Mind in Motion

by Barbara Tversky

  NUMBER AND NOTATION

  We turn now from maps, one of the most concrete of external representations, to math, possibly the most abstract. One of the earliest external forms of representing numbers (after fingers) was a tally. Like schematic maps, tallies seem to be a human universal; they were invented over and over by cultures dispersed in time and space.

  Tallies are abstractions, one mark for one thing, whatever the thing. No depiction, so you can’t tell what’s being tallied. Like maps, tallies appeared in many forms: incisions in bones, stones, or sticks, knots in strings, piles of pebbles. The one in Figure 8.6, two sides of the Ishango bone, is a tally and perhaps more.

  FIGURE 8.6. Ishango bone with tallies found in Congo, dating back at least twenty thousand years.

  The bone is the fibula of a baboon, found in a cave in what is now the Democratic Republic of Congo and dating back to at least 20,000 BCE. It can be seen in the Royal Belgian Institute of Natural Sciences in Brussels. The markings are grouped in curious ways, and that has led to speculation about what they represent. Prime numbers? Unlikely, as that would be astonishingly advanced mathematics. Lunar months? Also unlikely, as another bone found later in the same cave seemed to be arithmetic. Indeed, the consensus that emerged from the academic back-and-forth is that the groupings are arithmetic, perhaps used for calculation. One can’t help but wonder if the creators of the tally would understand what was represented many years later. Not infrequently, I am perplexed by some of the notes I wrote myself.

  Tallies, knots, and pebbles provide a visible, long-lasting record of counting, an enormous advantage over counting out loud or with the body. This one has lasted a remarkable twenty-two thousand years, far outliving its makers, and some are even older. But take note: tallies do not give a sum. You have to re-count the marks or line them up with whatever is being tallied. Tallies are one-to-one correspondences. Number names or symbols are needed for totals. At some point, preschoolers can count up a storm, but if you ask them how many, they are flummoxed. Until they make that leap from attending to each item to attending to an entire set of items, from one-to-one correspondences to cardinality, they can’t answer how many. The cognitive system that underlies counting a set of things is separate from the cognitive system that underlies giving a total for a set of things. This separation strikes us as odd because the systems are so well integrated in us adults.

  Yet the uses of tallies and totals are different. Tallies keep track of individuals: did each of my sheep return from pasture? Are there enough people to hold a meeting? Are there empty seats in the theater? Comparing individuals one-by-one to a tally answers those questions. Next time you ask the host if there’s a table for two, look at the chart the host uses; it’s usually a tally. It shows the tables and chairs in the restaurant, with marks where they are occupied. The host doesn’t count and the host doesn’t need to count—the total number of people in the restaurant isn’t relevant for the problem at hand. The host looks for a free table and then adds marks for your party. You’ve been tallied.

  Intriguingly, in many cultures, there are taboos against counting, especially valuables like people and cattle. For one thing, counting reduces individuals to numbers; for another, it can be seen as indication of riches and might then bring the evil eye. Using tallies of one form or another obviates the necessity of counting.

  Final counts, totals, though, are necessary for calculations. Notable among them: How much tax do you owe? Or, if each sheep is worth five shekels, how many shekels for twelve sheep? Accounting depends on calculations, as do engineering and architecture and science and mathematics. For complex calculations, two ingredients serve the notation system in wide use today: symbols for numbers, and position in space. Independently, cultures all over the world transformed tallies into numerals of varying utility for computation. Initially, most systems invented symbols for ones, tens (or twelves, depending on the base), hundreds, thousands, and so forth. You can already see problems with the “and so forth.” The representation of 7,846 would have 7 of the thousands symbol, 8 of the hundreds symbol, 4 of the tens symbol, and 6 of the ones symbol. This is called an additive system. It’s cumbersome. Neither easy to read nor easy to work with. Look how we do it now, using only nine symbols for numbers, zero, and spatial position for ones, tens, hundreds, thousands, and so forth from right to left. A multiplicative system. All you need to represent 7,846 is that. Easy to read, easy to work with.

