The Theology of Arithmetic

1. On the Monad.

The monad is the principle original: archē; the fundamental source or starting point of number, which does not have a spatial position. It is called "monad" because of its "remaining" original: menein. The author is using an etymological pun: monas (unit) is linked to menein (to stay/remain), suggesting the unit stays the same even when multiplied.; for the monad, to whatever number it is applied, preserves the same form. For example, once three is three, and once four is four. Observe how in these cases the monad, by approaching them, preserved the same form and did not change the number. For all things have been ordered from the monad, which contains all things potentially. For even if it is not yet so in actuality original: energeia; active existence, it nevertheless possesses "seminally" original: spermatikos; like a seed containing the potential of the whole plant all the proportions found in all numbers, and even those in the dyad the number two; duality.

It is both even and odd, even-odd, a line, a plane, and a solid—both cubic and spherical. It encompasses everything from a square up to an "infinite-angled" shape in the form of pyramids. It is perfect, abundant original: hypertelēs; a number whose factors sum to more than the number itself, deficient, and proportional. It is harmonic, prime original: prōtē; a number divisible only by itself and one, incomposite, secondary, and both the diagonal and the side This refers to the "side and diagonal numbers," a Greek method for approximating the square root of two.. It governs every relationship of equality and inequality, as has been demonstrated in the Introduction Likely referring to the Introduction to Arithmetic by Nicomachus of Gerasa, a standard textbook in antiquity..

In addition to what has been said, the monad appears as a point and an angle (along with every species of angle), as well as the beginning, middle, and end of the universe. Toward the "lesser," it defines the infinite division of continuous things referring to geometry, where a line can be divided infinitely; toward the "greater," it defines the corresponding increase of discrete things referring to arithmetic, where numbers can be added toward infinity. This is not something we have established by convention, but is the result of divine nature.

Each of these parts, then, corresponds and interchanges within it relative to the wholes, as was clarified in the lambda-shaped diagram The lambda-shaped diagram was a triangular arrangement of numbers (1, 2, 4, 8 and 1, 3, 9, 27) used by Platonists to explain the mathematical structure of the soul and the cosmos. at the beginning of arithmetic. For this reason, just as lengths that are doubled...