When surveying the history of mathematics, the impact of Euclid of Alexandria can hardly be overstated. His magnum opus, Elements, is the second most frequently sold book in the history of the world. For over 2,000 years, his work was considered the definitive textbook not only for geometry, but also for the entirety of mathematics. The Elements’ shelf-life comes with good reason: its rigor, comprehensiveness, and elegance are unmatched by any other ancient text. Even in modern times, various influential thinkers have been stirred by the magnificence of the work. Hobbes, Einstein, and Russell all praised the work–not only for its mathematical precision of planar geometry, number theory, and ancient algebra, but also for its aesthetic beauty and power to show these eternal, preexistent verities to the eye of the mind.
Already, one may start to realize that the ancients and early moderns saw mathematics quite differently than bona fide moderns like ourselves. Value-laden judgments and ascientific notions like beauty are strikingly absent in popular considerations. Mathematics has turned increasingly praxis-minded in contemporary teaching. Survey a hackneyed word problem one might find in standardized assessments:
If train A leaves Cincinnati for Chicago at 11:05AM, traveling at 75 mph, and train B leaves Chicago for Cincinnati at 12:20PM, travelling at 85 mph, what time will the two trains meet (assuming Chicago and Cincinnati are 300 miles apart)?
One needn’t have qualms about using word problems as a pedagogical device. In fact, I am confident that many young primary students experience a particular excitement at the prospect of a good “train leaving the station” problem. Rather, my qualm lies in looking at mathematics merely as a series of practical word problems to be solved. I fear that, due to the pedagogical ease of focusing on word problems, teachers fail to reveal the splendor of mathematics found so integral to ancients and early moderns.
Plato, Euclid, and even American heroes like Thomas Jefferson and Abraham Lincoln, saw mathematics as much more than a set of tools for solving practical problems. These men believed that the elegance and simplicity of mathematics underscored its primacy and power. For Plato and Euclid, this meant that the study of mathematics moved one beyond the realm of the material world and into the more pure realm of the Forms. Jefferson and Lincoln are less explicit in their mathematical ontology, but both consider the human mind’s mathematical capabilities awe-inspiring. Jefferson delighted in Euclid, noting that “no uncertainties remain on the mind, but all is demonstration and satisfaction.” Inspired by the clarity of mathematics, Jefferson pressed on to demonstrate his political philosophy with Euclidean certainty and precision. It is no small coincidence that the Declaration’s body begins with the axiomatic, “We hold these truths to be self-evident.” Lincoln, too, was impacted by Euclid’s certainty and precision, studying the first six books of the Elements until he mastered them all. He hoped that the intellectual power of Euclid would rub off, empowering him to argue with cogency and clarity.
To the aforementioned men, the truths of mathematics are unchanging, eternal, ordered, and aesthetically beautiful to the eye of the mind. The experience of “eureka!” moments after hours of mental exertion mirrored the experience of a divine revelation. The ancients and early moderns held that the study of mathematics trains the mind to look at eternal things above and beyond vain and earthly things. The study of mathematics prepares one to engage in the disciplines of philosophy and theology; which in turn, prepare one to gaze upon and contemplate God with the clear eyes of a soul that has been enlightened by a sublime mind. Tradition has it that above the entrance to Plato’s esteemed Academy, the words “Let no one ignorant of geometry enter here” were written. It was understood that the study of the trivium (grammar, logic, and rhetoric) followed by the quadrivium (arithmetic, music, astronomy, and geometry) would prepare one for the study of liberal arts par excellence (philosophy and theology). While the exact boundaries that confine these fields today differ from those that would have confined them 2,500 years ago, the general idea was that people must be well-versed in the seven fields of the liberal arts (found in the trivium and quadrivium) before they would be ready to study the more comprehensive and meta-level fields of philosophy and theology. Only the best prepared and most motivated minds were allowed to study these disciplines. They constituted the highest form of human thought, matching that of the divine mind. Yet, these disciplines could be most heinously perverted and used for ill. Throughout history, many atrocities have been perpetrated by the erudite in philosophy and theology. These members of society have been trusted to lead and instruct the people onto paths of truth, goodness, and beauty. Philosophy and theology had a more comprehensive definition in the ancient and into the early modern period; it was the philosophers and theologians who were to instruct the people about what it means to be human and progress towards flourishing.
At this point one may ask, “What does all this have to do with geometry?” To an ancient, or even a Jefferson or a Lincoln, the answer is: everything. Geometry is a rigorous discipline that takes intense study, mental sharpness, and clarity of thought and expression. There is little that can sharpen the mind more than constructing geometric proofs. The reasoning must account for any and all gaps, and in order for a proof to be considered true, each claim must logically follow from those already definitively demonstrated. Errors are exposed quickly in geometry they cannot be veiled as they can in other disciplines. Geometry directs the mind to think on truths that do not pass away with the whims of human opinion. These truths are eternal. These truths are sure. These truths are foundational in the very fabric of the world we inhabit and to the order we find throughout it. These benefits of studying geometry spurred Jefferson to give up reading newspapers in order to study classics like Euclid’s Elements. It was the study of Euclid that clarified the thoughts of a struggling Lincoln in the midst of his mid-life crisis. As a failed politician and doubting lawyer, Lincoln sought refuge in the precision and elegance Euclid exuded. Galvanized by such the rigor and assurance of truth laid out in the propositions, Lincoln successfully navigated this troubled point in his life, to becoming a more effective lawyer and one of the greatest politicians America has ever seen. Lincoln biographer Michael Burlingame puts it this way:
Lincoln in his early forties mastered Euclid, whose works he carried with him on business trips. This suggests a desire to strip away all superfluous mental baggage and get at the heart of the matter, psychologically as well as logically… When he emerged from his political semiretirement in 1854, he had formulated a basic critique of the proslavery and popular sovereignty cases, arguing with Euclidean coherence.
