alt Hacker NewsEveryone is capable of, and can benefit from, mathematical thinking
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I've been reading the author's book, Mathematica, and it's awesome. The title of this post doesn't do it justice.
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.
To my mind, the premature formalization of the math is the principal contributor to gas lighting and alienation of people from maths. The reduction of concepts to symbols and manipulation thereof, is an afterthought. It's misguided for them to be introduced to people right at the outset.
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
A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it. So is there some scientific study backing up the claim that all these people could easily learn it or are we just making it up because its a nice egalitarian thesis for a math popularization book?
I'm not a math teacher, but I do enjoy math, and I have helped several family members and friends with math courses.
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.
I’m far from being any kind of serious mathematician, but I’ve learned more in the last couple years of taking that seriously as an ambition than in decades of relegating myself to inferiority on it.
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.”
There is this element of abstract mathematical thinking that many young people get exposed to at some point in the educational system but just never "get it" and they disconnect. This is where it goes awry as the gap only widens later on and its a pity.
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 think for most people the issue is that they never even get to the fun stuff. I remember not really liking math right until university where we had set theory in the first semester, defined the number sets from scratch went on to monoids, groups, rings etc. That "starting from scratch" and defining everything was extremely satisfying!
In my high school we were basically only instructed to get good at applied math. Calculus. Which more often than not was simply "plugging it in". Most of that work is trivially automatible through Mathematica. When I reached a university, I took number theory and abstract algebra and it blew my mind that math was actually so beautiful in a way that defied explanation. When I took real analysis I finally saw the side of calculus that didn't seem like a waste of time.
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
"It's the economy stupid" is what I would say. Mental capacity is capacity. Most of us don't study math not because we don't want to but because we can't.
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.
The author: Would love to participate but account creation seems to be broken on Hacker News https://x.com/davidbessis/status/1859561768915173466
> the provocative claim
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...
Last year I read How to Solve It and the first half of one of Polya's other books - Mathematics and Plausible Reasoning. I certainly didn't commit them to memory, and I never systematically tried to apply them during self-study, but they do sometimes help give me a pointer in the right direction (i.e. trying to think of auxiliary problems to solve, trying to find a way to make the known & unknown closer together... etc.).
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.
Agree. I’ve been trying to learn ML and data for a few years now and, around 2021 I guess, realised Maths was the real block.
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.
If the article is in any way representative of the book, then I'm not sure what there is to be learned from the book. That mathematical skills can be honed through practice? That it happens at an intuitive, pre-rigorous level before it is ready to be written down on paper? How surprising. And I doubt he can disprove the genetical component of intelligence, only show that there are other components to mathematical productivity as well.
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.
I’m actually interested in the “can benefit from” claim in this title. I don’t particularly doubt that most people could become reasonably good at math, but I wonder how much of the juice is worth the squeeze, and how juicy it is on the scale from basic arithmetic up to the point where you’re reading papers by June Huh or Terry Tao.
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.
I studied math hard for several years in college and graduate school—purely out of interest and enjoyment, not for any practical purpose. That was more than forty years ago, but Bessis's description of the role of intuition in learning and doing math matches my recollection of my subjective experience of it.
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've lately been struck by people having a life difficulty due to "missing a clue" absent some experience. The person poorly conceptualizing and executing physical therapy, for lack of athletic experiences. The person poorly handling cognitive decline, for lack of a grasp of work processes. The person variously failing from having physical discomfort as an abort criteria, for lack of experiences normalizing its deferral.
"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?
I want to say yes, but I have two counters. One is that math nerds at school insisted on intimidating for the win and I just hated it.
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.
Not sure this article captured it for me.
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.
"mathematics is a game of back-and-forth between intuition and logic" I teach/guide Math at our school (we run a small school and currently have kids under age 10) and this is so so true.
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
Has anyone here self-taught themselves math in later life?
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?
I have an autistic friend with dyscalculia. They see numeric digits as individual characters (as in a story), each with their own personalities. Each digit has its own color, its own feelings. But they are not quantities; they don't make up quantities. Numbers are very nearly opaque to them. I wonder how this theory would apply to them. Do they still perform mathematical thinking? They're still capable of nearly all the same logic that I am, and even some that I'm not (their synesthesia gives them some color/pattern/vibes logic that I don't have)... just not math.
Two thoughts
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.
