29 ms·

Why do many math books have so much detail and so little enlightenment? (2010)

• alistproducer2 9y ago
  On this topic I recommend the "learn X in 20 minutes a day" series of math books for folks trying to get started in subjects they've had difficulty with in the past. I always struggled with Calculus until that book. The math is taught almost entirely by visuals and intuition hints. I came away understanding Calc, not just how to perform the steps. It helped me divine the purpose of calculus, not just the process. I've used them for Geometry and stats to similar effect. Highly recommend for anyone who share the sentiments of the OP.
  • sarang23592 9y ago
    Could you share any links where I can find these books. I tried google but couldn't find them.
    • theoh 9y ago
      Must be this publisher, Learning Express: https://www.goodreads.com/author/list/24061.LearningExpress https://www.goodreads.com/author/list/24061.LearningExpress
      • alistproducer2 9y ago
        Yes, this is the publisher. Thanks. I got the titles slightly wrong on my oc.
• wslh 9y ago
  In the top answer one of the last paragraph sounds like the real answer: > When I was a graduate student, we had a wonderful working seminar on Sunday mornings with bagels and cream cheese, where I learned a lot about differential geometry and Lie groups with my classmates. At the end, the students needed context and enlightment. We cannot underestimate other dimensions of the learning experience.
• BucketSort 9y ago
  Because they are written by mathematicians. In my case, when I have learned a mathematical topic, the intuition becomes obvious and the derivations/proofs seem to be much more important for gaining a complete understanding. I have gone up against texts with complete bewilderment, only to come back after gaining the intuition and found the extensions of the core premises and proofs provided by the text to be highly enlightening. Great math teachers understand the need to teach intuition. He wasn't a math teacher, but I think Richard Feynman is the pinnacle of this. See [1] to see how he expresses intuition about physics, and his Red Books[2] for how he teaches mathematical physics with all the qualities I believe makes a great maths text for students. Also, there's a linear algebra MOOC which also teaches great intuition before delving into proofs and heavy detail [3]. I mention these examples because they are exemplars of this idea of teaching intuition. [1] https://www.youtube.com/watch?v=4zZbX_9ru9U https://www.youtube.com/watch?v=4zZbX_9ru9U [2] https://www.amazon.com/Feynman-Lectures-Physics-Vol-Mechanics/dp/0201021161/ref=sr_1_2?ie=UTF8&qid=1494813362&sr=8-2&keywords=lectures+on+physics https://www.amazon.com/Feynman-Lectures-Physics-Vol-Mechanic... [3]https://www.edx.org/course/linear-algebra-foundations-frontiers-utaustinx-ut-5-05x https://www.edx.org/course/linear-algebra-foundations-fronti...
  • mathgenius 9y ago
    > Richard Feynman is the pinnacle of this There really needs to be a version of the Feynman lectures for mathematics. Although, this is what Arnol'd has to say [1]: "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap... In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences." [1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
    • cottonseed 9y ago
      > There really needs to be a version of the Feynman lectures for mathematics. I asked the question on math.SE: https://math.stackexchange.com/questions/62190/mathematical-equivalent-of-feynmans-lectures-on-physics https://math.stackexchange.com/questions/62190/mathematical-...
• ktRolster 9y ago
  He's talking about a fairly advanced math topic, abstract algebra. At some point, you graduate from "being taught" to "teaching yourself." By the time you get PhD, you need to teach yourself, because you're studying things no one has ever studied before.
  • mathgenius 9y ago
    But this whole mathoverflow post is professional mathematicians complaining about how difficult it is to "teach yourself". It's kind of sobering to realize that even the pro's have this problem.
    • skybrian 9y ago
      I'm not a mathematician, but it's my understanding from what I've read that part of it is becoming used to groping around in the dark, sometimes getting stuck, and accepting that as normal. Solving the really tough puzzles means getting stuck a lot. This might explain the relative lack of attention towards the user experience of non-mathematicians?
      • jordigh 9y ago
        I find it amusing that you call it "user experience", a very computery/hackery term for mathematics. There are so many computer-like expectations for mathematics that I see on HN. :-) Anyway, yes. Every mathematician I know acknowledges that frustration is the natural state of affairs. If you're not frustrated that's because you haven't been doing enough mathematics yet. There's always a bigger problem, a new concept to master, a new way to look at an old idea.
        • skybrian 9y ago
          Yes, to extend the metaphor, writers often pride themselves in offering a good "user experience." Bad writing does not flow smoothly and is unnecessarily difficult to read. Good games try to provide an optimal learning experience, providing just enough challenge to be interesting without players getting stuck. Play-testing is vital; if your players commonly get stuck in ways you didn't intend, it's a bug. There's a lot to be said for designing a learning experience to flow smoothly. We can admire the work that goes into making that happen. (It then seems strange that, by contrast, writers of math books often don't seem to be playing the same game.) One thing a well-designed experience doesn't do, though, is prepare you for being stuck and overcoming difficulty when you're not on an artificially smoothed path.
• hal9000xp 9y ago
  In the same way, I find most linear algebra books annoying. Throwing on you dull definition of matrix multiplication is completely useless unless you know where this definition comes from.
  • Chris2048 9y ago
    Yes! even a visual depiction of matrix multiplication, while helpful in some ways, isn't useful unless you relate it to an application.
• daxfohl 9y ago
  Why do holy books contain so much detail and so little enlightenment? Perhaps the details are what matters. Enlightenment is up to the beholder.
• thomastjeffery 9y ago
  The same can be said about programming. Reference and theory are very useful, but context is how they are tied together in implementation.
