Is mathematics an art? In this series of posts I will attempt to explore and answer this question. I encourage readers to also contribute their ideas in the comments.

The term *art* itself has undergone a metamorphoses over the centuries, so we should start by examining what humanity has meant by art and see if mathematics might fall under any of the definitions used over time. But beyond these definitions, we should examine whether we feel that mathematics is an art. Whether it is or not, what definition should art have in order to be consistent with the existence of mathematics? Only be examining these two angles will we gain insight into both art and mathematics, and how we as humans fit into both.

One of the earliest descriptions of art is the Platonic one, which says that art is imitation [1]. This view is echoed by Leon Battista Alberti (1404-1472), who thought that painting should be as faithful a reproduction or imitation of the real scene being constructed. Obviously this is a very limited definition by today's standards, but nonetheless worth looking at, as the germ of creating a reproduction of a scene using any media is still alive today, at least as a motivating factor to *create* art.

How would the people subscribed to this viewpoint consider mathematics? Is mathematics an imitation? Much of the early mathematics, and even modern mathematics, does contain a form of imitation: that is, the kind that describes the real world. The description is a depiction, just not a visual one. Differential equations, equations describing gravity, and such theories attempt to describe the real world. Unlike many visual depictions, they describe general situations rather than specific instances. However, that is not so much a difference in my mind, either. If one were to draw a scene of a mountain or a bicycle, it might be from a specific instance, yet there is also an element of generalization that takes place in that the depiction contains simplifications that simply cannot be represented exactly on the chosen medium.

Mathematics then has an element of imitation in it. On the other hand, we must be careful to consider where the imitation lies. If the imitation lies in the axioms, then mathematics is just a set of rules to make inferences from those axioms. The axioms themselves are a verbal or written description of rules that attempt to imitate reality. Whether or not those axioms are art, they are themselves not mathematics.

I don't want to dwell too much on the imitative aspects of math because the imitative definition of art seems too restrictive to apply it to our modern understanding of art. However, there is one point to consider. That is, whether or not the axiomatic description of the world is artistic, Plato and many other early humans believed that mathematics was a set of discoveries, and hence the entire process itself of mathematics is less of a creation of art but rather a series of explorations into another realm.

So if we accept the Platonistic view of mathematics, does that not rule it out as an art? Not necessarily. One could apply the same view to painting. There are only so many arrangements of molecules of paint that we can distinguish with our eyes. That number is large but also finite, and painting could also be said to be a discovery of a representation in a space of all possible paintings using the experience we have already gained from the interaction of our senses with the real world, even if the depictions are not true to any particular scene and even deviate significantly from it. So, the philosophical point of what mathematics is, whether it be discovered, a process, or a creation, does not really place evidence for or against math being an art. It may however, influence how individuals think of mathematics with regard to art.

That is enough for now, but in the next part we will continue to explore these ideas and new ones from what we considered as art over the ages.

## 2 Comments

Who said the arrangement of molecules of paint that we can distinguish with our eyes is finite? It is clearly infinite, since the very number line we use itself is infinite, has no ending or beginning. Everything is circular, thus, everything is infinitely interconnected. The combinations of painted depictions of any model of the world can go on indefinitely, since no two perceptions (via a subject) are the same.

I am not so sure about that. I mean, take one painting: if we take one molecule of paint (I am speaking loosely here since paints may not necessarily be a single chemical compound) and move it 10^(-12) meters upwards. It do not think anyone could distinguish that new painting with their eyes. At some point, differences will be so slight (such as in the best forgeries), that no person could tell the difference, even looking at the painting side by side in good light.

It is true that the number of mathematical lines that can fit within a finite space is infinite, but that is a mathematical model. I am talking about actual physical realizations of things that we can, under our unaided eyes, distinguish as different.

We can look at this also from a photographic point of view: imagine we are allowed to take one million 16MP photographs of the painting under any lighting, completely uniform—the lighting can be replicated exactly again and again. Surely, two paintings which are distinguishable by our eyes would also be distinguishable by a judicious choice of one million 16MP photographs taken in the exact same lighting? Some of these could be macro shots up close, from different angles, etc.

However, the sets of one million 16MP photographs is finite as well, even though that number is huge.

At some point, perhaps after Graham's number of paintings perhaps just to through a number out there, the human mind just won't have the perception to find something new to add to that. Of course, that number is far greater than civilization will ever reach in our extremely small lifespan as a civilization compared to that number.