Mathematics as it was thought in ancient Greece | Coffee and theorems | Science

We are all familiar with the mathematics of ancient Greece. In fact, if we were to ask a person if they know a mathematical theorem, they would most likely remember the Pythagorean theorem. However, what few people know is that Greek mathematics was deeply influenced by the mythological, magical and philosophical thought of the time.

Faced with 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 achieving absolute truth. For them, mathematics was above its obvious usefulness, it was a supreme form of truth and beauty. This idea appears reflected in the texts of Plato; for the philosopher, geometry is “the knowledge of what always exists”, and which “will draw the soul towards truth and will form philosophical minds which direct upwards what we now direct unduly downwards”. This is one of the texts collected in the book Mathematikós: lives and discoveries of mathematicians in Greece and Romepublished last year by Alianza Editorial, and commented by Antoine Houlou-Garcia.

Moreover, the Greeks made philosophical considerations about mathematical objects. They debated, for example, if number one was the building block that builds the world, or if it was the whole thing. In a fragment of the Marriage of Mercury and Philology, by Marciano Capela – also collected in Mathematikos, like all those referenced in this article, reflects on this: “If the monad constitutes the form inherent in the first being, whatever it may be, and its priority belongs to what it names and not to what is named, it is right that we venerate it even before what we call Principle. (…) it is from her that other beings were created; it alone contains the germ of all numbers (…) It is both the part and the whole, since it is found in everything; 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 Iamblique imagined the zero, as we know it today, his proposal fell into oblivion, because it was an idea that contradicted the conception of reality at the time. Aristotle concludes, in his text of Physics: “there is no proportion between nothingness and number (…) the void cannot have any proportion with the full”.

They treated the notion of infinity, although in a different way from ours. It was an enumerative vision, a quantity which, although it is finite at each instant, 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, though 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 it was a perfect number – that is, it equals the sum of its divisors minus itself – and they found a transcendental quality in its recurring appearance in the physical world. . In geometry, on the contrary, the two purest forms were considered the straight line and the circle.

Mathematics appears, personified, in Greek myths. For example, in another excerpt from The marriage of Mercury and Philology, Geometry teaches 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 mere contemplation of the mind 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 of the physical world into another referring to mathematical objects, which, according to the Greeks, gives mathematics a value superior to that of the other sciences. In this sense, Aristotle affirmed: “A science like arithmetic, which is not a science of the properties inherent in a substrate, is more exact and prior to a science like harmony, which is a science of the properties inherent in a substrate.

Agate A. Timón García-Longoria is coordinator of the mathematical culture unit of ICMAT

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 cultural expressions and remember those who marked their evolution and knew how to transform coffee into theorems. The name evokes the definition of Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.

Editing and coordination: Agate A. Timon G Longoria (ICMAT).

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