To modern mathematics the unit is the first issue of the natural series of whole numbers.
They are obtained starting from the unit and adding another unit later. The same is not true for arithmetic multiplication. In fact the same word, monad, indicated the arithmetic unit and the monad in the sense that today we would call metaphysical monad and the shift from universal to the duality is not as simple as switching from one to two by the addition of two units. In arithmetic, including multiplication, there are three direct operations: addition, multiplication and raising to power, accompanied by the three inverse operations. Now the product of the unit itself is still the unit, and a power unit is still the unit, then only the addition allows the passage from unity to duality. This means that to obtain two we must admit that there may be two units, which means having the concept of two, namely that the monad can lose its unique character, that it can stand out and that there may be a duplication of units or multiple units. Philosophically there is the question of monism and dualism, the metaphysical question of Being and its representation, the question of the biological cell and its reproduction. Now if we admit the intrinsic and essential oneness of the Unit, one must admit that another unit can be only an appearance, and that its appearance is an alteration of uniqueness that comes from the Monad operating a distinction in itself. Consciousness operates in similar manner a distinction between self and not self. According to Advaita Vedanta this is an illusion, and indeed is the great illusion, and there isn’t any choice but to get rid of it. It is not an illusion that there is this illusion, even if it can be overcome. The Pythagoreans said that the dyad was generated by moving away or separated from itself, which divided by two: and indicated that differentiation or polarization with different words: diaeresis, tolma.
In math a multiplication unit was not a number, but it was the principle, ἀρχή’s of all numbers, we say the beginning, not start. Once admitted strength of another unit and several units, then from unit, by addition, the two and then all the numbers result. Pythagoreans conceived numbers as drawn up or composed or variously depicted as points. Point was defined by Pythagoreans as a unit having position, while, in Euclid’s opinion, point has no parts. Unit was represented by the point (σημεῖον semeion sign or seed) or even, when written alphabetic system of numerals came in use, from the letter A or α, which was used to write the unit.
Once having admitted the possibility of the unit addition resulting in two, represented by the two end points of a segment line, we can continue to add units, and subsequently obtain all the numbers represented by two, three, four … points aligned. In that way we thus have the linear development of the numbers. Except the two that can be achieved only by the addition of two units, all integers can be considered either as the sum of other numbers, for example, five is 5 = 1 + 1 + 1 + 1 + 1, or also 5 = 1 + 4 and 5 = 2 + 3. One and two did not have this general property of numbers: and therefore, to the ancient Pythagoreans, as well as the unit two can’t be considered a number, but the principle of even numbers. This view was lost over time because Plato ( Parmenides, 143 d) speaks of the two as equal, and Aristotle (Ta Topikà, 2, 137) speaks of the two as the only even prime. The three in turn can only be considered as the sum of one and two, while all other numbers as well as being the sum of several units, are also both different from the sum of parts, some of them can be regarded as a sum of two equal parts in the same way that the two is the sum of two units, and are called even numbers because of their similarity with the pair, so for example 4 = 2 + 2, 6 = 3 + 3, etc.. are even numbers, while others, such as three and five which are not the sum of two parts or two equal addends, are called odd numbers. So the triad 1, 2, 3 has the mentioned properties which numbers greater than 3 don’t.
In the natural series of numbers, even and odd numbers take alternately place one after another; even numbers have in common with two the peculiarity above mentioned and then they can always be represented in the form of a rectangle (epipedo) in which one side contains two points, while odd numbers, like unity, don’t, and when they may be rectangular in shape, it happens that the base and height contain a number of points which in turn is an odd number. Nicomachus also reports a more ancient definition: excluding fundamental dyad, even is a number which is divisible into two equal parts or unequal parts that are either even or odd, or, as we should say, that have the same parity, while odd number can be divided only into two unequal parts, one even and the other odd, i.e. parts that have different parity.
According to Heath (A History of Greek Mathematics, I, 70 ) the distinction between odd and even goes back to Pythagoras of course, which is not hard to believe, and Reidemeister (Die arithmetic der Griechen, 1939, pag. 21.) says that the theory of even and odd is Pythagorean, and this concept makes explicit the mathematical logic of Pythagoreans as well as being the foundation of Pythagorean metaphysics. In Unequal Numbers, says Virgil, Deus Gaudet, God takes delight.
The Masonic tradition conforms to this recognition of the sacred or divine odd numbers, as indicated by the numbers expressing initiation ages, the number of lights, jewelry, brothers members of a factory etc.. Wherever there is a distinction, a polarity, there is an analogy with even and odd pair, and one can establish a correspondence between the two poles as well as equal and odd, so for Pythagoreans the male was odd and female peer, the right was odd and the left was equal ….
to be continued at Arturo Reghini Sacred Pythagorean Numbers. Part 2 .