Mind in Motion

by Barbara Tversky

  Ordering things by time is easy in contrast to ordering things by quality or preference or value. Time keeps happening whether we like it or not, and which came first can usually be determined—or measured objectively. Not so for qualities and preferences and values; we make those dimensions up. There’s no objective way to measure them. As a consequence, not only do people disagree with one another, they often disagree with themselves: Is Michelangelo better than da Vinci? Picasso better than Matisse? Beethoven than Bach? Although ordering is hard, comparing things that are far apart on an order is fast and easy. Is Picasso better than Renoir? A fast and easy yes. On most orderings, Picasso and Matisse are likely to be close and Renoir more distant. Comparing Picasso and Matisse causes more wavering and takes longer than comparing Picasso and Renoir. It’s parallel to it being faster to say San Francisco is farther from New York than from Salt Lake City. Placing things on a line of quality is quite like putting places on a line in space. Another example of anchoring abstract thought in spatial thought. Distance in space is real, but distance in quality or value is symbolic.

  Once symbolic distance had been demonstrated, people looked for it everywhere. It was easy to find. Geographical locations, letters of the alphabet, social status, size of animals, and, importantly, numbers. All with spatial underpinnings.

  Ordering isn’t just for humans

  Ordering things and making inferences from the orders are by no means limited to humans. Monkeys show a symbolic distance effect.

  Monkeys, along with other primates, birds, rats, and foxes, make transitive inferences. They know that if A dominates B and B dominates C, then A dominates C. Intriguingly, animals with more complex social relations are better at transitive inferences, but here it seems that social behavior is driving cognition. The implications of lining things up in an order are huge. Once order has been represented for one aspect of life, the abstraction, an ordered line, can be co-opted for so many other aspects of life.

  NUMBERS LINE UP

  The ultimate abstract ordering is number. Number, the great equalizer, is devoid of content. Number is both easy and hard. There are two number systems, the approximate number system (affectionately known as the ANS) that doesn’t actually have numbers, and the exact number system that does have numbers. The two systems are separable in development, in the brain, in evolution, and in cultural history. The approximate number system can answer: Which is more? Only the exact number system can answer: How many? The approximate number system depends on direct perception. The exact number system can act in memory as well as perception. Numbers summarize quantities and are an excellent mnemonic.

  The approximate number system

  Estimates turn out to be easy. Babies, primates, and pigeons can make approximate comparisons of quantities, by no means perfectly, but with decent accuracy. That means that some form of quantitative competence is deeply embedded in evolution, and that this competence doesn’t depend on counting or exact numbers. It depends on ordering, by amount. What’s hard are exact calculations. School math is hard; young and even older and former students struggle with multiplication and division, so-called simple arithmetic. The contrast between the two systems is revealing. Numerical accomplishments by organisms that do not speak or have other representational systems must be nonsymbolic; they can’t be verbal or symbolic; they must be unmediated and direct.

  The approximate number system bears strong similarities to the system that makes judgments about time, brightness, pleasantness, ferocity, and more. It is present in animals and present in humans. It is prone to error, especially as quantities get larger. Discriminating between large quantities or intensities is more difficult than discriminating between small quantities. Yet, it can do rudimentary estimation, addition, subtraction, and even multiplication and division. It can estimate over space, approximate number of items, and it can estimate over time, approximate number of events.

  Not surprisingly, the various brain areas underlying these estimates both overlap and are partly independent. In particular, all comparisons activate a broad network, including the intraparietal sulcus, an area generally involved in spatial thinking. Relative to the other comparisons, activation for numerical comparisons is especially strong in the left intraparietal sulcus and right temporal regions. The partial overlap and partial independence evident in behavior is—necessarily—reflected in partial overlap and partial independence in the brain.

  Implications of ordering

  Forming linear orders is a crucial skill, both social and cognitive. Creating an ordering requires abstracting a single attribute from a set of different things and ranking them on that attribute, ignoring myriad other attributes. Once done, ordering allows inferences fundamental to behavior and thought.

  Orders are only that—they do not carry exact numbers. They do come with several key features, each of which differs from exact numbers in significant ways. One is symbolic distance: comparing far instances is easier and faster than comparing near instances. For example, answering that 81 is more than 25 is faster than answering that 81 is more than 79. Next, semantic congruity: it’s easier and faster to compare small quantities for “less” or “smaller” and large quantities for “more” or “larger.” What’s more, the low end of the number continuum is associated with left and the higher end with right in languages where numbers and reading are ordered left to right, a phenomenon known as the spatial-numerical association of response, or SNARC, effect. For languages where numbers are ordered from right to left, the correspondence seems to reverse. We’ve noted another signature characteristic of ordering, transitive inference: if A is more/greater/less than B and B than C, then A is more/greater/less than C.

