Tikalon Blog by Dev Gualtieri

Math is Beautiful

December 9, 2019

While Cicero's orations have long since faded from my memory, I fondly remember a few pertinent Latin adages. One of these is "De gustibus non disputandum est," which is often rendered in the not precisely grammatical, "De gustibus non est disputandum." The translation of this is "about taste there can be no argument," which means that "one man's trash is another man's treasure." Abstract art is enjoyed by many, although some think it can be painted by a monkey. "Beauty is in the eye of the beholder" is a similar adage.

Sun through the curtains II by David Barrie

I've always enjoyed the abstract expressionalist works of Mark Rothko (1903-1970). However, our present era of near-perpetual copyright prevents me from showing any of his work.

Instead, this is a photograph entitled, "Sun through the curtains II," by David Barrie that's in Rothko's style. There's a Flikr group with nearly a hundred thousand photos of this type.

(Wikimedia Commons image.)

One pleasing aspect of art is its symmetry, and the symmetry present in many objects of science and mathematics is also considered to be beautiful. I discussed symmetry and the concept of beauty in science in a recent article (Is Science Simply Beautiful? August 26, 2019). In 2016, the BBC did an online poll to have the public select the most beautiful equation from a list of candidates proposed by physicists and mathematicians. The results, although not scientific, are still interesting; and, what is just as interesting, is that the equation for the Pythagorean theorem didn't make the list. At the least, we have a list of the twelve selected equations (see graph).

Ranking of beautiful equations in science and mathematics (BBC)

A beauty poll of twelve equations from physics and mathematics. Any physicist worth his salt should be familiar with at least ten on this list, but I must confess that I didn't know the Yang–Baxter equation until I checked Wikipedia. (Created using Gnumeric. Click for larger image.)

It appears that the online voters knew their topic area, since the winner by far was the Dirac equation. This equation, which describes massive spin-1/2 elementary particles such as the electron, is significant for two reasons. It was the result of the first theory that combined special relativity and quantum mechanics, and it predicted antimatter in the form of the positron. The originator of this equation, Paul Dirac (1902-1984), shared the 1933 Nobel Prize in Physics with Erwin Schrödinger (1887-1961), who has another equation on this list, the Schrödinger wave equation.

Although Pythagoras (c.570-c.495 BC) didn't make the list, Leonhard Euler (1707-1783) did, in the form of Euler's Identity,

e^iπ + 1 = 0

This equation combines five fundamental mathematical constants, 0, 1, e, i, and π, with the fundamental mathematical operations of addition, multiplication, and exponentiation. You obtain this equation by evaluating Euler's formula, e^ix = cos(x) + isin(x), at x = π.

While we recognize that something like an equation can be considered beautiful, we shouldn't equate beauty and truth. Massimo Pigliucci, a professor of philosophy at the City University of New York, has criticized the idea that truth can be recognized by beauty and simplicity.[2] For example, one motivation for string theory is the idea that beauty demonstrates truth even when experiment is not possible for validation; that is, when falsifiability is not possible.[2-4]

Mathematical beauty is less controversial, since many mathematicians see math as an art form comparable to composing music. There's also the fact that, unlike string theory, mathematical proofs can be inexpensively validated; and, simplicity in a proof is beautiful when it makes this validation easier to perform and easier for a student to see. Number theorist, G. H. Hardy (1877-1947), wrote in his 1940 book, "A Mathematician's Apology," that he was happy that his mathematics (like art) was not useful.

"I have never done anything useful. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

Godfrey Harold Hardy (1877-1947) and Paul Erdős (1913-1996)

Godfrey Harold Hardy (1877-1947), left, and Paul Erdős (1913-1996), right. Hardy was the mentor of number theory marvel, Srinivasa Ramanujan. Erdős was an obsessive mathematician who devoted every waking hour to mathematics. (Left image, and right image, via Wikimedia Commons and modified for artistic effect.)

The apparent rules for beauty in a mathematical proof, as compiled on Wikipedia, are as follow:

· A proof that uses a minimum of additional assumptions or previous results.
· A proof that is unusually succinct.
· A proof that derives a result in a surprising way (e.g., from an apparently unrelated theorem or collection of theorems).
· A proof that is based on new and original insights.
· A method of proof that can be easily generalized to solve a family of similar problems.

Supposedly ugly proofs involve laborious calculations (computer-assisted proofs probably fall into this category), approaches that are either very conventional or overly elaborate, and ones that require too many axioms or the results of previous ugly proofs. Eccentric, but eminent, mathematician, Paul Erdős (1913-1996), had an interesting idea, called Proofs from 'The Book'. "The Book" was the place in which God had written the best and most elegant mathematical proofs. It was high praise when Erdős said that a proof was from The Book.

Do ordinary people experience mathematics aesthetically like mathematicians and scientists? Three studies by scientists at the University of Bath School of Management and the Department of Mathematics of Yale University provide evidence that they do.[5-7] According to these studies, average Americans can relate to beautiful mathematical arguments in the same way that they can relate to art or music. Three hundred individuals were in approximate agreement about the particular way that four different proofs were beautiful.[6]

In his "Apology," Hardy specified some qualities of mathematical beauty, and these were generalized by the study authors into the categories of seriousness, universality, profundity, novelty, clarity, simplicity, elegance, intricacy, and sophistication.[6] According to one of the three studies, a high rating for elegance in both art and mathematical arguments closely predicted a high rating for beauty.[6] The three studies assessed the following:[5]

· The similarity of simple mathematical arguments to landscape paintings.
· The similarity of simple mathematical arguments to pieces of classical piano music.
· The commonality of reasons for the feeling of beauty for artworks and mathematical arguments.

In the first study, individuals were asked to match four mathematics proofs to four landscape paintings based on how aesthetically similar they found them. The proofs were the sum of an infinite geometric series, a method for summing a sequential list of positive integers attributed to a youthful Carl Friedrich Gauss, the pigeonhole principle, and a geometric proof of a Faulhaber formula.[6] The landscape paintings were "Looking Down Yosemite Valley, California" by Albert Bierstadt, "A Storm in the Rocky Mountains, Mt. Rosalie" by Albert Bierstadt, "The Hay Wain" by John Constable, and "The Heart of the Andes" by Frederic Edwin Church.[6]

A geometric determination of the sum of an infinite series, one of the four mathematical arguments used in the study.

(Created using Inkscape. Click for larger image.)

For the second study, the mathematical proofs were the same, and the classical piano music pieces were Franz Schubert's "Moment Musical No. 4, D 780 (Op. 94)," Johann Sebastian Bach's "Fugue from Toccata in E Minor (BWV 914)," Ludwig van Beethoven's "Diabelli Variations (Op. 120)," and Dmitri Shostakovich's "Prelude in D-flat major (Op.87 No. 15)."[6] The third study asked another group of participants to rate, on a scale of zero to ten, the beauty of each of the four artworks and mathematical arguments as well as scoring them in the nine categories listed above.[6]

The overall results demonstrated a considerable consensus among average people in the comparison of math and art, and somewhat less of a consensus in the judgment of classical piano music and mathematics.[7] As Samuel G.B. Johnson, a study author and a Lecturer at the the University of Bath School of Management, summarizes,

"Laypeople not only had similar intuitions about the beauty of math as they did about the beauty of art, but also had similar intuitions about beauty as each other. In other words, there was consensus about what makes something beautiful, regardless of modality...There might be opportunities to make the more abstract, more formal aspects of mathematics more accessible and more exciting to students... and that might be useful in terms of encouraging more people to enter the field of mathematics."[7]

Math is Beautiful

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