Everyone is capable of, and can benefit from, mathematical thinking

One of the highly generous mentors who dragged me kicking and screaming into the world of even making an attempt told me: “There are no bad math students. There are only bad math teachers who themselves had bad math teachers.”

Agreed with the above. Almost everyone can probably expand their mathematical thinking abilities with deliberate practice.

> But I do not think this is innate, even though it often manifests in early childhood. Genius is not an essence. It’s a state. It’s a state that you build by doing a certain job.

Though his opinion on mathematical geniuses above, I somewhat disagree with. IMO everyone has a ceiling when it comes to math.

Leibniz made that claim centuries ago in his critical remarks on John Locke's Essay on Human Understanding. Leibniz specifically said that Locke's lack of mathematical knowledge led him to (per Leibniz) his philosophical errors regarding the nature of 'substance'.

https://www.earlymoderntexts.com/assets/pdfs/leibniz1705book...

Whether that youthful immersion in math in fact benefitted me in later life and whether that kind of thinking is actually desirable for everyone as he seems to suggest—I don't know. But it is a thought-provoking interview.

At the very minimum, I ask people to always think of the distribution of whatever figure they are given.

Just that is far more than so many are willing to do.

[0]https://news.ycombinator.com/item?id=41962944

Yes, but it's higher than you think: https://www.justinmath.com/your-mathematical-potential-has-a...

As info explodes and specialists dive deeper into their niches, info asymmetry between ppl increases. There are thousands of specialists running in different directions at different speeds. Leaders can't keep up.

Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.

And while most specialists think this "sync" process happens through "education" and getting everyone to understand a complex ever changing universe, the truth is large diverse groups are kept in sync via status signalling, carrot/stick etc. This is why leaders will pay attention when you talk in terms of what increases clout/status/wealth/security/followers etc. Cause thats their biggest tool to prevent schisms and collapse.

Any healthy/able individual could learn to deadlift twice their bodyweight with sufficient training, but the vast majority of people never reach this basic fitness milestone, because they don't put any time into achieving it. There's a very large gap between what people are capable of theoretically and what they achieve in practice.

Who said it would be easy?

I've been working a lot on my math skills lately (as an adult). A mindset I've had in the past is that "if it's hard, then that means you've hit your ceiling and you're wasting your time." But really, the opposite is true. If it's easy, then it means you already know this material, and you're wasting your time.

Run into too many people like that, who I daresay are common in the field, and it's easy to see how people become dispirited and give up.

He was right on the nature of the universe, he was wrong on making a better world. I for one forgive him on the basis of time served.

I’ve tried a bunch of courses (MIT linalg, Coursera ICL Maths for ML, Khan etc etc) but what I eventually realised is my foundations were so, so weak being mid 30s and having essentially stopped learning in HS (apart from a business stats paper at Uni).

Enter a post on reddit about Mathacademy (https://www.mathacademy.com/). It’s truly incredible. I’m doing around 60-90 minutes a day and properly understanding and developing an intuition for things. They’ve got 3 pre-uni courses and I’ve now nearly finished the first one. It’s truly a revelation to be able to intuit and solve even simple problems and, having skipped ahead so far in my previous study, see fuzzy links to what’s coming.

Cannot recommend it enough. I’m serious about enrolling in a Dip Grad once I’ve finished the Uni level stuff. Maybe even into an MA eventually.

I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.

The author of the book has picked out mathematics because that was what he was interested in. The reality is that this rule applies to everything.

The belief that some people have an innate skill that they are born with is deeply unhelpful. Whilst some people (mostly spectrum) do seem have an innate talent, I would argue that it is more an inbuilt ability to hyper focus on a topic, whether that topic be mathematics, Star Trek, dinosaurs or legacy console games from the 1980’s.

I think we do our children a disservice by convincing them that some of their peers are just “born with it”, because it discourages them from continuing to try.

What we should be teaching children is HOW to learn. At the moment it’s a by-product of learning about some topic. If we look at the old adage “feed a man a fish”, the same is true of learning.

“Teach someone mathematics and they will learn mathematics. Teach someone to learn and they will learn anything”.

I've long thought that almost all have the capability to learn roughly high school level math, though it will take more effort for some than for others. And a key factor to keep up a sustained effort is motivation. A lot of people who end up hating math or think they're terrible at it just haven't had the right motivation. Once they do, and they feel things start to make sense and they're able to solve problems, things get a lot easier.

Personally I also feel that learning math, especially a bit higher-level stuff where you go into derivations and low-level proofs, has helped me a lot in many non-math areas. It changed the way I thought about other stuff, to the better.

Though, helping my family members and friends taught me that different people might need quite different approaches to start to understand new material. Some have an easier time approaching things from a geometrical or graph perspective, others really thrive on digging into the formulas early on etc. One size does not fit all.

One positive effect of having studied pure mathematics when young might have been that I became comfortable with thinking in multiple layers of abstraction. In topology and analysis, for example, you have points, then you have sets of points, then you have properties of those sets of points (openness, compactness, discreteness, etc.), then you have functions defining the relations among those sets of points and their properties, then you have sets of functions and the properties of those sets, etc.

I never used mathematical abstraction hierarchies directly in my later life, but having thought in those terms when young might have helped me get my head around multilayered issues in other fields, like the humanities and social sciences.

But a possible negative effect of spending too much time thinking about mathematics when young was overexposure to issues with a limited set of truth values. In mainstream mathematics, if my understanding is correct, every well-formed statement is either true or false (or undecided or undecidable). Spending too much time focusing on true/false dichotomies in my youth might have made it harder for me to get used to the fuzziness of other human endeavors later. I think I eventually did, though.

I met the guitar guy a few years later outside his house. He always had just one guitar but now owned something like 20, something like a hundred books about music. Quite the composer. It looked and sounded highly sophisticated. The dumb guy didn't exist anymore.

This reads like the foreword to the incel handbook.

I would argue something different. The "skill" angle is just thinly veiled ladder-pulling.

Sure, math is hard work, and there's a degree of prerequisites that need to be met to have things click, but to the mindset embodied by the cliche "X is left as an exercise for the reader" is just people rejoicing on the idea they can needlessly make life hard for the reader for no reason at all.

Everyone is familiar with the "Ivory tower" cliche, but what is not immediately obvious is how the tower aspect originates as a self-promotion and self-defense mechanism to sell the idea their particular role is critical and everyone who wishes to know something is obligated to go through them to reach their goals. This mindset trickles down from the top towards lower levels. And that's what ultimately makes math hard.

Case in point: linear algebra. The bulk of the material on the topic has been around for many decades, and the bulk of the course material,l used to teach that stuff, from beginner to advanced levels, is extraordinarily cryptic and mostly indecipherable. But then machine learning field started to take off and suddenly we started to see content addressing even advanced topics like dimensionality reduction using all kinds of subspace decomposition methods as someting clear and trivial. What changed? Only the type of people covering the topic.

Plato's Meno has Socrates showing that even a slave can reason mathematically.

It's not really math alone but modeling more generally that activates people's reasoning. Math and logic are just those models that are continuous+topological and discrete+logic-operation variants, both based in dimension/orthogonality. But all modeling is over attribution - facts, opinions, etc., and there's a lot of modeling with a healthy dose of salience - heuristics, emotions, practice, etc. Math by design is salience-free (though it incorporates goals and weights), so it's the perspective and practice that liberates people from bias and assumptions. In that respect it can be beautiful, and makes other more conditioned reasoning seem tainted (but it has to work harder to be relevant).

However, experts can project mathematical models onto reality. Hogwash about quantum observer effects and effervescent quantum fields stem from projecting the assumptions required to do the math (or adopt the simplifying forms). Yes, the model is great at predictions. No, it doesn't say what else is possible, or even what we're seeing (throwing baseballs at the barn, horses run out, so barns are made of horses...). Something similar happens with AI math: it can generate neat output, so it must be intelligent. The impulse is so strong that adherents declare that non-symbolic thinking is not thinking at all, and discount anything unquantifiable (in discourse at least). Assuming what you're trying to prove is rarely helpful, but very easy to do accidentally when tracking structured thinking.

As anti-intellectual as it sounds, you could imagine someone asking, is it worth devoting years of your life to study this subject which becomes increasingly esoteric and not obviously of specific benefit the further you go, at least prima facie? Many people wind up advocating for mathematics via aesthetics, saying: well it’s very beautiful out there in the weeds, you just have to spend dozens of years studying to see the view. That marketing pitch has never been the most persuasive for me.

This gives you vicious and virtuous cycles: Students' learning speed increases with time and past success. So "talented" students learn quickly and have extra time to further explore and improve, leading to further success. Students who struggle with the time constraint are forced to take shortcuts like memorizing "magic formulas" without having time to really understand. Trying to close that gap is very hard work.

You can even benefit from the thinking taught in good high school coursework (or studying online).

At an arithmetic, bookkeeping level you can better appreciate handling finances and the seductive pitfalls surrounding wagers (gambling, betting, risk taking).

* How many have a lower return on { X } through not understanding compound interest, tax brackets, leveraging assets, etc.

* How many have steady net losses through "magical thinking" wrt gambling, betting, hot stock tips.

One thing I'm anticipating from LLM-based tutoring is an adaptive test that locates someone's frontier of knowledge, and plots an efficient route toward any capability goal through the required intermediate skills.

Trying to find the places where math starts getting difficult by skimming through textbooks takes too long; especially for those of us who were last in school decades ago.

My first thought when the article got to the dialog between logic and intuition bit was that the same is true for school level physics.

If we spent 50% of time thinking productively - inborn thinking speed would matter. But in my estimate even 5% is generous.

So it matters far more what kind of feedback you have to filter out the wrong results, and how much time you spend thinking - than how quickly you can do it.

Also practice helps with speed.

Reminds me of my personal peeve that "studying" should not be "being taught", studying is pursuit of understanding, "being taught" is what happens in primary school (and I'm aware I'm simplifying here).

