Hi David, lovely post. It resonates deeply with something I've been working through - not just understanding how metacognitive habits differ, but actually changing them in practice.
Some months ago, I started thinking about the brain as fundamentally conservative, settling into stable attractor states. The brain can't run on nouns - it needs trigger → action → feedback. Some stimuli comes in, the brain responds with a practiced move, and keeps going until it gets relief (prediction error drops). Most of our daily actions - maybe 95%+ - are repetitions of moves we've done before given the same triggers. You can see this vividly in something like football. Watch players dribbling and their gaits are nearly identical match to match. The brain finds movements that work and defaults to them because it's metabolically cheap.
The same thing happens with thinking. When students hit confusion, they cycle through remarkably predictable policies: re-read, ask "why don't I get this?", check the answer key, or skip it. The feedback loop closes - uncertainty gets temporarily resolved - and the brain marks this as successful even though nothing was actually learned.
Looking back at my own learning, I realized every time I got unstuck after prolonged struggle, it was after some pivot or action - trying a diagram, computing a case, explaining to myself. Never did wheel-spinning in the same mode resolve confusion.
What I also noticed was that often it took me extended periods of time to get unstuck - not because the problem was that hard, but because that's how long it took for desperation to build up enough to lower my threshold for seemingly absurd action. Action that then, to my surprise, actually resolved the confusion. But I'd never systematized this or looked at what preceded breakthroughs.
What I settled on was: detect confusion early and immediately force a representational shift. Switch from algebra to geometry, symbolic to numerical, abstract to physical - whatever gets you out of the current stuck state.
This maps directly onto your point about secondary stimuli. The difference between someone stuck and someone learning isn't the problem itself but what they do internally when stuck. Re-reading the same passage generates almost no new neural activity. Switching representational modes - visualizing geometrically, computing specific examples, explaining out loud - activates completely different pathways.
But here's the implementation problem: at peak uncertainty, when you're most confused, your capacity for novel action is lowest. This is exactly when the brain defaults to familiar ineffective responses. Knowing you should shift representations changes nothing.
What actually worked was externalizing the choice. I made cards with specific moves: "Compute Something," "Extreme Cases," "Remove One Part," "Reverse/Rotate/Swap," "Sit With Confusion," "Make Prediction." When stuck, I draw a card. This removes the hardest part - committing to action under uncertainty. Any shift away from the stuck state generates information.
The second thing: permission to execute wrong. After some initial tries, I came to find there's usually this hesitation where you're evaluating "how exactly should I do this?" That evaluation overhead often kills the attempt. Slows you down. So instead: execute wrong on purpose! Don't know which numbers to compute? Use obviously wrong ones. Try the extreme case even if it seems absurd. I found wrong execution generates information fast - you learn why it's wrong in seconds, which reveals constraints and hidden structure.
I realized that confusion as a state is actually rather similar across different domains. The brain being what it is - a trigger-action-feedback system - if I could install a different state-action mapping at this inflection point, it should have outsized effects. If the new policy reduces prediction error better than the old one, it should outcompete it naturally over repetitions.
And it did. After some weeks of forced practice with the cards: detection latency dropped from 90 seconds to maybe 20-30, execution became nearly immediate, and subjectively I started feeling 'restless' when stuck and not shifting. The new policy's prediction-error-reduction is so much better that staying in one representation now feels 'wrong'. That restlessness is the experiential signature of attractor replacement.
What strikes me is the magnitude of improvement relative to effort. Problems I previously thought required checking textbooks or were just beyond me now resolve through sustained exploration. It takes longer than looking up answers, but I build actual understanding and the capability to do it again.
The old policy feels like doomscrolling - minimal cognitive load, no new information, just anxiety relief. The new policy feels like exercise - more effortful initially, but generating new neural activation with each attempt.
Your secondary stimuli insight is particularly apt here. Each representational shift isn't just "thinking harder" - it's generating different internal experience. Algebraic manipulation, geometric visualization, numerical calculation activate distinct neural substrates. High-frequency switching means high-diversity neural exploration, which should drive connectome reorganization much faster than passive re-reading.
Before this I was cognitively sedentary when stuck - burning mental energy on anxiety while generating minimal neural activity. Now there's constant motion: try this angle, doesn't work, try that, learn something, try another. The volume of distinct cognitive states explored per unit time has increased dramatically.
But I think the compounding goes deeper than just executing pivots more reliably. The brain being a pattern-matching machine, over time it comes to map certain kinds of cues - internal or external - with certain moves. It realizes some moves work better to resolve prediction error than others as it gathers experience. This is essentially what intuition is.
The increased information density per unit time, from this active learning strategy, means the brain gets vastly more data to identify underlying patterns. Each night when you sleep, the brain compresses this enormous amount of information, replays it, consolidates it. Over months and years, this might explain how people who naturally default to this stance appear to have better "smell" for what to do and when. It compounds exponentially.
It's really akin to what you described as going into "math mode" - and what you said elsewhere: "This bizarre, almost childish attitude is extremely hard to communicate to outsiders." That's exactly it. It's like a kid picking up a toy and figuring out all the funny ways they can play with it, learning about its properties in the process. The little moves are cheap, easy. If they don't work, it doesn't mean much - you just do something else.
I don't think any of this is particularly revolutionary from a pedagogy or neuroscience standpoint - I suspect it's implicit to how a lot of mathematicians and physicists actually work. But it was revolutionary to me personally, this shift in cognitive stance.
If intelligence lives in the connectome and connectomes reorganize in response to activity patterns, this high-frequency representational shifting should accelerate development. Not immediately, but compounded over months and years the divergence could be substantial.
I think what made this work wasn't just understanding the principle (intellectually) but the operationalization. The cards externalize the decision at exactly the moment when your brain is least capable of making it. Wrong execution removes the evaluative layer that causes hesitation. Together they bypass the exact bottlenecks that prevent people from using techniques they intellectually know about.
This seems to instantiate your conjectures directly. The cards operationalize a specific trainable habit at the exact inflection point that determines learning trajectories. Within weeks of deliberate practice, a completely different response pattern installed itself.
Motion precedes clarity. That's the stance you start to embody.
Being an optimist, I think you might be understating the potential magnitude. If the constraint on cognitive development is primarily metacognitive habits rather than genetic ceiling, and if simple protocols can shift those habits within weeks, the accessible improvement might be larger than the "20% full glass" suggests.
The protocol costs essentially nothing - some index cards and permission to execute badly - but it forces precisely the kind of "peculiar rumination techniques" you describe elite mathematicians practicing. It's a way to operationalize what you call quality of attention, but as concrete actions anyone can systematically train.
Most importantly: it's trainable in the sense of actually installable, not just intellectually understandable. You need a detectable trigger, an externalized action protocol, permission to execute imperfectly, and volume of practice. The policy that better reduces prediction error wins naturally. No willpower required once the initial pattern starts to dominate.
Thanks for articulating this framework so clearly. It's nice to see my experience map onto your educated guesses about secondary stimuli, metacognitive habits, and compounding neural differences.
i want to start using your method. “Compute Something," "Extreme Cases," "Remove One Part," "Reverse/Rotate/Swap," "Sit With Confusion," "Make Prediction.". this has 6 options so using a dice could be perfect! Can you explain each of these: “make prediction”, “reverse/rotate/swap” and “remove one part”. I dont quite get what they mean. Id be grateful. Awesome comment btw.
Sure. The Make a Prediction card is about forcing your vague model to commit. When you've been turning a problem over for a while, you've quietly built some picture of how things work — but it stays fuzzy until you make it pay rent. So you take whatever understanding you've accrued and force it into a concrete, checkable claim before you look anything up.
Example: "If I double the mass but keep force the same, acceleration halves." Now you can check it. If you're wrong, you've learned exactly where the model breaks. If you're right, you actually own it now — you didn't just recognise the answer, you generated it. I find this card works best after 4-5 others, once you've gathered enough data to make a reasonable guess.
Remove One Part is the simplest card in the deck. Take one ingredient out — a term in an equation, a component in a circuit, a constraint in a problem — and watch what collapses. Whatever breaks when you remove it is what that part was actually doing. Whatever survives didn't need it.
Example — capacitor circuit:
Remove the resistance → current spikes instantly to infinity
Remove the capacitance → no charging behaviour at all
Now you know what each element is actually for, not just that it's there. I use this constantly when an equation has an ugly term I don't understand — I delete it and solve the simpler version first. The understanding you gain from that almost always lets you go back and handle the original.
Reverse/Rotate/Swap is about turning the problem inside out to expose structure you can't see from the front. The three options are:
Reverse time or direction
Rotate the setup or coordinate frame
Swap labels, inputs, or particles
After each one, ask: what breaks? what stays the same? The answers tell you what's essential and what was just an accident of how you framed it.
Example — block sliding with friction: Reverse time → the heat doesn't flow back into the block and restart its motion. That asymmetry tells you dissipation is a real physical fact, not a modelling choice you made.
One note: I've since split this card into two separate cards, because having three options on one card reintroduced the decision paralysis the whole deck is supposed to fix. Each card needs to be concrete enough to get you moving immediately, but open enough that you can execute it your own way.
A few things worth knowing before you start:
Pick at random, at least initially. The point isn't to choose the optimal card — it's to break the freeze. Any motion beats the loop of staring at the problem waiting for clarity that isn't coming. Even executing a card badly produces new information.
After a while you'll find yourself reaching for them less, because the moves become natural. You'll also notice you gravitate toward some cards more than others — that's fine, just weight your deck toward the ones that feel unfamiliar, since those are the ones still expanding how you think.
The cards are optimised for physics but most of them translate directly to mathematics — the underlying logic is the same. If you were a pure mathematician you might swap a few out, but start with these and adjust once you know which ones aren't pulling their weight.
Intelligence boils down a brain that has optimized efficiency; each of you has described the process brilliantly.
Academics who are equally interested in athletics seem to trend towards optimization. An example: I am an equestrian, and my ecuyer was in the habit of reminding, “Read, ride, reflect.” Integrating the metal and physical aspects of an idea reliably leads to epiphany.
I think you end up with an extremely reasonable position. Sometimes when you push back against hereditarianism, you seem a bit starry-eyed about what we can accomplish by sheer force of will. In the end, though, you seem to admit that a lot of things are out of our control.
Personally, I still think all of those quotes--Newton, Einstein, Feynman, Grothendieck--are either disingenuous or incredibly naive. My own experience as a mathematician has not made me any less frustrated with these quotes. Quite the contrary, really. It feels as if geniuses feel obliged to remind everyone that actually they work very hard, as if we didn't already know. But I thought everybody knew that, just as they do for any other kind of excellence. Michael Phelps also had to work extremely hard to win all of those gold medals, but that doesn't mean that the rest of us can do it too if we just follow the same physical fitness regime that he did. Just because you have to work very, very hard to develop a gift doesn't mean that it isn't a gift.
