Beauty as Residue: What Math Taught Me About Code
I have been interested in both mathematics and computer programming since early years of my life. I have met several people who pursue math and/or write code daily.
Throughout these years, I have kept hearing about - “Beautiful Proof” , “Elegant Code”, etc. This has pushed me to think about what do people mean when they say these things? What are we trying to achieve.
What is Beauty?
“Beauty lies in the eyes of the beholder” - Margaret Wolff Hungerford.
Hence, beauty is subjective.
It should mean that there is some math which people find beautiful and some style of code which may suit to you but not to others.
If that is the case then why is the following beautiful to most mathematician?
\[e^{i\pi} + 1 = 0\]This signals to me that in the intellectual pursuit of doing math or writing code, we are rarely chasing the decoration of the craft.
Beauty in math and engineering means simplicity.
“Ockham’s razor - is a problem-solving principle that states the simplest explanation, with the fewest assumptions, is often the most likely to be correct”
Minimalism and Clarity
Simplicity evokes beauty because it reflects allows for the highest information gain by reducing noise.
Defining a problem
The idea of writing beautiful code often pushes young engineers to write esoteric code. In reality, Code that solves the wrong problem beautifully is like a perfect answer to a question nobody asked.
Beauty in engineering and math, first comes from defining a problem well.
There are often two steps -
- Define a problem
- Make it computationally tractable
This clears our objective.
For example, no such integer triples exist such that (Fermat’s Last Theorem) -
\[a^n + b^n = c^n, n>2\]This problem statement is beautifully simple. A well-defined problem.
Once the problem is clear, beauty shifts from definition to execution.
It’s worth noting that understanding doesn’t always precede expression. Often, it’s the other way around.
Mathematicians and programmers alike write to think - they formalize half-formed intuitions to see them clearly. The act of codifying a vague idea is itself a path to clarity. The point is to let beauty emerge(the beauty of expression), rather than chase it or force it.
Iteration loop is often a path to beauty.
Beauty scales.
Then comes the solution: minimal machinery with maximal insight.
Wiles proof spanned 150 pages and required machinery from elliptic curves, modular forms, and Galois representations. At first glance, this seems to contradict everything I’ve said about simplicity. But Wiles’ proof does one thing well: it proves FLT.
The proof revealed that elliptic curves and modular forms - two areas of math that seemed unrelated-are deeply connected. It proved the Taniyama-Shimura-Weil conjecture, showing that every elliptic curve is secretly a modular form in disguise.
This is elegant unification: revealing hidden symmetry. Beautiful!
Elegant solution
Math or Code in itself is not beautiful. The solution is!
In Math, a proof is beautiful or fascinating because it awes the reader with its construction and thought process. Often the proof is very minimal and has no bloat. It is cross-cutting. Ties various field of math. It seeks truth and solves a problem!
Similarly in Programming, the beautiful code is a bloat free implementation of the engineers solution.
Across domains, simplicity isn’t praised because it’s pretty, it’s praised because it works. Simple systems are:
- Verifiable, because it is built on top of fewer moving parts
- Maintainable, future can understand the building principles
- Robust
It is about conceptual clarity. Small, composable pieces moving in symphony. The Unix philosophy crystallized this in the 1970s: ‘Do one thing and do it well.’
Beauty often Emerges
Beauty is a byproduct of deep understanding - not the goal, but the residue.
Both Math and Programming have a formal structure. Beauty arises when a small, clean rule‐set yields rich behaviour. This allows code or proofs to be comprehensible and testable/verifiable by default.
Conclusion
Beauty, in both math and code, is not what you write or build - it’s what remains after everything unnecessary has been stripped away.
The pursuit isn’t decoration. It’s clarity. And clarity, counter-intuitively, is what happens when you stop trying to be clever and start trying to be understood.