This is overthinking it. People with power tend to be interested in outcomes. They can't evaluate all the reasoning of all their reports. It comes down to building credibility with a track record and articulating outcomes, when you want to advise decision makers.

That would be like strength training by carrying stuff home and believing that the point is to have a lot of stuff at home.

Intelligence is about being able to frame and analyze things on the fly and that ability comes from framing and analyzing lots of different things, not from memorizing the results of past (or common forms of) analysis.

Working with symbols, equations etc. feels like it should be more widely accessible. Its almost a game-like pursuit, it should not be alienating.

It might be a failure of educators recognizing what are the pathways to get the brain to adopt these more abstract modes of representing and operating.

NB: mathematicians are not particularly interested in solving this, many seem to derive a silly pleasure of making math as exclusive as possible. Typical example is to refuse to use visual representation, which is imprecise but helps build intuition.

I don't agree. First of all, ladder-pulling in math is observed at all levels, not only cutting-edge stuff. Secondly, it's in applied mathematics where pure math takes a queue onto where to focus effort. See how physics drives research into pure math.

Similar to systems thinking, just the ability to play out scenarios in your head given a set of rules is a very useful skill, one which programmers tend to either be good at because of genetics or because we do it every day (i.e. simulate code in our head). You can tell when someone lacks this ability when discussing something like evolutionary psychology. Someone with a systems thinking mindset and an ability to simulate evolution tend to understand it as obvious how evolutionary pressures tend to, and really must, create certain behavior patterns (on average), while people without this skill tend to think humans are a blank slate because it's easier to think about, and also is congruent with modern sensibilities.

This skill applies in everyday life, especially when you need to understand economics (even basic things like supply and demand seems elusive to many), politics etc.

And what's the shtick about girls? What are and were you looking for, love and a genuine relationship or attention to compensate for something? Personally I think your values and personality are what matter most and personality is usually what people fall in love with. Though charisma can help a lot to get the ball rolling. Most of what it takes is to treat people normally and nicely and you will have as much of a chance to find love as most people.

Though respect from peers and attention from women ideally shouldn't be your driving force. I think curiosity and passion are much better driving forces that don't involve such external factors and possibilities for insecurities.

Your post reads as if it expresses a frustration and a sense of entitlement. You may not be intrinsically entitled to the things you think you are. Think about that for a bit and try to be rational.

If I can earn an extra 1 million being 'dumb' and thus ensure quality healthcare, education, housing, is it smart to try to be smart?

This is the true tragedy of the commons (or the reverse tragedy, to be precise).

The second is notation. I had a snob teacher who insisted on using Newton not Leibniz and at school in the 1970s this is just fucked. One term of weirdness contradicting what everyone else in the field did. Likewise failure to explain notation, it's hazing behaviour.

So yes, everyone benefits from maths. But no, it's not a level playing field. Some maths people, are just toxic.

I think the only thing in autism that I'd call an innate talent is detail-oriented thinking by default. It'd be the same type of "innate talent" as, say, synesthesia, or schizophrenia: a side effect of experiencing the world differently.

I've tried night classes, tutors, activities. Nothing sticks.

Even the standard 12x tables I struggle at. I want to understand it but my brain just can't understand the non-practicality side of things.

A side effect for which there is a substantial, lifelong, and most importantly wide cost, even if it occasionally confers usually small, usually fleeting, and most importantly narrow advantage.

Nonsense!

The brain you are born with materially dictates the ceiling of your talent. A person with average ability can with dedication and focus over many years become reasonably good, but a genius can do the same in 1 year and at a young age.

We have an education system which gives an A Grade if you pass the course, but 1 person may put on 5 hours a week and the other works day and night.

Only an adult can look and see what that was - immature, insecure little boys, desperately trying to show off as bigger/more mature or kick down anybody showing any weakness or mistake. Often issues from home manifesting hard. Its trivial to look back without emotions, but going through it... not so much.

If my kids ever go through something similar (for any reasons, math nerds are just one instance of bigger issue) I'll try reasoning above, not sure if it will help though.

Or more specifically, two of my friends teach special needs children in the 50 to 70 IQ band. Who are we going to blame for them not becoming mathematicians? The teachers, for not unlocking their hidden potential? The kids, for not trying hard enough? Claiming that the only thing holding them back is choice seems as cruel as it is wrong, to me.

Yeah, we're probably not cultivating anywhere near the potential that we could, but I personally guarantee you I am not Ramanujan or Terence Tao.

That said, some people received a smaller starting container, and might need some help cracking it. That’s the work of those who think they’ve found a bigger one.

And this also gives the proponent (you in this case) an excuse to blame a person for not focusing hard enough or not being dedicated enough if they don't grasp the basics, let alone excel.

Do the stuff you're good at, provide for your family, earn the respect of your peers and forget about the rest.

I studied up to A level (aged 19) but honestly started hating math aged 16 after previously loving it.

It’s a big regret of mine that I fell out of love with it.

I self taught myself coding and Spanish and much enjoy self study if I can find the right material.

Any suggestions?

Who Is Fourier?: A Mathematical Adventure

https://www.amazon.com/Who-Fourier-Mathematical-Transnationa...

It started off as a bunch of non-math literate folks teaching themselves math from scratch, including trigonometry, calculus etc, and ending in Fourier series. It is a very approachable and fun book.

But when I started focusing on making money I've come to believe it's a bad heuristic for that purpose.

I just wrote about this. In fact, you can even see this at play in the video of the kids talking https://blog.comini.in/p/what-happens-in-math-class

- No, but the smarter kids might.

1) First, I had a vague vision of how I want to do mathematics on a computer, based on my experience in interactive theorem proving, and what I didn't like about the current state of affairs: https://doi.org/10.47757/practal.1

2) Then, I had a big breakthrough. It was still quite confused, but what I called back then "first-order abstract syntax" already contained the basic idea: https://obua.com/publications/practical-types/1/

3) I tried to make sense of this then by developing abstraction logic: https://doi.org/10.47757/abstraction.logic.1 . After a while I realized that this version only allowed universes consisting of two elements, because I didn't distinguish between equality and logical equality, which then led to a revised version: https://doi.org/10.47757/abstraction.logic.2

4) My work so far was dominated by intuition based on syntax, and I slowly understood the semantic structures behind this: the mathematical universe consisting of values, and operations and operators on top of that: https://obua.com/publications/philosophy-of-abstraction-logi...

5) I started to play around with this version of abstraction logic by experimenting with automating it, giving a talk about it at a conference, (unsuccessfully) trying to publish a paper about it, and implementing a VSCode plugin for it. As a result of using that plugin I realized that my understanding until now of what axioms are was too narrow: https://practal.com/press/aair/1/

6) As a consequence of my new understanding, I realized that besides terms, templates are also essential: https://arxiv.org/abs/2304.00358

7) I decided to consolidate my understanding through a book. By taking templates seriously from the start when writing, I realized their true importance, which led to a better syntax for terms as well, and to a clearer presentation of Abstraction Algebra. It also opened up my thinking of how Abstraction Algebra is turned into Abstraction Logic: https://practal.com/abstractionlogic/

8) Still lots of stuff to do ...

I would not be surprised if that is exactly the way forward for AIs as well. They clearly have cracked (some sort of) intuition now, and we now need to add that interplay between logic and intuition to the mix.

Who knows? You might also contribute meaningfully to these fields as you embrace your own unique path.

[1] https://dc.fandom.com/wiki/Batcomputer

My brother's mom tutors math. One of her insights with a former student was that they were in need of forming some number sense. She started by walking them both out to the street: "how many tires are there on this street of parked cars?" The student, already flummoxed, started panic guessing. So she started with counting.

For times tables, have you developed any intuition around it? For me, times tables are rectangles composed of unit squares and that helps with my intuition. Modern Common Core standards in the US focuses a lot on exposing different mental models to students. And after seeing the same 4x6 enough times your brain will automatically associate that with its solution. Instead of calculating, it is memorized.

My brain doesn't require car tires, geology, or other practical needs: it likes puzzles. I struggle with medical stuff and I can feel my brain switching to meh-mode and hardly anything sticks. I don't know how many times I have been told about the different kinds of sugar and how your body uses that energy and I would still have to look it up.

2 weeks ago I hired a professor to help me learn math again so I can attend University computer science.

I can tell you, you can and should.

I'm totally addicted to math, I work as a programmer once I finish my work for the day I spend all my free time learning math again.

I'm still going over the very basics like 9 th grade stuff but I can see already it's going to go fine! I'm enjoying it so much!

As a child of the 80s and 90s, "getting girls as a programmer" made me snort. Nerds do seem to have it a bit better now; the money/financial security of software development helps. But as a whole, we developers are still less socially capable than our sales/hr/marketing counterparts

- Maths: A Student's Survival Guide (ISBN-13 978-0521017077)

- Review Text in Preliminary Mathematics - Dressler (ISBN-13 978-0877202035)

- Fearon's Pre-Algebra (ISBN-13 978-0835934534)

- Introductory Algebra for College Students - Blitzer (ISBN-13 978-0134178059)

- Geometry - Jacobs ( 2nd ed, ISBN-13 978-0716717454)

- Intermediate Algebra for College Students - Blitzer (ISBN-13 978-0134178943 )

- College Algebra - Blitzer (ISBN-13 978-0321782281)

- Precalculus - Blitzer (ISBN-13 978-0321559845)

- Precalculus - Stewart (ISBN-13 978-1305071759)

- Thomas' Calculus: Early Transcendentals (ISBN-13 978-0134439020)

- Calculus - Stewart (ISBN-13 978-1285740621)

The main goal of learning is to understand the ideas and concepts at hand as “deeply” as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By “deeply” we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no “perfect” state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: “Young man, in mathematics you don't understand things. You just get used to them”, which I think really means that getting “used to” some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination.

Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap).

Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding.

Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory.

Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note.

Cross-reference. Don't read linearly. Instead, have multiple textbooks, and “dig deep” into concepts. If you learn about something new (say, linear combinations) — look them up in two textbooks. Watch a video about them. Read the Wikipedia page. Then write down in your notes what a linear combination is.

