It is easiest to read and absorb materials that are structured fractally. The general idea should be in the first paragraph. The main points should be summarized in the headings. Each subpoint should be the first sentence of a paragraph. Specific ideas and examples should inhabit those paragraphs. Much of nature around us is structured fractally, including the brain itself. We have evolved to absorb fractals and remember them. Fractally structured reading materials best facilitate efficient information transferrance (i.e.: they’re easy to read).
EXAMPLES OF FRACTALS
A fern is a perfect example of a fractal. Think of a document as a fern. First consider the overall shape of the fern; then notice its many leaves coming off the main stem, then notice each subleaf coming off of each leaf, ad infinitim… Most of us are satisfied after noticing only two or three levels of detail of a fern. Similarly, we are often satisfied after absorbing two or three levels of detail from a document. A reader should be able to absorb as many levels as desired without weeding through details.
When you go deeper in a fractal you get more detail. Back in the days of dial-up many photos were in a format called Progressive JPEG. A picture in this format appears on the screen very quickly, but in low resolution. The pixels are so big at first that sometimes you can’t tell what it is. After a few seconds (in the old slow days), the resolution would seem to double, and a little while after that it would double again, getting clearer and clearer, until the entire image had downloaded. Pictures in this format are not exactly fractals, but they share the important property of progressive levels of detail.
Suppose you have a question, and the answer you need is represented by a branch on a tree. You start at the trunk. As you move up the tree the trunk splits into several smaller trunks. You choose the one closest to your desired answer. You keep moving to successively smaller branches until you find your answer. Each successive choice provides an answer with greater and greater detail. From an Information-theoretic perspective, if any single branch can only split into two branches, each split represents one extra bit.
LANGUAGE AND THOUGHT
Is language the stuff of thought? Are thoughts defined in terms of language or is it the other way around? This may be relevant to showing that the brain and mind may be fractal.
The Sapir-Whorf Hypothesis purports that language determines or at least influences language. If you don’t have a word for something, you can obviously still think about it, but maybe that thought is only possible in your mind because of the structure built through learning and using language. Among the earliest to support this idea were Lenneberg and Brown. In a 1954 paper they put it beautifully:
Language is not a cloak following the contours of thought. Languages are molds into which infant minds are poured.
They believed that “Namable categories are nearer to the top of the cognitive deck,” i.e.: Having a name for something makes it easier to remember and distinguish from similar things. After many experiments, however, it was commonly thought that the Sapir-Whorf was wrong, or at best only true in its weak form.
There may, however, be evidence that strong Sapir-Whorf is plausible. Consider ‘speakers’ of the Nicaraguan Sign Language (NSL). A cluster of deaf language-less children in the 1970’s in Nicaragua were brought together by the creation of a new school. The school didn’t teach them sign language, or any language, so they created one out of nothing. Fascinated, Judy Shepard-Kegl decided to study them and attempt to interpret. At one point she gave these students a simple test. Imagine a boy who puts his toy under his bed and tells his little brother not to touch it. After leaving the room, the little brother takes the toy and hides it in the closet. Upon returning, where will the big brother look for his toy? Under the bed, obviously, because he still believes it’s there. English speakers pass this test with ease before the age of six. These students failed this test, even when they were adults.
Not having the right words may have impaired these signers. NSL was not as descriptive as other languages. For instance, it only had one word for anything to do with thought (e.g.: believe, think, remember, understand, etc.). Over the years, as younger deaf students came to the same school, NSL evolved into greater complexity. Younger signers had many words pertaining to thought. There is now a separate word for “I know something you don’t,” and “I know something you do know.” It turns out that these younger students passed the same test that their older alumni failed. Even more surpising was that just a few years later the tests were repeated. This time the older students had improved greatly. The younger students had graduated and were hanging out at the local deaf community center. They brought with them their richer dialect. It is possible that a more nuanced lexicon enables greater understanding [more].
If language and thought are tightly coupled, then perhaps thought is Context-Free. “Context-Free” (CF), refers to one level in Noam Chomsky‘s hierarchy of languages, a formal mathematical categorization of all possible types of languages. In short, CF languages are those where any string can be generated by replacing one symbol with one or more other symbols. The fact that only one symbol is considered in each replacement is why it’s called CF. For instance, one could start with the following skeleton of an english sentence: S V O, (AKA: Subject Object Verb). Each symbol could be replaced by english-appropriate rules and the sentence would still be english (e.g.: S→I, V→love, O→you). The sentence becomes “I love you.”
This topic deserves so much more detail, but the point here is that a finite set of simple rules can generate any english sentence. There has been some debate that Dutch, Swiss-German, and Bambara are not CF languages, but there are ways to get around this argument (If you really want to know, I think any string A^n B^n A^n can be generated by a CF grammar if we know the limit on n). Applying a replacement rule to a symbol is much like looking at a fern frond and generating the next level of detail.
Josh Tenenbaum has human learning all figured out. Without relying on linguistic examples he shows how people organize their knowledge in CF forms. He says that humans learn by example, but for a long time it has been believed that we learn too quickly. No one could explain how we learn to speak so well after being exposed to so little (by the age of three no human has been exposed to enough language for any of our computer models to generate correct language). There must be some inductive bias. He makes the case that the only inductive bias needed in order to explain human learning is this:
For instance, the ancients classified all living things in a linear Great Chain of Being. Now we use the Tree of Life. Both chains and trees can be generated by a CFGG. Instead of replacing a symbol by other symbols, consider replacing a node in a tree with a node that has two branch nodes. Using replacement rules one node can become either a chain or a tree. Tenenbaum identified other structures as well.
CFGG’s can be further generalized to include all forms. Such an idea has a clear application for fractals. Let a form (any object), be considered in a limited amount of detail. Greater detail can be generated by applying a replacement rule to a chosen subsection of the form. For example, imagine a fern with a single stem and some leaves. For greater detail, choose a single leaf and replace it with another stem with leaves. Get the picture?
If our brains are wired to understand in fractals, and much of nature is fractal, perhaps the best way to communicate is fractally. Language itself may be fractal, but just using language isn’t enough. Many documents and papers are dense and not prepared for efficient communication to a heterogenous set of readers. Those who want all the details can read every word. Those who want the gist should be able to glean it in a few seconds. Finally, Readers who want something between the gist and every detail should be able to get exactly what they want by naturally browsing the document in greater and greater detail. You have failed to write fractally when a reader must wade through copious details just to absorb a medium-level understanding. I hope I haven’t done that to you today.
As a final example, consider the archetypical newspaper article. It has a headline, byline, gist, detail, and more detail. A reader may stop reading whenever his or her particular desire for detail is sated.