Mathematics Teaches Us How to Think

The recent spate of articles on “woke mathematics” has raised the eyebrows of many people who thought that 2+2=4 was true no matter what race or ethnic background a person came from. I confess that the whole idea of mathematics being influenced by racial or cultural perspectives struck me as silly and even dangerous because it betrayed the nature of the subject. Yet, the more I read on this new wave, the more I realized that this debate isn’t about mathematics at all: It’s about mathematics education.

In the woke agenda, mathematics has been enlisted against white supremacy because the subject, taught in a traditional mode, is seen as inimical to racial justice. Woke thinking contends that since mathematics emerged out of a Western, patriarchal culture, it is culturally conditioned and therefore not of universal value. The rise of this strange way of looking at mathematics depends on what mathematics should be used for.

The problem lies in a common belief that one learns mathematics in order to use it for something else. In other words, people today see mathematics in utilitarian terms. When something is deemed worthy of attention because of its usefulness alone, it can more easily be dismissed as irrelevant if it seems to serve no useful purpose. Utilitarianism doesn’t worry about truth—only usefulness. When math is not studied for its truths, the only thing left is usefulness. Even math teachers try to motivate their students with utilitarian arguments: Mathematics allows us to build bridges and send rockets into outer space.

But using mathematics to do non-mathematical things, while necessary, is not the same as understanding what mathematics is. I write as a non-mathematician who is largely self-taught but who deeply appreciates the subject. I have come to see mathematics as a beautiful expression of human creativity and a form of knowledge that should be in everyone’s possession, at least to some degree. But because mathematics is generally valued only for its usefulness, most people view the subject as irrelevant.

I thought this way until I realized how profound and wonderful mathematics is. There are many reasons one could give for understanding mathematics besides utilitarian ones, but they all seem to come back to the same point: Mathematics teaches us how to think. Thinking is a precious commodity. I would venture that unless we learn how to think better, the future of our culture is in serious doubt.

How does mathematics foster better thinking? Let me suggest five ways.

Mathematics is fertile ground for teaching people how to work through problems patiently to arrive at conclusions. This is why the main textbook in the West for centuries was Euclid’s Elements; it is not the algebratized geometry of today. Euclid focused on reasoning from premises to conclusions. Geometry instructors have told me that they do not often teach proofs. But proofs are the heart and soul of geometry. Starting with precise definitions and axioms, students learn that one can derive a considerable amount of knowledge from very primitive starting points. They learn that those axioms (called postulates) are unproven starting points. This is important because all proofs in mathematics depend on unproven axioms. And this is a particular example of a more general method that all must use in any area of rational investigation. Studying proofs will produce better thinkers.

Secondly, we need to understand what kind of knowledge mathematics is and how it differs from the empirical sciences. This is important because most people study higher mathematics today to apply it in those fields. And yet mathematics and the empirical sciences are different kinds of knowledge. When Euclid proved that the interior angles of any triangle were equal to two right angles, he proved a truth that will never change. This is not the case in physics, chemistry, biology, or psychology. These sciences depend on empirical evidence to adjudicate truth claims. Therefore, as more evidence is accumulated, theories must be revised or discarded. Mathematical truths have an eternal quality about them that the empirical sciences do not. If people appreciated this difference, they would not use the word “proof” so lightly when talking about any physical science.

Mathematics can also open new avenues of thought by revealing how it is similar to other areas of life. Both mathematics and natural languages are highly structured systems. Many years ago, when I was teaching ancient Greek in a Christian seminary, I had a research assistant who had been a high school mathematics teacher. He was very gifted in explaining the subject. Watching him work, I began to grasp many things that language and mathematics had in common. In both subjects, there is an underlying principle that we can dub “variation under invariable laws.” There are multiple ways to say the same thing in English, such as active and passive sentences. The meaning remains the same while the syntax changes. Likewise, the law of triangles equaling two right angles remains the case despite variations in types of triangles (right, equilateral, scalene). Of course, there are many differences between the two systems. Still, both are highly patterned systems.

Perhaps the noblest reason for fostering mathematics is the window it opens onto higher metaphysical questions. Today’s academy studiously avoids the ultimate questions of life that many students would like to address. Above the door of Plato’s academy, it said: “Let him uninstructed in geometry not enter here.” Plato knew that precise reasoning was essential to answering the ultimate questions of life. To achieve the end of philosophy was “to praise the divine.” In this same spirit, the thirteenth-century Oxford Franciscan Roger Bacon contended that mathematics was the best preparation for the study of theology. He rightly perceived similarities between God and numbers because both are metaphysical entities.

Lastly, and perhaps most surprisingly, mathematics allows us to acquire virtue. For example, humility—which some moral and spiritual writers see as the root of all other virtues—can be defined as the honest evaluation of self in the face of a greater reality that cannot be controlled. Can mathematics foster this virtue? Indeed it can, because it contains puzzles and paradoxes that challenge us to evaluate how much we truly know. For example, Georg Cantor demonstrated that there were sets of numbers that could never be counted. Imagine that. A science whose business is counting encounters numbers that cannot be counted. And not just because we don’t have an infinite amount of time; even if we did, these number sets would still be uncountable. That ought to humble us. In the face of a reality we cannot control, we see our limited selves.

The fundamental question surfaces again: Why study mathematics? The question demands a choice. Either we study mathematics for pragmatic reasons alone, or we do so to foster higher human goods. If we continue to see mathematics in merely utilitarian terms, we will continue to be enmeshed in the debate around the woke teaching of mathematics. Teaching mathematics for utilitarian reasons alone can never escape the trap of cultural relevance. Studied for its truth value, mathematics can confer many more benefits. Understanding mathematics as a human endeavor will entail all the utilitarian reasons, but it will also foster the human intellect in a greater way and aid in the development of the virtues. Intellects trained in this humane way will be the antidote to wokeism or any other defective philosophy.

Kenneth J. Howell is academic director of the Eucharist Project and president of the Pontifical Studies Foundation. He is the author of God’s Two Books: Copernican Cosmology and Biblical Interpretation in Early Modern Science (University of Notre Dame Press). 

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Image by Raphael on Creazilla via Creative Commons. Image cropped.