This one really stuck for me: "focus on something special, then expand from there".
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 agree with this advice once you get to a sufficiently high level -- like, working on research problems.
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.
Meh? I think a more mercenary approach to learning prerequisites is fine, and frankly better. Use Mathematica or Alpha to show you how equations get from one place to another - those tools are great at the fundamentals. You can skim a calc book, use Mathematica/Alpha to fill in your gaps, and play with toy problems to convince yourself well enough to continue.
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.
There are some people who can pull that off, and you might be one of them, but they're extreme cases. For the vast, vast majority of students it's not a viable approach.
I really resonate with this. I've also struggled with perfectionist mindset when it comes to learning, especially in mathematics. Recently, I tried to meticulously add math problems to Anki for better retention, only to realize the time cost was too high. This perfectionist mindset often maximizes one variable (retention in my case) at the cost of all others (time as the most important one).
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.
I have a different advice for aspiring mathematicians. Learn in the way you want. If you enjoy learning the prerequisites or feel like you need to, by all means learn it carefully. There are enough mathematicians on either side of this advice and indeed of most advice. Don't take other people's opinions too seriously. Other people only know what worked or didn't for themselves, they have no idea what will work for you.
> 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.
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.
Passions, feeling special, finding the special one. Do I dare suggest that identifying (or equating(!!!)) these are not just urgent issues for young mathematicians, but young people in general? Or maybe even every one?
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.
Not a Mathematician myself, but I totally agree with him regarding the benefits of choosing Mathematics as the career.
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."
If your father is who I think he is then his fame is large in the mathematics community. His Ph.D. advisor was a terrible teacher as far as my experience with taking classes from him goes.
"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."
It’s not that surprising to me. For academic publications, it’s generally only the publisher who makes a significant profit — and even then it’s probably negligible in most cases.
I’ve even been personally sent PDFs of books by the author under the condition that I don’t tell anyone.
Reminds me a bit of Feynman, and his advice on working "fun" topics that aren't necessarily high-impact or important, but feel interesting to you - as they increase your self-confidence and joy.
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.
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.
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