The Geometry of Thought


Barbara Tversky, Image credit: Edge.org

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.

BARBARA TVERSKY is Professor Emerita of Psychology, Stanford University, and Professor of Psychology and Education, Columbia Teachers College. She is the author of Mind in Motion: How Action Shapes Thought. Barbara Tversky’s Edge Bio Page

Edge.org

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.

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