Numbers: History, Mathematics, Philosophy
Numbers are so common, so seemingly intrinsic to human life, that it’s baffling to recall that, for millennia, many humans did not use them at all. How, then, did the use of numbers come to be? Were they, in some sense, discovered? Are they qualities of the natural world or truths outside of time and space? Or, are numbers human creations—concepts devised to help us analyze, organize, and manipulate the world. If numbers “work,” are they therefore, in some sense, “real”?
In this course, we will examine the concept of number from various points of view: mathematical, historical, cultural, philosophical. We will consider the history and evolution of numbers and numeric systems: the prehistoric origins of the “natural numbers” (i.e., the positive integers) in the fundamental human activity of counting; the development of tally systems and numerals as symbolic/syntactic representations of numbers; and the use of different base systems, such as the “sexagesimal” (base 60) system used by the ancient Sumerians and Babylonians, the decimal (base 10) place-value representation of numbers that we use today, and the binary (base 2) system fundamental to modern computation. Students will look at the foundations of arithmetic, algorithms, and the branch of mathematics called “number theory.” And we’ll consider the expansion of the concept of number beyond the natural numbers: to negative numbers, rational numbers, irrational numbers, complex (or “imaginary”) numbers, and infinite numbers. As we proceed, we’ll constantly pose and repose the central question: what, really, are numbers?
Readings will include selections from Arithmetic by Paul Lockhart; Numbers and the Making of Us: Counting and the Course of Human Cultures by Caleb Everett; The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene; The Universal History of Numbers by Georges Ifrah; Number Words and Number Symbols: A Cultural History of Numbers by Karl Menninger; and The Foundations of Arithmetic by Gottlob Frege.
Course ScheduleClass will not meet the week of Thanksgiving (November 24th-30th).