The Perpetual Novelty of Sisyphus

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Wikipedia: Gustave Doré’s depiction of Canto VII of Dante’s Inferno turns the rocks that the damned hoarders and wasters are forced to move around, Sisyphus-like, into giant bags of gold, emphasising the reason for their punishment.

lbert Camus once said, “one must imagine Sisyphus happy”. That seems impossible to do if one feels deeply the meaningless monotony of struggling to roll the same massive boulder up the same hill each day, forever.

The myth of Sisyphus is a powerfully visceral metaphor for the concept of eternal recurrence (also known as eternal return and eternal repetition):

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.
Notes on the Eternal Recurrence — Vol. 16 of Oscar Levy Edition of Nietzsche’s Complete Works (in English)

The dread that is often evoked by the contemplation of eternal recurrence is that the universe, and everything and everyone in it, is doomed to repeat the same sequence of actions for eternity. Perpetual repetition seems to entail relentless boredom. It also evokes a feeling of meaninglessness. An act may seem profoundly meaningful when it first occurs, say a first kiss. But reliving that kiss forever seems to rob it of its uniqueness and therefore its meaningfulness.

Nietzsche was fascinated (obsessed?) with the concept of eternal recurrence. He even claimed that eternal recurrence was the fundamental idea of Thus Spoke Zarathustra. His approach to dealing with the dread of monotony and meaninglessness evoked by the contemplation of eternal recurrence was to embrace it. He called this embrace amor fati — love of fate:

My formula for greatness in a human being is amor fati: that one wants nothing to be different, not forward, not backward, not in all eternity. Not merely bear what is necessary, still less conceal it — all idealism is mendacity in the face of what is necessary — but love it.
Why I Am So Clever, Ecce Homo, section 10

For a long time, I’ve embraced amor fati in the face of the relentless sameness (“one wants nothing to be different…not in all eternity”) of eternal recurrence. I’ve embraced it in part because I too saw no way of logically escaping it: if the universe is finite (which I believe is the case) then there are only a finite number of permutations of its state. And if there are only a finite number of permutations, then over the course of eternity these permutations must infinitely repeat. (See Poincaré recurrence theorem.)

But recently I discovered a way out of eternal recurrence, or better, a way to aufheben (transcend) eternal recurrence. I found a simple logical way of ensuring that perpetual novelty emerges from eternal recurrence. Given that there can be perpetual novelty, I can not only embrace amor fati, I can also embrace amor novus — love of novelty; or better perhaps, neophilia. And if like me, Sisyphus loves novelty, one can easily imagine him happy if we can simply find a way for him to generate perpetual novelty from his eternally recurring labors.

Although the simple logical way of generating perpetual novelty from eternal recurrence is basically mathematical, I will explain it using Sisyphus’s labors to make it concrete. First note that like the universe, the world of Sisyphus is finite. Let’s make it as finite as possible to keep things simple. Let’s imagine that the only thing that can vary in the world of Sisyphus is the path he follows as he rolls the boulder up the hill. Nothing else varies: not the speed of his trip, not how he pushes, not the steps he takes, nor even the speed of his breathing. Nothing but the path can vary. And to make his world even simpler, let’s imagine there are only ten different paths up the hill. So the universe of Sisyphus can be in only one of ten different states.

In such a simplified world, we can immediately see that Sisyphus will tread those ten paths over and over again for eternity. In other words, the ten states of Sisyphus’s universe will repeat forever. How could there be any novelty in such a world, much less perpetual novelty? Let’s start with the most basic form of novelty and build up from there.

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Photo by Terry Vlisidis on Unsplash

As he begins his eternal labors, Sisyphus decides to make such a seemingly boring endeavor at least temporarily interesting by leaving up to chance the choice of the path he will take. Let’s complicate his world slightly by positing that he has some way of generating truly random numbers (of any magnitude) in his head; say in the way that people do when told, “Pick a number from zero to nine”. (I know most people say “one to ten”, but computer nerds would probably say “zero to nine”.)

So before taking his very first trip up the hill, Sisyphus picks a number from 0 to 9, randomly. Let’s say he picks 5. He is delighted by the novelty of unexpectedly taking a path he’s never been up before, because after all, he is a neophile. Before his second trip up, he again randomly picks a number, say 0, and is again delighted by the surprise of taking a new path for the first time. Before his third trip, he randomly picks a number, say 5. Now, for the first time, Sisyphus experiences boredom. “I’ve already had the surprise of taking path 5 for the first time”, he moans, “There’s nothing novel about taking it a second time”.

