Mind in Motion

by Barbara Tversky

  CHAPTER SEVEN

  Boxes, Lines, and Trees: Talk and Thought About Almost Everything Else

  In which we reflect on the ways simple geometric forms, dots, boxes, lines, and networks, capture thought about space, time, number, perspective, causality, and just about everything else.

  George is a deep thinker, but his thoughts rarely come to the surface.

  —TOM TORO, cartoon in The New Yorker, April 2, 2018

  TALK ABOUT SPACE

  Talk about space is as ancient as it is common. There is literal spatial talk: telling someone what a place looks like or where something is or how to get there. It’s an old neighborhood, the streets lined with four-story pastel stucco apartment buildings with bay windows, some with small shops and cafes at the corners. Your keys are in the right-hand pocket of your jacket. The symphony is one block west and one block south of City Hall. Go right on Broadway, down seven blocks, turn left, and cross the street to get to Central Park. There is figurative spatial talk: she’s on a path to success, he’s on top of the world, the government took a sharp turn to the right, those ideas are worlds apart. Underlying both kinds of talk, literal and figurative, is a simple structure: nodes for ideas and links that connect them. Nodes are boxes, they can be packed with people, places, things—any idea for that matter, singles or collections. Links connect them in myriad ways, sometimes specified, sometimes not. The architecture of the brain is like that, in the small, neurons linked to neurons. In the large, the hippocampus for ideas and the entorhinal cortex for arrays of ideas.

  Boxes and links turn out to underlie an enormous swath of talk and thought, maybe all of it. Links can fall into patterns: lines, trees, networks, circles, zigzags, spirals. Networks organize into clusters, hubs, and neighborhoods. Curiously, the boxes and links we create in the mind are mirrored by those in the world. Inside the body, the network of arteries and nerves. On earth, the network of paths of rivers, of traders, of automobiles, of airlines. Of phone lines and the internet. Then there are those that we put in the world, like maps and diagrams, to enlarge and enhance or instill those in the mind. Thoughts that we put in the world are the focus of the next chapter, Chapter Eight, but before the world, the mind.

  THE GEOMETRY OF THE MIND: FORMS

  We’ve had Cognitive Laws and Facts Worth Remembering. Now we add forms, abstract geometric structures that pervade talk and gesture and all the other ways thought is expressed. First, and central to this chapter, points and lines. Practical words that are spatial at their core and expand and enrich and enlighten every which way. We added boxes and networks and we’ll add arrows, circles, trees, spirals, and maybe a few more. Then we’ll need qualities of forms: center, periphery, symmetry, synchrony, repetition, pattern. Each has aliases that carry nuances. Points are otherwise known as dots or nodes or places or ideas; lines can be called links or paths or connections or relations; boxes can be termed regions or areas or containers. Center is also known as middle, focus, core, crux, hub, foreground. Yeats’s poignant line: Things fall apart; the centre cannot hold (from The Second Coming). Susan Sontag asks us to reflect on the difference between being in the center and being in the middle. Same geometry, radically different meanings. Center contrasts with periphery, another fundamental idea that comes in many forms. Some lines are edges: they can be boundaries, barriers, separating one set of things from another, or they can be seams where one set of things rubs up against another.

  These abstract geometric ideas—points, lines, arrows, boxes, circles, center, periphery, symmetry—are building blocks. They can be arranged to create forms or frameworks in the mind or in the world. Those frameworks represent structures of thought. These forms organize thought and especially organize carefully crafted thought, thought in the mind or put into the world with forethought in the form of language, gesture, diagrams, design, and art. It’s not just linguists and mathematicians who have been captivated by these forms. They have captivated poets, writers, artists, designers, architects, and others who closely observe their practices. They have inspired generations of mystics. Points and lines are the basic elements of drawings, of language, of thought, and of the brain. In language, subjects can be represented by points with lines linking them to predicates. Ideas can be represented by points and connections to other ideas by lines. Neurons linked to neurons. What could be more fundamental?

  Points and lines, and also arrows, boxes, circles, and the like, are simple geometric figures. Familiar patterns. Good gestalts. At the same time, they are abstractions, generalities representing the essences of a panoply of ideas. We’ll return to abstractions, but first, we consider some specifics. Thought does that, meander around from general to specific to general again.

  BOXES: CONTAINERS FOR STUFF AND IDEAS

  Before boxes and lines, there’s the stuff we put into boxes and connect with lines. People, things, places, events, ideas. Boxes are so much easier than continua, than dimensions, but they can obscure nuances that are meaningful and important. That First Law of Cognition again, no benefits without costs. Dots can be placeholders for just about anything you can imagine. Depending on the context of course. We don’t see the world as a helter-skelter array of dots, of people, things, places, events, and ideas. We make order out of them, we put them in boxes, string them along lines, hang them on trees. We connect the dots.

