24 ms·

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

• benreesman 2y ago
  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.”
  • thefaux 2y ago
    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.
    • benreesman 2y ago
      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.
    • Jensson 2y ago
      Isn't that a positive statement, that you can ask questions without worry since they aren't stupid?
      • cutemonster 2y ago
        Maybe, I guess it's easily misinterpreted though O.O
    • ckw 2y ago
      That’s a south park quote: https://youtu.be/wWfacULP1o0?si=ddBWXFuQMoxxY-Yu https://youtu.be/wWfacULP1o0?si=ddBWXFuQMoxxY-Yu
      • benreesman 2y ago
        That was a belly laugh on a tough day. Thank you Sir or Madame.
    • yimmothathird 2y ago
      [dead]
  • cchi_co 2y ago
    How much of math aversion stems from a chain reaction of ineffective instruction
    • benreesman 2y ago
      According to an excellent mentor: all of it minus epsilon.
      • cchi_co 2y ago
        [dead]
  • mr_mitm 2y ago
    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.
    • benreesman 2y ago
      Cantor gave his life to the Continuum Hypothesis, Hilbert gave much of his life to similar goals. You’re making an argument somewhat along those lines, but given that I didn’t stipulate a convergence condition your conclusions can be dismissed by me. If it were a valid argument then we’d need Gödel.
      • mr_mitm 2y ago
        Did you mean to reply to another post? I don't follow at all.
    • diffeomorphism 2y ago
      Doesn't follow. Bell curve in, shifted bell curve out. Ideally this also tweaks the variance a bit. In other words: Some students flourish despite their teachers, some flourish because of them.
      • mr_mitm 2y ago
        And how would you call the students in the left tail of the Bell curve if not bad students?
        • diffeomorphism 2y ago
          Below average students and as long as the average is high enough they are still very competent. A bad teacher instead gives you a bimodal distribution and just doesn't bother teaching those students.
• jyscao 2y ago
  > 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.
  • felideon 2y ago
    > 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-limit-but-its-likely-higher-than-you-think/ https://www.justinmath.com/your-mathematical-potential-has-a...
• limit499karma 2y ago
  > 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/leibniz1705book1.pdf https://www.earlymoderntexts.com/assets/pdfs/leibniz1705book...
  • vundercind 2y ago
    I haven't read his Human Understanding, but his Second Treatise is really weak in ways that can't really be blamed on lack of mathematical training (unless we're going with "all rigorous thinking is math") so there may be more to it in his case than just "he didn't study math enough".
    • limit499karma 2y ago
      Leibniz wasn't saying that "rigorous thinking" is only available to mathematically trained or that Locke's reasoning was not "rigorous". His critique of Locke was that one can not have a correct model of human understanding (or world model) based on purely philosophical means, and that the lack of exposure to certain aspects of modern mathematics (that was emerging at that time) was the basis of Locke's misunderstandings.
• tkgally 2y ago
  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.
  • Frummy 2y ago
    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.
    • tkgally 2y ago
      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.
      • 2y ago
• agtech_andy 2y ago
  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.
  • cen4 2y ago
    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.
    • namaria 2y ago
      > 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.
      • katzenversteher 2y ago
        I believe charisma, confidence and looks also play a huge role.
• DiscourseFan 2y ago
  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 https://news.ycombinator.com/item?id=41962944
  • sonabinu 2y ago
    I was actually thinking Jean Paul Satre when I read his answers
  • ai4eva 2y ago
    there is a debate between the intuitionists, formalists, and the symbolists nicely captured in the intro chapter of Heyting's Intuitionism. constructive mathematics is close to computation and programming. and many including myself have a natural feel or intuition for it. A majority of euclids elements, and galois's original proof are constructive in nature.
• tracerbulletx 2y ago
  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?
  • Jtsummers 2y ago
    > 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?
    • Jensson 2y ago
      It is easy to learn for some.
  • logicchains 2y ago
    >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.
    • paxcoder 2y ago
      [dead]
  • physicsguy 2y ago
    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.
• agnishom 2y ago
  Gentle Reminder that the author of this article used to have a wonderful math channel: https://www.youtube.com/c/pbsinfiniteseries https://www.youtube.com/c/pbsinfiniteseries
  • lupire 2y ago
    ??? "Mathematician Tai-Danae Bradley and physicist Gabe Perez-Giz offer ambitious content ... Previous host Kelsey Houston-Edwards "
• gsabo 2y ago
  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.