> everyone can, and should, try to improve their mathematical thinking — not necessarily to solve math problems, but as a general self-help technique
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.
Given the sentiment in these comments, I figure this crowd might be interested in the book "Measurement" by Paul Lockhart (the guy who wrote "A Mathematician's Lament")
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.)
Reading most of the answers here, I can only conclude most of you were home schooled or went to some fancy schools for gifted children.
An average human is unable to even write properly. Even basic mathematical operations like multiplication and division are too complicated from their perspective.
As a young child your brain is much more suitable to learn languages. You can make kids learn 4 languages effortlessly in the right context. When you grow up, slowly shift the focus to abstract thinking. And that shift can rely on building intuition using visualisation and experience.
Statistical (Bayesian) thinking is an extremely underrated way of thinking of almost everything.
Great to see so many reactions to my interview, thanks!
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.
Some very nice related works that dispel widespread math myths: (1) What Is Mathematics, Really? Hersh, 1997. https://books.google.com/books/about/What_is_Mathematics_Rea... (2) Where Mathematics Comes From. Lakoff and Nunez, 2000. https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
For example finance is such an important aspect of our lifes and you just need some understanding of math principles to understand how to make good financial choices.
This is pretty interesting. I did reasonably well in maths up to the A-levels, and then absolutely collapsed in university. I never got a grade better than B- in any maths-adjacent class. Discrete maths was my worst topic, I barely scraped a pass. And the irony was that I majored in CS and physics.
I should probably find a time machine and re-do everything.
- Hey teach, will I really need all these logarithms, derivatives and vectors in my adult life?
- No, but the smarter kids might.
I used to get very frustrated that others could not intuit information the way I could. I have a lot of experience trying to express quantities to leaders and policymakers.
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.
But, is that profitable? I'm both being sarcastic and real with this question.
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).
there's thinking mathematically and then there's being able to fluently read math articles on wikipedia as if they're easier than ernest hemingway. I can do the former and the latter I will insist until my grave is impossible for me.
I conducted an interview with leetcode 2 years ago while not doing any leetcoding before. Surprisingly, by just applying some math tricks i finished them and got into later rounds. So yes math tricks are helpful.
This guy is unbelievably French (I mean in his intellectual character). Here I was expecting a kind of rehash of the 20th century movements of pure math and high modernism[0], but instead we get a frankly Hegelian concept of math or at least a Hegel filtered through 20th and 21st century French philosophy.
That's a very important thougt and I belive the world would be better, if more people would connect with their mathematical side.
Shouldn’t that headline read “or can benefit from”. Not a shining example of good mathematical thinking. Oh the irony.
But what is the difference between math talent and plug-n-chug math talent? That seems to be the most significant filter.
And yet it is flame-bait to suggest that programmers benefit from mathematical thinking. I've not met a more passionate and divided crowd on the issue. Most traditional engineers wouldn't disagree that they use and benefit from mathematical thinking. Programmers though?
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.
To use an absolutist statement about human well bieng tied to the authors pet ideals,is a classic example of magical thinking, and a lack of understanding in general,tied to a character assination of the many many excellent non numerative humans living and thriving everywhere. Not nice at all.
It's a shame the title begins with 'Mathematica', makes one think the book is about the Wolfram software. That's the first thing I thought of when I saw the title. Hopefully Wolfram doesn't sue him for copyright violation or some such infringement.
This interplay between intuition and logic is exactly what makes the magic happen. You need intuition to feel your way forward, and then logic to solidify your progress so far, and also for ideas maybe not directly accessible via intuition only. I've experienced that myself, and it is even well-documented, because I wrote technical reports and such at each stage. My discovery of Abstraction Logic went through various stages:
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.
What is 13% of 91? I don't know. Do you? But I now 10% of 91 is 9.1 I got somewhere eh? Hey also I know that 1% of 91 is 0.91 Duh! lets triple that. 0.91 x 3 = 0.9x3 + 0.01x3 = 2.7 + 0.03 = 2.73 Now lets add 9.1.. 11.83 Weee! (Now my date is rolling her eyes and the waiter is stone faced)
Gentle Reminder that the author of this article used to have a wonderful math channel: https://www.youtube.com/c/pbsinfiniteseries
Careful there. They'll start voting logically..
I agree with the sentiment of this. I think our obsession with innate mathematical skill and genius is so detrimental to the growth mindset that you need to have in order to learn things.
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.