Mathematics is a valuable endeavor

Mathematics is the music of reason. — James Joseph Sylvester, English mathematician

“How in the world could anyone prefer to have this as a profession?” my fellow freshman was thinking to himself in a large lecture room while trying to catch up with the furious pace of a calculus professor and scribble down the last couple of formulas before they get erased from the chalkboard. “Can you imagine that? This professor has been wasting his entire life on this nonsense!” he was talking to me in the hallway after class, while I was nodding and chuckling at this exclamation. At that time I was not even realizing that I would spent subsequent years of my life trying to make sense out of mathematics and find its true meaning and place in myself and the world.

Is studying mathematics useful? What is the purpose of doing research in mathematics? Would one be able to find his own way to contribute to and make impact in this world through the foundation of math? I have been reflecting on these questions ever since I made the decision to double major in mathematics and computer science. Provided that I already had a strong programming background at the time of entering college, my dream was to combine mathematics and computers in powerful ways to produce something interesting and valuable. I believed that it was worth spending more time on mathematics rather than programming since math is so much more foundational and permanent; whereas coding looked like no more than just a tedious technical field whose trends were quite volatile and rapidly becoming obsolete. I felt that technical skills were easy to learn, but the study of math was deep and impossible without a teacher. In this post I would like to briefly share some of my reflections on the usefulness of math, and, hopefully, begin convincing you that not only is mathematics valuable and useful, but it is also a fabulous jewel in its own right.

First, I am absolutely convinced that we should not base the value of a particular field or research problem in mathematics on how applicable it is. Mathematicians are not responsible for providing rigorous reports on the application areas of their theories to physicists, chemists, and others, prior to starting to work on a new piece of mathematics. Mathematicians should not be treated as small children responsible for everything they do after all! To some extent mathematics can be considered as a peculiar art form, where there is a degree of freedom to what initial definitions and axioms we establish in a theory. Then, we start dancing with them according to the rules of logic while building layers and layers of results, which we know to be the absolute truth in the end – the kind of truth which does not even depend on the universe we live in, and the one that is completely devoid of biased opinions. In fact, neuroscience backs up that if you look at the brain activity of a mathematician contemplating a beautiful formula, you will see the same patterns of aesthetic pleasure as an ordinary person would feel, say, by relishing the beauties of Mona Lisa in Louvre. I think that the ultimate truths of mathematics are so infatuating for a lot of people exactly because they seem to exist in this abstract world of objective reality, whereas the world of humans is so much spoiled with biased opinions and other chaff. In this sense the world of mathematics transcends the materialistic layer of reality; it is more on the spiritual side of things. In fact, surveys shows that mathematicians tend to believe in God more than any other scientists. People get astonished by how naturally mathematical theories arise, how coherently they are built, and how glamorously the fields of mathematics intertwine one another in a way that suggests an inevitable role of the Creator in this.

Second, we need to realize that those mathematical theories are not just a set of symbols put on a paper. These seemingly very abstract ideas come together to describe the surrounding nature really well. Mathematical models are simplified abstractions of the world which turn out to be very useful in explaining phenomena as well as predicting how the world will behave after certain initial conditions are set. And when it comes to the mathematical research, just like with any other type of research, no one ever knows what you will encounter and how useful this work will turn out to be in the long run. In other words, a mathematician may not immediately conceive the applied usefulness and the potential groundbreaking consequences of his work. Sometimes, a momentous application of some mathematical work does not appear until after decades or even hundreds of years have elapsed from the time it was published. History has seen a multitude of examples of this. For instance, when Newton was developing calculus, he could not envision how revolutionary it would be in classical physics and other applied fields; when Riemann was creating Riemannian geometry he did not expect that his work would enable Einstein to formulate his general theory of relativity; when mathematicians were working on number theory, they had no idea that it would become the foundation of modern cryptography and computer security. This hints at how adept mathematicians are at injecting novel, far-reaching, and potentially very useful ideas into the world, and I believe that the community should be careful enough to not sweep aside these ideas immediately.

Third, engaging in mathematics is a great way to keep your mind in shape. Actually, solving problems is the sole mission of every mathematician. The ability to think is one of the most precious resources a human can have; it is also one of the most sought-after skills by employers.

Most people would die sooner than think. In fact they do so. — Bertrand Russell, British philosopher and mathematician

I believe that the thinking capacity of the brain is a muscle that can and should be trained. Imagine that mathematics is like an army boot camp, and mathematical problems are its core training routines. You go through the school of math once and come out of it being more fit to tackle the challenges of the world. Thus, the versatility of mathematicians comes from their ability to promptly pick up different subject areas and start working on solving specific problems immediately.

Last but not least, pure mathematics is not as far removed from everyday life and the cutting edge technology as people might mistakenly think it is: we deal with numbers in many situations in our lives; we use the principles of mathematical optimization in trying to minimize cost, time, or effort; the uncertainty of the future causes people to speak and make inferences about the future events in terms of probabilities; we use mathematical statistics on huge contemporary databases to understand a process and predict its future outcomes; etc. All other sciences make extensive use of mathematical tools too. For instance, the algorithms of mathematical optimization are the mainstay of machine learning and artificial intelligence (AI), which, in the last decades, have shown incredibly successful results in computer vision, speech recognition, robotics, recommendation systems, medicine, finance and numerous other application areas (actually, the topic of AI deserves another blog post from me).

In the end, I hope that this blog post has fulfilled its mission of refreshing your perspective on mathematics and its role in this world.