Even the mathematically averse among us today recognize the basic geometry that Radolph and Ragimbold failed to grasp, for we live in a numerate society, surrounded by countless manifestations of mathematics. Broadly defined as the ability to reason with numbers and other mathematical concepts, numeracy underlies our current information explosion. Its clichés dot popular speech: “do the math,” “crunch the numbers,” “figure the odds.” From birth to death, numbers track our lives institutionally and demographically. Some scorn such customs (think of Mark Twain’s “figures” of “lies, damned lies, and statistics”), but we all acknowledge numeracy as a cultural given, and agree that mathematics fuels the science, technology, and industry of our world.

Still, as the story of Radolph and Ragimbold suggests, it wasn’t always so.

Before the modern era, whose origins we often date from the Renaissance (circa 1250 to 1600), folks certainly counted and measured. But, bluntly put, the abstractions of numeracy and mathematics mattered not a whit in any practical sense.

Instead, literacy—not numeracy—ruled. For our progenitors, it was the invention of writing, probably between 3200 and 3100 B.C., that served the human impulse to order and manage the flux of experience and its “givens” or “data” (from the Latin, datum, “thing given”). Early scripts provided the first information technology, creating the first “information age”—that of literacy. Its apogee occurred with the phonetic alphabet, devised by the Greeks, who correlated the sounds of speech with individual letter symbols so that each symbol stands for a single vowel or consonant.

The alphabet provided the substrate, the symbols for framing nouns and adjectives, and thus the means of creating definitions, which connected thought to the objects and processes of the world. For Aristotle, science would organize and explain data taken in through the senses, abstracted into words, classified into general and specific categories, and bound together with the formal tools of logic.

The medieval world of Radolph and Ragimbold inherited this word-based technology and culture, assimilating the classifying temper into the curriculum of its new universities. Seven liberal arts anchored the course of studies—three linguistic (the trivium of grammar, logic, and rhetoric) and four mathematical (the quadrivium of arithmetic, music, geometry, and astronomy). The latter were taught primarily as stepping stones to contemplation of spiritual realities, for example the divine order that infused numerical proportions, musical harmonies, spatial beauty, and heavenly motion.

Medieval mathematics was everywhere “enchanted,” its numerology animated with allegorical, generally theological meanings. (Witness the bizarre practice of scapulimancy—divination according to the geometry of sheep shoulder blades.) Its categories, nonetheless, remained separate and distinct from one another. Arithmetic, the subject of discrete things, was incommensurable with geometry, which treated continuous things. And so, the topics of “thing” numeracy stayed tucked away in the cubby holes of literacy.

But then came the Renaissance centuries, and two dominant trends that would dramatically challenge the classifying temper and its embedded mathematics.

The first was an information explosion, begun earlier but powered after 1455 by that great engine of learning, the printing press. The mind-boggling proliferation of printed works (roughly 200 million books by the end of the 16th century) leant cheap paper and ink to spreading the humanists’ recovery of ancient texts, the New World discoveries, and the mounting harvest of information gleaned from nature. There was, in the words of Harvard’s Ann Blair, “too much to know,” too much to classify. The surfeit of new information swamped traditional classes, fractured categories, and overwhelmed the classifying temper. The world lay “all in pieces, all coherence gone,” intoned poet John Donne in 1611, gazing rearward at a more intellectually comforting age.

A second trend was intertwined with this overwhelming volume of facts: the advent of a new information technology. Arising largely from practical activities, new ways to encode information brought forth new and different ways of seeing, imagining, and analyzing nature. In each category of the quadrivium—arithmetic, music, geometry, and astronomy—these new means of data collection and processing laid the foundations of modern numeracy.

