For me, your blog posts cut deeper and provide a better explanation of your thesis than your (marvelous) book did. You get better at explaining your ideas over time. Perhaps you could do a blog post giving all the ways you know to develop math intuition. Having your ideas on that topic concentrated in one place rather than spread out through an entire book would be a huge benefit for many people. Instructors could then link to your post for students. A succinct blog post on this topic is likely to have a greater impact on students than an entire book they are less likely to read. My 2 cents ;-)
Thank you Travis! Yes, writing is like math, in that you continuously progress and gradually gain clarity.
Thanks also for the suggestion, which is definitely on my radar. Right now I'm laying the foundations, as I want to make sure that the more "general audience"/"easy" posts won't be brushed off by people who, a bit like Reader X, will underestimate the stakes. It's easy to dismiss as pep talk what it is actually a serious take on how mathematics function.
The interplay between the formalism of proofs and intuition is also a fascinating area. You've described the enormous difference between talking 1 on 1 with someone about a mathematical idea vs listening to talk vs reading a paper/book, and how it is so much more efficient to talk with someone 1 on 1. I wonder what information is being communicated 1 on 1 that isn't communicated in the formalism. Why couldn't the 1 on 1 communication be encapsulated in writing?
1 on 1 benefits for dynamic personalization: you see how the other person reacts, you notice the blank stare when they don't understand, you reuse the language they're comfortable, you find metaphors and examples that resonate with them... this is a game-changer.
I have to laugh, because I can't think of anyone who wouldn't benefit from reading this book, and I think anyone who is curious about how to better understand the world would enjoy reading it. I did.
I paint, and I hear all the time, "Oh, I could never paint like that. You are so talented!" Same old debate. It isn't talent. It is training.
Bonjour. Je suis pianiste et compositrice et je trouve que ce débat se retrouve de façon assez similaire en musique. J'ai beaucoup de plaisir à vous lire et à faire des parallèles. Bravo et merci. Hâte de lire le prochain message.
Personally, I'm happy you chose to become an author. Your content is consistently engaging and thought-provoking. Plus, you make considerable efforts to discuss your opinions with the public, which is unparalleled. I hope that you have a long and fruitful career as an author.
After reading your Substack posts, I found one point intriguing and would like to ask a question about it.
In your opinion, will changing the formal definition of math lead to concrete changes in what counts as math? For example, non-mathematicians might formalize their abstractions from their professional field and might study them through symbol manipulation and publish their findings. According to the "conceptualist characterization of math"*, will we regard them as mathematicians, or will we consider them doing math? If the answer is yes, what will distinguish math from pseudo-math? Actually I know such work already exist as part of interdisciplinary fields but I was wondering if what counts as professional math might have become broader due to lower barriers to entry.
If these questions don't make sense or are not answerable at this point, please feel free to skip them.
Thanks a lot.
*"a mental activity based on manipulating formal systems and imagining that they hold absolute and immutable truths about real objects."
For a complete evisceration of the view of nature vs nurture as a dichotomy, see Evelyn Fox Keller’s The Mirage of a Space between Nature and Nurture. She finds most of the discourse on the subject incoherent.
I loved your book. As an investor and a fan of Buffett, I think investing represents another instance of your argument against a pure genetic explanation of an outlier. Buffett has accumulated a net worth of the order $100 billion whereas most people are not even millionaires. He has written and spoken a lot about what explains this 100,000x difference. He attributes is to the magic of compounding over time. I think in investing the equivalent of honing "intuition" is: honing a tolerance of being against the crowd; honing the ability to see through the "noise" of volatile market price to focus on the intrinsic value of the invested asset; and honing the patience required to hold (sometimes for years!) before the investment finally delivers returns.
Here is my take. Mathematical ability, like most things in life, is a combination of genetics and environment. It is silly to deny that either one matters. Of course if you're an individual mathematician, you can't your genes at least not until we develop safe and effective ways to do CRISPR in adults. But you can change your environment. So it is more productive to focus on that.
