February 24, 2024

The science magazine of Imperial College

What makes a proof beautiful?

Olive Bradshaw discusses what makes a proof beautiful.

(By Olive Bradshaw on 29th November 2023)

“Mathematics, rightly viewed, possessed not only truth but supreme beauty – a beauty cold and austere, like that of a sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.” – Bertrand Russell

What makes a proof beautiful? This isn’t an unfamiliar question to those acquainted with the mathematical sciences. In fact, you can probably count on one hand the number of undergraduate maths students who didn’t include the phrase ‘the beauty of mathematics’ in their personal statement. I for one, having been through an undergraduate mathematics degree, believe wholeheartedly in this notion. That is, not only the beauty of mathematics as it manifests in physical phenomena that we can observe ourselves (the golden ratio, fractals etc.), but the internal feeling of joy and satisfaction one gets from a beautiful proof to an abstract problem. Despite this, I have never fully probed the idea of what exactly it is about certain explanations and solutions to mathematical problems that generates this feeling. Or, at least, I hadn’t until I came across a post on the publishing platform Medium detailing an intriguing tiling problem. The theorem concerned goes as follows:

Theorem 1: Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side.

(Taken from the paper Fourteen Proofs of a Result About Tiling a Rectangle by Stan Wagon

The statement is very simple and yet not immediately obvious, a sure-fire way to pique the interest of any young mathematician. And, after a brief search, a paper by Stan Wagon satisfied my curiosity. The paper, Fourteen Proofs of a Result About Tiling a Rectangle, walks through (you guessed it) fourteen proofs to Theorem 1, sandwiched by an introduction and conclusion, and punctuated with diagrams to accompany some of the proofs. As I went through each proof, I began considering how each of them made me feel.

I will pepper this post with some of the solutions from Wagon’s paper along with the thoughts I had when first reading through them. It is not important however, that you follow or understand the proofs presented. The point is to explore the emotional process of working through a proof, and how different methods evoke different responses. Having said that, if you are interested, I would highly recommend Wagon’s full paper here.

Although none of the solutions took up more than half a page, I was left by some feeling enlightened and pleasantly surprised while others felt unsatisfying, even taxing. To those unfamiliar with the experience of muddling through a mathematical proof it may seem ridiculous to have such emotional takeaways from what may seem like different ways to reach one logical conclusion. So, what does this to us? What makes a beautiful proof?

(Taken from the paper Fourteen Proofs of a Result About Tiling a Rectangle by Stan Wagon

Thoughts on Proof (1): This proof is very simple and makes the solution feel almost obvious to someone comfortable with a little complex analysis. Speed like this can be nice but can also feel a little abrupt, like the pacing isn’t right.

There are general attributes of proofs which earn them the title ‘beautiful’, but there is also something experiential which occurs as you work through a problem. A proof manipulates and breaks apart a problem into simpler claims which one can be more easily convinced of through logical reasoning or can accept based on previous known results. If initially a problem seems like a knotted mess, a good proof will mould its shape. Sometimes it feels like you are untangling it until it can be seen more clearly – like revealing a planar embedding of a graph. Other times it is as though you are folding it up over itself and manipulating its ins and outs until the whole thing lines up perfectly to reveal a familiar, satisfying shape.

(Taken from the paper Fourteen Proofs of a Result About Tiling a Rectangle by Stan Wagon

Thoughts on Proof (7): This is a lovely proof. Such a creative solution. It presents a simple set up and each subsequent step is clear and simple, proving the theorem before you even notice that it has. It feels very natural for such a unique way of solving the problem.

I have found that in general, for a proof to be considered beautiful or elegant it will fulfil one or more of the following criteria:

Simplicity – the proof uses few assumptions or previous results and comes together swiftly

Concision – the proof is surprisingly succinct; every part is effective and necessary

Unexpectedness – the proof derives a result in an unexpected or surprising way, often using an unrelated area of mathematics

Originality – the proof is based on an original or not-before-seen insight

Generalisability – the proof can be easily extended to a larger family of similar problems

Proofs which instead involve long, laborious calculations or rely heavily on many previous results may be described as ugly or clumsy. Strong, elegant claims may connect areas of mathematics which appear unrelated. Some are given these labels in retrospect because they have been found to have broad and far-reaching ramifications in unforeseen areas.

So, what makes a proof especially good? Is it simplicity? Originality? Is it concision? Unexpectedness? Generalisability? Or, to quote the Canadian Prime Minister Jean Chretien, “A proof is a proof, and when you have a good proof it’s because it’s proven!”.

There is another question – how much does beauty rely on the presentation of a proof? A beautifully written proof should read like a story with grammar, punctuation, a beginning, middle and end. Signposting, as in any other type of academic paper, can be essential in proofs which are constructive.  On the other hand, being left in the dark can lead to an enlightening feeling of discovery once the pieces fall into place and a solution is suddenly illuminated.

(Taken from the paper Fourteen Proofs of a Result About Tiling a Rectangle by Stan Wagon

Thoughts on Proof (11): I like this proof. Although it is the longest thus far, it comes together swiftly and neatly at the end after establishing the required set up. This can sometimes be slightly off-putting and distance you from the original problem. However, in this case you still retain visuals from the original problem and so it remains at the forefront of your mind.

Perhaps what we love about proofs is the same as what motivates us to study mathematics in the first place. For me this is unveiling connections and relationships which were previously hidden, for others it might be developing analytical or critical thinking. As Fields medallist William Thurston puts it:

The joy that comes from learning ways of thinking that explain and organize and simplify. One can feel this joy discovering new mathematics, rediscovering old mathematics, learning a way of thinking from a person or text, or finding a new way to explain or to view an old mathematical structure.”

Ultimately, I think hitting the five elements of a beautiful proof as denoted above is important, but finding beauty in a proof, as it is for finding beauty in art, music or people, is subjective. Beauty is in the eye of the beholder, as it were. Or, perhaps, it is an entirely elusive, unanswerable question. To leave you with some wisdom from the most prolific mathematician of the 20th century, Paul Erdös:

Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.”