  The Babylonians around 2000 BCE, then the Chinese around the beginning of the Common Era, and then the Maya around the third to fifth centuries independently invented efficient systems for representing numbers and using spatial position for calculation. Abacuses and counting boards use spatial position; they are at their core tables. The rudiments of the current system using ten symbols and spatial position for ones, tens, hundreds, and so on were developed in India around the fifth century and brought to Europe by the thirteenth century. Symbols for the arithmetic operations, addition, subtraction, multiplication, and division, came centuries later.

  Taking counting out of the mind and putting it into the world first helped counting and keeping records of counting. But once it was in the world, it became a tool of thought, and like so many cognitive tools, one that could be worked with, designed, and redesigned. Those brilliant inventions, symbols and spatial position, that seem so natural to us now depended on generations of transactions between the mind and the page, trial and error, give-and-take, with many false turns and dead ends. It is hard to overestimate the importance of transactions between the mind and the arrays in the world. Those transactions depended on putting math on a virtual permanent page—abacuses and counting boards count—so that there was something to look at, contemplate, and rearrange. Mental math and the hand and body weren’t sufficient. Taking counting and calculations out of the mind and into the world, placing symbols in spatial positions on a page, in turn allowed complex developments in society, agriculture, engineering, science, and math.

  Mathematics is regarded as the most abstract way both to reason and to represent reasoning. It works because of symbols and place in space. Landy and Goldstone said it in the title of one of their papers: “Formal notations are diagrams.” They found that people use spacing, even when irrelevant, as a clue to grouping when solving algebra problems. As they put it, “Algebra is a story about objects moving in space. Proofs tell a story about these objects.” Mathematical proofs are stories. Hmm.

  Math diagrams and culture

  A slight digression to cultural differences in math diagrams. The Asian street scene is rated as more complex than the Western street scene by both Western and Asian observers. Perhaps coincidentally, perhaps not, Asian societies are more interrelated, more socially complex than individualist Western societies. We wondered if the tendency for greater complexity in Asia would carry over to diagrams, specifically, math diagrams. We collected the first ten diagrams for each of the four arithmetic operations, addition, subtraction, multiplication, division, by entering those terms in Google Images and in Baidu Images (dominant Chinese browser). We took out any words and gave the images to European American and Chinese raters. We ran the study twice, a few years apart. The Chinese arithmetic diagrams were rated as more complex than the American ones by both groups of raters. One can’t help but wonder if the complexity of the street scene or the complexity of social and familial relations enable people to better comprehend complexity in other domains.

  As mentioned, math notation stands in stark contrast to another early cognitive tool, maps. Maps transform large spaces into small ones, shrinking distance and direction in the real world to distance and direction on the page. A direct mapping of space to space. Not so for math notation. For math, using place in space to code value in ones, tens, hundreds, and so forth is far more indirect and symbolic. Neither the marks, the symbols for the numbers, nor the spatial relations bear similarity to anything relevant in the world. Tallies do, if abstractly. Specifically, the more items,
the more marks, one-to-one. But there is nothing in the form of 9 that would indicate it represents 3 more than 6. Space in a map reflects space in the world, even if sometimes distorted. Space in math, the columns that we need to line up properly when we do arithmetic, are ordered, increasing right to left, and represent relations in a purely conceptual world, and a complicated mapping at that. Tallies are highly congruent with quantity, one-to-one, but they are highly inconvenient to use in calculation. For math notation, the two principles conflict and the Principle of Use overrides the Principle of Correspondence.

  Math notation has been and is hugely important in every aspect of our current lives. Remember from Chapter Seven that people, in common with many other animals, have an approximate number system that allows seat-of-the-pants estimations and comparisons when both sets are visible. However, that system is not only approximate but logarithmic, that is, the same difference for small numbers is more influential than for large numbers. This holds not only for judgments of numerosity but also for judgments of brightness, loudness, and more. The way to correct those errors is with an exact number system, a way to measure and compute. There are cultures without numbers to this day, and it took modern culture centuries to develop the sophisticated systems in use today, even by children in school. Perhaps it’s surprising that evolution did not wipe out these widespread systematic errors of the human mind’s estimation system. Undoubtedly, the approximate number system has served us in other ways. For better or worse—and truth is almost always like that—these are not the only systematic errors of the human mind. Some, but by no means all, can be corrected or reduced by measurement and calculations.

