How the ancient Greeks thought about math

We are all familiar with the mathematics of ancient Greece. In fact, if we were to ask a person if they knew any mathematical theorem, they would most likely remember the Pythagorean theorem. However, what not many people know is that Greek mathematics was deeply influenced by the mythological, magical and philosophical thinking of the time.

Faced with the mathematics developed by previous civilizations – such as the Phoenician or Egyptian –, the Greeks saw in this discipline the key not only to understanding the world, but also to reaching absolute truth. For them, mathematics was above its obvious usefulness: it was a supreme form of truth and beauty. This idea appears reflected in Plato’s texts. For the philosopher, geometry is “knowledge of what always exists”, and consequently draws “the soul toward truth and produces philosophical thought.” This extract from Plato’s Republic is one of the many compiled in the book Mathematikós: Vidas y hallazgos de los matemáticos en Grecia y Roma (or, Mathematikós: Lives and Discoveries of mathematicians in Greece and Rome), which was published last year by the Spanish publishing house Alianza Editorial, and features comments by professor Antoine Houlou-Garcia.

The Greeks also thought philosophically about mathematical objects. They debated, for example, if the number one was the elementary building block of the world, or if it was the whole. In a fragment of The Marriage of Mercury and Philology, by Martianus Capella – which is also included in Mathematikós, like all those referenced in this article –, the polymath reflects: “Before all things, let the monad be called sacred; numbers coming after it and associated with it have taught that before everything the monad is the original quickener. For if form is an accident of anything that exists, and if that which numbers is prior to that which is to be numbered, it is fitting to venerate the monad before that which has been called ‘the beginning.’ [...] The monad is everywhere a part, and everywhere the whole [...] For that which is prior to things existing and which does not disappear when they pass away, must be eternal.”

The Greeks’ philosophical conceptions of mathematics made them deny their own intuition. Thus, although Iamblichus devised the number zero, as we know it today, the concept made little impact, as it contradicted the conception of reality at the time. Aristotle concluded in Physics: “There is no ratio in which the void is exceeded by body, as there is no ratio of zero to a number [...] The void can bear no ratio to the full.”

The ancient Greeks did address the notion of infinity, although in a different way than we do. It was an enumerative vision, a quantity that, although finite at each moment, grows indefinitely. Aristotle considered: “Generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different.”

Mathematical ideas were also imbued with magical meanings: numbers thus became symbols representing different archetypes, such as femininity, masculinity and family. Of all the integers, 10 was considered a magic number. The Greeks knew that it was a perfect number – that is, it is equal to the sum of its positive divisors – and they found a transcendental quality in its recurring appearance in the physical world. In geometry, on the other hand, the straight line and the circle were considered the purest forms.

Mathematics was also personified in Greek myths. For example, in another fragment of The Wedding of Mercury and Philology, Geometry is a character who talks about her principles and those of her sister, Arithmetic, assuring that they are both “incorporeal.”

According to the ancient Greeks, it is the abstraction of math, which allows a problem from the physical world to be transformed into a mathematical one that makes it superior to other sciences. As Aristotle stated: “A science such as arithmetic, which is not a science of properties qua inhering in a substratum, is more exact and prior to a science like harmonics, which is a science of properties inhering in a substratum.”

Ágata A. Timón García-Longoria is the coordinator of the Mathematical Culture Unit at Spain’s Institute of Mathematical Sciences (ICMAT).