The Measure of Reality

Quantification and Western Society, 1250-1600

Alfred W. Crosby, Cambridge University Press, 1997


These notes provide a collage, primarily in the author's own words, of issues and evidence in the history of quantitative literacy adapted from "The Measure of Reality." The notes do not represent a complete or coherent summary of the book, but merely a selection of ideas relevant to quantitative literacy. (QL Home Page)

Europeans of the late middle ages inherited a profound change in mentalitéthat had been fermenting for centuries, a change from the ancient qualitative way of comprehending the world--what Crosby calls the "venerable model"--to a quantitative model that would soon dominate Western society and provide Europe with the power to dominate the world. Crosby's thesis is that in the late thirteenth century, beginning around 1250, a constellation of remarkable discoveries awoke European intellectuals and bourgeois alike to new ways of thinking--in "quanta"--that profoundly affected the subsequent cultural and political history of the entire world. In the space of less than a century just before and after 1300, Europe produced its first mechanical clock (which quantized time), marine charts and perspective painting (which quantized space), and double-entry bookkeeping (which quantized financial accounts).

The impact of these changes were recorded in the decades and centuries that followed through new words and new ideas that appeared as "sparks thrown off by the wheels of Western society grating against the edges of old ruts." In 1300 everyone thought of nature as heterogeneous, each quality with its own measure. Yet 250 years later Pieter Bruegel (in his 1560 painting Temperance)portrayed people engaged in visualizing reality as aggregates of uniform units (or quanta): leagues, miles, degrees, letters, guilders, hours, minutes, musical notes.

In the venerable model, all earthly things were ignoble and impermanent. Left to its own devices, fire rose up toward its proper home in the sphere of fire; stones, similarly motivated, fell straight down towards the earth. Bruegel shows the West changing its mind, deciding to treat the universe in terms of quanta uniform in one or more characteristics, quanta that are often arranged in lines, squares, and other symmetrical forms such as musical staffs, ledgers, platoons, or planetary orbits. What we now take for granted was an entirely new mentalitéin the late middle ages.


Seeding a New Reality

The old Europeans inherited a classical world-view that they rarely challenged. For Plato, mathematics is mental, measurement is material. The latter "is always becoming and never is," whereas the former "always is, and has no becoming." For Aristotle, the mathematician measures only after he strips away sensible features such as weight, hardness, and temperature, which the ancients did not see as quantifiable. Indeed, if you can imagine measuring heat before the invention of a thermometer, then why not also measure virtue, certitude, and grace? Before the invention of appropriate measuring instruments, the former are no less ephemeral than the latter. Intelligent people in the ancient world would not waste time trying to measure exactly anything as variable as material reality. The West's "distinctive intellectual accomplishment" in the late middle ages was "to bring mathematics and measurement together."

Consider the way Europeans marked the passage of time in the early middle ages. Because Europe did not straddle the equator, and because old traditions dictated twelve hours for each day and each night, Europeans developed a system of unequal "accordion-pleated hours that puffed up and deflated" so as to ensure a dozen hours for each daytime and each nighttime, winter and summer. Medieval Europeans were just as concerned with time as we are, but their way had more to do with symbolic values (of prayers) than with precision or consistency. The mechanical clock introduced in town squares in the early fourteenth century (e.g., Strasbourg, 1352) revealed that "invisible, inaudible, seamless time" was composed of quanta. Centuries later, faith in absolute time emboldened Kepler to see that planets sweep out equal areas in equal time.

As medieval time was what happened, medieval space was what it contained. Vacancy had no authenticity. The ranks of subjects painted were more important than their appearances. Alberti, "an inveterate measurer," suggested to artists that they hang between them and their subject a thin veil with bright threads woven to form a grid. By the fourteenth century, painters began to paint with a picture-unit, a quantum, in mind. Some of the rules of perspective drawing were developed by Ptolemy, whose work was rediscovered by Europeans around 1400 in the context of depicting a curved reality on a flat surface via a grid of lines. However, it was artists, not cartographers, who first made good use of these rules. More than any other method, perspective drawing satisfied the new craving for exactness and predictability.

