The Mathematical Cultures of Medieval Europe - Conclusions

There were clearly mathematical geniuses in all three of these medieval mathematical cultures, most of whom shared a common mathematical background of the Hindu-Arabic number system, the works of Aristotle, and Euclid’s Elements. Anyone with an interest in mathematics had certainly studied the Elements, and quite possibly knew other works of Euclid. Also, he was familiar with many texts of Aristotle and believed that any philosophical work had to contend with the great philosopher’s thoughts, either by attacking or defending them. Finally, it was the Muslims who brought the Hindu-Arabic system to Europe from the East; the Jews learned it from them; and the Catholics eventually mastered it as well, learning both from translations from the Arabic and from the material Fibonacci brought back from his travels to Muslim lands.

Yet starting with the same basic information, the mathematicians from the three cultures were interested in different mathematics.  Algebra, of course, had been developed in eastern Islam, but it seems that the only algebraic work available in western Islam was that of al-Khwārizmī from the ninth century. The more advanced Muslim algebraic work was not available in Spain, although Fibonacci discovered some of it in his travels. In any case, the Spanish Muslims were not apparently interested in algebra. But as we have seen, there was definite interest in geometry, both practical geometry (which was also of interest to the other cultures) and quite theoretical geometry. Muslim geometers had mastered the basic Greek techniques of proof and did not hesitate to prove all sorts of interesting results. And along with geometry, there was also trigonometry, important for astronomy, which in turn was necessary for religious purposes.

The Jews, too, were not interested in algebra, at least not until the late fourteenth century. Even then, the Hebrew work in algebra was basically limited to material found in al-Khwārizmī. Of course, just as in Islam, quadratic equations were solved earlier in the context of measuring areas and lengths, but the methodology was the older one of manipulation of geometric figures rather than the newer methodology of “things”. On the other hand, Jewish authors seemed to be very interested in geometry. There were quite a few authors who investigated advanced geometric topics, being careful to give strict Euclidean proofs. And there were also several investigations of topics in combinatorics, both intuitively and, in the case of Levi ben Gershon, with careful proofs. Levi and others also investigated some pure number theoretic problems. And, of course, trigonometry was studied, since, as for the Muslims, this was necessary for astronomy and therefore for calendrical questions.

Catholic Europe was interested in mathematics different from the kinds studied by the Jews and Muslims. First, there was more interest in developing algebra beyond al-Khwārizmī. Even Fibonacci had problems from later authors, and certainly Jordanus de Nemore in the thirteenth century developed additional material. Interestingly, however, many of the algebraic techniques developed in eastern Islam did not reach Europe during the medieval period. On the other hand, there was little interest in advanced geometry. Euclid was mastered, and there was some interest in the works of Archimedes, but there was nothing in Catholic Europe like the advanced geometry developed in Muslim Spain. In addition, even though astronomy was part of the medieval university curriculum, there was little development of trigonometry beyond what was already known in Greece. It was not until the work of Regiomontanus in the mid-fifteenth century that Catholic Europe had a trigonometric work comparable to the works written in Muslim Spain centuries earlier. Probably one of the reasons for this was that astronomy was not nearly so important for calculations involving the Julian calendar as it was for both the Muslim and Jewish calendar. Similarly, the subject of combinatorics, of interest to the Jews and to the Muslims of North Africa, was barely mentioned by Catholic mathematicians, although Jordanus de Nemore did display the Pascal triangle as part of a discussion of ratios. The most important mathematical topic studied in Catholic Europe – and not in Muslim or Jewish Europe – was the set of developments coming out of the study of Aristotle’s physical theories. In particular, as we have noted, mathematicians in Oxford and Paris were very interested in the ideas of motion, and it was the study of kinematics as well as mechanics that was crucial for the work of Galileo and others during the Renaissance.

Although mathematical geniuses existed in each of the three religious groups we have considered, men who could successfully attack any interesting problem, it seems clear that the culture in which they lived was crucial in their actual choice of problems to consider.  We can see in this study of mathematics in medieval Europe, as in other times and places, that mathematics is not, and indeed cannot be, a culture-free subject.

Acknowledgment

This article is a revised version of a plenary address given at the 2016 meeting of the International Study Group on the Relations Between History and Pedagogy in Montpellier, France.

About the Author

Victor J. Katz is Professor Emeritus of Mathematics of the University of the District of Columbia; founding co-editor (with Frank J. Swetz) of MAA Convergence; author of History of Mathematics: An Introduction (3rd ed., 2009), widely recognized as the definitive general history of mathematics text for professionals, instructors, and students; and editor of two sourcebooks for history of mathematics, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (2007) and Sourcebook in the Mathematics of Medieval Europe and North Africa (2016), the latter referenced in this article. - Janet Beery, Editor, MAA Convergence