  Perhaps the most important feature of orderings in the mind is that sensitivity is greater at the low end of the continuum than at the high end of the continuum. For numbers, we are more sensitive to the difference between one and two than the difference between eighty-one and eighty-two. We and other creatures are more sensitive to differences in weight for light objects than for heavy ones, to differences between dim lights than bright ones. The perceptual differences, weight and brightness, are embedded even in the peripheral nervous system. Relatively more neurons fire for increases in intensity at low levels of intensity than at high levels of intensity. The greater sensitivity to differences on the low end of a scale than differences on the high end of a scale is known as the Weber-Fechner function. We are more sensitive to the differences in sweetness of crackers than in sweetness of baklava, to the differences between small sums of money than between large sums of money. We talk that way: one or two, several, a few, and then jump to many, lots.

  Even highly educated people who make decisions about large sums of money show this bias. Formal numbers do not: the difference between one and two and eighty-one and eighty-two is always the same, one. The differences in distance between one mile and two miles and one thousand and one miles and one thousand and two miles are the same, and the same amount of gasoline will be consumed. People and other creatures have a fast and handy, broad and useful system for keeping track of and comparing amounts that is not based on formal numbers. Compared to numbers, which are indifferent to their place on the number line, the approximate number system distorts, conferring relatively greater weight to smaller quantities than to large ones.

  The Big Question hanging in the air is: Is this irrational? If so, why didn’t evolution correct it? Undoubtedly because it’s a fast, useful, easy kluge. What can correct this bias—but not always—and many others is cultural evolution, the slow development of systems for measurement, counting, and calculation.

  The exact number system

  Even if evolution didn’t correct the biases of the approximate number system, the exact number system (ENS) can. Numbers are indifferent to where they are on the number line. In budgets, every dollar counts equally. In building a bridge, every foot counts equally. The exact number system is necessary for counting and arithmetic and math and
engineering and science and the humanities and the arts and the uncountable number of artifacts, norms, laws, rules, conventions, inventions, and discoveries that depend on an exact number system. Without a system of counting, and importantly, a system for recording, nearly everything we depend on for our daily lives would not exist. Yet humanity survived many millennia without an exact number system, and many pockets of humanity lack one to this day. They can estimate, but not calculate.

  The exact number system is a cultural invention. In contrast to the approximate system, it has to be learned, in school or at home. Even the simplest mathematical task, counting, depends on representations for numbers, typically words. Surprisingly, there are communities today that speak languages that don’t have them. One such community, the Pirahã, live an isolated existence in the Amazon. They even lack a word for one. Nevertheless, they can accurately compare the magnitudes of two sets of things differing only by one object if those things are lined up so that one-to-one matching is easy to do. In other words, they understand one-to-one correspondence even if they cannot count. But if the comparison depends on memory or if the objects are not lined up, performance drops dramatically. That task can’t be done by one-to-one matching; it has to be done by counting. Another group indigenous to the Amazon, the Munduruku, has number words up to five. Although they are excellent at approximations, they cannot do exact arithmetic.

  Equally striking are the differences in neural substrates for the approximate and exact number systems. Patients with brain damage can lose one system and not the other. However, in intact brains, the two systems interact and cooperate. Although the approximate and exact systems are separate in evolution and in the brain, the systems get integrated. Children who are better at estimates turn out to be better at math. What’s more, training the approximation system to be more accurate enhances exact number performance.

  The development of an exact number system depends critically on developing a visible notation system, one external to the mind that the mind can use. Many cultures scattered across the world invented elaborate notation systems for counting and even calculating using objects such as tallies in stone or bone, knots, and pebbles. In fact, calculus is the Latin word for pebble. Many cultures used the body, especially the joints of the fingers, as instruments not only for counting but even for calculating. Hands were the first slide rules, if insufficient for square roots. In many languages, the body parts became the names of the numbers they represented. Digitus means “finger” in Latin. Many of us keep track and count with our fingers, even with ubiquitous paper and calculators. The body might work as an efficient calculator, but it doesn’t leave a record. Knots and tallies do leave records, but they are a clumsy way to represent numbers and even clumsier for calculations. Symbols for numbers, like those familiar to even preschoolers in literate societies, are more efficient, but a richer notation system is needed for calculations.

  It is accounting that drove the development of a notation system for numbers and thereby drove the development of writing by Sumerians living in Mesopotamia in the fourth millennium BCE. Keeping track of the number of sheep, cows, and the like of citizens was central to taxation, and taxation was needed for an organized society.

  Every schoolchild today knows + and − as well as numbers, and even 0, but these were hardly known only two thousand years ago. Our current notation system has taken thousands of years to develop and had many dead ends. Zero is an enlightening example. The Egyptians, Greeks, Romans, and Chinese built magnificent edifices without it. Mayans had a symbol for zero, but it didn’t leave Central America. Similarly, there seems to be a zero in Angkor Wat, dating from the seventh century; it, too, did not spread further. Zero seemed to catch on—if slowly—after it was borrowed from India and used in records of Arab traders in the ninth century. It was brought to Europe early in the thirteenth century by Fibonacci, a—you guessed it—number theorist.

