Why patterns?

"Since Newton, mankind has come to realise that the laws of physics are always expressed in the language of differential equations." – Steven Strogatz

This is a quote that's nested somewhere in the middle of a note, on my iPhone, that was created back on the 18th of June, 2020. I don't remember where exactly I read, or heard this, but I immediately felt it was profound. I was compelled to jot it down in this note where I housed similar, scattered thoughts; and it's stuck with me since.

Besides high-school level calculus and random Youtube videos, my understanding of differential equations at that point was... not so mathematically grounded. It was somewhat mystical; this idea that the evolution of an entire system could be described by an equation, that did not necessarily have a solution. I found it poetic. It would cross my mind while watching waves at the beach. Although there wasn't any equation you could solve analytically to accurately predict wave behaviour, the way the waves rose and fell in front of me was a solution to a differential equation playing out in real time.

Later on, I grappled with the language of differential equations on a deeper level through first and second year uni math courses. I was initially overwhelmed by the proofs and formalisms, but it quickly became one of my favourite topics, and something I’d look forward to learning about. I’m filled with awe whenever I think about how we’ve created (or discovered?) this elaborate language that can explain and predict all types of dynamic phenomena – like motion, heat transfer and even population growth.  

Differential equations are all about rates of change. And the beauty/fatal flaw of existence is that things are always changing. That’s where patterns come in.

The human brain can be thought of as a highly sophisticated pattern recognition instrument. Intelligent behaviour, in both biological and artificial systems, hinges on the capacity to: (a) reliably acquire/store data, (b) extract features, (c) find patterns, and (d) assess performance. Although most of this happens subconsciously, it drives conscious decision making.

For example, communication is riddled with patterns. Throughout development, we learn that communication involves turn-taking. We intuitively develop the skills to decode patterns from a symphony of cues: eye contact, tone, facial expressions, volume, contextual framing, word choice etc. These variables influence each other and vary over time, and somehow our brain can disentangle all this and respond accordingly.  

So, how are we so good at recognising such patterns? Are our brains somehow tuned in to the ‘language of differential equations’? Is there a mathematical expression that can describe communication, and if so what would the variables be? Is a pattern still a pattern if it’s only a pattern that you perceive? Where does the boundary lie between patterns and chaos? These are some questions that keep me up at night.

And hopefully now, you can kind of understand why I’m starting a blog on patterns. I love patterns – real and abstract. All good things follow patterns, like leaves and seashells and music. But it’s also fun when things don’t follow patterns, like weather and dreams and, also music. I hope to explore some cool and interesting patterns in this blog and take steps toward answering those questions.

PS. I only realised while writing this: Steven Strogatz (the goat behind the quote), is the same mastermind behind 'Nonlinear dynamics and Chaos'. It’s an absolute banger of a textbook I've been reading for a current research project. What a full circle moment :)