The Measure of Reality

What explains Western Europe’s rise from a cultural backwater at the end of the first millennium to its position of dominance in the centuries that closed the second millennium? The late Medieval blossoming, the Renaissance, the Scientific Revolution: why did these happen in Europe? Because Europeans acquired a quantifying mindset, according to Alfred Crosby. During the late Middle Ages, Europeans began to transition from viewing the world around them qualitatively to viewing the universe as quantified and quantifiable. It was this quantifying mentality that propelled their society forward.

Crosby first describes the classical and early medieval worldview, which he calls the Venerable Model. He then surveys various areas of culture, such as mathematics and art, pointing out the many ways in which the Venerable Model’s qualitative view of the world was giving way, during the late Middle Ages and Renaissance, to a view that everything could be quantitatively measured, a worldview he calls the New Model. His main thesis is that this increasing quantification went hand-in-hand with Europeans’ increasing drive to create accurate visualizations of the world around them, as reflected in such things as accurate navigational maps, perspective painting, musical notation, and mathematical notation. Such visualization both resulted from and promoted quantitative analysis.

Crosby manages to be mostly convincing without being heavy-handed. He is more intent to entertain and inform than to hammer his thesis at every turn. There are deep, overarching ideas in the book, but Crosby is mainly a storyteller here. He seeks to convey the flavor of how Europe transitioned from the Venerable Model to the New Model by telling sequences of connected anecdotes.

Roughly speaking, each chapter illustrates one aspect of the transition, for example how Europeans’ view of time changed, or how modern musical notation come into being. Most chapters are composed of a string of historical episodes and mini-biographies of people who either played an important role or in some way characterized the transition. The narrative is animated; Crosby is obviously having fun. Indeed, he frequently points out humorous idiosyncrasies of the person or episode under discussion. These short interludes sometimes have no direct bearing on his thesis or even on the main line of thought in the chapter, but they add color and give the reader a good chuckle.

Every so often the series of anecdotes is punctuated by a pithy synopsis that leaps off the page. These crisp summaries condense all that is significant in the preceding anecdotes, annunciating the general principle that the anecdotes were meant to illustrate. They are for the most part convincing. For readers who are not fully convinced or who for any other reason want to engage more fully with the material, copious pointers are provided to the literature, both ancient and modern. Almost all of the book’s footnotes consist of bibliographic data (as opposed to elaborating comments), and there are perhaps an average of two footnotes per page, a large number for a book written for a general audience.

Crosby uses the first half of the book to portray what he calls the “necessary but insufficient causes” for the transition from the Venerable Model to the New Model. That is, he surveys various cultural developments without which the transition could not have occurred, the “oxygen and combustibles.” Describing what he considers the “striking of the match,” the development that propelled the whole enterprise forward, he saves for the second half of the book.

To convey just how revolutionary it was for late Medieval and Renaissance Europeans to view the world as quantifiable, Crosby first describes the Venerable Model, the view of reality assumed by classical Greece and Rome and early medieval Europe. Reality consisted of qualities, not quantities. For example, Aristotle preferred qualitative, rather than quantitative, descriptions and analyses of things like weight, hardness, and temperature. How different is the modern mind. Indeed, things like weight, hardness, and temperature we sometimes denote as “quantities!”

“Medieval Europeans used numbers for effect, not for accuracy.” For example, when a large number was used to describe how many soldiers died in a battle, the intent was to qualitatively describe the size, intensity, or importance of the battle; the accuracy of the cardinal number was not the point. Even when numbers were used more accurately, the numbers did not merely convey quantity but also took on symbolic meaning. Three was the number of the Trinity, and so on. “Today we utilize numbers when we want narrow focus on a given subject and maximum precision in our deliberations. The old Europeans preferred broad focus and settled for imprecision in the hope of including as much as possible of what might be important. Often they were reaching not for a handle on material reality, but for a clue as to what lay beyond the scrim of reality.”

To help explain the miniscule impact mathematics had on the general culture, Crosby indicates that abstract mathematics and everyday measurement were, except for a few isolated episodes, completely separate endeavors. He also points to the lack of mathematical notation and to the difficulty of doing anything beyond basic calculation with Roman numerals. He notes that the counting board, which like the abacus enabled one to calculate with place value and zero without explicitly thinking about these advanced concepts, was absent from Europe from the sixth through the tenth centuries.

The move toward quantification can be seen in the emergence of the merchant class. Trade was mostly by barter in Europe after the Roman Empire. But around the tenth century, nascent kingdoms began minting coins. A merchant class arose. Soon everything had a price. Credit taught that even time has a price. But if time has a price, why not everything else? “Price quantified everything.”

Why did such an influential merchant class arise in Europe rather than elsewhere? Unlike other parts of the world at the time, late medieval Europe was resistant to political, religious, and intellectual centralization. “Kingdoms, dukedoms, baronies, bishoprics, communes, guilds, universities, and more — a compost of checks and balances.” Not even the pope had absolute power, especially once the beginnings of the Protestant reformation were underway. In other parts of the world, the few true power holders tended to band together to prevent any middle class from forming. In Europe, however, the nobility and clergy were so fractured there was little resistance to the burgeoning bourgeoisie.

