levels that this marvellous computer of becoming may operate and give infinite meaning to our lives.



From the original Kaufmann translation, obtained gratefully from the Athenaeum Reading Room:

1066
(March-June 1888)

The new world-conception.— The world exists; it is not something that becomes, not something that passes away. Or rather: it becomes, it passes away, but it has never begun to become and never ceased from passing away — it maintains itself in both.— It lives on itself: its excrements are its food.

We need not worry for a moment about the hypothesis of a created world. The concept "create" is today completely indefinable [This word is illegible.], unrealizable; merely a word, a rudimentary survival from the ages of superstition; one can explain nothing with a mere word. The last attempt to conceive a world that had a beginning has lately been made several times with the aid of logical procedures — generally, as one may divine, with an ulterior theological motive.

Lately one has sought several times to find a contradiction in the concept "temporal infinity of the world in the past" (regressus in infinitum): one has even found it, although at the cost of confusing the head with the tail. Nothing can prevent me from reckoning backward from this moment and saying "I shall never reach the end"; just as I can reckon forward from the same moment into the infinite. Only if I made the mistake — I shall guard against it — of equating this correct concept of a regressus in infinitum with an utterly unrealizable concept of a finite progressus up to this present, only if I suppose that the direction (forward or backward) is logically a matter of indifference, would I take the head — this moment — for the tail: I shall leave that to you, my dear Herr Dühring!—

I have come across this idea in earlier thinkers: every time it was determined by other ulterior considerations (—mostly theological, in favor of the creator spiritus). If the world could in any way become rigid, dry, dead, nothing, or if it could reach a state of equilibrium, or if it had any kind of goal that involved duration, immutability, the once-and-for-all (in short, speaking metaphysically: if becoming could resolve itself into being or into nothingness), then this state must have been reached: from which it follows—

This is the sole certainty we have in our hands to serve as a corrective to a great host of world hypotheses possible in themselves. If, e. g., the mechanistic theory cannot avoid the consequence, drawn for it by William Thomson [*], of leading to a final state, then the mechanistic theory stands refuted.

If the world may be thought of as a certain definite quantity of force and as a certain definite number of centers of force — and every other representation remains indefinite and therefore useless — it follows that, in the great dice game of existence, it must pass through a calculable number of combinations. In infinite time, every possible combination would at some time or another be realized; more: it would be realized an infinite number of times. And since between every combination and its next recurrence all other possible combinations would have to take place, and each of these combinations conditions the entire sequence of combinations in the same series, a circular movement of absolutely identical series is thus demonstrated: the world as a circular movement that has already repeated itself infinitely often and plays its game in infinitum.

This conception is not simply a mechanistic conception; for if it were that, it would not condition an infinite recurrence of identical cases, but a final state. Because the world has not reached this, mechanistic theory must be considered an imperfect and merely provisional hypothesis.

[*] First Baron Kelvin (1824-1907), British physicist and mathematician who introduced the Kelvin or Absolute Scale of temperature.

Nietzsche, Friedrich. The Will to Power. Ed. Walter Kaufmann. Transl Walter Kaufmann and R.J. Hollingdale. New York: Random House Vintage Books, 1967.
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