Sometimes I think you exaggerate the difference between intellectual and physical prowess. You use a 100 meter dash as an example, because we all agree that all of us could at least finish, even if we're pathetically slow. But there are other activities with threshholds. Lifting weights, for example. The vast majority of us will just never be able to lift 500 pounds, even if we are given a week to do it.
Despite my criticisms, I will say that the extremely valuable part of your book and this post is a sort of research program to try to understand how we might be better trainers of cognitive ability. You're right to point out that physical activities are much more straightforward to model. Cognitive patterns are much more hidden. As a professor I've tried to explain to my students, as far as I'm able, my actual stream of consciousness that occurs to me when I approach a problem. Sometimes this can be a bit frightening to students, probably because, in addition to being hidden, cognitive behavior can be highly idiosyncratic. Still, there's probably a lot to be gained from trying to figure out the common patterns in the cognition of highly effective intellectuals. I imagine neuroscience will play a big role in this.
Glad I'm ending up with an extremely reasonable position—I do think I'm an extremely reasonable person :) !
On the 500 pounds example: the success metric for weightlifting is how much you can lift. I'm not an expert on the matter, but I do suspect most people could, with adequate training, lift 100 pounds or even 200 and possibly more. This feels nowhere near the common perception of the math talent gap.
On this topic, I think you're missing the essential notion of conceptual compression—how things that initially seem unfathomably hard often become trivial once you develop a familiarity with the right conceptual framework. One of my favorite example is Hindu-Arabic numerals, through which you instantly "see" that 1,000,000,000 - 1 = 999,999,999, a computation that feels superhuman to someone who only knows Roman numerals (see this post based on a chapter from my book: https://davidbessis.substack.com/p/the-magic-of-mathematical-intuition)
There is no equivalent of conceptual compression for lifting 500 pounds and this is where, in my view, the analogy breaks down. Cognition isn't running or weightlifting, and this explains why insane inequalities can develop.
About what we can accomplish by sheer force of will — I am acutely aware of everyone's limits, yet I do think we should absolutely insist that people have a huge progression margin, because they do have one and often think they don't.
Maybe we had a different experience with mathematics — I do think most people are primarily blocked by their fears (and also their misconceptions of what is actually at stake). This certainly applied to me, which may explain I'm particularly adamant on the topic.
We certainly had a different experience with mathematics, which I think shapes a lot of the discussion (which is delightful, by the way).
My experience seems to be almost the opposite of yours in every way. I've known pretty much as long as I can remember that I loved mathematics, and I stood out in all of my classes from kindergarten onward. Far from having a fear of it, I almost found a sort of refuge in it. Perhaps I enjoyed it so much because I could see why things were true, without having to take my teachers' word for it. Other kids seemed to have the opposite experience. They couldn't see why any of it was true, they just learned rules. So I tried to explain it to them, but spent much of my childhood getting blank stares.
The difference between French and US education is a bit paradoxical. You would think that the French would have a much more egalitarian ethos, but when it comes to public schools, almost the opposite is true. We have nothing like "Classe Préparatoire," and on the whole I would say US public schools tend to focus on the median student, doing very little to foster exceptional talent. That was certainly my experience. So while I saw from a distance how much brilliant math students could do with exceptional training, I was left to teach myself, as you say many mathematicians do. I ended up in a pretty good university, but certainly had nothing like the boot camp that allows France to produce so many Fields Medalists.
Speaking of Fields Medalists, allow me to give a comparison from my own experience that will help explain why I hate those quotes by Newton and Einstein. My own research is in the same domain as P.-L. Lions (and maybe Cedric Villani and Alessio Figalli, to name two other Fields Medalists). Now, when I read a paper by Lions or hear his lecture, I can follow along--I know what he is doing. To use your metaphor, I can at least get to the same finish line as he does. But by the time I do, he has gone on to 10 other projects, which he will finish by the time I even start my own. I think I'm doing the same thing as he is, and I can grasp the same ideas on a similar intuitive level, but there is simply no way I can keep up with his speed. Saying I just need to work harder and I can be just like Lions or Villani or Figalli would not be encouraging, but rather soul-crushing. I bet many mathematicians just like me feel the same way.
I understand perfectly well that cognitive work is not like lifting weights, and that intellectual progress is a lot like capital accumulation--it can increase exponentially. But that's exactly my point with respect to my own experience. Even if Lions is actually only 2 or 3 times faster than I am, clearly over a lifetime this is going to yield an overwhelming difference in productivity. I cannot double my speed. There is no point in comparing myself with such giants. At this point, whether or not such capacities are "hereditary" becomes a mere technicality. That meme with the photo of von Neumann? Even if it is technically wrong because there is a lot more than genetics going on, it's still a very real pill that many of us have to swallow.
Now, I understand that in your book, you're more interested in getting people who currently have little to no grasp of mathematics to get some idea of what it's really about. I think that is an admirable goal. But, all I can say is, good luck with that. As I said, I've been trying all my life to try to explain to other people what goes on in my head, and I really get a lot of blank stares. And this was the point of my 500 pound weight example. There are some things you simply cannot do until you have passed a certain threshhold. Maybe the situation is not hopeless, but I do think it's quite a steep uphill battle.
" I've known pretty much as long as I can remember that I loved mathematics, and I stood out in all of my classes from kindergarten onward."
=> Interestingly, this is a part of your experience that I shared, but due to personal issues I had terrible years in my early 20s, where I "lost" my talent. This experience of having to "re-develop" my ability from a much more insecure perspective shaped my understanding of how math really works.
This thread actually implies that another conjecture may be useful to the theory—but it will need some tightening up, beyond how I am about to present it—a conjecture that suggests an additional parameter in apparent cognitive ability: the presence of personal conviction. My experience as a physics student, an engineer, a teacher, a mother and an artist has shown me that self doubt can (and often does) create false ceilings to our cognitive potential, and overcoming these requires the evasive quality that I am calling ‘personal conviction.’ Being that conviction is a loaded word, there may be a more general and effective way to phrase this idea. The issue with the word ‘conviction’ is that it usually conveys a rigid mindset, but in this case, the conviction must exist (or be developed) at a fundamental level of the student’s identity, such that flexibility and resiliency to challenges become attributes of this personal conviction. I suppose it is a similar concept to the ‘passionate curiosity’ that Einstein claims. Maybe this is already baked into one of your conjectures…. I’d have to read through again.
Indeed. In my experience, this is a shifting parameter, subject to self-reinforcing feedback loops — people "specialize" at being smart and not being destabilized by temporary setbacks, as part of their personal and social identity.
I view this as a part of the "mental habits" that helps overcome the inhibitive factors of Conjecture 6.
>They couldn't see why any of it was true, they just learned rules. So I tried to explain it to them, but spent much of my childhood getting blank stares.
You know, there are basically two conclusions you can draw from this. One is that the people will not understand the subject no matter how hard they try (or that they will require much more effort than is reasonable to understand it), and another is that the explanations you were providing were unsatisfactory in some way. Personally, I have always been able to improve my explanations when I talk to a student, address where exactly their misunderstanding is coming from, learn what base of knowledge they currently have, and try and build up a new explanation from there.
I'm not intending to be rude, but I am somewhat inclined to believe that the latter conclusion is more likely here, given that you later say "As I said, I've been trying all my life to try to explain to other people what goes on in my head, and I really get a lot of blank stares." Education isn't about what's going on in *your* head, it's about what's going on in *theirs*.
I think there's a contradiction in your position worth examining.
You write: "Just because you have to work very, very hard to develop a gift doesn't mean that it isn't a gift."
But if something requires development, in what sense is it a gift? The word "gift" implies you receive it without earning it. "Develop" implies you build it through practice. These seem mutually exclusive.
Your weightlifting analogy assumes there's some cognitive equivalent to "lift 500 pounds" - some specific mental operation the rest of us physiologically cannot perform. But what is it? Can you name the actual cognitive move that greater minds employ that lesser minds simply cannot execute?
The 100m dash seems more apt precisely because everyone can run - just at different speeds and efficiency. We can all put one foot in front of the other and cross the finish line. The question becomes: why are some so much faster?
I think you're conflating energy spent thinking with effective thought. Not all cognitive effort is equal. Using your weightlifting analogy: it's the difference between lifting with impeccable form (force efficiently transferred along the vertical axis) versus sloppy form where most effort dissipates without moving the weight.
Same total energy expenditure, vastly different outcomes.
The critical variable isn't effort quantity but the specific policies employed when stuck. When Einstein or Grothendieck hit confusion, what did they do? Not in vague terms like "work hard" or "be curious," but as concrete cognitive actions: Did they switch representations? Generate examples? Draw diagrams? Test limits?
You mention trying to explain your stream of consciousness to students. That's valuable, but I suspect the crucial difference isn't what thoughts occur to you, but what you do when your initial approach fails. Do you persist in the same representation or immediately pivot? That's a trainable habit, not a genetic gift.
If Phelps's advantage were purely genetic, we'd expect his training methods to be useless for others. But swimmers who adopt elite training techniques do improve substantially - they just don't reach Phelps's level.
The question is whether the gap is from physiological limits (like bone structure or muscle fiber composition) or from uncopyable aspects of practice patterns that compound over decades.
For cognition, what's the equivalent of bone structure? What's the hard ceiling? I'm genuinely asking, because I don't see it clearly articulated in hereditarian arguments beyond vague appeals to "processing speed" or "working memory" - terms that aren't well-grounded in neuroscience and often just redescribe the performance gap rather than explaining it.
If someone gives you $10,000 as a gift, and then you turn it into a small business through hard work and shrewd investments, it was still a gift. This is a very common sense idea, absolutely no contradiction.
Your analogy assumes the existence of a measurable "$10,000 head start" - but what is it, specifically?
If cognitive advantage were primarily biological like your gift analogy suggests, we should be able to identify and measure it. The history of trying to find physical correlates of genius has been remarkably unsuccessful. Gauss's brain sat mislabeled in a jar for over a century because it was so unremarkable. Einstein's dissected brain showed no convincing peculiarities.
More tellingly: genius is almost always domain-specific. Von Neumann was transcendent in mathematics but ordinary at music. Feynman was brilliant at physics but struggled with homotopy groups. If the advantage were a general biological gift - faster processing, better working memory, superior neural hardware - why wouldn't it transfer across domains?
Yet when exceptional minds apply themselves outside their domain of expertise, they're often no better than average. How does your $10,000 gift explain that? Shouldn't superior hardware help everywhere?
The business analogy actually undermines your point. Yes, $10,000 helps - but there are countless people who start with that amount or more who never build successful businesses. Meanwhile, some build empires from $100. At what point does the initial capital become less important than the strategies employed?
If someone turns $10,000 into millions through "hard work and shrewd investments," those two factors - the specific practices, the decision-making patterns, the strategies that compound over time - seem far more explanatory than the initial gift. Remove the shrewd investments and hard work, and the $10,000 likely dwindles. Remove the initial $10,000 but keep the shrewd strategies, and success still seems probable, just delayed.