  • Mathnerd314 9y ago
    By the Curry–Howard correspondence, programming is in fact (formal) theorem proving. Mathematics got a lot less interesting after I realized it amounted to a giant, informally-specified, mostly undocumented body of code designed to run on the human brain... from that perspective it's hard to see why one should prefer mathematics to a well-written software program that does the same thing.
    • mathgenius 9y ago
      > code designed to run on the human brain So it's legacy code that runs on the oldest hardware that we have.. Plenty of people are interested in updating this code so that it runs on more modern platforms. Actually, this is my favorite way to drill into a piece of mathematics: port it to python.
    • skybrian 9y ago
      Is Curry-Howard interesting if you're not programming with advanced type systems? It's my understanding that writing a function that returns an integer (for example) corresponds to proving that an integer exists. Whoop-de-do? This would imply that unless you're constructing a sophisticated return type, you're not proving anything interesting.
    • deleted 9y ago
      [deleted]
    • mrout 9y ago
      Yeah sorry, you don't at all understand mathematics.
    • bjl 9y ago
      That's not quite true. Curry-Howard only applies to constructivist mathematics, which is a very tiny subset of mathematics proper.
• jessriedel 9y ago
  The top-rated answer is either defeatist, or just rationalization for the sentiment "I had to go through this and figure out everything myself, so you should too". There is a huge amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed. For a famous philosophical treatise on this using the Euler characteristic as an example, see "Proofs and Refutations” by Imre Lakatos https://en.wikipedia.org/wiki/Proofs_and_Refutations https://en.wikipedia.org/wiki/Proofs_and_Refutations Part 1: https://math.berkeley.edu/~kpmann/Lakatos.pdf https://math.berkeley.edu/~kpmann/Lakatos.pdf Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject. It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. Unfortunately, the problem has been known for decades and there's not much reason to expect things to change. (Lakatos wrote the above in 1976.)
  • derefr 9y ago
    > This reflects a long process of trial-and-error as the field was constructed. A way to think about this process: "math" isn't quite a tower of abstract concepts; those only exist in any given mathematician's head, and die with them. Instead, "math" is a name for the novel pieces of language we invent, with the purpose of using them to talk about and (hopefully) share those internal abstract concepts, exposing to the rest of the world concepts that were entirely inaccessible (to everyone but the originator) before that language was introduced. Math is the attempt to communicate never-before-communicated epiphanies; to describe the shapes of never-before-pondered abstractions, with properties only describable using never-before-spoken words. One thing that this implies, is that the first piece of language that lets us even vaguely point at some particular idea so that we can get to work on analyzing it further, might stick around long after we come up with some clearer or more coherent language, because the former one now has the momentum of historical use behind it. Math, when done this way, then becomes a precarious pile of "just good enough to survive" conceptualizations, rather than a precise tower of "best-tool-for-the-job" conceptualizations. And that's no good for teaching.
    • 9y ago
• cft 9y ago
  "closed as no longer relevant ", I find that is a growing problem with StackOverflow. My colleagues and I personally asked several relevant questions (like this one, that has many upvotes on HN), that were shut down or deleted by StackOverflow heavy-handed mods as "off topic" , "not relevant" etc. This made me give up on contributing and treat it as a read-only resource.
  • sanderjd 9y ago
    I'm very curious about this: is there a reason the reductionists seem to win in communities like this? Wikipedia has a similar culture. Are there systemic forces that point this direction, or is it just an accident of the history of the specific people involved?
    • cft 9y ago
      There may be so much spam that they practice a shotgun banning approach without comprehension or real content evaluation. Somehow the best "spam banners" get elevated perhaps?
      • chadcmulligan 9y ago
        in all these things the pedants win, because every one else says whatever...and goes outside in the sun
        • reitanqild 9y ago
          One thing HN does well is having reasonable mods.
        • mrout 9y ago
          There should be a niwdoG's law: the longer an argument goes on, on the internet, the greater the likelihood that someone will accuse someone else of being a nerd that spends all their time indoors, while arguing on the internet themselves.
    • umanwizard 9y ago
• elchief 9y ago
  There are no hard concepts, only bad explanations
  • mrout 9y ago
    That's inane and false.
• sfRattan 9y ago
  I've historically had the opposite problem. I find attempts in both science and mathematics textbooks to provide some sort of real world context to be distracting and wasteful. I remember that impression going all the way back to high school, but the most recent memory comes from early in college. My discrete math book had some chapter that started with a page and a half about volcanoes. Volcanoes. Why are you wasting my precious time with this nonsense before getting to the meat? There are other things I need to read and study. There are problems I need to use these ideas to solve... and not only the ones in the textbook. But then, I can also recognize that such winding introductions to a subject or an idea might be helpful to people who "get bored" as the author describes. Though I suspect the remedy might be to acquire and deliberately practice study skills. It would be nice if publishers sold versions of their textbooks both with and without what the individual posing the original question would identify as "motivation." I tend to think of it as fatty prose.
  • santaclaus 9y ago
    Half-assed physical analogs can be a drag, especially when the analogy breaks down after any bit of scrutiny. Geometric intuition, on the other hand, I often find worth the weight of a thousand words. There have been times I haven't been able to make my way through a dense piece of mathematical text, only to pop open another book on the same topic to find a single picture that makes all of the pieces click.
    • mrout 9y ago
      You have to understand that there are many people that do not feel this way. Geometric 'intuition' does nothing for me. That's why there are two books. :)
  • eutectic 9y ago
    There is a big difference between 'practical motivation' and 'theoretical motivation'.
