The Unrecognized Art of Mathematics

Adarsh Kishore

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“What? An article on math?! I’m skipping it right away, I’ve had enough nightmares in my semesters already!”

I’m sure these are some of the thoughts going around in your head if you are reading this. But rest assured, this is going to be a completely different style of introduction than in your course textbooks, and it is written especially for those who think like what I’ve mentioned above. I’ll try to keep equations out of the way as much as possible, instead we’ll bring out the beautiful meaning behind those squiggles of text as we go on this journey together.

Were you ever told in kindergarten while being taught how to read and write that “this is important as it will help you in the future”? Consider drawing – in those creative classes of drawings, we go wild with our color pencils and paintbrushes. But again, I don’t think anyone is told “we’ll learn drawing today as this will help you in drawing diagrams in higher classes”.

But this one? “We’ll learn mathematics today because this has various applications in physics, economics, computer science, blah blah blah….”. I would be surprised if anyone said that they haven’t heard that. And that is the thing which makes all the difference. Too much focus on “learning this now, to be useful later” kills all the fun and joy out of it, reducing it into the mindless incomprehensible drill of exercises that we are so used to.

The Value of Real Art:

The point is, doing art does not need justification in how useful it is. Da Vinci did not make the Mona Lisa, or Beethoven did not compose those beautiful symphonies, because they were useful for society; far from it. They did it just because they enjoyed their art, and it allowed them a way to express themselves in different forms. It is the same with mathematics. It is an art and it is done because people enjoy doing it. Any by-product thus obtained is merely an added bonus.

Many play the piano for enjoyment, or make beautiful drawings for fun. These things are enjoyed not only by the one who’s doing it, but also by those around the person. You’ll probably be the star of the next family function if you can play instruments, or sing beautifully. Your paintings would be appreciated by your friends and family members, no doubt. But if you told someone, “Hey, I just found something new in mathematics!”, the chance of garnering similar levels of appreciation is slim.

Humans, consciously or sub-consciously, seek external validation. We want to be appreciated by society for what we have done, and this is natural. Music and paintings are a form of art which are recognized by society as such and therefore appreciated. Mathematics, unfortunately, is a form of art which is not recognized by society at all!

And without that external validation, the chance that someone would enjoy doing such a thing is, well, we all know how that can be. And without putting the effort in any art form, what is the chance of appreciating it as an art in the first place?

So, What is This Art Anyway?

So, you may ask me, what is the art of mathematics? Just like music has its musical notes, and paintings have their colors, mathematical art is made up of ideas. In the words of the mathematician,

“A mathematician, like a painter or poet, is a maker
of patterns. If their patterns are more permanent than
theirs, it is because they are made with ideas.”

G.H. Hardy

The real world is bound by the laws of nature, it is imperfect and inherently non-deterministic. But the mathematical world is something which exists inside our heads. We sometimes say about dreams that “since it’s my dream, I can do anything in it!” Similarly, since this world is in our head, we can do anything we want in it!

It is not bound by physical laws, we’re free to imagine anything and everything, things having a precise perfection to them, being simple and beautiful. When I say that I’m thinking of a circle, I’m certainly not thinking of a real-life example of a model of a circle, with all its atoms juggling around randomly. I’m thinking of a circle perfect up to the smallest detail. It’s our playground, and so we make the rules and play by it. If I want some event to happen, it will happen; there’s nothing like it being non-deterministic.

If there is one unifying aesthetic principle across all of mathematics, it is this – “simple is beautiful”. Now you may say that the theorems and definitions we study appear anything but simple to you. The problem is, our education system has got it backwards. Real discussions start with the simplest possible examples, then build on it, see some things occurring in a pattern, and make a theory around it. In music also, the most complex orchestra is built out of the simplest notes, and the richest painting is made out of simple strokes of the paint brush.

What we usually learn however, is the definitions first in their full generality and as disconnected from anything we could think up. Then there are the equally cryptic theorems, and then examples are left as an “afterthought”. This whole reversal of logic is the greatest reason why people get deterred from going further. In a programmer’s language I would say that we’re taught the API of the functions without any inside details of how they work. And trust me that’s boring. Gone is the joy and fun of thinking on our own.

Real mathematics, however, is anything like that. It is not a road to be followed inside a forest, it is when we make the road ourselves! Mathematics isn’t about following “guidelines or best practices for best results”. Mathematicians like to think of the simplest possible thing and build up from that. And the simplest things are imaginary.

An Example:

Geometry is the best source of beautiful examples. Say I’ve a triangle in a box, and I want to find out how much of the box it takes up. And notice that there’s no ulterior purpose for doing that; I’m doing it just because I enjoy thinking about it.

Is it two-thirds, does it depend on where the tip of the triangle lies on the box? These are questions waiting to be answered! And that’s a beautiful thing about mathematical objects – they talk back! You only have some freedom in allowing how objects can be, what we call definitions. Once you’ve got to a certain point, you can no longer make choices yourself, the objects themselves do the talking, what we dryly call theorems.

In this case, I had the freedom to put the triangle however I wanted inside the box. But no matter what, I cannot arbitrarily “choose” what its area is. It is something to be found out now, it is something which is hiding from us.

Now, how does one get this idea? How does a painter know which color to use, or a musician about the best tune for the next note? Experiment with ideas! Many of them do not work, but finally we stumble upon one which works, and the beautiful results we get then compensate for the hours of fruitless work we spent on it.

Notice that in this, I did not use any letters or symbols or numbers or whatever. It was just the triangle and the line. This emphasizes a very important point which is often pushed under the rug – the ideas are much more important than the symbols used to express them. Too much focus on symbols like A = ½bh undermine the beautiful visualization and thought process that goes on behind the scenes.

After all, books are written using letters, but it is not the alphabets alone which makes a novel interesting. It is the meaning behind those symbols, the emotion and visualization behind it, that conjures up the events that the story is about. Language is important, but it is not everything.

Conclusion:

I may have inspired you by now, or maybe not. Not everyone likes to compose their own songs or make their own paintings. Creativity is not restricted to these things only. Similarly, not everyone would like the form of art called mathematics. And that’s totally fine; if everyone liked the same thing then the world would be dry of variety. But at least love or hate mathematics for what it really is, and not the perverse mockery of it which is shown to us.

An IT undergraduate with an interest in all things computer and mathematics. Your average “nerdy geek”, he is also into novels, especially fiction written by Isaac Asimov and Brandon Sanderson. Likes to listen to English and instrumental songs while solving differential equations.