In arithmetic, from the 13th century forward, the growing presence of Hindu-Arabic numerals habituated Europeans to a new counting system. Employed initially by merchants, bankers, and accountants, it steadily crept into the procedures of mathematicians and natural philosophers, craftsmen and artisans, musicians, and artists. The new system showcased a much simplified, functional notation of nine ciphers, the numerals 1 through 9, in contrast to the cumbersome Greek scheme of 27 alphanumeric letters or to Roman numerals with their vertical strokes and letters. Further, it featured a place-value arrangement of numerals, our familiar decimal system for determining a numeral’s value by its place in the ones, tens, or hundreds column (and continuing). And the symbolic representation of zero as an empty placeholder greatly facilitated arithmetic computations. All these innovations contributed to perceiving numbers as abstract relations, not just collections of things or objects. By Shakespeare’s day, the character of Shylock in The Merchant of Venice was figuring his pound of flesh with the new, more efficient Hindu-Arabic numerals.

In the world of music, a newly invented and abstract notation accompanied the rise of polyphonic singing, which evolved from Gregorian chant. With their longs, breves, semibreves, minims, fusas, and other equivalents of modern musical notes, composers and musicians caged and managed as information the elusive, ephemeral data of rhythm and pitch. For the first time in human history, time itself was measured by means of an independent system of symbols—standardized units of sound (notes) and silence (rests) that corresponded to physical, acoustical reality. Here, too, Hindu-Arabic numerals described the fluid, irrational proportionalities and harmonies comprising the dynamics of musical tone, and gave rise to new tuning systems, including the equalized temperament of modern pianos and other instruments with fixed, musical intervals.

Turning to geometry, the discovery of linear perspective in the visual arts yielded new expression and shape to visual information. Novel geometric techniques offered alternatives to definitions as a means of seeing objects in the world. Perspective grids tied spatial proportionalities to the changing viewpoints of a world in motion. This was a world of the “winged eye” in the memorable phrase of the Renaissance man himself, Leon Battista Alberti. One-to-one mappings between objects and their representations (two-dimensional drawings or three-dimensional sculptures) provided a new context for situating objects in space. Forerunners of modern geometry’s graphs, these techniques led eventually to separating and analyzing the vertical and horizontal axes of motion.

And in astronomy, new technologies brought heavenly motion and eternal time down to earth, enabling the amalgamation of terrestrial and celestial branches of knowledge, physics and astronomy. Before it became money, time became information. Its “inaudible and noiseless foot” (Shakespeare) succumbed to the technological mastery supplied by that “fallen angel,” the oscillating mechanical clock. Pope Gregory’s solar calendar (1582) and subsequent linear chronologies put the capstone on “clock time” writ large. Time’s mystery was converted to time’s problem, at least in part, as time itself increasingly became an abstract variable expressed in mathematical formulas, such as in Galileo’s laws of free fall, pendulums, and projectiles.

Common to all these arenas—commerce and arithmetic, polyphonic sound and music, art and geometry, timekeeping and astronomy—a novel means of creating and managing information made its appearance. The phonetic letters of definitions, which were the foundation of literacy, gave way to numerals, notes and rests, grid lines, and linear chronologies that would undergird a new numeracy.

These were not the enchanted symbols of allegory associated with words and meanings. As new ways of abstracting, encoding, storing, and manipulating bits of information, these empty symbols—curlicue marks and lines on a page—were purely functional. They guided reasoning with their rules of combination and with their algorithms.

Strangely enough, these symbols even captured “nothing” and made it useful. The placeholding zero (null quantity) in arithmetic enabled “borrowing” and “carrying” amounts from column to column, the basis of adding, subtracting, and myriad further computations. The rest (absence of sound) in music made it possible for composers to develop and depict sophisticated, dynamical musical rhythms. The visual vanishing point in the gridlines of artistic perspective provided focus, thereby giving instructions for the spatial arrangements of objects. And in clock time, the instant (point of no time lapse) separated the flow of time past from that of time future, allowing one to plot events, large or small, on a linear continuum. Later, the 18th-century French philosopher Jean le Rond d’Alembert would summarily refer to the empty symbols of numeracy as “phantoms.”

