All of the traditional sciences of our day are founded on analytic arithmetic, a relatively

recent development. The ancient Greeks practiced a synthetic kind of arithmetic. Instead of

scalar magnitudes, they resorted to geometric line segments. The product of line segments

could be a rectangular area, or a volume, for example. Theirs was a geometric arithmetic. In

this paper, based on previous work, we report a new kind of science using non-ordered

synthetic number. We call them now numbers. The number system is binary with the two

elementary parts corresponding to the ancient ontological gender construct. It turns out that

there are four elementary now numbers corresponding to the binary genders fm, ff, fm, and

mm, reminiscent of the implicit gendering of the four classical elements