The importance of direction

Lately I’ve been reflecting on the impact that having a direction (or lack of it) has had in my life. For a long time now—since around the time I first set foot on a university campus—it’s been clear to me that I would be doing and writing about mathematics for the rest of my life. However, perhaps because I have never been particularly interested in thinking about money I never necessarily assumed that I would be paid to do mathematics. My expectation was that I might need to have a day job to support myself and my mathematical hobby.

Recently, I’ve been thinking that being paid to do mathematics would be great because it would allow me to devote more time to it. For example, I am quite proud of my latest paper—but it took a lot of time to write, and would have taken even longer if I didn’t already have a job as a postdoctoral researcher. One of my previous papers was written during evenings and weekends since at the time I was working full-time at an internship and it wasn’t ideal.

Thus, I’ve been doing the things necessary to apply to academic positions like putting together a research statement. During this process the importance of direction has been made abundantly clear to me and I can’t help but lament the amount of time I spent as an undergraduate and graduate student with unclear direction and ridiculous misconceptions.

For example, as an undergraduate student I studied the Ramanujan–Nagell equation and came up with a new solution for it that I hadn’t seen published anywhere. My thought was: this is so cool, I have to share this with the world. So I wrote a short report, gave a copy to one of my professors who I thought might be interested, and uploaded a copy to my website. But what to do beyond that? My thinking was: I don’t know how to get it published somewhere, but I do know how to make it accessible to anyone who wants to read it via the Internet. I told myself that 99% of the value comes from writing the report and I didn’t need the additional 1% “seal of approval” that comes from getting it published. Now I know this is totally backwards—at least when it comes to job applications almost all of a paper’s value is derived from being published.

Looking back, my advice to myself would absolutely be to try to get the report published. Maybe I could’ve gotten it published and maybe not, but either way it would have been a very valuable learning experience. Incidentally, the report does seem to be of value to some people: it’s been cited by two OEIS entries, two published papers, a PhD thesis, and Noam Elkies recently gave a talk referencing it (!!!):

We state several finiteness theorems, outline some of the connections among them, explain how a finiteness proof can be ineffective, and (time permitting) sketch Nagell’s proof and an even more elementary one discovered only 12 years ago by C. Bright.

Maybe I’ll come back to that result and get it formally published some day, but I already have more than enough papers on the table. At the very least it’s encouraging to know that I’m in no danger of running out of material to research and write about anytime soon. Even more importantly, I’ve learned how important direction is to achieving your dreams.

Five years of dancing

Five years ago I decided to start a new project which was totally different than anything I’d done before in my life: I decided to become a salsa dancer. The impetus for this was a drive to improve my overall well-being by deliberately pushing my boundaries—up until 2013, I spent the majority of my life on academic pursuits and the extent to which I focused on this had created a severe imbalance. In short, I didn’t have many friends, I was generally uncomfortable socializing with people, and even more to the point: Girls didn’t like me, at least not in “that” way.

I signed up for and took my first salsa class on May 28, 2013. I was absolutely terrified and struggled with the class but nevertheless committed to giving it my best effort for at least five years. It felt like a last-ditch effort: I wanted to improve at forming relationships but I had no idea what I could do to change. I didn’t know if salsa dancing would help, but I reasoned that the terror and incompetence that I felt while salsa dancing was temporary and would disappear once I had years of dancing under my belt. In short, I expected that dancing would eventually become second-nature if I kept at it—and then if nothing else I would be in a better position to improve my social relationships.

I kept that commitment I made 5 years ago, and since then the longest I’ve gone without salsa dancing has been the 1–2 weeks during the Christmas break. The result? Not only has my hypothesis that salsa dancing would eventually feel natural been confirmed, the effects on the rest of my life have been nothing short of transformative. I have a healthy social circle of friends, I’m much less shy to the point I enjoy socializing, and I’m much more comfortable using my physical body for things, even starting other hobbies like weight lifting. And yes, I started getting attention from girls.

For me, salsa dancing was a vehicle for changing one of the things most resistant to change but probably the thing I most needed to change: my identity. Previously my identity was the nerd who excelled at school but struggled at social relationships and as long as I thought of myself in those terms my ability to form social relationships was severely crippled. I had prioritized academics to the exclusion of everything else, and I was proud of it.

What I came to learn was that while taking actions that are at odds with one’s identity feels incredibly awkward and painful, it is possible to rewire your identity with consistent effort applied over an extended period of time. This was far from easy, as my identity resisted the change at every step of the way, and I would often slip back into my older more comfortable identity that I had built up over two decades. It would happen reflexively: One Thursday when I had been dancing for over a year I remembered that I would be going out that night to dance. A sudden wave of fear swept over me as I realized that I was going to have to ask girls to dance that night—immediately followed by a wave of relief when I remembered that I did that every week.

