The mathematician Israel Gelfand once said: “The most important thing a student can get from the study of mathematics is the attainment of a higher intellectual level.”1 We might twist this slightly to say that “the most important thing a human being can get from the study of mathematics is an understanding of oneself as developing intellectually”. Or, to put it another way, the most important thing one can learn in mathematics is the nature of the soul.
The idea that mathematics is key to understanding the soul has an old and distinguished pedigree. In Plato’s Meno, Socrates guides an uneducated child through a mathematical puzzle to uncover the origins of knowledge. Aristotle conceives of science on the model of geometry. Augustine appeals to numbers to show our capacity to transcend the sensible world and attain eternal truths.2
Is mathematics really so significant? If only we could find the answer in some timeless text. But Plato and the others were drawing on their own experience of doing mathematics, attending not only to mathematical problems but to the workings of their own minds. Unless we have that same experience and attentiveness, we cannot appreciate what they are talking about. One does not appreciate Shakespeare by avoiding his plays and reading Stephen Greenblatt’s scholarship. Nor does one grasp the significance of Bach by reading Christoph Wolff and never listening to a fugue. Similarly, to understand the significance of mathematics—and perhaps even ourselves—we need to work difficult problems with scratch paper at hand.
Mathematics and Mystery
Mathematics instills an authentic sense of mystery. A younger brother bent over his arithmetic homework may look at his sister’s algebra assignment and wonder, “How does one multiply letters?” In school arithmetic, addition and subtraction are concrete operations on specific numbers. Algebra, by contrast, is at present utterly inaccessible.
Yet the worlds of algebra, trigonometry, and calculus are not impossible to understand. They are impossible to understand at present. One has work to do, and the day will come when the letters become variables.
Mystery is something one can grow into through increased understanding. Mystery and the human soul belong together; neither can exist without the other. Mystery is often mistaken for the unintelligible, but that is false. The mysterious appears senseless only so long as one lacks knowledge.
Our intellectual nature is characterized by growth. Unlike God, we do not know everything. Unlike a rock, it is our nature to develop. Unlike a tree, our development extends beyond the physical. Our minds are capable of transformations more frequent and more radical than those of seed into tree or caterpillar into butterfly. To understand oneself in terms of one’s current achievements rather than one’s capacity for transformation is to overlook the spirit.
Transcending Appearance
In mathematics, we learn to move beyond the way things look to the way they ought to be understood. The fractions 1/1 and 10/10 are the same as 1. Adults can easily overlook how strange this is to a child, and so fail to appreciate the distinction between appearance and reality. A slightly harder example brings the point out more clearly.
1/1, 3/3, and 1.0 are different ways of writing the number 1. So is 0.999…. If this strikes you as implausible, then you already have some sense of the gap between appearance and reality, and a hint of how we move from the way things look to the way things are: by asking questions and answering them correctly.
In the case of 0.999…, the question is what its relation is to the number 1. It could be greater than 1, less than 1, or equal to 1. How can we determine what is really the case? We could try to use our imagination, adding more and more 9s:
1.0000000000000000… 0.9999999999999999…
And we might conclude that 1 and 0.999… not only look different, but are different. If we probe the question a bit further, however, we might see that the way things look and the way things are differ.
1/3 = 0.333…. 3 * 1/3 = 1 3 * 0.333… = 0.999….
There are different paths to the realization that 1 and 0.999… are the same number, and each leads to the same insight. This eureka moment is essential to understanding our own souls. It is a realization of our intellectual nature as genuine novelty emerges in our minds. It is the transition between, to use the philosophical terminology, the sensible and the intelligible.
Such eureka moments are easier to attain in mathematics than in other disciplines, for two related reasons. First, mathematics is an intrinsically simpler subject: there are simply fewer elements in play. Second, in other disciplines it is much easier to confuse the sensible with the intelligible. Do you understand a daffodil? (Have you mastered biochemistry? Has anyone?)
In mathematics, then, we can grasp those distinctive moments when we step across the threshold from the animal world—the audible, visible, tactile world—into the intelligible world, the realm of ideas.
This intelligible world is not accessible in the way a physical space is, as when one walks into a room or climbs a ladder. Even its existence remains unsuspected until one begins to seek an explanation. Its character remains hidden until one begins to form ideas and offer explanations.
By getting a feel for our intellectual operations—for their progress, and for the line that divides the sensible from the intelligible—we begin to have some clue to the meaning of words like “spirit” and “person.” But only a clue.
False Certainty
Mathematics in practice is characterized not by certainty but by disappointment. How often a promising solution turns out to be wrong! One quickly learns that feeling certain is not the same as being certain. The struggle toward an answer that could be right differs starkly from the work of checking whether a promising solution is in fact correct.
The value of such disappointment becomes more evident the more interesting the error turns out to be. We do not learn much from mere copy errors. The more interesting errors are those that reveal not a lapse in attention but a lack of mastery of the subject.
The Pythagoreans declared that all numbers were rational—whole numbers or fractions. And it is easy to see why. Suppose we have a number smaller than 1 but greater than one half. Is it not plausible that we can get closer and closer to it? Suppose we try 3/4 and find that it is too low, while 1 is too high. We then try 7/8, 15/16, 31/32.
Fractions, it would seem, allow us to be arbitrarily precise. If we imagine a length of yarn as 1 unit, it would seem that there is no point on it that we could not specify by increasing the denominator and adjusting the numerator. Try it, and you can see how the Pythagoreans could have been so certain.
The feeling of certainty, however, differs enormously from objective certainty. According to tradition, it was the Pythagorean Hippasus of Metapontum who discovered a proof that irrational numbers exist. For this discovery, legend has it, he was drowned at sea.
It is worth working through Hippasus’ proof yourself if you have never done so. The purpose in doing so is not so much the mathematics as attending to the difference within our own minds between developing a feeling of certainty and developing reasons for being certain that something is correct. It is this same general type of critical process by which we go from idealism to realism, not merely to what may be, but to what in fact is.
Mathematics and the Soul
In pursuing mathematics, we can get a feel for what it means to be a rational animal, a spiritual creature made in the image of God. It is not enough to acknowledge the opinions of great thinkers on how mathematics illuminates the soul. That illumination occurs within the person doing the work.
Our animality is evident. We look with our eyes, listen with our ears, smell with our noses, taste with our tongues, and feel with our fingers. But while our sense organs are very similar to those of other animals, how we use them differs enormously. Human sensation is imbued with a different quality, just as the gaze of a daydreaming student differs from that of Sherlock Holmes. We ask questions about the world we sense, and those questions arise from a wonder that serves no practical purpose but somehow aims at an ultimate end.
Why do mathematics? Because it is all too easy to confuse our animality with our rationality. We confuse what we imagine with what we understand, opinion with knowledge, and the ideal with the real. Even a little clarity about what it is to be a person—that is, a rational subject—is hard won. The giants of the intellectual tradition help, but they are no substitute for our own self-knowledge. Parroted definitions cannot substitute for that, even when the definitions are correct. Mathematics, by its accessibility and clarity, is particularly well suited to illuminate, however partially, the deep mystery of our person.
