Gödel, Escher, Bach

I just realized I've never actually discussed Gödel, Escher, Bach in detail here before, and then I went and made reference to it in my last post. Shame on me. It's awesome.

Gödel, Escher, Bach is a book by Douglas Hofstadter, whose subject cannot be really concisely explained. It's about everything. At it's core, you could say it's mostly about human consciousness. And music. And math. And art. (Kind of as implied by the title. You remember Kurt Gödel, right?) And puns, and palindromes, and Möbius strips, and vinyl records that destroy the record players that play them, and the holism/reductionism dichotomy, and computer programming, and artificial intelligence, and Charles Babbage's parable of Achilles and the Tortoise, and... and, it's amazing and you should go read it. Skim the parts that are too jargony for you, if you must, but keep going through it.

Let's talk about a construct that Hofstadter introduces, called a "strange loop". It's a shorthand way to talk about things that linearly relate to themselves. It'll probably be clearer if I discuss examples instead of trying to define it:

• You've seen M. C. Escher's lithograph of the endless stairs? (The artwork is called 'Ascending and Descending', by the way. The structure itself is called a Penrose staircase, after the fellow who actually invented it - if you can tell me who famously used the correct name, you get an imaginary cookie.) That's a strange loop. You start anywhere on the stairs, and do nothing but walk up - linearly, one direction - and you still return to the point you started from.
• A quine is a computer program that, with no input, produces its own exact source code as output. This is a strange loop - you proceed linearly down the generations of output, and each one is still the same code with which you began.
  
• The trigonometric functions are a strange loop. For the function y = sin(x), the rate at which y changes with respect to x changing is y' = cos(x). The rate at which y' changes is y'' = -sin(x). The rate at which y'' changes is y''' = -cos(x), and the rate at which y''' changes is y'''' = sin(x) = y. Applying one procedure over and over takes you back to the original result.
• The Liar Paradox - "I am lying right now" - is a strange loop. The thought process goes, if you are lying, then that statement is a lie, so it must not be true that you are lying right now, so you must not be lying. But if you're not really lying, then when you say you are lying you must be telling the truth, in which case you really are lying. Paradox.
  
• The way your mind itself works is a strange loop. This is the process called introspection - you are able to recognize that your brain is producing thoughts, and this recognition is itself one of those thoughts.

People tend to think of strange loops as tricksy, exotic, complicated, mind-blowing things, but in reality, they're everywhere. People just tend not to notice them, or to think too hard about them, when they encounter them in real life. It's both frustrating, and depressing.

I'll elaborate on the significance of this next week, but in the meantime, you can go to another fantastic strange loop by following this elegant and finely-crafted link.