The Power of Admitting Ignorance
Upon re-reading Mathematical Apocrypha, a collection of tales dealing with great mathematicians as well as their quirks and foibles, I thought about my own interactions with some of The Elders of Mathematics (from here on abbreviated as THEM) and what I would add to such a book. While I have a few interesting and potentially funny anecdotes to tell, one story stands out in my mind, because it had an extraordinarily positive impact on my life.
The first semester
When I started studying mathematics, I fell in with the wrong crowd, one could say. Most of them had received additional tutoring prior to even starting their studies1, they knew all the things about eldritch lore such as complex numbers2 and vector spaces. Most of them, through no fault of their own, came from very affluent and academic households, whereas my own origins are more down-to-earth and thrifty. So, to make a long story short, I was in awe of them: they seemed to know everything about mathematics already and they never confessed to misunderstand what we had been taught in lectures. In fact, it was almost a badge of honour to understand things much more quickly than the others.
Surviving the qualifying examinations
Looking back with some wisdom now, I groan inwardly at this sorry way of attempting to assert dominance. It certainly worked for me, though, as I was feeling increasingly out of place. Understanding all concepts took a sizeable amount of time and intellectual effort for me, and I was not in the-‘I-immediately-get-everything-when-being-exposed-to-it-for-the-first time-ever’ frame of mind. So, in short, I was miserable, but I continued to study, wondering sometimes whether I really was that much slower in comparison. Then the exams came—and much to my surprise, I did well. I was spurred on by this little victory and perplexed to discover that not all members of my ‘study group’ made it that far. They had failed this qualifying examination despite understanding everything3!
Topology with a cello and belt
I was happy to see the second term, though, and still hung out with this group at times. We were reduced in numbers, but the general tone was one of ‘smarter-than-thou’ still and not changing. Yet, things did change, mostly because of one course: linear algebra 2. It was given by Prof. Matthias Kreck, who was also the teacher of the first instalment of the course. But Prof. Kreck decided to go ‘off-script’ and started quickly veering off into the realms of algebraic and differential topology, trying to give us undergraduates a glimpse of these fascinating subjects. Suddenly, my group was in disarray: the subject was wrong, this was not supposed to be taught, this was not part of the script, and so on. Apparently, their tutoring had not prepared them for this!
For me, the subject was breath of fresh air: Prof. Kreck has a very idiosyncratic teaching style: at one point, he brought his cello into the lecture hall to play a song; in another lecture, he took of his leather belt, made it into a Möbius loop, and started using chalk to draw little arrows on it—this was all done to empirically show that this particular object cannot be oriented. I loved these glimpses of topology, and I started to be interested in the whole topic—I even started reading current research papers, and tried to understand them4. The good thing is that I also did not hang out with my previous study group any more, and started spending more time with cool people I met5.
A crazy plan
At some point, my ‘research’ made me stumble on the work by Grigori Perelman on the Poincaré Conjecture. This was my first contact with THEM. Thanks to the arXiv, I was able to browse away and discovered a treasure trove of papers. At some point, I even discovered a paper by none other than Terence Tao, who was providing a non-linear PDE perspective on the proof of Perelman.
Wading through all that literature was daunting—I had to look up almost every other concept (Ricci flow is not part of the standard undergraduate curriculum if you are in your second semester), but I was determined to go through with this, and a crazy plan formed: I wanted to fully understand the proof of the conjecture! But I needed more tools for that. So, at some point, I cornered Prof. Kreck after a lecture. He was always happy to answer questions of the students, and even though I found it daunting to be in the presence of THEM, he radiated a calm and almost beatific aura that made you feel at ease.
I screwed up my courage, approached the great sage, and asked him something along the lines of ‘Prof. Kreck, I really would love to understand the proof of the Poincaré conjecture. Do you have any tips for me?’ He looked at me calmly and replied something like ‘Not really, but best of luck to you; maybe you can it explain it to me once you understand it.’
I was absolutely flummoxed! Here is a member of THEM, confessing his own ignorance of a subject! Prof. Kreck expanded a little bit on this answer and basically explained to me that it might takes years of studying to finally grasp all the nuances of the proof and, since it was not directly within his realm of expertise in topology, he was just as clueless about certain concepts than I was. The major difference being that he was more experienced at feeling clueless, and knew more concepts for addressing this feeling.
Similar to the novice in many of the Zen koans, I truly was enlightened in this moment. If one of THEM can express ignorance about a topic, surely, ignorance is not that bad to begin with. I realised that the power of truly mastering a subject lies in realising that you do not necessarily understand everything—and being honest about it! My life was changed6, and I was gently steered towards being more intellectually honest.
There is a power in being as honest and outspoken as Prof. Kreck was. Here is this proficient and prolific member of THEM, and he could have just made up something on the spot to make me feel dumb. Instead, he chose the intellectually honest option, and made it clear that this is the normal state of affairs in mathematics (or any sufficiently complicated topic). I relish the fact that such a small action could have such a profound impact on one person, and I am grateful that I dared pose my question.
In the years since, in my own dealings with researchers, I never once feigned knowledge when I was not feeling sufficiently confident about it. I think it is important to be honest about what you know and what you do not know. Ignorance is not a moral blemish—pretending to be smarter than you are is (just as choosing to remain in a state of ignorance is).
So the moral of this story is: do not be afraid of not knowing or not understanding something. It even happens to THEM.
Whereas I did not even know that such a thing was possible, and started studies woefully under-prepared—or so I thought. ↩︎
It was the first term for me, and while I had heard about this in high school, it was being talked about in hushed voices, similar to how one would talk about certain things of an immoral nature. The fact that complex numbers somehow arose in equations that did not have a ‘real’ solution, made them all the more advanced to me at that time. ↩︎
It would only later dawn on me that they were probably just full of manure the whole time, but this wisdom was acquired over the years and not accessible at the time that I needed it. ↩︎
By that time, I was so accustomed to being ‘in over my head’ that there was really nothing that could knock me off balance any more. It was a mild surprise to see that I could follow along with some papers for at least a few paragraphs, or pages, until I started to drown in unfamiliar terminology and jargon. ↩︎
You know who you are! ↩︎
Again, I only properly realised this much later, but this certainly was a change point in my career! ↩︎