  • deleted 9y ago
    [deleted]
• ivan_ah 9y ago
  The conversation is about very advanced math topics (advanced undergraduate or graduate level). At some point in the levels of abstraction it becomes hard to show concrete applications of the ideas, or rather the applications of math is to do ever more advanced math. I find more deplorable the fact that even basic math topics are often covered in the same dry way, without discussing practical applications or introducing topics through real-world scenarios. Many math books take the attitude "You have to know this, because I say so." Whenever I write about math, I try to start with a concrete example or a useful application of the theoretical result—it's always possible to come up with something for most of first year stuff. Seriously, you'd be surprised how much better reading UX is if you start each chapter/section with a motivating example.
  • jordigh 9y ago
    I find that almost all books I read do discuss "applications". No mathematics is an island. But as you say, the problem is that a lot of those applications don't really count, because the domain of the application is as foreign as the original concept. For example, knowing that the snake lemma, a purely categorical statement, is most useful in homological algebra (such as, for example, simplicial or singular homology), is utterly useless if you don't already have an interest in algebraic topology. There really is almost no other motivation for the snake lemma, so now we're faced with the problem of trying to convince you that it's interesting by trying to convince you that algebraic topology is interesting. It can be done, and maybe we'll eventually bottom out in something like financial statements or bridge-building, or another topic that is widely recognised as "useful" and very far-removed from the snake lemma. Either way, it will be a long and arduous path, and I hardly think mathematicians can be blamed for this or be dismissed as elitists for the inherent difficulties of the subject. But even for "first year stuff", the applications are kind of pointless. Do you really want to learn calculus because of physics? That's the most obvious and most historical application, but calculus is so foundational that you might as well motivate addition of real numbers by saying that numbers are added in physics too. More likely for the HN crowd, you want to learn calculus because you want to know how a neural network's backprop algorithm works, but how is the first year teacher going to anticipate that this is your particular interest in calculus? At some point, I think there has to be a little "trust me, this is useful" and you just struggle through the subject until you can see on your own, after the fact, what the struggle was about. First wax on, wax off, Daniel-san. Then you will learn how you really were learning how to block karate blows.
• j45 9y ago
  I had a math teacher that explained that beyond simple and clear explanations, the other way to learn math was to do example questions. Lots of examples, to see all the patterns and subleties. The subject he'd most often quote was permutations and combinations - either you got it pretty much instantly or it would just be a struggle that you could only improve at so much by doing a lot of practice questions.
• oldmancoyote 9y ago
  The more closely you study something, the more defuse it becomes until when you totally understand it, it has completely disappeared from view. If you can no longer "see" a thing, you can not usefully explain it.
  • Qwertious 9y ago
    Relevant: http://lesswrong.com/lw/kg/expecting_short_inferential_distances/ http://lesswrong.com/lw/kg/expecting_short_inferential_dista...
• tnecniv 9y ago
  The best math class I ever took was probability, where the professor worked history of the field and famous historical examples into his lectures. It helped that probability has a colorful past, but the right motivation really brings things to life. Also, for self-teaching, I've found some math books that are setup to be in the form "Topic A with Applications to Topic B." If I care about B, such a book will typically does a good job motivating A, even if it isn't the purest introduction to the area. I can always read a book that is a more canonical intro to A later.
• EGreg 9y ago
  When I was teaching calculus, I made a fundamental decision of how to define limits: I used limits of sequences. In our university, we encounter limits in either a handwavy way in Calc I or an epsilon-delta way in Advanced Calculus. Sequences and series are introduced in Calculus II. Continuity is then described in the same terms. Or perhaps in terms of open sets if there is some topological topic. I think this is pretty terrible for intuition. Not just that, but it's not even general enough - it requires normed spaces, so then you have to generalize again. Much simpler to talk about ancient Greeks and Zeno's paradox. And then rigorously define limits of sequences, and define limits of real valued functions in terms of "for any sequence x_n that approaches x, y_n = f(x_n) approaches y". Simple and right away lets me show counterexamples beyond "left and right limits", like cos(1/x), and show students how to produce two sequences that converge to different numbers. There are similar ah-ha moments when discussing fundamental concepts of linear algebra, number theory, complex numbers etc. The two best books I have ever found on teaching Complex Numbers are: 1) Bak and Newman 2) Schaum's Outlines They actually give you the understanding and ceeling behind WHY analytic functions are the way they are, and derive holomorphic functions from that. Imho a more terrible approach is that of Serge Lang and proving everything the other way, with Taylor power series. Bottom line - make a directed graph of how you will teach your subject and then figure out the best entry point an direction for the greatest cohesion, as you would when telling a story. If you are curious, now I teach a course on "Thinking Matematically" and here are this semester's results of that approach: Numbers and Algebra: https://qbix.com/docs/mathematically1.pdf https://qbix.com/docs/mathematically1.pdf (https://vimeo.com/215335666 https://vimeo.com/215335666) Sets and Infinity: https://qbix.com/docs/mathematically2.pdf https://qbix.com/docs/mathematically2.pdf (https://vimeo.com/210500111 https://vimeo.com/210500111) Boolean Algebra: https://qbix.com/docs/mathematically3.pdf https://qbix.com/docs/mathematically3.pdf Logic and Probability (Coming up) The above videos were made with help from video exditors on upwork.com
• cf 9y ago
  I feel when it comes to motivation in math, I just want to know why people got so excited about a particular theorem in the first place. I'm ok with the answer "before this theorem we assumed all these different things we proved in similar ways were different. Now we know about their commonality and it allows us to borrow mathematical machinery and use it here". For example, take something like measure theory. A reasonable motivation for measure theory to me is remembering your introductory probability class how you had to learn a probability mass function for discrete spaces and a probability density function for continuous spaces. Now these two ideas are obviously nearly the same idea. I mean the notation used is a pretty big hint. But you need measure theory to have the right concepts to describe how they are the same. Where are all those kinds of motivation for things like topology and cohomology?