  • junto 2y ago
    > 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”.
    • ponderings 2y ago
      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.
• LAC-Tech 2y ago
  [flagged]
• te_chris 2y ago
  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/ 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.
  • namaria 2y ago
    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.
• magicalhippo 2y ago
  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.
  • cchi_co 2y ago
    Effort, combined with the right motivation, can overcome most perceived barriers
    • magicalhippo 2y ago
      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.
      • cchi_co 2y ago
        [dead]
      • gammalost 2y ago
        Well, isn't that a summary of most things? Most things worth learning are hard, but many things not worth learning are also hard. So we have to prioritize what hard things are worth learning. Math is low on the list for many people for (I think) understandable reasons.
• cchi_co 2y ago
  [dead]
• block_dagger 2y ago
  Statistical (Bayesian) thinking is an extremely underrated way of thinking of almost everything.
  • DiscourseFan 2y ago
    Frankly, its overrated. Now you can adjust your priors.
    • ai4eva 2y ago
      lol yea. bayesian thinking doesnt come to me naturally.. i have no intuition for it. seems forced. believe me - i have tried. but there are those who are swearing by it.
    • vundercind 2y ago
      I'm pretty down on it just by association. So many people who are super-into it seem to be doing a lot of that rigorously-wrong engineer-brain thing.
• w10-1 2y ago
  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.
• quus 2y ago
  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.
  • guerrilla 2y ago
    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.
    • quus 2y ago
      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.
      • guerrilla 2y ago
        It doesn't need to be for me to be right. These are false constraints you're trying to put on it. Mathematics in moderation can benefit everyone. This claim stands.
  • defrost 2y ago
    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).
• dboreham 2y ago
  Careful there. They'll start voting logically..
• kristopolous 2y ago
  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.
  • katzenversteher 2y ago
    I have a lot of trouble reading math formulas, implemented as code I understand most stuff though. Is there a good math book or something similar that teaches things using code or helps translating formulas to code?
    • vundercind 2y ago
      I realized some time in middle age that I have to convert formulas and equations to steps and things happening to something "passing through" each step—to algorithms. It's painful and slow and also the only way I stand a chance in hell of reading mathematical writing. That's probably why math writing largely makes me feel dyslexic, while programming came naturally. And why I hate Haskell and find it painful to read even though I understand the "hard" concepts behind it just fine—it's the form of it I can't deal with, not the ideas.
  • atribecalledqst 2y ago
    I used to judge myself for not understanding everything in math articles on Wikipedia, but as time has gone on I've realized that their purpose isn't really to be an introduction, but a reference. Especially as the topics become more esoteric. So they're not really there for you to learn things from scratch, but for people who already understand them to look things up. Which is why you'll sometimes see random obscure & difficult factoids in articles about common mathematical concepts. (don't have any examples on-hand atm, this is just my general perception after years of occasionally looking things up there)
    • kristopolous 2y ago
      I've heard that and I think it's silly. They handwave away why nothing should ever be explained. Wikipedia doesn't work like that for any other topic. You'll see something like a mathematical proof with no explanation and it's end of article. The edit history will have explanations aggressively removed. The equivalent would be the article for say, splay tree, to have no diagrams and just a block of code - feeling no obligation to explain what it is or if you looked up a chemical and it would just give you some chemical equation, some properties and feel no obligation to tell you its use, whether it's hazardous or where you might find it... Or imagine a European aristocrat and all that is allowed is their heraldry and genealogy. Explanations of what the person did or why they're important are forbidden because, it's just a reference after all. Nope, these math people are a special kind of bird and I'm not one of them.
• openrisk 2y ago
  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.
  • agentultra 2y ago
    I don't know how widespread this phenomenon is but in the book Do Not Erase [0] it seems that there are quite a few prominent mathematicians who do use visual representations in their work. [0] https://en.wikipedia.org/wiki/Do_Not_Erase:_Mathematicians_and_Their_Chalkboards https://en.wikipedia.org/wiki/Do_Not_Erase:_Mathematicians_a...
    • openrisk 2y ago
      There is this long-running (and quite fascinating I think) debate about the different "types" of mathematical thinking. Logical vs Intuitive, Geometric vs Algebraic etc. Can't recall where exactly but I remember reading about a 19th century mathematician that crowed they had not a single figure in their masterpiece. Visualization is probably not a silver bullet but a lot of people are visual thinkers so maybe it would help a few more to reach a higher level in mathematical thinking.
