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

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?

]]>The notion that mathematics is an art is extended to touch cultural influence, giving rise to an analogy between memes and mathematical axioms. This results in the argument that mathematical axioms and hence mathematical laws are susceptible to human biases as regards to their aims and intentions.

It has recently been calculated[1] that when the universe began, random gravitational waves were created by fluctuations of the inflaton field. With this, the handful of fine-tuned-parameters which support the weak anthropic principle stand to a test since the beginning of the universe lay in a random mess of particles. The current shape of earth, too, is the consequence of a string of random events. Even the evolution of life, like the evolution of a stock market, is random, purposeless and mindless on its own with the only aim to bring about a gene pool dictated by a contemporary natural selection pressure. It is incomprehensible to suggest that the basic building blocks of life were weighted against each possible combination of atoms to determine a best selection for the unit of inheritance.

Cultures evolved by the random interaction and copying of memes, transmitted through written, audio or visual media. This media, as it stands, cannot understand its information on its own. The contrary is akin to suggesting that a clock knows what time a current time is.

Randomness is inherently meaningless and purposeless. One might perhaps be able to throw an unbiased coin a million times and get straight heads. Of course, no person might be willing to bet their whole salary on it but that does not mean that the outcome in itself was dictated by an external entity, had a special purpose or is not random.

It does not necessarily follow that, for any unlikely consequence exhibiting itself, the outcome had a purpose, a significance, aim or an intention. This begs the question: do random events, which even gave rise to conscious life as we know it, reify the concept of our existence? An unlikely event, statistically, bears no weight per se. The black swans are given a weight. To take the words of James Christian[2], in general, humans reify values, symbols, classification systems and even reify the conceptualization of deep human experience. I might not get very excited about plasma physics but I stand with a civic duty not to announce that particularly to Plasma Physicists. Likewise, the symbol1might hold a different value for the writer and the reader. A topologist might place a donut and a mug in a similar classiffication whereas these two entities are entirely different for a classical geometer. A class may be a collection of sets in one Set Theory and, in another, simply it is simply a collection that is not ``small enough’’ to be called a set. To show that humans reify deep human experience, it is enough to note that each student might be inspired by a di¤erent assignment and perhaps take up a di¤erent line of research to become a mathematical physicist, an industrial analyst, a helping hand for data of, say, environmentalists or simply a person who likes to scribble on and on and has a romance with symbols and their associated ideas.

In reading these lines and perhaps possibly valuing one over the other, a reader has just reified a selection bias. An enlightened mind which can work with both without accepting either runs the risk of having an understanding of mathematics independently with no correspondence to reality, or of having a self-contradictory global vision of the unreasonable effectiveness of mathematics for the physical world. Is it not the case that in order to reconcile General Relativity with Quantum Mechanics, both of which stem from the same Set Theory, one says that Quantum Mechanics is merely a tool to compute probabilities whereas the universe obeys a Minkowski geometry of curve spacetime? Or that if these two need to be unified, there are more than 3+1 dimensions, e¤ectively dismissing Occam’s Razor?

Mathematical laws, which are used as a model for reality, rest on certain axioms. Whether these axioms are discovered or invented remains a controversy. If one accepts that mathematical laws are discovered, there is an inherent assumption about the existence of an entity or a structure ``beyond us’’. Could this entity have allowed for our existence only to be appreciated? This appreciation is directly integrated with the utility of mathematics for the physical world. Since these same laws model reality for us, it leads us to the opening paragraph of this essay, directly leading us to conclude that the evolution of the universe was random and, therefore, purposeless, discarding the existence of an entity, a contradiction.

If Mathematical laws are invented, one runs into the trouble of satisfactorily explaining how mathematical laws predicted the existence of a Higgs Boson, a black hole or even the bending of light. Even if an explanation exists, the existence of mathematical laws would then have a parallel with the evolution of a culture. The creation of the initial memes, whatever they were, was spontaneous and unintentional. Their utility was only realised later, with these then entering into a competition within one culture, until the useless ones died out, similar in manner to words disappearing from a dictionary. The invention of a meme itself takes place after a foundation is capable enough of supporting it. This leads us to suggest that mathematical laws are the product of human intervention, even the ones with dissimilar axioms. One or the other system so developed may not be of particular use and may not even take a deeper place in human experience but its existence, albeit not very useful, nevertheless cannot be denied.

Whatever the answer as regards to whether mathematics being created or invented, the consequence of them is a voyage, whether by a computer or a human, prompting any necessary refinement in the axioms and their results then explored further. The creation of Calculus, for example, faced criticism for its ghosts of departed quantities. Even with such questionable foundations, it predicted the position of heavenly bodies up to an extent which was acceptable during Newton’s time. Cauchy’s rigorous formulation of Calculus, hailed now in a modified form, was shaken by only one function introduced by a man named Karl Weierstrass[3]. Calculus in each of these di¤erent periods had a di¤erent life of its own, including a period in which smooth functions were just the same as continuous functions^{1}. Whatever shape Calculus has taken, the aims and intentions attributed to these Calculi were broadly the result of the axioms and their understanding thereof, cheered for their application to the physical world.

