...an Eternal Golden Braid

Continued from last week, when we were discussing strange loops!

Quick review: a strange loop is anything that linearly relates to itself - if you pick some one action to perform on it, you'll eventually return to exactly where you were when you began.

Now - per the last example last post, human consciousness is itself a strange loop, as shown whenever you think about thinking, or think about why you think or feel the things that you do. So it should be no surprise that philosophy, psychology, mathematics, art, and so on can all be shown to be complex, tangled hierarchies full of strange, looping structures. Here's a couple more examples, for fun:

• Storytelling. Structures which return the characters to the place they started from at the beginning - physically, mentally, emotionally - are common throughout great literature. The Hero's Journey is a perfect example - albeit an overly specific, misleading, and rigidly stratified one. In the Iliad, the Greeks go to war; in the Odyssey, the last of them finally comes home. (In the beginning of the Odyssey, Odysseus almost gets home but fails.) There and Back Again. Don Quixote. The answers are always in the place you started, but you need the intervening book in order to recognize them.
• Feynman diagrams. Antimatter and matter are always created together in nature, and annihilate each other when they converge again. Feynman diagrams are just how scientists keep track of their interactions. Particle creation and annihilation is happening constantly, everywhere, and usually the two particles that formed together also destroy each other shortly thereafter - the diagram looks like this (incoming energy waves convert into two particles with mass, which move apart, collide again, and re-release the energy). You'll notice, though, that the regular particle is marked with a forwards arrow, and the antiparticle is marked with a backwards arrow - because one possible interpretation of the math is that there is only one particle and it appears to sometimes be an antiparticle because at that point it has begun moving backwards in time. In other words, the particle is 'creating' and 'destroying' itself, in a single infinitesimal moment, forever.
  
• Fractals. You can zoom in indefinitely and get the same image you started with, yes? Well, not for all fractals - the famous Mandelbrot Set, for instance, is unimaginably more complex than that.
• Let's talk (briefly) about the Mandelbrot Set. Do you know the coordinate plane? With an origin and a defined axis, you can locate any point P(x, y), x units left or right and y units up or down. One of the many, many things we can do with this is to apply a formula to P to define a second point - call it P1. Let's say that if P is x units right and y units up, we'll start at the origin, and move (x squared minus y squared) units right, and (twice x times y) units up. Then, we'll move it x more units right, and y more units up, and call that P1. X and Y can be absolutely anything, and we'll always get another point... so let's say that we take what we just did to turn P into P1, and do it to P1 to get P2, then P3, P4... and so on. We'll do it forever. (Mathematicians have sneaky tricks to find out what happens if you keep doing something forever.) If we keep going forever, we'll find that either our point is now racing toward the edge of the coordinate plane, an infinite distance away - or it's still meandering about, passing by our original point P every so often. Every point that hangs about when you do this is part of the Mandelbrot Set; in this picture, these are the points in black (the non-black points are coloured based on how quickly they run for the horizon). But if the point keeps doubling back on itself when we keep doing the same thing... the Mandelbrot Set consists of all of the points that generate strange loops from a given formula. Awesome? Awesome.
  

Moving on.

Strange loops can be bewildering at times because our minds are adapted to a conventionally categorical, hierarchical way of thinking - the classic riddle of shallow philosophy, "the chicken or the egg", writ large. The two hands cannot actually be drawing each other, after all! That looping sort of cause and effect makes no sense to our ordinary perception.

So it's very helpful that Hofstadter addresses this. None of the loops we've discussed, he points out, are actually self-sustaining or self-perpetuating. Rather, we can restore part of our usual understanding of things by noticing that every point of the loop has both an internal cause - somewhere else in the loop - as well as an external cause, generally one single one for the loop as a whole. You stand on any step of the Penrose staircase by climbing there from a lower step - and because Penrose conceived of the staircase. Characters reach the fulfilment of their stories with the unnoticed aid of their author. A quine is produced by being run, but only because it was first run on a computer. (The bits that make up computer memory are also strange loops, tiny tiny circuit segments that continuously feed themselves their own voltage as new input.) The trigonometric functions, and fractals, and especially the Mandelbrot Set, were discovered by expanding upon math that was already known. Particle-antiparticle pairs form from errant energy waves nearby in the universe. The Liar Paradox needs someone to say it or there is no "I" to be lying and not lying.

You'll notice I skipped humans, because the human mind is more complex than any of these - it is a self-modifying strange loop. In this sentence, I am now making you aware of the fact that you are thinking about strange loops, such as this sentence and your thoughts. You've just added an additional metarecursion onto that cycle, and you can keep doing that to potentially infinite degrees, limited only by your boredom! But for the moment I'm going to tie whatever cycle you're currently on back to this sentence by noting that by thinking about that uppermost layer and the process by which your mind reflects on itself you've constructed a strange loop of strange loops. With my help, of course. This blog post provides initial external instigation. ;)

My point is, though, that your mind doesn't necessarily need the external instigation. I have no idea how many strange loops that cycle in the last paragraph, for instance, and you might have easily realised the loop of loops without my assistance. This is because your mind is a more complicated, tangled, crazy, and wonderful place than you may have ever realised before! :D

I'm not going to be able to finish this in only two posts. Next time: we have to go deeper.

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.