recurrence

The concept of recurrence (which sort of covers the idea of self-reference as well) pops up everywhere in science and mathematics. Here is a large selection of examples:

Such conceptual similarities make me worry. Is our mental tool kit really that small? I wonder how many times we will have to subtly repeat ourselves over the next couple of hundred years. Should the presence of such similarities be interpreted as a manifestation of the all-too-humanness that Nietzsche was pounding on?

Or perhaps it is not us but nature who is repeating the same theme over and over again? (i.e. Our models accurately reflect what is being modelled.)

Recurrence was a depressing idea for Nietzsche who put it in its ultimate form in "Gay Science":

What if, some day or night, a demon were to steal after you in your loneliest loneliness and say to you: “This life as you now live it and have lived it, you will have to live once more and innumerable times more; and there will be nothing new in it, but every pain and every joy and every thought and sigh and everything unutterably small or great in your life will have to return to you, all in the same succession and sequence—even this spider and this moonlight between the trees, and even this moment and I myself. The eternal hourglass of existence is turned upside down again and again—and you with it, speck of dust!”

If that has not arisen a mystical titillation in you, the following probably will. (Quotation is from an expository introduction to Bell's Theorem.)

The question is a variation on the old philosophical saw regarding a tree that falls in the forest with nobody there to hear the sound. A conflict between the assumption of reality and Quantum Mechanics has been suspected long before J.S. Bell. For example, in referring to the trajectory of the electron, in say the double slit experiment, Heisenberg stated "The path of the electron comes into existence only when we observe it."

People have long known that any measurement disturbs the thing being measured. A crucial assumption of classical sciences has been that at least in principle the disturbance can be made so small that we can ignore it. Thus, when an anthropologist is studying a primitive culture in the field, she assumes that her presence in the tribe is having a negligible effect on the behavior of the members. Sometimes we later discover that all she was measuring was the behavior of the tribe when it was being observed by the anthropologist.

Nonetheless, classically we assume a model where we, as observers, are behind a pane of glass where we see what is going on "out there." With Quantum Mechanics the pane of glass has been shattered. J.A. Wheeler suggests that we should drop the word observer entirely, and replace it with participator. He devised the following figure, whose caption is:

“Symbolic representation of the Universe as a self-excited system brought into being by ‘self-reference’. The universe gives birth to communicating participators. Communicating participators give meaning to the universe … With such a concept goes the endless series of receding reflections one sees in a pair of facing mirrors.”