On a more spiritual interpretation of things, Augustine looked to the study of geometry to promote the “discovery of objects of knowledge which are above the human mind.” In Augustine’s view, contemplating perfection which is beyond the physical world (such as geometrical perfection) is a crucial step in attaining pure contemplation of God. Once Augustine’s mind looks beyond the esteem of his peers and the fleeting moments of passionate lust, it looks upon that perfect peace and happiness that his soul has always longed for—that perfection and beauty and truth that lies in the eternal mind of God. To think these thoughts of pure geometry was, in a very real sense, to think the very thoughts of God.
Next time the audacious pupil probes, “Why do I need to learn this math I’ll never use?” answer, “Because it helps you participate in the good, true, beautiful, eternal, and never changing divine thoughts of God. Studying this rigorous discipline will train your mind to think on higher thoughts and prepare it for the spiritual ascent to participation in the divine nature.” The modern will likely find your oration quite eloquent, but your answer less than satisfying. Taking into account this disconnect between an ancient and popular modern understanding of mathematics, the rest of this essay will briefly sketch how we got from the ancient and pre-modern viewpoint to the modern viewpoint, and evaluate some positives and negative implications for us today.
There are two factors that radically changed how mathematics is taught and understood in pedagogical circles. Namely, the industrial revolution and the democratization of education. Both of these changes reframe the aim of education and the expectation of the student. Formerly, what we would know as secondary education was a privilege reserved for the wealthy. The wealthy had little need to learn practical skills such as medicine or engineering or vocational trades, so they could study the liberal arts without care of economic return on investment. The poorer needed to use their education to make themselves more marketable and economically reliable, and so invested more time into practical disciplines. Access to anything like the free compulsory education that we enjoy today was only a pipe-dream before the industrial revolution and the democratization of education. However, by the start of the 19th century, the place in society for the poor traditionally shifted, as did the needs of all in an industrial society. Educational reformers persuaded lawmakers that the ways of the past were obsolete, and a new approach was necessary for success in a modern society.
I do not find all these changes unfortunate. The modern view of mathematics has certain strengths. One such strength lies in the accessibility of free, compulsory education and the practicality acknowledged in the schooling of all children. There is a legend that the ruler of Egypt and founder of the Ptolemaic Kingdom, Ptolemy I Soter, wished to learn geometry, but found Euclid’s daunting Elements too challenging. Seeking a shortcut or an alternate road, he approached Euclid in person. Unsympathetic to his cry for help, Euclid is said to have quipped, “There is no Royal Road to geometry.” While the mastering of geometry may have no easy shortcuts, it seems evident that more elementary primers to geometry may be studied to provide students with sufficient knowledge on the subject. When schooling is offered indiscriminately, it must include a bit of practicality. Due to constraints in time and resources and other pressing educational matters, not everyone can graduate having mastered the Elements. Despite this constraint, that all children, regardless of socioeconomic status, have the opportunity to study mathematics should still be considered a victory for modernity.
Additionally, the advancement of technology has allowed for the use of applied mathematics in realms thought to be impossible mere decades ago, let alone millennia ago. Self-driving cars, 2,000 foot tall buildings, and trips to Mars all rely on the brainpower of highly skilled mathematicians, engineers, and physicists. The practical applications of mathematics are enough to boggle most minds and keep one busy for years on end. Modern schools that spur on the study of mathematics for practical ends are not all wrong. The modern world opens up the opportunity for marvelous inventions and innovations at the hands of talented applied mathematicians. Without them, the technology in our pockets and in our planetary orbit would be impossible.
Though separated by two and a half millennia, the planar geometry and number theory advanced by Euclid remains largely accepted by the mathematical academy. Whereas nearly all subjects of human inquiry have experienced seismic shifts, Euclid’s mathematics has experienced only minor tweaks. But while the mathematics itself has largely remained constant, the way that people look at and study mathematics has changed dramatically. Once seen as a participation in the divine, an act of metaphysical and spiritual significance, mathematics is commonly stripped of these majestic callings and left as a threadbare series of problems to be solved for the sake of saving money when buying paint. Though the modern popular understanding and practice of mathematics has brought about significant advances in STEM disciplines, the ancients understood something crucial about mathematics that moderns lack. Seeing mathematics as a spiritual discipline is more in tune with the nature of mathematics as a rich and cognitively transformative discipline. This view does not turn away from practical applications, but neither does it shy away from the majesty to which great thinkers throughout the ages have universally attested. Instead of emphasizing practical applications and satisfactory exam scores, we need to recover the rich spiritual tradition of mathematics. I think Euclid would agree.