As Kurt Godel showed, statements about numbers have coded within them higher-level statements that "talk" about themselves and the arithmetic system of which they're a part. Likewise, a long developmental process has resulted in neuronal movements having coded within them higher-level symbolic patterns that "talk" about the world and themselves.
This self-reference and criss-crossing of levels is why we're all strange loops, or, as Hofstadter also puts it, "hallucinations hallucinated by hallucinations."
That's the almost criminally abbreviated gist of this wonderful book, but there's much more in "I Am a Strange Loop." Included are discussions of why these high-level mind patterns are not necessarily dependent on the particular physical stuff of the brain, of how identities can be distributed throughout several brains, of gradual self-construction and degrees of self-awareness, and of ways we model others in our minds and thus give them a pale sort of life within us.
This all may seem quite esoteric, but questions like these lie near the surface of popular culture from the virtual world of "Second Life" to the surreal world of television news. I just read of a panel discussion held in Colorado, for example, whose subject was the number of levels of reality present if Stephen Colbert, anchor of the faux news "Colbert Report," were to interview Sacha Baron Cohen, creator of the characters Borat and Ali G.
Colbert is, one senses, a very nice guy, but he is also a comedian who pretends to be a self-centered, overbearing blowhard of a television pundit. Cohen is intelligent and thoughtful, but he is also a comedian pretending to be an ignorant, anti-Semitic homophobe. We sit at home watching the interview and forming little ancillary "I" symbols in our minds for each of these men as well as for their ancillary sub "I" 's.
This self-referential tangle, being indefinitely extensible and recursive, leads to strange psychological effects, one being that the characters played by Colbert and Cohen can be more truthful in disguise than they can if they present themselves straight.
That we can understand these various levels and personas, their interaction, and analogies to other situations is testament to how natural are some of the seemingly abstract ideas in "I Am a Strange Loop." Humor, in particular, calls on our ability to model others' personalities, understand their points of view, and stand outside ourselves.
Even the verbal and nonverbal cues (raising an eyebrow, changing one's tone of voice, winking) that are present in joke-telling are somewhat paradoxical. They say, in effect, "This is unreal," and they are more or less equivalent to the classic "I am lying," or, "This statement is false," which is true if and only if it's false.
Strange and loopy are we.
John Allen Paulos, a professor of mathematics at Temple University, has written such best-sellers as "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.