What is a "Portal"? (Why Math is Beautiful)

Many people have tried and failed to communicate how mathematicians feel about math. When someone asks a pure mathematician why they enjoy math, the mathematician will likely comment about math's intrinsic beauty, or how it is art as much as science. This is true: pure mathematicians, by and large, are motivated by curiousity and an artistic perspective. However, it's more difficult to express why a particular theorem is beautiful than to argue why it is useful, and so many people still don't understand why math is so alluring to its practitioners.

Though many people have failed, a few have managed to communicate effectively what mathematics means to a mathematician. The best articles are those written by Alon Amit on Quora and Terry Tao on his personal website. The article A Mathematician's Lament by Paul Lockhart also gives an amazing image of how mathematicians feel about math education. I don't come close to rivalling any of these mathematicians' accomplishments or teaching ability, but I still love math the way they do, and so I will try to express what mathematical beauty means to me.

Mathematical beauty is found in the creation or destruction of symmetry. Humans have an innate love for symmetry, and many objects in math are considered beautiful for their symmetry. The integers are beautiful because they add symmetry to the natural numbers (now the structure of subtraction is symmetric), the complex numbers are beautiful because they add symmetry to the real numbers (now the structure of exponentiation is symmetric), and so on. But in math we celebrate the unexpected, and so when symmetry is broken, it is also beautiful. For example, the automorphism group of the symmetric group of order n is always isomorphic to the group itself -- except when n is 2 or 6. Perhaps 2 makes sense (S₂ is the only nontrivial abelian symmetric group), but why should 6 be an exception? Why not 5, or 7, or an infinite family of exceptions? Why is there an exception at all? Even after learning the proof of this fact, it feels like a cosmic coincidence. The breaking of symmetry surprises us and upends our hypotheses, and so it too is beautiful.

Mathematical beauty is found in "portals" between different fields of math. Some of the most beautiful theorems act as wormholes, teleporting us from one field of math to another. For example, the Lindström–Gessel–Viennot Lemma (see Section 5) is an amazing link between graph theory/combinatorics and linear algebra. It gives us new ways to count lattice paths (thereby developing combinatorics and graph theory) but also gives an a geometric option whenever we consider determinant identities (for example, it can be used to easily prove the Cauchy-Binet formula). The proof requires no more complex math than typical high school algebra and a bit of matrix manipulation. For this reason, the Lindström–Gessel–Viennot lemma too is beautiful.

Mathematical elegance is found in arguments that act as keys to understanding, helping spread complex ideas with confidence and precision. Paul Erdős (perhaps jokingly, perhaps seriously) used to refer to "The Book" when explaining his search for elegance. The Book, according to Erdős, was where God kept the most elegant proof of every theorem. As Erdős told it, his job was to try to recreate the book, to discover the perfect solution to each problem he found. As such, an elegant argument was somehow innately valuable to him.

I have a different opinion. I believe that the value of a proof comes in its sharing, and so a great proof is one that illuminates a concept for a learner. Elegant arguments are easier to understand. They cut through the mess that often accompanies abstract math. Math is a community endeavour, and the best proofs enable that. This is not to say, of course, that difficult-to-understand or computer-assisted proofs like the Classification of Finite Simple Groups or the proof of the Four-Color Theorem aren't valuable; just that they gain their beauty from the result, not the proof.

In the end, math is beautiful for many reasons, some shared and some personal. But all these reasons have one thing in common: they appeal to our curiosity, our creativity, and our desire to share. In short, math appeals to what makes us human.