I'm sure that the best mathematicians in the world all have significantly above average polygenic educational attainment and IQ scores. But I'm very wary of claims like "people with an IQ of X cannot do/learn Y, there is a hard ceiling". Calculus was cutting edge a few centuries ago and only the best mathematicians did it, and now it can be taught to a lot of people.
Imagine being 7 years old and realizing your parents love you less because of the math grade on your report card.
For the rest of your childhood, this happens twice a year. On top of that you have a constant cycle of fear and stress about math homework and frequent tests which you can never quite manage to do. Your bad grades mean you go to a worse college and you're cut off from lots of high paying careers.
Well, this is very painful, but at least it's not your fault -- math ability is just innate, and you don't have it.
But then this guy David Bessis shows up, and he has this whole book titled Mathematica, the gist of which is -- it IS your fault!
If you had just had the proper virtuous attitude of curiosity, you wouldn't have struggled with math. Your whole childhood would have been more fun, you'd be more successful now, and every year your parents would be happy to see you at Thanksgiving. The only person to blame for all this pain in your life, was you -- you deserve it.
Obviously what I've laid out is a flagrant misinterpretation of your work, but it doesn't surprise me that it provokes irrational and angry criticism around the idea of innate ability. It's a highly emotional issue.
(For mathematicians and scientists who do have advanced mathematical skills and never had the childhood pain around math education, I think the emotional story is a little different -- the belief in innate ability helps to both cope with peers out-achieving you, and also manage anxiety about falling behind. Belief in innate ability is serving the same load-bearing purpose just in a different structure.)
"Imagine being 7 years old and realizing your parents love you less because of the math grade on your report card.
For the rest of your childhood, this happens twice a year. On top of that you have a constant cycle of fear and stress about math homework and frequent tests which you can never quite manage to do. Your bad grades mean you go to a worse college and you're cut off from lots of high paying careers."
I don't know about genetics although my mom was a chemistry and math major however, I can comment on curiosity. I've always been more curious than anyone I know (didn't stop me from blowing out of an all paid 4 year scholarship, in an honors program to an MS in 4, after 2 years. In old age I think what was different is "focus", "concentration". With this curiosity I lacked the focus to pursue one thing (as Curly said in "City Slickers"). With too many interesting things, wandering was the rule. It worked out fine. I wandered through a myriad of interesting jobs, was rewarded overall. Focus ... an unappreciated dimension.
The working title "Secret Math" may deserve its fate, because for an idea to be secret, the holder needs to intend to avoid disclosing it. A secret is clandestine, concealed, confidential, furtive, hidden, private, or surreptitious. The centuries of failed attempts to disclose this "secret" demonstrate its invisibility, not its secretness. It is overlooked, unnoticed, unseen, disregarded, inconspicuous, unobtrusive, or unperceived.
You write a careful long form post explaining how hereditarian myths make it nearly impossible to have a serious discussion about mathematical cognition. You know the subject quite well, because you're a former career mathematician who literally spent decades thinking about the topic. You wrote a serious and well-documented book that was published by rigorous academic press and endorsed by leading mathematicians.
You know that it's a sensitive topic, so you've been careful to acknowledge that genes play a role in mathematical capacity, reiterating your view that "There’s no doubt that natural aptitude in math, like natural aptitude in any other physical activity, isn’t equally distributed among individuals." (a citation from your book)
And yet, within 3 days, you're accused of blank slatism.
Substack is marginally better than X, though, because on X you get these comments within 10 minutes and they usually contain insults.
For what it's worth, I found your careful caveating, rebuttal and claims interesting enough to buy your book.
To frame where I'm coming from, vis a vis math, I was good enough at math to pick to a computer science degree and I'm curious enough now to bother any mathematicians I run into about their work (I've got a brother in law who's a topologist who hasn't gotten tired of it yet), but I wouldn't say I've ever seriously pursued math.
I've been pondering a lot recently on how I learn complicated subjects, often by rapidly creating and abandoning fuzzy abstractions until I get an intuitive sense that something worked, at which point I'll go back and find a post hoc objective mechanical justification. Over time some of those tricks keep on working, and the abstractions get sharper and more durable in my head. It's been a fair number of years since I've done that to learn math, but I do it all the time to avoid plateauing in other skills.
