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 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.
Thanks for sharing this! I have always found it so strange that people learn languages through gamified apps where you get thrown random vocabulary for hours on end. This article you shared makes total sense, of course!
Thanks for that very interesting piece. My training was in physics and I spent many years as a professor in related fields. One encounters two issues related to the discussion here.
In beginning graduate students one encounters two types as limiting cases. Some students can do math very capably. So long as they can translate the problem at issue into math, they do fine. Other students are not very good at math, but are highly developed conceptually. They can reason their way to the solution to the problem but can't do the math to quantify it. I used to tell my classes that our goal was to help them learn to think and do math at the same time.
The other issue arises whenever students begin research, whether as an undergraduate or as a third year graduate student. The latter group is at high risk, because while doing well in classes involves feeling smart a lot, doing research involves feeling stupid much of the time. This can be unpleasant enough that it drives many students out of research. It is much better for them to experience those differences as undergraduates.
Brilliant article. Read this across 3 days and could pick up right where I left off. This got me rethinking my abilities.
I'm in the trading profession. I put it that way because I trade without understanding fully what the hell I'm trading, instrument wise. Because when I go to study it and study things like the CFA, I feel im the stupidest person on earth. But I somehow make it work.
Maybe I'll give studying another shot now. Based on this article
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.
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.
Great article. “This system is extremely good at producing people who can operate fluently and be quickly promoted inside existing structures.” Totally agree - it reminds me of “Why Greatness Cannot be Planned” by Kenneth Stanley. He argues that the idea of a concrete objective is useful for small well-defined goals, but for anything truly creative, it’s counterproductive, and instead we should optimize for exploration, guided by what interests us the most. This path creates composable building blocks of knowledge/skill that seem completely orthogonal to the objective at first (ie super slow, useless) but eventually far outpace the direct approach.
I think our instincts for what’s interesting/exciting is surprisingly information-rich - a cognitive calculation that has probably been fine-tuned by evolution over eons.
And I think Stanley would agree with you that our current system falls prey to the tyranny of the objective: “We want you to be good at math! Mathematicians use techniques and write proofs! Therefore you must learn all the techniques and write lots of proofs!”
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.
Fascinating question regarding the nature of fluency! (And btw - I love ASL, learned it briefly myself, and it is really under-learnt as a global language!). I've thought about the fluency question for a while as I moved between French and English in my late teens. Given that you have correctly noted that any given native speaker can recognize it, there is a suggestion that it is an externally-validated condition. Internally, perhaps it's about order of linguistic preference in a random environment? Ease of communication? I'll keep thinking about this, because it's really fascinating!
For languages, you're absolutely right. We're linguistically receptive up to about age 12. I think you're right for other stuff, too. I took calculus in college more than 60 years ago. A year ago I got interested in learning it again, hired a tutor, & found I couldn't make any sense of it. Totally embarrassing.
It's funny to me that you call that embarrassing. I aspire to be the man who at 80 is willing to learn something as new and challenging as calculus and takes action to do so! This is heroic, and should be treated as much!
I really wish I knew more biology and neurology to know why age matters.
Here are some bets of mine that I think point that biological age (time from birth) is not the most important factor, although it likely is a factor
1- Imagine a child actively engaged in learning new languages from birth. every year or two after 5 they are introduced to a new language. by the time they are 15 they have learnt ~11 languages, and can speak 4 or 5 of them fluently. (This seems to be accessible to most humans, as we know of societies that are multilingual)
keep this going, but slow down the pace, they learn a new language every 5 or 6 years---and in addition they reflect on what the learning process is like.
when they are 70 they begin learning a new language.
will they learn it faster than a 9yo?
2- I think math is no different than this language issue--but like the @sineados wrote, we are actually making ourselves mathematically dumber in many ways. as we grow this gets harder and harder to unlearn. so when we are older, we might be worse at math BECAUSE we have cycled a pattern in our mind of how to understand certain math concepts.
its hard to test this one well, but I would bet that specialized math tutors for remedial math for adults would do better than remedial math tutors for students who are in a schooling environment that is more aggressively pushing they 'symbolic' manner of thinking. this would prove that age is less of a detriment than environment, but there are many many confounding factors here
But all that, I imagine that some brain deterioration happens biologically, and there's no way around it. I don't know when it starts, and I imagine there is some good research on this.
alas, I know too much about research to think that I could separate the wheat from the chaff on this topic in a time that is worth it.
