The Most Important Lesson of Modern Mathematics (Without Doing Any)

Did anyone else watch Square One TV on PBS (or reruns on Noggin) as a kid? It feels like a bit of a fever dream to remember, but it was a kids' show of short sketches, usually pop culture parodies, that introduced and applied concepts in mathematics. I distinctly remember one storyline of its Dragnet parody Mathnet where Detectives George Frankly and Kate Monday solved some mystery involving the Fibonacci sequence in tiles on a wall, and a parrot. Apparently the show was intended to "intended to address the math crisis among American schoolchildren." A 2018 Buzzfeed article by Anne Helen Petersen (whose Culture Study newsletter I subscribe to and enjoy) supports my suspicion that the show's impact was probably less remedial for those in "math crisis," and more inspirational for those already a bit nerdily inclined, like myself. "Square One ... wasn’t as long-running as Sesame Street or Mister Rogers’ Neighborhood and has never been the same kind of ubiquitous cultural touchstone," she acknowledged. "But like the best children’s television, it implanted itself — and its attitude — into millions of children’s minds. That attitude was pretty simple: Math is weird, and cool, and filled with secrets, and fun."

If math no longer seems weird, and cool, and filled with secrets, and fun, that's understandable. Going about our days, most people don't directly encounter much besides boring arithmetic and rarely-fun statistics. And most people would rather not deal with more of those, let alone calculus or other painful reminders of high school. But most people don't know there have been some mind-blowing developments in math, and in its even more esoteric-seeming sister field of formal logic, in recent decades that shake the very foundations of our culture's dominant understanding of reality. This isn't the math from high school. Hear me out: I'll introduce you to these developments and their implications, without actually using or requiring you to use any math. Contemporary logic and math are, for our purposes here, climactic events in a millennia-long story that involves everyone and everything, even if few know about these events outside (and even within) academia. You can be one of the few to know the cool secrets. I'll stick with math in this post and turn to logic in the next.

First, Some Context

You can probably skip this section if you're a How To Get Things Less Wrong regular (thank you!!!) or you've read the first post. The millennia-long story that involves everyone and everything is the subject of this newsletter. It goes like this:

1. Whenever human beings think about any thing, absolutely any thing at all (including people and categories), we all necessarily carry some underlying framework for what a thing is, what makes a thing a thing, what kinds of things the world is made of, what things are real, what real means, what really matters, and so on. These frameworks are usually implicit; when they're surfaced, we call that metaphysics, and make it intimidating. But it's not. You're always already doing metaphysics.
2. Everyone thinks their way of looking at reality is just, well, reality, but there's more than one way to do it and they're not equally or equivalently valid.
3. Ways to do metaphysics, i.e., to understand how reality works, are diverse but can be usefully arranged into two predominant directions. (3a. These two directions appear to align closely with the two predominant modes of our left and right brain hemispheres, respectively, suggesting we all have the capacity for both and it's a matter of how we trust and apply them.)
4. At one end is what's known generally as static or substance metaphysics. It takes the universe as made up of things that are separate, self-contained entities with timeless, determinate essences that can be measured against universal ideals. The combination of separateness, essentialism, context dismissal, and idealism puts everything—including people—into zero-sum competition for hierarchical mattering or realness. (Think "Real Americans.") This, in turn, promotes individualism, authoritarianism, oppositional binaries like nature versus culture and us versus them, and unchecked inequality. Existence outside the rigidly idealized order is deemed fundamentally disordered, even illegitimate or unreal. Because substance metaphysics dismisses messy context and complexity as less fundamental to "true" reality than its idealized hierarchy, flattening and abstracting everything it can into view-from-above certainty and thus control, it resembles a map-like view of the world. This map-metaphysics has long dominated Eur0-Western patriarchal culture. Various formulations have been crystallized by Plato (see the cave allegory), Aristotle, Descartes, Calvin, and others.
5. At the other end is the cluster we can call process-relational metaphysics. It takes the universe as co-constituted by things that are, ultimately, relational processes. Their properties are not intrinsic, but made real by other relational processes: internal and external, dynamic and stabilizing, and situated in specific contexts. Irreducible complexity, diverse plurality, contextuality, and interdependence are recognized as fundamental to the order of reality. And the future is not pre-determined, but mutually co-constituted and regenerated by all of reality's participating things, offering at least a baseline of inherent meaning to life. This mode is common in Buddhist and other Asian cultures, African ubuntu, and most Indigenous cultures, as well as in non-dominant strands of Euro-Western thought including pre-Socratics like Heraclitus, eco-feminist folk traditions, process philosophies, and some interdisciplinary academic domains.
6. Abundant, increasing, clear and converging evidence from dang near every domain of human knowledge confirms that substance metaphysics does not describe the actual territory of reality, only an approximate model or map of it. We conceptually separate, pin down, and essentialize things to bring them into our comprehension and control. That is useful, of course, and to some extent, our separations and definitions and hierarchies do become realized as we enact them. However, things themselves cannot be truly isolated from relations or time; their properties are not self-contained and determinate but emergent from complex relational processes; and mattering is not a universal hierarchy but pervasive, contextual, and interdependent. That's not opinion or belief but empirically verifiable fact, the same way plate tectonics and the existence of trans people are facts. Knowledge of that (meta)physical fact is just suppressed, primarily by the people and structures enforcing the dominant map.
7. The present and future of the story is where we find out whether we humans can steer ourselves back toward reality enough to ravel ourselves back together and avert the worst consequences of map-metaphysics' domination: fascism, ecological and climate collapse, and all-around global megacrisis.

