There’s a problem with this essay, that I want to clear up from the first. The issue is that I’m not fully sure that you and I understand Mathematics the same way. This might lead to some conflict down the line. The math being taught in high schools and many higher education institutes are skimming the surface of a very deep lake. I only got exposed to the mathematics I speak of in college. Read the essay with that in mind.

A good place to start is what mathematics felt like when I was in school.

Take a formula and shift it around to fit the new conditions, take a property that you read from the textbook to crack a problem. Proofs are delegated off to the sides as things that aren’t meant to be learned but nice things to have(but it won’t be asked in the exam, don’t worry wink wink.)

We start off our schooling by being taught certain ‘rules’.

• 1 + 1 = 2
• Here are the time tables
• Solve for x by moving x to the left and the rest to the right.

Schooling essentially boiled down to this:

Here are some rules and formulae, here are some problems, this problem is connected to this rule so figure it out, this formula if you look at it sideways is applicable to this other problem.

Then we come to being taught calculus, and I felt thoroughly lost. Calculus ultimately felt like a bag of tricks, especially Integration. I got no satisfaction from solving an integral. Reading the solutions always felt like a weirdly cheap hack that I wouldn’t really have thought about, but now I’m supposed to remember for the next time I see a similar problem.

It felt like there was no time to figure out what these arcane runes and incantations you are scribbling out on paper even mean. It’s all part of getting more and more answers, like we’re scrambling around in the dark with no real path. The epitome of ‘here are the rules, find the answer to these problems fitted around these rules.’

This is the picture of how high school maths looks to me. I don’t think anyone reading this will take issue with this picture, although they might either hate the subject entirely, or love this form of maths.

There are glimmers of beauty here to notice. In this kind of math, one thing will always lead to another, your answer can be rechecked soundly and there is no debate on whether the answer is right or wrong if the problem is well-defined and the thinking skills you obtain by doing well at this Math is very valuable.

I’d argue that these glimmers are because of the lake it’s skimming. There is a lot of depth being lost here, and most people will never know of these depths, but what they do know of is a small foggy image of a shadow of an inkling of the true grand scale that is mathematics.

So then the question arises…

What is Mathematics?

You’re Not Stupid If You Feel Stupid

A preface to the answer that needs to be elaborated here - there’s a subtle difference between feeling stupid and being stupid.

The difference lies in who’s saying it.

It is not stupid to not know something. In my opinion, it’s stupidity to not even be aware you don’t know something.

Good mathematics is supposed to make you feel stupid.

It should give you questions you can’t immediately answer, it should give you ideas that you can’t wrap your head around.

The evidence that you learned something hard is that you feel like you’re stupid. That stupidity is essential to the process, not of pushing at the boundaries of what is known, but simply of getting your mind to take in tools and ideas more abstract than it was ever meant to learn. Students need to know that this feeling is the norm when it comes to learning math. -The Centrality Of Stupidity In Mathematics by Dan Finkel

Onto answering the question.

I was watching Moulin Rouge, directed by Baz Luhrmann the other day. It’s a fantastic musical with spectacle and glitz, made in ways few movies care about . There’s this theater group at the center of the story, who recite these 4 common notions for what they stand for. It’s spread throughout the movie, like a recurring motif.

Beauty, Truth, Freedom and Love.

Mathematics, the motivation behind it, and the way it should be taught stands on these 4 notions.

Beauty

“Aesthetic beauty is as much about how and whether you look as what you see. From the quark to the supernova, the wonders do not cease. It is our attentiveness that is in short supply, our ability and willingness to do the work that awe requires.”-The Anthropocene Reviewed by John Greene

This brings up a larger discussion on what Beauty is. Is it innate to the senses that beauty exists? Is it part of being human that we regard certain things as beautiful and certain things as not?

With math, Beauty comes in the form of epiphany. That powerful euphoria when things click into place, when all that scaffolding you built in your mind constructs the right logical structure for you to finally understand the solution to a problem.

A crescendo of reason, that requires no one but your involvement in the work. A joy untethered from the falsehoods and troubles of real life.

There are other forms of beauty that is present within Mathematics if you know to look for it. Beauty in mathematics arises when you see two very disconnected fields suddenly come together to provide a result.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.-Bertrand Russel

Beauty also comes from elegance. A proof that’s simple, effective and somehow obvious is Beautiful.

Beauty comes from understanding, sudden and sharp… Like a flash of lightning. It must be experienced to be understood and it pains me to think that most of the world has never had the opportunity to witness this pure form of Beauty.

There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract.’-A Mathematician’s Lament by Paul Lockhart

Truth

Mathematics is the composition of logic. It’s taking certain truths within the world, reducing things to their core essentials and rebuilding them into much greater heights. It is about seeking Truth. It’s about understanding why. It’s understanding the cause and effect of the world to your limits.

Mathematics is also more than that.

No one ‘invents’ mathematics. An invention is human, which by definition means imperfect. Mathematicians don’t invent, they discover.

In other words, the Truth is out there, it’s our job to catch up.

Mathematics is a game of dominos that you play with the universe. Set up a few pieces, and you can then form an entire world surrounding the way those pieces fall. Take a few ‘truths’ or axioms and follow the implications of those axioms to their complete end.