The introduction to Europe of the mechanical clock is instructive. Medieval Europeans tended to mark time by noting events rather than by using any precise metric. Something might have happened “after the king left for the Holy Land” or “on a certain cold day after the harvest.” Time was viewed as an unsegmented flow; it was measured with things like water and sand. The mechanical clock was truly original. Before its introduction, daylight was divided into twelve hours that shrank and expanded with the seasons. (One of the delights of the book are the occasions when Crosby examines the etymology of a word. Our word “noon” comes from the Latin “none,” the ninth hour of daylight, that is, 3:00 PM. Over time, “none” migrated from mid-afternoon to midday, probably at different rates in different locations. One possible impetus for the migration was that during fasts monks of certain orders were not allowed to eat until “none.” Such was the fluidity of time.) After a few towns acquired clock towers in the late 13^th century, they became monuments to civic pride. Every town raced to build one. Because they were expensive, the merchant class was instrumental in their production. With mechanical clocks, hours were no longer fluid but of equal length. The clock taught people of all classes to think of time as quantized. Time, “like money, taught them quantification.”

But what, for Crosby, is the “striking of the match,” the main propeller of the transition to a quantified view of the world and all of the progress it entailed? In a word, visualization. In various disciplines, late medieval and Renaissance Europeans created accurate visual representations of the world around them. These visualizations were the product of the growing tendency toward quantification. But more importantly, they captured reality in a representation that could be seen all at once and therefore manipulated, that is, measured. The quantification and measurement of the world was then inevitable.

To see how a better visualization can revolutionize a discipline, consider literacy. Early medieval manuscripts contained words all running together with no punctuation. Reading was laborious, and the few who could do it read aloud. Monastery libraries were noisy rooms. As words became separated by spaces and punctuation came into vogue, manuscripts became a much better visualization of language. Reading became easier. It could be done silently. And quickly. Much could be learned in a short time. Libraries became places of enforced silence. Meaning was now located not in sounds but in the written word. Prior to 1300 “pictures showed God and His angels and saints always communicating with humans by speech. Shortly after 1300 an Anglo-French prayer book showed the Virgin Mary pointing to words in a book.”

Consider the development of music. Early music lacked sophisticated written notation. Gregorian chants were handed down orally through the generations; learning the entire repertoire of melodies required a long apprenticeship. In syllabic chant, “each syllable has one note, which is sung for as long as that particular syllable requires. That note is not necessarily an exact multiple or division of another note; it is as long as it needs to be. Gregorian chant provides as clear an example of time measured solely by its contents as we are likely to find.”

As chants became increasingly complex, a method superior to mere memory was needed to perform the songs, let alone to learn them. Polyphony and modern musical notation escorted each other onto the stage and helped alter Europe’s view of time and also the nature of music. “The musical staff was Europe’s first graph,” with pitch on the vertical axis and time on the horizontal. Thus time was viewed no longer as its contents but as “a measuring stick of independent existence with which you could measure things or even their absence.” Time had units. “Time measured its contents, not contents time.” Whereas musicians had previously been the guardians of music handed down, with time as their measuring instrument they now became composers, creators not only of sounds but of symbols on a page. Music was now a written, abstract, symbolic visualization. “Deaf Beethoven writing his last quartets became a possibility.” Some polyphony is so complex (with, for example, melodies playing forwards and backwards throughout a song), that it could be fully embraced only visually. “No ear can fully comprehend such complexity in time, only the eye.”

As one example of the impact of the new view of time that was promoted by modern musical notation, Crosby points to Kepler, a music devotee himself. The idea that time measures its contents and not the other way around enabled Kepler to develop his second law of planetary motion. The idea of equal areas in equal times could not have arisen from the old view of time as fluid. Thus visualization in music promoted science.

Crosby goes on to discuss both painting and bookkeeping as disciplines that adopted visualizations that promoted a quantitative view of the world. Renaissance perspective painters attempted to portray reality as it actually is, or at least as our eyes perceive it. In early medieval painting, size was determined by importance. A large Christ in the center of a painting would be flanked by a small saint on the side. In Renaissance perspective painting, size was determined by what the eye perceived, with largeness indicating closeness to the viewer. Modern musical notation and the accompanying understanding of time honored silence. Rather than time being defined by the sounds and syllables it contained, time was an external metric. Visualizing time as quantized allowed silences to have an identity. In a similar way Renaissance perspective painting honored empty space. (It is not only in mathematics that the idea of zero is difficult.) An early painting of a medieval town would typically cram all of the buildings together, giving the impression of the town’s tiny streets without explicitly portraying them. Renaissance perspective painters included the streets and other kinds of “absences.” They measured everything and had no problem portraying the empty spaces that their measurements clearly revealed.

In addition to perspective painting, double-entry bookkeeping was another revolutionary visualization. As trade grew, and business enterprises became more far flung, it became nearly impossible for a merchant to keep up with orders and shipments, inputs and outputs, credits and debits. The old way was to record some of these things in narrative form and attempt to keep the rest in memory. Never was the entire enterprise grasped at once. Double-entry bookkeeping reduced all of this complexity to sequences of numbers, the sum total of a merchant’s reality to a visualization that the eyes could take in all at once. Double-entry bookkeeping “taught us how to oblige grocery stores and nations, which are always whizzing around like hyperactive children, to stand still and be measured.”

Such powerful visualizations allowed Europeans to quantify and measure the world around them, thereby propelling them into the Scientific Revolution and the technological advantages and hegemony it entailed. Such is Crosby’s thesis. He tells the story well. Perhaps professional historians have quibbles (it is beyond the expertise of the reviewer to know), but the general reader will be entertained and enlightened.

David A. Huckaby is a professor of mathematics at Angelo State University.