Learning is a social activity, so maybe enroll in a community college course or find a local study group. I find it's especially important to have someone to discuss things with when learning math. I also recommend finding good public spaces to work in—libraries and coffee shops are timeless math spaces.

Pay graduate students at your local university to tutor you.

Take walks, they're essential for learning math.

Khan Academy is not enough. It has broad enough coverage, I think, but not enough diversity of exercises. College Algebra basically is a combination of Algebra 1, Algebra 2, relevant Geometry, and a touch of Pre-Calculus. College Algebra, however, is more difficult than High School Algebra 1 and 2. I would tend to agree that you should start with either Introductory Algebra for College Students by Blitzer or, if your foundations are solid enough (meaning something like at or above High School Algebra 2 level), Intermediate Algebra by Blitzer. Basically, Introductory Algebra by Blitzer is like Pre-Algebra, Algebra 1, and Algebra 2 all rolled into one. It's meant for people that don't have a good foundation from High School. I would just add, if it is still too hard (which I doubt it will be for you, based on your comment), then I would go back and do Fearon's Pre-Algebra (maybe the best non-rigorous Math textbook I've ever seen). Intermediate Algebra is like College Algebra but more simple. College Algebra is basically like High School Algebra 1 and 2 on steroids plus some Pre-Calculus. The things that are really special about Blitzer is that he keeps math fun, he writes in a more engaging way than most, he gives super clear—and numerous—examples, his books have tons of exercises, and there are answers to tons of the exercises in the back of the book (I can't remember if it's all the odds, or what). By the time you go through Introductory, Intermediate, and College Algebra, you will have a more solid foundation in Algebra than many, if not most, students. If you plan to move on to Calculus, you'll need it. There's a saying that Calculus class is where students go to fail Algebra, because it's easy to pass Algebra classes without a solid foundation in it, but that foundation is necessary for Calculus. Blitzer has a Pre-Calculus book, too, if you want to proceed to Calculus. It's basically like College Algebra on steroids with relevant Trigonometry. Don't get the ones that say “Essentials”, though. Those are basically the same as the standard version but with the more advanced stuff cut out.

Edit: I don't mean to argue that there aren't genetics involved in determining aptitude on certain tasks, of course, but the assumption that genius is born and never made feels like a very shallow understanding of the capacity of man.

(don't have any examples on-hand atm, this is just my general perception after years of occasionally looking things up there)

I'm not sure what you are trying to say.

I've taught math to psychology students, and they just don't get it. I remember the frustration of the section's head when a student asked "what's a square root?" We all know how many of our fellow pupils struggled with maths. I'm not saying they all lacked the capability to learn it, but it can't be the case they all were capable but "it was the teacher's fault". Even then, how do you explain the difference between those who struggled and those who breezed through the material?

Or let's try other topics, e.g. music. Conservatory students study quite hard, but some are better than others, and a select few really shine. "Everyone is capable of playing Rachmaninov"? I don't think so.

So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.

Auxiliary problems are something that always screwed me in college, when we were doing Baby Rudin, if a proof required a lemma or something first I usually couldn't figure out the lemma. Or in general, if I didn't quickly find the 'insight' needed to prove something, I often got frustrated and gave up.

This material seems like it would be good to actually teach in school, just like a general 'how to think and approach mathematical problems'. Feels kinda weird that I had to seek out the material as an adult...

One other thing I got out of the Polya books, was I realized how little I remember about geometry. So many of their examples are geometrical and that made them harder for me to grok. That's something I wish I could revisit.

You’re making an argument somewhat along those lines, but given that I didn’t stipulate a convergence condition your conclusions can be dismissed by me.

If it were a valid argument then we’d need Gödel.

I think a lot of maths is secretly a lot easier than it appears, but just missing an explanation that makes it easy to get the core idea to build upon.

For example, I've been meaning to write an explorable[0] for explaining positional notation in any integer base (so binary, hexadecimal, etc) in a way that any child who can read clocks should be able to follow. Possibly teaching multiplication along the way.

Conceptually it's quite simple: imagine a counter that looks like an analog clock, but with the digits 0 to 9 and a +1 and -1 button. We can use it to count between zero and nine, but if we add one to nine, we step back to zero. Oh no! Ok, but we can solve this by adding a second counter. Whenever the first counter does a full circle, we increase it by one. A full circle on the first counter is ten steps, so each step on the second counter represents ten steps. But what if the second counter wants to count ten steps? No problem, just add a third! And so on.

So then the natural question is... what if we have fewer digits than 0 to 9? Like 0 to 7? Oh, we get octal numbers. 0 and 1 is binary. Adding more digits using letters from the alphabet?

The core approach is just a very physical representation of base-10 positional, which hopefully it makes it easy to do the counting and follow what is happening. No "advanced" concepts like "base" or "exponentiation" needed, but those are abstractions that are easy to put on top when they get older.

I've asked around with friends who have kids - most of them learn to read clocks somewhere between four and six, and by the time they're eight they can all count to 100. So I would expect that in theory this approach would make the idea of binary and hexadecimal numbers understandable at that age already.

EDIT: funny enough the article also mentions that precisely thanks to positional notation, almost every adult can immediately answer the question "what is one billion minus one".

[0] https://explorabl.es/

Being interested in a subject is massively helpful to learning it. But interest arises circumstantially, it's an emotion. The grim reality that it would be really useful to you to learn a certain subject does not necessarily make you interested in the subject, unfortunately. (Perhaps "financially interested", but that's something else.)

I do broadly agree with your position that some people are going to excel where others fail. We know there trivially exist people with significant disabilities that will never excel in certain activities. What the variance is on "other people" (a crude distinction) I hesitate to say. And whatever the solution is, if there is even a solution, I'd at least like for the null hypothesis to be "this is possible, we just may need to change our approach or put more time in".

On a slightly more philosophical note, I firmly believe that it is important to believe some things that are not necessarily true -- let's call this "feel-good thinking". If someone is truly putting significant dedicated effort in and not getting results, that is a tragedy. I would, however, greatly prefer that scenario to the one in which people are regularly told, "well, you could just be stupid." That is a self-fulfilling prophecy.

It’s a well-established effect in pedagogics that learning vs. difficulty has a non-monotonic relationship, where you don’t learn efficiently if the material is either too hard or too easy compared to your current level. There is an optimum learning point somewhere in-between where the material is “challenging” – but neither “trivial” nor “insurmountable” – to put it that way.

It took me some time, but now it's a lot better -- like a little game I somewhat know the rules of. I now accept that mathematicians are often worrying about maximal abstraction or addressing odd pathological corner cases. This allows me to wade through the complexity without getting overwhelmed like I used to.

This sets them up for fantastic success very quickly. [1] shows a great example of this.

I'm fond of saying "You can do anything you want, but wanting is the hard part", because to truly be a grandmaster, genius-level mathematician, olympic athlete, etc, requires a dedication and amount of preparation that almost nobody can manage. Starting late, with emotional baggage, kids, and having to spend 5 years relearning how to learn? Forget it.

1. https://danielkarim.com/how-to-become-a-genius-the-polgar-ex...

There are some extreme cases of course but I’m not sure the general public needs to worry too much about those, most of us aren’t an Einstein nor do we have learning disabilities.

I respectfully, but strongly, disagree. There's a reason most NBA players are over 2 meters tall, and one does not become taller with time, dedication nor focus.

It might be different for intellectual skills but I am not that sure.

Almost anyone can become decent at almost anything though. Which is good already!

People need to speak in plain English [0]. To some mathematicians' assertion that English is not precise enough, I say, take a hike. One need to walk before they can run.

Motivating examples need to precede mathematical methods; formulae and proofs ought to be reserved for the appendix, not page 1.

[0] I mean natural language

I didn't fall in love with math until Statistics, Discrete Math, Set Theory and Logic.

It was the realization that math is a language that can be used to describe all the patterns of real world, and help cut through bullshit and reckon real truths about the world.

Thank you Sir or Madame.

As someone who went on to study mathematics at the graduate level, I was bored out of my mind in high school math and most subjects. What's missing from a lot of primary and secondary school education is context, and that's what makes it boring. Math wasn't easy because I was particularly good at it. It was easy because it was just blindly following formulas and basic logic.

Something is very wrong with our educational system because almost all math at the primary and secondary levels is basic logic. So when people with this maximum level of mathematics education say they're "bad at math" or "don't get math", it means that they lack extremely basic logic and reasoning skills.

In my mind, we need to teach mathematics in a contextual way (note that I don't necessarily mean applications) in a way to enrich the reasoning and exploring of it. This should include applications, yes, but not be fully concentrated on applications. Sometimes one needs to just learn and think without being tied to some arbitrary standard of it being applied.

I'm not so sure that adults always get it or rise about this. This happens in the workplace all the time.

The sin of modern mathematics is that it's meta language is so ill define that you need towers of software to manipulate it without contradiction. Rewriting all of it into s-expressions with a term rewriting system for proofs under a sequent calculus is an excellent first step to making it accessible.

We do not need to go back to the 16th century when men were men, an numbers positive. If people want to look at what math talks about instead of how it talks about let them pick up stamp collecting.

Not really. There's nothing inherently special about people who dedicated enough time to learn a subject.

> "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained.

What a bad comparison. So far in human history there were less than 200 people who ran 100m in less than 10s.

I think you're just reflecting an inflated sense of self worth.

I've found that the people who most believe in math being a genetic ability are the ones who do not work in the symbolic world of modern math, but in the semantic world of whatever the field the math describes is.

The two are rather different.

LLMs currently can't find efficient paths longer than 5 hops when given a simple itinerary. Expecting them to do anything but a tactical explanation of issues they have seen in training is extremely naive with something as high dimensional as math.

I'm saying, if it doesn't ruin lives to the point of preventing reproduction, then it stays in the gene pool.

Basically, I'm saying this:

> The trait is still here and still prevalent meaning people are still getting laid and starting families and presumably leading fulfilling lives.

