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Caravaggio & Dali

Set Theory and Beautiful Art

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It’s hard to explain why paintings are beautiful. Caravaggio and Dali are from different eras, paint for different reasons, and have vastly different techniques. Yet, they both evoke a response of awe and appreciation. They both produce beautiful art.

A good painting is a straddling. A delicate dance on the razor’s edge between order and chaos. The viewers should be able to impose beliefs and impressions onto a work — but only within the artist’s predefined parameters. A fixed parameter may be the general emotion of the piece, then the painted canvas is merely a guide towards feeling a certain way. This parameter, and others, are defined through the artist’s skillful application of shading, drawing, texture, shadow, etc. The beauty of the work is a result of them actualizing their skill.

Beauty is skill in action.


A set is a collection of things. We often regard this collection as a singular object. Suppose Q = {a,b,c,d}. Our set Q has four things (or elements), namely: a, b, c and d. One of the basic ideas in Set Theory is the Principle of Extensionality. If two sets have exactly the same members, then they are equal

Let’s define three sets:

a) The set of numbers 2,3,5,7

b) The set of prime numbers under 10

c) The solution to the polynomial: x⁴-17x³+1101x²-247x+210=0

The extension of all of these sets is the same: X={2,3,5,7}. In a), the pointing is straightforward — lacking originality or creativity. b) requires slightly more thought, as we must recall not only what a prime number is, but then determine the ones less than 10. c) exemplifies creativity, ingenuity and skill. The feat is twofold: the creator of the equation utilizes his knowledge of mathematics, but also demands something from us. We must solve the equation ourselves to come to the answer. The final goal is the same in each case, but the journey of getting there is vastly different.

But what does this have to do with art?

Why wouldn’t we consider a simple blue line drawing of a crying woman a good piece of art?

Suppose this piece of art aims to evoke sadness. A simple line drawing is akin to our above example of taking a short, unskilled route towards the set. Instead of crafting a complex polynomial, which takes both mastery and creativity — the artist simply says: look, someone is sad.

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Beautiful art contains myriad emotions, woven together in complex way, that is surely impossible to capture as a set. Perhaps it’s presumptuous to say that art has a purpose at all — perhaps it is just pure expression. Every person gazing at a piece will have a different response. However — good art (whatever that is) brings you from where you are now, somewhere else, then back again. Its beauty is a function of the depth and breadth of this journey. Its beauty seems to also be related to the skill of the artist. Perhaps the skill is not always technical, but the understanding of context. Releasing a performative piece at precisely the right time takes the same intuitive connection as painting a masterpiece. Perhaps a simple blue line drawing of a crying woman is exactly what the world needs to see. Regardless — this is for the artist to determine.

Art is transactional — neither totally objective, nor fully subjective. It is a give and take between an artist’s skill, and the viewers openness to truth and beauty. When you go to an art gallery, there is a onus placed on the artist — you hope they have taken the time to map out an emotion. Have taken the time to put out signposts, roadsigns, and maybe even paved a street on the rocky road from the mind to the heart.

Is the category of artistic genius even a tenable one? Of course it is. Is there even such a thing as good art? Definitely. No matter how far out the abstractions of set theory go, no matter the complex structures one can create in a subspace of R³, all mathematics boils down to axioms. A set of fundamental, unquestioned beliefs that form a foundation from which to build. No matter how far out humans go — our archaic axioms are built into our bodies. Our animal instincts, the Jungian Collective Unconscious, the limitations of our sense organs— there are certain things we simply cannot escape. As such, there are certain things that just look good, feel good, are good to the human animal — simply by virtue of being embodied in this primal form. I’m not using good as a moral judgement, but as an intuitive feeling, an unspoken understanding. As such, I think art can be good. And a genius artist is simply someone who creates good art at precisely the right time.

“And if there come the singers and the dancers and the flute players, — buy of their gifts also.
For they too are gatherers of fruit and frankincense, and that which they bring, though fashioned of dreams, is raiment and food for your soul.” — Khalil Gibran

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