Mapped onto various dimensions of experience, the new techniques of information coding framed new understandings of phenomena and transformed our ways of seeing. And they provided the tools for a new, logical layering of abstraction upon abstraction that became the higher reaches of mathematics.

By Galileo’s lifetime (1564–1642) these practices had mushroomed and coalesced into the information technology of numeracy. Henceforth, as Galileo wrote in a passage often cited, the book of nature would be understood increasingly as “written in the language of mathematics.” With his own experiments and investigations, this “father of modern physics” led a new generation of natural philosophers into the mathematical analysis of matter and motion. Fermat and Descartes followed with their analytical geometries, joining the discreteness of arithmetic with the continuity of geometry. A generation later the calculus of Newton and Leibniz united the forces and motions of matter. On the cornerstones of numeracy, laid in the Renaissance, would be constructed the entire edifice of modern science … and with it our numerate culture, our own phantom world.

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We use these units because some of the first people to make precise astronomical calculations, the Babylonians, utilized a base-60 (sexagesimal) number system that they inherited from a more ancient population in Mesopotamia, the Sumerians. The Sumerian base-60 system proved influential on Babylonian and Greek astronomers and, because of this influence, it was later used by Europeans to divide hours into 60 equal units. Contemporary U.S. culture has inherited, in a quasi-accidental series of events played out over millennia, this esoteric linguistic vestige of ancient Mesopotamia.

But surely, some might say, “hours” themselves are real, given to us by nature somehow. Yet these time units, too, are a linguistic remnant. When sundials were first developed in ancient Egypt, their creators relied on a base-10 (decimal) system like that of English wherein 10 serves as a recurring element within larger verbal numbers. (For instance “thirty-one”, “forty-one”, “fifty-one”, etc.) As a result, their sundials broke up the day’s shadows into 10 units. Egyptians added two units to represent the times around sunrise and sunset. The resultant 12-unit system was acquired by various cultures and eventually applied to both days and nights to yield a diurnal cycle with 24 major segments.

If all of this seems a bit arbitrary, that is because it is. There is an astronomical basis for dividing time into years and days. But most temporal units came into existence only because of the features of particular linguistic and mathematical systems. Time seems objective, as if it transcends our socio-cultural environment. But the ways we think of time depend profoundly on the place—and time—in which we live.

Temporal conventions are given to us so early in our development that a person may not remember his or her life before it was dissected into weeks, hours, and minutes. From infancy, linguistically contingent cognitive implements sculpt the way we experience the passing of time. The study of the world’s diverse cultures is demonstrating, more and more, just how much linguistic disparities impact human temporal experience.

The effect of linguistic conventions on the discrimination of time extend far beyond differences in time-related words—for instance, the words “minute” in English and “分钟” in Mandarin that differ in script and sound but refer to the same time span. More profoundly, the numbers that we use to keep track of time differ dramatically across languages and cultures. For example, number systems vary with respect to their bases. While ancient Mesopotamians relied on a sexagesimal system, most cultures have come to rely on decimal systems like the Egyptians’ or ours or, less frequently, base-20 (vigesimal) systems like that employed by the Maya.

The Mayan calendar had 20 names for days, in contrast to our seven, because of the vigesimal nature of Mayan numbers. The popularity of decimal and vigesimal systems owes itself to a non-temporal feature in nature, the quantity of our fingers and toes, but many other kinds of numbers exist, including the base-6 (senary) systems found in some languages of New Guinea. Had ancient Egyptians used a senary system, our days might have 16 hours instead of 24, since daylight could have first been divided into eight (6+2) units instead of 12.

The clock represents Mars and Venus, an allegory of the wedding of Napoleon I and Archduchess Marie Louise of Austria. Photo courtesy of Marie Lan-Nguyen/Wikimedia Commons.

Even the use of the same number base across cultures does not guarantee the same kind of time measurement. Decimal numbers, for example, don’t always yield decimal-oriented time. Napoleon famously abolished a decimal calendar used in post-revolution France, in which months were divided into segments of 10, rather than seven, days. (The expunged calendar had several flaws, a notable one being that laborers were only guaranteed one full day of rest every 10 days.)

