Recently I found myself watching a video that felt like going on a walk and talking with an old friend who I haven’t seen in a long long time. The video was called “Who is mathematics to you?” and it touched on many topics that resonated with me. My goal in this essay whatsoever is to come up with an answer: who is mathematics to me?

As it is the case with people, mathematics is complex, changing, and, as I attempt to convince you dear reader, full of contradictions. I have come across mathematics through different stages of life, through different people and for different purposes. Mathematics to me is then an amalgamation of different people through different contexts, where “me” is alo an ever boiling stew of places, memories, and emotions.

Growing up with(out) maths

Mathematics as a boogie man

I never thought I would find solace in mathematics. When I was little, maybe 4 or 5 years old, I struggled a lot with maths. During my first year of elementary school, I even had to go to some psychology sessions with the current math teacher because we could not get along. In retrospect, I don’t remember much of how I felt. I might have failed to see joy in solving little problems, or maybe I was getting bored and wanted to do something else. Regardless of the reason for me to struggle in that class, I am sure of one memory. I still remember the face of the maths teacher going to talk with the school psychologist right after me. She smiled at me with sympathy. A sympathy that then I couldn’t understand as I could swear she was angry at me. Her smile stuck with me, because she seemed scared too. I can now only imagine how difficult such a situation could be, when your livelihood depends on it. Moreover, she might have loved maths at some point and it would be confronting to get in trouble for doing your job trying to explain that which you love. I now wonder how she would react knowing that mathematics plays such an important role in my life. Little did that child know that he would grow up to love mathematics.

You will never have a calculator in your pocket all the time

Middle school was messy. I had to change schools, neighbourhoods, and all everything that comes with it. The first year I had to learn arithmetic and solve long arithmetic expressions. I don’t remember being particularly good or bad at it. I do remember being annoyed at the reason why I had to learn that: you will never have a calculator in your pocket. This argument did not make much sense to me, because even when going to small store people there had calculators to get the right price. I also got a small digital watch with a calculator which meant I always had a calculator with me. This, of course, got me in trouble again and I ended up having to not wear that watch during that course. I passed it with the lowest passing grade. Now we all have smartphones in our pockets, yet learning arithmetic is still important.

A two-headed beast: Care vs Appetite

One of the things the video touched on was mathematics as an art vs mathematics as a competition. Where one was about seeing beauty and self-expression and the other was about solving problems, solving the most problems, the most difficult problems, and becoming the best by some (not necessarily formal or reproducible) metric.This distinction reminded me of two teachers.

Let’s call the first one L. She was a caring teacher and always looking after her students. I didn’t have long interactions with her but she cared a lot for my sister. I remember her being passionate about mathematics. When I was leaving that school, she even offered me help to study.

The second one, let’s call him B., was an interesting guy. He was teaching algebra but he did so in a different way. It started with makings us read a book “Uncle Petros and Goldbach’s conjecture”- a nice book that I remember not so much for its mathematics or plot but more about the sense of achievement of being the first “book for grown-ups” that I ever finished. He wanted to introduce high-school algebra as an appetizer to proper abstract algebra. So we talked about the properties of operations and basic set theory. B.’s experimental approach to math education was interesting and now I probably agree with the spirit of it. However, its implementation was lacking. While it was good to elicit some curiosity, it failed to convey a cohesive story for it to be a proper introduction to abstract mathematics. At the same time, the abstract nature of this approach took us too far from attempting, let alone solving, the algebra problems that students at that age were expected to solve. So we were left hanging in this no man’s land between abstract math and high school algebra.

On a personal level, B. always gave me weird vibes. I felt I couldn’t really trust him. He was excessively friendly and chatty with the high-school administrators in a way that didn’t look genuine to me. As his flagship initiative, there was a program for the top students in mathematics to attend some introductory university lessons in maths. The top 5 students at the course he was teaching would be allowed to attend. I remember wanting to attend out of curiosity. For better or worse, I got the 6th or 7th position, close but no cigar.

When I saw the students who went there I saw different reasons. Most of them were already good students and their good habits applied to maths as well as to any other subject. It just happened to be the case that maths was taking them to university a bit earlier.

There were one or two of them who seemed hungry, but not for knowledge or learning. They wanted the prestige. I could see teachers fawning at those kids, always encouraging them. Their parents could not stop talking about them; they were proud with good reason. The kids themselves felt honored with a subtle air of superiority. After all who wouldn’t be impressed by a 8-10 y.o. taking university lectures.

Around one year after that, I was leaving that school to be home-schooled, while my sister was still studying at the same school. One day, I was at my old school with my family as there was an end of semester meeting for my sister. We came across L, who was my sister teacher. She told us nice things about my sister and asked me what I was going to study at uni, to which I answered engineering because it had mathematics. She smiled and said she was willing to answer any question I had about mathematics. I asked something about how hours are tallied using base 12 while minutes and seconds are using base 60. Asking that question came out of a place to keep face. Back then I thought of myself as someone not good in mathematics. The university program, other teachers, and other students had showed me that you needed to be either already good enough or struggling to ask for extra help. As I was neither, I felt ashamed and never took on her help. She was the first glimpse I had at people doing mathematics for the sake of it.

