There are living things we do not see.

I sense organizing behaviors — order — in what I do (when writing, painting, playing music, loving). Also in things I and many others participate: scientific research, political developments, war dynamics, gaming, financial markets. In nature, its sheer variety freezes judgement. Pigeons put a strong commitment to flock bringing to life an ever-surprising structure; stars do cluster in the night; atoms align; the ground is firm and thick.

Seeing this, I realize I am not neutral. One obvious difference is that while I’m immersed in human-involving phenomena, such a privilege is not available in natural ones. I do note coordination flowing through me because I’m part of the phenomenon, I can read the tape — whether as researcher, soldier, player, investor, lover or writer — something impossible to experience with hurricanes, a nebula, or a crystal (as one is not a water drop, a stellar body, or a molecule).

Tape-reading might provide a kind of insider advantage in experiencing what is like being inside of an invisible structure. So if I have to pick one specific human interaction tool to tape-read into, I’d choose one very specific form of mathematics-usage — algebra — because it carries information the simplest, unambiguous and most recognizable appearance.

How it does it.

This writing makes you and me interact. Algebra does the same. It sets it in terms of operational sequences instead. That makes it easier to spot how it makes people interact. A sequence of algebraic expressions tries to convey how certain things relate with another. Anyone who can read the formula and understands it, it’s free to apply it. But on top of that, with algebra, anyone with mathematical skill is also able to turn upside-down the formula, turning an input from the original sequence into an output it can compute to. And this new second sequence is true as long as the first holds. Depending on who uses it, such kind of postulation might range from a scientific prediction up to the answer to a production problem.

Interestingly, this process unintendedly aligns people in a loose way.

The process is like when Black says to White “the restaurant is sold-out for the entire night”. White may infer: “that means I should abandon all hope about dine there today”. This is equivalent to solve an equation: it explores the implications of the statement using close-knit deduction. It’s unnecessary that White shares her thoughts with Black expecting confirmation. Although if White does so it’s a way for both to verify conversation well-functioning, regardless even whether White actually wanted dining there or not, and whether it is true or not the restaurant is sold-out. It really means that both are on the same page, hence somehow connected (tied). In addition, at that point, White’s behavior ingrains something special. She could start looking for another restaurant or instead find ways to challenge Black’s assert (call by phone, ask another person, use a relative with insider info) because all those are no longer random. The important thing is: first, White herself reverses a third-party statement to give an answer to her question. Second, thanks to that, White’s actions as long as they are related with Black’s interaction will no longer be random. Third, imagine considerably more people performing the same action White made with Black — asking for reservation in a specific restaurant — though this time it involves each and any of all restaurants of the same franchise. Each with its own Mr. Black at the front door. A market pattern would take shape. Such a thing organizes what otherwise would be random walks.

In other words, solving for x is an act of coordination. It may well be part of a problem’s solution in a physics course or an actual production-planning problem in a semiconductor plant. Likewise, solving may lead to a scientific prediction.

If many use the equation C, with variables a and b, then solving for each of both will make experiments proliferate; they will want factual verification of solutions’ estimates. Despite the diversity, all of them point to the same origin: C. As a consequence, researchers stop thinking “random” and start being part of something larger, and invisible.

People using others’ algebra lead to a new type of unplanned organization of higher degree, regardless of what they’re saying and solving.

Take the hundred most cited physics papers in the last 40 years and think of their tens of thousands of researchers that reference them. Ever since, any single one of these do write using algebra those top authors once published. Because if a researcher formulates a statement about how to calculate the results of an experiment or observation interestingly enough (being hypothetical or not is irrelevant), a social structure that neither I nor anybody else by any means can control (or plan) arises. Algebra does that.

What algebra really is.

To me it’s a miracle how something could ever represent something else. Four alphabetical characters denote a ball even though it has nothing to do with that. The word “chair” isn’t a chair. And, as if weren’t enough, it propels behavior: certain colored marks on a surface can alter the trajectory of a body. That’s what language does with people and machines.

Figuring out new math always ends up one way or another as a finite phenomenon. Any numeric system must be so. Any formula comprises only a handful of elements. Any number is in practice limited; pi will always be substituted with a round number.