I think this is more like a consequence of Berkson's paradox - https://brilliant.org/wiki/berksons-paradox/. We have a multiselection with speed and hyperfocus so if a mathematician is "slower" they are going to be more likely to be hyoerfocused. June for example seems to be using most of his cognitive cycles for math whether he's sitting at a desk or not by taking everything else out of the equation.
Prior to the elite level it's a matter of more speed more better because there are enough people to select from that have that attribute and the bar for success is much lower. At the elite level, because the ingredients you need are so rare and success requires being out of distribution as much as possible you'll see someone peaky in one thing will likely be lacking in another. And the converse.
So being slow at math whether the first type of thinking or the second, unless you are a fields medalist, is probably correlated with less success. But at the fields medalist level it's anticorrelated so we get people like June sometimes. If they are slow and a fields medalist they have a other attributes which help with math success in which they are wildly out of distribution for.
I wonder if the distinction might actually go deeper than two styles of thinking. Could symbolic thinking be the kind of cognitive work that, because it’s just executing known procedures on well-defined structures, doesn’t really require understanding at all? (I’m reminded of Searle’s Chinese Room). Structural thinking seems different in kind, since orienting yourself toward an object whose shape you don’t yet know can’t really be reduced to rule-following. Does it worry you that our institutions increasingly reward exactly the capacities machines are best at?
Interesting! Sometimes i do indeed wonder if someone is thinking at all…! And I find it completely bizarre that AI is being designed to replicate what humans can already do. There are millions of other dimensions of “intelligence” that humans simply cannot access. Instead of making human intelligence multidimensional, we’re reducing the one dimension of human intelligence we already have…
Brilliant. As a small footnote, I don’t think that the affinity for art in many of the lives of mathematicians and scientists that make key discoveries is trivial. In the ability to grasp the broad scheme of things and in savouring subtleties, the practice of art and appreciation of beauty is unpaired. It is, in many ways, requiered to have a sharp aesthetic sense to make good discoveries.
One of the foundations you learn in composing music is it is fundamentally about whether you have a sound structure (pun intended) for your piece, and not only riffs of melodies. What are the core ideas of the composition, how do they evolve and remain invariant, and how are they linked in the structure? Not unlike proof-writing.
A great read. I'd like to expand on what makes pursuing a PhD in math, specifically, so difficult. (Full disclosure: I have an M.A. in Pure Math, but pivoted after realizing I could not achieve a doctorate). As you pointed out, not all fields are equally difficult. Theoretical physics and pure math are probably, objectively, the most challenging disciplines in existence. So what makes a math PhD so difficult? According to some sources, in the US, only 2% of adults have PhDs of any kind. And only 1500 math PhDs are produced each year, with over half of them going to non-US residents. To get a math PhD you must love and excel in all of the following:
(1) number sense, (2) elementary formal techniques, (3) logical proofs, (4) solving graduate-level math homework problems, (5) grasping complex notation, (6) reading/interpreting math texts written by others, (7) managing and organizing confusing collections of mathematical objects, (8) memorizing and understanding an endless list of mathematical definitions, (9) knowing how to perform math research, (10) expressing the discipline, intuition, and creativity required to solve new problems in isolation and without a support system.
Those who are not mathematically inclined are stopped dead in their tracks by (1) and (2). Many brilliant minds that make up the "STE-" in STEM are halted by (3). High school math teachers get stuck at (4). For me? I love proofs and did a lot of research, but (5), (6), (7), and (8) were a killer combo that signaled the end. Only a special few have what it takes, and the author is right to say that the issue isn't a "cognitive limit" or the lack of a genius intellect. Instead, the problem lies in the fact that obtaining a math PhD requires mastery of all 10 of the provided points. How many people can fit through such a sieve?
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!
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.
Thanks for sharing this! I have always found it so strange that people learn languages through gamified apps where you get thrown random vocabulary for hours on end. This article you shared makes total sense, of course!
Thanks for that very interesting piece. My training was in physics and I spent many years as a professor in related fields. One encounters two issues related to the discussion here.