Because I started from an identity of almost the polar opposite of a salsa dancer, my case essentially provides a lower-bound on the amount that it is possible to change. I’ve taught and assisted many people taking salsa classes and there are only a few I’ve seen that I would consider less skilled than myself when I started. Nowadays people don’t believe me when I tell them how much I struggled for years in ways which at the time felt scarring. As an example of how exceptionally incompetent I was when I started, my very first salsa social dance of my life was interrupted by a random bystander who asked if I was okay or needed help. 😂

In fact, I’m still struggling, because there is no end point you reach when you know it all. This is perhaps the biggest lesson I’ve learned from 5 years of dancing: the struggle itself is inherently meaningful, and I’ve learned to embrace the struggle as a purposeful and worthwhile part of learning and growing.


This is just a quick note to record a notable change in perspective I’ve had recently about initiating conversations. Previously, this is something I would not usually do, even around people I already knew. This lead to a lot of situations with awkward silence, for example if I was standing around with someone waiting for something.

My reasoning process was something along the lines of, “they must not want to talk, or otherwise they would say something”. Since I didn’t want to annoy them, I didn’t say anything. Recently, however, I have been going out of my way to start conversations in such situations. To my surprise, most people do not seem annoyed at all, in fact they seem genuinely pleased.

This stark contrast with my expectation and the reality makes me curious what was mistaken about my previous viewpoint, and how I came to adopt such a perspective in the first place. In retrospect, the obvious objection to my reasoning is that the other person could well be thinking the same thing as me. Perhaps they would even prefer to talk, but dislike the act of initiating; if one accepts this perspective then the initiator is even doing a favour by saying something—exactly the reverse of my previous view.

What could have caused me to get things backwards? It’s not as if there is a lot of conventional wisdom saying you shouldn’t initiate; the closest thing I can think of is feminists complaining about men overzealously initiating women (typically in a dating context). I have to wonder if this was a case where I didn’t want to do something so I rationalized a reason against doing it.

Leaving my head

The TED website hosts a large collection of inspiring talks on many different topics. The talk on how schools kill creativity is particularly fantastic; both funny and insightful, and is in fact the most-watched TED talk to date. During it Ken Robinson makes the following remark:

There’s something curious about professors in my experience — not all of them, but typically — they live in their heads.

Not to put too fine a point on it, this describes me perfectly. (Although I am a graduate student rather than a professor.)

This behaviour has both its strengths and weaknesses. For example, I have almost never felt lonely in my life, despite being alone a good portion of the time. Perhaps this is because by living in my head I am able to keep myself company. Indeed, my modus operandi when thinking about something could be described as a “conversation” with myself in which I try to make arguments for both sides of an issue.

However, like all human emotions, loneliness undoubtedly exists for a reason. In this case, it seems to be a trigger to make friends, which in turn improves one’s well-being. Without loneliness one has less short-term pain, but also less motivation to seek out ultimately beneficial connections.

In my case, I never gave much thought to the downsides, or even viewed them with a kind of perverse pride; the “cost of doing business” as it were. In short, I didn’t make much of an effort to ameliorate the absent-mindedness; in fact, I thought such traits were essentially unchangeable.

Early this year I decided to make a conscious effort to get out of my head. There were multiple reasons, but one is that almost every job I do involves interacting with other people in some way, so I figure that becoming more empathetic will actually improve the quality of my work.

How does one go about venturing out of their head? Well, I am still learning that. One step I’m taking is to learn—and remember!—the names of people I recognize from somewhere. I haven’t had a 100% success rate, but I’m already much better than I was previously, when I would be likely to forget someone’s name before my conversation with them had finished. Additionally, I will not ignore the fact that yes, I have a body, and to this end will pay more attention to things like exercise, diet, and appearance.

First post

So I took the plunge and decided to start up a blog, primarily as a way practising writing, something I’ve always enjoyed but never made a habit out of. I read a fair number of blogs, but to be honest I wouldn’t miss most of them if they shut down. I’ve found a few which consistently produce high-quality content; here are three of my favourites:

  • Paul Graham – A fantastic collection of essays. I like Graham’s writing style more than anyone else I’ve ever read, and don’t understand why the clear, matter-of-fact style that he employs isn’t very commonly used. Many of his essays are about entrepreneurship, which I am not especially interested in, yet I still find his writing engaging.
  • Scott Adams – Best known as the creator of Dilbert, his blog is a constant source of unique ideas. As a compulsive thinker who loves kicking around new ideas I look forward to reading his near-daily posts.
  • Eric Raymond – An open-source software advocate with an extremely strong ego. The unusual part about that is he actually has the accomplishments to back it up. He writes about a diverse number of things, and his writing is usually interesting even when I don’t care much about the topic.

My intention when starting this blog was to only write about things of interest to me. I had thought that this would necessitate being boring to everyone except in the case where a common interest is shared. Judging from the above list, in the optimal case that’s not actually true.