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This is the kind of thing that made me love math, and also made me suck at it.
https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/
https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/
Mathematical equality might seem to be the least controversial possible idea. Two beads plus one bead equals three beads. What more is there to say about that? But the simplest ideas can be the most treacherous.
Since the late 19th century, the foundation of mathematics has been built from collections of objects, which are called sets. Set theory specifies rules, or axioms, for constructing and manipulating these sets. One of these axioms, for example, says that you can add a set with two elements to a set with one element to produce a new set with three elements: 2 + 1 = 3.
On a formal level, the way to show that two quantities are equal is to pair them off: Match one bead on the right side of the equal sign with one bead on the left side. Observe that after all the pairing is done, there are no beads left over.
Set theory recognizes that two sets with three objects each pair exactly, but it doesn’t easily perceive all the different ways to do the pairing. You could pair the first bead on the right with the first on the left, or the first on the right with the second on the left, and so on (there are six possible pairings in all). To say that two plus one equals three and leave it at that is to overlook all the different ways in which they’re equal. “The problem is, there are many ways to pair up,” Campbell said. “We’ve forgotten them when we say equals.”