Mathematical Cultures and Their Adherents

August 1, 2009

Working alongside, and even preceding, the brilliant theoreticians of ancient Greek mathematics may have been another group of mathematicians: the practitioners, members of a professional guild of craftsmen who used a kind of rule-of-thumb mathematics to build monuments like the Parthenon (5th century BCE).

Book Review
Philip J. Davis

"The Two Cultures of Mathematics in Ancient Greece." By Markus Asper, pages 107–132, in The Oxford Handbook of the History of Mathematics, Eleanor Robson and Jacqueline Stedall, editors, Oxford University Press, Oxford, UK, 2009, 928 pages, $150.

When you think of ancient Greek mathematics, you are thinking, with high probability, of the works of the great names: Euclid, Apollonius, Diophantus. . . . Markus Asper, a professor of classics at New York University, one of whose specialties is ancient Greek science and technology, calls these men the theoreticians. But he points to another group, existing and working side by side with, and even prior to, the theoreticians: the practitioners. These were the people who went out and actually measured, surveyed, planned, designed, and constructed---and who took the praise or the lumps for what they had done. On the basis of exceedingly few but convincing written records, augmented by equally convincing insight into how ancient technology worked, Asper postulates a professional guild of craftsmen whose members, using a kind of rule-of-thumb mathematics suited to specific projects, built roads, canals, sewers, houses, temples, monuments. Notice that the Parthenon (5th century BCE) did not need to wait for the appearance of Euclid's Elements (around 300 BCE) for its structure to take shape.

The two cultures of Asper's title, then, are (1) the rule-of-thumb material, derived from long experience and passed down within a craft, and (2) the (remarkable!) theoretical material whose soul was invested in logical deductions from a set of definitions and axioms, a.k.a. "proof." Asper's article compares and contrasts these two coexistent types of mathematics.

The former, Asper writes, "derived from older traditions that ultimately originated in the ancient Near East," solved "real life" problems, communicated specific procedures from which general methods might be inferred. These men worked within an institutionalized guild structure and kept their procedures proprietary, using written texts only as a secondary means of knowledge storage and instruction. The theoreticians, by way of contrast, emerged in the 6th to the 5th centuries BCE and derived their inspiration only in part from a practical background. Theirs was a "game" with aesthetic implications that were removed from "real life," a game that derived general theorems about ideal geometric entities.

The two cultures differed sharply in rhetoric. The practical people produced recipes, often oral, while the theoreticians produced texts that often included labeled diagrams.

One further contrast drawn by Asper is a real eye-opener: The practitioners were lower-class craftsmen---often slaves---while the theoreticians were upper-class fellows whose work, Asper says, constituted an "epiphenomenon," i.e., secondary considerations in their lives. The social distance between the two classes was enormous.

Asper underlines the class difference by retelling an anecdote that I mentioned recently in another review. A student of Euclid's, having learned some theorem, asked what his profit would be for his learning. Turning to his assistant, Euclid said: Give the fellow a triobolon, for he needs to make a profit on what he learns. Asper comments:

"The triobolon, here probably synonymous with ‘small change,' was the day's wage of an unskilled worker in classical Athens, which would bring out the contempt for ‘work' on behalf of the mathematicians even better."

Fast forward now from classical antiquity, and realize that over the years a variety of popular mathematical practices have been put in place to serve the day-to-day needs of the technicians in society. Then ask about the mathematics underlying some specific projects: What mathematics did the Roman engineers employ in constructing the Pont du Gard aqueduct in Avignon? What about the Vikings, as they constructed their magnificent ships? And the guilds that built the medieval gothic cathedrals? In more recent times, what mathematics did Pierre l'Enfant use in 1791 in laying out Washington, DC? What was the mathematical underlay of the World War II B–29 bomber, or of the impressive Gateway Arch in St. Louis, designed in 1947 by the architect Eero Saarinen and the structural engineer Hannskarl Bandel? For the earlier constructions, it may be difficult to answer these questions, but they are not beyond conjecture; as we move toward the present, copious written records become available.

Facts are available in many cases. The mathematics underlying the design of the medieval cathedrals was referred to as sacred geometry, and Euclid (his axiomatic/deductive self totally cast aside) as its patron saint. Frederick II of the Hohenstaufen dynasty, himself knowledgeable in mathematics, ordained the octagonal design of the Castel de Monte (1240), which is replete with shape symbolisms. A fully developed aeronautical theory, together with wind tunnel and empirical flight data, was available in 1939 to Boeing's design engineers, who were able to produce the B–29 without the help of the electronic digital computer.

Every age has a substratum of mathematical lore and practices, informal, rule-of-thumb theory, more primitive than contemporary, that makes it possible to get things done. It can range from the "street arithmetic," untaught in schools, that is employed by young Brazilian kids who hawk stuff in the streets of Rio, to rules of thumb built into sophisticated computer algorithms for linear algebra. Yet the "standard" histories of mathematics neglect this layer almost entirely, preferring to detail the theoretical and ignore the practical.

The two cultures outlined by Asper morph roughly into today's pure and applied mathematics. Some of the snobbism cited by Asper persists between these communities, the former regarding itself as the upper crust and in possession of a monopoly of brains and creativity, working with ideas that are sharp and certain, and regarding the latter as peons who work with material that is vague and murky. I once visited the Gateway Arch, during a trip to St. Louis. Later, I asked a prominent St. Louis mathematician what mathematics he thought had gone into its design and construction. All he was able to tell me was that the arch was in the shape of a catenary. In my mind, he flunked. (Fortunately, his disdain for this real-world monument did not correlate with social class, as it might have in classical Greek times.)

How is mathematics pursued today within its various constituencies? The answer depends very much on which one you consider: logicians, set theorists, combinatorialists, computer scientists, physicists, biologists, economists, the individual investors or heads of fiducial institutions, the U.S. Office of Management and Budget. How does each of these constituencies develop its mathematical applications? Overall, we see in their work a variety of sources of inspiration and intuition; a wide range of assumptions; the use of older work; judgments of credibility, validation, and value; styles of computation and communication. We see also what the educational establishments in the various fields teach as currently important. But very little concerning the methods and the spirit of what is practical makes its way into most histories of mathematics. The mathematics of a subculture can be likened to a natural language or a dialect--including the possibility that it will disappear. It's a pity (but understandable) that such material is not memorialized and preserved in contemporary histories of mathematics.

Some scholars have made information about contemporary practices available. Among them is Julie Gainsburg, now of California State University Northridge, who over a period of several months and almost on a day-to-day basis, followed nine structural engineers across four major tasks. She recorded the formulas, reference books, computer programs they brought to bear on the tasks, along with the experiences they relied on. She noted the corners they were able to cut and remain within the local building code. One of the engineers, who was conducting a seismic analysis of an existing building, expressed his views on the role played by mathematics in his work:

"There's an allure to the completeness and the detailed level of our analysis. But at some point we have to rise above the math stuff. The math, at some level, is very much a means to an end. Engineering is, very . . . un-pure math. It's very expediency-driven math. So we're always having to step beyond that to say, ‘well, what does it mean?' "

In his thought-provoking article, Asper stresses two classical cultures. Today, there may be dozens of different mathematical cultures, and it's important that they exist.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at

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