From the Origin provides a forum for lively discussion of issues of importance to the mathematical community. The Michigan Section-MAA Newsletter solicits opinion pieces for publication in this column from anyone in the Michigan mathematical community. In addition, comments on pieces published in earlier issues are welcomed.
Items for From the Origin should be submitted to the editor by the beginning of October to be considered for inclusion in the December issue and by the beginning of February for the April issue. Main opinion pieces should be at most 1800 words long, and responses at most 400. The editors reserve the right to shorten responses, if necessary, in order to fit as many as possible within the available space.
A copy of the magazine Talking About Race landed on my doorstep in July 2000, three days after I returned from a mathematics conference in Granada, Spain. It was a special issue of The New York Times Magazine, capping that newspaper’s yearlong probe of race relations in America. “All across the country,” reported journalist Rubén Martínez in one column, “people of different races, ethnicities, and nationalities are being thrown together — and torn apart — by the churning forces of first the postindustrial and now the information economy. . . . It is a terrifying experience, this coming together, one for which we have as yet only the most awkward vocabulary.”
The mixing of different peoples and cultures can be terrifying, but there is another way to look at the issue. Recently, southern Spain has been torn apart by a wave of violent attacks against Moroccan immigrants in El Ejido and other towns. In the face of this, Alhambra 2000, the first European-Arabic Congress of Mathematics, brought 300 educators from Arab and European nations to Granada in July. In digging up the hidden history of fruitful interaction between the mathematics of these two cultures, but also in the living example it provided by unifying scholars from traditions and homelands that have sometimes been mutually hostile, Alhambra 2000 suggested that the most successful undertakings are those that tap into the cultural, linguistic, and ethnic diversity of the planet. I believe this lesson has important implications for higher education.
Alhambra 2000 was a satellite of the third quadrennial European Congress of Mathematics, a larger gathering in Barcelona organized by the Catalan Mathematical Society (SCM). These were the first important international math conferences held in Spain since the end of the Francisco Franco dictatorship (1939–75). Under Franco, the nation’s intellectual and academic life had been cut off from the rest of the world. SCM President Sebastià Xambó commented that in the intervening quarter-century, mathematical research in Spain has risen from next to nothing to world levels. In 1998 the Spanish Royal Mathematical Society finally joined 39 other organizations already grouped in the European Mathematical Society.
The core of Alhambra 2000 consisted of plenaries on the historical contributions of European and Arab cultures to current knowledge of mathematics. To promote greater interaction between historians and modern specialists, symposia were also devoted to current mathematical research in fields like computational mathematics, the geometry of submanifolds, orthogonal polynomials, and the representation theory of algebras. Most attendees came from countries of Europe and northern Africa and were native speakers of Spanish, French, or Arabic. Presenters were comfortable speaking English (official language of the conference) or, in a few cases, spoke in another language while projecting slides in English. I noticed that Hispanic colleagues using PowerPoint had a Windows operating system with a Spanish user interface — an important tool in a world in which computer use is rising rapidly among Spanish speakers.
Islamic Geometry and the Ideology of Inclusiveness
The opportunity to forge understanding from many languages and cultures has special resonance in Granada. Cultural mixing was a key reason why Spain rose to become the leading civilization in Europe in the Middle Ages. In stark contrast to the sieges, expulsions, and inquisitions that would later “ethnically cleanse” the peninsula, the Moors presided over a society in which Arabs, Berbers, Jews, and Christians coexisted. The resulting intellectual ferment greatly stimulated science and philosophy, as seen in commentaries like those of Ibn Rushd (Averroës) and Ibn Bajja (Avempace), both Muslim, and Ibn Ezra (Aven Ezra) and Ibn Maymun (Maimonides), both Jewish. The intermixture of ideas and techniques also shaped Iberian literature, agriculture, and art: Silk, for example, made by Arab craftsmen using Asian methods, became Granada’s leading export.
On a tour of the Alhambra, Granada’s palace complex, we were able to study the ingenious ways in which Moorish designers had tiled the palace walls. The abstract and intricate mosaics utilize symmetries of many kinds — rotations, reflections, translations, glides, and combinations of these. Conference speaker Rafael Pérez Gómez, following up on the work of Coxeter and Grünbaum, has shown that the Alhambra contains examples of all 17 types of symmetric tessellation of the plane. At no other cultural site in the world are more than a handful of these crystallographic isometries to be found.
Medieval Islamic artists and the geometers who advised them did not work out a whole theory of tessellations. They took up abstract ornamentation because of their reluctance to depict humans or animals, depictions they felt usurped the creative activity of Allah. Religious and intellectual impulses then pushed them to make an exhaustive search for patterns. Ensembles of repeated or interlocking figures were also a good way to broadcast the Islamic doctrine of altawhid, variously translated as “unification” or “unity in multiplicity”. According to this doctrine, all phenomena of nature and society are stamped by a single spirit that permeates everything. The boundless multiplicity of the universe is an expression of a more underlying unity, and each thing is a potential key to understanding the whole. In Moorish Spain, such beliefs encouraged the embrace of all knowledge and all people.