It’s easy to see the appeal of modeling reality on classically understood mathematics, building up from reductionist axioms and quantities deemed perfectly clear, certain, self-contained, and context-free. Math has thus tended to feed map-metaphysics thinking. Yet mathematics turns out to be more human than its reputation, and more excitingly dynamic and relational. The arithmetic, geometry, algebra, and perhaps calculus and statistics that most of us were taught are just the beginning. They're not even particularly representative of what math can be. More recently developed fields infuse relevance considerations, context integration, and other facets of process-relational reality.

Zoomed Out Math: Category Theory

[C]ategory theory is all about saying, we should not think about anything outside of a context, because things really change character depending on what context they’re in. And even numbers — I don’t mean “even numbers” — numbers themselves, change context .… [Equivalence, in category theory, is] a more flexible notion than equality, because it’s saying we’re going to make a choice in this context of which things we want to count as the same just for now to give us a point of view, and we are going to study what happens in the world if we treat those things as the same. And I think that that has huge lessons to teach us about the world as well….

- Mathematician Eugenia Cheng

At the cutting edge of mathematics, category theory, a "birds-eye view" of math, exemplifies this trend toward recognizing what's map and what's process-relational territory. Category theory "is all the rage. It's not just hot in the world of math. It's now being used in computer science, physics, engineering, chemistry, linguistics, and more. Though it began as an abstract branch of pure math, it's turning out to be a whole new way of thinking about lots of things, including situations in everyday life." It has even been proposed by proponents as a more useful subject to eventually replace calculus in high school. As summarized by mathematician Tai-Danae Bradley in perhaps more detail than needed here:

Different areas of math share common patterns/trends/structures. This becomes extraordinarily useful when you want to solve a problem in one realm (say, topology [note: see next section]) but don’t have the right tools at your disposal. By transporting the problem to a different realm (say, algebra), you can see the problem in a different light and perhaps discover new tools, and the solution may become much easier.

The bridges between realms are also provided by category theory. It explicitly identifies the realms’ common structures: each has objects in it (set theory has sets, group theory has groups, topology has topological spaces,...) that can relate to each other (sets relate via functions, groups relate via homomorphisms, topological spaces relate via continuous functions,...) in sensible ways (composition and associativity).

