Recently I’ve been reading Umberto Eco’s essay collection titled “From the Tree to the Labyrinth”. In it, he discusses the many attempts over history to cleanly organize and index the body of human knowledge. We have a natural tendency to try to impose order on the large amount of miscellaneous stuff we know, for easy access and for later reference. As typical with Eco, the book is equal parts fascinating insight, verbose pretentiousness, and meticulous historical detail. But I do find it fun to think about the overall shape of human knowledge, and how our visions of it have changed over the years.
It seems like most people organizing a bunch of facts start out by trying to group them into a “tree”. Mathematically, a tree is basically a structure that starts with a single node, which then links to sub-nodes, each of which links to sub-sub-nodes, and so on. On paper, it looks more like a pyramid. But essentially it’s the same concept as folders, subfolders, and sub-sub folders that you’re likely to use on your computer desktop. For example, you might start with ‘living creatures’, Under it you draw lines to ‘animals’, ‘plants’, and ‘fungi’. Under the animals you might have nodes for ‘vertebrates’, ‘invertebrates’, etc. Actually, living creatures are one of the few cases where nature provides a natural tree, corresponding to evolutionary history: each species usually has a unique ancestor species that it evolved from, as well as possibly many descendants.
Attempts to create tree-like organizations date back at least as far as Aristotle, who tried to identify a set of rules for properly categorizing knowledge. Later authors made numerous attempts to fully construct such catalogs. In later times, Eco points out some truly hilarious (to modern eyes) attempts to create universal knowledge categories, such as Pedro Bermudo's 17th-century attempt to organize knowledge into exactly 44 categories. While some, such as “elements”, “celestial entities”, and “intellectual entities” seem relatively reasonable to modern eyes, other categories include “jewels”, “army”, and “furnishings”. Perhaps the inclusion of “furnishings” as a top-level category on par with “Celestial Entities” just shows us how limited human experience and knowledge typically was before modern times.
Of course, the more knowledge you have, the harder it is to cleanly fit into a tree, and the more logical connections you see that cut across the tree structure. Thus our attempts to categorize knowledge have evolved more into what Eco calls a labyrinth, a huge collection with connections in every direction. For example, wandering down the tree of species, you need to follow very different paths to reach a tarantula and a corn snake, one being an arachnid and the other a reptile. Yet if you’re discussing possible caged parent-annoying pets with your 11-year old daughter, those two might actually be closely linked. So our map of knowledge, or semantic network, would probably merit a dotted line between the two. Thus, we don’t just traverse directly down the tree, but have many lateral links to follow, so Eco describes our real knowledge as more of a labyrinth. He seems to prefer the vivid imagery of a medieval scholar wandering through a physical maze, but in a mathematical sense I think he is referring more to what we would call a ‘graph’, a huge collection of nodes with individual connections in arbitrary directions.
On the other hand, this labyrinthine nature of knowledge doesn’t negate the usefulness of tree structures— as humans, we have a natural need to organize into categories and subcategories to make sense of things. Nowadays, we realize both the ‘tree’ and ‘labryrinth’ views of knowledge on the Internet. As a tree, the internet consists of pages with subpages, sub-sub-pages, etc. But a link on any page can lead to an arbitrary other page, not part of its own local hierarchy, whose knowledge is somehow related. It’s almost too easy these days. If you’re as old as me, you can probably recall your many hours poring through libraries researching papers back in high school and college. You probably spent lots of time scanning vaguely related books to try to identify these labyrinth-like connections that were not directly visible through the ‘trees’ of the card catalog or Dewey Decimal system.
Although it’s very easy today to find lots of connections on the Internet, I think we still have a natural human fascination with discovering non-obvious cross connections between nodes of our knowledge trees. A simple example is our amusement at puns, when we are suddenly surprised by an absurd connection due only to the coincidence of language. Next time my daughter asks if she can get a tarantula for Christmas, I’ll tell her the restaurant only serves steak and turkey. More seriously, finding fun and unexpected connections is one reason I enjoy researching this podcast, discussing obscure tangential links to the world of mathematics that are not often displayed in the usual trees of math knowledge. Maybe that’s one of the reasons you like listening to this podcast, or at least consider it so absurd that it can be fun to mock.
And this has been your math mutation for today.