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1. Industry cares more about concrete results, quick execution, and bias for action.
2. Academia cares more about positive results, quantity of published papers, and small achievable experiments over big experiments that might fail.
Where are the institutions that care about deep understanding?
Earlier quoted context omitted.
OP here -- that is a fantastic anecdote! Do you have an example of a "drawing" of a theorem, in this context? (I've seen these for fairly trivial theorems but not for more complex ones, so I'm curious.)
As someone who very much relates to the GP’s anecdote, I might suggest determinants as a good example. As an undergraduate studying maths, I encountered a standard theorem in one of my first courses, which says that about 947 different conditions are equivalent to a matrix having a determinant of zero. I dutifully memorised these. I also dutifully memorised the algorithm for how to calculate a determinant. I might ev…
I worry that the way modern society is structured disincentivizes deep understanding. 1. Industry cares more about concrete results, quick execution, and bias for action. 2. Academia cares more about positive results, quantity of published papers, and small achievable experiments over big experiments that might fail. Where are the institutions that care about deep understanding?
hopefully we see more companies go in this direction
To work on math, you need time, a peaceful place to think, and motivation. Even then, you can't do this for everything, because there is too much. Obsessing on something that's not urgent when there's more important stuff to do may not be good time management, depending on your priorities and other claims on your time. But you might do it anyway, depending on your interests.
Also, learning some other subject well may be less about thinking by yourself and more about going out and talking to people, or playing a lot of games, or challenging yourself in some other way. All that takes time too.
But there is a lesson here: knowing something in more than one way means you know it better. I see this especially in music, where there are multiple ways to memorize a piece and they reinforce each other. Auditory memory (being able to hear it in your head), muscle memory, knowing the chords, knowing the lyrics, even remembering where it is on the page can all help.
Great post. Personal anecdote: I don't think I really understood anything in school, but I was decent at going through the motions of carrying out certain methods and recalling certain facts when I needed to. I went on to study Maths at university, and for most of my first year, I had the same surface level "methods + facts" knowledge that got me through school. After some studying, I could recite definitions and the…
Excellent point.
1. A similar example: Feynman diagrams.
2. Another: Venn diagrams.
3. Longer example: On Navy nuclear-powered aircraft carriers, the officer of the deck underway (OOD) must have at least a basic understanding of how the engineering plant works. It's second nature for nuclear-trained OODs, of course, but non-nukes could sometimes have trouble. Back in the day, it turned out that an effective way to help non-nukes learn what they needed to know was to have them: (A) memorize a really-simple block diagram of the reactor and steam system, and also (B) memorize a chant, of modest length, that summarized how things worked. During slow periods while standing OOD watch, I'd make a non-nuke OOD trainee draw the diagram from memory; then I'd quiz him with "what if ..." questions (back then it was always "him"). If he got hung up on a question, I'd tell him, "chant the chant." That usually helped him figure out the answer in short order.
(U.S. submarines don't have that problem, AFAIK, because pretty much every officer who will stand OOD watches is nuclear-trained.)
IQ being the biggest life changer.
Consciousness is considered second and more importantly is considered trainable.
> This quality of “not stopping at an unsatisfactory answer” deserves some examination. > This requires a lot of intrinsic motivation, because it’s so hard; so most people simply don’t do it. It also requires self-confidence, persuasiveness, and social power. Without these traits, your attempts to really understand something will be dismissed as "overthinking things" or "trying to understand the universe". Those arou…
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Often, trying (and potentially failing) to do a task is the best way to learn about it. The key is to be very explicit about what parts of that task you actually understand and which ones you're pulling out of your ass.
This is especially true when creating software. It's super rare to have requirements that are concrete and detailed enough to form a comprehensive understanding of the final design before you start developing it. Instead there are usually parts that have clarity and others that are fuzzy. If you can enumerate those and keep them separate you can often leverage the parts you understand to make progress on those you do not. Writing placeholder/obviously-terrible code to stand in for the unknown parts just so that you can spin up a running system is a great way to do that. Along the way you'll see what patterns emerge, where you hit walls, etc, which is not always easy to imagine with raw abstract thought. And having "working" software that you can play with is a great way to find edge cases and otherwise make progress on those unknowns. Once you've gained a more complete understanding you can replace the placeholder junk with well-designed/actually-thought-out modules.
(Another obvious reason to do this is if your company will literally go out of business if you wait until you have a perfect understanding to launch a product/service, but I think most people here get that.)
I'm not sure how much this generalizes, but it also works well for me when writing. I usually start with a vague understanding of an idea I hope to communicate, then jot down disjointed sentences to capture parts of it. As I do so it gradually becomes clear how things are connected, where my reasoning is muddy, what I thought I knew but can't express so probably don't, etc, and I can use this gained understanding to iteratively rewrite and reshape my message until it becomes something coherent. Sometimes, anyway; other times I don't end up sending/publishing it at all because along the way I learned that the thing I was hoping to communicate was based on a faulty assumption or is not as straightforward as I thought it was. Which is great, because either way I've learned something.
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I guess I'd say it differently: "keep thinking and do the task".