  Mathematics and measurement begin with the body and in the world. The origins of hands to measure horses and feet to measure the ground are evident in their names. The simple act of counting is a sequence of actions, pointing to each item in turn or moving each item aside as it is counted. These actions create a one-to-one correspondence between items and number names. Notation systems allow calculations in the absence of the objects. Like language and graphics, notation systems, in this case for numbers, free us from the here and now. The notation system that eventually took over the world is inherently diagrammatic and spatial. Where a number appears in a sequence determines its value: 56 and 65 are not the same. The number on the left is multiplied by 10, so 56 is five tens and six ones. Arithmetic operations depend on getting the vertical columns lined up properly and on beginning the calculations with the right-most column for addition, subtraction, and multiplication, the left-most for division. The actions and the notation systems are at their foundation spatial, and the brain already knew that.

  BOUNDARY: ANOTHER KIND OF LINE

  Like so many useful words and marks, line has many senses. A sense that has had and continues to have enormous significance in history and politics is boundary, border. The contested borders of countries, the line in the sand, the metaphoric red line—crossing it will not be tolerated (or maybe it will). But borders can also be places where different things meet and interact. Crossing disciplinary borders makes interdisciplinary research. Crossing culinary borders creates fusion delights. Crossing subspecies borders can yield hybrid vigor.

  Boundaries can be subtle, even imaginary. The artist Fred Sandback stretched one or two strings from the ceiling to the floor. Museum goers often look at the space created by the strings and the nearest wall, but don’t enter it. That lone skinny string has become a barrier. When someone does enter the space, it gives permission to others to do the same, and many do. In a schoolyard or at a cocktail party, a group of people interacting creates a barrier. A line of people at a bus stop or a theater creates a beginning and an end, often one that keeps growing and sometimes endlessly. The line can even be tokens that represent the people, their backpacks or shopping carts. Those ordered lines are called queues.

  ARROWS: ASYMMETRIC LINES

  The first arrows are our eyes. They point in the direction of thought even when the thing we are thinking about is no longer in our field of view. The famous actor at a nearby table in the café, even when she’s left the café. In that case, as our parents told us, it wouldn’t be polite to point, but we can and ought to point when telling a stranger the direction of the nearest subway. And we can even use our entire hand. Much more on arrows (and boxes, lines, and trees) in the next chapter.

  PERSPECTIVE

  Perspective is another of those extraordinarily useful words that is consequently used with many senses. There’s near and far. There’s above and within, outside and inside. There’s global and local. There’s peripheral and focal. There’s yours and mine (go back to Chapter Three). Each is spatial, and each goes abstract. Many are encapsulated into epigrams, not necessarily consistently, as for all epigrams. The big picture. The devil is in the details. To see the world in a grain of sand (William Blake).

  Near and far

  We begin with an old, but nevertheless true, aphorism: Can’t see the forest for the trees. When we are close, we see the trees; only from a distance can we see the forest. From close, you see the details; from afar, you see the broad outlines. Now, which is better? The usual answer: it depends. But first, let’s examine the phenomenon that imagined distance affects thinking in a range of tasks. In particular, a distant focus is accompanied by generalities, abstraction, and greater certainty, whereas a close focus is accompanied by specifics, details, and greater uncertainty. Here are a few of the studies that are consistent with that analysis. People are faster to read words denoting certainty, like sure, when they are located close in a drawn scene and faster to read words expressing uncertainty, like maybe, when the words are placed at a distance in a scene.

  When they imagine the dist
ant future, people judge that others and they themselves will be more consistent than when they imagine the near future. This implies that we are more likely to get out of ourselves when we take a distant perspective on ourselves. According to the fundamental attribution error, we see our own behavior as more dependent on external influences, so more variable and uncertain, but we see others’ behavior is more dependent on traits, so more consistent and predictable. Distancing ourselves from ourselves makes us see our own selves like selves of others. People use more abstract words to describe their distant past than their close past.

  Together, the studies show that taking a distant spatial perspective induces people to think more abstractly. This suggests that taking a distant spatial perspective should abet creative problem solving, and in fact, children and adults are more likely to solve insight problems after they have been primed with a distant perspective.

  But distance is only one-dimensional, and space has three dimensions, though it’s often flattened to two in the mind and on the page. Instead of thinking along a single line, let’s now pop up overhead.

  Above and within; outside and inside

  Networks and lines are simple structures that contain and connect ideas. Networks are surveys, they provide outside perspectives from above, like maps. Lines are routes, we’re on them, they give us inside perspectives. Surveys are spaces, routes are sequences of actions. Inside perspectives along routes vary in imagined distance, with consequences. People who imagine themselves on the East Coast judge the distance between San Francisco and Salt Lake City to be smaller than people who imagine themselves on the West Coast. Differences in distance between near things are exaggerated relative to distances between far things, which get minimized, a phenomenon you now know is characteristic of the approximate estimation system.