| Biology in the 18th century was all about taxonomy. The staggering diversity of life made it hard to draw conclusions about how it came to be. Scientists first had to put things in their proper order, grouping species according to shared characteristics — no easy task. Since then, they've used these grand catalogs to understand the differences among organisms and to infer their evolutionary histories. Chemists built the periodic table for the same purpose — to classify the elements and understand their behaviors. And physicists made the Standard Model to explain how the fundamental particles of the universe interact.
In his book The Order of Things, the philosopher Michel Foucault describes this preoccupation with sorting as a formative step for the sciences. "A knowledge of empirical individuals," he wrote, "can be acquired only from the continuous, ordered and universal tabulation of all possible differences."
Mathematicians never got past this obsession. That's because the menagerie of mathematics makes the biological catalog look like a petting zoo. Its inhabitants aren't limited by physical reality. Any conceivable possibility, whether it lives in our universe or in some hypothetical 200-dimensional one, needs to be accounted for. There are tons of different classifications to try — groups, knots, manifolds and so on — and infinitely many objects to sort in each of those classifications. Classification is how mathematicians come to know the strange, abstract world they're studying, and how they prove major theorems about it.
Take groups, a central object of study in math. The classification of "finite simple groups" — the building blocks of all groups — was one of the grandest mathematical accomplishments of the 20th century. It took dozens of mathematicians nearly 100 years to finish. In the end, they figured out that all finite simple groups fall into three buckets, except for 26 itemized outliers. A dedicated crew of mathematicians has been working on a "condensed" proof of the classification since 1994 — it currently comprises 10 volumes and several thousand pages, and still isn't finished. But the gargantuan undertaking continues to bear fruit, recently helping to prove a decades-old conjecture that you can infer a lot about a group by examining one small part of it.
Mathematics, unfettered by the typical constraints of reality, is all about possibility. Classification gives mathematicians a way to start exploring that limitless potential.
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The first mathematical classification we learn about in grade school involves categorizing numbers — into positive and negative numbers, or into numbers that can be written as fractions (the rationals) versus those that cannot (the irrationals). In a recent Quanta feature, Erica Klarreich describes how it can be incredibly difficult to prove that a given number is irrational, even if mathematicians suspect that it is. And there are plenty of other types of numbers that mathematicians like to study, too.
In other fields, mathematicians classify objects based on whether they're "equivalent" in some sense. In topology, two shapes are the same, and therefore belong to the same class, if one can be stretched or squeezed into the other without breaking or tearing. A doughnut is the same as a coffee cup, but different from a sphere. But it can get prohibitively difficult to tell whether more complicated (and high-dimensional) objects are the same. Mathematicians are still trying to figure out if all shapes in certain dimensions must be equivalent to a sphere, for instance, or whether more exotic forms are permitted. "After centuries of concerted effort," Kevin Hartnett wrote in this topology rundown, "mathematicians aren't even close to finishing."
Similarly, classification has played an important role in knot theory. Tie a knot in a piece of string, then glue the string's ends together — that's a mathematical knot. Knots are equivalent if one can be tangled or untangled, without cutting the string, to match the other. This mundane-sounding task has lots of mathematical uses. In 2023, five mathematicians made progress on a key conjecture in knot theory that stated that all knots with a certain property (being "slice") must also have another (being "ribbon"), with the proof ruling out a suspected counterexample. (As an aside, I've often wondered why knot theorists insist on using nouns as adjectives.)
Classifications can also get more meta. Both theoretical computer scientists and mathematicians classify problems about classification based on how "hard" they are.
All these classifications turn math's disarrayed infinitude into accessible order. It's a first step toward reining in the deluge that pours forth from mathematical imaginings. | 
