# Guest Editorial: Diving into the Unknown

November 17, 2022

As a toddler, I moved from the United States to Taiwan and then back to the United States. I remember struggling with language, trying to manage a mix of English, Taiwanese and Mandarin. Throughout the tumultuous international moving, math remained a constant: the numbers I saw in the United States were the same numbers I saw in Taiwan. In my world filled with unknowns, I found myself drawn to the promise of certainty brought by the algorithms and theorems of math. It felt like a place where I could find answers and be sure of their truth. I also loved that it was a pragmatic subject to study. I would often hear of the limitless “applications” of math to the world. After all, calculus would pave the way to the question of the century: how fast does the top of the ladder really slide down a wall?

But my relationship with math and my understanding of it changed when I got to college. There’s nothing like going from getting straight A’s in math classes to receiving a 60% on your first college math midterm to give you a wakeup call. The once constant assurance that math gave me suddenly vanished. And I must confess that even though I often tell my students to never give up, I gave up — on more than one occasion, in fact. I went to college knowing I would major in math, but what I did not know was that I would end up nearly dropping the major more times than I can count on one hand. If you’re like me and are not good at “math,” that’s more than 5 times.

But the subject recaptured my interests through mathematical logic. The theorems that first truly mesmerized me were Kurt Gödel’s Incompleteness Theorems. These theorems essentially tell us that math will never fully know everything. They absolutely fascinated me; the tantalizing power of logic to know what exactly was unknowable intrigued me deeply. But they also unsettled me: I felt betrayed by my previous assumptions of math. There was, in fact, no certainty. Math itself had somehow dealt the finishing blow on my dream of a stable and certain foundation. As I approached my graduate studies, it became more and more apparent that this is the life of a mathematician: to essentially live with uncertainty. The thing they are trying to prove could be true, but they just haven’t tried hard enough. Or it might be false, and all their effort would come up empty. But now, I guess there’s even the third option that it’s neither true nor false…

During my time exploring logic, an unexpected chain of events led me to be under the guidance of my then advisor and one of his graduate students. With their support, I found my interests diverge away from the incompleteness theorems and converge on another branch of logic called model theory. As time passes, I may forget how difficult it all was, but I will never forget how hard I fell back in love with math through model theory. My favorite theorem in all of mathematics is a theorem called Morley’s Categoricity Theorem. Of course, the central reason I love this theorem is all the precious memories it holds for me. But there is another reason I love it: its proof. The proof of Morley’s Categoricity Theorem showed me that the seeming dissonance in the math world ultimately coalesces together to form harmony. As beautiful to me as Morley’s theorem is, I must confess that I cannot remember the last time I used it outside of the math world. But to me, that’s ok. It is beautiful, and that’s enough.

I am a math teacher, but I never feel that I do a good enough job of making math feel relatable or relevant to my students. So, I’m hoping that my story might humanize the subject and give it that relatability.

Working mathematicians fully understand the math one learns as an undergraduate. But past that stage, they tackle the frightening unknown. It’s the quintessential question one must confront when in the midst of any challenge: have I not achieved this because I haven’t worked hard enough, or have I not achieved this because it was impossible in the first place? It is a rather terrifying prospect to put all your effort into something, not knowing if it’ll work out. But mathematicians do it anyway.

The fact that I am Mr. Chien today and not Dr. Chien tells you how the pursuit of my PhD ended. And while I once wanted to hide this mark of failure from others, the more I teach, the more I feel it is important that my students know my story. Students often approach a math problem feeling like the answer should immediately jump out at them, and if not, then that’s the end of the story. They simply can’t do it. But nothing could be farther from the truth. I mean, even the professional mathematicians often wrestle with problems that stump them.

Similar to the student paralyzed by the difficult math problem, every year many students, and seniors in particular, are paralyzed by the uncertainty of where they’re going to go for college. Confronted with the unknown of whether they will get into their “perfect” school, they often spiral into a panic. They start catastrophizing every point they miss on an assignment. They convince themselves that some universities are simply out of the question because it does not fit the picture-perfect image they have painted in their minds. There are two things that bother me the most about all this: first, that they start trying to do anything and everything that will add another star to their resumé, all the while forsaking things and activities that might not look as impressive. Everything becomes one big calculation of what is pragmatic and what isn’t. Second, the moment they get into college, all but the barest amount of effort is left. This isn’t particularly surprising: If everything is about the destination, then once you get there, nothing else remains.

But consider a different approach. For just a moment, set aside your sense of pragmatism and step into my ideal world. What if you were a little more like a mathematician and embraced the uncertainty, finding joy in it even? Not knowing if the problem in front of you is one that can be settled but nonetheless enjoying your attempt to settle it. What if you realized that uncertainty is just a part of the process, inescapable even, and that it is not a sign to simply give up? A mathematician often values the exploration that might lead to a theorem as much as the theorem itself. What if you valued your time at Harvard-Westlake, the memories you made, the connections you formed and the identity you built as much as the college acceptance letter at the end? And above all, what if you did things you truly loved and enjoyed instead of things you thought some college admissions officer thinks you should do? So, here’s my invitation to you: go and learn that language if you want or make pottery if you’d like. Ask that question, both in and outside the class, simply because you’re curious. You never know what path it might lead you down.

When I started doing math, I was drawn to the certainty and the practical applications. My love for the subject burned out but was rekindled by the uncertainty and the beauty. And along the way, I stumbled, fell flat on my face and developed quite a few mental scars. But it was precisely because of all this that I developed the foundations of my principles and identity. I also met two of the most inspirational people I will probably ever meet in this lifetime. And from them, I picked up the habit of wearing bowties. It was my dream to become a logician. I failed. As I reached for that goal, I can’t remember a day when I didn’t ask myself, “will I actually be able to do this?” “Do I actually belong here?” “Is any of this worth it?” To the first and second question, the answer ended up being a resounding “no.” And to the third question, maybe model theory wasn’t the most pragmatic thing to study. Maybe it would have been more practical to study something that would be more useful down the line. But because of model theory, I got to meet the mentors of a lifetime. I was taught by them, guided by them, molded by them and lifted by them to see a view that I would otherwise never even dream to see. What a view it was. And there is not a single day where I don’t feel the reverberations of their impact. So, “is any of this worth it?” The answer is yes, and the proof can only be left as an exercise to the reader.

Joshua Helston • Nov 18, 2022 at 6:45 pm

Very well written! Loved your article Mr. Chien!