  Boxes in the world and in the mind: Kinds

  Here’s one way to organize the stuff in the world and the stuff in our minds. Put similar things together. Those are the categories of Chapter Two. The brain and the mind do love boxes; they simplify the world by putting a multitude of different things together. We begin at home, in our bedrooms and kitchens. Socks in one drawer and sweaters in another. Plates on one shelf and glasses on another. All too often we procrastinate putting the stuff into the boxes and leave them in stacks, sometimes organized by type. It even happens in department stores, though usually they are organized, clothing in one department organized by type, bedding in another. Same for online stores. In a zoo, monkeys in one cage, giraffes in another. In counting money, bills and coins, each sorted by amount, on a table or in our wallets. We organize the activities of our lives as well as the things in our lives, sleep at night, work and dine in the day, go out or zone out in the evenings. There are boxes in time, night and day, week and month, the seasons.

  The mind declutters and organizes in the same way. Socks and shirts, glasses and plates, monkeys and giraffes, eating and working are categories, useful ones because the items in them share appearance or function or both so they are easy to identify and to group. These categories are known to a community, so we can call them names. I just did that. We label the boxes: clothing, food, tools. The labels are informative: they tell you what the things inside look like and act like, and how they relate to you. In the world, we literally put one kind of thing in one box—or drawer or shelf or, sadly, cage—and another kind in another box. Those boxes are inside larger ones, a drawer for socks and another for sweaters inside a chest of clothing, a shelf for plates and another for pots and pans inside a kitchen cabinet. That’s in the world. Boxes inside boxes not only store our stuff but also represent a taxonomy, a hierarchical arrangement of kinds. Kinds and kinds of kinds.

  Boxes in the world and in the mind: Places and parts

  Here’s another way to organize the stuff in our minds and our stuff in the world. We mix up the categories. Not randomly, but for a purpose. Bedrooms have beds and bureaus and closets and night tables. Kitchens have stoves and refrigerators and cupboards. Bathrooms have sinks and tubs and toilets. We sleep and dress in bedrooms, prepare food in kitchens, bathe and brush teeth in bathrooms. These are places, places that contain different kinds of objects that are appropriate for certain activities. We can call the whole complex, places with objects selected for certain activities, themes. Rooms are also boxes inside a larger box, a home. Furniture stores might arrange furniture by categories, but our homes arrange furniture by themes. Of course, wi
thin the themes we have categories; we just noticed that we organize closets and bureaus by kinds of clothing and kitchen cabinets by kinds of utensils. More themes: supermarkets have aisles and packages of food and checkout counters (who knows how long that will last?), theaters have box offices and seats lined up in rows; parks have grass and benches and swings and slides. Disparate kinds put together to serve a common purpose. These boxes inside boxes also form hierarchical networks, not of kinds but of parts, partonomies. You may remember them from previous chapters (Two and Three). The partonomy most familiar to us is our own body. And partonomies, like taxonomies, form trees, trees composed of a hierarchy of nodes and links. The analogy to the living tree is direct.

  TREES: BIG IDEAS DIVIDE INTO PARTS OR KINDS

  Trees

  The tree idea, the tree visualization, the tree name, come from the world: the trunk of a tree, embodying a whole, the large branches splitting into smaller branches, literally embodying parts and parts of parts. The parts and parts of parts emerge from the whole through a biological process. That process isn’t evident to the eyes, but the thick, stable trunk and the thinner and thinner branches are visible in trees large and small, wide and narrow. That abstraction, trunk and branches, has been borrowed to represent origins and branches of thought since ancient times and proliferates today.

  Real trees are everywhere. They teem with life, providing fruit and seeds and remedies and shade and fuel and boards and homes for birds and beauty. Insects, too. Independently, faiths across the world have conferred trees with mystical and mythical powers. They have been called the Tree of Life for abundant reasons, for reasons of abundance. Although trees have long had rich symbolic meanings, using the branching of trees to represent the branching of knowledge seems to have originated with Aristotle and was explicit in the writings of the third-century Greek philosopher Porphyry. No diagrams remain from Porphyry’s writings, but his description of Aristotle’s categories and subcategories was sufficient for later philosophers to construct a tree diagram of the scales of being, which came to be known as the Porphyrian tree. Tree diagrams, often with new content, became a standard for study and memory in the high Middle Ages and thereafter.

  The process that generated the branching is not always clear. Some seem to be partonomies, some taxonomies, many a mix, some neither. Family trees seem to have originated in medieval representations of the genealogy of Jesus and, later, monarchies and dynasties of rabbis. Bacon, Descartes, Linnaeus, da Vinci, and Darwin are among the many philosophers and scientists who relied on trees to organize, understand, and explain their investigations. Rivers form trees, as does the circulatory system. The brain is a tree, both macroscopically and microscopically. Macroscopically, the branching of the major structures and their functions. Microscopically, the branching of neurons and their interconnections. Freud was a neuroanatomist before he was an analyst, developing staining techniques for microscopic scrutiny and drawing what he saw, essentially neurons and their branching. Freud’s drawings of neurons and his insights from the drawings were crucial not only to his later theories but also to the work of the great neuroanatomist and drawer Ramón y Cajal.