  • drostie 9y ago
    I asked the ##math chat room on Freenode about cohomology recently, more precisely I expressed that it was for me the scariest math-word-that-I-don't-know-yet, and I received the explanation "it isn't that scary once you get used to it; it's just a way to repair exactness of sequences." Maybe that helps? For topology, I feel like there's a sort of meeting-of-two-different-things; one starts with being very frustrated with the delta-epsilon-definition of "limit" and its one-dimensional nature, and the limit-definition of "continuous" and its clumsiness. The other starts from wanting to play with spheres and Möbius strips and knots and the like. When you're playing with these shapes a bijective mapping between two surfaces is not a fine-grained-enough idea because it is not continuous; adding continuity gives "homeomorphisms" which also aren't a fine-grained-enough idea because they do not make reference to the space an object is embedded in; wrap a torus about itself in a pretzel knot and you have something which is homeomorphic to a torus but in 3D you can't get there without tearing part of the surface through the other one, but in 4D you can. So finally we come to the idea of an isotopy, which bumps "continuous" to the next level by saying "Just like you can have a continuous path of points in space, you can have a continuous path of homeomorphisms from one to another," and that's where the pretzel knot becomes finally distinct from the torus in 3D, there is no continuous path from the homeomorphism of the pretzel knot to the torus, to the identity homeomorphism of the torus to itself. Or something like that. So this path is then an "isotopy" and then certain things are nicely isotopy-invariant and so forth.
• hyperpallium 9y ago
  > And of course it is historically backwards; groups arose as people tried to solve problems they were independently interested in. Reminds me of the joke that philosophy students university learn the work of people who didn't go to university. Would a text written by the discoverer of a field be more interesting? I've found important papers easier to read than textbooks covering them... OTOH the original paper can be too close to their own motivation, and not treat - or yet be aware - of its general significance. A motivating problem doesn't have to be practical or relatable... it just must lack obvious solution. A puzzle.
  • mrout 9y ago
    Good example of something (although it's perhaps a bit trivial) that is a puzzle that seems to make even non-mathematicians curious is the 'justification' for greco-latin squares. The idea is that you have 6 files of 6 men. Each is of a different regiment, each is of a different rank. Is it possible to arrange them so that no row and no file have two men of the same rank or from the same regiment? You can mention this in your first lecture on algebraic geometry (or, for that matter, combinatorial geometry i.e. matroid theory) and then come back to it when you talk about projective geometries.
• khazhoux 9y ago
  If a dry presentation of groups doesn't excite you, then there's plenty of books that are full of examples and motivations. Personally, I prefer the dry stuff (e.g., Herstein) because that's the abstract-est abstraction. Rotations and matrices are groups? Don't care. I just wanna see the strange hidden properties of abstract structures reveal themselves.
  • gunnihinn 9y ago
    > Rotations and matrices are groups? Don't care. I just wanna see the strange hidden properties of abstract structures reveal themselves. ... for example by noting that rotations and matrices form a group?
• dzdt 9y ago
  I think this, as much of higher mathematics, will change soon. Today's technology in natural language processing is reaching the point where it will be possible to marry a natural language processing system to an automated theorem prover and have it generate formally verified proofs from math prose proofs. Once this technology can readily process the textbooks making up the PhD curriculum, I think there will be a culture shift. Quickly there will be a new standard that math results should be formally verified. The hallmark of math, after all, is that it can be proven correct! But with an increased role for computers will come an increased appreciation for the things that only humans can provide. Motivation and explication will be more valued when the technical aspects of theorem-proving are automated.
  • xyzzyz 9y ago
    If what you are saying is indeed going to happen, then the "problem" will become even worse. The formally verified proofs tend to be much more unintelligible than the human generated ones, and when they stop being so, the humans will become deprecated in general. Practically speaking though, working mathematicians do not care that much about formally verified proofs. Working mathematicians are more interested in insight and understanding, and not necessarily in being completely sure of every detail. Formally and automatically verified proofs are much better suited for programming, as the automatic verification of the correctness of the program is after all _the_ best regression test. So, while interesting in principle, I doubt formal methods will change much in how we do mathematics. Hopefully they'll change how we do software engineering though.
    • skybrian 9y ago
      Perhaps it would mean that theorems are sometimes known to be true or false before anyone understands why. Digging through automatically generated proofs looking for interesting insights seems like a rather different experience than groping in the dark, not knowing whether a proof is possible?
      • mrout 9y ago
        Yeah and wait a second, isn't that the problem people are talking about in this thread? Aren't people basically complaining that things are discussed and proven without being justified? I don't really see how some fantastical process of formally verifying all of mathematics would have any relevance to teaching, anyway.
• Animats 9y ago
  Mathematics traditionally has a macho ethos. This goes all the way back to Euclid, and his "Let no one ignorant of geometry enter" sign. The training process primarily involves solving puzzle-like problems. Cambridge University still has Wranglers and Senior Wranglers [1], chosen for success at solving puzzle-like problems, not originality. (Hardy once wrote that this set English mathematics back a hundred years.) This history infects mathematics books. The other traditional problem with mathematics is terse and obscure notation. There are many implicit assumptions embedded in published mathematical papers and books. This is not helpful. It's like reading code snippets without the declarations. [1] https://en.wikipedia.org/wiki/Wrangler_(University_of_Cambridge) https://en.wikipedia.org/wiki/Wrangler_(University_of_Cambri...
