A Catholic Reflection on Nicomachus’ Arithmetic, Book I, Chapter 7
What can the even and the odd teach us about truth? To modern minds, such a question may seem trivial—fit for grade school, not for theology. But to Nicomachus of Gerasa, and to the philosophical tradition he represents, even and odd are not mere labels. They are principles of being, woven into the very structure of creation. And as he unfolds their definitions in Chapter 7 of Arithmetic, he reveals a spiritual harmony that lies beneath the most basic of mathematical concepts.
As Catholics, we are called to contemplate creation—not just with curiosity, but with reverence. For God’s invisible attributes, says St. Paul, are “clearly seen from the creation of the world, being understood from what he has made” (Romans 1:20). Nicomachus’ reflection on number invites us to pause, to attend to what may appear simple, and to find in it the pattern of divine wisdom.
Number: A Limited Multitude of Units
Nicomachus begins with a foundational definition: number is a limited multitude, a combination or flow of quantity made up of units.
This reflects the classical understanding of number, not as an abstraction from nothing, but as a unity built from ones. Number begins when multiple units are joined together with measure and intention.
St. Thomas Aquinas echoes this view when he writes that “number is the first expression of quantity.” It is, in essence, a relationship of unity and difference—several things, gathered into one order.
God, who is Himself One, created a world of many things, and yet all things together form a single cosmos. So number, in this light, becomes a way of describing the unity of multiplicity, the very principle by which the One creates the many.
The First Division: Even and Odd
From this starting point, Nicomachus introduces the first and most fundamental division of number: the even and the odd.
He first offers the ordinary definition:
The even is that which can be divided into two equal parts, without anything left over.
The odd is that which, when divided, leaves a unit intervening—it cannot be split perfectly.
Already here we glimpse something profound. The even speaks of symmetry, balance, and equality. The odd introduces a kind of difference, a disruption of symmetry. Yet both are necessary. Both are real. And their interplay is what gives number its vitality.
In Scripture, the theme of balance and difference runs throughout creation. God separates light from darkness, water from land, male from female. These are not contradictions, but complementary opposites. Even and odd numbers reflect the same creative balance of differences that we see throughout creation.
The Pythagorean Insight: Contraries in Harmony
Nicomachus then presents a deeper, more symbolic definition drawn from the Pythagorean tradition.
According to this Pythagorean view:
The even is that which can be divided into both greatest and smallest parts in the same operation—“greatest in size and smallest in quantity.”
The odd is that which does not allow for such a division; it yields unequal parts.
What does this mean?
This definition highlights the natural tension between size and number. When an even number is divided, its parts may be equal in size, but they can still serve different purposes or functions. For example, two equal halves may play different roles within a greater whole. This shows that even numbers contain a kind of built-in harmony—not because everything is the same, but because different parts work together in balance. It is a harmony formed through contrast, not sameness. St. Augustine would later develop this idea in moral terms, arguing that order in creation consists in “giving each thing its due place.” The even, then, reflects this moral principle: it is a symbol of justice, proportion, and peace.
The odd number, unlike the even, cannot be divided into two equal parts—it always leaves one unit that sets the halves apart. This makes it a symbol of imbalance or tension. But this kind of difference is not a flaw—it’s essential. Without difference, there would be no growth, no contrast, no unfolding of events. Just as a story needs conflict to move forward, the odd introduces the possibility of movement, change, and development. It reminds us that contrast gives life its shape and meaning.
This harmony of opposites is at the heart of Catholic metaphysics. All things exist in relation—substance and accident, form and matter, act and potency. The even and the odd become symbolic of this greater reality: that creation is held together not by sameness, but by ordered difference.
The Dyad: The Root of Evenness
Nicomachus notes that the dyad, or the number two, is the elementary form of the even. And he draws attention to a mystery here: the dyad, while even, can only be divided one way—into two equal parts. No other division is possible.
Because the number two can only be divided in one way—into two equal parts—it serves as a kind of model or archetype of clarity and simplicity. It shows us only one kind of relationship: equality. But this simple division also marks something profound. It is the first moment when unity gives way to plurality—when ‘one’ becomes ‘two.’ And with that, the possibility of relationship is born. Two things can face one another, respond to one another, and exist together in harmony or tension. In this sense, the dyad—though simple—represents the very beginning of all relationship and interaction.
In Catholic theology, two often represents the first expression of love and relationship. God said, “It is not good for man to be alone” (Genesis 2:18). So He made woman—not a copy, but a complement. From the dyad comes the possibility of communion, and from communion, fruitfulness.
