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Kaj_Sotala comments on The Cognitive Science of Rationality - Less Wrong
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More accurately, the map should worry about mapping its future states, to plan the ways of setting them up to reflect the world, and have them mapped when they arrive, so that knowledge of them can be used to update the map (of the world further in the future, including the map, and more generally of relevant abstract facts).
(More trivially, there are quines, programs that know full own source code, as you likely know.)
I don't think being able to quine yourself really has anything to do with fully understanding yourself. I could get a complete printout of the exact state of every neuron in my brain, that wouldn't give me full understanding of myself. To do something useful with the data, I'd need to perform an analysis of it at a higher level of abstraction. A quine provides the raw source code that can be analyzed, but it does no analysis by itself.
"The effects of untried mutations on fifteen million interacting shapers could rarely be predicted in advance; in most cases, the only reliable method would have been to perform every computation that the altered seed itself would have performed... which was no different from going ahead and growing the seed, creating the mind, predicting nothing." (Greg Egan).
In any case, if we are talking about the brains/minds of baseline unmodified humans, as we should be in an introductory article aimed at folks outside the LW community, then XiXiDu's point is definitely valid. Ordinary humans can't quine themselves, even if a quine could be construed as "understanding".