The Theology of Arithmetic

...[according to the ratio] which is double original: diplasios; a 2:1 ratio, or the ratio to the term after it, which is sesquialter original: hemiolios; a 3:2 ratio (literally "a whole and a half"). In Pythagorean music theory, these are the fundamental intervals of the octave and the fifth.. This number is the root of ratios that proceed infinitely in each direction, such that the same nature belongs to both multiples Numbers like 2, 4, 8... and superparticulars original: epimorion; a ratio where the larger number contains the smaller plus one part of it, like 3:2 or 4:3..

Furthermore, the Dyad is an element of the composition of the universe, standing in opposition to the Monad. For this reason, it is subjected to the "persistence" of the Monad, acting as Matter original: hyle does toward Form original: eidos.

Hence, since Form is that which encompasses "being" and "eternal existence," while Matter encompasses opposites—specifically those things that are entirely similar and identical in name, which is to say, squares—the Monad is their cause. This is not only because when successive odd numbers In Pythagorean "gnomon" math, adding successive odd numbers to 1 always creates a square: 1+3=4, 1+3+5=9, etc. are placed around it as a gnomon original: gnomon; an L-shaped tool. In arithmetic, it refers to the numbers added to a square to make the next larger square. (they being the "formal" expressions of the Monad), they complete squares through cumulative progression, extending further and further to infinity; but also because each side—like a runner going from the starting gate original: hysplex of the Monad to the goal post original: nyssa of the Monad—contains within itself the composition of the journey out and the journey back, producing the square from its own nature.

Conversely, for those things entirely dissimilar—that is, the oblong original: heteromekes; "other-lengthed" numbers where the sides differ by one, such as 2x3=6 or 3x4=12. numbers—the Dyad is the cause. This is not only because when even numbers (which are the "formal" expressions of the Dyad) are placed around it as a gnomon, these too are completed cumulatively; but also because, in that same image of the turning-post original: kampter, the goal, and the starting line, the Monad appears to provide the origin in the same way (being the cause of identity and simple permanence), while the Dyad receives the "destruction" and the journey back in a varied way compared to the former, acting as a kind of material substance capable of receiving all corruption In this philosophy, the Monad represents the eternal and unchanging, while the Dyad represents the world of matter, which is subject to birth, change, and death ("corruption")..

Since "multitude" is conceived of according to the Triad The number Three., and that which is viewed in opposition to multitude is according to the Monad, the Dyad would be the middle ground original: metaichmion; the space between two opposing armies. between them. For this reason, it possesses the properties of both at once.

For the property of the One, as the starting point, is to make something greater by addition original: synthesin than by multiplication original: enkrasin; literally "blending" or "intermixing.". For example, one plus one is more than once times one 1 + 1 = 2, whereas 1 x 1 = 1.. But the property of multitude, as the result, is the opposite: it makes more from multiplication and less from addition; no longer...