“At the time, I was infuriated by these inadequate proofs, but I was under time pressure to just learn the operations so that I could answer exam questions because the class needed to move onto the next thing.
And since you actually can answer the exam questions and mechanically perform calculus operations without ever deeply understanding calculus, it’s much easier to just get by and do the exam without really questioning the concepts deeply -- which is in fact what happens for most people. (See my essay on education.)
How many people actually go back and try and understand this, or other such topics, in a deeper way? Very few. Moreover, the ‘meta’ lesson is: don’t question it too deeply, you’ll fall behind. Just learn the algorithm, plug in the numbers, and pass your exams. Speed is of the essence. In this way, school kills the “will to understanding” in people. “
Great insight. I’ve said similar before and people don’t often understand how there could be a difference between acing a Calc III exam and having a conceptual understanding of it. Many people understand math only through the lens of rote algorithms and techniques; some people understand other subjects through this perspective as well.
Even if you try to understand them , in todays real world you will be lagging behind in the workplace and you would be stuck like the others. Also when money is involved you would be unable to pay for anything
Back when I was studying STEM subjects for my A-Levels, this mindset of not accepting anything until I've understood it was really rewarding.
But I went on to study law and jesus fucking christ did I find myself in a rabbit-hole from which I have yet to emerge. I couldn't even convince myself that central governments and legislatures were necessary for order, although I probably should've satisfied myself with Hobbes' argument and moved on. I thought legal concepts were "transcendental nonsense". Reasoning from precedent seemed extremely shoddy. Judges seemed all too human, value judgments abound disguised in "legal reasoning". There were hardly any "first principles" one could speak of. These were, in short, massive obstacles I had to overcome (or bracket off) before I could justifiably move on to actual doctrinal study.
But a deep understanding of law really requires understanding so many things, such that I could never reach that point where I could say, "Okay, I've got it, now let's use this groundwork to start thinking about legal doctrine". Examples: What even is law? Legal philosophy. Indeterminacy of language and interpretation? Philosophy of language. Distinguishing right and wrong? Moral philosophy, then moral psychology once it became clear to me that morals have to do with groups and emotions. How about understanding society, the realm of law's application and origin? Sociology/Social theory. Not to mention the huge body of "postmodern" work that has grown up around it which sees everything as a social construct, an effect of discourse/power, humans themselves as lacking the kind of agency assumed in law and liberal political philosophy.
And despite everything I have written, I don't feel like I've gotten a grasp on things. (Social) reality is such a massive, incomprehensible beast where everything is chaotically intertwined, arguably unlike mathematics where you can advance your understanding in little, isolated chunks at a time, with reasonable confidence that you've made sense of it. Some days I feel like I've wasted all my energy trying to analyse all these underlying dynamics of law and society, that I've entered a field where the "will to know" never pays off (you never see the full picture) and simply rips your common sense worldview apart without offering anything in return.
Thanks for the essay, it allowed me to explore my thoughts in the preceding paragraphs :)
Thank you! I think it's worthwhile for both -- some of the very best non-fiction I've read has unforgettable specifics (see John McPhee's book on Oranges, for example...)
That said, non-fiction often deals in abstractions and concepts, where this is maybe less important overall.
My wife says she was the top of her class for Calculus and 10 years later couldn't tell you a single thing about it. She excelled at school because she was good at memorizing, not because she understood things well -- she had good hardware, but never learned to load in the software she really needed to be an expert. I guess it doesn't matter. Her field doesn't require Calculus. Makes me wonder if we're wasting people's time on subjects that don't actually matter instead of doubling down on the ones that will for them.
Truly fantastic read. I feel like it has put into words many thoughts I’ve had for so long.
I would like to add that, even though I agree that first-hand experience is an amazing vehicle for understanding, it is inherently limited and should be taken with a grain of salt. Besides the concern that human senses can fool us, I think it is even more concerning that our own human experience is very limited in size, speeds, and time scales. We humans are unable to experience first-hand the very small, the very fast, and processes that take several millions of years to happen. Hence this empirical-data approach can sometimes limit us to exclusively think about concepts that are inherently tied to the human experience.
I would imagine that a being that is several orders of magnitude larger than us and lives over many millions of years instead of mere decades will be inclined to think about physical systems that operate on these scales.