  • vundercind 2y ago
    Lots of people seem to get permanently lost right around when operations on fractions are introduced. Other places, too, but that seems like the earliest one where a lot of people get lost and never really find their way back. Factoring was another that lost a lot of folks in my class. Lots of frustration around it seeming both totally pointless and the process involving lots of guessing, several classmates were like "well, fuck math forever I guess" at that point, like if they'd been asked to dig a ditch with a spoon and then fill it back in.
• fifteen1506 2y ago
  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).
• ggm 2y ago
  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.
  • jajko 2y ago
    > 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.
    • bmitc 2y ago
      > 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. I'm not so sure that adults always get it or rise about this. This happens in the workplace all the time.
  • dennis_moore 2y ago
    > One is that math nerds at school insisted on intimidating for the win and I just hated it. For me the worst part was the teachers that encouraged that behavior and did the same.
  • Tainnor 2y ago
    The different notations in calculus are unfortunately a historical accident and you're going to see all of them if you do anything with it (it's certainly not true that everyone in the field uses the same notation). I agree that it's frustrating but so are irregular verbs in Spanish/French/other languages - you can't really change it.
• KevinMS 2y ago
  But what is the difference between math talent and plug-n-chug math talent? That seems to be the most significant filter.
• LoganDark 2y ago
  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.
  • tartoran 2y ago
    I don't have dyscalculia but behind the numbers I have my own intuitive system(s) that I jump to sometimes when doing arithmetic. I think we all do since the early days arithmetic in school or what not, perhaps the dyscalculia folk missed making some connection at some point. I feel that arithmetic with numbers without that intuitive system is rote memory..
    • LoganDark 2y ago
      Indeed, I do also have an intuitive system for this; that's what I mean by "make up a quantity". When something makes up a quantity for me, then I can perform math on it. I have to wonder if my friend with dyscalculia doesn't have a "quantity" system like I do. They can't really perform multiplication or division but can technically perform basic addition and subtraction, though it takes them a while. I think it may be the rote memory you mention doing that for them.
• jonplackett 2y ago
  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?
  • sriram_malhar 2y ago
    Try this remarkable book: Who Is Fourier?: A Mathematical Adventure https://www.amazon.com/Who-Fourier-Mathematical-Transnational-College/dp/0964350408 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.
  • AntoniusBlock 2y ago
    Check out Susan Rigetti's guide: https://www.susanrigetti.com/math https://www.susanrigetti.com/math
  • webdev1234568 2y ago
    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!
  • rnewme 2y ago
    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.
• ChaitanyaSai 2y ago
  "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 https://blog.comini.in/p/what-happens-in-math-class
  • rnewme 2y ago
    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.
    • ChaitanyaSai 2y ago
      Tiring because the answer wasn't revealed? That was the whole point :) It's the path, not the summit.
    • GrantMoyer 2y ago
      Does the quip have to do with someone asking what the quip is?
• jojobas 2y ago
  - Hey teach, will I really need all these logarithms, derivatives and vectors in my adult life? - No, but the smarter kids might.
• practal 2y ago
  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 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/ 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 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 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-logic/2/ 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/ 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 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/ 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.
• sureglymop 2y ago
  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!
  • yodsanklai 2y ago
    totally agree! in high school, lots of things were vaguely defined. I remember, I didn't fully understand what "f o g" was until I was given the definition of a monoid. Also limits and derivations, once you get the proper definition, you can pretty easily derive all the formulas and theorems you use in high school. Also in high school, we mostly did calculations and simple deductions, but at university we were proving everything. Nice change of perspective.
