Why We Study Math
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Is math an important part of education? This is the first question I ask my geometry students every year. I do not think they believe I am asking in earnest. Surely such a question from the math teacher must be contrived and intended to elicit a particular response, some trite justification of her job and our required attendance. But the students are good-natured and play along. The average reply is that it is important to study math because it is useful.
“Certainly, it’s useful,” I argue, “for computing taxes and gas mileage, but what about synthetic geometry? What about calculus? Could a person live a full life not knowing that the diagonal of a square with integer-length sides is always an irrational quantity (Plato would say “no”)? Will the future music major in my class ever need to compute a derivative or understand the implications of the Mean Value Theorem?” At this point the students’ true opinions emerge, and they are not always favorable towards intensive mathematical study.
The truth is that many students have already learned all the math they will ever apply directly by the time they enter my classroom, so why has classical education always placed such great emphasis on its study?
Let no one ignorant of geometry enter here.
-Inscription at the entrance to Plato’s Academy
Pushups are at least part of the answer. Pushups are notoriously difficult for some of us, easier for others – usually with training – and beneficial to all. Described as “the perfect exercise,” pushups work core muscles together with the extremities to improve muscle tone, balance, and overall endurance.
Studying math is like doing pushups. It is a coordinated workout across multiple areas of the brain, improving connections between cortical regions, and strengthening overall endurance for higher-level thought. Math teaches creative problem-solving, pattern recognition, and critical thinking, which generalize to all areas of study. And just as most of us do not perform pushups for pushup’s sake, but for the sake of attaining some athletic prowess or improved physique, so we do not all study math because we intend to make a career out of it. Yet we reap the intellectual benefits of the workout.
For some students this is enough to justify their required attendance of my class, but actually this argument does not go deep enough. After all, the “math exercises the brain” argument is just a variant of the ubiquitous “math is applicable” argument. It is not math’s applicability but the reason behind this applicability that makes it an essential component of education.
Math is essential to education because the universe itself seems to be inherently mathematical. We couch our physical theories in the language of equations which often unite abstract areas of mathematics developed entirely for their own sake with observed physical phenomena to make startlingly accurate predictions. This is what Eugene Wigner referred to as “the unreasonable effectiveness of mathematics in the natural sciences.”
Middle school students know, for example, that x^2=-1 has no solutions within the real numbers, but later they learn that, through an entirely consistent extension of our number system, we can define a new object, i, so that i^2=-1. The students are usually baffled that such a definition is allowed within the confines of a respectable institute of learning, but I (try to) persuade them that the complex numbers formed by the introduction of the imaginary unit i are a fascinating new dimension of number with many interesting properties. Moreover, these numbers are an integral part of our current models of electromagnetism and space-time. And this is far from the only example of a pure mathematical concept being the exact tool needed to model physical reality. There is something about the universe that is fundamentally mathematical.
Education, then, as the practice of orienting the soul towards Truth, ought to be highly concerned with the exploration of mathematics. That math is useful for building bridges, launching rockets, and predicting market fluctuations is a consequence of its being “the language with which God has written the universe,” and certainly these are useful consequences worthy of our time, but they are not the primary focus of classical education. Rather, its aim is to understand “that which is.” In our studies we will learn what is of practical use, but we will not be satiated by it. We desire the entire cosmos and its underlying principles, and mathematics seems to point the way. This is why we study math: to find the patterns in creation that direct us towards Truth. We will settle for nothing less.