Operators are verbs, not nouns. Each one is a geometric act that identities perform on
each other.
Plainly
In ordinary arithmetic class, + and ×
and ÷ get taught as little machines that
grind numbers into other numbers. That is true, but it
misses what they actually are.
Each operator is a verb. It is an act. It
takes two identities — two named things, x and y — and
does something to them. The kind of thing it does is the
same regardless of whether those identities are numbers,
cars, sound waves, players, frames, or pixels. The
geometry of the act is the same. Only
the names change.
And every act either keeps you on the same dimension,
lifts you up one, or drops you down one. There is no
fourth option. That is why so few operators are needed.
Sharper
An operator is a function of two identities,
op(x, y) → z, where z is itself an identity
(page 4). Operators partition into three geometric roles:
Same-dimension: +, −, mod. Movement along
the line of x.
Ascending: ×. Two inputs collapse into one output that lives one
dimension higher.
Descending: ÷. One input explodes into the parts that compose it,
exposing the dimension below.
Comparators: =, <, >. They do not
produce a new identity; they yield a relation between two existing identities. They are how the
system queries the manifold.
+
ADD · accumulate
Move further along the same dimension as x.
geometry: translation · z lives on the line of x
−
SUBTRACT · retreat
Move backward along the same dimension as x.
geometry: translation reversed · z lives on the line of x
×
MULTIPLY · gather
Two identities collapse into one whole, one dimension up.
geometry: saddle z = x · y · ascends
÷
DIVIDE · explode
One whole opens into the parts it was composed of, one dimension down.
geometry: hyperbolic z = x ÷ y · descends
mod
MODULO · cycle
Wrap x around the period y; expose where in the cycle x stands.
geometry: closed loop · z lives on the ring of y
=
EQUAL · identify
Ask whether two identities are the same identity.
geometry: query · yields relation, not new identity
<
LESS · order below
Ask whether x lies before y on their shared dimension.
geometry: query · yields relation
>
GREATER · order above
Ask whether x lies after y on their shared dimension.
geometry: query · yields relation
Three roles, no others. Comparators query these geometries; they do not change them.
Operators are domain-blind
Because an operator is a geometric act on identities and not
on numbers, it applies identically across domains.
The same × that builds a plane out of two
lines builds a colour out of hue and luminance, and builds
a chord out of two pitches. Different identities, same
act, same geometry.
What this rules out
Code that treats + and × as
interchangeable "math". Operator overloading that breaks
geometric role (a × that does not collapse,
a ÷ that does not expose parts). Domain-specific
"operations" that duplicate a verb already provided by the
eight actors. Each is a violation of the geometric law.