Advice to Young Mathematicians [video]
youtube.com
youtube.com
I have time and again fell victim to fulfill all the prerequisites before I begin to attempt to understand a topic. This is a mistake I have made repeatedly. I now understand why this is tempting to do and why it is a mistake.
It is tempting to do so because you feel things will come easier to you if you fulfill the prerequisites first. But the problem is that there is just not enough time. AND, it actually may not be even necessary.
It is a mistake to do so because you are wasting time and ultimately it may not be necessary after all.
Even in a field such as pure mathematics (I have an MS in pure Math), it is okay to skim through some of the background material and understand it intuitively or even non-rigorously, while focusing on what you want to actually learn.
It took me a while to learn that and I am glad it is being repeated here by such an accomplished professor.
I think this is a lesson I am learning. I am going through the book PRML throroughly (which means attempting every single exercise) and writing out the solutions.
I also did the prereqs somewhat rigorously.
It has been incredibly time consuming.
I've grown to really like this Richard Feynman quote: “Study hard what interests you the most in the most undisciplined, irreverent and original manner possible.” It feels like I've been finally granted permission to embrace my natural, messy way of learning.
HOWEVER I think there's more nuance: there is tremendous value in having a baseline level of foundational knowledge in whatever domain you're in.
For instance, if someone hasn't learned math to an undergraduate level, and they try to sink their teeth into a math research problem, then they're probably going to spend all their time flailing around and being confused. Waste of time.
A better use of time would be for them to develop a baseline level of mathematical knowledge through a structured curriculum, and then take the leap once they've got their fundamentals down.
Same thing with machine learning. If you want to work on a research problem and you have your ML fundamentals down, then yeah, go ahead and jump right into the research problem. But if you don't even know how gradient descent works, or what an eigenvalue is, how to work with continuous probability distributions, etc., then you're woefully underprepared and you'd be better off just shoring up your foundations first.
The basics are the ones with the most different approaches developed to teach/use them, so they can most easily be skimmed by with cheap modern trickery.
I was a math major and I used to be adamant about reading the chapter perfectly before starting my problem sets.
I mentioned this to my math professor and he’d always say to not be afraid to jump into the problems, that my time was better spent that way.
My take on it is that a) it’s true and b) it works because reading with a purpose is very different than reading as a way of surveying something before you jump into a problem.
My father is a Mathematician with small fame. He was obsessed with Mathematics when he was a kid. In fact, Mathematics was the single thing that helped him to move through the political turmoil during his first 30 years of life.
I'll also quote a paragraph from Cixin Liu's "Ball lighting". It's related.
> "Of course I know!" Dad took another half glass of wine, then turned to me, "Actually, son, living a wonderful life is not difficult. Listen to Dad: choose a universally recognized world problem, preferably a mathematical one that only requires a piece of paper and a pencil, like the Goldbach Conjecture or Fermat's Last Theorem, or a pure natural philosophy problem that doesn't even need paper and pencil, like the origin of the universe. Dedicate yourself entirely to it, focus solely on the process, not the outcome. In the unconscious focus, a lifetime will pass. What people often refer to as a 'pursuit' is just this. Or, conversely, make earning money your sole goal, always thinking about how to make money without considering what to do with it. By the time you die, you can, like Grandet, hold a pile of gold coins and say, 'Ah, so warm...' Therefore, the key to a wonderful life is what you can become passionate about. For example, me—" Dad pointed to the small watercolor paintings placed all around the room. They were all very traditional in technique, well-painted but lacking any real inspiration. The paintings reflected the electric light from outside the window, like a group of flickering screens, "I became passionate about painting, even though I know I can't become Van Gogh."
But his early experience did reflect who you though he is. They also work in the same sub domain.
https://en.wikipedia.org/wiki/Michel_Talagrand
A surprising tip from his personal page.
"If you are desperate to get my books and your library can't afford them, try to type the words "library genesis" in a search engine. I disagree with piracy, but this site saved me many trips to the library, which unfortunately does not carry electronic versions of older books."
I’ve even been personally sent PDFs of books by the author under the condition that I don’t tell anyone.
The example he used from his own work escapes me, but he had spent years working on topics quite unrelated to quantum mechanics, just out of pure interest and curiosity.