I'm excited to read your book and see if it gives me language to talk about those internal mental processes and maybe some tricks to get better at learning. Maybe learn some more about math on the way too.
Yeah I think that dismissing your views as “blank slate-ism” is very silly.
As someone with pretty strongly hereditarian views, including on controversial things like group differences, I actually don’t find much to disagree with in what you write. No question most things are a combination of genetics and environment, mathematical output no exception.
Thanks for your comment. Thanks also for reminding me that some of my readers have strong hereditarian views, which is essential for me to keep in mind (especially since I’ll be coming back to these subjects in the near future.)
Ha, a fair critique of my snide comment. Its true you're not actually promoting blank slatism but I have to agree with the reviewer and many of the like-minded commenters online that your foray into disproving innate mathematical ability is as ill-conceived as my comment and probably does detract from your book (but certainly aids visibility). From reading your previous posts on X and substack I can see that you seem to misunderstand the overwhelming (plurality at minimum, most likely majority) innateness of the non-shared environment, and severely range restrict your sample (those publishing in math journals) to the tails of global mathematical ability to disprove the polygenic nature of mathematical ability. Tao, Von Neumann, Gauss, Ramanujan, etc. have/had innate mathematical talent far beyond most of humanity and that played a large role in their success.
For me, your blog posts cut deeper and provide a better explanation of your thesis than your (marvelous) book did. You get better at explaining your ideas over time. Perhaps you could do a blog post giving all the ways you know to develop math intuition. Having your ideas on that topic concentrated in one place rather than spread out through an entire book would be a huge benefit for many people. Instructors could then link to your post for students. A succinct blog post on this topic is likely to have a greater impact on students than an entire book they are less likely to read. My 2 cents ;-)
Thank you Travis! Yes, writing is like math, in that you continuously progress and gradually gain clarity.
Thanks also for the suggestion, which is definitely on my radar. Right now I'm laying the foundations, as I want to make sure that the more "general audience"/"easy" posts won't be brushed off by people who, a bit like Reader X, will underestimate the stakes. It's easy to dismiss as pep talk what it is actually a serious take on how mathematics function.
The interplay between the formalism of proofs and intuition is also a fascinating area. You've described the enormous difference between talking 1 on 1 with someone about a mathematical idea vs listening to talk vs reading a paper/book, and how it is so much more efficient to talk with someone 1 on 1. I wonder what information is being communicated 1 on 1 that isn't communicated in the formalism. Why couldn't the 1 on 1 communication be encapsulated in writing?
1 on 1 benefits for dynamic personalization: you see how the other person reacts, you notice the blank stare when they don't understand, you reuse the language they're comfortable, you find metaphors and examples that resonate with them... this is a game-changer.
Wonderful.
I find myself reflecting on chess which has very similar debates about talent and process.
A key skill in chess is calculation. But I strongly suspect that the whole way it is discussed is wrong.
I have to laugh, because I can't think of anyone who wouldn't benefit from reading this book, and I think anyone who is curious about how to better understand the world would enjoy reading it. I did.
I paint, and I hear all the time, "Oh, I could never paint like that. You are so talented!" Same old debate. It isn't talent. It is training.
As the saying goes: "the more I practice, the luckier I get!"
Bonjour. Je suis pianiste et compositrice et je trouve que ce débat se retrouve de façon assez similaire en musique. J'ai beaucoup de plaisir à vous lire et à faire des parallèles. Bravo et merci. Hâte de lire le prochain message.
Mais oui! Bon dit!
Merci :)
Unfortunate, but you could always resort to modern internet discussion methods, like, "Why don't you write one, smarty pants?"
I can appreciate your perseverance. I'm still working on distilling everything in the world into individual 3 x 5 cards. Wish me luck.
Personally, I'm happy you chose to become an author. Your content is consistently engaging and thought-provoking. Plus, you make considerable efforts to discuss your opinions with the public, which is unparalleled. I hope that you have a long and fruitful career as an author.
