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Absurdism uses The Myth of Sisyphus to give a metaphor for the unending human need to attribute meaning to life.

What would be the most well-understood (or established) word that means: "a fundamentally un-endable task that people are compelled to pursue"

The best example might be the search for truth; all humans love to participate it but we are still infinitely far from unveiling "ultimate truth", even though every day we take steps towards it.

Another example I can think of could be (as first mentioned), each generation's self-assigned task to search for a meaning for their lives.

Have any philosophers coined words to describe these? Or is any word already in common usage for this, or is any word's common usage close enough that it could be appropriately reused for this?

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An idiom used in literary and philosophical contexts derives from the myth of Sisyphus itself (the Greek original, not Camus' version), Sisyphean task or labor. Camus gave the myth an additional connotation related to the conception of life's absurdity in existentialism.

Etymonline traces its English use back to 1590-s (spelled "Sisyphian" originally). Here is a description from Idioms Online:

"A Sisyphean task is a pointless, fruitless, and unrewarding task that must be repeated over and over again; an endless task... This rarely used idiom derives from the Greek myth of King Sisyphus, who ruled over Ephyra. He had offended the Gods due to his clever schemes, such as cheating death by fooling Thanatos, kind of the dead, into binding himself with his own chains, thus not only escaping death himself but preventing the death of all others. This angered the God of War since battles had little color without the specter of death!...

In the end, Zeus himself sent Sisyphus to Tartarus, which was the lowest part of the Underworld and basically the modern equivalent of hell. There, Sisyphus was punished by being compelled to spend eternity rolling a huge boulder up a hill. This labor was difficult enough, but when he did manage to succeed, the boulder would roll down the other side of the hill, forcing him to start all over again. Thus, a never-ending and fruitless task came to be known as a Sisyphean task or a Sisyphean labor."

For earlier philosophical uses with a more positive take see Simon, The Myth of Sisyphus: Renaissance Theories of Human Perfectibility:

"The myth of Sisyphus symbolizes the idealization of human excellence as a perpetual process of becoming over the impossibility of absolute achievement. In Stoic philosophy, the writing of the Early Church Fathers, and in its allegorical interpretations in medieval and renaissance mythologies, Sisyphus is the archetypal model of human perfectibility. This Sisyphean archetype is a principal theme in renaissance theories of astral magic in the works of Pico, Ficino, Reuchlin, Paracelsus, Agrippa, and Dee... Sisyphus illuminates the sacred mysteries of life in the works of Philo Judaeus, Plato, Nicholas Cusanus, and Ficino... and the tribulations of the unrequited lover in the works of Petrarch, Ronsard, and Sidney."

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Cyclic cosmologies, like the dharmachakra and the bhavacakra of Buddhist thought, imply or illustrate this, although in the context of pointing beyond to a different way to be.

Antinatalism is arguably one response to perception of life as an endless treadmill. You could point to consciousness as at odds pr potentially so, to the Darwinian requirement that units of selection that replicate more will predominate, and that may have no meaning at all.

The dialectic could be a candidate for a progressive developing truth which demands continual dialogue with it to develop.

The idea of any 'ultimate truth' or meaning being found already embody a host of assumptions about truth that many philosophers don't share, such as absurdists. For me, I look to that style of thinking as coming from an implicit model of thinking as a mathematical logic, of axioms that are fundamental and build into necessary truths. Since Godel we know that doesn't work. Foundationalist approaches get caught on the horns of Munchausen's trilemma. So I look to strange-loops and tangled hierarchies, to picture how minds have this capacity to just begin where they are, building an emergent structure to integrate themselves with their experiences. This article about philosophers using a strange looping tactic illustrates the power of this.

This framework applied to understanding consciousness as originating poinrs to awareness awareness as fundamental, and only then calculation or iteration of rules/structures; so as able to put a set of those 'down', step outside given structures of thought and encapsulate them as elements in a new structure, or break them down in new way - like with Hofstadter's 'tangled hierarchies'. Using consistency & other values or heuristics acquired separate to specifics, to guide the process of how to integrate our experiences. This can be linked to understanding what creativity is, and Poppers observations about hypothesis generation in science.

In a peer-to-peer reality with no mind-of-god as arbiter of what is objective, as quantum mechanics points to, truth and reality are properties of an interacting network, as illustrated by Indra's Net.

The search for meaning is a search to integrate experiences. The Integrated Information Theory approach sketches how that might describe the direction towards 'more conscious' or 'more awake'.

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