In beginning graduate students one encounters two types as limiting cases. Some students can do math very capably. So long as they can translate the problem at issue into math, they do fine. Other students are not very good at math, but are highly developed conceptually. They can reason their way to the solution to the problem but can't do the math to quantify it. I used to tell my classes that our goal was to help them learn to think and do math at the same time.
The other issue arises whenever students begin research, whether as an undergraduate or as a third year graduate student. The latter group is at high risk, because while doing well in classes involves feeling smart a lot, doing research involves feeling stupid much of the time. This can be unpleasant enough that it drives many students out of research. It is much better for them to experience those differences as undergraduates.
Brilliant article. Read this across 3 days and could pick up right where I left off. This got me rethinking my abilities.
I'm in the trading profession. I put it that way because I trade without understanding fully what the hell I'm trading, instrument wise. Because when I go to study it and study things like the CFA, I feel im the stupidest person on earth. But I somehow make it work.
Maybe I'll give studying another shot now. Based on this article
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.
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.
Great article. “This system is extremely good at producing people who can operate fluently and be quickly promoted inside existing structures.” Totally agree - it reminds me of “Why Greatness Cannot be Planned” by Kenneth Stanley. He argues that the idea of a concrete objective is useful for small well-defined goals, but for anything truly creative, it’s counterproductive, and instead we should optimize for exploration, guided by what interests us the most. This path creates composable building blocks of knowledge/skill that seem completely orthogonal to the objective at first (ie super slow, useless) but eventually far outpace the direct approach.
I think our instincts for what’s interesting/exciting is surprisingly information-rich - a cognitive calculation that has probably been fine-tuned by evolution over eons.
And I think Stanley would agree with you that our current system falls prey to the tyranny of the objective: “We want you to be good at math! Mathematicians use techniques and write proofs! Therefore you must learn all the techniques and write lots of proofs!”
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.
Fascinating question regarding the nature of fluency! (And btw - I love ASL, learned it briefly myself, and it is really under-learnt as a global language!). I've thought about the fluency question for a while as I moved between French and English in my late teens. Given that you have correctly noted that any given native speaker can recognize it, there is a suggestion that it is an externally-validated condition. Internally, perhaps it's about order of linguistic preference in a random environment? Ease of communication? I'll keep thinking about this, because it's really fascinating!
maybe everything can be taught, but age matters!
For languages, you're absolutely right. We're linguistically receptive up to about age 12. I think you're right for other stuff, too. I took calculus in college more than 60 years ago. A year ago I got interested in learning it again, hired a tutor, & found I couldn't make any sense of it. Totally embarrassing.
It's funny to me that you call that embarrassing. I aspire to be the man who at 80 is willing to learn something as new and challenging as calculus and takes action to do so! This is heroic, and should be treated as much!
I really wish I knew more biology and neurology to know why age matters.
Here are some bets of mine that I think point that biological age (time from birth) is not the most important factor, although it likely is a factor
1- Imagine a child actively engaged in learning new languages from birth. every year or two after 5 they are introduced to a new language. by the time they are 15 they have learnt ~11 languages, and can speak 4 or 5 of them fluently. (This seems to be accessible to most humans, as we know of societies that are multilingual)
keep this going, but slow down the pace, they learn a new language every 5 or 6 years---and in addition they reflect on what the learning process is like.
when they are 70 they begin learning a new language.
will they learn it faster than a 9yo?
2- I think math is no different than this language issue--but like the @sineados wrote, we are actually making ourselves mathematically dumber in many ways. as we grow this gets harder and harder to unlearn. so when we are older, we might be worse at math BECAUSE we have cycled a pattern in our mind of how to understand certain math concepts.
its hard to test this one well, but I would bet that specialized math tutors for remedial math for adults would do better than remedial math tutors for students who are in a schooling environment that is more aggressively pushing they 'symbolic' manner of thinking. this would prove that age is less of a detriment than environment, but there are many many confounding factors here
But all that, I imagine that some brain deterioration happens biologically, and there's no way around it. I don't know when it starts, and I imagine there is some good research on this.
alas, I know too much about research to think that I could separate the wheat from the chaff on this topic in a time that is worth it.
Interesting. Are you multilingual?
I'm bilingual (English and Hebrew), and I have very weak Arabic.
My mother knows three languages fluently (farsi, English, Hebrew) and has learnt basic Arabic and French with decent success (more than me).