Today we continue to rely on an ancient sexagesimal system to tell time, and it would take much effort to overturn this usage. Yet our decimal system also influences our time-telling. When great precision is required, we measure time in tenths and hundredths of seconds, or even in thousandths (milliseconds) and billionths (nanoseconds). At the other end of the scale, we measure years in decades, centuries, and millennia.

Some languages rely on restricted number systems without any bases at all. These include the “one-two-many” systems of some populations in Amazonia and Australia. Some hunter-gatherers do not use any precise numbers. Research by a number of cognitive scientists has shown that such numberless adults do not exactly differentiate quantities greater than three. Instead, they rely on the approximation of most quantities in their day-to-day lives. Such approximation methods are quite useful for most tasks, but do not enable cultures to tell time in precise ways. To do so, they need to innovate or adopt numbers. The development and refinement of linguistic numbers allowed humans to reach into the amorphous, abstract temporal dimension and begin to shape it into things like hours and minutes.

“One-two-many” cultures are not atavistic holdouts from the Paleolithic, but the ways they experience the passing of time do seem to reflect more clearly the ways that most people experienced time for the majority of our species’ existence. Minutes and seconds did not really influence European life until the usage of accurate clocks in church towers became widespread in the 15th and 16th centuries. Pendulum-based clocks and spring-loaded watches were invented and refined in the 17th and 18th centuries, bringing both minutes and seconds to the masses.

These inventions facilitated the coordination of labor that proved critical to the Industrial Revolution and enabled better navigation. Arguably, though, they also made our perception of time less natural. Our construal of time came to revolve around quantitatively based cultural conventions like minutes and seconds, becoming less centered around natural rhythms like the diurnal cycle. Regardless, this new focus on temporal units facilitated the Industrial Revolution that, in turn, led to developments like the creation of time zones in the 19th century. Time zones were initially used to streamline rail travel in North America, though they now influence air travel and many other aspects of our lives. Precise measurement of time also advanced science, eventually expanding our understanding of time itself: Einstein’s proof of the relative nature of elapsed time was based on the constancy of the speed of light, which he knew to be about 300,000 kilometers per second.

Perhaps the most interesting cultural differences surrounding time-sense are not related to numbers, but to how we turn time into space. I have alluded to the “passing” of time, but this passing is also a culturally conditioned idea. English speakers often speak of past events as though they are “behind” the speaker while future events are “ahead” of them. In contrast, speakers of Aymara in the Andes refer to the future as being behind them, while the past is in front. (This makes sense, in a way, since we can more clearly “see” what happened in the past.)

The Yupno of New Guinea refer to the past as being downhill, the future as uphill. Such diverse perspectives surface in gestures, too: When English speakers talk about past events they often point backwards, while Aymara speakers point forwards. The Yupno point downhill when discussing past events, regardless of the direction they are facing while speaking. The Kuuk Thaayorre, indigenous to the Cape York Peninsula in Australia, may point east when speaking of earlier events. Some people do not seem to use space at all when speaking or gesturing about time, for instance speakers of the Amazonian language Tupi-Kawahib. These people do not refer to the future or past as being anywhere in space, unlike speakers of English, Aymara, or just about any other language. Unsurprisingly, perhaps, the Tupi-Kawahib language also does not offer precise ways of measuring time.

Being human certainly does not require the usage of precise temporal measurements, nor does it require that we even think of time in the same ways when we are not measuring it. Time is fundamental to our lives but discriminated in culturally dependent ways.

The radical variability in how humans construe time illustrates well the extent to which communicative conventions can profoundly impact our lives. More and more, cross-cultural research on time and other basic facets of life is demonstrating that the human experience is more varied than is often assumed. The exploration of cultural and linguistic variation is critical to advancing our understanding both of others and of ourselves. The continuation and expansion of this exploration is, therefore, well worth our time.

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