> Moreover, I have noticed that these ‘hardware’ traits vary greatly in the smartest people I know -- some are remarkably quick thinkers, calculators, readers, whereas others are ‘slow’. The software traits, though, they all have in common -- and can, with effort, be learned.
On the other hand, natural selection being what it is, if you’re not quick enough, you may be deemed unworthy of the opportunity to spend so much time learning something. Social norms or practical needs limiting your study time are effective mechanisms to defeat any intellectual ambitions in people with insufficiently powerful hardware. They can be combined with bullying to increase the need to work on one’s physical prowess—if one is even allowed to do so—further extinguishing any hope of devoting substantial time to intellectual pursuits in anyone but the strongest. Should this be promoted? I think the Spartans were certain that it should.
> It’s also so easy to think that you understand something, when you actually don’t.
It’s also easy for higher-status people to further lower your status by denying that you understand something when you actually do. They just have to not respect you and to not listen to you, and the audience will follow their lead. And they’re likely to be very good at these games because they’ve spent about as much time perfecting their skills as you have studying your favorite subjects.
> Related to this is honesty, or integrity: a sort of compulsive unwillingness, or inability, to lie to yourself.
This will get you into a lot of fights when you refuse to swallow other people’s bullshit answers, not realizing they mean, “Shut up. I can beat you up; don’t make me get my hands dirty”.
> Malcolm Gladwell on his father:
[...]
>> So if he doesn’t understand something, he just asks you. He doesn’t care if he sounds foolish. He will ask the most obvious question without any sort of concern about it...
If you’re not high status enough, _you_ may still not care whether you sound or look foolish, but other people won’t hesitate to kill two birds with one stone, saving effort and further raising their status at your expense by answering with something obvious that doesn’t address your doubt, or by outright shaming you for not already knowing the answer or for caring about it.
>> ‘I don’t understand. Explain that to me.’
Indeed, you’d better be in a position where you can afford to talk to them in the imperative like that.
> Most people are not willing to do this -- looking stupid takes courage, and sometimes it’s easier to just let things slide.
If you’re aware of the social dynamics at play, it takes courage for good reasons, and often there’s simply nothing to be gained by asking questions nobody else is more interested in than in the status games. If you’re socially clueless, it doesn’t take any courage—you just blindly cross the line and suffer the consequences without understanding what’s going on.
> It turns out that my misgivings were right, and that the Leibniz notation is basically just a convenient shorthand and that you more or less can treat those things “as if” they are fractions, but the proof is super complicated etc.
In other words, you just needed a chance to take a look at a slightly more advanced calculus textbook. But, collectively, we’re more interested in social games like denying youngsters such opportunities and shaming parents who encourage those interests as child abusers.
> It is also good advice for understanding things. When in doubt, go closer.
Ho, ho, only to be told you’re missing the forest for the trees.
I use a similar process when I try to sort through my semantic explorations - trying to understand words and their meanings, the conceptual frameworks built around them and the implicit aspects people include when they use the word.
This kind of nitty-gritty exploration always requires the kind of tenacity that you describe. What does this mean? How about that? And that? You keep going until you know it inside out and there are no loose ends and outstanding questions.
It takes ages and can get very abstract but it really sorts it out when you get to a satisfactory end.
>understanding is not a binary “yes/no”. It has layers of depth.
Amen.
Case in point: I first "fell in love" with optimization as an undergrad decades ago and went on to get a PhD in it. Fast forward to *last week*: I understood something new about linear programming that I wasn't aware of all this time.
On the one hand, it is a bit embarrassing that I had not understood this earlier :). On the other hand, it is a beautiful example of the fractal nature of knowledge.
p.s. Loved the article! The only thing I would add: since we don't have the time to reason up from facts and first principles for *every* thing, we need to "pick our battles" carefully.
“At the time, I was infuriated by these inadequate proofs, but I was under time pressure to just learn the operations so that I could answer exam questions because the class needed to move onto the next thing.
And since you actually can answer the exam questions and mechanically perform calculus operations without ever deeply understanding calculus, it’s much easier to just get by and do the exam without really questioning the concepts deeply -- which is in fact what happens for most people. (See my essay on education.)