So you know how hard it is to learn a language as an adult.
I think this is more like a consequence of Berkson's paradox - https://brilliant.org/wiki/berksons-paradox/. We have a multiselection with speed and hyperfocus so if a mathematician is "slower" they are going to be more likely to be hyoerfocused. June for example seems to be using most of his cognitive cycles for math whether he's sitting at a desk or not by taking everything else out of the equation.
Prior to the elite level it's a matter of more speed more better because there are enough people to select from that have that attribute and the bar for success is much lower. At the elite level, because the ingredients you need are so rare and success requires being out of distribution as much as possible you'll see someone peaky in one thing will likely be lacking in another. And the converse.
So being slow at math whether the first type of thinking or the second, unless you are a fields medalist, is probably correlated with less success. But at the fields medalist level it's anticorrelated so we get people like June sometimes. If they are slow and a fields medalist they have a other attributes which help with math success in which they are wildly out of distribution for.
I wonder if the distinction might actually go deeper than two styles of thinking. Could symbolic thinking be the kind of cognitive work that, because it’s just executing known procedures on well-defined structures, doesn’t really require understanding at all? (I’m reminded of Searle’s Chinese Room). Structural thinking seems different in kind, since orienting yourself toward an object whose shape you don’t yet know can’t really be reduced to rule-following. Does it worry you that our institutions increasingly reward exactly the capacities machines are best at?
Interesting! Sometimes i do indeed wonder if someone is thinking at all…! And I find it completely bizarre that AI is being designed to replicate what humans can already do. There are millions of other dimensions of “intelligence” that humans simply cannot access. Instead of making human intelligence multidimensional, we’re reducing the one dimension of human intelligence we already have…
I really felt inspired while reading this, make you think about the limitations we put on ourselves.
Holy shit. this is exactly what I'm working on.
i am researching how to make sure people have structural thinking because intelligence is a practice, not an object.
Anna Sfard and Richard skemp (and lots of math research) talks exactly about what you are talking about
Math circles are a pedagogical system thst focuses only on the real thinking.
socratic sessions teach interlocutors to be grounded in their own understanding, and not symbols.
Exactly this! Wonderful read, thank you :)
Brilliant. As a small footnote, I don’t think that the affinity for art in many of the lives of mathematicians and scientists that make key discoveries is trivial. In the ability to grasp the broad scheme of things and in savouring subtleties, the practice of art and appreciation of beauty is unpaired. It is, in many ways, requiered to have a sharp aesthetic sense to make good discoveries.
Love this. Thanks for this thoughtful addition!
One of the foundations you learn in composing music is it is fundamentally about whether you have a sound structure (pun intended) for your piece, and not only riffs of melodies. What are the core ideas of the composition, how do they evolve and remain invariant, and how are they linked in the structure? Not unlike proof-writing.
A great read. I'd like to expand on what makes pursuing a PhD in math, specifically, so difficult. (Full disclosure: I have an M.A. in Pure Math, but pivoted after realizing I could not achieve a doctorate). As you pointed out, not all fields are equally difficult. Theoretical physics and pure math are probably, objectively, the most challenging disciplines in existence. So what makes a math PhD so difficult? According to some sources, in the US, only 2% of adults have PhDs of any kind. And only 1500 math PhDs are produced each year, with over half of them going to non-US residents. To get a math PhD you must love and excel in all of the following:
(1) number sense, (2) elementary formal techniques, (3) logical proofs, (4) solving graduate-level math homework problems, (5) grasping complex notation, (6) reading/interpreting math texts written by others, (7) managing and organizing confusing collections of mathematical objects, (8) memorizing and understanding an endless list of mathematical definitions, (9) knowing how to perform math research, (10) expressing the discipline, intuition, and creativity required to solve new problems in isolation and without a support system.
Those who are not mathematically inclined are stopped dead in their tracks by (1) and (2). Many brilliant minds that make up the "STE-" in STEM are halted by (3). High school math teachers get stuck at (4). For me? I love proofs and did a lot of research, but (5), (6), (7), and (8) were a killer combo that signaled the end. Only a special few have what it takes, and the author is right to say that the issue isn't a "cognitive limit" or the lack of a genius intellect. Instead, the problem lies in the fact that obtaining a math PhD requires mastery of all 10 of the provided points. How many people can fit through such a sieve?