We are all familiar with the mathematics of ancient Greece. In fact, if we ask a person if he knows any mathematical theorem, he will 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.
Compared to the mathematics developed by previous civilizations —such as the Phoenician or the 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 that “will attract the soul towards the truth and form philosophical minds that direct upwards what we now unduly direct downwards”. This is one of the texts collected in the book Mathematikós: Lives and Findings of Mathematicians in Greece and Romepublished last year by Alianza Editorial, and with the comments of Antoine Houlou-Garcia.
In addition, the Greeks made philosophical considerations about mathematical objects. They debated, for example, if the number one was the elementary brick that builds the world, or if it was the whole. In a fragment of The Marriage of Mercury and Philology, by Marciano Capela –also collected in Mathematikós, like all those referenced in this article–, reflects on it: “If the monad constitutes the form inherent to the first being, whatever it may be, and its priority belongs to what it names and not to what is named, it is fair that we will venerate it before even what we call Principle. (…) it is from her that other beings have been created; she alone contains the seed of all numbers (…) It is both the part and the whole, since it is found in anything; it cannot, since it is prior to beings and will not disappear with their destruction, cease to be eternal.”
The philosophical conceptions that the Greeks had of mathematics made them deny their own intuition. Thus, although Iamblichus devised the zero, as we know it today, his proposal fell into oblivion, since it was an idea that contradicted the conception of reality at the time. Aristotle concluded, in his Physics text: “there is no proportion between nothing and number (…) the void cannot have proportion to the full”.
They did handle the notion of infinity, although in a different way than we do. It was an enumerative vision, a quantity that, although it is finite at each moment, grows indefinitely. Aristotle considered that “in general, the infinite has such a way because what is taken in each case is always something different, and what is taken is always finite, although always different”.
Mathematical ideas were also imbued with magical meanings: numbers thus became symbols representing different archetypes: femininity, masculinity, family… Among all numbers, ten was considered a magic number. The Greeks knew that it was a perfect number—that is, it is equal to the sum of its divisors less than itself—and they found a transcendental quality in its recurring appearance in the physical world. In geometry, on the other hand, the two purest forms were considered to be the straight line and the circle.
Mathematics appears, personified, in Greek myths. For example, in another snippet of The wedding of Mercury and Philology, Geometry tells us about its principles and those of its sister, Arithmetic, ensuring that both are incorporeal. However, numbers and lines are “both corporeal and incorporeal, since what we perceive by the mere contemplation of the spirit is one reality, and what we see through the eyes is another.”
It is precisely this abstraction, which makes it possible to transform a problem from the physical world into one referring to mathematical objects, which, according to the Greeks, gives mathematics a value superior to that of the other sciences. In this sense, Aristotle stated: “A science like arithmetic, which is not a science of properties as inherent to a substratum, is more exact and prior to a science like harmony, which is a science of properties inherent to a substratum” .
Agate A. Timón García-Longoria is coordinator of the ICMAT Mathematical Culture Unit
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.
Edition and coordination: Agate A. Timón G Longoria (ICMAT).
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