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!
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.
> 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 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”.
Caveat here is that "talent" and "dedication" is linked to speed at least in the beginning. For instance, any student can learn calculus given enough time and advice even starting from scratch. However, the syllabus wants all this to happen in one semester.
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.
>I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
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.
So you're saying success at maths isn't an inbuilt ability. Instead, it depends on an (inbuilt) ability to hyper focus... Which you are just born with?
>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.
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.
I've had some success converting people by telling them others had convinced them they were stupid. They usually have one or two things they are actually good at, like a domain they flee to. I simply point out how everything else is exactly like [say] playing the guitar. Eventually you will be good enough to sing at the same time. Clearly you already are a genius. I cant even remember the most basic cords or lyrics because I've never bothered with it.
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.
> Whilst some people (mostly spectrum) do seem have an innate talent
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 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 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.
I saw a lot of this when I went to college for engineering, some professors had this ability (or willingness) to make hard things simple, and others did the opposite, it was the same with the books, I dreaded the "exercise for the reader" shit, I don't think I ever did any of those, so those were all proofs I never got.
I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
> If it's easy, then it means you already know this material, and you're wasting your time.
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.
As a kid I was also terrible at maths, then later became obsessed with it as an adult because I didn't understand it, just like OP. It was the (second) best thing I've ever done! The world becomes a lot more interesting.
It's funny because I've had the opposite heuristic most of my line: the things I want to do most are whatever is hardest. This worked great for building my maths and physics skills and knowledge.
But when I started focusing on making money I've come to believe it's a bad heuristic for that purpose.
> ...my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
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
"A loser in societal view"... What does that objectively mean? That only reads like you had or have a low sense of self worth. It must've been your perceived definition of what society is because how could you have come to such a conclusion? I think I'd actually subconsciously tend more to viewing someone as "a loser" if they made such a statement because it comes off as self victimization (without an apparent explanation to an outside observer).
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.
Amazingly, I believe that today, with the myriad of tools available, anyone can advance in sciences like mathematics at their own pace by combining black-box and white-box approaches. Computers, in this context, could serve as your personal “Batcomputer” [1]. That said, I would always recommend engaging in social sciences with others, not working alone.
Who knows? You might also contribute meaningfully to these fields as you embrace your own unique path.
"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.
This was a very tiring blog post for me. And I have a quip about posts that open with questions but close without obvious definite answer, no matter how simple it is.
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.
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?
That certain countries both now and in the past have had significantly higher mathematical ability among the general population and much higher proportions going on to further study suggests that ability isn’t innate but that people don’t choose it. In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun.
Why would ability not be innate just because some people with the ability don't use it?
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.
We are not really taught (thought) to think, we are taught to memorize. Until one actually tries to think, you really can't tell if they're able to do it.
>A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it
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.
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.
One size doesn't fit all is what I believe Common Core math is attempting. The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all
It sounds like trivial insight, but at least in my experience many adults and even teachers have this "it's hard so it's ok to not want to do it" attitude towards math. And I think that is very detrimental.
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.
- 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.
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.”
Sadly, when I was a postdoc, an eminent mathematician I was working under once shared a story that he found amusing that one of his colleagues was once asked a question in the form: "This might be a stupid question, but..." and the response was "There are no stupid questions, only stupid people."
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.
I think we can recognize Pauli for his identification of one of the few magic gadgets we accept around spin statistics without accepting his educational philosophy: “Das ist nicht einmal falsch.”
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.
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'.
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.
Pure math is probably not worth the squeeze. I think more important to everyday life is systems thinking and a bit of probability/stats, mainly bayesian updates. "Superforecasting" was an eye-opening book to me, I could see how most people would benefit massively by it.
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.
I'll second guerrilla - you can absolutely benefit from mathematical thinking without pushing into territory higher than undergaduate studies.
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).
My claim isn’t really that there’s no benefit or utility to math — that’s obviously false — but that maybe its benefits to regular people are more modest than the cheerleaders want to admit.
Is it worth it to be able to think better, have a growth mindset and learn how to learn? Yes. Everyone can benefit from that. Pushing on into higher math? No, very few people can benefit from that.
Math doesn’t seem to me the only source of thinking clearly, or learning how to learn, etc. And if I’m searching for an aesthetic high, there are definitely better places to look — and ones that don’t require such a long runway.
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 also studied it and got several degrees, but I don't think that it actually benefited me. I think high school math is incredibly important to be able to think clearly in a quantitative way, and one university-level statistics course, but all the other university math... I dont think it helped me at all. I am disappointed by it because I feel that I was misled to believe that it would be useful and helpful.
Have you ever ascribed numbers to real life personal problems?
I find that managing to frame something bothersome into a converging limit somehow, really dissolves stress.. A few times at least.
