This post is actually one of the comments in my Mathematics is an Art post. I thought it would be a waste to just leave it in the comments section. It’s very informative and well written so I’m sharing it here. That way more people will have a chance to read it. It’s from the writer of one of my favorite blog Research in Practice, mr. benblumsmith. Here’s what he says about mathematics and art.
I want to add three defining characteristics (to me they are the defining characteristics) to the definition of art that you are constructing. I think all three help illuminate why math can be an art. I’m basing the choice of these characteristics on my own experience as a musician. Other artists might have other definitions, but actually I speculate that these are pretty commonly held as important characteristics of art.
1) Art is creative and expressive.
All the arts are about making things. Songs, paintings, plays, performances, etc. Crafts and industry are also about making things, but what distinguishes art is that the act of creation is an act of expression for the person doing the creating. Creating the painting, playing the sonata, singing the song, etc. are all ways of taking a part of yourself and giving it to others.
(Why math is creative and expressive: every time you figure out how to solve a problem you’ve never solved before, or how to prove a conjecture you have, you’ve made something new that expresses how your mind works to others.)
2) Art engages the imagination.
All the arts stimulate you to see, hear, feel things that aren’t part of the material world. You read a novel or see a movie and you imagine the world it creates. You hear a symphony and you may visualize all kinds of things, or you may just feel them. Either way, your mind and spirit are sent off in totally new directions.
(Why math engages the imagination: if not for math, would I ever have tried to visualize a 4-dimensional torus? The Riemann surface of the sine function? The real projective plane? The strange images in my head that I use to visualize concepts like exact sequences or field automorphisms?)
3) Art is driven by aesthetics.
Artists try to make their creations aesthetically captivating: beautiful, haunting, winsome, grotesque, tragic, delightful, etc.
(Why this applies to math: it’s different for every math person but we all have a sense of aesthetics – what makes math beautiful. For me and for many, a simple argument that proves a powerful result is elegant, for example Euclid’s proof of the infinitude of the primes. Even more beautiful is a theorem that describes a deep connection between apparently very different mathematical realms. For example the Pythagorean theorem as it connects “right anglyness” with “sum of sqaresiness,” the Fundamental theorem of Calculus as it connects “speed” with “area,” or Galois theory as it connects “groups” and “field extensions.”)
Two semi-classic pieces of writing on math as an art that you might find add to this conversation –
Paul Lockhart’s A Mathematician’s Lament.
G. H. Hardy’s A MathematiciansApology.
Both Lockhart and especially Hardy aggravate me (Lockhart with his blanket disdain for math educators and Hardy with his attitude that mathematical talent is an intrinsic attribute that declines with age), but both write eloquently and passionately about how math is a creative art.
Don’t forget to click the links he recommended. Very good read indeed.
, and e are not only difficult to work with, they are also difficult to understand conceptually.
where p and q are integers and q 
0. Irrational numbers are all the numbers that cannot be expressed in the form of 
is an irrational number, how do we show them that it fits the definition without resorting to indirect proof or proof of impossibility? What I am saying here is it is not pedagogically sound to start with definitions because definitions are already abstraction of the concept. I would say the same for all other mathematical concepts.