What happened to the students who attended university lectures? Well, many years later, I got to know that the program was not really a success. It took maybe 15 students and only one of them pursued higher education in mathematics. As for B., he ended up pulling up a scam related to some books and was never seen at the high-school again. This taught me that mathematicians are all sorts of people: from caring mentors to sly scammers.

Ready or not you gotta choose

That’s deep and all, but… Does it make money?

Later the video introduced another way to approach mathematics: as a tool. A collection of gadgets and little machines to solve more and more complicated problems. This reminded me a lot of how maths were taught in engineering. They were just a way to get to a solution. Was it the right problem to solve? What can you learn from the solution? Does it mean anything? were all questions besides the point. What only mattered was to get the correct solution. This instrumental approach had analogies in other areas of life: mathematics as a mean to gain prestige, and mathematics as a mean to make profitable decisions.

Thankfully this education was not all utilitarian. I came across different beautiful results. The one I remember the most was how dimensionless numbers showed up in describing different and apparently unrelated physical phenomena. The first time I saw that it hit me as something deep. Why would completely different areas be described by the same language? It is like writing a story on one part of the world and then realizing someone else thousands of kilometers away completely unaware of your existence is writing the same story with the same characters, but just has different names for them. It was a vivid experience but still I didn’t make much of it.

Years later, and one country after, I was doing research in automation of chemical plants. We had to use different mathematical and computational tools. To this aim, I had the opportunity to take an optimization course and a mathematical modeling course. I still remember the lecture in the optimization course when they showed the conditions for a point to be an optimal point with a proper proof. I had seen mathematical proofs before, like that of the infinity of primes or the irrationality of the square root of two. They seem to me like cute gadgets, but none of these related to life in a meaningful way. This proof about optimality was different. People were using these results to design systems that affected my everyday life. These abstract results had a concrete effect on the world!

Another core memory about mathematics happened during the mathematical modelling course. They showed how the same language of mathematics can describe different physical (and non-physical) areas. This stirred lots of curiosity in me: how can a language be so powerful and expressive?

Beyond these experiences, I also noticed how other researchers in engineering were quite good at mathematics and how they used it to get surprisingly deep insights using tools that didn’t lie beyond a first year calculus class. One of them told me informally: “if you think about it, most of the mathematical tools we are using are from the 1800’s. We are barely using modern mathematics” This provocative comment made me wonder about what modern mathematics are.

I could see for myself that even if you want to use mathematics for profit, you still need a solid conceptual understanding. While proofs are far from direct applications, they lay the foundation for all future applications. That maths is not solving problems, it is a language. A language to describe the world. A language that describes itself, while building itself.

Once you’re in there is no way back

The questions about mathematics from my days in engineering never left. Many events and yet another country later, I was studying mathematics on my own. I began first learning about machine learning as I needed a job - as one does. At the same time, I couldn’t run away from my inquisitive nature so I was trying to understand why and how these tools worked. A finite sequence of questions of the form “Why something” converged to me studying pure mathematics. This time I was diving into the abstract and started getting a taste for how mathematics is an art where you see the same things in a different way. This was the first moment in my life that I have heard people saying things like “This is a beautiful result”. I was enthralled by the deep connections between fields, how you could express one thing in different ways depending on what you wanted to prove or what you wanted to do with the result.

Beauty drew me in but I stayed for the craftsmanship, as mathematics requires lots of skill. It is not just solving problems, it is about seeing problems creatively, coming up with the right questions and being critical of one’s own arguments. Good mathematics lies in that - lagom, as the swedes would say - goldilocks zone , a proof must be logically sound and should be written without overwhelming the reader with details; yet it should display enough details for the reader to fill in the missing gaps. From this point onwards, it becomes a craft. The goal is not the solution of a specific problem but constant improvement regarding depth, breadth, understanding, elegance, and articulation of mathematics. To me, this is more akin to art than science.

Wait! Are you saying a human pursuit is also influenced by human institutions?

As my MSc came to an end, the honey moon with pure mathematics was coming to an end too. There was a battle between wanting to sill dive into the abstract, and wanting to find a stable career where I could apply some of this abstract knowledge. I reached out to many mentors and people in and out mathematics: researchers, former researchers, people who wanted to do research but couldn’t and people who never wanted to do research to begin with. The conclusion was to make a compromise. This lead to another field, another country.

I was now doing theoretical computer science. The biggest shock was to learn how this area approached problems and how they communicated the solutions. I was used to questions such as “can you show a solution exists?” but now I was getting hit by “can you solve it quickly enough?”. There is certainly a lot of overlap between the disciplines of maths and computer science. Sadly, this is not reflected in their respective communities.