I dont disagree with the premise that mathematical thinking can benefit anybody, but its absurd notion that everything abstract is teachable and learnable to all is a fantasy of a distinctly left-wing variety, who would have you believe that everything is just social conditioning and human beings dont differ from one-another.

Absolutely. There's also a pernicious idea that only young people can master complex maths or music. This is a self-fulfilling prophecy - why bother try if you're going to fail due to being old? Or perhaps it's an elitist psy-op, giving the children of wealthy parents further advantage because of course no-one can catch up.

Handsome men are largely irrelevant.

I've tried different approaches. 4x3 being 3x4

But somehow I end up miscounting and giving the answer for 4x4 or be off a digit every-time.

A good example was that I was at a brewery last night.

They didn't do pint glasses but glasses of: 1/4, 1/2 and 2/3rd's.

I thought 1/2 was more than 2/3rd's so I ordered a 1/2 rather than thinking I was getting more.

However I was unable to visual how 2/3 is more than 1/2 when 1/2 is half a pint, or half a glass.

My visual capabilities are great at others but just couldn't formulate the equation of 2/3 is more than 1/2.

Very simple stuff, but it just doesn't meld.

While it’s indisputable that terse symbolic formalisms have great utility, one can capture all the same information verbally.

This is perhaps most evident in formal logic. It’s not hard to imagine a restricted formalized subset of natural language that is amenable to mechanical manipulation that is isomorphic to say modal logic.

And finally, for logic at least, there is something of a third way. Diagrammatic logical systems such as Existential Graphs capture the full power of propositional, predicate, and modal logic in a way that is neither verbal nor conventionally symbolic.

Being tall isn't a skill. I suspect you could be skillful enough at basketball to overcome the hight disadvantage. However, I think most people who might become that skillful see the high disadvantage (plus the general difficulty of becoming a pro basketball player) and take a different path through life. It's also possible that the amount of time that would be needed to grow your skill past the height disadvantage is too long, so it's not feasible to do it to gain a position in the NBA.

"You didn't work hard enough." People really blame you for that, not for lacking talent.

> So far in human history there were less than 200 people who ran 100m in less than 10s.

And many millions have tried. There may be 200 people who can run it under 10s, but there are thousands that can run it under 11s, and hundreds of thousands that can run it under 12s. Unless you've got clear evidence that those people can actually run 100m in less than 10s if they simply try harder, I think the experience of practically every athlete is that they hit a performance wall. And it isn't different for maths, languages, music, sculpting (did you ever try that?), etc. Where there are geniuses, there also absolute blockheads.

That's not to say that people won't perform better when they work harder, but the limits are just not the same for everyone. So 'capable of mathematical reasoning' either is some common denominator barely worth mentioning, or the statement simply isn't true. Clickbait, if you will.

And I didn't mention genetics. Nature is complicated.

Neither trivia has anything to do with being good at mathematics as done by mathematicians.

https://softwarefoundations.cis.upenn.edu/

People who aren't even working in a field that's STEM-adjacent have even less use for stuff past simple algebra and geometry (the latter mainly useful just for crafting hobbies and home-improvement projects) and a handful of finance-related concepts and formulas.

I expect to go to my grave never having found a reason to integrate something, at this rate.

The result is that any time I try to get back into math (because I feel like I should, I guess?) it's not really motivated by an actual need. The only things that don't bore me to tears for sheer lack of application ends up being recreational math problems, and even that... I mean, I'd rather just read a book or do almost anything else.

[0] https://en.wikipedia.org/wiki/Do_Not_Erase:_Mathematicians_a...

I don't think there's a single answer as to why many dislike it so much. Some folks view it as a way to gate-keep programming. Others view it as useless ("I've been a successful programmer all my life and I've never used math").

On the other side of the coin there are many who view our craft as a branch of applied mathematics -- informatics if you will.

Why does curiosity being natural necessarily mean some people are born without it? It could also mean everyone (or every average human) is born with it, and overtime it gets pushed out of people.

Factoring was another that lost a lot of folks in my class. Lots of frustration around it seeming both totally pointless and the process involving lots of guessing, several classmates were like "well, fuck math forever I guess" at that point, like if they'd been asked to dig a ditch with a spoon and then fill it back in.

I cannot overstate enough how consistently and extremely this has turned my kids off from math. 3-for-3 on absolutely hating this. Having to solve the same thing five different ways just pisses them off, and, like... yeah, of course it does. They want to finish the work and go play and it feels like you're just fucking with them and disrespecting their time by making them solve the same problem several times, even if that's not the intent.

bayesian thinking doesnt come to me naturally.. i have no intuition for it. seems forced. believe me - i have tried. but there are those who are swearing by it.

Before high school, math is just a grind of memorization and unmotivated manipulation of numbers.

Many students (ok, me, but I expect the same was true of others), get turned on to math for the first time when they encounter proofs in high school geometry and also actual applications in high school physics.

It's a revelation to students that math can be a way to go from one truth to another and thus find new truths. It's a way of thinking and that can be very exciting.

Tragically, many students disengage before this can happen because of sheer boredom and the tedium of endless math drills. Once they develop a gap in their knowledge it becomes difficult to progress unless those gaps are addressed. For lots of students, it all ends with fractions. You'd be surprised how many adults don't really understand fractions. For others it ends with algebra, and for the college bound it ends with calculus.

Only math majors and a minority of engineering/science/CS folks get past the "standard sequence" of math courses in college and gain an appreciation for the really interesting stuff that comes AFTER all that.

He shows that math skill is almost more like a sports talent than it is knowledge talent. He claims this based on the way people have to learn how to manipulate different math objects in their heads, whether treating them as rotated shapes, slot machines, or origami. It's like an imagination sport.

Also, he inspired me to relearn a lot of fundamental math on MathAcademy.com which has been super fun and stressful. I feel like I have the tetris effect but with polynomials now.

Rather, learning ability is a continuum. people have varying degrees of ability to learn mathematics. Couple this with environmental factors and society generates a huge variability in mathematical ability that crosses income levels and other demographics.

This view is rejected by many because it is against the push for equality.

That's probably why math writing largely makes me feel dyslexic, while programming came naturally. And why I hate Haskell and find it painful to read even though I understand the "hard" concepts behind it just fine—it's the form of it I can't deal with, not the ideas.

We were given the exact text of the final exam weeks in advance, and were allowed to do anything at all to prepare, including collaborating with the other students or asking other professors (who couldn't make heads or tails of it). The goal was to be able to answer 1 or 2 out of the 10 questions on the exam, and even if you couldn't you got a B+ at minimum.

I wish I had a better memory, but I believe one of the questions I successfully answered was to prove Post's Theorem using Turing machines? The problem is, I never used the knowledge from that class again, but to this day I still think about it. It would be amazing to go back and learn more about that fascinating intersection of philosophy and computer science.

What I loved the most was that it combined hard math with the kind of esoteric metaphysical questions about mathematics which many practitioners despise because they feel like it undermines their work. It turns out, when you go that deep it's impossible not to touch on the headier stuff.

constructive mathematics is close to computation and programming. and many including myself have a natural feel or intuition for it. A majority of euclids elements, and galois's original proof are constructive in nature.

I hate new textbooks because they're so built around instant gratification. They just come out and tell you how to solve the problem without building the solution up in any way. Maybe afterward they take a swipe at telling you how it works, but that's just completely the wrong way around IMO. It robs me of the chance to mull things over, try to anticipate how this will all come together in the end, and generally have my own "aha" moments along the way.

But, getting back to what you say, I think that it also engenders this tendency to reduce math to symbol manipulation. Because if they give you the formula in the first paragraph, then all subsequent explanation is going to end up being anchored to that formula. And IMO that's just completely wrong. Mathematical notation is at its best when it's a formalizing tool and mnemonic device for cementing concepts you already mostly understand. It's at its worst when it's being used as the primary communication channel.

(It's also an essential tool for actually performing any kind of symbolic reasoning such as algebraic manipulation, of course, but I'm mainly thinking about pedagogical uses here.)

So the minimum might not be zero, but it isn’t some fixed quantity.

Careful with that "all", even the most highly intelligent humans still have peaks and deficits in different domains.

Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.

The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.

Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).

When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.

As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.

In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.

There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.

I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.

Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.

A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.

In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.

I was having a problem agreeing with this subthread, and I have you to thank for putting it into words that I can finally formulate my disagreement against.

Have you never met one of those people for whom they did not need to "want"? They could literally phone it in and still do better than anyone else, no matter how dedicated they were. Even should practice/study be necessary for them, they benefited from it to some absurd proportion that I couldn't even guess to quantify. I've known more than one of these people.

I think most believe they don't exist for two reasons. The first is the ridiculous number of television shows and movies that depict motivation as being the key to success. We're just inundated with the (unsupported by evidence) that this is the means to extraordinary genius. Second, I would say that this is the most comforting theory. "Why yes, I could have been a gifted whatever or a talented something-or-other if I had put the time in, but I chose this other thing instead."

Maybe some would say we all need to believe this, that a society that doesn't believe in it is harsher or more unkind.

1. Everyone (young, old, poor, rich) thinks that maths is interesting and fun and beautiful and important. Not important "to get a good job" or "to go to a good college" or "to be an impressive person", but rather important because it's fun and interesting. And maybe it also helps you think clearly and get a good job and all these practical things, but they're secondary to the tremendous beauty and wondrousness of the domain.

2. Everyone believes that barring actual brain injuries people can learn mathematics to a pretty high level. Not Ramanujan level, not Terrence Tao, not even a research mathematician at one of the smaller universities, but a level of extreme comfort, let's say a minimum level of being able to confidently ace the typical types of exams 17 and 18 year olds face to finish secondary school in various countries.

Would you claim that in that world - people think maths is great, and that anyone can learn it - we'd see similar levels of ability and enjoyment of mathematics?

My claim is that we don't live in "Math-World", as described above, but "Anti-Math-World". And further, that anyone suggesting things have to be the way they are in Anti-Math-World is not only wrong, but also fundamentally lacking imagination and courage.

