On Geometry

The faculty of Hagia Sophia always takes delight in seeing a student relate material learned in one class to that learned in another; the wider the apparent gulf, the better. Recently, one student related Euclid’s The Elements—a treatise on geometry—to St. Basil the Great’s On the Holy Spirit; in particular, he noticed that both made use of the axiom that “if two things are equal to a third, they are equal to each other.” Such a connection is pleasurable since it helps confirm what we the faculty suspect and Classical Education assumes, namely, that all subjects of learning, no matter how disparate they may seem, are essentially united.

This assumption is born out in the structure of Classical Education itself and its division into various arts, all which relate to one another and form a pathway to the highest knowledge possible to man: the knowledge of God. It is thus no accident that one finds relations between geometry and theology, for the latter assumes the former and can be considered in many ways its fulfillment. Such a belief, that geometry is fulfilled by theology, is difficult to grasp given the relative isolation in which these subjects are often taught. Mathematics is commonly thought only to apply to those subjects that can use mathematical tools in a practical way so as to obtain some result. Mathematics is hardly considered to be a theoretical study, except in the highly advanced realm of number theory, a subject that seems to have little import for the general student.

However, the Ancient Greeks, in whom Classical Education finds much of its foundation, believed mathematics to be more than a practical study directed at obtaining results in quantitative problems. Rather, mathematics—a word derived from a Greek word meaning “that which is learned”—was considered to be learning par excellence. Through mathematics, one was thought to learn the essential building blocks of knowledge and their various relations. Indeed, Pythagoras, a Greek mathematician and philosopher in the 6th century B.C., considered the cosmos to be essentially composed of numbers; and Plato thought geometry so essential to the educated mind that on the doorway to his Academy he placed the following inscription:

Let none ignorant of geometry enter here.

This “numerology” of the Ancients might strike us as an eccentric aberration of otherwise reasonable minds; however, one should not simply dismiss this apparent fascination with numbers and shapes as mere pagan mysticism.

A quick perusal of Euclid’s foundational treatise The Elements—a book which laid the groundwork for all future geometry and mathematical reasoning—will reveal not simply a text devoted to the discovery of the various relations among geometrical shapes. The book is rather, at its heart, a discovery of logic itself, an investigation into the deductive or axiomatic method of knowledge. Though the book speaks in the language of shapes, the relations it ultimately teaches are those more universal relations with which all minds discover and discuss the truth.

The Elements is truly elemental in that it teaches the elementary logic of all rational thought. Its goal is not this world of sensible shapes, but rather a higher world, a world of theory. The book begins with a series of formal definitions that immediately reveal this elementary concern with theory. Points are “that which have no part”; lines are “breadthless length.” Such entities must defy the world of the senses, for any point we draw invariably has some part, and any line some breadth. These perfect, geometrical entities exist only in the “eye of the mind,” and must be accessed by thought alone. Using sensible diagrams as guides and starting points, our mind is led upwards to the more perfect realm of “pure” shapes, unadulterated by the imperfections and aberrations of sensible media. Speaking of this upward path Nichomachus of Gerasa writes:

For it is clear that [mathematical] studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls. (Introduction to Arithmetic I.3)

And Plato, in his Republic, has Socrates defend the study of geometry as follows:

Therefore…[geometry] would be something that draws a soul towards truth and works up philosophic thinking to get onto an upward track what we now keep on a downward one where it doesn’t belong. (Republic 527B)

We begin thus to see the importance that the Ancients attached to geometry and arithmetic, and to understand that they believed mathematics to be not simply an art of practical calculation, but the first step on the ladder to truth. Though the good life is not simply theoretical, it is necessary to see truth, to behold truth—theory literally means “what we see.” We affect this theoretical aim of education by opening our eyes to the activity of the mind and its ability to know, and this opening of the eyes is the primary and elementary aim of arithmetic and geometry. Classical Education, sharing this belief, thus places mathematics firmly at the door of learning; yet, not as some standard or bar of excellence, but rather as a means and pathway to wisdom, that is, as the road to the knowledge of God.

Comments are closed.