Recently, I’ve read the book Mathematica: a secret world of intuition and curiosity, and it blew my mind.

This blog series is an attempt to help distribute its content! In this part, I reflect on my own path to discovering the beauty of mathematics.

Elementary School

When I was in elementary school, I absolutely loved arithmetics. Once you figure out that multiplying 13 by 8 is as easy as 80 + 24 which is as easy as 100 + 4, homework gets incredibly easy and people start to encourage you. If you know you can solve a math puzzle, every one you solve merely becomes a little pad on the shoulder. Just encouraging you to go further. Especially as a kid. At the same time, I would also take my mistakes very serious. Imagine a 9 year old debugging his own thinking in order to understand even the slightest errors in his answers out of pure curiosity.

Back in those days, there were just a few people in my class that were doing those calculations with the same level of intensity as me. And I was fiercely competitive, without getting carried away too much.

Social Pressure

However, at the end of elementary school, around age 11-12, things started to change. Social status became more and more important, and it became slowly more important to fit in than to get good grades. So I felt the urge to take it down a notch. When I entered Dutch high school at the age of 12, I was less motivated to really understand my mistakes.

High School

With me being less engaged in mathematics, I noticed that slowly it became more and more difficult. And halfway into high school, I was more and more focusing on passing in general. Math became more difficult, although I never really had to work overtime to catch up. I just remained at a comfortable 65/70%.

Much later I realized that high school mathematics is often more focused on learning to apply some sort of tricks, rather than building up thorough intuition. Basic calculus like derivatives and integrations are mostly presented as a series of executable steps to be learned, rather than through its fundamental axioms.

The Turning Point

After high school, I started studying cellular biology, physics and chemistry at the technical university of Delft (TU Delft). So that means proper calculus classes, linear algebra, programming and differential mathematics. Being part of the Physics department when it comes to mathematics, I strongly believe the teachers were excellent. And their motivation still strongly impresses me even to this day. They really did an amazing job in sharing some of the wonders of mathematics, but still it did not really click for me there. During the courses, we would be introduced to mathematical theorems and axioms, whereupon we would together go through a couple of example exercises. The homework would then consist of the same exercises, with different numbers and slight variations. That means, the focus is on applying the mathematical tools rather than understanding the tools thoroughly.

The Epiphany

After one year of studying in Delft, I decided that the future work field was not for me. Simply because I did not envision myself working in a laboratory. I did enjoy the mathematics to some extent; hence I decided to move the Dutch city of Maastricht in the south and study Econometrics and Operations Research. That is, the mathematics behind probability theory, statistics and modelling & solver technologies (constraint optimisation). There, rather than applying mathematical tricks, mathematics was taught through proving the theorems behind the basic axioms. We did not go as far as Hardy’s Principa Mathematica or the foundational wizardry work by Grothendieck on abstract algebra, but we were required to show our ability to understand the underlying mathematical theorems and their corresponding proofs. We were required to learn how to read and reason in the mathematical language.

From that moment on, mathematics was never the same. Rather than learning the particular derivation and integral rules, like for example $\frac{d}{dx} \sqrt{x} = \frac{1}{2\sqrt{x}}$, we learned how to obtain any derivative through its raw definition of limits and sequences. Imagine whiteboards full of mathematical symbols coming to live in front of your eyes. Every exercise, every transformation further establishing your central understanding, further shaping your thinking, your imagination and your creativity.

Off course this takes years, and not everybody goes the same through the process or even through the same process. It’s an highly individual experience heavily on introspection that is hard to put to words. The benefits however can be incredible. Like viewing the world through the perspective of probabilities and statistical inequalities, or seeing shapes through their mathematical geometric definitions, or understanding the content of an academic paper on any topic just by glancing at the formulas.

I also love the problems at Project Euler, and sometimes I wish I had more time to work on a few of them. At the same time, I don’t hold any illusions of me being able to solve some of the difficult ones. What I love about these problems often contain solution spaces that are too large to brute force. Rather, you are required to map the problem into a solvable problem. So you’re pushed to be creative and look at the same problem from different perspectives. I absolutely love this, and I still use this way of thinking in my work a lot. Take a step back, re-approach the problem from a different perspective and continue to learn. Bring in more perspectives, push for active open-mindedness and people sharing their ideas. Make mistakes, admit them, correct them, and improve together.

It can sometimes be a truly indredible, yet individualistic experience. At the same time, for outsiders it can look like black magic. When I was studying for exams and filling paper after paper with endless formulas and derivations to display them all over the kitchen table, my parents would stop asking what I was studying exactly. Again, this is the trouble of mathematics. The journey is more important than the destination; as it remains very difficult to explain mathematical concepts in layman terms to somebody who does not speak the language.

Finally, I wrote my thesis on a Bayesian cluster-based estimation method of a time-series of variance-covariance matrices, which I explained to friends and family as predicting the bubbles in a loaf of bread. Rarely would they show more understanding after that explanation only.