Throughout human history, finite beings have grappled with one of philosophy's most profound questions: how do we comprehend infinity? Whether inspired by nature, philosophical inquiry, spiritual contemplation, or mathematical discovery, the concept of infinity has captivated human thought for millennia. This course offers a historical and conceptual exploration of infinity within mathematics. We will examine the ancient Greeks' breakthrough discovery of irrational numbers and Zeno's celebrated paradoxes; Aristotle's foundational distinction between "actual infinity" and "potential infinity"; the contentious historical debates surrounding infinitesimal numbers in the development of calculus; and the seemingly paradoxical nature of infinite sums.
Our central objective focuses on exploring the remarkable beauty of "the paradise that Cantor created for us": Georg Cantor's groundbreaking theory of infinite sets, developed in the 1880s. We will engage with one of mathematics' most profound proofs: Cantor's ingenious "diagonalization" argument. This elegant demonstration reveals that the set of real numbers is uncountably infinite, meaning that in a precise mathematical sense, the real numbers constitute a larger infinity than the integers, or counting numbers. Cantor's argument further establishes that an infinite hierarchy of infinities exists, each transcending the last in magnitude and complexity.
We will use David Foster Wallace's Everything and More: A Compact History of Infinity and William Dunham's Journey Through Genius: The Great Theorems of Mathematics as guides to this intellectual history. These will be supplemented with additional readings exploring the mathematical, historical, and philosophical dimensions of infinity, including primary texts by Cantor, David Hilbert, and Kurt Gödel. These foundational works collectively illuminate how mathematicians came to understand the infinite.
Note: There is no mathematical prerequisite for this course; only a willingness to engage with challenging abstract concepts is required.
This course is available for "remote" learning and will be available to anyone with access to an internet device with a microphone (this includes most models of computers, tablets). Classes will take place with a "Live" instructor at the date/times listed below.
Upon registration, the instructor will send along additional information about how to log-on and participate in the class.