That’s an interesting approach. I don’t think I’ve done that myself, but I can see how it could be helpful.
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.
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 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.
> One is that math nerds at school insisted on intimidating for the win and I just hated it.
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.
> 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.
I totally agree! The barriers many of us face with math are less about ability and more about how we've been taught to approach it. All it took was for me to change my math teacher at school, and boom. Love, but at second sight. And curiosity and persistence can unlock more than just numbers
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.
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 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.
Waste of time. Just talk in terms of what they want to hear. They are just interested in the payoffs (not in the details).
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.
> Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.
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.
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.
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.
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.
Too often people think of learning as accumulating knowledge and believe blockers are about not enough knowledge stored.
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.
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!
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.
> 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 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”.
Caveat here is that "talent" and "dedication" is linked to speed at least in the beginning. For instance, any student can learn calculus given enough time and advice even starting from scratch. However, the syllabus wants all this to happen in one semester.
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.
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>I strongly believe that the average human being can be exceptional in any niche topic given enough time, dedication and focus.
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.
So you're saying success at maths isn't an inbuilt ability. Instead, it depends on an (inbuilt) ability to hyper focus... Which you are just born with?
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>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.
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.
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I've had some success converting people by telling them others had convinced them they were stupid. They usually have one or two things they are actually good at, like a domain they flee to. I simply point out how everything else is exactly like [say] playing the guitar. Eventually you will be good enough to sing at the same time. Clearly you already are a genius. I cant even remember the most basic cords or lyrics because I've never bothered with it.
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.
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> 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.
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.
> Whilst some people (mostly spectrum) do seem have an innate talent
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.
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> 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 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.
I saw a lot of this when I went to college for engineering, some professors had this ability (or willingness) to make hard things simple, and others did the opposite, it was the same with the books, I dreaded the "exercise for the reader" shit, I don't think I ever did any of those, so those were all proofs I never got.
I think the ML people want to get (a narrow band) of stuff done and ivory towered people want to understand a prove things. ML is applied mathematic. Both are needed.
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> If it's easy, then it means you already know this material, and you're wasting your time.
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.
As a kid I was also terrible at maths, then later became obsessed with it as an adult because I didn't understand it, just like OP. It was the (second) best thing I've ever done! The world becomes a lot more interesting.
I haven't been able to grasp maths as a kid nor as an adult.
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.
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It's funny because I've had the opposite heuristic most of my line: the things I want to do most are whatever is hardest. This worked great for building my maths and physics skills and knowledge.
But when I started focusing on making money I've come to believe it's a bad heuristic for that purpose.
How have you been working on it? Asking for a friend ;)
When I was a young adult, i spent a lot of time on math and physics.
I was initially celebrated for the mathematical talent.
But as life progressed, I my family started seeing me as an academic loser.
Basically, no girls would be interested in me because "mathemetical talent" doesn't help you with that.
And i seen handsome men had more respect from society than spending countless time on math.
So, i later gave up because my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
Funny enough, I was still a loser in societal view doesn't matter I started clearly half a million a year.
So most people don't try hard at math because math is not rewarding, for most people.
It's much better to build physique, music talent, comedic talent, this helps you get girls and respect from peers.
Most people don't try hard at the gym. Most people don't try hard at music. Most people aren't comedians.
This reads like the foreword to the incel handbook.
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> ...my family kept pressuring me to attain real success, girls, money and car and i became a programmer.
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
"A loser in societal view"... What does that objectively mean? That only reads like you had or have a low sense of self worth. It must've been your perceived definition of what society is because how could you have come to such a conclusion? I think I'd actually subconsciously tend more to viewing someone as "a loser" if they made such a statement because it comes off as self victimization (without an apparent explanation to an outside observer).
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.
You will stay a loser as long as you care about what some fictional mystical society thinks of you.
Do the stuff you're good at, provide for your family, earn the respect of your peers and forget about the rest.
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This perspective has discouraged so many people from exploring their potential
Amazingly, I believe that today, with the myriad of tools available, anyone can advance in sciences like mathematics at their own pace by combining black-box and white-box approaches. Computers, in this context, could serve as your personal “Batcomputer” [1]. That said, I would always recommend engaging in social sciences with others, not working alone.
Who knows? You might also contribute meaningfully to these fields as you embrace your own unique path.
[1] https://dc.fandom.com/wiki/Batcomputer
easy_things -> comfort_zone
"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
This was a very tiring blog post for me. And I have a quip about posts that open with questions but close without obvious definite answer, no matter how simple it is.
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.
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?
That certain countries both now and in the past have had significantly higher mathematical ability among the general population and much higher proportions going on to further study suggests that ability isn’t innate but that people don’t choose it. In the Soviet Union more time was spent teaching mathematics and a whole culture developed around mathematics being fun.