How many people actually go back and try and understand this, or other such topics, in a deeper way? Very few. Moreover, the ‘meta’ lesson is: don’t question it too deeply, you’ll fall behind. Just learn the algorithm, plug in the numbers, and pass your exams. Speed is of the essence. In this way, school kills the “will to understanding” in people. “
Great insight. I’ve said similar before and people don’t often understand how there could be a difference between acing a Calc III exam and having a conceptual understanding of it. Many people understand math only through the lens of rote algorithms and techniques; some people understand other subjects through this perspective as well.
Even if you try to understand them , in todays real world you will be lagging behind in the workplace and you would be stuck like the others. Also when money is involved you would be unable to pay for anything
Sometimes, I too feel that we tend to skip a lot of things due to time constraints.
Back when I was studying STEM subjects for my A-Levels, this mindset of not accepting anything until I've understood it was really rewarding.
But I went on to study law and jesus fucking christ did I find myself in a rabbit-hole from which I have yet to emerge. I couldn't even convince myself that central governments and legislatures were necessary for order, although I probably should've satisfied myself with Hobbes' argument and moved on. I thought legal concepts were "transcendental nonsense". Reasoning from precedent seemed extremely shoddy. Judges seemed all too human, value judgments abound disguised in "legal reasoning". There were hardly any "first principles" one could speak of. These were, in short, massive obstacles I had to overcome (or bracket off) before I could justifiably move on to actual doctrinal study.
But a deep understanding of law really requires understanding so many things, such that I could never reach that point where I could say, "Okay, I've got it, now let's use this groundwork to start thinking about legal doctrine". Examples: What even is law? Legal philosophy. Indeterminacy of language and interpretation? Philosophy of language. Distinguishing right and wrong? Moral philosophy, then moral psychology once it became clear to me that morals have to do with groups and emotions. How about understanding society, the realm of law's application and origin? Sociology/Social theory. Not to mention the huge body of "postmodern" work that has grown up around it which sees everything as a social construct, an effect of discourse/power, humans themselves as lacking the kind of agency assumed in law and liberal political philosophy.
And despite everything I have written, I don't feel like I've gotten a grasp on things. (Social) reality is such a massive, incomprehensible beast where everything is chaotically intertwined, arguably unlike mathematics where you can advance your understanding in little, isolated chunks at a time, with reasonable confidence that you've made sense of it. Some days I feel like I've wasted all my energy trying to analyse all these underlying dynamics of law and society, that I've entered a field where the "will to know" never pays off (you never see the full picture) and simply rips your common sense worldview apart without offering anything in return.
Thanks for the essay, it allowed me to explore my thoughts in the preceding paragraphs :)
Phenomenal! Thank you so much!
I love this essay so much. Thanks for writing this.
Hi Nabeel,
Enjoyed this!
Question: Why is going closer only good fiction writing advice? How is it different with non-fiction?
Thank you! I think it's worthwhile for both -- some of the very best non-fiction I've read has unforgettable specifics (see John McPhee's book on Oranges, for example...)
That said, non-fiction often deals in abstractions and concepts, where this is maybe less important overall.
"People who have not experienced the thing are unlikely to be generating truth. More likely, they’re resurfacing cached thoughts and narratives."
Seems like LLMs. 🙂
That was the case but we're slowly moving away from that paradigm: https://nonint.com/2024/06/03/general-intelligence-2024/
My wife says she was the top of her class for Calculus and 10 years later couldn't tell you a single thing about it. She excelled at school because she was good at memorizing, not because she understood things well -- she had good hardware, but never learned to load in the software she really needed to be an expert. I guess it doesn't matter. Her field doesn't require Calculus. Makes me wonder if we're wasting people's time on subjects that don't actually matter instead of doubling down on the ones that will for them.
Truly fantastic read. I feel like it has put into words many thoughts I’ve had for so long.
I would like to add that, even though I agree that first-hand experience is an amazing vehicle for understanding, it is inherently limited and should be taken with a grain of salt. Besides the concern that human senses can fool us, I think it is even more concerning that our own human experience is very limited in size, speeds, and time scales. We humans are unable to experience first-hand the very small, the very fast, and processes that take several millions of years to happen. Hence this empirical-data approach can sometimes limit us to exclusively think about concepts that are inherently tied to the human experience.