You model the things within the real world by defining as minimal a set of axioms as possible and deriving the systems within the real world. I’d like to amend my statement, mathematics isn’t even fully concerned with the real world. The ‘real world’ or our perception of it is simply one of the many consequences of mathematics.

This is the part that surprises me the most. Mathematics by pure implication alone, has built up systems of logic that go outside our intuitive understandings of the universe. The implications and the systems developed in exploring implications has widespread use in all fields. The computer was first devised by a mathematician, before we even had the technology for it.

Does the sheer scale of this not bring you down in awe? The universe expressed through sheer logic, imagination and minds working at the height of their powers?

In Math, you build up abstract objects and define Truths about them and assume nothing else, then you follow each logical next step to build out the entire structure. From pure logic, a clear mind and determination, you get the same consequences every time.

What this means is that because of the sheer scale of how this is done i.e. rediscovering everything through pure logic, this means that no matter where you are in the universe, if you burned down every Math textbook in existence… The same Mathematics will still apply. You could leave a civilization for 20 millenia and pick up their textbooks to find the same Math in those textbooks.

I want to take a second to highlight just how significant this is. Can you say the same thing will occur for our languages? Our religions? Our cultures?

Our mathematical objects will not change, shockingly there is no case where mathematics changes. Every piece of research simply adds to mathematics. Even science cannot say the same, there are constant revisions to the ways we learned science from the beginning as we understood the universe more and more. Quantum mechanics in the early 1900s upended everything we understood about science.

Mathematics is full of mysteries, but unlike most things… These mysteries are built around empowering you to understand them not this vague notion that humanity ‘understands’ the thing.

Mathematics is built out of reason… there aren’t any answers you can’t find by developing a solid argument and deriving insight.

It’s difficult, brutal at times. You are constantly left feeling like an idiot, constantly inadequate and stuck with no way out, then things settle in, you deeply understand the structure and properties of the problem you’re solving, you find moments where you merge with what you’re solving and you understand the problem so deeply that it swallows you whole, and out of this process a solution that is both simple and obvious in retrospect emerges.

This is what keeps me going.

I’d argue that this is the only way to even do things. Inquiry must be answered with pure reason alone. Of course, it’s unfortunate that reality is not forgiving enough to give us enough information so that we can assume as little as possible, nevertheless mathematics equips us with rigid tools and frameworks for thought that would improve your conclusions and understanding.

Freedom

The world is bizarre and chaotic. The circumstances of our lives change in an instance. The world’s variables are so out of sync with anything that we can grapple with that we pretend to understand what’s going on to retain some semblance of control. Here’s a line I think about, spoken by The Merovingian in The Matrix Reloaded :

Beneath our poised appearance, the truth is we are completely out of control. Causality. There is no escape from it, we are forever slaves to it. Our only hope, our only peace is to understand it, to understand the why. ‘Why’ is what separates us from them, you from me. ‘Why’ is the only real social power, without it you are powerless.

From this, real power comes from understanding why. Knowing why is what gives you any semblance of Freedom. Mathematics is built on the foundation of understanding the why of all things.

We are not free. We are chained by our circumstances. We are a small element in the Rube Goldberg machine that is the universe. Consciousness is an illusion of free will.

The universe is only contained by math, and we can only attain Freedom through our understanding of it.

Love

At the risk of invoking cliche, Mathematics is an expression of love.

There’s a joy in the process of a problem refusing to crack open to you and working days or even weeks to weave the concepts so thoroughly into your fabric of understanding that the problem becomes trivial. The process changes you, and you do it again. You move with precision, taking nothing for granted and you build out this large beautiful structure to swallow the problem whole.

Why would anyone subjugate themselves to this constant never-ending confusion?

It’s love. Love for the pursuit of an answer. Losing sleep over an answer. Fighting for understanding at all costs. That’s only spawned by love

A Conclusion And A Personal Anecdote

This essay is given structure, by imposing a 2001 musical’s catchphrase onto it. Some among you might consider that a very strange choice. I think that it’s simply another consequence of framing. There are endless versions of this essay, written by different people, different structures, different cultures and different arguments. I find this artificial external framing to be far more interesting than a mere wandering essay that gets to its point.

I hope this essay at the very least, allowed you to appreciate the scale of the infinite insurmountable structures that Mathematics is capable of building, and understand why I’m so deeply consumed by it.

I’ve found myself thinking with a significantly higher amount of clarity about the things surrounding my life the more and more I’m exposed to the clean logical through-line that Mathematics forces you to take. I am now part of a small group within my college known as The Journal Society, which is focused on exploring Mathematics in such a rigorous fashion. Feel free to check out the notes taken during those sessions here.

Mathematics is gradually becoming a significant part of who I am and I would never have foreseen this happening even a year ago but such is the way of things. It is my hope that at the very least I have piqued some of your interests in this subject and you won’t disregard it so offhandedly as I had until a year ago.

Dive deep into the poetry of reason and the language of the universe, and find joy in the knowledge that you can understand it if you simply learned to submerge yourself in its embrace. Find joy in holding truth that will never waver, beauty that will never fade, freedom that is purely yours and love that will only grow.

With this, I leave you. I wish you luck on your journey. The seas of reality are treacherous, may the anchors of reason hold you steadfast towards where you want to go.