After reading your Substack posts, I found one point intriguing and would like to ask a question about it.
In your opinion, will changing the formal definition of math lead to concrete changes in what counts as math? For example, non-mathematicians might formalize their abstractions from their professional field and might study them through symbol manipulation and publish their findings. According to the "conceptualist characterization of math"*, will we regard them as mathematicians, or will we consider them doing math? If the answer is yes, what will distinguish math from pseudo-math? Actually I know such work already exist as part of interdisciplinary fields but I was wondering if what counts as professional math might have become broader due to lower barriers to entry.
If these questions don't make sense or are not answerable at this point, please feel free to skip them.
Thanks a lot.
*"a mental activity based on manipulating formal systems and imagining that they hold absolute and immutable truths about real objects."
Changing the definition is about changing the expectations: what is essential in the experience of doing math, and what teaching should care about.
For students, this perspective can be a game changer. Based on my own experience, I appreciate it a lot.
I was probably thinking like a biologist. Change a piece of DNA and many unexpected effects will occur.
For a complete evisceration of the view of nature vs nurture as a dichotomy, see Evelyn Fox Keller’s The Mirage of a Space between Nature and Nurture. She finds most of the discourse on the subject incoherent.
I loved your book. As an investor and a fan of Buffett, I think investing represents another instance of your argument against a pure genetic explanation of an outlier. Buffett has accumulated a net worth of the order $100 billion whereas most people are not even millionaires. He has written and spoken a lot about what explains this 100,000x difference. He attributes is to the magic of compounding over time. I think in investing the equivalent of honing "intuition" is: honing a tolerance of being against the crowd; honing the ability to see through the "noise" of volatile market price to focus on the intrinsic value of the invested asset; and honing the patience required to hold (sometimes for years!) before the investment finally delivers returns.
Yes. Wealth is the most famous example of a Pareto distribution (and the original context in which it was discovered by Pareto.)
Here is my take. Mathematical ability, like most things in life, is a combination of genetics and environment. It is silly to deny that either one matters. Of course if you're an individual mathematician, you can't your genes at least not until we develop safe and effective ways to do CRISPR in adults. But you can change your environment. So it is more productive to focus on that.
I'm sure that the best mathematicians in the world all have significantly above average polygenic educational attainment and IQ scores. But I'm very wary of claims like "people with an IQ of X cannot do/learn Y, there is a hard ceiling". Calculus was cutting edge a few centuries ago and only the best mathematicians did it, and now it can be taught to a lot of people.
Imagine being 7 years old and realizing your parents love you less because of the math grade on your report card.
For the rest of your childhood, this happens twice a year. On top of that you have a constant cycle of fear and stress about math homework and frequent tests which you can never quite manage to do. Your bad grades mean you go to a worse college and you're cut off from lots of high paying careers.
Well, this is very painful, but at least it's not your fault -- math ability is just innate, and you don't have it.
But then this guy David Bessis shows up, and he has this whole book titled Mathematica, the gist of which is -- it IS your fault!
If you had just had the proper virtuous attitude of curiosity, you wouldn't have struggled with math. Your whole childhood would have been more fun, you'd be more successful now, and every year your parents would be happy to see you at Thanksgiving. The only person to blame for all this pain in your life, was you -- you deserve it.
Obviously what I've laid out is a flagrant misinterpretation of your work, but it doesn't surprise me that it provokes irrational and angry criticism around the idea of innate ability. It's a highly emotional issue.
(For mathematicians and scientists who do have advanced mathematical skills and never had the childhood pain around math education, I think the emotional story is a little different -- the belief in innate ability helps to both cope with peers out-achieving you, and also manage anxiety about falling behind. Belief in innate ability is serving the same load-bearing purpose just in a different structure.)
This should be an indictment of school,
"Imagine being 7 years old and realizing your parents love you less because of the math grade on your report card.
For the rest of your childhood, this happens twice a year. On top of that you have a constant cycle of fear and stress about math homework and frequent tests which you can never quite manage to do. Your bad grades mean you go to a worse college and you're cut off from lots of high paying careers."
