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 mathematical system. This ‘Arabic system’ included the mathematical use of zero and positional numbers indicating units. Al-Khwarizmi's ‘al-jebr’ (literally the reducing and recombining of parts), with the simple procedure of changing both sides of the equation by the same amount, allowed complex relationships to be quantified and unknown variables (‘x’) to be determined in terms of other variables. At last, Pythagoras’ dream of a mathematical description of nature was realizable.

These cumulative accomplishments marked the zenith of Arab science. In the 12th century, Muslim science was smothered by the growing consensus that all worthwhile knowledge can be found in the Koran. Science survived through serendipity: after nourishing the flame of science throughout the millennium of anti-science ‘Dark Ages’ in Europe, the Muslim passed it back to Europe just when a cultural revival there was beginning to crave it.

The medieval cultural revival of the 12th century began a rediscovery of the most basic scientific foundations. The Catholic Church, sole source of schools and learning, was the epicenter. For example, Peter Abelard used religious reasoning to rediscover the connection between nature and human logic: the universe is logical and ordered because God made it that way; humans were created in God’s image so they can decipher the universe’s logic. In his book Sic et Non [1122 A.D.], he argued against religious dogmatism and for personal critical evaluation: “All writings belonging to this class [of scriptural analysis] are to be read with full freedom to criticize, and with no obligation to accept unquestioningly. . . These questions ought to serve to excite tender readers to a zealous inquiry into truth and so sharpen their wits. The master key of knowledge is, indeed, a persistent and frequent questioning. . . By doubting we come to examine, and by examining we reach the truth.”

The scientific renaissance began in the 12th-century cathedral schools, particularly the School of Chartres [Goldstein, 1988]. By the early 13th century, the surge of knowledge had moved to the first universities, such as those in Paris, Oxford, and Salerno. Yet, in the brief period surrounding the construction of the cathedral of Chartres, its school made several impressive innovations:
 * establishment of the natural sciences as areas of study at least as important as liberal arts;
 * creation of the first substantial library of science since Roman times, with a particular emphasis on collecting ancient scientific writings;
 * reintroduction of the Pythagorean idea of a mathematically ordered structure of the universe; and
 * search for causality throughout nature, based on the idea that “nature is intelligible for the human mind precisely because both proceed according to the same inherent rational law” [Goldstein, 1988].

The architects of the new science at the School of Chartres were Thierry of Chartres and his student William of Conches. Thierry laid the groundwork by establishing religious justifications for the study of nature. He asked, “Given God, how do we prove it?” and he encouraged scientific contribution to this goal. William of Conches [~1150 A.D.] was less cautious: “To seek the ‘reason’ of things and the laws governing their production is the great task of the believer and one which we should discharge together, bound by our curiosities into a fraternal enterprise.”