I think we can all see where this is very quickly going. Shortly, he will have exhausted the novelty of traveling up each of the ten paths for the first time, and so forever after he will be perpetually bored. But Sisyphus is much too clever for this to happen. Once he realizes that the novelty of traveling up a single path for the first time will soon cease, he decides to create a novel pattern of paths to take up the hill. To build up the complexity of our story incrementally, let’s assume he decides to chose between all possible combinations of just two trips up the hill. We’ll call traversing such a multi-trip pattern taking a journey. Since there are ten paths he can take on the first trip, and the same ten paths he can take on the second, there are 100 (10x10) novel two-trip patterns to choose from, i.e., 100 novel journeys he can take.

So now Sisyphus picks a random number between 0 and 99 before each journey. Let’s say he picks 25. This means for this journey he will follow path 2 on his first trip up the hill and then he will follow path 5 the second time. He is delighted with the novelty of taking this journey, since he has never taken it before and he had no idea he was about to take it. After completing this journey based on a short pattern of composed of just two trips, he picks another number between 0 and 99, say 22. Even though he has already traveled up path 2, he has never before taken this particular journey, i.e., traversing this particular two-path pattern — following path 2 up the hill twice in a row. How wonderfully novel, he thinks!

As you can easily see, it will take much longer for him to become permanently bored. In fact, he will have to take 519 journeys on average (see the coupon collectors problem for how to calculate this average) before he will have experienced all 100 patterns for the first time. Note this means that Sisyphus we be temporarily bored on 419 of these journeys, because he will be following the same pattern two or more times. But as long as there is still one pattern that he has not yet traversed, Sisyphus is filled with the hope of future novelty. That’s how much of a neophile he is!

But inevitably, he will have journeyed (for the first time) over all 100 two-trip patterns and his future now holds only perpetual repetition with no hope of novelty — he will be perpetually bored. But all is not lost, Sisyphus need only move on to three-trip patterns, then four-trip patterns, and so on… Using this approach, Sisyphus will eventually be choosing journeys based on patterns of 500 trips up the hill. This will give him 10⁵⁰⁰ novel journeys, which will keep a neophile like Sisyphus happy for quite a while. (Assuming a single such journey takes 500 days, he can still expect novelty well beyond the time when all the stars of the universe are extinguished.) But of course it won’t keep him happy forever.

The only way Sisyphus can be happy eternally, the only way to generate perpetual novelty, is if he can forever extend the length of the patterns of paths he creates for his journeys. As long as he can do that, he will be perpetually surprised by randomly choosing to take longer and longer journeys based on ever longer patterns.

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Photo by Mika Baumeister on Unsplash

To abstract away from Sisyphus, note that these (eventually) infinitely long patterns of paths up the hill are just concrete instantiations of infinite numbers. Specifically, each pattern of length n (i.e., a combination of n paths constituting each pattern) can be represented by a randomly chosen decimal numeral of length n. So the perpetual novelty of Sisyphus’s labors can be reduced to the perpetual novelty of creating ever longer decimal numerals and then randomly choosing among them. This may seem like a trivial degree of novelty, but it is novelty nonetheless — and it can be generated perpetually.

Once we have this tiny sliver of perpetual novelty, we can apply it to a much more complex world than the radically simplified world of our bare-boned Sisyphus and his ten paths. To do this, let’s first understand what a 30-digit decimal numeral like


represents in the world of Sisyphus. It represents a 30-path pattern that he inevitably journeyed at some point in this infinite labors. In fact he has repeated this pattern myriad times in his infinite labors. It is a also a partial history of the states of his universe. At some point in his labors he went up path 1, followed by going up path 8, then path 2, 0, 8 again, 2 again, and so on.

Note the degree of repetition even in this short 30-trip history: He went up path 1 three times. He went up path 8 six times. And so on. If we look at sub-paths of length two, Sisyphus repeated the pattern 82 (going up path 8 followed by going up path 2) five times. How tedious! The 3-trip pattern 827 is repeated three times. And finally, the 4-trip pattern, 8275, is repeated twice.

So this one 30-path pattern, represented by a 30-digit decimal numeral, and representing a brief slice of the history of Sisyphus’s labors, is composed of repeating elements at several different scales. That’s why I meant when I said above that perpetual novelty emerges from eternal recurrence. The 30-digit numeral is generated by repeating the digits 0–9 over and over again. It is also generated by repeating the two digit pattern 82 six times, and three-digit patterns, and four-digit patterns, etc.