Infinities, however, keep making sense. Despite proving to be impracticable and unthinkable, they are indispensable in algebra’s use of letters.

Yet at the same time I note they are invaluable in simplifying to draw, understand and manipulate math formulas. It means that anyone familiar with mathematics can efficiently reload an equation, and work on it maintaining information integrity in the process. In other words, letters in mathematics let you employ a lot less symbols and time to say the same while mostly erasing the risk of misunderstanding.

How it starts.

Imagine a coordinate system. Certain know-how makes it possible. And probably it also has its own working literature with its own algebra in it. The algebraic spark happens when the writer identifies the input that most influenced the device outcome. The degree in which does impact final values’ variability is what makes it of literary interest. The more the influence over end-values the better. It’s recognizable because altering it makes the output to respond in a telling way. Ideally, it should lead to the perception that it controls output levels.

But the abstraction is just starting. It’s desirable to establish a clear relationship between variables. Ideally using a mathematical function. It’s so because it tests whether there’s something meaningful to tell. In and of itself is a measure of diligence. In mathematics a function is for an equation what cause-effect is for explanation. Absence of an arrow saying that values at one letter unambiguously patterns in another letter is akin to rambling — one question must have one and only one answer, not two or three.

It’s important to note that the function of choosing to speak about a coordinate system has very little to do the coordinate system itself. One can gloss over something ignoring how it actually works. It bears no relation and provides no clue about the nature of the subject of study.

But besides that, students will benefit from finding this function in the literature anyway because it’s unambiguous. Each value one plugs into the independent variable (the input I chose before), it will calculate one and only one end value on the other side. This kind of analysis increases persuasive power to the writing piece.

Before continuing, couple words regarding curves, shapes, geometries and algebras.

Some of these structures uncannily resemble those that exist in the most familiar part of the real-world.

First, as soon as I apply math know-how a mathematical reality appears. Second, some math realities show clear patterns. Third, of these, some resemble those I observe in Nature. Seeing things this way the answer is rather simple and even self-evident: a full moon and a circle they look alike, analogous to two persons who resemble each other because they are brothers however different. That’s why sometimes an enemy’s enemy can be automatically an ally, and planets’ orbit can be predictable — by similarity. And why all water drops look alike, why snowflakes are so visually geometric and some crystals of pyrite are so close to a platonic cube: worlds-apart different with a slight resemblance among them. Mathematics is order by its own right (when so). It is an existing reality as those of sunlight, a grain of salt, or a thunderclap. With a difference grounded in that we are involved in its production. In the same manner I produce a laser beam a right triangle appears on the paper.

This is remarkable because it means there’s a way where a pattern and an independent reality arises in the instrument I mainly use to speak about numerical relations.

Let me explain from another angle. Think two pebbles that look alike. Because they are similar I couldn’t say that one pebble does explain the other or one is the essence of the other. Now, imagine this time a geometrical shape is one of these stones, being both similar-looking. I would repeat exactly what just said before, that one object in no way explains the other. Much less would believe it is its essence.

For the record: I can’t find a good reason why I have to accept mathematics is different from stones. The only real difference among the mathematical shape and a pebble with similar appeal is that I can see the former inside-out as I create it. It’s something which, of course, causes perplexity.

Having said that, a reminder: knowing how to do algebra and algebra itself are different things. Only one can be put in writing.

Speaking of which, and back to how algebraic literature develops: if it’s true that a writer makes more persuasive arguments if uses a function, it’s also true that, for practical reasons, will want to add the benefits of using an equality to the piece. By introducing similar changes on both sides simultaneously, it can alter the function’s original operational sequence while still honoring its initial assumptions. This way, anyone can widen the subject.

At this point, I invite you to do the following exercise: imagine tens of thousands of people exchanging information. Moreover, see all of them communicating through equations. Constantly transforming, redrawing, replying to sender or forwarding to another. Such flow creates massive waves of coordination articulated with the (almost) solely use of algebra.

So, let me remind what I think algebra is: an interaction device we see inside-out that switches people’s actions to non-random. It’s objective (though essentially inexplicable by physics) and at the same time observable from within (similarly as when one sees the interiors of a cathedral).