Kids are told week in week out that maths is stupid, that they are stupid, that their parents themselves are stupid, that the parents hated maths, that the teachers are stupid, and then when they end up doing poorly, people say: "ahhh, some kids just aren't bright!"

Parents who like things like learning and maths and reading and so on, have kids that tend to like those things. And parents that don't, usually don't. Saying that this somehow tells us something concrete and inalterable about the nature of the human brain is preposterous.

It's a card that's used by grown-ups who are terrified by the idea that our education systems are fundamentally broken.

I should probably find a time machine and re-do everything.

I bet if you asked in a survey of people that if you were given a UBI that covered all your expenses and needs what would you do? It would be perfection of the self or art. Both of these are what is practicing and learning math.

An average human is unable to even write properly. Even basic mathematical operations like multiplication and division are too complicated from their perspective.

You'll see something like a mathematical proof with no explanation and it's end of article. The edit history will have explanations aggressively removed.

The equivalent would be the article for say, splay tree, to have no diagrams and just a block of code - feeling no obligation to explain what it is or if you looked up a chemical and it would just give you some chemical equation, some properties and feel no obligation to tell you its use, whether it's hazardous or where you might find it... Or imagine a European aristocrat and all that is allowed is their heraldry and genealogy. Explanations of what the person did or why they're important are forbidden because, it's just a reference after all.

Nope, these math people are a special kind of bird and I'm not one of them.

I would say focusing on mindfullness(like vipasaana) can go a long way in this. But mindfulness is not an intellectual exercise, one has to live it. Do multiple hours of meditation a day and it gets you somewhere good in a few months.

If I was a multi-millionaire, learning lots of math on my free time would be one of the things I would pursue while chilling at my beach house.

They're just quick. But the ones I've met, at least, are quick to make associations. When I really dig and ask them to explain themselves or a concept, they usually make analogies to things they know, but I don't. Then I have to go learn that thing. Then they try the analogy again, but I haven't fully learned it from years of making analogies about it.

Years of grad school experience was painful like this, until I got to a point 10 years after grad school, after a PhD, and well into research, that I "just got" things (in my subfield) as well. It's these experiences that made me feel that it's 80% preparation and perspiration (both of which are dominated by time), and 20% "other" mythology. Don't get me wrong, that 20% is what makes a 2 year old read earlier than others, and getting started reading at 2 (and continuing it!) for 4 years before starting school will make you light years ahead of your peers. The same goes for chess, math, etc etc. There is something legendary about Oppenheimer learning enough dutch in 6 weeks to deliver a lecture. Or perhaps learning to translate his lecture and memorizing it. Who knows.

Do we really believe there's a magical "genius" such that they can do anything? No, so what are the limits to their genius? The limits are defined by what they are a genius at. This is a tautological definition.

I'm not saying "Anyone at any time can become a genius at anything". I'm saying "If you take a kid, start early, and cultivate them just right so that you have time to realize compounding effects, - you can let them grow into basically anything" (probablistically speaking - there are learning disabilities and physical issues etc).

Once you do this, you will soon develop Attention(Concentration) and Equanimity(Inner Calmness in the face of tremendous external circumstances).

Soon you start realizing some inescapable facts:

1. Your current moment is ever changing.

2. One attaches more of the self(the ego, I me and mine) with the past and tries to predict his state in the future and ends up miserable. Don't think with the I, you are bound to suffer. The right action is timeless and free of the I. This is the reduction of the ego.

3. There is tremendous joy in focusing on being present in the moment. If you are into running and all you are doing is taking joy in your feet, breath and posture all the time for 5,10 miles you know what I mean.

This is the key to everything. No amount of book reading on self improvement can get this to you

Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out!

Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not.

I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day informing me of its arrival. Thank you for confirming that it's worth it!

Agreed, but from my observation mathematics is often taught with a rigor that's more suited to students with a highly mathematical and or scientific aptitude (and with the assumption that students will progress to university-level mathematics), thus this approach often alienates those who've a more practical outlook towards the subject.

Mathematics syllabuses are set by those with high mathematical knowledge and it seems they often lose sight of the fact that they are trying to teach students who may not have the aptitude or skills in the subject to the degree that they have.

From, say, mid highschool onwards students are confronted with a plethora of mathematical expressions that seemingly have no connection their daily lives or their existence per se. For example students are expected to remember the many dozens of trigonometrical identities that litter textbooks (or they did when I was at school), and for some that's difficult and or very tedious. I know, I recall forgetting a few important identities at crucial moments such as in the middle of an exam.

Perhaps a better approach—at least for those who are seemingly disinterested in (or with a phobia about) mathematics—would be to spend more time on both the historical and practical side of mathematics.

Providing students with instances of why earlier mathematicians (earlier because the examples are simpler) struggled with mathematical problems and why many mathematical ideas and concepts not only preceded but were later found to be essential for engineering, physics and the sciences generally to advance would, I reckon, go a long way towards easing the furtive more gently into world of mathematics and of mathematical thinking.

Dozens of names come to mind, Euclid, Descartes, Fermat, Lagrange, Galois, Hamilton and so on. And I'd wager that telling students the story of how the young head-strong Évariste Galois met his unfortunate end—unfortunate for both him and mathematics—would never be forgotten by students even if they weren't familiar with his mathematics—which of course they wouldn't be. That said, the moment Galois' name was mentioned in university maths they'd sit up and take instant note.

Yes, I know, teachers will be snorting that there just isn't time in the syllabus for all that stuff, my counterargument is that it makes no sense if you alienate students and turn them off mathematics altogether. Clearly, a balance has to be struck, tailoring the subject matter to suit students would seem the way with the more mathematically inclined being taught deeper theoretical/more advanced material.

I always liked mathematics especially calculus as it immediately made sense to me and I always understood why it was important for a comprehensive understanding of the sciences. Nevertheless, I can't claim that I was a 'natural' mathematician in the more usual context of those words. I struggled with some concepts and some I didn't find interesting such as parts of linear algebra.

Had some teacher taken the time to explain its crucial importance in say physics with some examples then I'm certain my interest would have been piqued and that I'd have showed more interest in learning the subject.

> art same as entertainment

Could you volunteer me how much time you spend on it? And how is your day job?

> my pursuit of other things like cryptography, better algorithms, and understanding physics is limited by my primitive understanding of mathematics.

Could you volunteer why you would want to learn these subjects? Is it your day job or is it something you would like to pursue in the future.

> If I was a multi-millionaire, learning lots of math on my free time would be one of the things I would pursue while chilling at my beach house.

I said UBI

At least I know that David Bessis's mathematical work is not as shallow as this. His twitter thread on the process https://x.com/davidbessis/status/1849442592519286899 is actually quite insightful. I would guess this also made it into the book in some longform version, but I don't know whether I would buy the book just for that.

https://www.lines.com/guides/average-height-nba-players/1519

General intelligence also seems to have been trending downward since the 1970s (the reverse Flynn Effect)[3]. It has been measured in the US and Europe.

So, while it is true that the education system and other factors have an influence, the idea that "everybody is capable of X" is wrong and harmful. It's the equivalent of "nobody needs a wheelchair" or "everybody can see perfectly." People are different. A lot of nerds only hang out with other nerds, which screws up their perception of society.

[1]: https://www.thenationalliteracyinstitute.com/post/literacy-s... [2]: https://leo.blogs.uni-hamburg.de [3]: https://www.sciencedirect.com/science/article/pii/S016028962...

If someone's smart enough to get into a psychology class they are smart enough to be thought basic undergrad math. It wasn't your teaching failure necessarily, but it was someone's teaching failure at some point.

Not everyone can play Rachmaninov like Lugansky or do math like Terence Tao, but there is absolutely no doubt that almost all people are magnitudes away from their latent potential in almost all domains. I'm fairly certain you could teach any average person how to play Rachmaninov decently. You could bring any person to a reasonably high mathematical level. You can get any person to lift a few hundred pounds.

Most people today read at a 7th grade level, can't do basic math, and are out of air after 3 flights of stairs. "Everyone can do everything" is maybe not literally right but directionally right given how utterly far removed we are from developing practically anyone's potential.

https://awrestaurants.com/blog/aw-third-pound-burger-fractio...

The book, as I understand it, is about the life changing power of mathematical thinking. Quanta's mission is to make deep mathematics and mathematicians a "sexy" "human interest" topic, by making it as non-mathematical as possible while keeping a superficial veneer of mathematics.

"Mathematician Tai-Danae Bradley and physicist Gabe Perez-Giz offer ambitious content ... Previous host Kelsey Houston-Edwards "

"We can't on one hand believe in the plasticity and retrainability of the mind, while simultaneously believing that speed is something only a few are born with."

Sure we can, the initial orientation of neurons differs between people, so some people need less "plasticity and retrainability" to be good at a task. Plasticity is physical characteristic like height and varies between people.

Initial speed usually isn't that important, but speed of learning is important and makes the difference between possible and impossible within a human lifetime.

I see that many people are confused by the interview's title, and also by my take that math talent isn't primarily a matter of genes. It may sound like naive egalitarianism, but it's not. It's a statement about the nature of math as a cognitive activity.

For the sake of clarity, let me repost my reply to someone who had objected that my take was "clickbait".

This person's comment began with a nice metaphor: 'I cannot agree. It's just "feel-good thinking." "Everybody can do everything." Well, that's simply not true. I'm fairly sure you (yes, you in particular) can't run the 100m in less than 10s, no matter how hard you trained. And the biological underpinning of our capabilities doesn't magically stop at the brain-blood barrier. We all do have different brains.'

Here was my reply (copy-pasted from my post buried somewhere deep in the discussion):

I'm the author of what you've just described as clickbait.

Interestingly, the 100m metaphor is extensively discussed in my book, where I explain why it should rather lead to the exact opposite of your conclusion.

The situation with math isn't that there's a bunch of people who run under 10s. It's more like the best people run in 1 nanosecond, while the majority of the population never gets to the finish line.