  The enormity of the influence of tree diagrams on the accumulation and dissemination of knowledge has not been fully recognized. Trees, knowledge, brain. By now their uses are uncountable and their visualizations myriad. The Big Bang, phylogenetic trees, corporate trees, occupational trees, decision trees, diagnostic trees, linguistic trees, knowledge trees, probability trees, family trees, the list goes on. And on.

  Networks

  Trees are a special kind of network that emanate from a single source. Conversely, networks can be thought of as decentralized trees. Networks have no origin. Nevertheless, networks are often referred to as trees. Typically, networks don’t link every node directly to every other node; if the nodes were all interconnected, there would be no reason to create a representation. Network representations are useful when some connections are direct and others indirect. Your link to your parents and siblings and offspring might be direct, but your links to your MD and dentist are not; you need to go through their administrators to make an appointment and through their assistants when you arrive. To fly from where you are to somewhere else, you have to go through certain airports, hubs, and you have to get to and from the airports. Even driving from one place to another is constrained by the networks of roads; normally, you can’t just drive directly, the way a bird can fly directly. But even birds don’t fly directly; their actual paths depend on the network of winds, a network fundamental to sailors as well. To get to the entertainer you adore or the boss of the company you are applying to or the president of your country might require many links, one of the raisons d’être for social networks such as Facebook and LinkedIn.

  Social networks: Six degrees of separation

  How many links separate one random person from another? In research that became a meme, the influential social psychologist Stanley Milgram conceived of an experiment that would reveal how closely Americans are interconnected. He sent packages to people in the Midwest and asked them to get a letter to an individual in Boston through people they knew on a first-name basis. A postcard was sent back to Milgram at each link in the chain. Of course, many chose not to play the game and many letters never arrived, so the results remain controversial. Nevertheless, the average number of links in successful chains was around six, a number that has replicated. The meme, however, seems to have come from John Guare’s 1990 play with that name, Six Degrees of Separation, in which one character ruminates on the phenomenon, the ultimate closeness of each of us to each other. Psychological research on Broadway! The field, the analysis of social networks, has proliferated, not without controversy, using more direct ways of tracing social (and other) networks.

  When asked to diagram their social networks, people do interesting things. They quite naturally put themselves in the center. Center. They quite often put their parents above, and their siblings and friends sideways or below. Their diagrams aren’t copies of traditional family trees; for one thing, all the links emanate from the person in the center. Everyone in the network is directly connected to the person doing the diagram. The lengths of the lines reflect closeness, shorter lines to those the participant feels closer to. The ease and naturalness of the task show clearly that social relations are readily thought of as spatial relations. Behavior, too, not just thinking. We stand and sit closer to those we feel closer to, and we use the proximity of others to infer their social relations.

  The applications of networks are as diverse as they are vast. Their proliferation and complexity pose challenges to visualization, providing work for many clever scientists, journalists, and designers, among others. How can you visualize a family tree of thirteen million people? Researchers have done that.

  LINES: PUTTING IDEAS IN ORDER

  Lines in the mind and in the world

  Simpler than trees to both eye and mind are lines. The books on our shelves alphabetically or by topic or size; events in history and events in our lives by time. We form lines at bus stops and in supermarkets; in supermarkets, we leave our carts as placeholders when we go to grab an item we’d forgotten. Those lines are spatial as well as temporal. We line people up by height or age or status. Wines and washing machines by quality. Countries by population or GNP. Movies and video games by sales. The landmarks along a route. The steps of a process. Again, nodes and links, places and paths, dots and lines, but now arrayed on a line. Lines for time, for quantity, for size, for cost, for preference, for any dimension on which things can be ordered.

  Ordering things on a line requires abstraction. The line selects a single dimension along which disparate things can be ordered, no matter how different they are from each other in other ways. Nota bene: These are only orders. They don’t have values or amounts. The dots along the lines represent the order of things, they don’t represent the distances between the things, only what comes before or after what. They aren’
t metric, they don’t come with numbers that indicate exact values. Impressively, ordering things and even judging the relative magnitude of pairs of things are things other animals can do. You don’t need language. More on that soon.

  Time on a line

  One of the earliest concepts to be placed on a line is time. Depictions of sequences of events in time, like making cheese or planting, growing, and harvesting wheat, lined the walls of tombs more than three thousand years ago in ancient Egypt. Aztec and Maya codices depict their histories, step-by-step, along a line.

  Jumping from space to time is easy. Space has two, or really three, dimensions; time has only one. A line. Not a plane or a volume. Because of Einstein, we know, even if we don’t understand, that space and time trade off; time is the fourth dimension of space. We talk about events in time using the language of moving in space. We treat a landscape of events in time like a landscape of places in space. After the Washington Monument we come to the Reflecting Pool; after Xmas we come to New Year’s. We are looking forward to seeing the Washington Monument; we are looking forward to the new year. We put events on calendars as though they were books on a shelf. We arrive in time and in place. Life is just one damn thing after another (source disputed). We passed quickly through the mall. Time passes quickly when you’re having fun. We’ve moved past the worst; the best is yet to come. We raced through the summer; the summer raced past us. This from the New York Times: “With Thanksgiving behind us, the country is rapidly slipping towards the year’s end.” Note the change in perspective.