The Modern Math Academy with Dr. W

For thousands of years, mathematics has been an esoteric and secretive venture reserved only for the initiated. In the 6th century BC, the Divine Brotherhood of Pythagoras kept disclosed, under quiet confidence, the existence of the twelve-faced dodecahedron which, to them, represented the aether permeating all of space (because it broke their correspondence of the four elements of Fire, Air, Earth, and Water with the tetrahedron, octahedron, hexahedron, and icosahedron respectively). In AD 1637, Rene Descartes remarked in La Géométrie that he had erased all his laborious analytic geometrical calculations “so as to grant the reader the enjoyment of discovering them for himself.” When non-Euclidean geometry was being simultaneously discovered in Russia and Hungary during the 19th century by Nikolai Lobachevsky and János Bolyai, its earth-shattering and paradigm-shifting consequences had already been discovered, collecting dust on Karl Friedrich Gaus’s shelf from years prior.

Today, in the 21st century, mathematics has never been more accessible, more available, and in plainer view of the public eye. Yet, paradoxically, it is more mysterious and hidden than at any point in history. Everyone is required to learn math in primary and secondary school. They believe they know what mathematics is, and how it can be utilized in the real world. Even the staunchest “mathphobe” will pay lip service to what physicist Eugene Wigner coined as “the unreasonable effectiveness of mathematics in the natural sciences.” But if there is anything that they know about mathematics it is that they are “not a math person.” This is often the greatest stumbling block for the uninitiated in their study of mathematics.  But this discipline is not something that falls out of the sky, and it was not developed in a vacuum; mathematics is a traditional subject that is handed down from one generation to the next. Its problems, constructions, solutions, theorems, and notations are guided and gifted to us by history in a way which, when properly understood, humanizes a cold, dead subject of brute facts and calculations. The ability to excel at mathematics is not a genetic disposition: it is a practice undertaken through careful supervision by a trained practitioner and keeper of its secrets.

My philosophy of teaching and learning mathematics, shared with students on the first day of class, is derived from the tradition of Euclid of Alexandria, Isaac Newton, and John Von Neumann. In Alexandria, Egypt circa 200 B.C., Euclid was tutoring King Ptolemy I. When the king complained that mathematics was too difficult, he asked Euclid if there was an easier way to learn. Euclid famously responded: “Sire, there is no royal road to geometry.” I warn my students to steer clear of tutors promising to make mathematics easy for them. They are, almost surely, trying to take their money.  Mathematics must be studied the same way Isaac Newton studied La Géométrie of Rene Descartes. Newton explained: “I opened La Géométrie and read it until it didn’t make any sense. Then, I started from the beginning and read it again until it didn’t make any sense. I repeated this process until I understood the entire thing.” One must fight the text and material. Scrutinize it, question it, try to prove it wrong. But more than anything, one must force the material into their mind, repeatedly, until their cognition is conformed to the shape of the truths under investigation. This is the most necessary part of learning mathematics. As John Von Neumann teaches us: “You do not understand anything in mathematics; you just get used to it.” Because mathematics is esoteric and a purely mental experience that one must be initiated into, there is no reason to be surprised that students are confused or do not understand something. What is at first dark and mysterious becomes routine and subconscious through repeated exposure. What happens in between these two experiences is inexplicable and frustrating and shapes our character in the process. We study mathematics so that we can become better people.

In the 15th century, a debate was had about whether mathematics, as a subject, should be taught at university at all. It was argued that students should only be taught what qualified then as science (based on Aristotle’s definition, science required more than formal deduction). Mathematics, a purely deductive art, did not rest its foundations on empirical knowledge. Although empirical observation has always played a pivotal role in the discovery of theorems, our sensory perceptions often prove to be a stumbling block through the treacherous rigor of formal logic. (For example, the first Proposition in Book 1 of Euclid’s Elements has a conspicuously hidden logical gap which is filled in by a subtle and “obvious” picture). We can only imagine an alternate universe where mathematics was removed from the University system. Those who argued to keep mathematics in the University clearly won the day. They believed that, though not necessarily fitting the Aristotelean definition of “science,” studying mathematics trains the mind to think scientifically, deductively, and methodically. Mathematics is an indispensable tool and field of study for the formation of virtuous and scientific scholars.

The word mathematics, coming from ancient Greece, can be best translated as “the lesson” or “the study”. As I warn my students “if you are not studying, you are not doing mathematics.” It is fundamentally the responsibility of the professors to impart “the lesson” to their pupils. Mathematics does not come out of nowhere. It is also the pupil’s responsibility to work through the exercises given to them because mathematics is not something we learn through our ears and eyes. Mathematics is learned through the hand. Countless problems must be mastered to achieve any success in this art. However, they can’t be just randomly generated problems from some textbook. By carefully choosing specific problems, an instructor can set up all the logical traps for the students to be caught in to help them build immunities from fallacies and see what is “really” going on. Sometimes, this requires doing an extremely difficult problem to not hallucinate any weird coincidences (often called “the law of small numbers”). Sometimes, this requires doing an extremely simple problem which has a subtle detail that students will overlook completely. It’s the responsibility of the instructor to know when to use each kind of problem to prepare their students for the logical trials and traps found in their everyday lives.

I define mathematics as that which makes sense, something that can be understood. A mathematician makes sense of the world, preferring reason over opinion. If what I am communicating does not make sense and if I am not being understood, then I am failing at my duty to be a maker of sense, a mathematician in the fullest sense. It is my responsibility, as a bearer of the tradition handed down once to Euclid, to Descartes, to Newton, to Gauss, and to Von Neumann, to present the whole picture, the whole approach, the whole method of initiation into the beautiful art of mathematics.