As analogized more vividly by Inna Zakharevich, a mathematician at Cornell University: "There are lots of things we think of as things when they're actually relationships between things… The phrase 'my husband,' we think of it as an object, but you can also think of it as a relationship to me. There is a certain part of him that's defined by his relationship to me."

Category theory thus gets right at defining things, including even mathematical objects, in a context- and relevance-dependent way. Boundaries are less rigid and absolute, and humans don't merely observe them but participate in drawing them. Even something like "equal" is no longer a yes/no binary.

Naturally, philosophers have taken notice. The Stanford Encyclopedia of Philosophy states:

[I]t should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics. Similar remarks can be made when we turn to ontology, in particular formal ontology: the part/whole relation, boundaries of systems, ideas of space, etc.

Category theory has practical implications galore, enabling new understandings of our world where earlier math hit limits. The point here is really just that the field of category theory exists; that it exhibits the same metaphysical trend as we've seen in physics, biology, economics, and elsewhere; and that knowing that should update our prior thinking about how the world works.

The Math of Spaces, Shapes, and Relations

As the sketch of category theory above makes plain, math is much more than numbers and their relationships. Sometimes it deals in shapes and forms. It turns out geometry works even better with a dash of process-relational thinking, too.

The first and in many cases only geometry many of us learn in school is Euclidean, though we may not know it as such. Much of Euro-Western thought has tended to view the world in terms of a fixed and regular three-dimensional Cartesian coordinate system that is just a sort of container for objects and their (inter)actions. This static background geometric view is limited in its ability to account for how space and time are shaped by the things that are happening "in" them and how they are being observed or measured, making it easier to maintain essentialist defining lines around or between things. The smooth, idealized curves of Newtonian calculus can also be seen as situated in this kind of geometry. This classic math obviously has its uses. However, using it exclusively and treating it as fully descriptive of the world misses opportunities to get things less wrong. Even short of diving into the mathematical details, concepts from other kinds of thinking about the shapes and relations of things can be very useful conceptual tools for everyday application.

Fractals, a term coined by mathematician Benoît Mandelbrot in 1975, beautifully illustrate beyond-Euclidean thinking. It’s not only mathematicians who find fractals helpful in finding order in complexity and in dealing with the relations of things at different scales or levels of being. While there is a great deal of nuanced terminology, I'll use the term "fractal" a bit loosely here to describe what happens when things retain or recursively repeat relevant attributes or patterns like roughness across different scales, whether those scale dimensions are spatial, organizational, temporal, and/or topological.

Topology, in turn, describes spaces and things dynamically in terms of relations, connections, folds, inclusions, exclusions, boundaries, and knots or tangles. It is often used in studying nodes and connectivity in networks. In comparison to classical spatial geometries, topological definitions of spaces tend not to depend all that much on physical distances or "absolute" positions in spatial dimensions; stretchy "deformations" in these attributes are temporarily disregarded.

You’re familiar with fractals. They’ve been everywhere for billions of years before Mandelbrot put a term to them in the disco era, though they were actively resisted by earlier mathematicians who liked their nice, neat calculus. Fractal pattern repetition or self-similarity may be relatively smooth like a spiral galaxy or shell. Or fractals may repeat shapes or degrees of roughness at discrete intervals of one or more dimensions, like streams converging into rivers, or like a fern frond’s shape being made of parts that are similar to the whole. Some fractals may have a fine structure down to arbitrarily small scales, making them literally unmeasurable with traditional Euclidean geometry, like the length of a coastline: you can measure an outline zoomed out on a map, but out in the actual territory, your measurements would differ depending whether you were using a ruler, a fabric tape measure, or some even finer tool to account for all the microscopic details of rocky roughness in a zoomed-in perspective. Yet foregrounding the patterns and relations is all it takes to make fractals much more manageable. Fractals also give us a whole new way of thinking about dimensions that don’t have to be whole numbers at all: it means something to say the coastline of England has a 1.25 fractal dimension, in reference to what amounts to its repetition pattern or roughness zooming in and out.