Why would ability not be innate just because some people with the ability don't use it?
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.
> So is there some scientific study backing up the claim that all these people could easily learn it [emphasis added]
Who said it would be easy?
It is easy to learn for some.
We are not really taught (thought) to think, we are taught to memorize. Until one actually tries to think, you really can't tell if they're able to do it.
I do not think that Bessis's argument is entirely "made up"
>A nice sentiment but clearly a large % of people never do learn even basic mathematical thinking and seem very confused by it
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.
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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.
One size doesn't fit all is what I believe Common Core math is attempting. The part that it misses is that a student should probably be fine demonstrating one modality instead of having to demonstrate them all
Effort, combined with the right motivation, can overcome most perceived barriers
It sounds like trivial insight, but at least in my experience many adults and even teachers have this "it's hard so it's ok to not want to do it" attitude towards math. And I think that is very detrimental.
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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?
Try this remarkable book:
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.
I was the same In high school.
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!
Check out Susan Rigetti's guide: https://www.susanrigetti.com/math
List of good books, sorted by difficulty:
- 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.
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.”
Wouldn't it then follow that all students of the same teachers end up with the same skill level in math? Not sure that's the case.
Sadly, when I was a postdoc, an eminent mathematician I was working under once shared a story that he found amusing that one of his colleagues was once asked a question in the form: "This might be a stupid question, but..." and the response was "There are no stupid questions, only stupid people."
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.
Isn't that a positive statement, that you can ask questions without worry since they aren't stupid?
I think we can recognize Pauli for his identification of one of the few magic gadgets we accept around spin statistics without accepting his educational philosophy: “Das ist nicht einmal falsch.”
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.
How much of math aversion stems from a chain reaction of ineffective instruction
According to an excellent mentor: all of it minus epsilon.
> 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...
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.
Pure math is probably not worth the squeeze. I think more important to everyday life is systems thinking and a bit of probability/stats, mainly bayesian updates. "Superforecasting" was an eye-opening book to me, I could see how most people would benefit massively by it.
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.
I'll second guerrilla - you can absolutely benefit from mathematical thinking without pushing into territory higher than undergaduate studies.
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).
My claim isn’t really that there’s no benefit or utility to math — that’s obviously false — but that maybe its benefits to regular people are more modest than the cheerleaders want to admit.
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Is it worth it to be able to think better, have a growth mindset and learn how to learn? Yes. Everyone can benefit from that. Pushing on into higher math? No, very few people can benefit from that.
Math doesn’t seem to me the only source of thinking clearly, or learning how to learn, etc. And if I’m searching for an aesthetic high, there are definitely better places to look — and ones that don’t require such a long runway.
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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 also studied it and got several degrees, but I don't think that it actually benefited me. I think high school math is incredibly important to be able to think clearly in a quantitative way, and one university-level statistics course, but all the other university math... I dont think it helped me at all. I am disappointed by it because I feel that I was misled to believe that it would be useful and helpful.
Have you ever ascribed numbers to real life personal problems? I find that managing to frame something bothersome into a converging limit somehow, really dissolves stress.. A few times at least.
That’s an interesting approach. I don’t think I’ve done that myself, but I can see how it could be helpful.
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.
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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.
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.
> One is that math nerds at school insisted on intimidating for the win and I just hated it.
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.
> 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.
> IMO everyone has a ceiling when it comes to math.
Yes, but it's higher than you think: https://www.justinmath.com/your-mathematical-potential-has-a...
I totally agree! The barriers many of us face with math are less about ability and more about how we've been taught to approach it. All it took was for me to change my math teacher at school, and boom. Love, but at second sight. And curiosity and persistence can unlock more than just numbers
But what is the difference between math talent and plug-n-chug math talent? That seems to be the most significant filter.
Statistical (Bayesian) thinking is an extremely underrated way of thinking of almost everything.
Frankly, its overrated. Now you can adjust your priors.
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.
[0]https://news.ycombinator.com/item?id=41962944
I was actually thinking Jean Paul Satre when I read his answers
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 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.
Waste of time. Just talk in terms of what they want to hear. They are just interested in the payoffs (not in the details).
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.
> Their job is to try to get all these "vectors" aligned toward common goals, prevent fragmentation and division.
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.
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.
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.
- Hey teach, will I really need all these logarithms, derivatives and vectors in my adult life?
- No, but the smarter kids might.
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.
Too often people think of learning as accumulating knowledge and believe blockers are about not enough knowledge stored.
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.
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).
Careful there. They'll start voting logically..
Gentle Reminder that the author of this article used to have a wonderful math channel: https://www.youtube.com/c/pbsinfiniteseries
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