There is a mathematics of similarities and directions that’s different from the mathematics of numbers.

“You don’t learn to walk by following rules. You learn by doing, and by falling over.”
― Richard Branson

“The idea is to go from numbers to information to understanding.”
– Hans Rosling, physician

I’m no mathematician and, chances are, you aren’t either.

Some conspiracy theories are more likely than others, and I think most have some truth. One of the least plausible conspiracy theories is that forces have conspired to make us mathematically stupid. Yet here we are: mathematically stupid.

Maybe it’s more about thinking and less about math. We are thinking-disabled, and there definitely is a conspiracy keeping us so. In that plan, math is hyper-logical thinking pressed on children to keep them from developing.

Mathematics has little to do with numbers. It has a lot to do with equations, but not the kind of equations you think. It’s not about establishing equality in formulas; it’s about establishing the equality of concepts. Numbers are just the nuts and bolts of mathematics. Nuts and bolts won’t make that happen if you can’t envision the relationships between things.

Numbers

“If you don’t know your numbers, you don’t know your business.”
– Marcus Lemonis, businessman, television personality and philanthropist

Numbers measure. If you want to make precise comparisons, you need to measure. You need to measure if you want to establish absolute equality. But if you’re more interested in similarity and growth, you don’t.

Math is taught as a machine for measuring. If ideas were architecture, then math is taught as the craft of driving nails. And for twelve years in school, children are drilled, over and over, on how to drive nails.

In the nineteenth century, children were taught to shovel coal. In the twentieth century, they were taught to add numbers. Now, in the early twenty-first century, school still teaches children how to drive ideas, not how to design them.

Now, in the early twenty-first century, school still teaches children how to drive ideas, not how to design them. This form of teaching is increasingly irrelevant. As artificial intelligence makes clear, it’s no longer building machines that’s important, it’s designing them. Math will play a role, but at a higher level.

Ideas

If you’re going to think, then it’s imperative that you’re able to compare ideas. That’s because small differences can lead to big changes, and if your discernment is poor, and your awareness is dull, then you’ll be mixed up. There is definitely a conspiracy against thinking. We see this in every organization: the media, entertainment, work, law, advertising, and social trends.

Maybe our dominant political structures will continue to need non thinking workers. Outside-the-box thinking is discouraged in most animal colonies, and human colonies are rarely an exception. Religion is an example of constrained human thinking that’s for the betterment of all… except when it’s not. And the more we need non-thinkers, the more mathematics will continue to be mis-taught.

To be clear, mathematics is not taught badly, it’s taught wrongly. Students are intentionally misdirected. That’s the conspiracy part of it, but even if you don’t believe that, realize that what’s taught is wrong. Mathematics is not about numbers, it’s about ideas.

Metaphor

I’m a metaphorician, but since there’s no science of metaphor, there’s no such thing. There are similarities between ideas, and these similarities form bridges. The metaphorical connections between ideas are often paths to their extension. That’s because we invent all our ideas and they all come out of the same soup.

Mathematics offers many useful metaphors, but you wouldn’t know it if all you know about math is numbers and equations. I want to introduce some of these metaphors. This does not involve numbers. The equalities are metaphorical.

Math puts numbers to qualities, and refines qualities into categories like shape, texture, direction, progression, and destination. We can categorize shapes by the number of significant holes they have. We can describe textures by the size and shape of extensions. To compare directions, we must define a reference system. And to describe a destination, we need to say whether things will change on our way to it.

But to ascribe a quantity to something that does not have it, you need imagination. A metaphor will connect things that are otherwise dissimilar. We can say a car is like a submarine that travels through air, or that a towel is like a sheet covered with a forest of tiny fur trees.

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Equality

Equality is the foundation of measure, judgment, and mathematics. Let there be hard and soft equalities and a range of in-betweens. Equality is just the counterpart of difference and, if you know the difference, then you can restate the equality.

We rarely see things as equals because we’re more attuned to differences. Our nervous systems are designed to notice differences and to amplify them. Most of what we perceive as large differences are small ones, but the closer things get to equal, the more we refine our sense of difference.

We set our expectations based on the largest differences we see. Maybe that’s why people find thriller and horror movies relaxing. By stretching our sense of difference, they make the differences of normal life demand less of our attention, and we relax.

Linear and Logarithmic Measures

While we mostly see in terms of lines—direct proportions, linear progressions, ratios, and changes in the same direction—we mostly perceive things in terms of logarithms. Logarithms measure changes in terms of powers of 10, not in terms of equal measure.

We cook using equal measures—liters, pounds, cups, and grams—but we see distances in terms of things that are 10, 100, and 1000 times farther away. We call it “perspective” but it’s really logarithmic. The same with illumination. What we perceive as being twice as bright really provides 10 times more surface radiance (or lux). What we see as four times as bright, radiates 100 times more light.

We might say, “At this rate, that will take forever.” And this is a perfect example of how much better our thinking would be if we were more adept at measuring things. Something that takes “forever” is more likely a progression in which each step takes longer to complete than the last.

We could say, “A solid relationship takes years to develop.” Or we could say, “A solid relationship takes a dozen steps where each step takes twice as long as the one before it.” The first description doesn’t tell us much and isn’t optimistic. The second description says much more about relationships and also reflects something about ourselves.

Progression: A Vector

A vector is a number with a direction. The number gives speed, size, or amount. The direction could be in time, space, emotion, or anything you like. Progression is so common we don’t bother to specify it, but once you do measure progressions you can combine, separate, and say things about them.

People are clusters of vectors with different attitudes pointing in different directions. Partnerships don’t develop between people who lack common interests. They may approach those interests differently, even oppositely, but the common interests form common values.

