The Purpose of Mathematics in a Classical Education
By Dr. Thomas Treloar
Seventeenth-century philosopher and mathematician René Descartes made the following observation concerning the mathematics of his own day:
I afterwards bethought myself how it could be that the earliest pioneers of Philosophy in bygone ages refused to admit to the study of wisdom anyone who was not versed in Mathematics… I was confirmed in my suspicion that they had a knowledge of a species of Mathematics very different from that which passes current in our time.
Nearly three hundred and fifty years ago, Descartes lamented a lost knowledge in mathematics. He understood that the mathematics passed down from his predecessors was only a shadow of what it had formerly been.
That observation is as true today as it was in the time of Descartes, and has led to a resurgence of interest in classical education. But while there is a clear vision available for the purpose of the study of humanities in classical education, the purpose of mathematics remains unclear.
How is mathematics to be approached? As with all other disciplines, we must look to history for advice. If we study the Egyptian and Babylonian civilizations, we see that they offered a very practical approach to mathematics that rarely extended beyond what was necessary to operate in daily life. These civilizations developed an elementary arithmetic, notation, some early algebra, and basic empirical formulas in geometry.
The goal of a classical education is to bring the mind out of the shadows of the cave and into the reality of the light.
It was the Greeks who removed mathematics from its practical underpinnings. They recognized that mathematics — in numbers and geometric figures — can be dealt with in the abstract. Mathematical results became a chain of propositions, arising out of preliminary assumptions and advanced by reasoning.
Probably the best illustration of this is Euclid’s Elements. In approximately 300 B.C., Euclid brought together much of what was known in mathematics up to that point in 13 volumes. He systematically organized this material, beginning with a short list of first principles and piecing together a body of knowledge as an extended chain. The Elements became the standard textbook in geometry for the next twenty-two hundred years. It is only in the last one hundred years that it has been discarded as required reading for all educated people.
With the strengthening of the connection between mathematics and reasoning by the Greeks, mathematics would next become closely tied to philosophy, theology, and the natural sciences. Consider the following from Plato’s Republic (Book VII):
The knowledge at which geometry aims is knowledge of the eternal, and not of anything perishing and transient. Geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down. Therefore, nothing should be more sternly laid down than that the inhabitants of your fair city should by all means learn geometry.
After the Greek contributions, there was very little advancement in mathematics until the Renaissance and the Enlightenment. Nevertheless, mathematics retained its importance throughout the Middle Ages, as the Quadrivium demonstrates. The Quadrivium consists of four mathematically-based subjects and forms the basis of the continuing medieval liberal arts education after the study of the Trivium. The Quadrivium consists of arithmetic, music, geometry, and astronomy which were described by Proclus in the following way:
- Arithmetic as the study of quantity
- Music (harmonics) as the relationship between quantities
- Geometry as the study of magnitudes at rest, and
- Astronomy as the study of magnitudes in motion
In the sixteenth and seventeenth century, figures such as René Descartes and Pierre de Fermat, pushed further by the desire to discover order in the universe, brought geometry out of abstraction and pure reasoning by introducing what is known today as the Cartesian coordinate system. Coordinate systems allowed more readily for measurements of natural events in the universe and paved the way for such greats as Newton and Leibniz who further developed these ideas in Calculus.
It was this period that helped usher in the “mathematization” of science, an ever-increasing attempt to discover order in the physical universe through the language of mathematics. This led to a tremendous growth in knowledge in both mathematics and science, but these advancements have come at a cost.
The honing of reasoning skills — a critical component of a liberal education — is often downplayed or completely neglected in mathematics education. The cohesive structure of mathematics is often overlooked to get to the “useful” results. But these “useful” results are often useless because this language of mathematics is without its cohesive structure and, therefore, the order of the universe remains unclear. Thanks to this emphasis on usefulness, it may be that modern mathematics education is much closer in spirit to the Egyptians than the Greeks.
A classical upper mathematics curriculum must fight these utilitarian tendencies and be faithful to Greek ideals. It must provide the opportunity for students to hone reasoning skills. It will promote an understanding of order and harmony in the universe. Mathematics, as a language, reveals this order and harmony, yet it should also be lifted from this concrete foundation and brought into the world of the abstract.
In terms of content, students would explore Greek mathematics and mathematics in the seventeenth and eighteenth centuries. Algebra I allows us to take our knowledge of elementary arithmetic and further organize it. Geometry hones our reasoning by building its propositions from an initial set of postulates. Algebra II and trigonometry allow us to place our understanding of geometry in an analytical framework in the spirit of Descartes, developing a description of motion and other questions of order in the universe.
The classical curriculum should also include discussions of the original settings which led to the development of mathematical concepts. These could include projective geometry, which arose as a tool for artists in depicting the real world and spurred innovation in mapmaking. The mathematical concepts which followed from the study of sound led to an increased understanding of light and electrical current. The plethora of calculators in modern times has meant a change in the use of logarithms, but it is still incredibly worthwhile for students to understand how and why they were originally created. Bringing this context into the classroom allows students to see the interrelatedness of various ideas in mathematics and science and better understand the creativity involved in the discovery of these concepts.
The goal of a classical education is, to use an image from Plato, to bring the mind out of the shadows of the cave and into the reality of the light. Mathematics, more than any other subject, has the power to transform and enlighten the mind in this way. As Morris Kline puts it in Mathematics for Liberal Arts:
The philosophers pointed out that, to pass from a knowledge of the world of matter to the world of ideas, man must train his mind to grasp the ideas. These highest realities blind the person who is not prepared to contemplate them. He is…like one who lives continuously in the deep shadows of a cave and is suddenly brought out into the sunlight. The study of mathematics helps make the transition from darkness to light…[it] purifies the mind by drawing it away from the contemplation of the sensible and perishable and leading it to the eternal ideas.
In a world of increasing materialism and utilitarianism, mathematics is seen as the quickest and most efficient road to success. Rightly understood, however, it is the beginning of the longest and most challenging road — the road to the love of the eternal and the immutable that education, at its best, strives to give.
Dr. Thomas Treloar is associate professor of mathematics at Hillsdale College. This article was based on a lecture titled “Classical Math: Euclid, Pythagoras, and Newton,” available in its entirety here.
