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> “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it t

> “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.” This was interesting to me. My math profs always admonished us to ensure foundations are completely watertight before advancing to the next thing in tiny increments. I've absorbed this to the point where I perhaps get stuck filling in inconsequential gaps at roughly the same level, kitting out base camp as fully as possible but postponing the ascent. I have no dreams of becoming a professional mathematician, but maybe I'd have quicker insights and more creative ideas if I tried this approach too, tackling something impossible and working backwards.

Chris2048 9y ago
I think the trick is figuring when underlying details are just details, and when you miss an important understanding in the underlying foundations that might spawn misunderstandings. I guess a good lecturer is one who can tell you - insisting on learning everything is inefficient, although less work for your lecturer..
kefka 9y ago
> My math profs always admonished us to ensure foundations are completely watertight before advancing to the next thing in tiny increments. Of course someone who gets paid to teach would highly stress learning tidbits slowly and excruciatingly. That's their economic incentive. Schools also stress learning detritus for "learning's sake", even if the very people who teach it can't properly explain what it's actually for. I also have learned compsci with his similar methods of finding interesting areas and digging in. I know my programming knowledge has holes, and I fill them in as I come to them. I like to know how things fit together, even if they are cross-domain and seemingly disparate. I come on in, and go "see these two areas are pretty similar, let me show you what can be done". And then I look like a miracle worker, because I see the generalities. Frankly, professors would be more useful to me, if I could purchase their time by the hour over issues I don't understand. I can teach myself most things. Sometimes, a professional helps with the jump-start to get a good grasp.
- pklausler 9y ago
  How do you identify the "holes" when you reach them? How do you even know whether you've reached one? "Aha! This is clearly a situation in which a monad would be the best approach. Time to go learn about monads!" Just doesn't seem like a reasonable method to me. Some things you just to to learn well before you can even recognize when you need to remember them later.
  - kefka 9y ago
    That's not a valid criticism if you have intent and will to learn. As for your monads example, getting into functional programming via things like CLisp, Erlang, Haskell, and the like will expose you to lambda calc pretty darn quick. And learning how monads work is near the beginning of that path. And just being inquisitive leads to a whole lot of areas that give indications on what to learn. For example, doing computer vision enforces you to learn how linear algebra works. Machine learning teaches a great deal of how statistics works. Finite State Machines have their own really interesting niches to work with. Working a crummy operator job teaches how to do automation (on the sly!). It really depends on how you approach learning. If you're just slowly grinding away because you have to, going through a 4 year BS degree is probably better.
    - pklausler 9y ago
      Not all knowledge sits on the frontier of yours.
      - kefka 9y ago
        What's that supposed to mean? If one wants to learn anything in the sciences, we things like MIT Courseware, Arxiv, Libgen, SciHub, and "canihaspdf" on twitter. Yes, these are primarily pirate options - so what? I can publicly see the course projections for any arbitrary degree, along with class titles. And many have book lists linked, so I can hunt for the books online using less legal methods. The only difficulty with some STEM learning paths is they require laboratories - those are hard/impossible to do at home and thusly necessitate academic environments. Computing, on the other hand, is easy to learn even at a Starbucks with a laptop and a phone. What's stopping people from learning what they wish is primarily time and the will to (and the fact that school does a great job at beating the will to learn out).
ethan_g 9y ago
The thing about this approach--where you learn by assimilation rather than structured study--is that it you need to have amazing intuition for it to work. One of the benefits of structured study is gradually building intuition of the definitions and theorems. For someone of Scholze's caliber, the intuition is already there before any study. Structured study of linear algebra probably wouldn't have done much for him other than assigns names to theorems and definitions that he already intuitively understands. I might make an analogy to studying music: for the majority of people, it takes a lot of structured study to develop a good ear (i.e. being able to write down melodies and harmony after hearing it). For example, you'll study intervals, chords, and inversions, and extensive practice identifying them on hearing--just as you learn theorems and definitions in math class and do problem sets to practice applying them. But some people innately have a very good ear (e.g. perfect pitch) and don't need a course to teach them to identify intervals and chords. Even though they might not yet know the names of chords and intervals, they already "understand" them.

