Irrational numbers such as the square root of two—numbers that cannot be expressed as ratios between integers—were first encountered in ancient Greece in 500 BC. Comprehensive mathematical accounts, providing systematic ways to specify irrational numbers, to compare them, and to combine them arithmetically, were finally achieved in the late 19th century. But, unfortunately, these accounts, though meeting certain key mathematical requirements, presented the number system as a purely formal construct, satisfying required axioms, but bearing no particular relationship to objects in the world. In historical terms, this development marked a major turning point in man’s view of mathematics, a final abandonment of a reality-oriented conception of mathematics.
Properly conceived, the real numbers (i.e., rational and irrational numbers) are tools of measurement. Drawing from his book, now published as "Mathematics is About the World", Robert Knapp argues that real numbers specify quantitative relationships among magnitudes. Mathematical relationships among real numbers express and reflect relationships between the magnitudes that they measure. Offering a geometric perspective on magnitudes and their mathematical relationships, Dr. Knapp rehabilitates the mathematical contributions of the late 19th century to provide a reality-based approach to specifying, comparing and arithmetically relating rational and irrational numbers within the real number system.
Mathematics is about the world, a means of identifying relationships in the world. The immediate goal of this course is to explain irrational numbers as they relate to the world. But the broader objective is to provide a model of how one should think about mathematics.
This course was recorded at the 2011 Objectivist Summer Conference in Fort Lauderdale, FL.