  • vanderZwan 2y ago
    Yes, I agree! And also that a lot of the fun stuff is hidden behind historically opaque terminology. Although I'm also sympathetic to the fact that writing accessible explanations is a separate and hard to master skill. Once you understand something it can be really hard to step back into the mindset of not understanding it and figuring out an explanation that would make the idea "click". I think a lot of maths is secretly a lot easier than it appears, but just missing an explanation that makes it easy to get the core idea to build upon. For example, I've been meaning to write an explorable[0] for explaining positional notation in any integer base (so binary, hexadecimal, etc) in a way that any child who can read clocks should be able to follow. Possibly teaching multiplication along the way. Conceptually it's quite simple: imagine a counter that looks like an analog clock, but with the digits 0 to 9 and a +1 and -1 button. We can use it to count between zero and nine, but if we add one to nine, we step back to zero. Oh no! Ok, but we can solve this by adding a second counter. Whenever the first counter does a full circle, we increase it by one. A full circle on the first counter is ten steps, so each step on the second counter represents ten steps. But what if the second counter wants to count ten steps? No problem, just add a third! And so on. So then the natural question is... what if we have fewer digits than 0 to 9? Like 0 to 7? Oh, we get octal numbers. 0 and 1 is binary. Adding more digits using letters from the alphabet? The core approach is just a very physical representation of base-10 positional, which hopefully it makes it easy to do the counting and follow what is happening. No "advanced" concepts like "base" or "exponentiation" needed, but those are abstractions that are easy to put on top when they get older. I've asked around with friends who have kids - most of them learn to read clocks somewhere between four and six, and by the time they're eight they can all count to 100. So I would expect that in theory this approach would make the idea of binary and hexadecimal numbers understandable at that age already. EDIT: funny enough the article also mentions that precisely thanks to positional notation, almost every adult can immediately answer the question "what is one billion minus one". [0] https://explorabl.es/ https://explorabl.es/
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• atribecalledqst 2y ago
  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.
• trialAccount 2y ago
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    I just published the comment above by mistake and couldn't find a way to delete it.
• revskill 2y ago
  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.
• penguin_booze 2y ago
  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
  • bmitc 2y ago
    What does premature formalization mean and when does it occur? Do you mean formal in the sense of using formulaic, rote manipulations or formal in the sense of proofs and rigor? As someone who went on to study mathematics at the graduate level, I was bored out of my mind in high school math and most subjects. What's missing from a lot of primary and secondary school education is context, and that's what makes it boring. Math wasn't easy because I was particularly good at it. It was easy because it was just blindly following formulas and basic logic. Something is very wrong with our educational system because almost all math at the primary and secondary levels is basic logic. So when people with this maximum level of mathematics education say they're "bad at math" or "don't get math", it means that they lack extremely basic logic and reasoning skills. In my mind, we need to teach mathematics in a contextual way (note that I don't necessarily mean applications) in a way to enrich the reasoning and exploring of it. This should include applications, yes, but not be fully concentrated on applications. Sometimes one needs to just learn and think without being tied to some arbitrary standard of it being applied.
  • llm_trw 2y ago
    Mathematics is the conversion of a large number of object languages in to a single meta language that lets us talk about all of them. The sin of modern mathematics is that it's meta language is so ill define that you need towers of software to manipulate it without contradiction. Rewriting all of it into s-expressions with a term rewriting system for proofs under a sequent calculus is an excellent first step to making it accessible. We do not need to go back to the 16th century when men were men, an numbers positive. If people want to look at what math talks about instead of how it talks about let them pick up stamp collecting.
• fHr 2y ago
  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.
  • Malidir 2y ago
    Ironically, the ones who don't do well at school (inc maths) are the ones who then become trades like builders/plumbers etc or run small businesses like a shop. And so are regularly working with numbers via estimates and billing.
• lugu 2y ago
  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.
• Relic0935 2y ago
  That's a very important thougt and I belive the world would be better, if more people would connect with their mathematical side.
• musgravepeter 2y ago
  https://www.smbc-comics.com/comic/grind https://www.smbc-comics.com/comic/grind
• agentultra 2y ago
  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.
• trialAccount 2y ago
  The author: Would love to participate but account creation seems to be broken on Hacker News https://x.com/davidbessis/status/1859561768915173466 https://x.com/davidbessis/status/1859561768915173466
  • macintux 2y ago
    If someone still on X could send them the hn@ycombinator.com support address, that'd be useful.
• mbbbackus 2y ago
  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.
  • sourcepluck 2y ago
    > rotated shapes, slot machines, or origami Or gears (like Seymour Papert), or abacus beads, or nomograms, or slide rules, etc etc. Anyone have any more, throw them out! Is mathacademy good? I have been thinking of giving it a month of a try. You say "stressful", which I'm not sure is a mis-type or not. I ordered Mathematica at my local library by the way, and can now forget about it until I get an SMS one day informing me of its arrival. Thank you for confirming that it's worth it!
