This symbolic vs structural distinction explains so much about why people plateau in unexpected places. I've watched this exact thing happen in tech, where engineers who crush leetcode interviews completley fall apart when asked to design systems that don't exist yet. That insight about modern careers optimizing for early symbolic performance hits differntly when you realize we're basically training people to be really good at the wrong thing for the problems that actually matter.
Thanks for reading, and yes. When you realize we're being trained so badly for the world we live in, combined with the fact that AI will eventually be able to do anything "symbolic", you realize we're screwed!
In symbolic thinking we mirror other people's compressions, often and typically without fully understanding ourselves. This is the more common way of thinking, see Gabriel Tarde's "Laws of imitation"
Cognitively speaking it is lower effort; it can be viewed as a survival mechanism, efficient use of limited resources
Love this way of thinking about it! As a mirroring of compressions. In an age of information overload, perhaps this is ultimately a survival mechanism at the individual level, perhaps at the cost of annihilation at the societal level!
As a PhD is applied mathematics (specialty probability and statistics) I can tell you that "confusion" is an integral part of my thought processes. I work in healthcare, so most of my daily work consists in transforming clinician's views and experience and data into actionable formal structures. This is where "changing the shape of the solution" comes very useful. Looking at stochastic structures like martingales and recognising a real-world pattern helps you understand the maths and the World better.
If I may apply this to another area: Warren Buffett is acclaimed as a uniquely successful investor; his career follows an arc within which his focus or style or method changed over the years. He undoubtedly knows the 'rules' of investing as well as anybody yet nobody else following 'rules' succeeded as he did. When asked for advice to other investors he says: "invest in a S&P based ETF." I think what he is saying with that advice is: all the rules about PE ratios and cash flow and internal rates of return etc., etc., didn't really put me where I am so I cannot advise that spending your days delving into those things is worthwhile. I got where I am by leaps of intuition and I know you cannot duplicate those so here is an alternative that will at least not do you harm.
Super interesting, love that! A highly intuitive investor. Although we are (worryingly) increasingly in a world where “rules” in the markets vis-a-vis PE ratios, cash flows, etc no longer seem to connect with prices! And maybe this is why we are living in an era of “dumb money”….
best essay i’ve read in a while! thank you. as someone who tapped out of math early, in favor of majoring in CS, i wish i had taken the pure math route, getting at least an MS or dropping out of a math phd.
I’ve considering doing a online math bs or ms in person but not sure if i have the time or money? we’ll see. i’m happy to do math proofs on my own in my spare time, but i think you can only safely self study math at the grad school level. before that you might need actual peers. Idk.
sorry for the terrible punctuation. but i genuinely loved this article and it speaks to me. i am a pure math lover and id love to have pure math fundamentals way more than CS fundamentals. Alas! Maybe an MS in math is still in the books for me. We’ll see.
Thank you for reading Darius! It's a complex question, whether it's worth the time (or money), and whether it's a career (or passion) opportunity. I wish you luck in thinking about this and whatever path you decide! S.
I’ve been trying to tell a version of this to friends for a few years now.
I ask friends with degrees in CS, physics, engineering etc, can they still solve a simple quadratic equation. Almost all can, of course!
then I ask them to give me a real world problem they would put in an exam that needs solved by one. more than 90% of them couldn’t it.
Interestingly, it depends on what shape it takes in your mind, as you say! My natural assumption was to try and prompt people into thinking of physical 2D planes, like painting a room - what else could it be?
Yet my only two friends who gave a correct answer both gave it in terms of acceleration, which wouldn’t have occurred to me in a million years.
The “wouldn’t have occurred to me in a million years” is where I am consistently in awe of how differently our thinking processes contribute to problem solving!
Badically, you’re my new fav sub. Reminds me of one of my stats professors in grad school who told me that if you are not confused or reaching higher levels of confusion, you are not advancing.
Read it and it is a gem, ofc. I sent it to the curator of my favorite museum in Bucharest who introduced me to Malevich. Let s see. I sensed he was massively relevant to how all broke at the beginning of the century in painting, but was never able to articulate it in such a coherent way.
Thank you for this thoughtful suggestion. I have just read about him and his tabla playing. You see! Another great example of creativity first, problems later. Also since his mother was a math prof, I wonder what impact this had on his early comprehension of what “doing math” entailed.
As he sets out in light of linguistics research, the best way to learn a language is the opposite of how we typically go about it: First immerse yourself in its structure, so that you ultimately understand instinctively what does and doesn't feel 'right'. Then you can add on the rote-learning of grammar etc.
Fantastic and really interesting article, thank you for sharing!
