If you've taken a first-year college history course - or read through a basic history textbook - you may have noticed a small gap. It's only a thousand years or so.
For a long time, the history of Western culture was told like this: around the fifth century BCE, math, philosophy and science developed, thanks to the hard work of some very smart Greeks such as Thales, Plato, Archimedes and Aristotle. Then Rome took over Greece, and Rome fell, and things went dark for a thousand years or so. Then the Renaissance came along, and thinkers like Galilei and Johannes Kepler took up where the Greeks had, in effect, left off.
Thanks to new historical research - and broader awareness of non-Western countries and of the very rich intellectual cultures being developed east of the Urals - this picture of the history of math, philosophy and science is changing, slowly. But still, teachers tend all too often to skip over one of the most interesting stories in intellectual history - the way that math and logic, including the best insights of Greek logicians, became the property of Muslim countries during the long twilight period, from Rome's fall to the Renaissance, when most Europeans could no longer read Greek.
Without the work of these great Muslim scholars, math today might be a very different animal. The Islamic Arab Empire, beginning in the eighth century, was a world intellectual capital, and Arabic became a language of learning to rival Latin. Some of the best mathematical reasoning in the world was done here.
We may as well start with Muhammad ibn Musa al-Hwarizmi (9th century), a Persian astronomer deeply learned in the mathematical lore of ancient India. From his name (in its Latin form) we get the word algorithm, and from one of his book titles we derive algebra. It's appropriate that he should be associated with the history of algebra - after all, his books preserved most of what the ancient world knew about algebra (as well as his own brilliant innovations), and his works helped to spread the use of Arabic numerals (the numbers we know and use today) to the West, thus making algebra a good deal more feasible. (To understand why this is important, imagine trying to do algebra problems while using Roman numerals: XIIa times XXVb equals c? No, thanks.)
Then there's Al-Karaji, who around 1000AD invented the proof by mathematical induction - one of the most basic logical maneuvers in math. Poet Omar Khayyam, writing in the twelfth century, laid the groundwork for non-Euclidean geometry. During this period, Muslim mathematicians invented spherical trigonometry, figured out how to use decimal points with Arabic numerals (though the decimal itself had long been invented by Hindu mathematicians), and developed cryptography, algebraic calculus, analytic geometry, among other things.
As important as any of these contributions, though, was the rescue of Aristotle's texts from obscurity by Arab scholars. For long periods during the middle ages, Aristotle was considered by Western intellectuals as one of the world's great thinkers - but most of them hadn't read him. The few of his works that had survived the twin falls of Greece and Rome were available in sometimes poor, or rather freehanded and inaccurate, Latin translations, and many of his most important works weren't available at all. Here and there a Greek manuscript survived, but almost nobody, at this point, could read Greek. (Widespread teaching of Greek had to wait for the Renaissance - even famously learned scholars such as the poet Petrarch struggled over it.)
The same went for such seminal works as Euclid's Elements, the greatest known treatise on geometry. During this long period, when it was thought that these brilliantly logical works were gone forever, Islamic scholars kept their own copies and translations. When European scholars began traveling to Spain and Sicily (then under Muslim rule) during the 12th century, these works and others were rediscovered in the West, leading to great intellectual ferment, including the theology of Thomas Aquinas - and to an understanding of logic that helped the discipline of mathematics to survive and, slowly, thrive again in the Western countries.
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