    • Shosty123 2y ago
      I've had a MathAcademy subscription for some time and it's quite good. I'd say it's best at generating problems and using spaced repetition to reinforce learning, but I think it falls short in explaining why something is useful or applicable. I don't know, most math education seems to be "here's an equation and this is how you solve it" and MathAcademy is undoubtedly the best at that, but I wish there were resources that were more like "here's how we discovered this, what we used to do before, why it's useful, and here's some scenarios where you'd use it."
      • Nevermark 2y ago
        I have so wanted such resources for years. I have found some and should make a list. The first time the difference between understanding some math, and understanding what the math meant, was after high school Trig. The moment I started manually programming graphics from scratch, the circle as a series of dots, trigonometry transformed in my mind. I can't even say what the difference was - the math was exactly the same - but some larger area of my brain suddenly connected with all the concepts I had already learned. While ordering the "Mathematica: A Secret World of Intuition and Curiosity" I came across these books, which looked very promising in the "learning formal math by expanding intuition" theme, so I bought them too: Field Theory For The Non-Physicist, by Ville Hirvonen [0] Lagrangian Mechanics For The Non-Physicist, by Ville Hirvonen [1] The Gravity of Math: How Geometry Rules the Universe, by Steve Nadis, Shing-Tung Yau [2] Vector: A Surprising Story of Space, Time, and Mathematical Transformation, by Robyn Arianrhod [3] [0] https://www.amazon.com/dp/B0CN7HMTJN https://www.amazon.com/dp/B0CN7HMTJN [1] https://www.amazon.com/dp/B0CN7HMK38 https://www.amazon.com/dp/B0CN7HMK38 [2] https://www.amazon.com/dp/1541604296 https://www.amazon.com/dp/1541604296 [3] https://www.amazon.com/dp/0226821102 https://www.amazon.com/dp/0226821102 Excited to read each (based on their synopses & ratings), and if I will get compounding fluency across both math and physics between all five books.
• delta_p_delta_x 2y ago
  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.
• itissid 2y ago
  "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.
  • nradov 2y ago
    Come on, get real. If people had all of their needs covered a lot more of them would sit around getting high and playing video games than perfecting their art.
    • itissid 2y ago
      And how far would that get one doing that? 1, 5, 10, 40 years?
    • ahoka 2y ago
      Perhaps some, but why is that a problem? There are already people who do this.
    • intelVISA 2y ago
      Is that not better than the current alternative where they're forced to become grifters?
      • nradov 2y ago
        No one is forced to become a grifter.
    • manvillej 2y ago
      I would perfect my video gaming art. Teenagers in their basement would fear me after school.
    • deleted 2y ago
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• XajniN 2y ago
  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.
  • cutemonster 2y ago
    Not sure about schools for gifted, but I think lots of people here a bit live in bubbles. Maybe industrialized countries, a bit better off neighborhoods, STEM classes in high school, and then university? (Not everyone but ... Most here?)
  • denkmoon 2y ago
    God that's one of the most pessimistic views of humanity I've seen on the web, and I'm pretty pessimistic. Developed nations have a 99.2% literacy rate per wikipedia. 69% of OECD students meet "level 2" mathematical proficiency (https://www.oecd.org/en/topics/sub-issues/mathematics-literacy.html https://www.oecd.org/en/topics/sub-issues/mathematics-litera...), which indicates an understanding above the basic mathematical operators like multiplication and division. FWIW I did not go to any fancy school, I was educated in the state system.
