Arithmetic as the First of Sciences

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Classical Arithmeic - Engraved image of an elderly philosopher in robes studying a large open book with focused intensity.

A Catholic Reflection on Nicomachus’ Arithmetic, Book I, Chapter 4

Why should arithmetic come first? Among the sciences of number, shape, music, and motion, what gives arithmetic its primacy? In Book I, Chapter 4 of his Arithmetic, Nicomachus of Gerasa offers an answer that is not merely logical—it is theological.

He explains that arithmetic is not only first in time and understanding but is also first in the order of being. It stands as a root and mother of the other sciences. More than that, it reflects the mind of God, who used number as a divine exemplar in the creation of the world.

For Catholic readers, Nicomachus’ words draw us into profound reflection. If number exists first in the eternal mind of God, then arithmetic is no idle study. It becomes a window into divine wisdom. Through this “mother of sciences,” the soul is drawn upward, from the visible world to the eternal pattern by which all was made.

Number in the Mind of the Creator

Nicomachus opens this chapter by asserting that arithmetic is first among the sciences because it existed first in the mind of the creating God—as an archetype, a model, a divine design.

This echoes a deep truth in Christian philosophy: God created all things with order, measure, and proportion. As the Book of Wisdom declares, “You have arranged all things by measure and number and weight” (Wisdom 11:20). St. Augustine, reflecting on Genesis, said that God created the world according to eternal numbers—immutable principles in His divine reason.

The Catechism likewise teaches that creation is not a product of chance, but is “ordered,” reflecting the mind of God (CCC 299). Therefore, when we study arithmetic, we are not merely handling tools—we are contemplating patterns that first existed in God Himself.

Nicomachus puts it beautifully: arithmetic is like the universal and exemplary plan by which the Creator established the cosmos. It is the blueprint according to which all things were made and set in motion.

Why Arithmetic Is Naturally First

Nicomachus then offers a philosophical proof: arithmetic is ontologically prior—that is, it comes first in the order of being.

He uses a principle from classical logic: that which can exist without depending on something else is older and more foundational. Consider these examples:

  • “Animal” is prior to “man.” If “animal” does not exist, “man” cannot exist. But the reverse is not true.

  • “Man” is prior to “schoolteacher.” One must first be a man before he can be a teacher. But a man does not cease to be a man if the teaching role is removed.

So too, says Nicomachus, arithmetic is prior to geometry, music, and astronomy. Without number, these other disciplines cannot exist. But arithmetic can exist without them.

This aligns with St. Thomas Aquinas’ teaching that being is known through its principles. What is first in being is first in knowledge. If we remove number, we cannot speak of triangle, rhythm, proportion, orbit, or any other scientific concept. But we can speak of number without reference to anything else.

Therefore, arithmetic is not just chronologically first—it is logically and metaphysically first.

The Hierarchy of Concepts and the Nature of Science

Nicomachus explains this using an elegant hierarchy:

  • If you remove “animal,” “man” disappears.

  • If you remove “man,” “schoolteacher” disappears.

  • Likewise, remove number, and geometry collapses.

Geometry speaks of triangles, squares, octagons. But each of these terms presupposes number. A triangle has three sides—remove the number three, and the very concept of triangle vanishes.

Music speaks of intervals—fourths, fifths, octaves. Remove the numbers, and harmony itself dissolves. Astronomy counts cycles—days, months, years. But without number, time has no structure.

So Nicomachus shows that number is the principle that sustains all these sciences. It is not just another branch of knowledge—it is their foundation.

This concept finds deep resonance in the Christian intellectual tradition. In The City of God, St. Augustine says that the soul finds God more readily in eternal truths than in sensory things. Number, being abstract, changeless, and intelligible, draws the soul to contemplate truth without confusion—a reflection of the eternal Logos.

The Independence and Sovereignty of Number

Nicomachus then emphasizes a crucial point: number can exist without reference to other sciences, but the reverse is not true.

He writes that while we can speak of 3, 4, or 8 without needing triangles or rhythms or star cycles, we cannot speak of “triple,” “eightfold,” or “octave” without number. These are dependent ideas.

Thus, number is independent, sovereign, and self-sufficient. It does not rely on the other sciences for its existence, but they all rely on it.

This idea reflects a deeper theological principle: truth is one and ordered, and the higher truths are not dependent on the lower. The study of number, because it is so abstract and universal, lifts the soul closer to the divine.

St. Thomas teaches that higher sciences deal with higher causes. Since arithmetic deals with first principles—number in itself—it is a higher and more universal science. Geometry, music, and astronomy are beautiful and necessary, but they flow from this first and noblest source.

Arithmetic and the Structure of Creation

It is no wonder, then, that Nicomachus speaks of arithmetic as the “mother and root” of the other disciplines. From it arise all proportions, measurements, motions, and structures.

We see this reflected in creation itself:

  • The harmony of the heavens follows numerical patterns.

  • The design of the human body reflects symmetry and ratio.

  • The timing of seasons and days follows precise cycles.

  • Even justice and morality, as Aristotle observes, depend on proportionality and balance.

All of these emerge from number. And number, says Nicomachus, existed first in the mind of God.

To contemplate number, therefore, is to contemplate a reflection of the divine mind. In doing so, we begin to think God’s thoughts after Him.

Conclusion: The First Step Toward Wisdom

For Nicomachus, arithmetic is not just a subject to be learned—it is the first step toward wisdom. It is the beginning of a philosophical ascent that moves from the material to the immaterial, from sense to reason, from time to eternity.

In the Catholic tradition, this ascent is called contemplation. It is the highest activity of the soul, a participation in the life of God. Arithmetic, because it is stable, eternal, and immaterial, becomes a fitting training ground for contemplation.

As St. Augustine put it, “The good man, though he may not be a mathematician, cannot despise mathematics, for its truths come from the One who made all things in number.”

Let us then regard arithmetic not as mundane or mechanical, but as a sacramental discipline—a visible sign of invisible order. Let us teach it with reverence. For in its simplicity lies the gateway to the eternal.

Mr. William C. Michael, O.P.
Headmaster
Classical Liberal Arts Academy

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