Another “soft post”. It has been a pretty long time…
I’m not really sure how to get started here, so let me just start with an observation. As I’ve mentioned many times in my posts, I tend to be very nocturnal; although I sometimes manage to “normalize” my sleep schedule, it never lasts for more than a week. For example, it is currently 6:00 in the morning. When I have classes to attend, or other daytime responsibilities, I usually compensate by sleeping twice a day for half as long.
Fortunately, my plight is shared by a fair number of mathematics students (although admittedly it seems to be shared by even more computer science students). The result is that I spend a lot of time talking to those people (mostly undergrads in lower years) about mathematics. You all know who you are.
What I find ironic is that lower-year students seem to assume they’re wasting my time, or that I will find their thoughts and questions boring. This couldn’t be further from the truth; in fact I’ve found all such conversations quite intriguing and useful. I feel like I probably learn as much from these people as they do from me. For example, I kind of understand the implicit function theorem now! And it’s been almost 3 years since I took Calculus 3. Anyone who knows me will agree that unless I’m honestly rushing to finish some work, I’m always happy to think about a problem, or explain some concept.
The other thing I want to mention is the burden of correcting others when they make a false statement. I choose the word “burden” carefully, because it is usually a feat (especially for people like me, whose body language tends to be pretty arbitrary) to execute this favour without rendering the atmosphere uneasy, or provoking a defensive reaction. The smallest difference in the tone of your voice could make the difference between the person laughing and thanking you, or becoming irritated as if you had stood up and straight out called them an imbecile.
Alas, such is the untrained human ego. Over the years, I’ve grown completely accustomed to being flat out wrong. It’s unavoidable. Mathematics is done with such acute precision, and such calculated deliberation, that recalling everything correctly all the time is practically impossible, and trust me, there are diminishing returns on trying to achieve that kind of perfection. Often I’ll state a flawed version of something I remember reading in a book, or miss an edge case in a definition. The whole point is that there are usually enough people in the room to tell me I’m wrong.
If you were the only person left in the world, doing mathematics would be utterly pointless. As they say: “a proof in a vacuum is no proof at all.” Others are rendering you an enormous service by correcting your mistakes. Whenever the primal ego-goose inside you starts ruffling its feathers, just give it a swift slap of rationality, and remind yourself of this fact. The alternative is that people remain silent and thereby leave you to spoil in the cesspool of your own ignorance for that much longer. Is that really desirable?
One thing you need to understand about doing mathematics is that you are working in a veritable exosphere of abstraction. You are standing on a magnificent, skillfully crafted mountain, and should you ever choose to trek down to its base, you will discover that this mountain was merely rooted on yet another mountain, double its size, and so on and so forth. It’s way too difficult to remember how to prove every single fact on the “dependency chain” to where you are now. That’s like remembering each and every stone you saw on your way up the mountain. Try to remember as much as you can, but don’t forget there are plenty of pretty yellow photo albums of all those stones anyway, should you ever forget one of them.
Okay, I think that’s all I wanted to say. I’ll try to type Chapter 2 of Wei Xi’s quantum calculus saga soon. Until next time, cheers.
Also, it looks like representation theory got slightly nerfed due to the fact that we now only have one course for both group and ring theory: the material on the Sylow theorems and semidirect products has been moved to rep theory instead of in a group theory course where it actually belongs (and where I saw it). Sigh… less time for actual rep theory. The old system was better; upper-year courses are being watered down more and more.