Belts of Interchange
“Unity in multiplicity” seems equally relevant to our own multicultural society. To the extent that we try to blot out any people or culture, various groups will be arrayed against one other and society will be weakened as a result. In light of this, it needs to be pointed out that Western scholars have too often been dismissive of the accomplishments of Arab and other non-European scientists: “The Arabs made no significant advance in mathematics. What they did was absorb Greek and Hindu mathematics, preserve it, and . . . transmit it to Europe.” (Morris Kline, Mathematical Thought from Ancient to Modern Times) “The general impression left is that the Arabs were quick to appreciate the work of others — notably of the Greek masters and of the Hindoo mathematicians — but, like the ancient Chinese and Egyptians, they did not systematically develop a subject to any considerable extent. Their schools may be taken to have lasted in all for about 650 years, and if the work produced be compared with that of Greek or modern European writers it is, as a whole, second-rate both in quantity and quality.” (W. W. Rouse Ball, A Short Account of the History of Mathematics)
As the Alhambra 2000 conference documented, these Eurocentric views are no longer tenable. Much of arithmetic, algebra, trigonometry, combinatorics, and the analysis of functions were developed by scholars in the medieval Arab world. A presentation by Christian Houzel showed that these scholars had developed a key concept of calculus — the derivative of a polynomial function — over 400 years before Fermat, Newton, and Leibniz. Far from simply preserving and transmitting ancient Greek and Indian learning, the Arabs had richly extended these and even pushed mathematics in some wholly new directions.
Of course, mathematics and science do not magically leap cultural and political boundaries by themselves. Historically, East/West contact bore the most fruit wherever Arabs and Europeans lived or worked together. Arabic numerals, for example, and the simplified arithmetic they made possible, were first encountered by Gerbert d’Aurillac at Arab-influenced monasteries in Spain and Italy, and by Fibonacci of Pisa at his father’s trading post in Algeria. Today, a satellite-based information superhighway speeds the cross-border flow of data, but the biggest breakthroughs still hinge on humans working together “on the ground”. Conference speaker Michel Balinski, for example, a math professor in Paris, collaborated with the Turkish Ministry of Education to invent a graph-theoretic algorithm for accomplishing more fairly and efficiently what the Turkish government had long carried out by hand: the assignment of students to universities based on their exam scores and areas of specialization.
Cultural Mixing in the Classroom
“College students who experience the most racial and ethnic diversity in classrooms and during interactions on campus become better learners and more effective citizens,” General Motors vice chairman Harry J. Pearce pointed out recently, explaining the company’s decision to file an amicus brief in defense of affirmative action policies at the University of Michigan. On one level, GM management is simply acknowledging demographic reality. By 2015, the College Board projects, the proportion of nonwhite students at U. S. college campuses will increase to about 38 percent, accounting for some 80 percent of growth in the student population.
The removal of artificial barriers is relevant not only to admissions policies but also to course content itself, wherever various cultures and disciplines have been excessively walled off from one another. At Alhambra 2000 we were inspired by the presentation of Nathalie Aime, who initiated at her school in Réunion a curriculum that is team-taught by teachers in mathematics, physics, French, Spanish, and art. With symmetry as unifying thread, they study crystallography, mosaics and friezes, musical canons, novels, and architecture. The approach is broad: One week they might learn HTML and computer graphics to create artwork and Web pages, while another week they might visit a local sugar factory to study the crystallization process.
In the United States, a multidisciplinary approach also guides Mathematics Across the Curriculum (MATC) at Dartmouth College. The NSF-funded project has brought together nearly 200 faculty members at Dartmouth and collaborating institutions, from such disciplines as mathematics, literature, history, philosophy, art, biology, geology, and engineering. MATC has produced over 75 modules, books, and videos, including some on the use of pattern and symmetry in African, Islamic, and European cultures. Also of note is a new book from the MAA edited by Victor Katz, Using History to Teach Mathematics: An International Perspective, based on papers presented at international conferences in Seville, Spain, and Braga, Portugal, in 1996.
When our coursework delves into the global diversity of human culture, we open up a range of opportunities for “breaking down walls”. We help link what is learned in a particular discipline to broader questions of science and culture, showing that all knowledge is interwoven, and that wielding this knowledge has practical consequences on an international scale. We also foster a sense of mutual respect and understanding when students learn of the contributions made by every people on the planet — an understanding that is sorely needed now in our fractured and fractious world.
In the quotation cited at the beginning of this essay, Rubén Martínez expresses fear of the gut-wrenching experiences of diverse people being thrown together by the forces of the new economy. Racial strife in places like southern Spain, not to mention Michigan, is real enough. But discussion that focuses one-sidedly on the “terrifying” aspects of cultural mixing does not move things forward. Granada teaches a powerful lesson: The last Moorish stronghold in Europe, it finally fell to the armies of Ferdinand and Isabella in 1492, the same year they dispatched Columbus westward in search of gold and spices. The Islamic library was torched, non-Christians were hounded from the city, and the silk industry collapsed. The beginning of Spain’s Age of Gold was in fact the beginning of its long, deep decline. Alhambra 2000 helped make clear to me that if there is something “terrifying” about race, it is that the failure to see our diversity as a rich resource would doom us, as well, to stagnate as a society.
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