Many of the points you made reminded me of a lot of what @davidepstein talks about in his book Range Widely! I’m not sure if you’re familiar with this book, but essentially he argues that specialization is overrated in today’s world of increasingly complex problems, and that when you look deeper at the backgrounds of high performers you find that a lot of them switch careers or fields of study or took a winding road to find their calling where they perform best. He lays out that these unique experiences and diverse knowledge sets allow them to become high performers because they bring a unique and creative perspective to their specialized fields.
He also contends (and this is where y’all’s ideas really link up!) that in wicked environments where the rules may be unknown or change without warning or where feedback may be misleading or indiscernible, generalists as he calls them, are essential because their unique experiences allow them to think structurally about the problem and relate the underlying structure to other fields to draw unintuitive analogies that help them solve the problem! This obviously relates to your points about structural thinking, especially if you consider that high level mathematics is in some ways a wicked environment. He contrasts that to a kind environment where the rules are well understood and don’t change and feedback is often immediate and trustworthy. An environment where symbolic learning would thrive!
Very interesting parallels between this article and his work!
Interesting but I'm not sure any of this can be taught. Let me give a personal example. I speak three languages & have a 4th I haven't used in 50 years but probably could work up. My son teaches finance, is a fine musician, & speaks three languages with native fluency (note that adj). One of my languages is ASL, my wife & many of our friends are Deaf, I've worked & lived in ASL for decades, yet my son is more fluent in ASL than I am even though he has almost no vocabulary. In fact, the first time he tried to sign with wife he made a bilingual pun. How do you define "fluency"? Any native speaker of a language can recognize it.
Could you seriously read this and let me know your thoughts
We are not voting: If Congress already knows who's going to win, we are not voting. The same people would be in office if nobody casted ballots, and we gave the office to the one with more money 90% of the time. Would be the same results.
This symbolic vs structural distinction explains so much about why people plateau in unexpected places. I've watched this exact thing happen in tech, where engineers who crush leetcode interviews completley fall apart when asked to design systems that don't exist yet. That insight about modern careers optimizing for early symbolic performance hits differntly when you realize we're basically training people to be really good at the wrong thing for the problems that actually matter.
Thanks for reading, and yes. When you realize we're being trained so badly for the world we live in, combined with the fact that AI will eventually be able to do anything "symbolic", you realize we're screwed!
In symbolic thinking we mirror other people's compressions, often and typically without fully understanding ourselves. This is the more common way of thinking, see Gabriel Tarde's "Laws of imitation"
Cognitively speaking it is lower effort; it can be viewed as a survival mechanism, efficient use of limited resources
Love this way of thinking about it! As a mirroring of compressions. In an age of information overload, perhaps this is ultimately a survival mechanism at the individual level, perhaps at the cost of annihilation at the societal level!
As a PhD is applied mathematics (specialty probability and statistics) I can tell you that "confusion" is an integral part of my thought processes. I work in healthcare, so most of my daily work consists in transforming clinician's views and experience and data into actionable formal structures. This is where "changing the shape of the solution" comes very useful. Looking at stochastic structures like martingales and recognising a real-world pattern helps you understand the maths and the World better.
If I may apply this to another area: Warren Buffett is acclaimed as a uniquely successful investor; his career follows an arc within which his focus or style or method changed over the years. He undoubtedly knows the 'rules' of investing as well as anybody yet nobody else following 'rules' succeeded as he did. When asked for advice to other investors he says: "invest in a S&P based ETF." I think what he is saying with that advice is: all the rules about PE ratios and cash flow and internal rates of return etc., etc., didn't really put me where I am so I cannot advise that spending your days delving into those things is worthwhile. I got where I am by leaps of intuition and I know you cannot duplicate those so here is an alternative that will at least not do you harm.
Super interesting, love that! A highly intuitive investor. Although we are (worryingly) increasingly in a world where “rules” in the markets vis-a-vis PE ratios, cash flows, etc no longer seem to connect with prices! And maybe this is why we are living in an era of “dumb money”….
best essay i’ve read in a while! thank you. as someone who tapped out of math early, in favor of majoring in CS, i wish i had taken the pure math route, getting at least an MS or dropping out of a math phd.
I’ve considering doing a online math bs or ms in person but not sure if i have the time or money? we’ll see. i’m happy to do math proofs on my own in my spare time, but i think you can only safely self study math at the grad school level. before that you might need actual peers. Idk.
sorry for the terrible punctuation. but i genuinely loved this article and it speaks to me. i am a pure math lover and id love to have pure math fundamentals way more than CS fundamentals. Alas! Maybe an MS in math is still in the books for me. We’ll see.
Thank you!
Thank you for reading Darius! It's a complex question, whether it's worth the time (or money), and whether it's a career (or passion) opportunity. I wish you luck in thinking about this and whatever path you decide! S.
Basically we should not confuse education, credentials, and intelligence.