I would imagine that a being that is several orders of magnitude larger than us and lives over many millions of years instead of mere decades will be inclined to think about physical systems that operate on these scales.
> Moreover, I have noticed that these ‘hardware’ traits vary greatly in the smartest people I know -- some are remarkably quick thinkers, calculators, readers, whereas others are ‘slow’. The software traits, though, they all have in common -- and can, with effort, be learned.
On the other hand, natural selection being what it is, if you’re not quick enough, you may be deemed unworthy of the opportunity to spend so much time learning something. Social norms or practical needs limiting your study time are effective mechanisms to defeat any intellectual ambitions in people with insufficiently powerful hardware. They can be combined with bullying to increase the need to work on one’s physical prowess—if one is even allowed to do so—further extinguishing any hope of devoting substantial time to intellectual pursuits in anyone but the strongest. Should this be promoted? I think the Spartans were certain that it should.
> It’s also so easy to think that you understand something, when you actually don’t.
It’s also easy for higher-status people to further lower your status by denying that you understand something when you actually do. They just have to not respect you and to not listen to you, and the audience will follow their lead. And they’re likely to be very good at these games because they’ve spent about as much time perfecting their skills as you have studying your favorite subjects.
> Related to this is honesty, or integrity: a sort of compulsive unwillingness, or inability, to lie to yourself.
This will get you into a lot of fights when you refuse to swallow other people’s bullshit answers, not realizing they mean, “Shut up. I can beat you up; don’t make me get my hands dirty”.
> Malcolm Gladwell on his father:
[...]
>> So if he doesn’t understand something, he just asks you. He doesn’t care if he sounds foolish. He will ask the most obvious question without any sort of concern about it...
If you’re not high status enough, _you_ may still not care whether you sound or look foolish, but other people won’t hesitate to kill two birds with one stone, saving effort and further raising their status at your expense by answering with something obvious that doesn’t address your doubt, or by outright shaming you for not already knowing the answer or for caring about it.
>> ‘I don’t understand. Explain that to me.’
Indeed, you’d better be in a position where you can afford to talk to them in the imperative like that.
> Most people are not willing to do this -- looking stupid takes courage, and sometimes it’s easier to just let things slide.
If you’re aware of the social dynamics at play, it takes courage for good reasons, and often there’s simply nothing to be gained by asking questions nobody else is more interested in than in the status games. If you’re socially clueless, it doesn’t take any courage—you just blindly cross the line and suffer the consequences without understanding what’s going on.
> It turns out that my misgivings were right, and that the Leibniz notation is basically just a convenient shorthand and that you more or less can treat those things “as if” they are fractions, but the proof is super complicated etc.
In other words, you just needed a chance to take a look at a slightly more advanced calculus textbook. But, collectively, we’re more interested in social games like denying youngsters such opportunities and shaming parents who encourage those interests as child abusers.
> It is also good advice for understanding things. When in doubt, go closer.
Ho, ho, only to be told you’re missing the forest for the trees.
Whew, this is a wonderful read and brought to mind a Shane Parrish’s essay on reading well.
Love this essay! What an exploration!
I use a similar process when I try to sort through my semantic explorations - trying to understand words and their meanings, the conceptual frameworks built around them and the implicit aspects people include when they use the word.
This kind of nitty-gritty exploration always requires the kind of tenacity that you describe. What does this mean? How about that? And that? You keep going until you know it inside out and there are no loose ends and outstanding questions.
It takes ages and can get very abstract but it really sorts it out when you get to a satisfactory end.
>understanding is not a binary “yes/no”. It has layers of depth.
Amen.
Case in point: I first "fell in love" with optimization as an undergrad decades ago and went on to get a PhD in it. Fast forward to *last week*: I understood something new about linear programming that I wasn't aware of all this time.
On the one hand, it is a bit embarrassing that I had not understood this earlier :). On the other hand, it is a beautiful example of the fractal nature of knowledge.
p.s. Loved the article! The only thing I would add: since we don't have the time to reason up from facts and first principles for *every* thing, we need to "pick our battles" carefully.
Nice piece.
Did Nielsen find the equals essay?
Loved the article; will to think is a powerful concept. I plan to incorporate in my daily routine .
I love this writing! Thank you!
To be like water is to understand your natural, authentic self.
——Be Water，My Friend (written by Shannon Lee)