The question isn't whether some people have advantages. It's whether those advantages are the primary variable or whether the compounding effects of different practices over decades are doing most of the work. Your analogy seems to assume the former without establishing it.
As a long-time startup consultant, I've found that many successful entrepreneurs *did* found an empire with $100...an empire that grew to nearly $387 based on reselling candy to their third grade class. (Classic pattern: under legal threat from regulatory bodies such as Mrs. Baxter, they'd shut down and reopen after a while with Pokemon cards.) This behavior is typically serial. We talk about the risks of backing a "first-time founder," but the truth is, most of those first-time founders have a couple-three $100 empires under their belt.
I don't know whether the initial interest is mostly innate or is developed. I've seen attempts to develop it: a trend for many upper-middle-class Black families in Atlanta is to teach their kids entrepreneurialism by having them open a business. But the results are generally the same as giving the kid ballet lessons or soccer: keeps them occupied, nothing much comes of it usually.
Curiosity and a questioning attitude provide the fuel needed to relentlessly ponder a confusing topic. I can understand things in my 50s that put me to sleep in my 20s, even though I had an "interest" in learning them in my 20s. Back then, I didn't have the sufficient quality of curiosity that lights one's brain on fire. Also, not giving up and not expecting to understand in 5 minutes, an hour, a day, a week, a month or a year is also key. I wouldn't know how to teach the curiosity that I feel now.
I enjoyed reading your essay and it has really improved my idea extensively on the subject,here is my initial thoughts and ideas on the same subject-but purely from an epistemological point of view
One thing we all have to agree on—and a solid starting point—is that the brain you are born with and the environment you are exposed to play a massive role in how IQ and ability manifest over time. This is not controversial; it is undisputed.
Let me take an extreme vantage point by comparing three types of individuals. There are people who are born mentally impaired relative to the average population. Some children are born without full brain development, missing vital regions responsible for computing, abstraction, and the interpretation of complex internal and external feedback. No matter which teaching technique is applied, such an individual’s “hard drive”—the brain itself—does not have the capacity to execute certain tasks. The difference between an average individual and this person is clearly rooted in biological limitations of brain development.
That same principle applies when comparing highly gifted individuals to the average person. The very brain structures that separate the average individual from the impaired one are also what separate geniuses—such as Isaac Newton or Terence Tao—from the general population. Even with years of hard work and intentional cognitive optimization, an average individual given the same techniques would find it extraordinarily difficult to reach the same level of achievement without the necessary neurological hardware to support such complexity.
The second major factor is environment, and I’ll support this with a practical scenario. I come from Africa, and when one compares global intellectual contributions to those of the West, a clear difference emerges. If Albert Einstein had been born in Africa, it is highly unlikely he would have achieved what he did—not because of lack of intelligence, but because of conceptual and environmental limitations.
There are two environments that shape cognition: the internal environment of ideas and the external physical and cultural environment. Both provide feedback to the brain. A brain with sufficient capacity can use this feedback to refine information, generate abstractions, and create entirely new ideas. The probability that Africa has never produced someone with the raw cognitive potential of Einstein or Newton is close to zero. However, the lack of complex environmental architecture to augment and utilize such minds prevented full optimization of that potential.
Three hundred years ago, an individual with Einstein-level ability born in an African village would likely have become a master of traditional medicine, an exceptional storyteller, a wise village elder, or an ingenious local engineer inventing boats or tools. The domain of available knowledge directly shapes how the brain organizes, structures, and ultimately expresses its power. Intelligence does not operate in a vacuum; it is amplified—or constrained—by the richness of the environment it is allowed to interact with.
Take a genius and place him in an environment with little to no development in sophisticated foundational knowledge, and you will observe him rise only slightly above the intellectual ceiling of that environment. Without access to the conceptual architectures of mathematics, physics, biology, or formal logic, even a highly gifted mind lacks the scaffolding required to compound insight at scale. A genius without exposure to fundamental intellectual tools is like powerful hardware running without advanced software—capable, but constrained.
Without being handed a structured body of foundational knowledge to build upon, the mind of a genius will most likely reach a plateau far below what is theoretically possible. Compared to individuals with the same neurological hardware who are embedded in information-rich environments, the difference in outcome is dramatic. The limitation is not intelligence itself, but the absence of a framework that allows intelligence to recursively build upon prior discoveries.
Genius does not emerge in isolation. It requires raw cognitive capacity and a domain that provides deep, layered abstractions to interact with. When those abstractions are missing, the mind is forced to reinvent basic concepts rather than extend them, slowing progress exponentially.
Therefore, my argument is straightforward: cognitive excellence is shaped by two dominant forces—the biological capacity of the brain and the richness of both the internal and external intellectual environment. Remove either, and even extraordinary potential is capped. I have taken a very practically observable view on intelligence.
This is so intriguing. I'm wondering about this passage: 'This is the fundamental reason why educational interventions so often fail to move the needle. While they deterministically alter the primary stimuli, their impact on the secondary stimuli is always indirect and contingent to uncontrolled factors.' Could you give an example of a 'secondary stimulus' to clarify it a bit? And of uncontrolled factors?
Thank you James for your feedback. I will edit this passage as it definitely deserves an example. Here is one:
When you read a book, the primary stimulus is the ink on the page, the secondary stimuli are the mental imaginary and the train of thoughts that are prompted by the primary, and may linger on for minutes, hours, days, years.
This condition of thoughts lingering for minutes, hours, days, years reminds me of the cognitive step function between studying a subject (say calculus or probability) to demonstrate proficiency in problem solving through homework or exams vs studying a subject to obtain a level sincere mastery including history, nuance, edge cases, interpretation, context, questioning assumptions, practical usage, etc.. that may end up going down long rabbit holes into other topics and learning. The former is what our educational system encourages, while the latter seems more consistent with the cognitive development arc described in this work.
Makes me think of Steve Jobs in Sweden, "A computer is a delivery vehicle for software, just like a book is a delivery vehicle for its own kind of software." I think the software of a book is the secondary stimuli. So the book is the primary stimulus and the ideas or the feelings engendered, which as you say can linger for years, are the secondary. You read something and a light goes on!
I appreciate the connectome hypothesis and the emphasis on secondary stimuli. But I think there’s a missing layer: what constrains divergence? If intelligence is primarily a compounding developmental process, what prevents runaway amplification? Biology isn’t a free market. It’s a regulated dynamical system. So where is the physical regulator?
Intelligence is *by-product* of compounding cognitive elaboration.
Sometimes the process goes awry (eg delusions, paranoia, cults...)
I don't fully understand your question about physical regulator. In my view there are physical limitations to all this, and clear suppression mechanisms (my conjecture 6).
I think we may be talking past each other. I was asking a structural question, not about failure cases like delusions.
If intelligence is a compounding nonlinear process, then from a dynamical systems perspective the key issue is boundedness. Nonlinear feedback systems generically amplify. So what prevents runaway cognitive amplification?
In other words, what is the stabilizing invariant? What physical mechanism enforces bounded cognitive elaboration?
According to Conjecture 6, if I’m correct, inhibition acts as a damping term against recursive elaboration. My question then becomes: what constrains the inhibition mechanism itself? In nonlinear systems, the regulator must also be bounded. Is the stabilizing invariant metabolic, informational, thermodynamic, or environmental? I’m trying to understand where the global constraint lives.
There is a video on Youtube about a Japanese guy who tried the following experiment: What if someone kicks a ball 1 Million times.
He described himself as someone who was always the worst player in every team he played at. He also stopped playing for 7 years.
At around 100k kicks he managed to get a contract at a Polish second division team, and anyone who has ever played football knows how impressive it is to become pro.
It is very interesting to see how his skills develop. For a very long time it looks awful, but there is a tipping point where his kicks become elite.
Not only does he show that elite skills are achievable for a below average talented guy, but it is achievable by repeating simple drills.
Would that work in math? Can we collect 100k problems that would unlock elite skill? Can we do this for other things?
I am reading you with extreme interest. I am coming to similar conclusions about [some forms of] mental illness. I am starting to believe (though, like you, I am a million miles away from being able to prove) that the question of exactly who becomes mentally ill, and how bad it gets, is going to turn out to depend in very small part on genetics, and in very small part on environment, and in very large part on a third thing which is habits (in particular, habits around paying attention). The driving force is good habits in the case of mathematics, but harmful habits in the case of [some forms of] mental illness. Changing bad habits to good habits can drag you out of [some forms of] mental illness.
I think that one important thing to notice about habits is that their effect is recursive and hence can grow out of control and result in all sorts of surprising things. Like cancer cells reproducing all out of proportion to other cell types -- or like the growth of twigs on a tree and other fractals found in nature -- recursion is an incredibly powerful idea. I'm not a math head at all, but I do take some inspiration from a couple of mathy sources. Every geek's favorite book "Godel Escher Bach" stresses the importance of recursion, and so does another book I've been learning from recently, which is titled "The Computational Beauty of Nature" (Gary William Flake). I think this post of yours, like other things you have written, touches on this idea, but you don't seem to make much use of the words "recursion" or "recursive."
Recursion is probably important enough that it deserves to be understood as foundational to a lot of developmental processes. Instead of "nature or nurture?" perhaps someday people will ask "nature, nurture, or re-cur?"
Thank you Kent — yes, I agree that the model’s natural scope includes most behavioral traits, including psychopathology (but it should probably be extended to account for neurotransmitter mix and the activitiy of non-neural brain cells — brain “habits” are likely to also be materialized in non-cognitive manners — this also matters in the cognitive aspects I’m discussing, but becomes even more relevant for psychopathology.)
‘The brain is a learning device, not a computing device.’ That single sentence is the ultimate reality check for the IQ-obsessed. As an analyst, I find your Conjecture 5 ter—that extreme talent is the outcome of unusually effective metacognitive approaches—to be the most plausible explanation for the 'Pareto' distribution of achievement. We aren't looking at different hardware; we're looking at High-Fidelity Internal Modeling. If 'Attention is all we have,' then cognitive inequality is a reflection of how we've invested that limited resource over decades. It’s not a lottery; it’s a capitalization process.
Re the Raven’s Progressive Matrices example: “I have no idea how normal people approach a question like this one”. Let me try to explain.
I got the right answer (5, I think), but it probably took me about a minute. Hugely more cognitive load than 132+37.
And I realised as I read the paragraphs below it that I had taken a quite different approach. I **articulated in words** what the structure was. “There are three bars over the shapes in the first line…seem to be there in the second line too…they have different shading ….the bars are the same direction throughout each line…the shapes in the different lines seem similar too,, are they actually the same…”
I have to do all this. I do not have **any** eidetic perception as you describe. And my way seems much more dependent on working memory: I need to remember the story I am telling.
