Equal doesn’t mean same.

The art of doing mathematics consists in finding that special case which contains all the germs of generality. — David Hilbert

Equal doesn’t mean same. Equivalence is the art of making things interchangeable — the idea that two things can be swapped out without changing the essence of what they’re a part of. Like swapping a red Lego brick for a blue one, the color changes but the structure remains intact. Different inputs can produce identical results, and there is usually more than one way to solve a problem. Equivalence as a model helps us simplify complex systems by focusing on the essentials instead of getting lost in details. It also reminds us not to fixate on apparent differences; look for the underlying equality of experiences if you want to connect with others.

One of the ways equivalence is most useful is when our traditional solution to a problem is no longer viable. We know we must now do things differently, yet we wish to achieve an equivalent result. This model shows that multiple paths can lead to the same outcome.

In Mathematics

In math, one of the most basic equivalency concepts is “If A = B and B = C, then C = A.” We can infer that A, B, and C need not be the same — after all, they are represented by different symbols. But for the purposes of comparison, in at least one aspect they are equivalent. Different symbols can provide an equal answer to a question. This idea that distinct representations can be swapped without loss captures the heart of equivalence.

As mathematics progresses beyond numbers into shapes, curves, and patterns, we discover many more ways in which things can be “the same.” We even learn to navigate different levels of sameness. Two triangles might be congruent — exactly the same shape and size — or merely similar, sharing angles but not side lengths. Flip a triangle over and, depending on context, it may still count as the same shape (a slice of pizza remains a slice of pizza) or it may not (a “p” flipped becomes “q”). In everyday life we already handle such gradations effortlessly: every handwritten “a” is recognizably the same letter even though no two are identical, while a computer font’s identical “a”s can look artificial precisely because they lack variation.

One of the most famous recently solved mathematical problems, the Poincaré conjecture, relies on homotopy equivalence — a kind of sameness where a doughnut (a ring torus) is equivalent to a coffee cup with a handle. The hole in the doughnut maps to the hole in the handle; the rest is flexible. Under homotopy equivalence, a square and a circle are also the same. Yet a square is clearly not the same as a circle if you need a wheel, though it might serve just as well as a circle if you’re baking a cake — though with the practical downsides that a square tin is harder to wash and its corners tend to overcook.

The insight is not simply to declare things “same” or “not same,” but to understand the entire grey area in between and choose the level of equivalence appropriate for our purpose.

Human Universals

The world is full of things that seem different yet are in some way equivalent. Take the case of human universals. We are, as a species, unimaginably diverse. Despite this, cultures across the world often solve the same problems in equivalent ways. According to anthropology professor Donald Brown, these universals include taboo language, a distinction between how people behave when they are in full control of themselves and when they are not, making promises, rules surrounding inheritance, attempts to predict and influence the weather, and bodily adornments. While these features and behaviors manifest in different forms, they serve equivalent purposes across cultures.

Historical Recurrence

Sometimes things recur in equivalent yet different ways. Historical recurrence is the phenomenon wherein seemingly equivalent events happen more than once at varying points throughout history. It’s a cliché that history repeats itself, but the similarities can be uncanny — as in the assassinations of Lincoln and Kennedy, or the invasions of Russia by both Napoleon and Hitler. People in similar situations facing similar incentives are likely to behave in similar ways, producing outcomes that are functionally equivalent despite the different contexts.

History, we know, is apt to repeat herself, and to foist very old incidents upon us with only a slight change of costume. — George Eliot

Of course, equivalence has its limits. Not everything is interchangeable. You can’t swap out a car’s engine for a hamster wheel and expect the car to run. The art is in knowing where equivalence applies and where it doesn’t — recognizing the essential differences that matter, and the superficial differences that don’t. This is precisely where the logical fallacy of false equivalence lies: using superficial similarities to claim two things are equal on deep or critical dimensions, when in fact they differ in ways that make them anything but interchangeable. The next time you’re faced with a complex problem, try thinking in terms of equivalence. Look for the underlying patterns. See if there are components you can swap out or simplify, but always check whether the chosen level of sameness genuinely preserves what you need. You might just find a solution that’s been hiding in plain sight all along.