As the spiral, river, plant, coastline, and network examples make clear, fractals and topological patterns are everywhere in reality. They just may have been beneath our awareness, lacking a conceptual term to tie them together and a framework providing reason to use the concept. But this way of thinking can make a difference for getting things less wrong, as fields from geography to physiology to cognitive science (where declining fractal complexity in neuron networks is generally a bad sign) to social transformation for sustainability are finding.

Beyond academia and medicine, people often use spatial metaphors for social, economic, and other ideas, and the standard Euclidean map can carry over to those core metaphors. However, when a conceptual "space" is dynamic, tangled, and contested, it can be worthwhile to recall that there are different ways to see and intentionally define such spaces. For instance, "upward" or "forward" or "leftward" or "near" or "above" may not matter as much as "patterned" or "interconnected" or "included." Just as importantly, when we understand that we live in a highly fractal world, we can appreciate how the patterns we practice at our individual level recur similarly at organizational, community, and larger system levels, and we can start to disrupt and shift harmful systemic patterns from wherever we are. Our actions matter. It’s math!

Math for the Rest of Us

One conclusion to draw from the preceding sections is that there is no such thing as a social realm that is separate from the realm of mathematical insights. People do not just observe math from the outside, no matter how "pure" it may seem or how much some may feel excluded from mathematical pursuits. Divisions between nature and culture, between what is knowable and how we know it, are contingent and not predetermined. These divisions are mutually shaping, dynamic, and participatory. Math shapes society, and society shapes math, inseparably.

This is crucial when it comes to mathematical models of our world, from science to economics to polling. As mathematician and former Wall Street quantitative analyst Cathy O’Neil explained in her bestselling (and Euler Book Prize winning) 2016 book Weapons of Math Destruction, "models are opinions embedded in mathematics." Despite the veneer of mathematical "objectivity," human assumptions—including implicit metaphysical frameworks—shape models and algorithms in ways that are frequently, well, destructive, as well as inaccurate. As reviewer Cory Doctorow summarized, "One highly visible characteristic of WMDs is their lack of feedback and tuning. In sports, teams use detailed statistical models to predict which athletes they should bid on, and to deploy those athletes when squaring off against opposing teams. But after the predicted event has occurred, the teams update their models to account for their failings." That a thing is mathematical doesn't insulate it from needing its defining lines intentionally redrawn to account for context. 

Critical math pedagogy is a developing movement helping to bridge divides, tune models, and redraw the lines of what mathematics is. Deriving in part from the work of influential Brazilian educator and educational theorist Paulo Freire, this approach teaches through applying mathematical tools (e.g., data analysis, graphing and modeling) to explore specific situations of real-life relevance to communities, rather than treating students as passive recipients of knowledge content. It obviously benefits students, enhancing their grasp of concepts as well as their understanding of the world around them. But awareness of the mutually shaping interactions between math and culture can also lead to better math and math-based algorithms. Data deficiencies can be flagged and improved, and moreover, the formulas applied to such data inputs can be adapted to be more accountable to reality. We can "interrogate the world with mathematics, and mathematics with the world."

Conclusion

I hope this foray into modern math has been valuable, despite or because of the absence of real mathematical examples. There's no homework. But maybe try to question the faith put in old-school mathematical "objectivity" when you see it. Especially, ahem, in economics. Math isn't something we observe from outside, a neutral and value-free descriptor of a universal Truth. It's something we participate in. It's a tool that models a process-relational world from chosen perspectives. I think that's what makes it weird, and cool, and filled with secrets, and fun.

Stay tuned for a Recent-ish Reading roundup and then a post on the better, more nuanced logic systems that everyone should be aware of but few are! And please comment below, even just to say hi. I'm trying to treat writing as practice, valuable in itself, but it's definitely more fun if anyone out there is reading.