Now the context. I am an actuary, not a mathematician. I have always been painfully aware that my mathematical ability is poor, even for an actuary. I know this because I have written papers and done consulting work with much better mathematicians – not geniuses, but people in with maths PhD’s, including academics in university maths departments and two former IMO medallists.
I never took a course with a title like “Analysis” or “Linear Algebra “ or “Group Theory”. I did A-level (age 18) mathematics as a series or recipes. And then first year university “maths for social scientists”, again largely recipes. I passed these with high marks, but that was the end of my maths (or I now realise, not really maths) education.
On the practice point: despite my terrible pure maths skills, I can do compound interest arithmetic in my head. An investment grows 6x in 10 years, quick, what rate of growth is that? I think something like this: “6 factorises as 2x3, so apply the rule of 72 and the rule of 120 and add the answers….72/10 is 7.2 and 120/10 is 12, so that’s 7.2 plus 12 is 19.2% per annum.” Calculator says 1.192^10 = 5.79, so close enough to 6, and I got the answer while the other guy was still searching for a calculator!
Thank you Guy. An interesting question would be whether you could train to visualize the permutation matrices and symmetries of the picture. (Maybe with someone who sees them coaching you), and whether this would accelerate your response time to other matrices on the test.
I think the question of variable cognitive abilities only appears puzzling because of an implicit assumption—namely, that everyone is orienting their mental habits toward acquiring intelligence. Within that frame, differences in outcome do look surprising. But that’s only true if we assume equal commitment to that particular cognitive style, along with the changes in perception and daily thought it entails (or is at least suspected to entail).
But developing acuity in any domain requires inhabiting a certain regime of commitments, a way of allocating attention, effort, a willingness to label and parse experiences in certain ways. It is this framework that dictates how much of which mental muscle is used. Not all people think like mathematicians when they leave the office, many take that hat right off. And not everyone is willing to even enter, let alone sustain, the costs of being heavily committed to this (or any one) regime.
So what looks like a failure to train attention is better understood as indifference or refusal. People who reject it don’t reject attention, but a package deal: the broader form of life that this kind of attention supports.
It’s similar to people trying to become “health ambassadors” to lose weight, still talking in moralized language, craving cheeseburgers, having cheat days, etc. A genuine commitment regime is much more ordinary from the inside. My healthy disposition doesn’t feel like a habit, or like disciplined resistance against something more “naturally” appealing. I don’t feel like a moral hero because my body and mind are (luckily) in a state where good food tastes good, and fast food just doesn’t sound appetizing, so it doesn’t even enter my thoughts.
I suspect something similar is going on here. It’s not so much a failure to train attention, as a lack of commitment to enter and stay in the stable attractor of the kind of life in which that attention would actually take hold.
Coming from a neuroscientist, this was a great article, I'll probably be archiving it to reread.
I actually study learning impairments for a living - there are definitely genetic mutations/environmental influences/neurological stimuli/nootropic interventions that can improve learning capacity, just as there are those that can tank it.
Nonetheless, I think your theory is likely correct on its foundations, though I might quibble with it in some places (e.g. I don't know how innate differences in mental habits and metacognition might be, nor how vastly different they are even in adulthood; the fMRI studies, while interesting, have their own reproducibility issues and, furthermore, have no real metric for "magnitude" of difference). I think your proposed experiments are valuable means for testing these ideas.
One such metacognitive difference I can think of readily, that might vary wildly (perhaps on a curve) is the presence and frequency of an internal monologue. I am an individual on the "lower end" of this spectrum - I often don't have a monologue without significant cognitive effort, and so it's presence and frequency are low - but I have friends on the "higher end", whose monologues describe everything they see, do, think and feel at every moment of every day. No stillness, in other words. This 'style', for me at least, has meant I get sharp detail perception, low "latency", a great working memory, quickly grasp concepts, differences in certain executive functions (I'm almost impulsive, not indecisive). But, I have impairments, too - my imagination can feel a little weaker, my visual memory and abilities don't seem nearly as good as others, I am *very* bad at discovering "simple" solutions to practical problems, and so on. I'm fascinated by these differences between people and what impacts they might have in society, which is why I find your fundamental premise convincing - genetics are not the "it factor" in making geniuses, even if it can be a big contributer (a comparison might be you can make good swimmers, but Michael Phelps has genetic contributions that make him especially advantaged).
Definitely a lot to think about. Thanks for writing it!
-much of what is perceived as intelligent behavior is the serious internalization of the things one learns, real and genuine letting go and assumption of the role expert.
-spatialization of memory helps a lot and many wealthier geniuses of history were fortunate to literally see more places and learn more things than the typical person. also they were typically well versed in multiple languages and music and other artistic forms so they were exposed with sufficient regularity and depth that they could properly "map" things out in their brains such that these knowledge and capabilties are essentially first nature.
-as such id basically argue that what we consider intelligence is primarily a collection of behaviors that facilitates the production of novel, useful information for the wider populace. this is why many describe it in this way, there is no special mechanism in one brain versus the other, there is simply the behavioral repetition of activities which over time enable careful practitioners with enough time and quiet to see the cracks in the zeitgeist that they inherited.
It’s a one-two punch and, no, those punches are not equal.
Genetic traits determine connectivity and neurotransmitter profiles, which in turn leads to different (and relatively static) cognitive profiles. For example, huge differences between people great at forever tweaking their generative world model, even consciously, aka “curiosity” and “thinking” in the Einstein sense. And people who don’t but who in turn are great at absorbing the given structure of the world (instead of finding its deeper principles) and then work with extreme high “executive functioning”. Two almost incompatible existences that no secondary stimuli will meaningfully change without some major cognitive friction that is too costly to maintain over time anyway. And that’s ok.
But secondary stimuli are extremely important. An Einstein without the rigorous math and physics training could have never unlocked the worlds that he did. It’s a huge problem in the humanities. Lots of “generative thinkers” but since they have never encountered rigor nor a theory that actually works, they never manage to develop theories that unify fields or topics. They could cognitively but their education just doesn’t lay the foundations to do so.
Anyway, great first pass at a theory. I suggest maybe asking about the phenomenality of other people’s experiences and distilling the different cognitive types. I’d personally be super interested in your inner experience of doing math as someone in the “middle of the gap” compared to the top talent. Do you perhaps talk about your own inner experience somewhere else? In your book?
Read your book. It was absolutely wonderful, thank you for this!
I once had a brilliant mathematical physics prof tell me to stop thinking visually, even though it was exactly that which allowed me to me to follow and contribute to his lectures, and I have felt a little insecure about it ever since. Not anymore :).
The right advice might have been to learn to navigate back and forth between linguistic and visual thinking. Many young visual thinkers struggle with this, and this may have been what your teacher had noticed—but the advice was phrased in the worst possible manner!
Hi David, great post! The question of defining intelligence being that complex, it is nice to see an account both wide and simple. I found the focus on metacognition on point and refreshing. A few quick notes from a psyc/neuro perspective:
Conjectures 1&2 and Conjectures 3&4 seem to align with the notion of fluid vs crystallized intelligence, an alternative account to the g-factor. In fact, given the wide variety of definitions of intelligence (factor g, dual-factors model, triarchic model; plus the folk notions of emotional intelligence, social intelligence; and the notion of general intelligence used to characterize AI capacities), you may need to include other dimensions of information processing to account for the diversity of manifestations of intelligence.
There has been once a Cortical Minicolumn theory explaining that the higher neural density in local cortical columns compared to that of long-distance projections could account for strong performance in narrow capacities and poor performance in capacities that mobilize broad, diverse networks. Extrapolated, the consequence would be genius-level performance in narrow tasks but poor performance in wide tasks. Depending on societal needs, someone with strong narrow competence but poor generalizable skills could be evaluated as more intelligence than someone with strong broad competence but poor expertise skills - or the other way around.
Experiments in topological connectivity seem to suggest that more important than the number of connections, is ensuring efficiency via a) redundancy of signals, b) diversity of signal propagation solutions (e.g., permitting selective routing vs broad network diffusion; sometimes both at the same time), and c) temporal unfolding of neural cliques through the construction and deconstruction in time of network simplices (2d - > 3d - > nd - > 3d - > 2d).
Although it is a general statement, I think the best account of intelligence in your post is that "there are multiple neurodevelopment pathways, and each child follows a singular trajectory shaped by a unique combination of genetic, cultural, and emotional factors, and also freak events—with long-term consequences on their cognitive style.". Similarly, I think that "the talent gap is so wide that this isn’t a well-defined spot." summarizes the best the current state of discussions on intelligence we have.
If we could hypothetically ‘rewind’ an individual to the same initial state and expose them repeatedly to the same novel task, due to the stochastic nature of neural activity, we would likely observe a distribution of possible responses. In other words, although each child’s cognitive development follows a unique trajectory, every moment would contain branching possibilities.
The multiple realizability problem explains why we tend to observe in individuals diagnosed with different syndroms similar phenotypes and neural patterns. This is equally true for neurotypical people, where the means to achieve the same score on a benchmark are diverse.
Cognitive inhibition likely emerged alongside the development of executive functions. As brain volume expanded (fueled by increased protein intake) new cortical folding may have enabled the emergence of new cognitive functions. One of those would a counterbalance to our instinctive drives toward immediate, nearby rewards. By tempering spatial and temporal impulsivity, inhibition and planning allow us to prioritize long-term goals over short-term gratification. Olaf Sporns wrote a great deal about cortical development and brain network analysis.
Overall, I find the notion of intelligence pretty elusive. It needs to be addressed carefully and with operationalized means; at the same time it seems to better grasped with unoperationalized statements like that of Einstein you started with.
I'm writing a series on the notions of intelligence and cognition. I've released a first post where I distinguish both of them: intelligence as the capacity to solve narrow/general problems, and cognition as the organization that allows intelligent processes: https://substack.com/@valentinguigon/p-185651959
My point in the article is that cognition might not be the real concept we're interested in when we talk about intelligence. Check it out, it might bring water to your mill. The other posts will be about whether AI systems are cognitive systems.
The observation that cognitive performance in practice diverges much more widely than putative differences in hereditary IQ is neatly explained by the Dickens and Flynn hypothesis: the larger your (small) inborn advantage in some domain, the more you will be stimulated to develop that specific advantage through practice, thus amplifying the difference with others: Dickens, W. T., & Flynn, J. R. (2001). Heritability estimates versus large environmental effects: The IQ paradox resolved. PSYCHOLOGICAL REVIEW-NEW YORK-, 108(2), 346–369.
Hi David, lovely post. It resonates deeply with something I've been working through - not just understanding how metacognitive habits differ, but actually changing them in practice.
Some months ago, I started thinking about the brain as fundamentally conservative, settling into stable attractor states. The brain can't run on nouns - it needs trigger → action → feedback. Some stimuli comes in, the brain responds with a practiced move, and keeps going until it gets relief (prediction error drops). Most of our daily actions - maybe 95%+ - are repetitions of moves we've done before given the same triggers. You can see this vividly in something like football. Watch players dribbling and their gaits are nearly identical match to match. The brain finds movements that work and defaults to them because it's metabolically cheap.