  • xyzzyz 9y ago
    The other traditional problem with mathematics is terse and obscure notation. It's only a problem if you are unfamiliar with the subject area, and if you are, you have very little chance to understand the paper or book anyway. Mathematics is just difficult, way more difficult than most other stuff people might be doing. Here's an example: consider this classical paper[1], "Vector Bundles Over An Elliptic Curve" by M. Atiyah. The author is a well known and regarded mathematician, and the paper itself has around 1000 citations. Sections 1. and 2., "Generalities" and "Theorems A and B", just recall the basic notions and theorems that are absolutely necessary to have any grasp whatsoever as to what is going on in the paper. If you know nothing about algebraic geometry, the classical way to learn these is Robin Hartshorne's famous textbook "Algebraic Geometry". It is famous for being both good, self-contained to a large degree, and having very well chosen exercises, but also for being quite terse and often difficult to follow. Here's an Amazon link[2]. You can look at the table of contents. To really fully understand these two sections from Atiyah's paper, you need to have very good understanding of Chapter 2. "Schemes", and at least first 5-6 sections of Chapter 3. "Cohomology". This is 200 pages of pretty terse mathematics. At a fast understanding pace of 4 pages a day, it will take you two months to even have some basic toolset to understand the Atiyah's paper. But, if you try to understand the Harthshorne's textbook, you'll quickly find out that it also has some prerequisites of its own. Also, the "4 pages a day" pace is only possible if you've already spent 2-3 years learning how to learn mathematics. I encourage anyone to try to understand even first 2-3 sentences of the "Generalities" section. Google and Wikipedia the unfamiliar terms, you can also try to look it up in Harthorne's textbook, or Vakil's lecture notes, or any other source. The notions used in these first 2-3 sentences are basic to anybody working in the field, and yet one needs to spend hours to fully understand these when starting from scratch. Compare this to other famous paper from other field, "The Market for Lemons" by G. Akerlof[3]. This is also a very famous paper by a well regarded economist, who received a Nobel Memorial Prize In Economic Sciences for it. It is a much easier read, precisely because the economic sciences do not operate on nearly the same level of complexity as mathematics. Once you know what are some common sense notions like, supply, demand, utility etc., and some very basic calculus, you can easily follow the argument without too much training. My point here is that it's not that it's difficult to read mathematics just because it uses terse and obscure notation. It all is just genuinely difficult and complex, and it is impossible to invent better notation that will transfer days and weeks worth of understanding straight to the reader's brain. I would love it to be the case, but then it would cease to be as fun and rewarding to really understand. [1] - https://math.berkeley.edu/~nadler/atiyah.classification.pdf https://math.berkeley.edu/~nadler/atiyah.classification.pdf [2] - https://www.amazon.com/Algebraic-Geometry-Graduate-Texts-Mathematics/dp/0387902449 https://www.amazon.com/Algebraic-Geometry-Graduate-Texts-Mat... [3] - https://www.iei.liu.se/nek/730g83/artiklar/1.328833/AkerlofMarketforLemons.pdf https://www.iei.liu.se/nek/730g83/artiklar/1.328833/AkerlofM...
• tps5 9y ago
  People always say that math students don't learn about applications of math. This was never the case for me. When I learned trig ratios, I always understood some basic things that trig ratios could be used for. The teacher always introduced some applications, we always had a lot of word problems, and I could fill in the gaps myself. Same for calculus. When I learned calculus, I always understood some things that calculus could be used for. So I understood how those things could be applied to general, everyday sorts of problems. What was missing, though, was that I had nothing to which I could apply those techniques, besides homework. Learning math (and reading STEM papers) has become easier for me since I now have actual problems to solve. Don't get me wrong: I'm not solving particularly challenging problems or using particularly advanced math. Nothing that tens of thousands of people haven't done before me. But I do need to understand the problems, solutions, and some of the context in order to successfully implement them. This provides a motivation that was always missing before. I suspect this general narrative is true for a lot of people: that having an actual problem to solve is almost necessary to get a student to really learn the material, instead of just coasting along for a grade.
  • jacobolus 9y ago
    High school trigonometry and introductory differential & integral calculus are not the kind of books being described in this discussion. The example in the original post is books about group theory (or the group theory sections of abstract algebra books more generally). I can attest that this subject is very rarely described in textbooks with clear examples shown before definitions and theorems; usually the presentation is entirely abstract, following a pure definition–theorem–proof kind of structure. But many other areas of pure mathematics at the undergraduate level and above are presented in a similar fashion. (I recommend Nathan Carter’s book Visual Group Theory for a lovely counter-example to the prevailing trend, which starts with the concrete, and is very accessible. http://web.bentley.edu/empl/c/ncarter/vgt/ http://web.bentley.edu/empl/c/ncarter/vgt/)
    • ianai 9y ago
      We used to "run through" books like that. Their reasoning was to prove a theorem you only need the definitions/axioms. They really wanted us to be able to grasp the truths of a logical system from just its theorems and definitions. It was horrifically difficult. (Not all professors taught that way there.) I feel that a lot of blame lies at modern academias curriculums. They feel every student needs to graduate in X number of years with a pretty long list of courses. It leaves little time for students who need or want more time with topics.
