The Riemann sum

 

Images from another era rise in my mind—those noisy, light-filled halls of the Faculty of Mathematics and Informatics. I remember the chalk dust, the creaking on the blackboard, and the lecturer’s voice guiding us through the labyrinths of the discrete. Back then, everything seemed like a dry architecture, like the skeleton of a reality we had yet to clothe in flesh. But today, years later, these formulas return to me not as tools for calculation, but as deep spiritual principles that have always been there, waiting for me to transform information into wisdom.

The Riemann sum. Once, it was simply a sum of rectangles under a curve—an exercise in precision, a technical step toward defining the integral. Today, it is almost a prayer.

Because what are we really doing when we try to “approximate” the integral? We acknowledge our limitation. Truth is not given directly; it is not a fruit we can simply pluck and fully possess in its continuity. We, finite beings, do not have eyes for the infinite in its absolute form. Instead, we approach it—through small, sometimes clumsy, imperfect steps. Like life. Like breathing. Like a series of attempts. And in this quiet night, I begin to feel that our entire human journey is precisely a vast Riemann sum unfolding through time.

We divide our existence into small intervals—days, hours, moments. In each of these intervals, in our little Delta x, we are called to choose a “point”—an experience, a reaction, a thought. And then we multiply that choice by the “width” of our presence. Sometimes we are awake, fully present with our whole being, and then our rectangle touches the sky of Truth. Other times we are distracted, tired, completely absent from ourselves, and then our contribution to the sum is flat, devoid of height. And in the end… we sum it all. But what we obtain is never entirely exact. It is never the true area under the curve of life. There is always a difference. There is always that small gap, that “less than 1,” reminding us that we are creatures—created, not the Creator Himself.

And precisely in this lack, in this small error of approximation, I begin to sense something sacred. Mathematics teaches us that the limit exists—that it is there, unshakable and majestic. But it is not reached through the finite. It is approached infinitely, quietly, humbly. This is our tension between dust and stars—we are rectangles that long to become a curve, yet always remain fragmented. And perhaps here lies the deepest spiritual lesson: that the human being is not called to possess Truth, but to approach it—with every action, every pain, every mistake.

Here the psychoanalytic perspective also emerges, whispering through the shadows of the room. In psychoanalysis, as in the Riemann sum, there is no “final answer,” no magic formula that suddenly makes us “whole.” There is a process—a slow, fragmented unfolding. The unconscious does not reveal itself all at once like a panoramic landscape; it appears in fragments, in symbols, in dreams, in repetitions. It is like a sum of our past traumas and longings that we try to integrate. And the more we work with ourselves, the finer our “partitions” become (our self-observation), the more precise the rectangles of our self-knowledge. But we never reach the full “integral value” of who we are. And that is precisely what saves us. Because if we could be fully calculated, fully determined, fully complete—there would be no room for freedom. Freedom is that small, elusive remainder, that difference between the sum and the integral that cannot be captured in any algorithm.

Now I understand why error is allowed. It is not merely a defect in calculation; it is the space in which we learn to be human. If our rectangles were perfect, we would have no need for faith. Faith is exactly this—to continue summing our imperfect moments, believing that somewhere, in the infinite, they merge into a meaningful flow. Error exists so we can learn. It is the signal that we have chosen a “point” that does not reflect the height of our soul. But life does not stop because of error. The sum continues.

And here my anxiety returns, but now more clearly defined. What does artificial intelligence do? It is a master of optimization. It is designed to minimize error, to find the best approximation as quickly as possible, to serve us the result “on a platter.” But in this process, it performs a quiet killing—it removes the path. It removes the wandering. It removes those inaccurate, painful, yet necessary “rectangles” through which we actually grow. When AI gives us the “optimal solution,” it deprives us of time for error. And without time for error, we have no time to mature. A baby in the birth canal does not need “optimization” to be instantly delivered; it needs the very effort of passing through, because that effort activates its vital functions.

Algorithms do us a disservice by offering a world without resistance. They give us the answer before we have understood the question. And so we become lazy—not only physically, but lazy in our adaptation to life. Just as pharmaceuticals sometimes conceal symptoms instead of healing the cause, algorithms conceal the complexity of moral and emotional effort. But pain is the strongest voice of health. It is the sign that our “rectangle” is too far from the curve of Truth. When we are deprived of the chance to see error signals, we stop correcting ourselves. We stop moving toward God.

From a spiritual perspective, God is the Integral—the Wholeness, the Fullness, the Continuous presence. And we are those who move through sums. Every prayer, every suffering, every small victory over the ego—these are our finite steps. And in the fact that we always remain “a little below 1,” there is immense comfort. It means there is always more. More meaning, more depth, more possibility for transformation. If we had reached “one,” we would be static. But life is movement.

I remember Manev’s words about discreteness… how strange that it is precisely through discrete steps that we try to understand the continuum. This is the paradox of our existence: we are fragmented, broken into thousands of moments, yet we long for unity. And error is not a deviation from the path—it is the path itself. It is that “friction” with reality that reminds us we are alive. Perfect accuracy belongs to infinity, while we belong to the process. We belong to the effort to become better approximations of the divine image within us.

Today, in this world of instantaneous solutions, I choose to be a “slow Riemann sum.” I choose not to fear my inaccuracies. I choose to accept that my answers will always be partial. Because in this imperfection, in this quiet difference between my “now” and that “always,” lies the place for Grace. Grace is what fills the gaps between our rectangles. It is what makes our sum whole, even though mathematically it never truly is.

Life is not an integral to be solved, but a Riemann sum to be lived. With attention to every interval. With acceptance of every error. With faith that the limit exists—and that it is Love. When we replace this difficult, fragmented process with the smooth, soulless solution of the algorithm, we commit a kind of spiritual suicide, because we renounce the very birth of meaning.

Now, as I write these lines, I feel my thoughts settling. I am no longer afraid that I am not “one.” I am no longer afraid of the “epsilon” of my error. It is my signature. It is my proof that I am not a machine. My identity lies in the potential for future transformation, in my right to be inaccurate today so that I may be closer to Truth tomorrow.

I am grateful to those who taught me to see overdetermination—that saturation with meaning which turns mathematics into poetry, and poetry into truth. We are co-creators of our own integral. And at the end of the day, when we close the notebook of our life, what will matter is not how precisely we calculated the result, but how much love and patience we placed into each of our small, imperfect rectangles beneath the sky of Eternity.

The night is fading. The light returns—slow, continuous, whole. And I remain here, within my approximation, quietly and humbly awaiting the next moment, the next interval, the next breath. And in this imperfection… lies the very soul.