The same thing happens with thinking. When students hit confusion, they cycle through remarkably predictable policies: re-read, ask "why don't I get this?", check the answer key, or skip it. The feedback loop closes - uncertainty gets temporarily resolved - and the brain marks this as successful even though nothing was actually learned.
Looking back at my own learning, I realized every time I got unstuck after prolonged struggle, it was after some pivot or action - trying a diagram, computing a case, explaining to myself. Never did wheel-spinning in the same mode resolve confusion.
What I also noticed was that often it took me extended periods of time to get unstuck - not because the problem was that hard, but because that's how long it took for desperation to build up enough to lower my threshold for seemingly absurd action. Action that then, to my surprise, actually resolved the confusion. But I'd never systematized this or looked at what preceded breakthroughs.
What I settled on was: detect confusion early and immediately force a representational shift. Switch from algebra to geometry, symbolic to numerical, abstract to physical - whatever gets you out of the current stuck state.
This maps directly onto your point about secondary stimuli. The difference between someone stuck and someone learning isn't the problem itself but what they do internally when stuck. Re-reading the same passage generates almost no new neural activity. Switching representational modes - visualizing geometrically, computing specific examples, explaining out loud - activates completely different pathways.
But here's the implementation problem: at peak uncertainty, when you're most confused, your capacity for novel action is lowest. This is exactly when the brain defaults to familiar ineffective responses. Knowing you should shift representations changes nothing.
What actually worked was externalizing the choice. I made cards with specific moves: "Compute Something," "Extreme Cases," "Remove One Part," "Reverse/Rotate/Swap," "Sit With Confusion," "Make Prediction." When stuck, I draw a card. This removes the hardest part - committing to action under uncertainty. Any shift away from the stuck state generates information.
The second thing: permission to execute wrong. After some initial tries, I came to find there's usually this hesitation where you're evaluating "how exactly should I do this?" That evaluation overhead often kills the attempt. Slows you down. So instead: execute wrong on purpose! Don't know which numbers to compute? Use obviously wrong ones. Try the extreme case even if it seems absurd. I found wrong execution generates information fast - you learn why it's wrong in seconds, which reveals constraints and hidden structure.
I realized that confusion as a state is actually rather similar across different domains. The brain being what it is - a trigger-action-feedback system - if I could install a different state-action mapping at this inflection point, it should have outsized effects. If the new policy reduces prediction error better than the old one, it should outcompete it naturally over repetitions.
And it did. After some weeks of forced practice with the cards: detection latency dropped from 90 seconds to maybe 20-30, execution became nearly immediate, and subjectively I started feeling 'restless' when stuck and not shifting. The new policy's prediction-error-reduction is so much better that staying in one representation now feels 'wrong'. That restlessness is the experiential signature of attractor replacement.
What strikes me is the magnitude of improvement relative to effort. Problems I previously thought required checking textbooks or were just beyond me now resolve through sustained exploration. It takes longer than looking up answers, but I build actual understanding and the capability to do it again.
The old policy feels like doomscrolling - minimal cognitive load, no new information, just anxiety relief. The new policy feels like exercise - more effortful initially, but generating new neural activation with each attempt.
Your secondary stimuli insight is particularly apt here. Each representational shift isn't just "thinking harder" - it's generating different internal experience. Algebraic manipulation, geometric visualization, numerical calculation activate distinct neural substrates. High-frequency switching means high-diversity neural exploration, which should drive connectome reorganization much faster than passive re-reading.
Before this I was cognitively sedentary when stuck - burning mental energy on anxiety while generating minimal neural activity. Now there's constant motion: try this angle, doesn't work, try that, learn something, try another. The volume of distinct cognitive states explored per unit time has increased dramatically.
But I think the compounding goes deeper than just executing pivots more reliably. The brain being a pattern-matching machine, over time it comes to map certain kinds of cues - internal or external - with certain moves. It realizes some moves work better to resolve prediction error than others as it gathers experience. This is essentially what intuition is.
The increased information density per unit time, from this active learning strategy, means the brain gets vastly more data to identify underlying patterns. Each night when you sleep, the brain compresses this enormous amount of information, replays it, consolidates it. Over months and years, this might explain how people who naturally default to this stance appear to have better "smell" for what to do and when. It compounds exponentially.
It's really akin to what you described as going into "math mode" - and what you said elsewhere: "This bizarre, almost childish attitude is extremely hard to communicate to outsiders." That's exactly it. It's like a kid picking up a toy and figuring out all the funny ways they can play with it, learning about its properties in the process. The little moves are cheap, easy. If they don't work, it doesn't mean much - you just do something else.
I don't think any of this is particularly revolutionary from a pedagogy or neuroscience standpoint - I suspect it's implicit to how a lot of mathematicians and physicists actually work. But it was revolutionary to me personally, this shift in cognitive stance.
If intelligence lives in the connectome and connectomes reorganize in response to activity patterns, this high-frequency representational shifting should accelerate development. Not immediately, but compounded over months and years the divergence could be substantial.
I think what made this work wasn't just understanding the principle (intellectually) but the operationalization. The cards externalize the decision at exactly the moment when your brain is least capable of making it. Wrong execution removes the evaluative layer that causes hesitation. Together they bypass the exact bottlenecks that prevent people from using techniques they intellectually know about.
This seems to instantiate your conjectures directly. The cards operationalize a specific trainable habit at the exact inflection point that determines learning trajectories. Within weeks of deliberate practice, a completely different response pattern installed itself.
Motion precedes clarity. That's the stance you start to embody.
Being an optimist, I think you might be understating the potential magnitude. If the constraint on cognitive development is primarily metacognitive habits rather than genetic ceiling, and if simple protocols can shift those habits within weeks, the accessible improvement might be larger than the "20% full glass" suggests.
The protocol costs essentially nothing - some index cards and permission to execute badly - but it forces precisely the kind of "peculiar rumination techniques" you describe elite mathematicians practicing. It's a way to operationalize what you call quality of attention, but as concrete actions anyone can systematically train.
Most importantly: it's trainable in the sense of actually installable, not just intellectually understandable. You need a detectable trigger, an externalized action protocol, permission to execute imperfectly, and volume of practice. The policy that better reduces prediction error wins naturally. No willpower required once the initial pattern starts to dominate.
Thanks for articulating this framework so clearly. It's nice to see my experience map onto your educated guesses about secondary stimuli, metacognitive habits, and compounding neural differences.
i want to start using your method. “Compute Something," "Extreme Cases," "Remove One Part," "Reverse/Rotate/Swap," "Sit With Confusion," "Make Prediction.". this has 6 options so using a dice could be perfect! Can you explain each of these: “make prediction”, “reverse/rotate/swap” and “remove one part”. I dont quite get what they mean. Id be grateful. Awesome comment btw.
Sure. The Make a Prediction card is about forcing your vague model to commit. When you've been turning a problem over for a while, you've quietly built some picture of how things work — but it stays fuzzy until you make it pay rent. So you take whatever understanding you've accrued and force it into a concrete, checkable claim before you look anything up.
Example: "If I double the mass but keep force the same, acceleration halves." Now you can check it. If you're wrong, you've learned exactly where the model breaks. If you're right, you actually own it now — you didn't just recognise the answer, you generated it. I find this card works best after 4-5 others, once you've gathered enough data to make a reasonable guess.
Remove One Part is the simplest card in the deck. Take one ingredient out — a term in an equation, a component in a circuit, a constraint in a problem — and watch what collapses. Whatever breaks when you remove it is what that part was actually doing. Whatever survives didn't need it.
Example — capacitor circuit:
Remove the resistance → current spikes instantly to infinity
Remove the capacitance → no charging behaviour at all
Now you know what each element is actually for, not just that it's there. I use this constantly when an equation has an ugly term I don't understand — I delete it and solve the simpler version first. The understanding you gain from that almost always lets you go back and handle the original.
Reverse/Rotate/Swap is about turning the problem inside out to expose structure you can't see from the front. The three options are:
Reverse time or direction
Rotate the setup or coordinate frame
Swap labels, inputs, or particles
After each one, ask: what breaks? what stays the same? The answers tell you what's essential and what was just an accident of how you framed it.
Example — block sliding with friction: Reverse time → the heat doesn't flow back into the block and restart its motion. That asymmetry tells you dissipation is a real physical fact, not a modelling choice you made.
One note: I've since split this card into two separate cards, because having three options on one card reintroduced the decision paralysis the whole deck is supposed to fix. Each card needs to be concrete enough to get you moving immediately, but open enough that you can execute it your own way.
A few things worth knowing before you start:
Pick at random, at least initially. The point isn't to choose the optimal card — it's to break the freeze. Any motion beats the loop of staring at the problem waiting for clarity that isn't coming. Even executing a card badly produces new information.
After a while you'll find yourself reaching for them less, because the moves become natural. You'll also notice you gravitate toward some cards more than others — that's fine, just weight your deck toward the ones that feel unfamiliar, since those are the ones still expanding how you think.
The cards are optimised for physics but most of them translate directly to mathematics — the underlying logic is the same. If you were a pure mathematician you might swap a few out, but start with these and adjust once you know which ones aren't pulling their weight.
Intelligence boils down a brain that has optimized efficiency; each of you has described the process brilliantly.
Academics who are equally interested in athletics seem to trend towards optimization. An example: I am an equestrian, and my ecuyer was in the habit of reminding, “Read, ride, reflect.” Integrating the metal and physical aspects of an idea reliably leads to epiphany.
Interesting- I love the phrase "metabolically cheap"- reminds me of "thrifty evolution" from
George Lakoff and Srini Narayanan’s The Neural Mind.
I think you end up with an extremely reasonable position. Sometimes when you push back against hereditarianism, you seem a bit starry-eyed about what we can accomplish by sheer force of will. In the end, though, you seem to admit that a lot of things are out of our control.
Personally, I still think all of those quotes--Newton, Einstein, Feynman, Grothendieck--are either disingenuous or incredibly naive. My own experience as a mathematician has not made me any less frustrated with these quotes. Quite the contrary, really. It feels as if geniuses feel obliged to remind everyone that actually they work very hard, as if we didn't already know. But I thought everybody knew that, just as they do for any other kind of excellence. Michael Phelps also had to work extremely hard to win all of those gold medals, but that doesn't mean that the rest of us can do it too if we just follow the same physical fitness regime that he did. Just because you have to work very, very hard to develop a gift doesn't mean that it isn't a gift.
Sometimes I think you exaggerate the difference between intellectual and physical prowess. You use a 100 meter dash as an example, because we all agree that all of us could at least finish, even if we're pathetically slow. But there are other activities with threshholds. Lifting weights, for example. The vast majority of us will just never be able to lift 500 pounds, even if we are given a week to do it.
