One of the chief aims of mathematics has always been to reveal and describe an order in the natural world. 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… A resurgence of interest in classical education has been evident in recent years. This has been due, in part, to a number of influential writings on regaining “lost” knowledge in our culture which have, in turn, inspired an increasing number of schools founded on a classical model. When surveying the landscape of classical education, it becomes evident that there is a clear vision available for the purpose of the study of humanities. What does not seem as clear, though, is the nature of mathematics in a classical education. How is mathematics to be approached? Is mathematics a science? Is it a set of skills to be memorized? Can the study of mathematics be more deeply integrated into a classical education? If so, is this necessary or desirable? Nearly everyone would agree that the study of mathematics belongs in a classical education, but the purpose of this study is not always clear.
The study of mathematics should instill in students an ever-increasing sense of wonder and awe at the profound way in which the world displays order, pattern, and relation. Mathematics is studied not because it is first useful and then beautiful, but because it reveals the beautiful order inherent in the cosmos. The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics. Just as the eye was made to see color and the ear to hear sounds, so the human mind was made to understand quantity.
If we look back to the early days of mathematics, say four thousand to five thousand years ago, we will see what the Egyptian and Babylonian civilizations offered in mathematics. They offered a very practical approach to mathematics answering questions that rarely extended beyond what was necessary to operate in daily life. During this time, rudimentary arithmetic and algebra were built up to answer questions in commerce and agriculture. The useful purposes for which they employed mathematics dealt with: monetary exchange, simple and compound interest, computing wages, expressing weights and lengths, dividing inheritances, and determining volumes of granaries and areas of fields. Their mathematics was also used to study astronomy, making it possible to create calendars to accurately predict natural occurrences such as floods, something necessary for agricultural purposes. Accurate calendars could also be used for purposes of religious ceremony, such as building temples so that the sun would shine on the altar at the appropriate time. These civilizations developed an elementary arithmetic, notation, some early algebra, and basic empirical formulas in geometry.
Whereas the Egyptians and Babylonians produced a fairly crude and very practical mathematics based on experience, the Greeks removed mathematics from its practical underpinnings. A major step in the advancement of mathematics was the recognition that mathematics—in numbers and geometric figures—can be dealt with in the abstract. This was not a small step in human thinking, and this initial step was attributed to the Pythagorean School of ancient Greece.
In the sixteenth and seventeenth century, figures such as Rene 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 a new setting, 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.
We must endeavor that those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind only; … Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible and tangible objects into an argument.