Summary

A personal reflection on my journey from foundations of math and physics towards industry.

What makes us human?

What makes us human? Is it our capacity to reason? To show love and empathize with one another? Or is it the way we can collectively organize ourselves? Is it that we exhibit passions that no one has? Or that we express ourselves in ways that manifest our reach for the abstract and the unnamed? The answer to all of the above is in the negative, since all of these traits are found in varying degrees in many other species on earth.

I believe what makes us uniquely human is our ability to pass ourselves through the following chain: we can frustrate ourselves, fall back on the reasons why we do that in the first place, and find a way out of it, via some happy end or otherwise. For me, that is exactly what mathematics is all about. Every calculation, every proof that we attempt or accomplish, is all about how we find our way out of ourselves. For me, mathematics really is all about the human condition, and finding ourselves through it, by carving our very own personal paths to get a glimpse of its beauty.

It is then no surprise that my area of research interest can be broadly classified as foundations of mathematics and physics. That is because my love for mathematics comes from our ability to frame very important questions about ourselves into very formal terms. The title of this research area, however, is a loaded term, and one that mathematicians generally frown upon. Instead, I played my part in little pieces of mathematics which manifest these questions in one way or another.

My mathematical career has been guided by my relationship with mathematics. I tell students that as human beings, we’re merely trying to probe a formalism that is trying to tell us something. It’s okay to say that it doesn’t make sense, and so it’s okay to make mistakes on your journey. That’s because a majority of you will stumble, your understanding will falter at times, you will come across roads which unmask your façade of mastery, and you will fall, but what will set you apart is how much you have prepared for moments like these, and how much tried to get back on your feet after your encounters.

As a student myself, I was good at calculating numbers, and I naturally found math enjoyable. The more I learn, the more I realize what my shortcomings are, and the more I understand what I don’t know. Being not able to understand something at first sight is not just an incredibly humbling experience, but a very human experience.

And yet such experiences do not generally diminish my love I have for math. However, my first love-hate relationship with mathematics was developed from the shock of learning about Godel’s Incompleteness Theorems. To date, every piece of math that I do has Godel in the back of my mind. I have yet to graduate from such `unpractical’ mathematics, since there are always more shocks that follow: our ability to formalize our manner of thinking – or our rationale, our way out of our frustrations – via classical logic also falls short in its breakdown of the distributive law with the real world. Quantum Logic, a logical system deduced by the lattice of closed subspaces of Hilbert Spaces, does not obey the distributive law. Determined to explore more on why the distributive law fails, I chose Quantum Computing as an area of research for my undergraduate thesis. That only pushed a rabbit hole with more questions than answers and led me to choose the area of Foundations of Mathematical Physics as my dissertation for my MPhil, and even that opened pandora’s box for me. This brought me to my current research area of Machine Learning, which I now explain.

There are deep reasons why we have unanswered questions, which can be made precise: part of the reasons stem from the equivalence of lambda calculus (think classical logic) with cartesian, closed categories, and of the equivalence of linear logic (which is a further step away from classical logic) with closed symmetric, monoidal categories. Both categories are examples of Chu Spaces, a cousin of the Dialectica Category (a category named after Godel’s work on intuitionistic logic). Machine Learning paradigms obey all the nice properties of Dialectica Categories, called Lenses.

Machine Learning, therefore, hides within it parables of what makes intuitionistic logic click. What excites me about Machine Learning is not that it is mostly about how we live our lives and carry ourselves in this data age; I am also not most excited about how machine learning is a big black box that we must probe; I am also not interested in machine learning because it stems from the assumptions that we make about the world around us (should this behavior be modelled as a linear function, a polynomial or otherwise?), and also not because it tries to mimic the human brain, but mostly because our relationship with ourselves – the way we reason out our frustrations – is being modelled in a very pragmatic way, and turns the question back on its head: what makes us human, and how do we tell that to a machine?