Another signal of the new mentalitéwas the shift from aural to visual language for important documents. Between 1220 and 1270 both the Vatican and England's Royal Chancery increased enormously the use of written records. In the early middle ages (500-1000) writing and reading was laborious. Writing was just speech on a page without spaces or punctuation. so most people read aloud. Reading then was like "walking on stilts." The introduction of spaces and punctuation (quanta) transformed writing and reading. By the end of the thirteenth century reading had become silent and swift, thus more informative, private, and "perhaps heretical." From then on, literacy--verbal or quantitative--was visual.


Arithmetic and Mathematics

The greatest difference between medieval and renaissance thinking is in the designation of quantity. Medieval Europeans used numbers for effect, not for accuracy. For us, numbers are utterly neutral, free of all moral and emotional value. Not so for the old Europeans: they thought of numbers as qualitative and symbolic as well as quantitative. For them the number 3 may be 1 + 1 + 1 as well as the square root of 9, but it is just as likely to also be a reference to the Trinity. Now we use numbers whenever we want to narrow focus and achieve precision. The old Europeans accepted imprecision in order to grasp better what they believed to be important--the meaning of numbers. Often they were reaching not for a handle on reality but for a clue as to what lay beyond. Medieval Europeans were often "as poetic about numbers as about words."

In the Middle Ages and Renaissance, numbers seethed with messages. Even in the hands of an expert--especially in the hands of an expert--numbers were the source of extra-quantitative news. Christian number-smiths started down the path to mathematics as an expression of awe. Bacon, for example employed patterns in numbers to predict the downfall of Islam. The strong belief in the symbolic meaning of numbers undoubtedly contributed to a lag in Europeans' development of practical mathematics.

Hindu-Arabic numerals, algorism (strategies for calculation), place value, and zero (cipher) appeared in Europe in the twelfth and thirteenth centuries, but to be effective they needed to be adopted as a single package. Reluctance to accept zero as a digit--because it was not the count of some thing--hindered the spread of the entire system. In the fifteenth century, businessmen found themselves drowning in a quicksand of fractions (typical account balance: 3345312/4320864). They were rescued by the decimal system, which itself took three hundred years to develop (from the early thirteenth to the early sixteenth century).

The advance of practical mathematics was slowed also by the lack of clear and simple mathematical expressions. For example, there were no universal signs for plus, minus, equal, or root; Roman numerals prevailed in most written records. Although the abacus (or counting board) was known in ancient times, it mysteriously disappeared from the written and archeological record of Europe for 500 years from 500 to 1000 A.D. (Europeans never saw the oriental abacus; if they had, "they may never have adopted the Hindu-Arabic numerals." They were stuck with stones arranged on lines drawn in the sand, not nearly as efficient as beads on wires.)

Algebraic notation remained a mishmash of words, abbreviations, and numbers until French algebraists in the late sixteenth century began using single letters to denote quantities--vowels for unknowns, consonants for knowns. In the next century, Descartes shifted the tradition to beginning and ending letters of the alphabet.

Parallel with advances in symbology was a change in the perception of the meaning of mathematics. We now see numbers as symbols of quantities devoid of qualities, which is why they are so useful. But the old view of numbers as symbols of qualities was deeply entrenched. "It is simplistic but false to believe that number mysticism retreated as practical mathematics advanced." (Millenialists of today show that the transition is not yet complete.)


Music, Money, and Maps

Music. Although we now think of quantification as more scientific than artistic, it most probably first appeared in European thought through elaboration of Gregorian chant. "Immaculately nonmensural," Gregorian chant employs notes of arbitrary length, not exact multiples of any other note; they are as long as they need to be. Chant provides as clear an example of "time measured solely by its contents" as we are likely to find.

Polyphony, which emerged in the late twelfth and early thirteenth century, required a rhythmic measure (or quanta) to synchronize the several parts. To help choirs learn this more complex music, choirmasters introduced musical notes and measures (quanta), symbols for rests (silences, or ciphers, symbols for something that is nothing), and the musical staff (Europe's first graph). (Inexplicably, Europeans waited seven centuries before exploiting this device to represent physical phenomena.) In contrast to chant, notes were precise multiples or divisions of each other. Time became the measure of sound, as well as of its omission--a yardstick to measure things or their absence. Thus did time become abstract. Sounds in abstract time--that is, sounds written on paper--could be inverted, reversed, divided, and repeated. In the span of one century the change in music from purely aural and nonmensural to visual and quantized was so profound that a fourteenth century writer commented about musical procedures that are "more easily seen than heard."