These are 3 completely different entities only loosely correlated.
I’ve been trying to tell a version of this to friends for a few years now.
I ask friends with degrees in CS, physics, engineering etc, can they still solve a simple quadratic equation. Almost all can, of course!
then I ask them to give me a real world problem they would put in an exam that needs solved by one. more than 90% of them couldn’t it.
Interestingly, it depends on what shape it takes in your mind, as you say! My natural assumption was to try and prompt people into thinking of physical 2D planes, like painting a room - what else could it be?
Yet my only two friends who gave a correct answer both gave it in terms of acceleration, which wouldn’t have occurred to me in a million years.
The “wouldn’t have occurred to me in a million years” is where I am consistently in awe of how differently our thinking processes contribute to problem solving!
I love Bird’s lament! Thank you for the great article!
Ha! It’s fantastic. Thanks for reading!
Wow! And thank you! This answer a very real and present issue for me.
Then I hope you continue going, unlike me who stopped! 🫡
Badically, you’re my new fav sub. Reminds me of one of my stats professors in grad school who told me that if you are not confused or reaching higher levels of confusion, you are not advancing.
Ha, very kind, thanks. That is very helpful advice! I hope to continue “getting dumb” into my old age!
Btw, forgot to mention that Malevic was an absolute GOAT. Hilma af Klint too.
Absolutely! In case you missed it, here’s my piece on Malevic!
https://open.substack.com/pub/butthistimeitsdifferent/p/kazimir-malevich-and-the-art-of-staying?r=4eaetr&utm_medium=ios
Read it and it is a gem, ofc. I sent it to the curator of my favorite museum in Bucharest who introduced me to Malevich. Let s see. I sensed he was massively relevant to how all broke at the beginning of the century in painting, but was never able to articulate it in such a coherent way.
Will read it. I was amazed when checking him how ahead of his times he was.
Would add Fields medalist Manjul Bhargava to your list of mathematicians.
He is a Sanskrit scholar and learnt tabla under Ustad Zakir Hussain.
Thank you for this thoughtful suggestion. I have just read about him and his tabla playing. You see! Another great example of creativity first, problems later. Also since his mother was a math prof, I wonder what impact this had on his early comprehension of what “doing math” entailed.
Fascinating, thank you. It brought to mind this piece by @colingorrie on language learning. https://www.deadlanguagesociety.com/p/why-people-fail-at-learning-languages
As he sets out in light of linguistics research, the best way to learn a language is the opposite of how we typically go about it: First immerse yourself in its structure, so that you ultimately understand instinctively what does and doesn't feel 'right'. Then you can add on the rote-learning of grammar etc.
It feels like there's an analogy there.
Fantastic and really interesting article, thank you for sharing!
Many of the points you made reminded me of a lot of what @davidepstein talks about in his book Range Widely! I’m not sure if you’re familiar with this book, but essentially he argues that specialization is overrated in today’s world of increasingly complex problems, and that when you look deeper at the backgrounds of high performers you find that a lot of them switch careers or fields of study or took a winding road to find their calling where they perform best. He lays out that these unique experiences and diverse knowledge sets allow them to become high performers because they bring a unique and creative perspective to their specialized fields.
He also contends (and this is where y’all’s ideas really link up!) that in wicked environments where the rules may be unknown or change without warning or where feedback may be misleading or indiscernible, generalists as he calls them, are essential because their unique experiences allow them to think structurally about the problem and relate the underlying structure to other fields to draw unintuitive analogies that help them solve the problem! This obviously relates to your points about structural thinking, especially if you consider that high level mathematics is in some ways a wicked environment. He contrasts that to a kind environment where the rules are well understood and don’t change and feedback is often immediate and trustworthy. An environment where symbolic learning would thrive!
Very interesting parallels between this article and his work!
This happened to me but in high school introductory physics.I didn’t know how to think (at least structurally).
Interesting but I'm not sure any of this can be taught. Let me give a personal example. I speak three languages & have a 4th I haven't used in 50 years but probably could work up. My son teaches finance, is a fine musician, & speaks three languages with native fluency (note that adj). One of my languages is ASL, my wife & many of our friends are Deaf, I've worked & lived in ASL for decades, yet my son is more fluent in ASL than I am even though he has almost no vocabulary. In fact, the first time he tried to sign with wife he made a bilingual pun. How do you define "fluency"? Any native speaker of a language can recognize it.
Could you seriously read this and let me know your thoughts
We are not voting: If Congress already knows who's going to win, we are not voting. The same people would be in office if nobody casted ballots, and we gave the office to the one with more money 90% of the time. Would be the same results.
https://open.substack.com/pub/mrnobody99999/p/we-end-up-with-the-will-of-the-people?utm_campaign=post-expanded-share&utm_medium=web