Despite my criticisms, I will say that the extremely valuable part of your book and this post is a sort of research program to try to understand how we might be better trainers of cognitive ability. You're right to point out that physical activities are much more straightforward to model. Cognitive patterns are much more hidden. As a professor I've tried to explain to my students, as far as I'm able, my actual stream of consciousness that occurs to me when I approach a problem. Sometimes this can be a bit frightening to students, probably because, in addition to being hidden, cognitive behavior can be highly idiosyncratic. Still, there's probably a lot to be gained from trying to figure out the common patterns in the cognition of highly effective intellectuals. I imagine neuroscience will play a big role in this.
Dear Jameson,
Glad I'm ending up with an extremely reasonable position—I do think I'm an extremely reasonable person :) !
On the 500 pounds example: the success metric for weightlifting is how much you can lift. I'm not an expert on the matter, but I do suspect most people could, with adequate training, lift 100 pounds or even 200 and possibly more. This feels nowhere near the common perception of the math talent gap.
On this topic, I think you're missing the essential notion of conceptual compression—how things that initially seem unfathomably hard often become trivial once you develop a familiarity with the right conceptual framework. One of my favorite example is Hindu-Arabic numerals, through which you instantly "see" that 1,000,000,000 - 1 = 999,999,999, a computation that feels superhuman to someone who only knows Roman numerals (see this post based on a chapter from my book: https://davidbessis.substack.com/p/the-magic-of-mathematical-intuition)
There is no equivalent of conceptual compression for lifting 500 pounds and this is where, in my view, the analogy breaks down. Cognition isn't running or weightlifting, and this explains why insane inequalities can develop.
About what we can accomplish by sheer force of will — I am acutely aware of everyone's limits, yet I do think we should absolutely insist that people have a huge progression margin, because they do have one and often think they don't.
Maybe we had a different experience with mathematics — I do think most people are primarily blocked by their fears (and also their misconceptions of what is actually at stake). This certainly applied to me, which may explain I'm particularly adamant on the topic.
We certainly had a different experience with mathematics, which I think shapes a lot of the discussion (which is delightful, by the way).
My experience seems to be almost the opposite of yours in every way. I've known pretty much as long as I can remember that I loved mathematics, and I stood out in all of my classes from kindergarten onward. Far from having a fear of it, I almost found a sort of refuge in it. Perhaps I enjoyed it so much because I could see why things were true, without having to take my teachers' word for it. Other kids seemed to have the opposite experience. They couldn't see why any of it was true, they just learned rules. So I tried to explain it to them, but spent much of my childhood getting blank stares.
The difference between French and US education is a bit paradoxical. You would think that the French would have a much more egalitarian ethos, but when it comes to public schools, almost the opposite is true. We have nothing like "Classe Préparatoire," and on the whole I would say US public schools tend to focus on the median student, doing very little to foster exceptional talent. That was certainly my experience. So while I saw from a distance how much brilliant math students could do with exceptional training, I was left to teach myself, as you say many mathematicians do. I ended up in a pretty good university, but certainly had nothing like the boot camp that allows France to produce so many Fields Medalists.
Speaking of Fields Medalists, allow me to give a comparison from my own experience that will help explain why I hate those quotes by Newton and Einstein. My own research is in the same domain as P.-L. Lions (and maybe Cedric Villani and Alessio Figalli, to name two other Fields Medalists). Now, when I read a paper by Lions or hear his lecture, I can follow along--I know what he is doing. To use your metaphor, I can at least get to the same finish line as he does. But by the time I do, he has gone on to 10 other projects, which he will finish by the time I even start my own. I think I'm doing the same thing as he is, and I can grasp the same ideas on a similar intuitive level, but there is simply no way I can keep up with his speed. Saying I just need to work harder and I can be just like Lions or Villani or Figalli would not be encouraging, but rather soul-crushing. I bet many mathematicians just like me feel the same way.
I understand perfectly well that cognitive work is not like lifting weights, and that intellectual progress is a lot like capital accumulation--it can increase exponentially. But that's exactly my point with respect to my own experience. Even if Lions is actually only 2 or 3 times faster than I am, clearly over a lifetime this is going to yield an overwhelming difference in productivity. I cannot double my speed. There is no point in comparing myself with such giants. At this point, whether or not such capacities are "hereditary" becomes a mere technicality. That meme with the photo of von Neumann? Even if it is technically wrong because there is a lot more than genetics going on, it's still a very real pill that many of us have to swallow.
Now, I understand that in your book, you're more interested in getting people who currently have little to no grasp of mathematics to get some idea of what it's really about. I think that is an admirable goal. But, all I can say is, good luck with that. As I said, I've been trying all my life to try to explain to other people what goes on in my head, and I really get a lot of blank stares. And this was the point of my 500 pound weight example. There are some things you simply cannot do until you have passed a certain threshhold. Maybe the situation is not hopeless, but I do think it's quite a steep uphill battle.
Thanks for sharing this.
" I've known pretty much as long as I can remember that I loved mathematics, and I stood out in all of my classes from kindergarten onward."
=> Interestingly, this is a part of your experience that I shared, but due to personal issues I had terrible years in my early 20s, where I "lost" my talent. This experience of having to "re-develop" my ability from a much more insecure perspective shaped my understanding of how math really works.
This thread actually implies that another conjecture may be useful to the theory—but it will need some tightening up, beyond how I am about to present it—a conjecture that suggests an additional parameter in apparent cognitive ability: the presence of personal conviction. My experience as a physics student, an engineer, a teacher, a mother and an artist has shown me that self doubt can (and often does) create false ceilings to our cognitive potential, and overcoming these requires the evasive quality that I am calling ‘personal conviction.’ Being that conviction is a loaded word, there may be a more general and effective way to phrase this idea. The issue with the word ‘conviction’ is that it usually conveys a rigid mindset, but in this case, the conviction must exist (or be developed) at a fundamental level of the student’s identity, such that flexibility and resiliency to challenges become attributes of this personal conviction. I suppose it is a similar concept to the ‘passionate curiosity’ that Einstein claims. Maybe this is already baked into one of your conjectures…. I’d have to read through again.
Indeed. In my experience, this is a shifting parameter, subject to self-reinforcing feedback loops — people "specialize" at being smart and not being destabilized by temporary setbacks, as part of their personal and social identity.
I view this as a part of the "mental habits" that helps overcome the inhibitive factors of Conjecture 6.
(See also https://davidbessis.substack.com/p/why-genes-cant-explain-genius for the skewed distribution of talent, that is indicative of feedback loops.)
>They couldn't see why any of it was true, they just learned rules. So I tried to explain it to them, but spent much of my childhood getting blank stares.
You know, there are basically two conclusions you can draw from this. One is that the people will not understand the subject no matter how hard they try (or that they will require much more effort than is reasonable to understand it), and another is that the explanations you were providing were unsatisfactory in some way. Personally, I have always been able to improve my explanations when I talk to a student, address where exactly their misunderstanding is coming from, learn what base of knowledge they currently have, and try and build up a new explanation from there.
I'm not intending to be rude, but I am somewhat inclined to believe that the latter conclusion is more likely here, given that you later say "As I said, I've been trying all my life to try to explain to other people what goes on in my head, and I really get a lot of blank stares." Education isn't about what's going on in *your* head, it's about what's going on in *theirs*.
I think there's a contradiction in your position worth examining.
You write: "Just because you have to work very, very hard to develop a gift doesn't mean that it isn't a gift."
But if something requires development, in what sense is it a gift? The word "gift" implies you receive it without earning it. "Develop" implies you build it through practice. These seem mutually exclusive.
Your weightlifting analogy assumes there's some cognitive equivalent to "lift 500 pounds" - some specific mental operation the rest of us physiologically cannot perform. But what is it? Can you name the actual cognitive move that greater minds employ that lesser minds simply cannot execute?
The 100m dash seems more apt precisely because everyone can run - just at different speeds and efficiency. We can all put one foot in front of the other and cross the finish line. The question becomes: why are some so much faster?
I think you're conflating energy spent thinking with effective thought. Not all cognitive effort is equal. Using your weightlifting analogy: it's the difference between lifting with impeccable form (force efficiently transferred along the vertical axis) versus sloppy form where most effort dissipates without moving the weight.
Same total energy expenditure, vastly different outcomes.
The critical variable isn't effort quantity but the specific policies employed when stuck. When Einstein or Grothendieck hit confusion, what did they do? Not in vague terms like "work hard" or "be curious," but as concrete cognitive actions: Did they switch representations? Generate examples? Draw diagrams? Test limits?
You mention trying to explain your stream of consciousness to students. That's valuable, but I suspect the crucial difference isn't what thoughts occur to you, but what you do when your initial approach fails. Do you persist in the same representation or immediately pivot? That's a trainable habit, not a genetic gift.
If Phelps's advantage were purely genetic, we'd expect his training methods to be useless for others. But swimmers who adopt elite training techniques do improve substantially - they just don't reach Phelps's level.
The question is whether the gap is from physiological limits (like bone structure or muscle fiber composition) or from uncopyable aspects of practice patterns that compound over decades.
For cognition, what's the equivalent of bone structure? What's the hard ceiling? I'm genuinely asking, because I don't see it clearly articulated in hereditarian arguments beyond vague appeals to "processing speed" or "working memory" - terms that aren't well-grounded in neuroscience and often just redescribe the performance gap rather than explaining it.
If someone gives you $10,000 as a gift, and then you turn it into a small business through hard work and shrewd investments, it was still a gift. This is a very common sense idea, absolutely no contradiction.
Your analogy assumes the existence of a measurable "$10,000 head start" - but what is it, specifically?
If cognitive advantage were primarily biological like your gift analogy suggests, we should be able to identify and measure it. The history of trying to find physical correlates of genius has been remarkably unsuccessful. Gauss's brain sat mislabeled in a jar for over a century because it was so unremarkable. Einstein's dissected brain showed no convincing peculiarities.
More tellingly: genius is almost always domain-specific. Von Neumann was transcendent in mathematics but ordinary at music. Feynman was brilliant at physics but struggled with homotopy groups. If the advantage were a general biological gift - faster processing, better working memory, superior neural hardware - why wouldn't it transfer across domains?
Yet when exceptional minds apply themselves outside their domain of expertise, they're often no better than average. How does your $10,000 gift explain that? Shouldn't superior hardware help everywhere?
The business analogy actually undermines your point. Yes, $10,000 helps - but there are countless people who start with that amount or more who never build successful businesses. Meanwhile, some build empires from $100. At what point does the initial capital become less important than the strategies employed?
If someone turns $10,000 into millions through "hard work and shrewd investments," those two factors - the specific practices, the decision-making patterns, the strategies that compound over time - seem far more explanatory than the initial gift. Remove the shrewd investments and hard work, and the $10,000 likely dwindles. Remove the initial $10,000 but keep the shrewd strategies, and success still seems probable, just delayed.
