Infinity: Mathematics, History, Philosophy
247 West 37th St, 5th Floor
New York, NY 10018
“From the paradise that Cantor created for us, no-one can expel us.” —David Hilbert, “On the Infinite,” 1926
How can we, as finite beings, grasp the concept of infinity? Yet humans have been contemplating infinity for millennia, whether inspired by nature, philosophy, spirituality—or mathematics. This course is a historical and conceptual approach to the latter realm, the mathematics of infinity.
Topics will include the ancient Greeks’ discovery of irrational numbers and Zeno’s paradoxes; Aristotle’s distinction between “actual infinity” and “potential infinity”; debates about infinitesimal numbers in the history of calculus; and the seeming paradoxes of infinite sums.
But our main goal will be explore the beauty of “the paradise that Cantor created for us”—the theory of infinite sets created by Georg Cantor in the 1880s. We will grapple with one of the great proofs in the history of mathematics, Cantor’s famous “diagonalization” argument, which shows that the set of real numbers is uncountably infinite–meaning, in a precise sense, that the real numbers constitute a larger infinity than the integers (the “counting numbers”). In fact, Cantor’s argument establishes that there is an infinite hierarchy of infinities!
As a guide to this intellectual history, we will read portions of David Foster Wallace’s Everything and More: A Compact History of Infinity and William Dunham’s Journey Through Genius: The Great Theorems of Mathematics in conjunction with additional readings on the history, mathematics and philosophy of infinity, including primary texts by Cantor, David Hilbert and Kurt Gödel.
Note: There is no mathematical prerequisite for this course, just a willingness to grapple with the concepts!
Course ScheduleThursday, 6:30 - 9:30 pm
October 19 — November 09, 2017