# Operators: Harbingers of the Zeitgeist

I’m talking about mathematical operators here, knowing the title may spark thoughts of co-conspirators, as in many language games and “operator” is a somewhat mysterious fixer, engaged in “special operations” which may be bad.

We also have surgeons who “operate” in “operating rooms”.

As for Zeitgeist, I know that term is overused, or let’s just say overdetermined, as to its meaning. “Harbingers” sound a bit like “valkyries” in riding toward us over the horizon, bearers of war news. Omens.

Let’s remember that “operator” comes from “opus” which comes from “work”. An author writes a “magnum opus” (major work). An operator, then, is a worker (one who operates).

The workers bees of arithmetic are its operators. Numbers just lie around, inert (“numb ers”) until taken up in operations, initially the Major Four: addition and multiplication, subtraction and division.

We know we’re doing maths when these operators show up and start asserting their various powers over a number field. They’re like tractors, combines, that pull numbers from the dirt, singly (unary) or in pairs (binary), to create yet more numbers. They’re our number crunchers, these operators.

In the early years, these operator symbols are instructions to us, the computers, the people who do computations. They tell us what to do next, and in what order. As we grow older, we take these operators to be instructions to our devices, our machines.

We’ve learned what these operations entail, from the inside, by doing them manually. Now we’re allowed to order up vast numbers of computations, by writing programs that do them for us. Rites of passage suggest themselves.

One operator that hangs out on the fringes, hoping for a speaking part, often through many years of elementary school, is the modulus operator, used for dividing a modulus into a number as many times as it will go, then obtaining the remainder as the sought-for result.

That the “modulo operator” as it’s also named, should have to struggle so hard to gain attention, is sometimes attributed to politics.

Many computer languages use % for “modulo” and by means of computer languages, % is making a comeback. But why had it fallen by the wayside? How much “clock arithmetic” did you learn in school, and was it ever linked with cryptography?

The story I’ve heard is that Number Theory, associated with Friedrich Gauss especially, the German genius, was thereby tainted as Germanic. This would prove problematic.

Voters around Woodrow Wilson’s time now included women (1920) and Prohibition was coming into force (1919). Germans, the enemy, were also big beer drinkers, adding to the negative PR campaign.

Anglo culture was in a tug-o-war for American hearts and minds and needed Germany to be the bad guy. Wartime propaganda took its toll.

The public elementary school mathematics curriculum would lionize Euclid and keep the German stuff safely out of the limelight. That’s the story I heard, from professor Milo Gardener:

Certainly political forces do have an impact on the everyday curriculum, in terms of what they insist be left out, and kept in.

However political forces will encounter countervailing forces, so it’s often difficult to work backward from any eventual hodgepodge to determine the original agendas of the various interest groups (assuming many at work).

Historians need primary materials if wanting to assert conscious goals.

Now let’s talk about another operator, popularized since the Dot Com boom and bust (the high tech bubble, contemporaneous with the Clinton administration): the dot.

That’s right, the “dot” (denoted by a period) is officially an operator in its own right and means “to reach inside of” as if into a container. These containers are known as “objects”.

The reason the “dot operator” doesn’t appear in elementary school mathematics, in the form of noun.adjective or noun.verb conjugations, is that this operator is associated with “coding” and that’s computer science.

Whereas the arithmetic operators stem from abacus days, from bead counting, the “dot” started making sense (in its current sense) only in the age of the transistor. Calculators wouldn’t need it though. Arduinos (programmable chips aimed at kids) would come later.

The dot and its containerized mereological paradigm (“object oriented”), remains on the other side of a border wall, or fence, so long as mathematics and computer science succeed in maintaining it.

What may be giving the dot operator and its subcultures a second wind, and new relevance to elementary school mathematics teachers, is a pending Renaissance around polyhedrons.

Once a rhombic triacontahedron, a container, is named RT, then it makes sense to write RT.v + RT.f = RT.e + 2, meaning vertices plus faces, equals edges plus two. True for all polyhedrons that aren’t donuts, or worse.

In other words: polyhedrons (A) have properties (attributes) and (B) they do stuff (have verbs). They’re also containers. What if polyhedrons become the paradigm “math objects” of a next generation of computer-savvy curriculum? Will “dot notation” piggy-back on this opportunity?

In the curriculum I’ve been teaching, at the 6th and 7th grade level, we don’t start out with polyhedrons (which are 3D) but with 2D “sprites” (cartoon figures).

Their motions against a “canvas” (stage, background) may be controlled, and dot notation is the way to do it.

We say: monster.goto(10, -10) for example.

Or monster.set_scale(2).

Translation, rotation, scaling: the “big three” of animation. We’re there, with the dot.

Another change in thinking may be consequent to the advent of dot notation, although it’s on the subtle side. Numbers become objects too, in the sense that they now “interiorize” the operations. We “reach into” a number to retrieve its native ability to add with another. Numbers, as nouns, now have their own attributes (such as default base) and behavior (“I, a number, can now compute my own roots and powers”).

In the background, in higher mathematics, Category Theory and Type Theory (intrinsic to computer languages) are already shaking hands. A new Zeitgeist is taking shape.

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