Let’s try to get through this discussion without defining art, a neat trick. We’ll treat art as a basic notion and allow our intuition to fill in the blanks.
Mathematics Possesses Great Beauty, But Is It Art?
I earned a Bachelor of Science degree in mathematics. Most schools at the time offered a Bachelor of Arts mathematics degree. Which of these classifications avoided being wrong. My personal conceit categorizes mathematics as a science practiced by artisans, who frequently rise to the level of artist through ingenuity and sense of style. My degree designation is the correct one.
One asks what distinction was being made for placing a mathematics degree within the arts spectrum? It seems that perceived utility was the criteria for choosing one or the other. In those days, the arts were not deemed practical, and likewise, pure mathematics was not seen to move the needle on the practical meter. Now we understand things more deeply and can see much of pure mathematics having the utility to reveal things about reality that our senses could never perceive. Cosmology, particle physics, quantum physics, and cryptology, among others, are practical beneficiaries.
This characteristic of utility is not in the least incompatible with art. In fact, art that tells us about reality and about ourselves ranks at the highest levels of expression. Then why shouldn’t one consider mathematics itself as art? One argues so because mathematics springs from necessity, rather than from inspired acts of free will, causing mathematics to stand fundamentally apart from art, akin to science. Artists’ ‘DNA’ is evident in their art, making identifiable and often unique contributions to the human experience. In mathematics, only the ‘DNA’ of logic is evident; the artist only contributes a mind capable of constructing from that blueprint.
For over a century, mathematicians and philosophers have argued the necessity of our primitive mathematical objects, as defined by primitive notions and axioms. Some maintained that our choices are arbitrary, that we are free to choose different axiomatic bases for mathematics. Were this so, it could perhaps be argued that mathematics is art.
In their classic 1941 text What is Mathematics?, Courant and Robbins cast a worried eye on this potential tendency to view the axioms and definitions of mathematics as possessing many degrees of freedom, limited only by a need for internal consistency. They point out that if such a viewpoint were operative, “mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal”. They go on to argue that only intrinsic necessity, guided by responsibility to the organic whole, will achieve results of value.
Such intrinsic necessity has powered mathematics for its entire existence. Practitioners of mathematics almost never begin with a clean piece of paper. Over two thousand years of prior convention clutter every practitioner’s initial canvas. Terms like milieu or movement, typically used to characterize art, seem never applicable in discussing mathematics or science. Sometimes, schools of mathematics are referenced, but only with regard to the fundamental choices of axioms and logic where there is some freedom of a clean slate. Sometimes, mathematical styles may be evident, as with the Bourbaki formal style or a naive style, but these distinctions mainly refer to how much of the logical machinery can be omitted without rendering an unaesthetic or insufficiently substantiated result. If mathematics were admitted to the pantheon of the arts, it would seem to be equivalent to admission of a painting by the numbers.
Art is sometimes confused with aesthetics. Mathematics offers beautiful proofs, but it is not art, just as a beautiful sunset is not art. A mathematical aesthetic may, in some way, manifest as in the arts. Graphs of complex surfaces or fractals or algebraic numbers become computer-generated visual ‘art’. The beauty of an elegantly crafted proof is analogous to artful use of language. The surprise of a sudden glimpse of the profound sends a chill down the spine.
In other ways, our mind’s eye discovers an appeal unique to a product of mind. Possible characteristics of such a purely mathematical aesthetic are cleverness, revelation/understanding, simplification/clarity, symmetry, unification, closure. Sometimes, an untrained mind might react emotionally to an equation, such as Maxwell’s Laws or Euler’s Identity, dazzled by a mysterious depth of meaning from such a slight symbolism. The trained mind, however, will see the meaning as literal and straightforward, yet pleasing in its simplicity and symmetry.
Mathematics is fertile with inspiring examples of creative mental powers in pursuit of deep results. In some sense, the ability of mind to conquer the hardest problems must qualify as beautiful. For example, the initial choice of foundational assumptions impresses us through the power of mind to distill and capture the essence of great complexity. Some heightened aesthetic appreciation is reserved primarily for the creating artist, those Eureka moments when intuition hits a home run, those ‘you heard it here first’ moments.
Mathematics is similar to musical arts in one aesthetic respect. The untrained mind will at best appreciate a small fraction of the inherent beauty of either mathematics or music. This may strike an elitist tone, in the sense that only a few people will have the luxury of free time to spend training their minds. But that is the nature of the beast. When the attraction is sufficiently strong to overcome any issues of circumstance, priority, interest, energy, and relevance that stand in the way, beauty is the reward.
Aside: Those motivated by the beauty offered by mathematics may find it increasingly tough slogging through the ‘generalized abstract nonsense*’ that is evolving as mathematicians come ever closer to merging all mathematics within one formal meta-shell.
*Mathematicians occasionally poke fun at themselves.