Beyond Quantity: Nicomachus' Introduction to Arithmetic

The Lost Qualities of Numbers

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Illuminated medieval manuscript page of Nicomachus (right) and Plato.

My online Great Books reading club is discussing Nicomachus’ Introduction to Arithmetic this week. It’s the first text on mathematics that we’ve read together, and I anticipate it will be an intellectual “stretch” for most of us. To that end, I’ve prepared the following study guide for my seminar mates and paid subscribers.

Biographical Introduction

Introduction to Arithmetic is an ancient mathematical text that explores the properties and philosophical nature of numbers. It was written by the Greek mathematician and philosopher Nicomachus of Gerasa who lived circa 60-120 AD and may have studied in Alexandria. Not much is known about Nicomachus today, but his writings reveal that he was a devoted adherent of Pythagorean philosophy. In addition to his treatise on arithmetic, he wrote an Introduction to Geometry, Introduction to Harmonics, and a Life of Pythagoras.

Introduction to Arithmetic was a popular textbook throughout late antiquity and the Middle Ages, the latter using a Latin paraphrase by Boethius. Unlike a more famous text, Euclid’s Elements, Nicomachus’ treatise is no longer considered a useful aid for teaching because it treats mathematical problems that have been largely neglected since the Middle Ages.¹ Whereas Euclid took a rigorous, axiomatic approach to mathematics, Nicomachus focused on the mystical and philosophical qualities of numbers believed to be inherent in the universe.

In contrast, our “scientific age”—with its monomania for measurement—has reduced numbers to mere quantities, making Nicomachus’ treatise somewhat difficult for modern readers to approach. This is not to say that modern mathematicians are not still concerned with the various properties of numbers identified by Euclid and Nicomachus (like odd-even, prime etc.), but that our modern tendency moves away from the cosmological significance of numbers towards the utility of measurement.

“For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding.”
Nicomachus, Introduction to Arithmetic I, III.6

Rediscovering the Qualitative Properties of Numbers

Why study Nicomachus’ Introduction to Arithmetic today then? According to Aristotle, the Pythagoreans believed that learning about numbers taught them about the elementary properties of all things, including their constituent parts. They considered numbers the foundation or principle of reality, and the study of philosophy which treated numbers (later known as the Quadrivium, i.e. arithmetic, geometry, music, and astronomy) as necessary for a happy life. Mortimer Adler writes,

Mathematics to the Greeks was not only a useful method of calculating quantitative relations. It was also a divine science, almost a religion, having to do with eternal things. It was the way to wisdom, to knowledge of the eternal patterns in and behind all things.²

Nicomachus invites us, then, to untether our minds from the complacency of mere quantitative thinking in order to (re)discover how numbers figure into the integral nature of the universe. This is about studying numbers for their own sake, not for their application (counting money, measuring distances, etc.) Therefore, each number for Nicomachus (who considers only natural numbers, i.e. whole and positive) is a real and distinct thing, with its own properties and classification. His investigation includes:

• The classification of numbers (odd/even, prime/composite, perfect numbers, etc.).

• The relationships between numbers, including proportions and ratios.

• The philosophical significance of numbers in the Pythagorean tradition.