Scientists through the ages have noted, often with some astonishment, not only the remarkable success of mathematics in describing the natural world, but also the fact that the best mathematical formulations are usually those that are the most beautiful. And almost all research mathematicians pepper their description of important mathematical work with terms like "unexpected," "elegance," "simplicity" and "beauty."
Some selected opinions
British mathematician G. H. Hardy (1877-1947), pictured below, expressed in his autobiographical book A Mathematician's Apology what most working mathematicians experience: "Beauty is the first test; there is no permanent place in the world for ugly mathematics."
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.
Henri Poincaré (1854-1912), often described as a "polymath," wrote, in his essay Mathematical Creation, that ignoring this subjective experience "would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility."
While a very few very applied mathematicians view such ideas as a waste of time, the mathematics community is almost unanimous in agreeing with Poincare.
Physicists are just as impressed by the beauty of mathematics, and by its efficacy in formulating the laws of physics, as are mathematicians. Mathematical physicist Hermann Weyl (1885-1955) declared, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful." This was fully reflected in his own career, when he first attempted to reconcile electromagnetism with relativity.
His work was initially rejected (by Einstein and others), because it was thought to conflict with experimental results, but the subsequent formulation of quantum mechanics led to a renewed acceptance of Weyl's work. In other words, the "beauty" of Weyl's work anticipated its final acceptance, well before the full scientific facts were known.
Nobel physicist Paul Dirac (1902-1984), shown below and described by Niels Bohr as the strangest man, made his most impressive discoveries or predictions, such as that of the positron, largely from demanding elegant, simple mathematical descriptions.
He further elaborated on mathematical beauty in physics in these terms:
[The success of mathematical reasoning in physics] must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature's scheme...
What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. The theory of relativity introduced mathematical beauty to an unprecedented extent into the description of Nature.
a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me the austere beauty of the four statues representing "Day," "Night," "Evening" and "Dawn" which Michelangelo has set over the tomb of Giuliano de'Medici and Lorenzo de'Medici.
Why is this so?
In February 2014, a team of British researchers, including two neurobiologists, a physicist and a mathematician, published a groundbreaking study in Frontiers in Human Neuroscience on the human experience of mathematical beauty. This study is nicely summarized in a BBC Science report.
These researchers employed functional magnetic resonance imaging (fMRI) to display the activity of brains of 16 mathematicians, at a postgraduate or postdoctoral level, as they viewed formulas that they had previously judged as beautiful, so-so or ugly. The results of this analysis showed that beautiful formulas stimulated activity in same field, namely field A1 of the medial orbito-frontal cortex (mOFC), as other researchers have identified as the seat of experience of beauty from other sources.
This is an entirely satisfactory result. It gives an experimental validation of the mathematicians' intuition that they are experiencing the same qualitative states (qualia) as are experienced in other modalities from architecture and sculpture, to poetry and music.
So what exactly is the source of mathematical beauty? All aesthetic responses seem in part to come from identifying simplicity in complexity, pattern in chaos, structure in stasis. In the arts, "beauty" can be accounted for, at least in part, by well-understood harmonies, distributions of colors or other factors.
But what about mathematics? Aesthetic responses, as Santayana in The Sense of Beauty (1896) has argued, require a certain distance:
When we have before us a fine map, in which the line of the coast, now rocky, now sandy, is clearly indicated, together with the winding of the rivers, the elevations of the land, and the distribution of the population, we have the simultaneous suggestion of so many facts, the sense of mastery over so much reality, that we gaze at it with delight, and need no practical motive to keep us studying it, perhaps for hours altogether. A map is not naturally thought of as an aesthetic object...
And yet, let the tints of it be a little subtle, let the lines be a little delicate, and the masses of the land and sea somewhat balanced, and we really have a beautiful thing; a thing the charm of which consists almost entirely in its meaning, but which nevertheless pleases us in the same way as a picture or a graphic symbol might please. Give the symbol a little intrinsic worth of form, line and color, and it attracts like a magnet all the values of things it is known to symbolize. It becomes beautiful in its expressiveness.
This captures the aesthetic in mathematics: balancing form and content, syntax and semantics, utility and autonomy. The 2007 book Mathematics and the Aesthetic is dedicated to exploring "new approaches to an ancient affinity." Formed around nine essays, three by practitioners, three by philosophers and three by mathematical educators, it contains a chapter by one of the present bloggers.
Why it matters
As the Economist put it, in a fine essay on the changing notion of mathematical proof, Proof and Beauty (2005):
Why should the non-mathematician care about things of this nature? The foremost reason is that mathematics is beautiful, even if it is, sadly, more inaccessible than other forms of art. The second is that it is useful, and that its utility depends in part on its certainty, and that that certainty cannot come without a notion of proof.
Some argue that mathematical principles are experienced as "beautiful" because they point directly to the fundamental structure of the universe. Physicist Max Tegmark argues further that the reason that mathematics works so well, and so elegantly, in physics is because the universe (or, more properly, the multiverse) is, ultimately, just mathematics -- mathematical structures and the relations that connect them constitute the ultimate irreducible "stuff" of which our world is made. See our recent article on Tegmark and his new book, Our Mathematical Universe.
Few researchers are willing to go as far as Tegmark. But the widely sensed experiences of mathematical beauty, and the astonishing applicability of sophisticated mathematics in the natural world, still beg to be fully understood.
Understood or not, tapping the aesthetic component of mathematics is a crucial and neglected component of mathematical education. See Simon Fraser mathematical educator Nathalie Sinclair's 2006 book Mathematics and Beauty: Aesthetic Approaches to Teaching Children. Given that basing mathematical education on utility and importance has not worked very well, perhaps introducing the aesthetic is past overdue.