• matt_j 9y ago
  I enjoy math books like: - e; The story of a number (Eli Moar) - An imaginary tale; The story of -1 (Paul Nahin) - The Poincare Conjecture (Daniel O'Shea) - The man who loved only numbers (Paul Hoffman) - Prime Obsession (John Derbyshire) They're part biography, part history, and give a little colour to the subject that isn't available from your typical college textbook.
• onmobiletemp 9y ago
  Its because math is used as test material. Thats all it is in american educations and many others i might add. Its a fucking crime. All of science and almost everying else is treated similarly. They build tests into the material you are to learn. At the end of the day you get a monstrosity. I say that if we are going to try to hide iq tests inside the material then we should have no qualms about simply giving everyone iq tests separately from learning material. Yes there are flaws with that proposal but it would still be better than what we have now where all of the material is molested. Imagine if people actually got a thorough education. No more grtting to the job only to learn that they didnt teach you what you need to know. In computer science this problem is fucking atrocious.
• graycat 9y ago
  I agree with the sentiment of the OP. For an answer, there have been various influences: (1) Whatever math was before 1900 or so, by the time of the Russell paradox and its fix with axiomatic set theory, the style of the fix was to be close to Russell-Whitehead (if I have that right) idea that proofs could be checked essentially mechanically by just symbol substitution and manipulation. E.g., in those days there was a book on the natural numbers, that is, 1, 2, 3, ... that apologized for numbering the pages before the natural numbers had been carefully defined! There was even a name given to this style of writing, telegraph style. (2) Of course, the books written on axiomatic set theory easily fell into the telegraph style. Even there the writers were getting into trouble: They gave names for the various axioms; they didn't explain why the names they gave were appropriate, and I never could discover why. But I was eager to get out of the sub-sub-basement of axiomatic set theory ASAP so just did a f'get about it. (3) When books were written on abstract algebra, e.g., basic set theory, construction of the main number systems -- the naturals, integers, rationals, reals, complex -- and then went on to the main algebraic systems defined with axioms -- groups, rings, fields, vector spaces -- it was easy to stay with the telegraph style. E.g., it was tough to find a book on abstract algebra that also discussed group representations and its applications to quantum mechanics and molecular spectroscopy. (4) Long calculus was often done with a lot of intuition and nothing like some carefully done definitions, theorems, and proofs as in, say, W. Rudin, Principles of Mathematical Analysis. And physics and engineering kept drawing diagrams with "little interval dx" etc. So, when abstract algebra was proving theorems, the calculus authors also wanted to be careful at least about delta-epsilon arguments. Then including a lot of physics, mechanical engineering, touching on the heat equation or fluid flow, was considered off-topic. Bummer. (5) The series of astoundingly carefully written books, close to telegraph style, by the team Bourbaki was influential. (6) During the Cold War and the Space Race, US math was awash in grant money and essentially turned its back on the physical science motivations and applications. Some of the funding people started to get angry about that, and we got the Tom Lehrer song and joke about abstract math being about "the analytic algebraic topology of locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some such. But, sure, especially in analysis, for a good proof, there's often a good picture and if can see the picture then can construct the proof easily. E.g., for positive integer n, the set of real numbers R, and convex f: R^n --> R, that f is continuous has a really cute picture. Same for Jensen's inequality. In linear algebra, the polar decomposition says that each square matrix is just (A) a rigid motion, rotation and/or reflections followed by (B) moving a sphere into an ellipsoid by stretching and/or shrinking on mutually orthogonal axes. One or more of the axes goes to zero if and only if the matrix is singular. Etc. Nice picture. Currently, then, there is an opportunity for math authors to include motivations for their subject, definitions, and theorems, intuitive descriptions and helpful pictures, applications, issues, open questions, etc. Uh, when reading a proof, for each of the assumptions, check off where it was used in the proof! The definitions, theorems, and proofs can still be fully precise and solid. Some students of, say, analysis have long tried to find and draw pictures that would clarify what was going on in some definitions, theorems, and proofs. I would advise new students to do that also.
• pzh 9y ago
  I thought "God Created the Integers" by Stephen Hawking was a pretty good high-level view of the history and progression of mathematics. It did give me some 'aha' moments about things I knew but never realized how they were developed.
• letitgo12345 9y ago
  Might be just laziness. Far easier to put down everything that you know in the final form than try and figure out how to present it in a way that will be most intuitive.
• wmnwmn 9y ago
  I've wondered the same thing. Moreover, math books tend to prove the most general case of every theorem first, rather than work up with the special cases which were the reason anyone thought about the theorem to begin with. It is strangely possible to learn how to prove quite abstract and difficult theorems without actually learning much about what the theorems mean. In the process of doing this one's ability to really think about mathematics or formulate new questions worth asking, is at best not developed and at worst stunted.
  • tome 9y ago
    > math books tend to prove the most general case of every theorem first If only! Many times during my studies I was hoping that someone had proved a result I needed, only to find textbooks that proved a slightly more special-cased version that wasn't enough for my uses.
• petters 9y ago
  Peter Lax wrote a graduate text on functional analysis where every other chapter consists of "applications". Even though the quotation marks are certainly warranted, I liked that concept.
  • ice109 9y ago
    name of book?
• godmodus 9y ago
  Because they assume a false premise (hah! Does this count as a pun?) and take a high road. Best series of books i read about matg tgst really helped me were the "for the practical man" series of books! Another book i recommend that helped me was "the architecture of math" by pierre basieux. These books really assume nothing beyond the ability to read and do a great job!
