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DSimon comments on Nature: Red, in Truth and Qualia - Less Wrong
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You might be able to end this recursion problem ("I know about knowing things, but I don't yet know about knowing about knowing about things, and when I do know, then I'll have to go and learn to know about knowing about knowing about knowing...") by eventually reaching a point where each level is so similar to its predecessor that it and all its successors can be described with a quine.
The problem is that unpacking the quine requires computation, and you can experience the feeling of knowing from the results of a computation, e.g. by doing math.