  FIGURE 8.7. A Euler diagram for reasoning about sets of entities.

  NOTATION: LOGIC AND PHYSICS

  So much I’ve had to leave out. Fascinating developments in notation for math, for logic, for physics, chemistry, statistics, and many other domains. Geometry, a mixed system, part literally spatial, part abstractly spatial, part symbolic. Interestingly, for Euclid, the proofs were in the diagrams; the text simply annotated them. Unfortunately, the original diagrams were lost.

  For logic, one tool is Euler diagrams, circles that represent sets of things, with overlapping circles representing partially overlapping sets, separate circles indicating separate, independent sets, and inclusive circles representing inclusion. Even the language is the same. Think of how many inferences can be made from the simple Euler diagram in Figure 8.7. Each circle represents a set of things, for example, Artists and Poets. The overlap, called the intersection of two sets, indicates those who are both Artists and Poets. What’s outside the circles are those who are neither.

  Here are a few inferences readily apparent from the diagram: some Artists are Poets, some Poets are Artists, some Artists are not Poets, not all Poets are Artists, and so on. Those relations are transparent in the diagram. The diagram enlivens those lifeless propositions directly, so much easier than imagining them from the statements. It has been well documented that people can construct spatial mental models from clear (well-designed!) prose but doing so takes time and effort. Euler diagrams save that effort.

  FIGURE 8.8. Feynman diagram that physicist Mark Wexler imagined twisting with his fingers to “prove” a theorem.

  A priori arguments aside, empirical support for the superiority of Euler diagrams over statements in reasoning has been mixed. It is likely that the representation interferes with the reasoning process, which is sequential and in parts. The visualization shows all the relations simultaneously; for large sets of relations, it is difficult to separate into parts the separate propositions that would be used in an argument. A list of statements does that—it separates the entire set of relations into discrete parts. This seems to be another case where the Principle of Use overrides the Principle of Correspondence.

  The ease of reasoning from well-designed diagrams has encouraged new fields to blossom, endeavors to make mathematics, logic, physics, and computer science diagrammatic, yet rigorous, in order to capitalize on our extraordinary abilities to see spatial relations and to reason about them. The rationale is the same, that diagrams use the power of spatial-motor reasoning for abstract reasoning. Mark Wexler, now a cognitive scientist working on perception and mental imagery, used to be a physicist. When he was a physicist, he was working with the Feynman diagram in Figure 8.8. Each gray blob represents a separate universe. For the universes to be coordinated, the twist in the lower blob had to be undone. He imagined grabbing each of the lower ellipses with his thumb and index finger and twisting them in opposite directions, a bit like Cat’s Cradle. Doing that made him realize that untwisting the lower one twists the upper. The only way to remove the twist is to cut one of the attachments. This conclusion has implications for spacetime and quantum gravity, but that’s beyond me and thankfully beyond the scope of the book. His intuition turned out to be right, as he later showed in a rigorous line-by-line proof.

  Feynman diagrams are admittedly abstruse, as is the physics they represent, but once they are learned, like all effective visual spatial representations, they become a powerful thinking tool.