  • Rzor 9y ago
    I can't find any Architecture of Math authored by that person. Are you sure that's the author?
    • godmodus 9y ago
      https://www.amazon.de/Die-Architektur-Mathematik-Denken-Strukturen/dp/3499611198 https://www.amazon.de/Die-Architektur-Mathematik-Denken-Stru... it might not have an english translation, but there you go man here's a good reads link with all his books http://www.goodreads.com/author/show/745580.Pierre_Basieux http://www.goodreads.com/author/show/745580.Pierre_Basieux the original is in french, it seems the only options to read it is to either know german, or french!
• Melchizedek 9y ago
  Perhaps it is also the case that many mathematicians are far to the autism side of the spectrum, and find it difficult to verbalize "soft" aspects such as motivation and intuition, even though those are extremely important in a textbook. It's almost as if they partly lack theory of mind, and cannot imagine that somebody else does not know what they know, and consequently give explanations that are only suitable for someone who already knows the material.
  • mrout 9y ago
    There's no shortage of 'intuitive' books on physics or programming though, and those fields have no shortage of, to use your terminology, 'autists'. Mathematics is hard. Many people find it hard, and blame their lack of understanding on the material they are given. That is not really the fault of mathematicians.
  • gjulianm 9y ago
    Sorry but this is extremely misguided. You are making assumptions that make you look like you haven't talked to a mathematician as if they were a person, but a machine or a misterious magician. Mathematicians do not find difficult to verbalize motivation and intuition because they (we) are autists. We struggle to do it because it is really difficult sometimes. You can find that you have an intuition that you cannot put into useful words (sometimes the intuition is "it should be like this"), or that it relies on more advanced concepts (something that clicks only when you get to know concepts that rely on it), or that you know that the idea in your head is not useful to others and putting that down in a book would make things worse. I have not written any book, only some notes for other students, and with that I already know it is difficult to write a good motivation and intuition to abstract concepts even when you completely dominate them. Moreover, most of the mathematicians learn to develop their own intuition and motivations and it is not as important to write yours (which are subjective most of the time) down.
  • Broken_Hippo 9y ago
    I'm an artist. I've been doing art since I was a child: I started oil painting when I was 8 or something, and remember drawing and coloring being one of my favorite activities as a young child (before kindergarten). I can't properly explain how I see things the way I do. I can't really explain why I choose some colors in non-realistic pieces other than "they look good", although I know that learning color theory from the color wheel helps explain that and helps folks learn. It took me some time to learn to explain this stuff to others. I'm not autistic, though I likely have a mild case of dyslexia. I always imagine that skilled mathematicians are somewhat the same. It isn't that it only works for someone that already knows, it is more that they are writing the books for folks whose mind simply has that sort of slant to it. If we change all the books to help someone like me - I generally did poorly in algebra, trig, and calculus but found geometry fairly straighforward - that's going to let the other folks down because their mind's logic simply doesn't work on the same wavelength. This isn't autism, but simply human variation.
• stared 9y ago
  It goes back to Gauss and "no self-respecting architect leaves the scaffolding in place after completing the building" - about intentionally leaving all motivation, heuristics etc aside. Though, compare it with https://terrytao.wordpress.com/ https://terrytao.wordpress.com/, full of insight.
  • Chris2048 9y ago
    Does no self-respecting software dev leave comments/documentation behind? A software program is the equivalent product to a blueprint, not a building. And an architect might leave guidelines and annotations on those blueprints (they would not dismantle scaffolds, that's a builder).
• ianamartin 9y ago
  This is a general problem with textbooks. Textbooks are not written for students. They are written by experts in the field, and the audience is other experts in the field. This isn't necessarily a problem. It's a good thing to have experts working together to codify the current state of affairs, but it isn't necessarily the best teaching device. Even when you have people writing books that are intentionally targeted at people new to a topic, it's easy to forget how much you know and how much beginners don't know. I'm working on a Music Theory book with an audience of total beginners, and it's extremely difficult for me to put myself in that mindset. In doing research for that book, I've been going over many of my college music theory textbooks, and re-reading some of the treatises about the subject going back to Pythagorus and Plato. Things that were infuriatingly opaque to me as an undergrad are now--after decades of study--just plainly obvious and practically self-evident. You forget how little you knew after a while. If there were one thing I could go back in time and tell myself when I was younger, it would be to make it clear that textbooks are not there to explain it to me like I'm five. Textbooks are scholarly works intended to be as precise as possible about the current consensus of experts in the topic area. Approach a textbook the same way that you would approach a philosophical treatise. You aren't going to get it all the first time around. And I do tell that to my violin students who are now starting to go to college. Textbooks are designed to be paired with an instructor who can interpret the book and guide the students. They are not actually designed to be self-sufficient courses of study that you can simply read and learn from. The reason for this is that this is the model of education that's been pretty much generally accepted. The university model really hasn't changed a whole lot since medieval times and arguably going back to greco-roman times. At least in the West. I don't know enough about Asian culture or history to comment about that. You have a master and students and some texts laying out the way of things. And in order to become the master you have to jump through a certain set of hoops. It's an almost religious sort of process, in some ways. There's a tradition of a certain type of study with anointed masters, there are holy texts that you don't understand but must adhere to. And if you put in enough time and survive certain types of hazing rituals, you get the privilege of becoming anointed yourself, making very little money, and possibly contributing to a holy text yourself. Out of the many things that get cited as disruptive with respect to the technology revolution, one thing that doesn't get talked about a lot is the way that the internet has disrupted the education market. An "explain it like I'm 5" post on reddit is far more practically useful than any textbook on any topic, if your goal is to help a person learn about a topic. I spent the vast majority of my life studying violin, and music theory. And I was a professional performer for 20 years. When I got burned out on performing and traveling and never making enough money, I got online and started learning, practically, how to code. At first just enough to get a junior job. And I kept reading, and I got better. It's been 11 years now, and I haven't looked back. I can write code and play the violin when I feel like it, and work on a book about music theory. All that as a college dropout. It's a life I'm deeply grateful for. A generation ago, changing careers would've meant going back to school, probably taking on some debt, finishing up a degree that I never really wanted anyway, and then going to start something else. Which is a pretty big pause in your life when you're in your late 20s. I didn't have to do any of that. At the same time, rigorous textbooks certainly have a place in the world. The same way that Plato's Timmeus or Descartes Compendium musicae have a place. The biggest problem with textbooks is that people misunderstand their intended purpose and audience. They are tools created by academics to help create more academics. Nothing more; nothing less. If you want to participate in the academic system, you have to play by its rules. That means that you deal with these textbooks and the professors who guide you through them until you yourself are a professor, and you can in turn, guide others. For those of us who don't want to do that particular type of work and would rather be educated practitioners, there's the internet. I'm glad we have both options available now.