I don't know about genetics although my mom was a chemistry and math major however, I can comment on curiosity. I've always been more curious than anyone I know (didn't stop me from blowing out of an all paid 4 year scholarship, in an honors program to an MS in 4, after 2 years. In old age I think what was different is "focus", "concentration". With this curiosity I lacked the focus to pursue one thing (as Curly said in "City Slickers"). With too many interesting things, wandering was the rule. It worked out fine. I wandered through a myriad of interesting jobs, was rewarded overall. Focus ... an unappreciated dimension.
The working title "Secret Math" may deserve its fate, because for an idea to be secret, the holder needs to intend to avoid disclosing it. A secret is clandestine, concealed, confidential, furtive, hidden, private, or surreptitious. The centuries of failed attempts to disclose this "secret" demonstrate its invisibility, not its secretness. It is overlooked, unnoticed, unseen, disregarded, inconspicuous, unobtrusive, or unperceived.
Blank slatism in 2025, embarrassing.
And here we go again!
You write a careful long form post explaining how hereditarian myths make it nearly impossible to have a serious discussion about mathematical cognition. You know the subject quite well, because you're a former career mathematician who literally spent decades thinking about the topic. You wrote a serious and well-documented book that was published by rigorous academic press and endorsed by leading mathematicians.
You know that it's a sensitive topic, so you've been careful to acknowledge that genes play a role in mathematical capacity, reiterating your view that "There’s no doubt that natural aptitude in math, like natural aptitude in any other physical activity, isn’t equally distributed among individuals." (a citation from your book)
And yet, within 3 days, you're accused of blank slatism.
Substack is marginally better than X, though, because on X you get these comments within 10 minutes and they usually contain insults.
The smarter you are, and the more groundbreaking your contribution to knowledge, the more likely you are to be misunderstood and dismissed as stupid.
Sadly, truly stupid people are also likely to be dismissed as stupid. Oh well.
For what it’s worth, I think your piece is smart as hell and deserves to be read far and wide.
Thanks 🙏!
For what it's worth, I found your careful caveating, rebuttal and claims interesting enough to buy your book.
To frame where I'm coming from, vis a vis math, I was good enough at math to pick to a computer science degree and I'm curious enough now to bother any mathematicians I run into about their work (I've got a brother in law who's a topologist who hasn't gotten tired of it yet), but I wouldn't say I've ever seriously pursued math.
I've been pondering a lot recently on how I learn complicated subjects, often by rapidly creating and abandoning fuzzy abstractions until I get an intuitive sense that something worked, at which point I'll go back and find a post hoc objective mechanical justification. Over time some of those tricks keep on working, and the abstractions get sharper and more durable in my head. It's been a fair number of years since I've done that to learn math, but I do it all the time to avoid plateauing in other skills.
I'm excited to read your book and see if it gives me language to talk about those internal mental processes and maybe some tricks to get better at learning. Maybe learn some more about math on the way too.
Thanks! Hope you’ll like the book :)
Yeah I think that dismissing your views as “blank slate-ism” is very silly.
As someone with pretty strongly hereditarian views, including on controversial things like group differences, I actually don’t find much to disagree with in what you write. No question most things are a combination of genetics and environment, mathematical output no exception.
Thanks for your comment. Thanks also for reminding me that some of my readers have strong hereditarian views, which is essential for me to keep in mind (especially since I’ll be coming back to these subjects in the near future.)
Ha, a fair critique of my snide comment. Its true you're not actually promoting blank slatism but I have to agree with the reviewer and many of the like-minded commenters online that your foray into disproving innate mathematical ability is as ill-conceived as my comment and probably does detract from your book (but certainly aids visibility). From reading your previous posts on X and substack I can see that you seem to misunderstand the overwhelming (plurality at minimum, most likely majority) innateness of the non-shared environment, and severely range restrict your sample (those publishing in math journals) to the tails of global mathematical ability to disprove the polygenic nature of mathematical ability. Tao, Von Neumann, Gauss, Ramanujan, etc. have/had innate mathematical talent far beyond most of humanity and that played a large role in their success.