The Geometry of Thought [1]


[2]I do get obsessed. One recent obsession is lines, I see them everywhere. The lines that the eye creates when there are none, connecting the dots so that we perceive whole objects even when they’re occluded. The lines that connect our thoughts in our minds, the neurons in our brains, and our paths in the world. The lines the hand draws. Lines aren’t necessary straight, but they need to connect. Then there are the lines we create when we arrange our stuff in our homes and in the world, walls and shelves and streets. Another obsession, not unrelated, is perspective: yours or mine, and neither yours nor mine but rather from above, a map. Maps are a major feat of the mind: the mind can imagine a map of a large place we’ve explored even if we’ve never seen a map of it or seen it from above. You can find ancient maps carved in stone or painted on walls of caves all over the world. Map-like perspectives, overviews or surveys, capture arrangements of lines or paths, and landmarks or points, not just in real space, but also in imaginary space, a social landscape or a political one. That’s a longer story for another time.

It’s really structure that I keep circling back to (note that word: circle). How do we structure our moving, changing thoughts and how do we structure the world we design and move and act in?

The venerable view of the movement of thought is association; thought is associative. Sure, but a three-year-old would ask, "Where do the associations come from?" They’re not random, they’re organized, and in many ways, and three-year-olds have long begun to form them. Chair-table, both in the category furniture. Or the theme, dining room. Early on, we form categories: stuff we eat, stuff we wear, stuff we play with. More formally: food, with sub-categories like fruit and cheese and bread; clothing, with sub-categories like shirts and pants and pajamas; toys, with sub-categories like cars and blocks and dolls. There are also themes, stuff that gets used together, like bathtubs and sinks and towels, or pots and pans and dishes, and refrigerators and stoves, or paper and pencil and scissors and glue. Typically, we arrange our homes around both categories and themes. Food is in the kitchen, fruit in one place, cheese in another, together with pots and pans and refrigerators. Toys are in a bedroom (or more realistically for three-year-olds, all over the house) along with books and clothing and beds. Think now of word associations, a standard measure: do we respond "chair" to "table" because they’re in the same category or because they’re used together, they’re in the same theme? For years, cognitive and developmental psychologists thought that categorical associations were more sophisticated than thematic ones. That view is being challenged, and surely we need both.

So far, we have categories and themes, in our minds and in the world. Your mind might have jumped back to lines: we arrange towels and dishes and toys in lines on shelves. That seems by necessity—after all, there’s gravity. But we line up windows in rows in apartment and office buildings; surely, gravity doesn’t require that. Buildings are lined up on streets. Streets are traditionally lined up in grids, not just in the west but also in the east. Also not required by gravity, more likely by our desire to organize. I did say lines are not necessarily straight, though there are huge advantages to straight lines, but you might be thinking: what about the curved streets and paths common in US suburbs and Chinese gardens? Answers: first, almost nothing is always (note the hedge, I added “almost”). Second, the curved streets aren’t so much designed to disorient you, though they do that, but to give a feeling of mystery and discovery. And perhaps to slow down traffic. There’s aesthetics too. Some people like curves, others like lines.

Our arrangements in space go far beyond lines and categories. We create hierarchies of categories, plates on one shelf, bowls on another, arranged by size. Books arranged by topic, then ordered alphabetically by author. Table settings arranged in 1-1 correspondences, everyone gets a plate and a glass and a napkin and cutlery. There are also symmetries and repetitions and recursions in both the outsides and insides of buildings. There are orderly arrangements in time as well as space. It’s beginning to sound like programming.

We’ve begun an answer to the question: how do we structure our thoughts and our world? Many ways, not just one, and they mirror each other. We put our minds in the world. The mind in the world creates common ground for our collective thoughts. It informs us, tells us what it is, it directs our thoughts and it directs our actions. Think how different our world looks, where nearly everything is designed, from the world of our hunter-gatherer ancestors. Think of what that organization, an organization by abstractions, does to our minds and to our bodies, even tiny minds and bodies.

Let me jump back in time to say how I got entangled in all this thinking about structure I’ve called the Geometry of Thought. Undergrad school at the University of Michigan was liberating, both in the classrooms and outside of them. Outside, there was music and theater and art and demonstrations, there were poets and writers, there were students from all over the world. Tom Hayden edited the paper, SDS (Students for a Democratic Society) began there, the student union was a haven for the counter-culture. Bobby Dylan walked in one day, Ed White and I often chatted at the same table, though neither will remember me. I co-organized the world’s first Teach-In on the Vietnam war and brought a petition for the Peace Corps to John Kennedy as he ran for president. Academically I tried just about everything, but getting invited to do research, soon my own research, in a psych lab after a single course in psych my sophomore year did the trick. I stayed on for graduate school. Those years were heady: the shackles of behaviorism were broken, setting the mind free and attracting scads of creative minds. I ended up marrying one of them.

At that time, language was king. We were nurtured on Chomsky’s elegant transformational grammar and on Miller’s demonstrations of it in behavior. We were awed by rigorous propositional logic in philosophy, and its corollary, that the brain was a proposition machine. Then there were our own introspections about thinking, when we become aware of those thoughts, they seem to come in words. The hegemony of language was over-determined.

