Using tetration for recursion: more movement for brain science
The problem of conceptualizing movement is one of the oldest human problems. Conceptual models are first static, and only grudgingly achieve movement. In human history, it took thousands of years to be able to accurately model movement, which is why Isaac Newton is so championed by scientists and philosophers alike.
Recursion is a form of movement popular in mathematics and computer science. Nature has used recursion to bring us broccoli. Mandelbrot taught us that broccoli is recursive because it exhibits the same pattern through several different scales. He grew so excited by broccoli that he figured out a method for inventing/discovering self-similarity through infinite scales, aka fractals.
Nearly exact self-similar fractal forms do occur in nature, but I’d never seen such a beautiful and perfect example until, sometime after moving to Switzerland, I came across a chou Romanesco like the one [below] in a grocery store. This is so visually stunning an object that on first encounter it’s hard to imagine you’re looking at a garden vegetable rather than an alien artifact created with molecular nanotechnology. But of course, then you realize that vegetables are created with molecular nanotechnology, albeit the product of earthly evolution, not extraterrestrial engineering.
For a long time, recursion in mathematics was confined to three families of operations: addition, multiplication and exponentiation. Tetration – the iteration of exponentiation – was little studied before advances in dynamical systems and easily available computer power made such study feasible. The peculiarity of the tetration among these operations is that the first three (addition, multiplication and exponentiation) are generalized for complex values of n, while for tetration, no such regular generalization is yet established; and tetration is not considered an elementary function.[i]
I don’t know of any use of tetration in consciousness science or neuroscience. I am presenting it here because it’s cool and because it is another way of forming patterns of great beauty and subtlety.
Some fun for the feeble minded. Start with a simple English sentence: Maybe they will be of use to you. Make it recursive: Maybe they will be of use to you or maybe they will just be of use to you or maybe they might just be of use to you or maybe they might just possibly be of use to you or maybe they might just possibly somehow be of use to you or maybe they might just possibly somehow someday be of use to you or maybe they might just possibly somehow someday for some reason be of use to you.
Recursive sentences bring up interesting questions. Are all sentences (let’s keep it to English) infinitely recursive? Are there classes of infinitely recursive English sentences? If either of these is true, can they be proved?
And, of course, why has the brain been built to use recursion?
[i] Wikipedia contributors, “Tetration,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Tetration&oldid=563835966 (accessed July 24, 2013).