Mathematical revolutions

Growing up, I thought the plural of the word ‘mathematics’ was based on algebra and geometry, the two branches of elementary mathematics I had studied before I had the freedom to say no to maths and to make the biggest mistake of my life. It turns out the English and the French plural noun harks back to the ancient Greek plural based on how Aristotle understood maths: all things mathematical, ta mathēmatiká, all things learned, from máthēma, the object of learning and knowledge.

In ancient times, no culture was more mathematically minded than that of the ancient Greeks. Axioms, notation, idiom, all goes back to Pythagoras, Eudoxus, Euclid, Archimedes and their successors. Yet, it’s remarkable that while the Romans made Greek knowledge their own, they didn’t much advance the field. The stagnation passed on to the Western medievals. At least for a while.

In the Middle Ages, the most mathematically minded were the scholars dealing with astronomy and computus, the science behind the calculation of Easter day, the most complicated algorithm in Christendom. But otherwise, maths was about studying the same texts the Romans had studied, and achieving as much as Roman mathematicians had achieved, which was not much. The change came around the 11th century.

The weakness of Western mathematics up to that point was due to two main factors: numerals and tradition. On the one hand, algebra was crippled by the use of Latin numerals. Looking at mathematical manuscripts from before the 11th century (and to some extent after that), one is filled with endless pity for the scribes who got lost in copying calculations and algebraic expositions. How much is MCCXXXIV minus CCXLV, quick?

Maths was taught and studied in the second cycle of medieval education known as quadrivium using books that hadn’t been updated for centuries. Early medieval maths was very limited compared to what was going on outside the West, especially in the Muslim world. From the 11th century onwards, Europeans came into contact with Arabic scholarship in Muslim Spain, first, and then in the Holy Land in the wake of the Crusades. The result was an upsurge in interest for Arabic mathematical texts. Generations of European scribes and scholars translated and studied the mathematical knowledge of the East and transmitted it to a bewildered and excited West, which made it its own and quickly added to it. Even Ancient Greek texts which hadn’t reached the West but had been translated into Arabic were once again circulating in Europe, albeit in often strange Latin translations. It didn’t matter, the ball was spinning.

On the back of this renewed interest in mathematical study and scholarship came the adoption by the West of the so-called Arabic numerals, which were not as Arabic as one would think. Developed in India, the numerals we call Arabic today were adopted by the Western Arabs of Muslim Spain (Al-Andalus) and Maghreb before they entered Western Europe to become quasi-universal.

The mathematical revolution of Europe knew no boundaries, as it gave rise to modern science and to a type of rationality that set the West on a unique course in history.