Highly-heritable polygenic traits like height follow a Gaussian distribution because this is what you get through linear expression of many random variations. There is no genetic pathway to Pareto-like distribution like what we see in math — they're always obtained through iterated stochastic draws where one capitalizes on past successes (Yule process).

When I claim everyone is capable of doing math, I'm not making a naive egalitarian claim.

As a pure mathematician who's been exposed to insane levels of math "genius" , I'm acutely aware of the breadth of the math talent gap. As explained in the interview, I don't think "normal people" can catch up with people like Grothendieck or Thurston, who started in early childhood. But I do think that the extreme talent of these "geniuses" is a testimonial to the gigantic margin of progression that lies in each of us.

In other words: you'll never run in a nanosecond, but you can become 1000x better at math than you thought was your limit.

There are actual techniques that career mathematicians know about. These techniques are hard to teach because they’re hard to communicate: it's all about adopting the right mental attitude, performing the right "unseen actions" in your head.

I know this sounds like clickbait, but it's not. My book is a serious attempt to document the secret "oral tradition" of top mathematicians, what they all know and discuss behind closed doors.

Feel free to dismiss my ideas with a shrug, but just be aware that they are fairly consensual among elite mathematicians.

A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.

In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes. They may have a hard time putting words to it, but they all have a very clear memory of how they got there.

I want to learn those subjects because I enjoy learning and understanding. Life should be lived with knowledge applied through wisdom.

> Could you volunteer me how much time you spend on it? And how is your day job?

On entertainment? I can't tell you, I like to watch a movie or a tv show whenever I have time for it. There are more enjoyable pursuits in life, and most worthy pursuits involve adversity and require perseverance.

Because it turns out that, in the NBA, height does not correspond with ability! This of course makes sense, because all the players are filtered by being NBA professional basketballers. A shorter player simply has more exceptional ability in another dimension, be that dodging reflex, ability to visualize and then hit a ball trajectory from the three point line, and so on. Conversely, a very tall player is inherently useful for blocking, and doesn't have to be as objectively good at basketball in order to be a valuable teammate.

Despite this lack of correlation, when you look at an NBA team you see a bunch of very tall fellows indeed. Simpson's Paradox.

We see the same thing in intellectual pursuits. "I'm not nearly as smart as the smartest programmer I know, but I get promoted at work so I must be doing something right. Therefore anyone could do this, they just have to work hard like I did". Nope. You've already been selected into "professional programmer", this logic doesn't work.

[0]: https://en.wikipedia.org/wiki/Simpson's_paradox

Visualization is probably not a silver bullet but a lot of people are visual thinkers so maybe it would help a few more to reach a higher level in mathematical thinking.

The interesting finding is that this universal correlation is strong, real, and durable. Of course people in general have cognitive domains where they function better or worse than their g factor indicates, and that's before we get into the fact that intellectual task performance is strongly predicated on knowledge and practice, which is difficult to control for outside of tests designed (successfully, I must add) to do so.

Yes - there will be 10x-ers. And that group will have a 10x-er iside it, and so on given exponential dropoff of frequency of talent. Bobby Fisher is a few std dev above even the best, perhaps.

Generally speaking, "You can do anything you want, but wanting (enough, and naturally) is the hardest part" might need a three standard deviation limit.

Have you heard the phrase: Being average among those who practice makes you 9X% among the population? I think that's what I'm saying - you can be a top performer if you dedicate yourself, especially early enough, but almost nobody will.

A paid promotion of International Median is not Average Association.

In other words: Some students flourish despite their teachers, some flourish because of them.

Bad example, it's much more likely to create a musical prodigy by providing early and appropriate guidance. Of course this is not easy as it assumes already ideal teaching methods and adequate motivation to the youngling, but even those with some learning difficulties have the potential to excel. The subtypes of intellect required to play complex music and absord advanced abstract math subjects are quite different, former requiring strong short-term memory (sightreading) the latter fluid intelligence -I think almost everyone is familiar with these terms by now and knows that one can score high/low on certain subtypes of an IQ test affecting the total score-.

BTW IDK if the Rachmaninoff choice was deliberate to imply that even the most capable who lack the hand size won't be able to perform his works well yeah, but there are like 1000s of others composers accessible that the audiences appreciate even more. Attempting to equate music with sports in such manner is heavily Americanized and therefore completely absurd. Tons of great pianists who didn't have the hand size to interpret his most majestic works and of others. Tons of others who could but never bothered. There have been winners of large competitions who barely played any of his works during all stages of audition or generally music requiring immense bodily advantage. Besides, it's almost 100% not a hand size issue when there are 5 year old kids playing La Campanella with remarkable fluidity.

And even in this case this isn't even the point. Most conservatory alumni today are 100x skilled than the pianists of previous generations... yet they all sound the exact same, their playing lacks character/variability, deepness, elegance to the point where the composers ideas end up distorted. And those can be very skilled but just have poor understanding of the art, which is what music is, not the fast trills/runs, clean arpeggios, very strict metronomic pulse.

> So no, unless you've placed the bar for "mathetical skill" pretty low, or can show me proper evidence, I'm not going to believe it. "Everyone is capable of..." reeks of bullshit.

Well the vast majority of people in the Soviet Union were very math literate, regardless of what they ended up working as (although indeed most became engineers) and in quite advanced subjects. This is obviously a product of the extensive focus of primary and secondary education on the sciences back then.

So the point isn't to make everyone have PhD level math background and I heavily dislike the dork undertones/culture that everyone should love doing abstract math on their freetime or have to have some mathematical temperament' . But let's not go the other way and claim that those not coming close to achieving the knowledge those in the top % of the fields possess, they are chumps.

Come on, how often are kids exposed to such stupid talk? I suspect very infrequently.

My grandmother, who wasn't stupid by any means but who knew only basic arithmetic, would never have uttered such nonsense.

And I'd stress, like many of her generation and background, her knowledge of mathematics was minimal, if she'd been ask what calculus was she'd likely have been perplexed and probably have guessed it to be some kind of growth on one's foot.

For me the worst part was the teachers that encouraged that behavior and did the same.

Absolutely correct. And that begins with getting their interest, thus their attention; and that's a whole subject in and of itself.

His critique of Locke was that one can not have a correct model of human understanding (or world model) based on purely philosophical means, and that the lack of exposure to certain aspects of modern mathematics (that was emerging at that time) was the basis of Locke's misunderstandings.

I think that's also a trap. Even professional athletes spend a little bit of their time doing simple drills: shooting free throws, fielding fly balls, hitting easy groundstrokes.

Sometimes your daily work keeps up the "easy" skills, but if you haven't used a skill in a while, it's not a bad idea to do some easy reps before you try to combine it with other skills in difficult ways.

Hm? That is exactly why UBI works, no? As the name indicates, the bare minimum is taken care so that you can work towards "the best life possible for our loved ones" without worrying about starving, sickness or homelessness.

In contrast, if you were wrong and people would be content with the bare minimum, then UBI would be a bad idea. Though then they could just commit some felonies and be content with having a cell, bread and water for the rest of their lives, no?

Cool, come and have a coffee with me :) I have older and younger siblings and was the one randomly blessed.

Whereas most recognised talents are associated with hard work and so there is then this visible link, I am a good example as I did the bare minimum throughout education (and beyond...).

The way my brain processes and selectively discards/stores the information it receives is very different to majority of the population. I have no control over it.

I take zero credit for any of my achievments - I regularly meet intelligent people near to retirement who have been to a tier 1 university, may have PHDs, worked 60 hours a week since they were born, been on course and what not and cannot reach the levels I can.

My nurturing was no different to siblings/peers (and was terrible!)

Note: I have my weaknesses too, but as a whole, I am exceptional. Not through effort!! Completely random - neither of my parents are intelligent and nothing up the ancestary tree as far as I know.

Likely so, but is suggest that personality, drive and motivation are also very important factors. I know from experience that stuff I had little interest in as a youngster and that I've still little in I still know little about.

Yes, my interests have grown and broadened over the years but simply I regard some stuff so irrelevant to my life that it's not worth a second thought and I am much better off applying my limited number of neurons to matters of greater importance and enjoyment.

Of course, no one has the luxury of just learning about what one finds interesting and or enjoyable, life's knocks and experiences along with utilitarian-like imperatives force one to learn stuff they'd rather not know about.

Maybe visualize splitting a pint with a friend. If you split the pint into 2 equal parts and each of you gets 1 of those 2 parts you each get the same amount.

Then visualize splitting it instead into 3 equal parts. You get 1 of those parts and your buddy gets 2. There's no fractions there so it should be easier to visualize that your buddy got twice as much as you did.

Comparing those two visualizations might make it easier to see that someone who gets 2/3 of a pint gets more than someone who gets 1/2 of a pint.

A bad teacher instead gives you a bimodal distribution and just doesn't bother teaching those students.

This issue becomes very clear when you see how many ways there are to express a simple concept like linear regression. I've had the chance to see that for myself in university when I pursued a bunch of classes from different domains.

The fact that introductory statistics (y = a + bx), econometrics (Y = beta_0 + beta_1 * X) and machine learning (theta = epsilon * x, incl. matrix notation) talk about the same formula with quite different notation can definitely be confusing. All of them have their historical or logical reasons for formulating it that way, but I believe it's an unnecessary source of friction.

If we go back to basic maths, I believe it's the same issue. Early in my elementary school, the pedagogical approach was this: 0. only work with numbers until some level 1. introduce the first few letters of the alphabet as variables (a, b, c) - despite no one ever explaining why "variable" and "constant" are nouns all of a sudden 2. abruptly switch to the last letters of the alphabet (x, y, z), two of which don't exist in my native language 3. reintroduce (a,b,c) as sometimes free variables, and sometimes very specific things (e.g., discriminant of a quadratic equation) 4. and so on for greek letters, etc.

It's not something that's too difficult to grasp after some time, but I think it's a waste to introduce this friction to kids when they're also dealing with completely unrelated courses, social problems, biological differences, etc. If you're confused by "why" variables are useful, why does the notation change all the time, and why it sometimes doesn't - and who gets to decide - this never gets resolved.

