4 ms·

A really fascinating corollary I've observed since I've gotten into more advanced maths, or even doing actual research as a PhD student, is that there's nothing

A really fascinating corollary I've observed since I've gotten into more advanced maths, or even doing actual research as a PhD student, is that there's nothing special going on at the higher levels. You're just working with different materials. Materials that require more time and effort to 'get', but once you get them they are just another tool at your disposal. I was similar to the author in that, throughout high school and undergrad, I presumed that the mind that could comprehend advanced math or do novel research (in any field) was truly unknowable. Like there was this x-factor they had that wasn't there for me. I've long enjoyed puzzle games (like The Witness or Stephen's Sausage Roll). It turns out that problem solving in non-trivial domains is never terribly different than problem solving in those games, or any other domain really. Like my brain isn't doing anything different than the usual tree-search algorithm that any chess player performs when they are projecting moves ahead into the future. Its just iterating on concepts that seem abstruse to most people. But at the end of the day, deep problem solving in math or AI research tends to be the same moving-shapes-around-in-my head that I would do if I was trying to move an awkwardly shaped couch through a narrow doorway.

dwaltrip 2y ago
That’s a really cool observation, thanks for sharing
almostgotcaught 2y ago
> Materials that require more time and effort to 'get' They really don't. Each rung in the ladder takes about as long to step through as the previous. The challenging/annoying/discouraging aspect is that a lot of research (especially in pure math) has so many rungs to step through. The truth is many of those aren't useful at all - distinctions invented purely for the sake of being able to say you invented something. Even when I was doing really "deep" research I could explain it to anyone by just dropping all the technical jargon and using goal oriented language instead. When I would do this most people would naturally try to reason about it in exactly the same way "researchers" would. The proof is they'd come up with the exact same solutions to the problems that were legit solutions, they had just already been discovered ie the low-hanging in fruit solutions.
atomicnature 2y ago
The central thing to me is - "quality of efforts" more than "quality of results". For anything sufficiently complicated or "difficult" - one needs to persevere. There's nothing magically special in any place, like a university, except perhaps a few understanding people around to egg you on (that can make a big difference in the right conditions). But end of the day, it is the individual's effort that matters. Effort can make up for many, many deficiencies. More quality effort held steady over a long-enough time should produce at least tolerable results, if not exceptional results.
atulatul 2y ago
But at higher you also learn how to solve problems. Suppose you see a problem which is quite unfamiliar, you have tools and you know how to use tools- combine tools, separate those and use for separate parts, invent tools, etc.
- otteromkram 2y ago
  This is a good argument for, "why do I need to learn math/science? I'll never use it." Maybe you won't need to factor polynomials daily, but knowing how to solve that kind of problem can be applied to other scenarios outside of Algebra II. Great post!
bonoboTP 2y ago
To see the brilliance of advanced ideas, you need to be quite advanced yourself. What you wrote is analogous to saying that playing football in the Premier League is nothing special compared to me and my weekend buddies playing in the yard, it's still human beings kicking a ball, just with better coordination and strategy. Yeah. Coming up with new math at a research level is also just manipulating conceptual objects in your mind. Except this insight doesn't make the average person capable of either. For some reason people have a big hangup around admitting that some people are more talented at abstract thought and advanced math than others, even though this is absolutely obvious even to a schoolkid. The existence of the Premier league in no way diminishes the value of kicking the ball with friends over the weekend. Math can be fun in the same way, but not everyone will reach the same levels.
- captainclam 2y ago
  Hmmm...this wasn't at all my point, but I understand why I my comment could be read this way (the failure to communicate is mine). I am not suggesting an equivalency between all efforts at all levels, or that innate talent doesn't exist. I was not at all speculating about the nature of preternatural talent. The expertise-gap I was invoking was more like early undergrad to late grad school levels, not pick-up football to Premier League. The concept I'm getting at is that of a mental block I have observed in myself, and I suspect resides in others, which is really subtle but ultimately quite limiting. I'd have to think harder than I want to at the moment if I were to try to articulate it more clearly, but I do want to be clear that my comment wasn't about innate talent.
  - bonoboTP 2y ago
    Yes, maybe I have experienced something similar in my research area in machine learning. Fundamentally, the best people of the best labs use the same tools in roughly the same ways as I do, there's no set of magical secret tools on that tier. They also aren't really using any math I don't know about. They just comprehend papers quite fast, understand what's relevant and what is not with great intuition, have a lot of prior work in their memory, have gone through lots of projects so they know how to approach the new problem, they are aware of what actually are interesting open questions, they have refined "research taste", (and of course stuff like understanding the academic system and how to make the most of it etc). But the day to day activity is not fundamentally different than for lower tier researchers. They just have better ideas, better overview, and better execution. When they explain it all in hindsight it appears simple and as if you could have done it too. And of course they also run up against bugs and hardware issues and big messes but often just plow through and grind it out because they can see the light at the end of the tunnel well. It's kind of a bummer in a way. Life is just less magical than we imagine. When I was much poorer as a kid than now as an adult, I used to imagine that being rich must be so different. But actually people are just people, there is no discontinuity anywhere. I've flown business class and been to fancy dinners, it's not fundamentally different from taking a bus trip or throwing a student party at the dorm. The people talk about somewhat different subjects, or are more polite, but overall it's nothing that a poor person wouldnt be able to comprehend. It also reminds me how there's no "arrival" in life. There's alsays a next thing to do. But for example as a student you think graduation is "making it" but then you realize it continues in somewhat different form, but it's now about acing job interviews and you think landing a job will be the goal line, but then you see it goes on and now you chase promotions and good evaluations and so on. Similar things about private life.