Another back-track. I come from a family of statistician/business people on my father’s side and artists on my mother’s. I toyed with art, became obsessed with drawing, but would soon graduate and needed to support myself; in fact, I was already doing that. Graduate school promised fellowships and a clear career path. Perhaps it was that interest in art, architecture, and design that prompted me to see how or whether propositions could account for memory for the visual-spatial world. Naturally, the propositionalists thought that was a cinch. In fact, there was research claiming that representations of and memory for the visual spatial world was propositional—the shorter the description, the better the memory. At that I balked. I also come from a household that reflexively challenged any idea that anyone put forth. If someone said X, someone else was sure to say not-X. Contrarian thought runs deep inside me. I thought, what about faces? We recognize thousands of them in a split second but we are hard put to describe them precisely enough for others to pick them out. Spatial thinking is prior to language, I thought, both developmentally and evolutionarily. Spatial thinking must have its own logic, not necessarily that of language. If anything, language had to be built on spatial thinking, not the reverse.

That set my path, first an intuitive meandering one with many detours. Language was always there, I rather like it, and it serves as a comparison to spatial thought, usually explicitly. About a quarter-way through that journey, the direction of the meandering path became clear. I was systematically going from space to space. We did research on how people represent, organize, and think about the objects we interact with in the world, the space of the body that acts on objects and moves in space, the space around the body in reach of hand or eye, and the space of navigation, too large to be seen from one place, so pieced together from actual experience, descriptions, maps, and more. Then we turned to the spaces we create in the world to expand the mind—the mind is too small. That is, cognitive tools, like maps and abacuses and diagrams and sketches, but also gestures. So far as I know, no one has found a chimp or a bonobo gesturing or making a map in the sand to show another how to get from here to there or how to do something. Putting complex thought in the world seems to be a distinctively human ability, where "human" must now include the mistakenly maligned Neanderthals, who apparently put thought on the walls of caves. In every case, for every space we inhabit or create, we have found that people’s perception and action in that space bias memory and judgment. Modes of perception and action also guide our mental as well as worldly representations of those spaces, how we think about them, how we think with them, how we communicate them.

We used a variety of ways to uncover the mind, the standard reaction times and correct responses, those the butt of jokes, but also judgments and maps and sketches and diagrams and notation systems and gestures and ancient and modern expressions of thought from many cultures. I had excellent companions on this route, many talented graduate students and post-docs, as well as companions everywhere in the world and in many fields from whom I learned, we collaborated even if no publications came out of it.

Slowly, the significance of spatial thinking is being recognized, of reasoning with the body acting in space, of reasoning with the world as given, but even more with the things that we create in the world. Babies and other animals have amazing feats of thought, without explicit language. So do we chatterers. Still, spatial thinking is often marginalized, a special interest, like music or smell, not a central one. Yet change seems to be in the zeitgeist, not just in cognitive science, but in philosophy and neuroscience and biology and computer science and mathematics and history and more, boosted by the 2014 Nobel prize awarded to John O’Keefe and Eduard and Britt-May Moser for the remarkable discoveries of place cells, single cells in the hippocampus that code places in the world, and grid cells next door one synapse away in the entorhinal cortex that map the place cells topographically on a neural grid. If it’s in the brain, it must be real. Even more remarkably, it turns out that place cells code events and ideas and that temporal and social and conceptual relations are mapped onto grid cells. Voila: spatial thinking is the foundation of thought. Not the entire edifice, but the foundation.

The mind simplifies and abstracts. We move from place to place along paths just as our thoughts move from idea to idea along relations. We talk about actions on thoughts the way we talk about actions on objects: we place them on the table, turn them upside down, tear them apart, and pull them together. Our gestures convey those actions on thought directly. We build structures to organize ideas in our minds and things in the world, the categories and hierarchies and one-to-one correspondences and symmetries and recursions.

Now a sideways jump to another obsession, really a rant. I’ve said something about the complexity of the mind, the myriad structures the mind builds and puts into the world. There are more, I haven’t finished. But cognitive science, behavioral science, other sciences too, are rife with one-liners. An old friend called it one-bit thinking in a two-bit world. Two-bits doesn’t begin to capture the intricacies of the mind or the world. Here’s one making the rounds now. The brain is a prediction machine—you will remember that not too long ago it was regarded as a proposition machine. There are one-liners, memes, in other fields, that people maximize utility, that selection in evolution eliminates irrationality and other undesirable traits. That the key to explaining and understanding and persuading is stories. Maybe I’m just getting cranky, but I do believe that it’s the youthful contrarian in me that objects to one-liners.