Chris2048 9y ago

I think the trick is figuring when underlying details are just details, and when you miss an important understanding in the underlying foundations that might spawn misunderstandings. I guess a good lecturer is one who can tell you - insisting on learning everything is inefficient, although less work for your lecturer..

kefka 9y ago

> My math profs always admonished us to ensure foundations are completely watertight before advancing to the next thing in tiny increments. Of course someone who gets paid to teach would highly stress learning tidbits slowly and excruciatingly. That's their economic incentive. Schools also stress learning detritus for "learning's sake", even if the very people who teach it can't properly explain what it's actually for. I also have learned compsci with his similar methods of finding interesting areas and digging in. I know my programming knowledge has holes, and I fill them in as I come to them. I like to know how things fit together, even if they are cross-domain and seemingly disparate. I come on in, and go "see these two areas are pretty similar, let me show you what can be done". And then I look like a miracle worker, because I see the generalities. Frankly, professors would be more useful to me, if I could purchase their time by the hour over issues I don't understand. I can teach myself most things. Sometimes, a professional helps with the jump-start to get a good grasp.

pklausler 9y ago
How do you identify the "holes" when you reach them? How do you even know whether you've reached one? "Aha! This is clearly a situation in which a monad would be the best approach. Time to go learn about monads!" Just doesn't seem like a reasonable method to me. Some things you just to to learn well before you can even recognize when you need to remember them later.
- kefka 9y ago
  That's not a valid criticism if you have intent and will to learn. As for your monads example, getting into functional programming via things like CLisp, Erlang, Haskell, and the like will expose you to lambda calc pretty darn quick. And learning how monads work is near the beginning of that path. And just being inquisitive leads to a whole lot of areas that give indications on what to learn. For example, doing computer vision enforces you to learn how linear algebra works. Machine learning teaches a great deal of how statistics works. Finite State Machines have their own really interesting niches to work with. Working a crummy operator job teaches how to do automation (on the sly!). It really depends on how you approach learning. If you're just slowly grinding away because you have to, going through a 4 year BS degree is probably better.
  - pklausler 9y ago
    Not all knowledge sits on the frontier of yours.
    - kefka 9y ago
      What's that supposed to mean? If one wants to learn anything in the sciences, we things like MIT Courseware, Arxiv, Libgen, SciHub, and "canihaspdf" on twitter. Yes, these are primarily pirate options - so what? I can publicly see the course projections for any arbitrary degree, along with class titles. And many have book lists linked, so I can hunt for the books online using less legal methods. The only difficulty with some STEM learning paths is they require laboratories - those are hard/impossible to do at home and thusly necessitate academic environments. Computing, on the other hand, is easy to learn even at a Starbucks with a laptop and a phone. What's stopping people from learning what they wish is primarily time and the will to (and the fact that school does a great job at beating the will to learn out).

ethan_g 9y ago

The thing about this approach--where you learn by assimilation rather than structured study--is that it you need to have amazing intuition for it to work. One of the benefits of structured study is gradually building intuition of the definitions and theorems. For someone of Scholze's caliber, the intuition is already there before any study. Structured study of linear algebra probably wouldn't have done much for him other than assigns names to theorems and definitions that he already intuitively understands. I might make an analogy to studying music: for the majority of people, it takes a lot of structured study to develop a good ear (i.e. being able to write down melodies and harmony after hearing it). For example, you'll study intervals, chords, and inversions, and extensive practice identifying them on hearing--just as you learn theorems and definitions in math class and do problem sets to practice applying them. But some people innately have a very good ear (e.g. perfect pitch) and don't need a course to teach them to identify intervals and chords. Even though they might not yet know the names of chords and intervals, they already "understand" them.