The question isn't whether some people have advantages. It's whether those advantages are the primary variable or whether the compounding effects of different practices over decades are doing most of the work. Your analogy seems to assume the former without establishing it.
As a long-time startup consultant, I've found that many successful entrepreneurs *did* found an empire with $100...an empire that grew to nearly $387 based on reselling candy to their third grade class. (Classic pattern: under legal threat from regulatory bodies such as Mrs. Baxter, they'd shut down and reopen after a while with Pokemon cards.) This behavior is typically serial. We talk about the risks of backing a "first-time founder," but the truth is, most of those first-time founders have a couple-three $100 empires under their belt.
I don't know whether the initial interest is mostly innate or is developed. I've seen attempts to develop it: a trend for many upper-middle-class Black families in Atlanta is to teach their kids entrepreneurialism by having them open a business. But the results are generally the same as giving the kid ballet lessons or soccer: keeps them occupied, nothing much comes of it usually.
Curiosity and a questioning attitude provide the fuel needed to relentlessly ponder a confusing topic. I can understand things in my 50s that put me to sleep in my 20s, even though I had an "interest" in learning them in my 20s. Back then, I didn't have the sufficient quality of curiosity that lights one's brain on fire. Also, not giving up and not expecting to understand in 5 minutes, an hour, a day, a week, a month or a year is also key. I wouldn't know how to teach the curiosity that I feel now.
I could have written every word of this! (including the ages)
I enjoyed reading your essay and it has really improved my idea extensively on the subject,here is my initial thoughts and ideas on the same subject-but purely from an epistemological point of view
One thing we all have to agree on—and a solid starting point—is that the brain you are born with and the environment you are exposed to play a massive role in how IQ and ability manifest over time. This is not controversial; it is undisputed.
Let me take an extreme vantage point by comparing three types of individuals. There are people who are born mentally impaired relative to the average population. Some children are born without full brain development, missing vital regions responsible for computing, abstraction, and the interpretation of complex internal and external feedback. No matter which teaching technique is applied, such an individual’s “hard drive”—the brain itself—does not have the capacity to execute certain tasks. The difference between an average individual and this person is clearly rooted in biological limitations of brain development.
That same principle applies when comparing highly gifted individuals to the average person. The very brain structures that separate the average individual from the impaired one are also what separate geniuses—such as Isaac Newton or Terence Tao—from the general population. Even with years of hard work and intentional cognitive optimization, an average individual given the same techniques would find it extraordinarily difficult to reach the same level of achievement without the necessary neurological hardware to support such complexity.
The second major factor is environment, and I’ll support this with a practical scenario. I come from Africa, and when one compares global intellectual contributions to those of the West, a clear difference emerges. If Albert Einstein had been born in Africa, it is highly unlikely he would have achieved what he did—not because of lack of intelligence, but because of conceptual and environmental limitations.
There are two environments that shape cognition: the internal environment of ideas and the external physical and cultural environment. Both provide feedback to the brain. A brain with sufficient capacity can use this feedback to refine information, generate abstractions, and create entirely new ideas. The probability that Africa has never produced someone with the raw cognitive potential of Einstein or Newton is close to zero. However, the lack of complex environmental architecture to augment and utilize such minds prevented full optimization of that potential.
Three hundred years ago, an individual with Einstein-level ability born in an African village would likely have become a master of traditional medicine, an exceptional storyteller, a wise village elder, or an ingenious local engineer inventing boats or tools. The domain of available knowledge directly shapes how the brain organizes, structures, and ultimately expresses its power. Intelligence does not operate in a vacuum; it is amplified—or constrained—by the richness of the environment it is allowed to interact with.
Take a genius and place him in an environment with little to no development in sophisticated foundational knowledge, and you will observe him rise only slightly above the intellectual ceiling of that environment. Without access to the conceptual architectures of mathematics, physics, biology, or formal logic, even a highly gifted mind lacks the scaffolding required to compound insight at scale. A genius without exposure to fundamental intellectual tools is like powerful hardware running without advanced software—capable, but constrained.
Without being handed a structured body of foundational knowledge to build upon, the mind of a genius will most likely reach a plateau far below what is theoretically possible. Compared to individuals with the same neurological hardware who are embedded in information-rich environments, the difference in outcome is dramatic. The limitation is not intelligence itself, but the absence of a framework that allows intelligence to recursively build upon prior discoveries.
Genius does not emerge in isolation. It requires raw cognitive capacity and a domain that provides deep, layered abstractions to interact with. When those abstractions are missing, the mind is forced to reinvent basic concepts rather than extend them, slowing progress exponentially.
Therefore, my argument is straightforward: cognitive excellence is shaped by two dominant forces—the biological capacity of the brain and the richness of both the internal and external intellectual environment. Remove either, and even extraordinary potential is capped. I have taken a very practically observable view on intelligence.
This is so intriguing. I'm wondering about this passage: 'This is the fundamental reason why educational interventions so often fail to move the needle. While they deterministically alter the primary stimuli, their impact on the secondary stimuli is always indirect and contingent to uncontrolled factors.' Could you give an example of a 'secondary stimulus' to clarify it a bit? And of uncontrolled factors?
Thank you James for your feedback. I will edit this passage as it definitely deserves an example. Here is one:
When you read a book, the primary stimulus is the ink on the page, the secondary stimuli are the mental imaginary and the train of thoughts that are prompted by the primary, and may linger on for minutes, hours, days, years.
This condition of thoughts lingering for minutes, hours, days, years reminds me of the cognitive step function between studying a subject (say calculus or probability) to demonstrate proficiency in problem solving through homework or exams vs studying a subject to obtain a level sincere mastery including history, nuance, edge cases, interpretation, context, questioning assumptions, practical usage, etc.. that may end up going down long rabbit holes into other topics and learning. The former is what our educational system encourages, while the latter seems more consistent with the cognitive development arc described in this work.
Makes me think of Steve Jobs in Sweden, "A computer is a delivery vehicle for software, just like a book is a delivery vehicle for its own kind of software." I think the software of a book is the secondary stimuli. So the book is the primary stimulus and the ideas or the feelings engendered, which as you say can linger for years, are the secondary. You read something and a light goes on!
I appreciate the connectome hypothesis and the emphasis on secondary stimuli. But I think there’s a missing layer: what constrains divergence? If intelligence is primarily a compounding developmental process, what prevents runaway amplification? Biology isn’t a free market. It’s a regulated dynamical system. So where is the physical regulator?
Intelligence is *by-product* of compounding cognitive elaboration.
Sometimes the process goes awry (eg delusions, paranoia, cults...)
I don't fully understand your question about physical regulator. In my view there are physical limitations to all this, and clear suppression mechanisms (my conjecture 6).
Thank you for the response David.
I think we may be talking past each other. I was asking a structural question, not about failure cases like delusions.
If intelligence is a compounding nonlinear process, then from a dynamical systems perspective the key issue is boundedness. Nonlinear feedback systems generically amplify. So what prevents runaway cognitive amplification?
In other words, what is the stabilizing invariant? What physical mechanism enforces bounded cognitive elaboration?
According to Conjecture 6, if I’m correct, inhibition acts as a damping term against recursive elaboration. My question then becomes: what constrains the inhibition mechanism itself? In nonlinear systems, the regulator must also be bounded. Is the stabilizing invariant metabolic, informational, thermodynamic, or environmental? I’m trying to understand where the global constraint lives.
There is a video on Youtube about a Japanese guy who tried the following experiment: What if someone kicks a ball 1 Million times.
He described himself as someone who was always the worst player in every team he played at. He also stopped playing for 7 years.
At around 100k kicks he managed to get a contract at a Polish second division team, and anyone who has ever played football knows how impressive it is to become pro.
It is very interesting to see how his skills develop. For a very long time it looks awful, but there is a tipping point where his kicks become elite.
Not only does he show that elite skills are achievable for a below average talented guy, but it is achievable by repeating simple drills.
Would that work in math? Can we collect 100k problems that would unlock elite skill? Can we do this for other things?
Hi David,
I am reading you with extreme interest. I am coming to similar conclusions about [some forms of] mental illness. I am starting to believe (though, like you, I am a million miles away from being able to prove) that the question of exactly who becomes mentally ill, and how bad it gets, is going to turn out to depend in very small part on genetics, and in very small part on environment, and in very large part on a third thing which is habits (in particular, habits around paying attention). The driving force is good habits in the case of mathematics, but harmful habits in the case of [some forms of] mental illness. Changing bad habits to good habits can drag you out of [some forms of] mental illness.
I think that one important thing to notice about habits is that their effect is recursive and hence can grow out of control and result in all sorts of surprising things. Like cancer cells reproducing all out of proportion to other cell types -- or like the growth of twigs on a tree and other fractals found in nature -- recursion is an incredibly powerful idea. I'm not a math head at all, but I do take some inspiration from a couple of mathy sources. Every geek's favorite book "Godel Escher Bach" stresses the importance of recursion, and so does another book I've been learning from recently, which is titled "The Computational Beauty of Nature" (Gary William Flake). I think this post of yours, like other things you have written, touches on this idea, but you don't seem to make much use of the words "recursion" or "recursive."
Recursion is probably important enough that it deserves to be understood as foundational to a lot of developmental processes. Instead of "nature or nurture?" perhaps someday people will ask "nature, nurture, or re-cur?"
Thanks for continuing to write on this stuff.
--Kent
Thank you Kent — yes, I agree that the model’s natural scope includes most behavioral traits, including psychopathology (but it should probably be extended to account for neurotransmitter mix and the activitiy of non-neural brain cells — brain “habits” are likely to also be materialized in non-cognitive manners — this also matters in the cognitive aspects I’m discussing, but becomes even more relevant for psychopathology.)
‘The brain is a learning device, not a computing device.’ That single sentence is the ultimate reality check for the IQ-obsessed. As an analyst, I find your Conjecture 5 ter—that extreme talent is the outcome of unusually effective metacognitive approaches—to be the most plausible explanation for the 'Pareto' distribution of achievement. We aren't looking at different hardware; we're looking at High-Fidelity Internal Modeling. If 'Attention is all we have,' then cognitive inequality is a reflection of how we've invested that limited resource over decades. It’s not a lottery; it’s a capitalization process.
Re the Raven’s Progressive Matrices example: “I have no idea how normal people approach a question like this one”. Let me try to explain.
I got the right answer (5, I think), but it probably took me about a minute. Hugely more cognitive load than 132+37.
And I realised as I read the paragraphs below it that I had taken a quite different approach. I **articulated in words** what the structure was. “There are three bars over the shapes in the first line…seem to be there in the second line too…they have different shading ….the bars are the same direction throughout each line…the shapes in the different lines seem similar too,, are they actually the same…”
I have to do all this. I do not have **any** eidetic perception as you describe. And my way seems much more dependent on working memory: I need to remember the story I am telling.