Not to mention how arbitrarily things are presented, no explanation of how things came to be or why we learn them, and every other problem that schools haven't tackled since my grandparents were kids.

You probably have a narrow definition of “most people” (probably some motivated high school or undergraduate student) and too loose with what it means to “understand mathematical concepts abstractly”.

Take an analogy: imagine professional musicians saying that most people should be able to take a piece of music and understand its harmonic structure, then apply it to a new setting to generate a new piece. Most people will reject this idea as absurd.

It reminds me of programming, that moment when new code starts to really sync up and code goes from being a bunch of text to more intuitive structures. When really in the zone it feels like each function has its own shape and vibe. Like a clean little box function or a big ugly angry urchin function or a useless little circle that doesn't do anything and you make a note to get rid of. I can kinda see the whole graph connected by the data that flows through them.

I add third (okay, 2b) - because the pain of coming up with the fact other people are better than you at a deep, fundamental level is too overwhelming.

The first time the difference between understanding some math, and understanding what the math meant, was after high school Trig. The moment I started manually programming graphics from scratch, the circle as a series of dots, trigonometry transformed in my mind. I can't even say what the difference was - the math was exactly the same - but some larger area of my brain suddenly connected with all the concepts I had already learned.

While ordering the "Mathematica: A Secret World of Intuition and Curiosity" I came across these books, which looked very promising in the "learning formal math by expanding intuition" theme, so I bought them too:

Field Theory For The Non-Physicist, by Ville Hirvonen [0]

Lagrangian Mechanics For The Non-Physicist, by Ville Hirvonen [1]

The Gravity of Math: How Geometry Rules the Universe, by Steve Nadis, Shing-Tung Yau [2]

Vector: A Surprising Story of Space, Time, and Mathematical Transformation, by Robyn Arianrhod [3]

[0] https://www.amazon.com/dp/B0CN7HMTJN

[1] https://www.amazon.com/dp/B0CN7HMK38

[2] https://www.amazon.com/dp/1541604296

[3] https://www.amazon.com/dp/0226821102

Excited to read each (based on their synopses & ratings), and if I will get compounding fluency across both math and physics between all five books.

There's a difference between what your human brain would become accustomed to (which you'd be right, it'd scale up and up forever) versus what would allow the base level of health and opportunity. As in, not having to worry about eating the next day.

And because you're right that human brains strive for more wealth, UBI should grant you the opportunity to pursue it without fear of failure.

https://a.co/d/fZnWUU8

Speaking only for myself, I'm not confused at all. Rather I vigorously disagree with this statement, and think that stumping for this counterfactual premise leads to cruel behavior towards children (in particular) who plainly do not have what it takes to learn, for example and in particular, algebra.

> In other words: the people who run the mathematical 100m in under a second don't think it's because of their genes.

This is not their subject of expertise, and they are simply wrong. Why? Simpson's Paradox, ironically.

UBI would relieve stress for the lowest earning people, but it won't result in pursuit of passion for most people. economically speaking, because most people can afford certain things (like rent) the price of those things will go up, things are priced based on what potential consumers are willing to pay. If rent costs $1000 for a specific type of unit, but suddenly everyone on UBI can afford that easily, the landlord would raise the rent, the cost of things won't remain static when wages rise for a large portion of the population. Increased demand without increased cost is loss of potential revenue. The quality of life for people on UBI would be barely surviving, and UBI would need to increase constantly to keep people from becoming homeless or starving.

This is the "Cobra effect" embodied. It provides a perverse incentive. healthcare in the US is out of control for this reason. health care providers keep increasing cost, because the patient is not the client, the insurance company is, so long as everyone is getting insurance, the cost of care is the maximum reliably predictable pay out by the insurance company. Not increasing cost is just bad business. You will have to also force all kinds of businesses from raising prices if it can work, and even when it works UBI will result in subsidizing low-wager workers for businesses, because they'll still have to work some job to afford anything outside of food,shelter and the basics.

A practical alternative to UBI is a local tax on businesses, kind of like a property tax but this tax is based on an inverse of an assessment of wages, rent, welfare pay out and other social conditions in the area. the higher wages are, lower rent is,etc.. the lower the tax is, it might even result in a credit. An inverse of a perverse incentive like UBI. Unemployment would also be partially funded through this, the unemployed would forever get a UBI like pay out so long as they are pursuing education or work of some kind based on what is in demand in their area. Businesses get a healthy talent pool to choose from and cost of living is balanced.

>You probably have a narrow definition of “most people” (probably some motivated high school or undergraduate student)

I was thinking "3-4 out of 5 people you pick on the street at random".

>too loose with what it means to “understand mathematical concepts abstractly”.

Enough that they could recognize whether a mathematical concept is applied correctly (e.g. if I have a 2% monthly interest, should I multiply it by 12 to get the annual interest? Why, or why not?) and conversely to correctly apply concepts they already understand to new situations, as well as to leverage those concepts to potentially learn new ones that depend on them.

>imagine professional musicians saying that most people should be able to take a piece of music and understand its harmonic structure, then apply it to a new setting to generate a new piece. Most people will reject this idea as absurd.

Okay, but we're arguing about what is the case, not about which idea has more popular support. Since most people don't understand thing 1 about composition, why should their opinion matter? A skilled composer's opinion on the matter should have more bearing than a million laymen's.

This is how I got back into learning maths... through necessity and immediate application.

This is not really true is it? There were not that many standardized testing globally to measure such claims. Many people were in poverty and did not get tested, did not go to schools, or finished schools very early (5, 9 years). Many more kids go to school these days.

> In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun

It is just wrong. It was the same as now, except it was critical for people to show results because otherwise you had grim perspectives in the life, there was little "fun". People wanted to get into universities to get better jobs and to get better apartments, to be able to leave their parents. You could not just buy places, but a good position in some public body would guarantee you a nice place. FYI engineers could earn more in comparison to other jobs, not to mention if you could get into defense industry.

One of the interesting things is being able to exactly describe how something is an anti-pattern, because you have a precise language for describing constraints.

The story is much more complex than "if it's not genetic then everybody should get it". It's quite cruel to assume that if you don't get math today you'll never get it, and there are tons of documented counter-examples of kids who didn't get it at all who end up becoming way above average.

If you think that Descartes, Newton, Einstein, Feynman, Grothendieck (to just cite a few) are all equally misled on their own account because of Simpson's Paradox, which statistical result will to bring to the table to justify that YOU are right?

By the way, Stanislas Dehaene, one of the leading researchers on the neuroscience of mathematical cognition, is also on my side.

Your best example is the decimal system in contrast to roman numerals. I think that explains the point well. The zero is one of those tricks, and most people know it now, but that wasn't true until very recently.

I mean to say, just applied scientific thinking is important. Even if you never get into pure math or computer programming, applying concepts like "variables", "functions" or "proofs" is universally useful.

I have heard it is the ego protecting itself by rejecting something outright rather than admitting you can't do it. It still happens to me all the time. My favorite technique was one I heard from a college professor. He starts reading while filling a notepad with sloppy notes, once a page is filled he just throws it away. He claimed it was the fastest way to "condition his brain to the problem space". More than the exercise I like the idea that your brain cannot even function in that space until it has been conditioned.

But be warned, it gets progressively harder, along with the concepts, so unless you're conversant with tensors, at some point you will have to put on your thinking cap.

The reviews on Goodreads – including my own – are worth reading to get a flavour: https://www.goodreads.com/book/show/202104095-vector

He's of the opinion that math should be taught not as jumping through hoops for "reasons", but as an art, enjoyable for its own sake, and that this would actually produce more confident and capable thinkers than the current approach. (I think the argument applies to almost all education but his focus is just on math.)

Total time on site (gathered from a web extension): 40h 30m Total days since start: 32

Total XP earned: 1881

Since "1 XP is roughly equivalent to 1 minute of focused work", I "should have" only spent 31 hours. I did the placement test and started at ~30%, and now I'm at 76%. I'd say 75% is stuff I learned in HS but never had a great handle on, 25% I never knew before.

Overall, I'm quite happy with the course. I'm learning a lot every day and feel like I have stronger fundamentals than I did when I was in school. The spaced review is good but I do worry I'll lose it again, so I'm thinking of ways I can integrate this sort of math into my development projects. It's no Duolingo, you really do have to put in effort and aim for a certain number of Xp per day (I try for 60 XP rather than time).

As for my own half-baked opinion, I want to say that the Church-Turing Thesis and Chomsky's innate theory of cognition have something to add to the picture. Homo sapiens as a species essentially has the capacity to think analytically and mathematically; I want to argue this is a universal capacity loosely analogous to the theory of universal Turing machines. So what matters is people's early experiences where they learn how to both practice and, critically, to play, when they learn difficult ideas and skills.

Also, as an amateur pianist, most people don't know that modern piano teaching emphasizes not fixed limits of the student but that many students learn the wrong techniques even from well-meaning piano coaches. Just the other day I was watching a recent YouTube Julliard-level masterclass where the teacher was correcting a student on her finger playing technique, presumably this student had been taught the wrong technique since childhood. With music or sports a coach can visually see many such technique problems; with math teaching it of course harder.

One day, I went back to my high school and spoke to my computer science mentor back then [1]. I passionately asked him why we were never exposed to group theory. The answer, he said, was the SAT. None of that stuff is on the SAT, so it can't be justified teaching.

[1] The great Andrew Merrill

Track and swimming are where innate physical attributes have the most obvious benefits. Michael Phelphs had the perfect body for swimming. There is no amount of trainingg that 99.999% of the population could do to get close to what Usain Bolt ran. Most humans could not train to run under 4 minutes in a mile or under 2:30 in a marathon. They just don't have the right muscular and cardiovascular physiology.

Team sports are of course more complicated as other qualities come into play that aren't as directly physiological.

When I say it's not primarily genetic, many people wrongly assume there's an entirely explainable and replicable way of accessing this state. There isn't.