• hilbert42 2y ago
  "High school students are often unhappy with math, because they think it requires some innate things that they don’t have,” Bessis said. “But that’s not true; really it relies on the same type of intuition we use every day." Agreed, but from my observation mathematics is often taught with a rigor that's more suited to students with a highly mathematical and or scientific aptitude (and with the assumption that students will progress to university-level mathematics), thus this approach often alienates those who've a more practical outlook towards the subject. Mathematics syllabuses are set by those with high mathematical knowledge and it seems they often lose sight of the fact that they are trying to teach students who may not have the aptitude or skills in the subject to the degree that they have. From, say, mid highschool onwards students are confronted with a plethora of mathematical expressions that seemingly have no connection their daily lives or their existence per se. For example students are expected to remember the many dozens of trigonometrical identities that litter textbooks (or they did when I was at school), and for some that's difficult and or very tedious. I know, I recall forgetting a few important identities at crucial moments such as in the middle of an exam. Perhaps a better approach—at least for those who are seemingly disinterested in (or with a phobia about) mathematics—would be to spend more time on both the historical and practical side of mathematics. Providing students with instances of why earlier mathematicians (earlier because the examples are simpler) struggled with mathematical problems and why many mathematical ideas and concepts not only preceded but were later found to be essential for engineering, physics and the sciences generally to advance would, I reckon, go a long way towards easing the furtive more gently into world of mathematics and of mathematical thinking. Dozens of names come to mind, Euclid, Descartes, Fermat, Lagrange, Galois, Hamilton and so on. And I'd wager that telling students the story of how the young head-strong Évariste Galois met his unfortunate end—unfortunate for both him and mathematics—would never be forgotten by students even if they weren't familiar with his mathematics—which of course they wouldn't be. That said, the moment Galois' name was mentioned in university maths they'd sit up and take instant note. Yes, I know, teachers will be snorting that there just isn't time in the syllabus for all that stuff, my counterargument is that it makes no sense if you alienate students and turn them off mathematics altogether. Clearly, a balance has to be struck, tailoring the subject matter to suit students would seem the way with the more mathematically inclined being taught deeper theoretical/more advanced material. I always liked mathematics especially calculus as it immediately made sense to me and I always understood why it was important for a comprehensive understanding of the sciences. Nevertheless, I can't claim that I was a 'natural' mathematician in the more usual context of those words. I struggled with some concepts and some I didn't find interesting such as parts of linear algebra. Had some teacher taken the time to explain its crucial importance in say physics with some examples then I'm certain my interest would have been piqued and that I'd have showed more interest in learning the subject.
• generationP 2y ago
  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 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.
  • lupire 2y ago
    This topic is probably the worst possible topic for Quanta. The book, as I understand it, is about the life changing power of mathematical thinking. Quanta's mission is to make deep mathematics and mathematicians a "sexy" "human interest" topic, by making it as non-mathematical as possible while keeping a superficial veneer of mathematics.
• davidbessis 2y ago
  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.
• KameltoeLLM 2y ago
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  • zb3 2y ago
    No
• what9001 2y ago
  What an incredibly neuro-typical thing to claim
• johnp314 2y ago
  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.
  • tombert 2y ago
    I guess I assumed it was a reference to Alfred Whitehead's and Bertrand Russell's book Principia Mathematica, which predates Wolfram himself by several decades.
• exprofmaddy 2y ago
  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_Really.html?id=R-qgdx2A5b0C 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 https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From
• sethpurcell 2y ago
  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.)
• aleden 2y ago
  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
  • BigGreenJorts 2y ago
    Eh, I mean it's not on the SATs, but why isn't it? In Canada, we had a similar calculus based curriculum up to the first year of university. A little bit of linear algebra thrown into the mix. Why is that? Well you need calculus to do any form of engineering, physics, certain domains of chem/bio, stats, certain domains of economics, etc etc. Math in society is first and foremost a tool. I say this as a person who majored in Pure Math and focused on algebra and number theory. For the vast majority of students, it truly is about the practicality. Math just has the layer of abstraction that makes it hard to enjoy without deliberate framing unlike the sciences or humanities.
    • aleden 2y ago
      Many people dismiss mathematics because they aren't interested in it. I definitely wasn't interested in doing dozens and dozens of error-prone problem sets that mainly boiled down to performing arithmetic. I don't need to do that. My number theory professor was a brilliant mind, someone who had spent lots of time at the Institute for Advanced Study, and he absolutely sucked at adding/subtracting/multiplying numbers. And that was something he freely admitted. It wasn't important to his work, and it isn't important to mine.
  • sn9 2y ago
    Calculus isn't on the standard SAT or even the Math Subject II tests [0], at least not in the early 2000s. They taught us calculus because it's a prereq for engineering and physics and that's been important since the space race. [0] https://en.wikipedia.org/wiki/SAT_Subject_Tests https://en.wikipedia.org/wiki/SAT_Subject_Tests
• elmerfuddwalz 2y ago
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• javier_e06 2y ago
  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)
• plasticeagle 2y ago
  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.
• mncharity 2y ago
  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?
• carlysagan 2y ago
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• metalman 2y ago
  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.
• etherealG 2y ago
  Shouldn’t that headline read “or can benefit from”. Not a shining example of good mathematical thinking. Oh the irony.