In fact, the very concept of a pattern entails a finite set of repeating elements arranged to constitute the pattern. So all patterns, even novel ones, even completely random ones, entail repetition in their composition. Thus, repetition isn’t something to be dreaded. It is something to be desired, for without it we could not generate novelty in the form of novel patterns. Furthermore, there is no need to fear eternal repetition/recurrence/return, for as long as time is infinite and there is an element of randomness as to how repeating sequences follow one another, the world will generate ever longer and therefore ever novel patterns containing ever longer repeating patterns.

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Photo by NASA on Unsplash

Now the 30-digit numeral above, representing a sequence of 30 states of Sisyphus’s universe only seems a trivial form of novelty because his universe is so simple. We designed the state of Sisyphus’s universe to be so simple that its state (i.e. which path he follows up the hill) could be represented by a single decimal digit from 0–9.

But there’s no reason, in principle, that we can’t use digits to represent a universe with a much larger set of possible states. All we have to do is to pick a larger base than 10. If we picked base-64, we could represent a universe with 64 different states represented by 64 different digits. How diverse a set of digits would we need to represent all the possible states of our universe? In other words, what base-b would we need to represent the b possible states of our universe? Well, at least one estimate puts the number of possible states of our universe at 10¹²⁰. So we would need to use base-10¹²⁰. That’s a lot of digits! But in principle, we could still represent a sequence of 30 successive states of our universe using a 30-digit base-10¹²⁰ numeral just as we did for Sisyphus’s universe. Except in this case, our 30-digit number would have digits that could be any one of 10¹²⁰ different symbols.

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Possible digits for Base-64 numerals (

The point of this very unrealistic, but still theoretically possible, representation of thirty successive states of our universe by a 30-digit base-10¹²⁰ numeral is simply to show, as we discussed above, that we can apply Sisyphus’s tiny sliver of perpetual novelty to a much more complex universe. Given infinite time, our universe will journey through ever longer patterns of its possible states. Though the individual states of the universe will repeat incessantly, like the single digits representing the states of Sisyphus’s universe (and so will the two-state patterns; and three-state patterns; and the 500 state patterns; and so on), as long as time allows the possibility of ever-longer patterns, such patterns will be new forms of novelty, until they themselves begin to repeat within even longer novel patterns, forever and ever.

So while Nietzsche claimed one achieves greatness by facing eternal repetition over infinite time and nonetheless embracing amor fati, I claim that one achieves greatness by understanding that out of eternal repetition emerges perpetual novelty and thereby embracing amor novus — neophilia. To recast Nietzsche’s claim: One wants everything to be different — forwards and backwards — for all eternity.

This perpetual novelty thought experiment makes me happy. And now I have no doubt whatsoever that Sisyphus is happy as well.

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Although I’ve looked diligently for a similar discussion of the relationship between novelty and repetition, I have only found one instance:

The recurrence implied by these models is a very irregular one, with varying periods of time between one approximation and the next, according to the phenomenal or numerical limit taken as a standard. One could say that this recurrence is still an eternal one, in that the approximations will continue to go on throughout an infinite time. Nevertheless, it is not an “eternal recurrence” in the Nietzschean sense. This is not just because of the absence of a constant “Great Year,” because that is not of great importance by itself. What matters to Nietzsche is the recurrence not of single states of affairs, but of whole sequences of them, such as those constituting a particular human life. He is not alone in this, for the astrologers Oresme is arguing against would have understood the doctrine in the same way. Its deterministic or fatalistic implications depend on assumptions about the identities of particular persons, and these in turn refer to certain sequences of states of affairs, which establish patterns of experience or behavior. If such sequences undergo disruption, the patterns in question change, and the basis for asserting the identity of anything more than a momentary state is lost.
Incommensurability and Recurrence: From Oresme to Simmel, Journal of the History of Ideas, Vol. 52, №1 (Jan. — Mar., 1991), pp. 121–137,

This passage seems to be making a point similar to mine. If the repetition of patterns of any length is never periodic, but always recurs at random intervals, then such a universe will not manifest the kind of eternal (predictably periodic) recurrence that Nietzsche was drawn to. Random recurrence is a form of eternal recurrence, but one that is perpetually novel due to the random intervals between repetitions. Without randomness, in a finite universe perpetual novelty is not possible. Randomness is the foundation of novelty.

I’d be delighted to hear of any other such discussions that I may have missed.

I am an Ironist currently exploring new career paths.

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