Before leaving pure mathematics, I was warned and forewarned and warned again of the instability that comes with an academic/research career. You will need to move countries many times, learn new cultures both inside and outside the workplace, and be already on the lookout for a new position almost at the same time you are settling to a new position. I was also told about that grants, contacts, and publications were the metrics. This meant that ideas were only valuable if and only if they could secure a grant, a collaboration, or a publication. Seeing this shifted mathematical research in my head from a craft to an enterprise. This was shocking to me, and it was only made more shocking by noticing the quicker pace in computer science compared to that in pure mathematics. All my life people told me there was no such thing as a proofs factory, yet there I was.

It took me until the end of my PhD to understand how proofs made money. A quick and dirty summary goes as follows. Institutions and people get money from grants. Researchers are the ones who bring grants to their institutions. These grants are used to buy equipment, carry out experiments, go to conferences, pay publication fees and hire PhDs/ postdocs. A funding application is more or less awarded on the relevance of the proposal, the quality of the researchers filing the application and its feasibility. The relevance depends on what the funding body wants and what the community is doing, and so it is very important to attend conferences and events when one can see these developments first hand. The quality of the researcher is measured by the number of publications, and successful grants they have achieved. All these factors feed on each other, having better publications helps getting better contacts and gives higher chances to be awarded grants which in turn can strengthen your collaborations and lead to better publications. What you do has to make money somehow.

The humanity of it all also came in other flavors. Maths seemed as something that anyone from anywhere could get into. In practice, there was much more nuance to it. For recreational and general topics, it was open for everyone. The deeper you went the more important other conditions became. Your future in mathematics didn’t only hinge on your talent and discipline, but also on your location, the institution you attended, and who your advisor was. The location limits available institutions (both for grants and education), which then limits the topics studied and who is doing research there. Then your advisor becomes key in what is going to happen to you. First through the problem they give you. If you were lucky your might get a problem that would take off in a few years. But you might also be unlucky and land a topic that will become unfashionable in a few years. Your supervisor and their community are the audience for your results. As researchers from one community have a difficult time evaluating contributions from other communities, switching areas involves additional friction. Moreover, your reputation in the community carries a lot of weight as academic positions usually require a recommendation letter. While it is still possible to make it through an academic career without that support, in my experience, the most prolific academics I have met have continued to collaborate with their mentors. Maybe this is worth looking into more scientifically.

Prestige was also a huge currency. Highly regarded academics have more chances to be published and to get funding. This meant that people, whether consciously or not, tried to pander to the interests and quirks of these academics. Their word carried lots of weight, which was not related to the degree of involvement in the day to day research activities. In this way we have that curiosity is bound by the interests of senior academics, who are themselves bound by funding opportunities which are in turn constrained by political agendas. While this exact interaction is not clear in all jobs, it is a reality that happens to an extent in all fields and industries.

Academia was not all systemic issues and money. I had the joy of teaching and showing people how beautiful mathematics can be and how you can be creative. One of the best things about having done academic work was seeing how people’s mindset changed by thinking about mathematics in a different way. I took it with great pride to see how by the end of the course my students were writing better arguments, asking deeper questions, and being more willing to engage with critical thinking.

Another source of joy was talking to colleagues from different fields. I always found it delightful to see how they tackled a problem, why they found certain problems interesting, and how did their mind work. It was engaging and nurturing. Many times a short example over a coffee talk gave me a clearer idea than many hours of just listening to lectures. It was also nice to see how someone’s passion for a topic triggered me to become curious about the same topic and learn more about it. This human aspect is probably what I found most fulfilling from academic work. It is what I romantically reminisce about, when people ask me about academia.

Who is mathematics to me then?

This convoluted trip has shown a push and pull between different forces such as internal curiosity vs external rewards, understanding vs prestige, beauty vs applications, and so on. These different aspects of mathematics throughout my life have taken form through different people, books, lectures, topics, and anecdotes. From the highest highs understanding a deep result and sharing it with other people, to the lowest lows struggling to prove a result that could make or break my thesis and get me fired; from the anxiety of trying to understand a concept before a deadline, to the peace of solving problems at my own pace, mathematics has been there. A silent, yet loud, companion. A ruthless one since the logic is either sound or not, but also compassionate one giving you a supportive community and all the time in the world to figure out things on your own.

Of all the contrasts in and around mathematics, the most striking one to me is that between its enduring almost eternal nature and its human emotional core. Mathematical statements are the closest we have to something truly transcendental. If a statement is true, it is so no matter what anyone says and it is always the case. At the same time, the activity of mathematics is deeply human both in the emotions it evokes and how it can channel creativity as well as its context within human interactions: from informal talks among peers to institutional and political directives. To be human is to be ephemeral, yet mathematics is as human as it is beautiful, expressive and immutable.

So, who is mathematics?: Mathematics is a friend with whom I hang out some times, to catch up about old friends, to keep my brain busy, I guess… and to not be bored.