If we think of our parents, they will have similarities and differences. We might think of each as having many dimensions, and sharing similarities in some and not in others. Partnerships develop between people who are either alike or different in aspects that are important to each.

This is an extensible description. We can add new components and we can add more detail to the components. It is an ironic property of people that they are classifiable, and I think this is because we tend to think alike. We feel included by lining up with others and this limits us. Before too long, we are not much more than how other people describe us.

Projection: The Dot Product

We can compare two expressions that have something in common. We can ask if they’re leading in the same direction, and how similar they are. If our progressions have magnitude and direction, we can project one onto the other. If expressions were vectors, then a measure of their similarity is called their dot product.

In the above picture, two vectors radiate from the same point. Their dot product is the shadow of one on the other. It’s a measure of how much they have in common.

Men and women may adopt different gender roles, but men and women who form partnerships share similar values. People who engage in play take different roles, but the roles they take have a place in a shared theme, even if those roles are antagonistic.

For example, players in role-playing games are allies or adversaries in imaginary contexts. They have measured qualities that determine the outcomes of conflicts. This is a contrived example of the dot product because these qualities are the same, only the quantities are different.

This is also what makes these games poor reflections of real life. Comparisons in real life are more like the comparisons between vectors because, in real life, different qualities can have similar effects. It’s not the physically strongest person who wins the battle, it’s the person who defines the battle to make the best use of their strength. It’s the person who aligns their skills with the situation.

In these situations, partners or players may share similar powers to different degrees, or different powers to similar degrees. They may also be powerful in opposite or unrelated ways, like physical strength versus ingenuity, or skills in seeing specifics versus the skill in appreciating the whole. With enough clarity, we can reduce our emotional comparisons to geometry.

Distortion: The Tensor

A tensor is an abstract mathematical object unlimited in dimension, value, or effect. We have to dumb it down in order to apply it. Think of a tensor not as a thing, but as a description of the forces that act on a thing.

If you have a cube of sugar and a spoon, and you press the cube until it breaks, then a tensor describes the forces inside the cube: down from the top and up from the bottom, outward on the four sides.

More generally, our personality is a multidimensional sugar cube, or maybe a multidimensional cube of pudding. We have certain resiliencies and limits. We can bend but we can also break. What’s more, many of our aspects are interdependent.

Consider safety and happiness, or fidelity and connection. Not only does a force on one of our qualities deform it, but that force can distort other qualities, too.

Anger and love distort many feelings. Our physical health affects our mental health. At this moment, I’m taking an antibiotic that makes me depressed. I marvel at how my depression is a chemical artifact. That realization helps me to disengage from it, but the feeling lingers like a damp chill. Lucky for me, I know things will brighten next week when I finish these pills.

We experience forces of engagement and indifference with wildly different results. A tensor would be the object that describes how one force applied to one aspect of our personality affects our other aspects.

Extension: The Wedge Product

Mathematics gets odd when you extend it to represent objects with size, shape, direction, and movement. These are not the old objects we’re familiar with, like blocks, they’re new logical objects that follow new rules.

Much of this is abstract mind bending waiting to give us new ways to solve problems. At a basic level, we can apply these as metaphors to circumstances in our lives. The wedge product is such a concept.

The vector is simple and easy to imagine; it’s just an arrow. The tensor seems reasonable enough; it combines forces from different directions. The wedge product, made from vectors, is something few people encounter outside abstract math, but we can use it here.

We create wedge products by taking any two vectors, arrows if you will, duplicating each to get four, and moving tails to heads to make a four-sided parallelogram. This results in something that has directions and also has an area: in 2-dimensions it’s a flying diamond. Applying old rules to new things lets us add, subtract, multiply, and divide these flying diamonds.

Returning to the mother and father metaphor, think of the wedge product as their combined powers. If both move in exactly the same direction, then there is nothing new between them and their wedge product has no volume and is zero. But if they are perpendicular to each other—not opposite but entirely different—than their combination encloses all that’s between them.

This is not a simple answer to anything, it’s just a tiny concept of extension: the greater the difference in magnitude and direction of the two partners—represented by their different arrows—the greater the area that’s included by their combination. That’s assuming they remain partners, which implies collaboration. That doesn’t mean “good,” it just means “together.”

It may be useful to think of how directions combine, and whether there is some algebra to our personal combinations. What does it mean to combine similar or different talents, and is the result something that stands on its own, to be further combined with other things?

These metaphors may be common sense, or they might offer something more. The math offers more, but here we’re just concerned with scale and symmetry. In defining these objects, we’ve overlooked the equals sign. That returns when we do things with these objects, and then we want to know what results.

You can imagine equations that combine magnitudes of things, things with direction, their projections, distortions, and extensions of and with each other. Numbers are unnecessary. We can create conceptual objects and ask conceptual questions. We don’t have to answer the question on the blackboard, we can make up our own questions. We can make up our own language. We can invent our own concepts.

A Need for Evolution

One thing that mathematics avoids is disruption. There is a kind of mathematical segregation. Things that are not equal are kept apart, and things that evolve must stay within the container. Mathematics starts with fundamental objects and makes things from them. It has many domains, but specialties remain within them.

Humans famously break rules. Most of the time, they do so for bad reasons and with poor results. Mathematics doesn’t like wrong answers, but evolution is all about doing what doesn’t work for lack of anything more expedient.

This is my fundamental dislike of mathematics. It feels sterile. I don’t want to build inanimate objects, mechanized humans, or paintings made by artificial intelligence. There will always be a need for errors, mistakes, misunderstandings, and absurd assumptions.

With more important problems, we need more insightful solutions. That requires more creativity. You don’t need to be a mathematician to use these concepts for your own purposes. Take these ideas and build richer building blocks. Richer building blocks will give you greater insight.