First the prediction machine, and that will, perhaps surprisingly, take us to stories. I’ll skip one-bit economics and evolution for now. The brain doesn’t only predict what will happen next as we act in the world. It also stores information, puts stuff into categories and themes and hierarchies and more, often information that has no immediate use and may never have a use. Kids store an enormous amount of information they have no immediate use for. Think of the alphabet song, what do two-year-olds understand about abcd, elided together into a single "word"? Nursery rhymes and songs, "Humpty Dumpty" or "Ring Around the Rosy." None of it makes sense, but parents seem to love teaching them and kids seem to love learning them. The kid who sat next to me in 3rd grade knew all the train schedules in the US. Baseball statistics were the passions of others. We mature grown-ups are a little pickier about what we put in our heads, but we still put a lot of information inside that lacks apparent use, now or in the future: developments in politics, sports, economies, music, art, recipes, literature, the lives of movie stars and rappers and scientists and kings and presidents and gangsters and friends and friends of our friends, people we are unlikely to ever meet. Gossip. And yes, RBIs. (And I know some of you contrarians are finding uses for those, especially gossip.) The brain is far more than a prediction machine.

Now to stories. Stories of course are another kind of structure the mind uses to organize events in time or place. I got into stories years ago when I began working on memory distortions, the sort that happen in eye witness testimony where you confuse what you actually saw with suggestions that interrogators later provide. I began to wonder if we could distort our own memories by the ways we ourselves described them afterwards. The way our personal earthquake stories grow better over time—I watched that happen at Stanford after a large one dumped my library on the floor, luckily not on me. Beth Marsh and I devised experiments showing that we could change your memory in real time by the perspective we gave you to recollect them. Then we got interested in the stories people, in this case, students tell each other in casual conversation and how they alter them. Most of them acknowledged editing the stories they told, adding piquant details, omitting others, exaggerating or minimizing still others. Despite that, they said they weren’t misrepresenting. Speakers, as well as listeners, seem to tolerate some distortions, they know when you say you had four papers and three exams and haven’t slept more than four hours the past week, not to take that literally, that what you’re saying is "I’m stressed out." Aren’t we all?

In fact, stories had always intrigued me. When I was seven, my grandfather sent me a kids’ book on anthropology. Each chapter described the fascinating customs and creation stories of a different culture. I saw quickly that each culture had a different creation story, that all of them couldn’t be true so none of them was true, that what was true was that people needed stories. So they made them up.

After we published our research on stories, I was asked to speak about them to different audiences, psychologists for sure, but also computer scientists, linguists, journalists, philosophers. We did say that stories have become a meme. That got me reading the literature. So many disciplines have written about stories, claim them as their own. That was fun. Here’s what I came away with. Stories contrast with other forms of discourse, in other words, with other ways of organizing thought. There’s description, a state of affairs in space or time. Like a map or a schedule. Descriptions don’t have a prescribed order, a beginning, middle, and end. Explanations do. They also add causality. You start one place, something happens, and you end up somewhere else. How to get from A to B. How to make a cake. How to assemble a piece of furniture. How photosynthesis happens. Richer still are stories. That’s why talk about science anthropomorphizes, but then we can’t help but think that electrons attract protons. The philosophers I met insisted that what characterizes a story is a narrative voice, a point of view. Filmmakers and literary types and even psychologists add other ingredients, like characters, emotion, suspense, and surprise. Morals, take-away-messages. How this applies to your life. Of course, these discourse forms aren’t pure: stories can have descriptions and explanations. But not everything is a story.

Then there’s conversation, two or more talking, no one in charge, the flow of thoughts is shared, created on the fly, it wanders. It’s interactive, another popular concept, especially in the online world. There’s argument, where stuff is selected and integrated to make a point, a proof in logic or a case in a courtroom or propaganda. All of those structures are paths from one thought to another.

The more I think, and I can’t stop thinking, the more kinds of structures I see. There are cycles, like the seasons, or doing laundry, or seeds to flowers. Interestingly, we’ve found that people are inclined to linearize cyclical phenomena, to think about cyclical processes as having a beginning, middle, and end. Time doesn’t return to itself, often to our dismay. What about other domains like music or dance? For those, emotion and pace, rhythm, seem central, and those should change and vary, usually in synch, light, fast, and cheerful, or heavy, slow, sad, and so much more—but the mood, the emotion has to keep changing. Of course good stories do that too, conflict, suspense, contemplation, comic relief. Not just stories—sermons, too, change pace and emotion, going from poignant tales to life lessons, the stories are parables.

What intrigues me just now is collage. Collage in art mixes media and mixes ideas in an artful way, but it’s a jumble, there’s no start and no end. A bit like description but not organized spatially or temporally or organized at all. We’ve had a century of seeing collage in art, and there’s something wonderful about looking and seeing and looking and seeing new things every time we look. After discovering and analyzing the multitude of systematic errors in so-called cognitive maps—people upright titled regions like the Bay Area, they line up large bodies like North and South America, they think Jacques’ house is closer to the Eiffel Tower than the Eiffel Tower to Jacques’ house—we drew the conclusion that cognitive maps were really cognitive collages, multi-modal and disorganized, but perhaps more beautiful. And certainly more true. We bring together whatever we know and try to make sense out of it. Maybe collage describes all thinking. Not all, as I said, nothing is all, but perhaps a lot.

So now I’m obsessed with structure, dots and lines and boxes and networks and categories and hierarchies and cycles and spirals and descriptions and linear if zig-zag explanations and arguments and conversations and stories and chaotic intriguing collages. Spatial structures that we create by actions of the mind and put into world.