Now the context. I am an actuary, not a mathematician. I have always been painfully aware that my mathematical ability is poor, even for an actuary. I know this because I have written papers and done consulting work with much better mathematicians – not geniuses, but people in with maths PhD’s, including academics in university maths departments and two former IMO medallists.
I never took a course with a title like “Analysis” or “Linear Algebra “ or “Group Theory”. I did A-level (age 18) mathematics as a series or recipes. And then first year university “maths for social scientists”, again largely recipes. I passed these with high marks, but that was the end of my maths (or I now realise, not really maths) education.
On the practice point: despite my terrible pure maths skills, I can do compound interest arithmetic in my head. An investment grows 6x in 10 years, quick, what rate of growth is that? I think something like this: “6 factorises as 2x3, so apply the rule of 72 and the rule of 120 and add the answers….72/10 is 7.2 and 120/10 is 12, so that’s 7.2 plus 12 is 19.2% per annum.” Calculator says 1.192^10 = 5.79, so close enough to 6, and I got the answer while the other guy was still searching for a calculator!
Thank you Guy. An interesting question would be whether you could train to visualize the permutation matrices and symmetries of the picture. (Maybe with someone who sees them coaching you), and whether this would accelerate your response time to other matrices on the test.
I think the question of variable cognitive abilities only appears puzzling because of an implicit assumption—namely, that everyone is orienting their mental habits toward acquiring intelligence. Within that frame, differences in outcome do look surprising. But that’s only true if we assume equal commitment to that particular cognitive style, along with the changes in perception and daily thought it entails (or is at least suspected to entail).
But developing acuity in any domain requires inhabiting a certain regime of commitments, a way of allocating attention, effort, a willingness to label and parse experiences in certain ways. It is this framework that dictates how much of which mental muscle is used. Not all people think like mathematicians when they leave the office, many take that hat right off. And not everyone is willing to even enter, let alone sustain, the costs of being heavily committed to this (or any one) regime.
So what looks like a failure to train attention is better understood as indifference or refusal. People who reject it don’t reject attention, but a package deal: the broader form of life that this kind of attention supports.
It’s similar to people trying to become “health ambassadors” to lose weight, still talking in moralized language, craving cheeseburgers, having cheat days, etc. A genuine commitment regime is much more ordinary from the inside. My healthy disposition doesn’t feel like a habit, or like disciplined resistance against something more “naturally” appealing. I don’t feel like a moral hero because my body and mind are (luckily) in a state where good food tastes good, and fast food just doesn’t sound appetizing, so it doesn’t even enter my thoughts.
I suspect something similar is going on here. It’s not so much a failure to train attention, as a lack of commitment to enter and stay in the stable attractor of the kind of life in which that attention would actually take hold.
Coming from a neuroscientist, this was a great article, I'll probably be archiving it to reread.
I actually study learning impairments for a living - there are definitely genetic mutations/environmental influences/neurological stimuli/nootropic interventions that can improve learning capacity, just as there are those that can tank it.
Nonetheless, I think your theory is likely correct on its foundations, though I might quibble with it in some places (e.g. I don't know how innate differences in mental habits and metacognition might be, nor how vastly different they are even in adulthood; the fMRI studies, while interesting, have their own reproducibility issues and, furthermore, have no real metric for "magnitude" of difference). I think your proposed experiments are valuable means for testing these ideas.
One such metacognitive difference I can think of readily, that might vary wildly (perhaps on a curve) is the presence and frequency of an internal monologue. I am an individual on the "lower end" of this spectrum - I often don't have a monologue without significant cognitive effort, and so it's presence and frequency are low - but I have friends on the "higher end", whose monologues describe everything they see, do, think and feel at every moment of every day. No stillness, in other words. This 'style', for me at least, has meant I get sharp detail perception, low "latency", a great working memory, quickly grasp concepts, differences in certain executive functions (I'm almost impulsive, not indecisive). But, I have impairments, too - my imagination can feel a little weaker, my visual memory and abilities don't seem nearly as good as others, I am *very* bad at discovering "simple" solutions to practical problems, and so on. I'm fascinated by these differences between people and what impacts they might have in society, which is why I find your fundamental premise convincing - genetics are not the "it factor" in making geniuses, even if it can be a big contributer (a comparison might be you can make good swimmers, but Michael Phelps has genetic contributions that make him especially advantaged).
Definitely a lot to think about. Thanks for writing it!
Thank you!
really enjoyed reading this. a few thoughts:
-much of what is perceived as intelligent behavior is the serious internalization of the things one learns, real and genuine letting go and assumption of the role expert.
-spatialization of memory helps a lot and many wealthier geniuses of history were fortunate to literally see more places and learn more things than the typical person. also they were typically well versed in multiple languages and music and other artistic forms so they were exposed with sufficient regularity and depth that they could properly "map" things out in their brains such that these knowledge and capabilties are essentially first nature.
-as such id basically argue that what we consider intelligence is primarily a collection of behaviors that facilitates the production of novel, useful information for the wider populace. this is why many describe it in this way, there is no special mechanism in one brain versus the other, there is simply the behavioral repetition of activities which over time enable careful practitioners with enough time and quiet to see the cracks in the zeitgeist that they inherited.
It’s a one-two punch and, no, those punches are not equal.
Genetic traits determine connectivity and neurotransmitter profiles, which in turn leads to different (and relatively static) cognitive profiles. For example, huge differences between people great at forever tweaking their generative world model, even consciously, aka “curiosity” and “thinking” in the Einstein sense. And people who don’t but who in turn are great at absorbing the given structure of the world (instead of finding its deeper principles) and then work with extreme high “executive functioning”. Two almost incompatible existences that no secondary stimuli will meaningfully change without some major cognitive friction that is too costly to maintain over time anyway. And that’s ok.
But secondary stimuli are extremely important. An Einstein without the rigorous math and physics training could have never unlocked the worlds that he did. It’s a huge problem in the humanities. Lots of “generative thinkers” but since they have never encountered rigor nor a theory that actually works, they never manage to develop theories that unify fields or topics. They could cognitively but their education just doesn’t lay the foundations to do so.
Anyway, great first pass at a theory. I suggest maybe asking about the phenomenality of other people’s experiences and distilling the different cognitive types. I’d personally be super interested in your inner experience of doing math as someone in the “middle of the gap” compared to the top talent. Do you perhaps talk about your own inner experience somewhere else? In your book?
Thanks!
“Do you perhaps talk about your own inner experience somewhere else? In your book?”
=> Yes, the inner experience of mathematical cognition is the central topic of my book.
Read your book. It was absolutely wonderful, thank you for this!
I once had a brilliant mathematical physics prof tell me to stop thinking visually, even though it was exactly that which allowed me to me to follow and contribute to his lectures, and I have felt a little insecure about it ever since. Not anymore :).
Great to hear that!
The right advice might have been to learn to navigate back and forth between linguistic and visual thinking. Many young visual thinkers struggle with this, and this may have been what your teacher had noticed—but the advice was phrased in the worst possible manner!
Hi David, great post! The question of defining intelligence being that complex, it is nice to see an account both wide and simple. I found the focus on metacognition on point and refreshing. A few quick notes from a psyc/neuro perspective:
Conjectures 1&2 and Conjectures 3&4 seem to align with the notion of fluid vs crystallized intelligence, an alternative account to the g-factor. In fact, given the wide variety of definitions of intelligence (factor g, dual-factors model, triarchic model; plus the folk notions of emotional intelligence, social intelligence; and the notion of general intelligence used to characterize AI capacities), you may need to include other dimensions of information processing to account for the diversity of manifestations of intelligence.
There has been once a Cortical Minicolumn theory explaining that the higher neural density in local cortical columns compared to that of long-distance projections could account for strong performance in narrow capacities and poor performance in capacities that mobilize broad, diverse networks. Extrapolated, the consequence would be genius-level performance in narrow tasks but poor performance in wide tasks. Depending on societal needs, someone with strong narrow competence but poor generalizable skills could be evaluated as more intelligence than someone with strong broad competence but poor expertise skills - or the other way around.
Experiments in topological connectivity seem to suggest that more important than the number of connections, is ensuring efficiency via a) redundancy of signals, b) diversity of signal propagation solutions (e.g., permitting selective routing vs broad network diffusion; sometimes both at the same time), and c) temporal unfolding of neural cliques through the construction and deconstruction in time of network simplices (2d - > 3d - > nd - > 3d - > 2d).
Although it is a general statement, I think the best account of intelligence in your post is that "there are multiple neurodevelopment pathways, and each child follows a singular trajectory shaped by a unique combination of genetic, cultural, and emotional factors, and also freak events—with long-term consequences on their cognitive style.". Similarly, I think that "the talent gap is so wide that this isn’t a well-defined spot." summarizes the best the current state of discussions on intelligence we have.
If we could hypothetically ‘rewind’ an individual to the same initial state and expose them repeatedly to the same novel task, due to the stochastic nature of neural activity, we would likely observe a distribution of possible responses. In other words, although each child’s cognitive development follows a unique trajectory, every moment would contain branching possibilities.
The multiple realizability problem explains why we tend to observe in individuals diagnosed with different syndroms similar phenotypes and neural patterns. This is equally true for neurotypical people, where the means to achieve the same score on a benchmark are diverse.
Cognitive inhibition likely emerged alongside the development of executive functions. As brain volume expanded (fueled by increased protein intake) new cortical folding may have enabled the emergence of new cognitive functions. One of those would a counterbalance to our instinctive drives toward immediate, nearby rewards. By tempering spatial and temporal impulsivity, inhibition and planning allow us to prioritize long-term goals over short-term gratification. Olaf Sporns wrote a great deal about cortical development and brain network analysis.
Overall, I find the notion of intelligence pretty elusive. It needs to be addressed carefully and with operationalized means; at the same time it seems to better grasped with unoperationalized statements like that of Einstein you started with.
Thank you Valentin, very interesting! It’s intimidating to read comments by people who are evidently more competent than I am on the topic ;)
Sure. Hope I didn't dump too much.
I'm writing a series on the notions of intelligence and cognition. I've released a first post where I distinguish both of them: intelligence as the capacity to solve narrow/general problems, and cognition as the organization that allows intelligent processes: https://substack.com/@valentinguigon/p-185651959
My point in the article is that cognition might not be the real concept we're interested in when we talk about intelligence. Check it out, it might bring water to your mill. The other posts will be about whether AI systems are cognitive systems.
The observation that cognitive performance in practice diverges much more widely than putative differences in hereditary IQ is neatly explained by the Dickens and Flynn hypothesis: the larger your (small) inborn advantage in some domain, the more you will be stimulated to develop that specific advantage through practice, thus amplifying the difference with others: Dickens, W. T., & Flynn, J. R. (2001). Heritability estimates versus large environmental effects: The IQ paradox resolved. PSYCHOLOGICAL REVIEW-NEW YORK-, 108(2), 346–369.