dwaltrip 2y ago

That’s a really cool observation, thanks for sharing

almostgotcaught 2y ago

> Materials that require more time and effort to 'get' They really don't. Each rung in the ladder takes about as long to step through as the previous. The challenging/annoying/discouraging aspect is that a lot of research (especially in pure math) has so many rungs to step through. The truth is many of those aren't useful at all - distinctions invented purely for the sake of being able to say you invented something. Even when I was doing really "deep" research I could explain it to anyone by just dropping all the technical jargon and using goal oriented language instead. When I would do this most people would naturally try to reason about it in exactly the same way "researchers" would. The proof is they'd come up with the exact same solutions to the problems that were legit solutions, they had just already been discovered ie the low-hanging in fruit solutions.

atomicnature 2y ago

The central thing to me is - "quality of efforts" more than "quality of results". For anything sufficiently complicated or "difficult" - one needs to persevere. There's nothing magically special in any place, like a university, except perhaps a few understanding people around to egg you on (that can make a big difference in the right conditions). But end of the day, it is the individual's effort that matters. Effort can make up for many, many deficiencies. More quality effort held steady over a long-enough time should produce at least tolerable results, if not exceptional results.

atulatul 2y ago

But at higher you also learn how to solve problems. Suppose you see a problem which is quite unfamiliar, you have tools and you know how to use tools- combine tools, separate those and use for separate parts, invent tools, etc.

otteromkram 2y ago
This is a good argument for, "why do I need to learn math/science? I'll never use it." Maybe you won't need to factor polynomials daily, but knowing how to solve that kind of problem can be applied to other scenarios outside of Algebra II. Great post!

bonoboTP 2y ago

To see the brilliance of advanced ideas, you need to be quite advanced yourself. What you wrote is analogous to saying that playing football in the Premier League is nothing special compared to me and my weekend buddies playing in the yard, it's still human beings kicking a ball, just with better coordination and strategy. Yeah. Coming up with new math at a research level is also just manipulating conceptual objects in your mind. Except this insight doesn't make the average person capable of either. For some reason people have a big hangup around admitting that some people are more talented at abstract thought and advanced math than others, even though this is absolutely obvious even to a schoolkid. The existence of the Premier league in no way diminishes the value of kicking the ball with friends over the weekend. Math can be fun in the same way, but not everyone will reach the same levels.

captainclam 2y ago
Hmmm...this wasn't at all my point, but I understand why I my comment could be read this way (the failure to communicate is mine). I am not suggesting an equivalency between all efforts at all levels, or that innate talent doesn't exist. I was not at all speculating about the nature of preternatural talent. The expertise-gap I was invoking was more like early undergrad to late grad school levels, not pick-up football to Premier League. The concept I'm getting at is that of a mental block I have observed in myself, and I suspect resides in others, which is really subtle but ultimately quite limiting. I'd have to think harder than I want to at the moment if I were to try to articulate it more clearly, but I do want to be clear that my comment wasn't about innate talent.
- bonoboTP 2y ago
  Yes, maybe I have experienced something similar in my research area in machine learning. Fundamentally, the best people of the best labs use the same tools in roughly the same ways as I do, there's no set of magical secret tools on that tier. They also aren't really using any math I don't know about. They just comprehend papers quite fast, understand what's relevant and what is not with great intuition, have a lot of prior work in their memory, have gone through lots of projects so they know how to approach the new problem, they are aware of what actually are interesting open questions, they have refined "research taste", (and of course stuff like understanding the academic system and how to make the most of it etc). But the day to day activity is not fundamentally different than for lower tier researchers. They just have better ideas, better overview, and better execution. When they explain it all in hindsight it appears simple and as if you could have done it too. And of course they also run up against bugs and hardware issues and big messes but often just plow through and grind it out because they can see the light at the end of the tunnel well. It's kind of a bummer in a way. Life is just less magical than we imagine. When I was much poorer as a kid than now as an adult, I used to imagine that being rich must be so different. But actually people are just people, there is no discontinuity anywhere. I've flown business class and been to fancy dinners, it's not fundamentally different from taking a bus trip or throwing a student party at the dorm. The people talk about somewhat different subjects, or are more polite, but overall it's nothing that a poor person wouldnt be able to comprehend. It also reminds me how there's no "arrival" in life. There's alsays a next thing to do. But for example as a student you think graduation is "making it" but then you realize it continues in somewhat different form, but it's now about acing job interviews and you think landing a job will be the goal line, but then you see it goes on and now you chase promotions and good evaluations and so on. Similar things about private life.