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.
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 every 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.
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!
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.
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.