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

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.

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.

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'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 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.”

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.

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

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.

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...

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.

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.

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.

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.

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.

"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?

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.

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.

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

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?

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.

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.

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.)

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

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

- No, but the smarter kids might.

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.

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).

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

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.

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.

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.