Money. One of the "shattering simplifying" ideas of all time, money quantifies everything. Around 1300 Europeans took another giant step towards abstraction by introducing the notion of "money of account"--a currency used for keeping books but not necessarily for actually exchanging money. (The euro is like that now.) Money of account provided for finance what measures did for music or minutes for time-- a common unit of measurement. It made possible double entry bookkeeping, "a mirror in which the adept sees both himself and others." Double entry books, introduced at the beginning of the fourteenth century, record assets and liabilities separately. Ever since, bookkeeping has had a pervasive influence on the way we think, especially on our practice of dividing things into black and white, good and evil, this or that. "In the past seven centuries bookkeeping has done more to shape the perceptions of more bright minds than any single innovation in philosophy or science."

Maps. In the thirteenth century maps offered not a representation of geography but information on what was deemed to be important and unimportant. Maps of that era were for sinners, not navigators; they were more an expressionist portrait than a scale drawing. The idea of drawing maps in accordance with a gridwork of lines existed in Western Europe in the early fourteenth century, but the gridwork served only as an aid to reproduce mariners' sketches. It took the re-entry of Ptolemy (through his Geographia) in 1400 to treat the gridwork as coordinates on the surface of the earth, calculated in relation to fixed stars. By 1494, just after Columbus' discovery of the New World, Spain and Portugal mapped boundaries in the high seas by measurements calculated by degrees. (In practical terms, distances on water can be measured only in degrees.)


The Power of Quantification

Just how did descendents of the dark ages manage to conquer the world in a span of just five centuries? One clue may lie in the West's flexibility. Compared with the more advanced Arab, Indian, and Chinese civilizations, western civilization in the middle ages lacked firmness in political, religious, and cultural authority. Among the great civilizations, it was unique in its stubborn resistance to centralization and standardization. Western Europe was a warren of competing jurisdictions; no authority had effective political, religious, or intellectual control. Thus artisans and merchants who developed techniques that challenged the old order quickened the climate of change. "The elites of cathedral or palace could not suppress the merchants because they required the skills of this cocky meritocracy."

Teachers of philosophy and theology were the most influential intellectuals of the middle ages. They did not believe they had to invent or discover wisdom, only rediscover it. As compilers and weavers of approved opinion, they faced the daunting task of organizing the massive bequests of classical, Islamic, and Christian thought. In the process of working through contradictory texts these scholars, epitomized by Thomas Aquinas, reinvented rigor and logic, carefully climbing ladders of syllogisms from premise to conclusions. The next step beyond logic is mathematics. But the fourteenth century scholars never took that step because they did not think in terms of measured quantities. They made great progress in mathematics by geometrizing qualities such as velocity and temperature, but with no measurements. These scholars were "mathematicians without being quantifiers."

According to Crosby, quantification's greatest ally is vision, a "martinet" that encroaches on the other senses. The greatest advantage of vision is its compatibility with measurement in terms of uniform quanta. Visualize an idea on paper and you can divide it into equal quanta; you can then count the quanta and think rigorously. "Record events on paper and you have a time machine that can be read forward of backward. Record accounts, you can work backwards to find mistakes. Record music, and you can play the piece backwards." Written records possess an independence that words rarely do: they can contradict your fondest wishes (e.g., as they did to Kepler, forcing him to abandon the idealism of nested Platonic solids for more rigorous planetary laws).

Quantification and visualization together "snap the lock" on the rationalistic, precise, punctual character of modern culture. They introduced to the West a faith that lasted for centuries, a faith that mankind was capable of an intimate understanding of their universe. By the fifteenth century, the West had a greater proportion of individuals who understood wheels, levers, and gears than any other region on earth. In the sixteenth century few societies equaled the West in the ability to project power over long distances, to improvise institutions, and to create new commercial opportunities.

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