Thus, the dyad is not only mathematical—it is spiritual. It is the image of otherness united in equality. It is the root of all peaceful division.
The Odd: Interwoven and Inseparable
By contrast, the odd number, Nicomachus says, always divides into unequal parts. Yet in that division, it reveals both species of number—the greater and the lesser—never one without the other.
Here again, we see a philosophical and even theological insight: the odd is not disorderly. It is complex. It reflects the interwoven nature of creation, where no element exists entirely on its own.
This is deeply consistent with the Christian view of society and the Church. St. Paul writes of the Body of Christ that “the eye cannot say to the hand, ‘I do not need you’” (1 Corinthians 12:21). Each part is different, some greater, some lesser, but all are bound together in one harmonious whole.
The odd number, then, symbolizes this intermixture of difference within unity. It is not “imperfect,” but complex and ordered.
Definitions by Comparison: The Odd and Even in Relation
Nicomachus concludes with a final insight: even and odd define each other by difference of one.
The odd is that which differs from the even by a unit—either greater or lesser. And the even differs from the odd by the same.
This reveals a profound truth: even and odd are interdependent. You cannot know one without the other. Their opposition is not a contradiction—it is a relationship.
Here we see a principle that runs through Catholic thought: truth is relational. God is not merely a monad (one) but a Trinity—a communion of Persons. Likewise, an individual man is made for relationship: with God, with others, with creation. The even and odd remind us that difference is meaningful only in light of unity, and unity is only known through distinction.
Conclusion: The Spiritual Meaning of Number
In Chapter 7 of Arithmetic, Nicomachus teaches us that when we study number with reverence and careful attention, we are not just learning how to count—we are uncovering something much deeper: the very structure of reality itself. The categories of even and odd may seem simple at first, but they are not random or meaningless. They symbolize profound truths about how creation is ordered. The even represents balance and equality, while the odd represents contrast and unevenness. Together, they express a pattern of unity and difference—of simplicity on one hand, and complexity on the other. These two types of number reflect the way God has woven harmony into the fabric of the world, using both sameness and variety to create order, beauty, and meaning.
But there’s something even deeper: the even and the odd show us that creation is not dull or uniform, as if everything were the same. Instead, creation is rich with texture and depth. It is carefully woven together—like a beautiful tapestry—using both similarity and contrast. What holds it together is not sameness, but the careful balance of differences. These differences do not cause division; they serve a higher purpose. They are arranged in such a way that they support one another and move all things toward unity and harmony. In other words, the world is not made of disconnected parts, but of varied parts working together under a single divine order.
For Catholic readers, this lesson goes far beyond the world of mathematics. It invites us into something sacred. It reminds us that number is not just a tool for measuring things or solving problems—it is part of the divine language with which God created the universe. To study number with understanding and reverence is, in a way, to join in worship. Every mathematical truth—every equation, every balanced ratio, every distinction between even and odd—can become a gentle echo of God’s eternal wisdom. These patterns are not random; they are glimpses of the divine order that governs all things. In this sense, mathematics becomes almost liturgical. It is a form of participation in the harmony and beauty of God’s creation, and it leads the soul closer to the One who holds all things together by measure and number and weight.
And so, as we study the even and the odd, may we be reminded that God has written order into creation—not only for calculation, but for contemplation. For through these simplest patterns, He invites us to know Him.
Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy
References
Aristotle. Nicomachean Ethics. Translated by W. D. Ross. https://www.perseus.tufts.edu/hopper/text?doc=Aristot.+Nic.+Eth.
Catechism of the Catholic Church. 2nd ed. Libreria Editrice Vaticana, 1997. https://www.vatican.va/archive/ENG0015/_INDEX.HTM
Holy Bible. New American Bible, Revised Edition (NABRE). United States Conference of Catholic Bishops, 2011. https://bible.usccb.org/
Nicomachus of Gerasa. Introduction to Arithmetic. Translated by Martin L. D’Ooge. https://archive.org/details/NicomachusIntroToArithmetic
Plato. Timaeus. Translated by Benjamin Jowett. https://www.perseus.tufts.edu/hopper/text?doc=Plat.+Tim.
St. Augustine. De Musica.
St. Thomas Aquinas. Summa Theologica. https://www.newadvent.org/summa/
Wisdom 11:20. New American Bible. https://bible.usccb.org/