Mathematics and Culture: Geometry and Everything Else
Euclid's Elements is the most influential textbook in the history of western civilization, serving as a model of reasoning not only in mathematics but in philosophy, theology, and politics. But Euclid's geometry rests on assumptions, and one of the assumptions -- even from the beginning -- didn't seem self-evident. People kept trying to prove that assumption, and the ways they tried tell us a lot about the relationship between mathematics and society. Meanwhile, the unchallenged authority of the Euclidean ideal was used by people like Newton, Voltaire, Euler, and Lagrange to support the Enlightenment world view. But in the nineteenth century, suddenly there were new non-Euclidean geometries. They challenged the authority of mathematics, undermined received ideas in philosophy and culture, and had a hand in the birth of modernism. Changes came not only from people like Gauss, Lobachevsky, Helmholtz, and Einstein, but also artists and philosophers. Looking at all of this will illustrate both how culture helps shape mathematics and how mathematics has shaped the modern world.