  NOTATION: MUSIC AND DANCE

  Like the alphabet, music notation maps sound to sight. Like written language and math notation, there have been many systems developed across the world in ancient and modern times. Music has deep cultural as well as personal significance. Singing unites people; its rhythms keep us moving together for joyous occasions like weddings as well as agonizing ones like forced marches. Good notation helps the creation, the learning, the remembering, and the performance of music. The system that so many children learn all over the world maps music primarily in two dimensions, convenient on a page. Increasing pitch is mapped to the vertical, a natural mapping both because lower notes are produced lower in the vocal tract and because higher notes have higher sound frequencies, though the latter fact would not have been known when music notation was invented. Time is mapped horizontally, consonant with the way most cultures think about time. Pitch is categorized into discrete notes, resonating both with the way music is perceived by the brain and with the way it is performed on most instruments. Standard music notation, then, conforms to both principles: correspondence and use. The mapping of pitch vertically and time horizontally is only the rudiments; music notation has been and continues to be enriched to capture subtler aspects of music.

  Dance notation has been far harder. Ballroom dancing is intended to be performed by amateurs, so it does not demand the skills and knowledge of ballet and other dance performance forms. Ballroom dancing is primarily small, simple movements of the feet on the floor and those are relatively easy to portray, exactly in that way. But many of the kinds of dance that elate us involve elaborate movements of arms, legs, head, and torso as well as complex tempos. Add to that duos and trios and ensembles. Creating a system to annotate dance has attracted many people across the world. The dancer, choreographer, and dance theorist Rudolf Laban invented what came to be called Labanotation in the early part of the twentieth century. Bodies are divided into parts with horizontal lines, much like a musical staff, with stick figures depicting the positions of each part. The system has been adapted to show other kinds of movement. However, it is difficult to learn and difficult to use and cannot capture many aspects of dance, so it is not a standard part of practice. The same for similar systems such as those developed by the Beneshes in England and Eshkol and Wachmann in Israel as well as groups in other countries. Dance remains primarily in the Homeric age, passed down from choreographers and dancers to choreographers and dancers, but with the increasing help of video.

  TIME

  Clocks follow us everywhere, on the streets, on walls, on our appliances, on our wrists, on our ever-present devices. We check the time constantly. Everything we do and everything we understand seems to depend on the time. All our actions are predicated on their predictability and their consequences, and those causal predictions are keyed to time. Merging with traffic on foot o
r in a car, catching a ball, entering or ending a conversation. Comedy, timing and all that.

  We can’t directly estimate time the way we can directly (if erroneously) estimate space, by eye. Time isn’t visible. We can only measure the effects of time, processes that occur in time and that go hand-in-glove with the passing of time. We can count seconds silently: one one thousand, two one thousand, but we wouldn’t use that heuristic for hours much less for days. Sundials, water clocks, hourglasses, the burning of oil (remember the eight days of Hanukkah) or a candle (“out, out brief candle”) have all been used to measure time by observing the visible consequences of processes that occur in time. T. S. Eliot’s J. Alfred Prufrock: “I have measured out my life with coffee spoons.” More recently, the decay of radioactive substances. What’s especially elegant for these is that they are self-measuring. Time is correlated with visible changes we can read. In Egypt, obelisks were not only monuments to the sun god but also sundials, providing the approximate time of day and year to those at a distance. Undoubtedly, because of the significance of agriculture in our lives, Stonehenge and Mayan temples are aligned with the solstices and equinoxes. Capturing the movement of water, sand, or shadows captures time. These are self-illustrating instruments; they visualize time directly. On a larger scale, the phases of the moon, the rotation of stars, and the angle of the sun all can be used to indicate time.

  In contrast to sundials and water clocks and hourglasses and grandfather clocks, calendars are not self-illustrating. They needed human creators. The history of calendars is a surprising mix of cognition, astronomy, agriculture, religion, and politics, unfortunately too much to relate here. Recall from Chapter Seven that cultures all over the world think about time, talk about time, and gesture about time linearly, typically horizontally in the order of reading and writing. Although seasons are cyclical, spring and winter return each year, our own lives do not repeat, they are linear, each spring is a new one. So are calendars, with some exceptions. The stunning Aztec calendar wheel is the highlight of the impressive archaeology museum in Mexico City. Maya, Aztec, and other Mesoamerican calendars were sun-based and round, representing the cyclicity of the year. Who they served and how is unknown. By contrast, oracle bones depicting Chinese calendars dating from 1400 BCE were linear tables.