Monday December 9th, 2019

# What Are the Liberal Arts and How Do We Teach Them? (5) Arithmetic and Geometry

Grammar, logic, and rhetoric are the foundations upon which all further learning is grounded. The three together constitute the trivium; the four subjects that traditionally followed the trivium were arithmetic, geometry, astronomy, and music—collectively they form the quadrivium.

The goal of arithmetic is to learn how to calculate. That is simple enough. But there are greater truths that arithmetic reveals: the world is ordered, objective, and intelligible. What is more, arithmetic teaches us that there is a stratum of truth that is impervious to our opinions and emotions—it doesn’t matter what we think or feel, 2 + 2 = 4. Truth pre-exists and supersedes us. If we are wise we will learn it and learn to live in harmony with it and not think we can ignore truth or bend it to our whims. This recognition is essential to the development of wisdom.

Geometry, like arithmetic, reveals to us a world of unchangeable truth. However, unlike arithmetic geometry involves more than simple calculation—it involves the rational deduction of conclusions from premises. In this way geometry has more in common with logic than arithmetic.

Indeed, traditionally geometry was the precursor to and partner of logic. “Let no man ignorant of geometry enter,” so said the sign above Plato’s Academy. Plato believed that a person untrained in geometry could not study philosophy. Likewise, later Christians believed that a grounding in geometry was essential to the study of theology.

The studies of arithmetic and geometry have changed far less than any of the other Liberal Arts. For example, a student would have an incomplete historical and literary education if they stopped in the 4th century BC, but a student could have a solid education in arithmetic and geometry if they stopped then. And yet, there have been changes. We now have two great tools that the ancients lacked—Calculus and the use of Arabic numerals. Classical as we may be, we heartily take advantage of both!