The contribution of the Pythagoreans has become known not
only in the field of philosophy, but also in the field of mathematics,
especially the Pythagorean Theorem. In one way or another, I think that their
group has positively influenced people down the century on the significance of
numbers. As they say, everything in this world is number, and is numerical. This
means that things around us are indeed composites of numbers, and they are
countable through numbers. But how Protagoras understands the concept of
number? I will show three things his notion of number in relation to reality.
1. In the abstract level, number has two elements – odd numbers
and the even numbers. The odd number is described as the unlimited, and the
even number is the limited. How did Protagoras come about that idea? It is not
specifically explained.
2. In the concrete level, numbers are assigned to cosmology.
He claims that the world and all it contains are described as the limited
cosmos being surrounded by the unlimited cosmos, which is the air. Since air is
everywhere, the world inhales it. But his limited cosmos is composed of the
unlimited cosmos. Thus the limited cosmos is a mixture of both limited (even) and
unlimited (odd) numbers.
3. Now, how are we to understand the all things are numbers?
Protagoras explains number spatially. That is why his idea of numbers earns the
label “geometry.” This is how he expounds it that one is the point, two is the
line, three is the surface, and four is the solid. When taken together (the
point, line, surface and solid), it constitutes the idea of number. In short, all things are
composed of points, lines, and surfaces. Every material body is an expression of
the number four or solid. Thus things are both numbers and numerable.
4. Regarding the shapes of things, he expresses that it is either
oblong numbers or square numbers. When we start with one object and add odd number
of objects, we get the square shape. And when we start with two objects and add
even number of objects, then we get the oblong shape. This explains the shapes
of things. (The “Tetraktys” is treated
highly as a sacred and perfect figure. It composed of ten dots, and it is the
sum of one, two, three and four integers.)
The implication of Numbers
Protagoras discovers that everything is numerical, and that
is, it can be counted. Nothing escapes the reality of numbers. And he observes
that even in music, its interval is also expressed in numerical. (For those who
know the fundamental of note reading could relate this matter.)
In the classical music, one can easily observe the presence of
harmony. The rhythm of the music is well-arranged and coordinated. This is
attributed to existence of numbers because each note in the bar measure is equally
divided according to its assigned time signature. We can intuit that the
employment of numbers creates harmony or chaos.
This idea of harmony
applies to the material things as well. If one observes how our human bodies
function, they are well-regulated and well-designed, and all other things too. We
can say that the harmony of the universe is numerical too. The intervention of
man to our body and our world puts them in disorder.
