Strange loops

In 'The Barber Paradox' and 'The Pinocchio Paradox', I gave examples of self-reference paradoxes. Douglas Hofstadter, in 'Godel, Escher, Bach' introduces the notion of a 'strange loop', saying:

'The "Strange Loop" phenomenon occurs whenever, by moving upwards (or downwards) through the levels of a hierarchical system, we unexpectedly find ourselves back where we started.' (GEB, p.10).

'Strange-loopiness' is best expressed in the paradoxes mentioned above, but it also crops up elsewhere, Hofstadter points out, in art and music. Here are some more examples:

1. The next sentence is false
2. The previous sentence is true

In this developed version of 'The Liar Paradox', consideration of each sentence in turn seems to send us in a logical loop. The truth of the first sentence depends on the falsity of the second, which entails the falsity of the first, which depends on the truth of the second, and so on ad infinitum.

So too with Escher's impossible waterfall:

And the rather more famous never-ending staircase:

Hofstadter gives a musical example: a particular Bach canon, which modulates upwards by one key at a time, seemingly getting further and further away from its original key, until finally, it suddenly ends up back where it started, in its original key.

The common feature to all these 'strange loops' is the notion of moving through a hierarchical system, step by step, and but always ending up back where one started, on the first level of the system, and thus inviting an infinite loop. Some philosophers have suggested that time could exhibit the same sort of 'strange loopiness', though this is a notion that is rather hard to conceptualize. Hofstadter himself applies the notion to human consciousness, in an original and inventive attempt to explain why conscious experience seems so radically different from everything else the human mind has been able to comprehend; developed from 'Godel, Escher, Bach', his work on this crazy theory is called, appropriately, 'I Am A Strange Loop.'