All mathematical ideas have flourished to one extent or another, taken up a life of their own, refined themselves and were then replaced with a broader generalisation. Geometry and Algebra are two good examples. In some cases, the resulting refinement of mathematical ideas have followed a Kuhnian pattern: Set Theory had to undergo constant revision; Category Theory was once called abstract nonsense and Symplectic Geometry is recently trying to address its foundational issues. With these examples in mind, it might be heretical but not outlandish to suggest that mathematical axioms follow the meme rule.

To suggest that Mathematical laws stand consistent, one should be able to answer with certainty the following: might some famous mathematician of the distant past be able to master the mathematics of today, if he did not have access to a pedagogical history of the evolution of a concept, as in [4]? The tracing of the evolution of mathematical ideas, to stand on the shoulders of giants, is akin to evolution nevertheless dictated by external factors. Modify the definitions or axioms and you modify the laws. The axioms of mathematics are themselves mindless not only because they stand merely as a string of symbols but because they are powerfully arbitrary. To see why this equates to randomness, consider n mathematicians, each equally clever, each with one set of different axioms. The systems which are so developed might have similar ideas but who is to tell? Weierstrasses are hardly dense in number, especially with the current dismal state of education. At any rate, this is especially applicable if a proof of a law differs or if the tools are entirely different.

Even with mathematical literature piling up, some mathematicians yet are not inclined to include 0 as a natural number. Notwithstanding the null set of disparities of notations and thus ideas, the inuence of one set of axioms over another is dictated by the approval of a number of mathematicians, which are further dictated by contingencies. For example, the exact moment of having thought up such an essay topic is a collection of a string of (unremarkable) events, the interaction of various memes, leading right up to the intersection of so many butterfly e¤ects. The essay might hardly garner support because of its unorthodoxy but its life is a result of a spectrum of bristling experiences. What do mathematicians fully agree on? Logic? Notwithstanding Gödel’s Incompleteness Theorems, Hilary Putnam has put that to question, as well with his probing that Logic is perhaps governed by empirical forces[5]. Most mathematics, as taught, is based on Classical Logic. Thanks to Brouwer, humanity now has a respectable standing of constructive mathematics, based on intuitionistic logic and the world may well be on its way to see Mathematics based on Fuzzy Logic. How would mathematics look like if it was based entirely on a nonassociative, noncommutative logic, like Quantum Logic? What would the Hilbert Space Structure be, in this case?

Even if a set of axioms are universally agreed upon, the disagreements between mathematicians can still not be curtailed even with mechanical aid. One only needs to cite the Frequentist and Bayesian divide as justification^{2}. The same disaccord can be invoked for the aims and intentions of mathematics for the physical world, if mathematical symbols have a correspondence with reality. The past success of mathematics for the physical world exists because humans, unintentionally perhaps, evaluate it to be that way. Laplace’s idea of determinism, for example, comes to the fore. Any such success, dictated by a zeitgeist, is illusory and certainly of no comfort when one has to talk about an infinite sequence of coe¢ cients of a Fourier sequence.

The absence of evidence of the existence of a good Theory of Everything corresponding to physical reality may not be evidence of absence but is rather evidence of looking for a theory, based on arbitrary ideas, for an arbitrary world. The extraction of meaning in the mindless string of mathematical symbols only reflects our desire to look for order amidst chaos to look for data where it does not exist.

Mathematics is not just the God of Spinoza for whom there is a one-sided amour, but a god which we can murder and replace with a better god. Regarding the aims and intentions (prayers and rituals, if you wish) which are closely iterated and exchanged with the underpinnings of mathematics, one must appeal to Wittgenstein: “Whereof we cannot speak, thereof we must be silent.”

[1] S. Antusch, F. Cefalà, and S. Orani (2017). Gravitational Waves from Oscillons after Inflation.Physics Review Letters. 118 011303 [2] J. L. Christian (2007). Philosophy: An Introduction to the Art of Wondering. Belmont, CA: Wadsworth Cengage Learning; 10th Ed. pp 88–89. [3] A. Kucharski (2014) Math’s Beautiful Monsters, retrieved from http://nautil.us/issue/11/light/maths-beautiful-monsters [4] M. Raman-Sundström (2015). A Pedagogical History of Compactness. The American Mathematical Monthly, 122(7), pp. 619-635. doi:10.4169/amer.math.monthly.122.7.619 [5] H. Putnam (1969) Is logic empirical? Boston studies in the philosophy of science. Springer Netherlands. pp. 216241.

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