The 20,000 hours rule is a bit misleading, because who gets to invest 20,000 hours into something? How do you create this drive, this trajectory? You must have a good hope that it'll yield something worth the effort.

This is why the injunction to "work harder" so often misses the mark.

However, even if only a tiny fraction of the population will end up becoming a "genius", it's very important to debunk the myth, because the real story has valuable lessons for everyone: it gives concrete and pragmatic indications on what one should be on lookout for.

It's not fully teachable up to genius level, but the directionality is teachable and extremely valuable.

@faangguyindia, I hope everything will be ok one day :-) Or maybe it already is? You wrote "was".

I might be wrong but I think if you (@faangguy) manage to create a life that makes you feel happy, the women will notice that you're happy, and that goes a long way. But you'll also need to be somewhere where there are some women around? (obviously) If you're in a FAANG in SF maybe for the moment that's not so easy

My number theory professor was a brilliant mind, someone who had spent lots of time at the Institute for Advanced Study, and he absolutely sucked at adding/subtracting/multiplying numbers. And that was something he freely admitted. It wasn't important to his work, and it isn't important to mine.

> A good number of Abel prize winners & Fields medallists have read my book and found it important and accurate. It's been blurbed by Steve Strogatz and Terry Tao.

Sounds like people mostly living in different bubbles? What do they know about the world?

They aren't hanging out with the kids who fail in school because maths and reading and writing is to hard, and then start selling drugs instead and get guns and start killing each other.

> [they] don't think it's because of their genes

Do you think someone would tell you, if he/she thought it was?

I mean, that can come off as arrogant? Wouldn't they rather tend to say "it was hard work, anyone can do it" and prioritize being liked by others

> Pareto-like distribution like what we see in math

Unclear to me what you have in mind. If there's a graph it'd be interesting to have a look? I wonder whats on the different axis, and how you arrived at the numbers and data points

1) It's tragic that being "bad at math" is often positioned as some kind of badge of honour.

2) It's definitely not the case that everyone is capable of mathematical thinking. Having spent a certain amount of time trying to teach one of my kids some semblance of mathematical thinking, I can report confidently that his ability in this area is almost non-existent. His undeniable skills lie in music and writing, but definitely not in maths.

Yes, music and maths have some things in common. But musical thinking is not mathematical thinking.

In short, it is silly to expect UBI to be a means by which people would work only if they want to work, and they would pursue their passions and dreams instead. That kind of a society I think is possible, but it would also have to reach a level of wealth where money itself is not required (think star trek).

"I have this clue at hand" can have broad impacts. Software development's emphasis on clarity, naming, and communication protocols, helped me a lot with infant conversation. Math done well, can be a rich source of clues, especially around thinking clearly.

There's an idea that education should provide more life skills (like personal finance). And another, that education should have a punch list (as in construction), of "everyone at least leaves with these". Now AIish personalized instruction will perhaps permit delivering a massive implicit curriculum, far larger than we usually think of as a reasonable set of learning objectives. Just as a story can teach far more than the obstensible moral/punchline of the story, so too might each description, example, question and problem, dynamically tuned in concert. So perhaps it's time to start exploring how to use that? In the past, we worked by indirection - "do literary criticism, and probabilisticly obtain various skills". And here, from math. Perhaps there's a near-term opportunity to be more explicit, and thorough, about the cluefulnesses we'd like to provide?

Happy to hear you're enjoying it, makes me even more confident that I should read it :)

At least anecdotally, many people around me now have more children, on the other hand.

Well, they do know something about math — in particular that it requires a certain "attitude", something that no-one told them about in school and they felt they only discovered by chance.

Starting from Descartes and his famous "method", continuing with Newton, Einstein, Grothendieck all these guys insisted that they were special because of this "attitude" and not because of what people call "intelligence". They viewed intelligence as a by-product of their method, not the other way around. They even wrote books as an attempt to share this method (which is quite hard to achieve, for reasons I explain in my book.)

Why do you bring "kids who fail in school" and "start selling drugs" into this conversation? What does it have to do with whether math genius is driven by genetics or idiosyncratic cognitive development?

And why would a mathematician be disqualified from discussing the specifics of math just because they're not hanging out with lost kids? Are you better qualified? Did you sequence the DNA of those kids and identified the genes responsible for their learning difficulties?

>> [they] don't think it's because of their genes

> Do you think someone would tell you, if he/she thought it was?

Well, an example I know quite well is mine. I was certainly "gifted" in math — something like in the top 1% of my generation, but not much above and definitely nowhere near the IMO gold medallists whom I met early in my studies.

A number of random events happened to me, including the chance discovery of certain ways to mentally engage with mathematical objects. This propelled me onto an entirely different trajectory, and I ended up solving tough conjectures & publishing in Inventiones & Annals of Math (an entirely different planet from the top 1% I started from)

My relative position wrt my peer group went through a series of well-delineated spikes from 17yo (when I started as an undergrad) to 35yo (when I quit academia), associated with specific methodological & psychological breakthroughs. I'm pretty confident that my genes stayed the same during this entire period.

And as to why I was initially "gifted", I do have some very plausible non-genetic factors that might be the explanation.

I don't claim this proves anything. But I see no reason why my account should be disqualified on the grounds that I'm good at math.

Usually, competency in one domain is presumed to make you a bit more qualified than the random person on the internet when it comes to explaining how this domain operates. Why should math be the exception?

Sure they don't. Most extremely successful people (want to) think that the main reason of their success is their commitment and hard work. It runs completely contrary to the findings of modern biology and psychology, most of our intellectual potential at adulthood is genetic.

The floor and ceiling you will operate on in your life is decided the moment of chromosomal crossover.

Weirdly, accurately throwing a ball never got me a good grade on a math test.

Maybe industrialized countries, a bit better off neighborhoods, STEM classes in high school, and then university? (Not everyone but ... Most here?)

What he is saying is the default hypothesis based on our understanding of biology and psychology. If you have variability in genes you'll get variability in characteristics that are connected to them - height, bone structure, mental capacity, etc.

It is on you to prove that there is an arbitrary cut-off when it comes to this variance from which point it doesn't matter in regards to e.g. cognitive and mathematical ability.

> Enough that they could recognize whether a mathematical concept is applied correctly (e.g. if I have a 2% monthly interest, should I multiply it by 12 to get the annual interest? Why, or why not?) and conversely to correctly apply concepts they already understand to new situations, as well as to leverage those concepts to potentially learn new ones that depend on them.

No it doesn't if they do not have the abilities to comprehend it. I think you're living in a bubble of at least average-smart people and don't get that probably millions if not billions of people around the globe (based on average IQs) won't really get that concept.

We are extraordinarily fortunate to be taken seriously enough as a community for the primary sources on stories like this to come talk to us directly. I'd ask, as a neighbor of yours in this community, that you not (rhetorically) chase those people away.

Thanks!

FWIW I did not go to any fancy school, I was educated in the state system.

In practice your views result in stinting access of non-existent (in your opinion) talented children to a faster education track. They don't exist therefore they don't need different treatment (finer points get lost when the idea disseminates). Quite a hot theme in American education two (or so) years ago.

Honestly speaking I think this is a wrong way to teach people to think about Math. Math is just one of those things which feels hard because people struggle to hold long trials of manipulations in their head. Especially if they are manipulations to something very large, evolved slowly over hundreds of steps. People are not coming short, its just how the human mind works.

IMO, the right way to teach Math is to teach people that its just base axioms, manipulation rules. And after that its how you evolve the base axiom using rules. People need to be taught how to make one valid change at a time. Of course this means tons of paper work and patience. But that is what Math actually is. Its taking truth and rules, to make new ones.

Im teaching this to my kid, and she often goes like this is it?? its really just laborious paper work??

Im using this method and LLM help at times these days to learn Algorithms and Data Structures. When you start working things from base conditions and build from there. A lot of Algos that otherwise seem like the domain of novel inventions just seem to follow from the manual steps you just worked, and then translated into a program.

When you remove all the fluff, Patience and Paper work is all there is to Math.

They don't though - there are benefits all the way up.

Giving a few examples of benefits from low order mathematical thinking (understanding concepts of compounding, etc) does not equal a statement that these are the only benefits.

I see you're familiar with Ryan North.

I think it is misleading to call that scenario UBI and that the rest of your post also shares barely any resemblance to actual UBI plans, goals or side effects.

Then you're agreeing with me. The thing all of those have in common is that they follow normal distributions. The shortest recorded adult and the tallest recorded adult are quite far apart, yet the vast majority of adults are between 150-200 cm tall. That's precisely what I was saying; the outliers of mathematical skill are very very far apart, but most people are roughly equally capable.

>I think you're living in a bubble of at least average-smart people and don't get that probably millions if not billions of people around the globe (based on average IQs) won't really get that concept.

What I'm saying it that it even someone with below-average IQ could do it, if taught properly. Mathematics is less about being smart and more about being rigorous.

I think the reason you disagree is that it sounds like you’re teaching your child to be good at math class (a perfectly valid and good thing to do). Being good at math class requires being good at rational/logical thinking and computation. It also has only glancing similarities to anything that the author would recognise as mathematics.

Nah, these are the same things. Trying to make Math look like is for people who are 'geniuses' i.e people with massive capabilities of holding large thought trials and changelogs in their head is how you arrive at making people look stupid doing math and eventually make them hate the subject.

Math is paper work. Approach it that way and all of a sudden doing a 100 page proof is within everyones reach. If you ask people to hold a 100 page proof in their head, and more importantly make changes to that in random places and fix the entire changelog trial, probably 2 - 3 people on earth will be able to do it, and you will just convince everyone else its not for them.

I have a hunch that big mathematical breakthroughs in history have happened around and after renaissance era due to paper getting cheap and ubiquitous. There is only that much you can do in your brain alone.

It's tragic and all, but were there attempts to teach non-preselected children/adolescents interesting stuff first? What are the results?

People do studies on weeks old babies, it seems unlikely curiosity has been restrained at that point.