Why Math? Beauty — Gal Gross

Perhaps the most famous proponent of mathematical beauty, G.H. Hardy has argued for the intrinsic value of mathematics by drawing analogies to (other?) art-forms:

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

Putting aside the issue of whether (à la Oscar Wilde) he might have gone too far, his point is well-taken—math is beautiful. However, what does beauty mean in this context? The truth is, the concept of mathematical beauty is multifarious and subtle and perhaps beautiful in its own right (leading us to recursive beauty with mathematics at its root). In this post, I would like to concentrate on two of the most common conceptions of mathematical beauty which are interestingly at odds with each other.

Intricate Patterns
Math is inextricably linked to patterns; in fact, patterns often loom large in many attempted definitions of mathematics. It has been consistently argued for some time now, that the human brain has evolved specifically to identify and notice patterns. In fact, sometimes it does it too well, finding patterns when none are there. It is not surprising, therefore, that patterns play a significant role in our aesthetic appreciation. Surveying the history of art, one cannot help but notice how patterns and regularity are favoured. From the simple majesty of the Parthenon, to the extravagant elaborations of Gothic architecture. Or how about the Alhambra, its intricate arabesques tapestries are a systematic architectural investigation of symmetry. Architectural marvels indeed, but let us not forget to mention the central role geometry and perspective has played in the visual arts; entire books are still being written on these subjects.

Pythagoreans first peeked famously argued this short video

Elegance
“Refinement,” “dignified grace,” “restraint,” “exactness and precision.” These words, taken from the definition, try to capture the flavour and the texture of elegance. At the beginning of the post, I have mentioned that the concept of mathematical-beauty is versatile. The aesthetics of elegance is quite different from the beauty of patterns. Here’s Bertrand Russell:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of 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.

Notice especially the contrast Russell is drawing between math and painting or music. “[B]eauty cold and austere,” “stern perfections” — this is the language of elegance. In poetry, which Russell favourably compares to math, elegance is achieved through economy of language. In mathematics, elegance is achieved through economy of thought. To wit, most of the theorems in the Elements were already well-known in the ancient world, but Euclid was the first to deduce all of them from 10 axioms. By compressing 13 books and hundreds of deep and powerful theorems into a mere 10 axioms, Euclid has turned himself immortal – the patron saint of modern mathematics. It is for this amazing compression, this hitherto unparalleled economy of thought that we say Euclid Alone Has Looked on Beauty Bare (in the words of Edna St. Vincent Millay).

The Twain Shall Meet
Mathematical beauty is truly awesome—in the original sense of awe-inspiring (though the other sense applies as well). Feelings of awe stem, I think, from size comparison. Seeing something immense is awe-inspiring because we compare it to ourselves. Seeing something immensely complex is awe-inspiring because we compare it to our puny powers of understanding. The recurring motif is our limited ability to grasp, our inability to behold the entire thing with our own eyes, or our inability to hold it all in our mind, to comprehend. Thus, the stupendously intricate patterns of mathematics are awe-inspiring.

While gargantuan and gargantuanly complicated things are almost always awe-inspiring, it is precisely when we are able to comprehend them that they become truly beautiful. Thus, it is not ridiculously complicated patterns per se which we’re after; what we truly crave are ridiculously complicated patterns arising from laughably simple rules. The beautiful patterns are easy enough to find—mathematics is rife with them. These pique the curiosity, but are not capable of sustaining it. The driving force behind mathematical research are not the patterns, but the promise that behind them we can find a fantastically elegant theory—the simple rules that give rise to these patterns (this is closely related to the idea of mathematical insight).

Far from being two contrasting conceptions of mathematical beauty, intricate patterns and elegance are important parts of a bigger whole. Uniting them together, Lady Mathematics adorns herself in magnificence and splendour.

Granada Alhambra. [Image credit: Wikipedia.]

[Imported from my now defunct blog 10,000 Hours of Mathematics.]