• fpoling 9y ago
  There are absolutely non-boring mathematical books by Russian authors. In addition to already mentioned "Mathematics: Its Content, Methods and Meaning" by by A. D. Aleksandrov, A. N. Kolmogorov, M. A. Lavrentiev I would add "Elements Of Applied Mathematics" by YA. B. Zeldovich, A. D. Myskis [1]. The latter is really good introduction into math when one needs to apply it in various settings. [1] https://archive.org/details/ZeldovichMyskisElementsOfAppliedMathematics https://archive.org/details/ZeldovichMyskisElementsOfApplied...
  • deleted 9y ago
    [deleted]
  • inlineint 9y ago
    And also there is a more introductory book by YA. B. Zeldovich and I. M. Yaglom "Higher Mathematics for Beginners" [1] that is a fun to read book about calculus. [1] https://archive.org/details/ZeldovichYaglomHigherMathematics https://archive.org/details/ZeldovichYaglomHigherMathematics - for some reason the book at archive.org has extension .pdf , but in fact it is a djvu document, so it has to be opened with DjView instead of a PDF viewer.
  • agumonkey 9y ago
    The figure on the first page reminds me of SICM
• jansho 9y ago
  Try Roger Penrose's "The Road to Reality." He takes the approach of assuming that his readers can range from absolute beginners to mathematicians, and that if you prefer, you can skip the equations and the exercises. But if you attempt to understand them, then your picture of the Universe will become richer that no other pop science book can promise. It's an enticing thought, and that and his gentle prose are just about his only tricks to push his readers through his enormous tome. I haven't finished it yet but it does feel rewarding that much of the maths that I've come across (Alevels and engineering degree) are there, laid out in a new framework of meaning, the Universe. P.S. There's also an eBook version, and an online resource that gives solutions to the exercises.
  • 77pt77 9y ago
    It's a very good book for people of a technical background to get a taste of theoretical physics (probable the only one) but let's not exaggerate, it's not a math book.
  • pbhjpbhj 9y ago
    Penrose says the same in "Emperor's New Mind" - that you can skip the equations. Perhaps you can but IMO you'd miss an awful lot, I found it too difficult to just jump over the equations until I'd studied enough to have met the Schrodinger equation and from then one can pretty much grasp all the physics/maths. Perhaps he does better at keeping the flow going in "Road to Reality" for those trying to read without deeper knowledge?
  • anatoly 9y ago
    My advice is to not try Penrose's book. The vast majority of its contents will be utterly incomprehensible to someone without a PhD level background in physics. Technically it is self-contained and nothing beyond high school knowledge is needed to understand it; but while technically true, in reality it's a cruel deceit. If you try to learn it without advanced training in physics and math, you'll run into limitations related to mathematical maturity (https://en.wikipedia.org/wiki/Mathematical_maturity https://en.wikipedia.org/wiki/Mathematical_maturity) very quickly, within the space of the first few chapters. The idea of this book seems noble and uplifting, but it cannot be done in this fashion. It doesn't matter that every theorem and assumption are clearly stated and follow each other in an orderly sequence. There's a huge number of new ideas, mathematical and physical, which require proper acquaintance to settle in your mind, not the 1-2 pages Penrose is able to give each of them. If someone is serious about learning physics to that level "properly" and is willing to invest the time and effort, they're much, much better off working through conventional college-level (and later grad-level) textbooks in particular subjects, taking care of math prerequisites before and during such study. That's very very hard to do, but is in fact possible, whereas studying solely from Penrose's book, I think, is not. It functions, sadly, as a sort of trap for impressionable bright people very eager for advanced knowledge - they will try to struggle through it, inevitably give up rather early on, and blame themselves.
• basicplus2 9y ago
  One of the biggest problems I think is that most experienced mathematicians seem to forget that the written forms of mathematical systems contain significantly shortened symbology which requires significant understanding of what is NOT written explicitly​. The most basic example I can think of is "ab" meaning "a x b" but the "times" function has been deleted but still needs to be understood. I find when reading mathematical books I cannot follow examples and progressions to the next level then the next level etc as at every progression there are significant pieces of information that are missing yet text books rarely if ever mention what is not written, so how can the novice possibly figure it out.
• Naa4 9y ago
  The first time I really grasped the very basic concepts of linear algebra instead of just being able to calculate things like eigenvalues was after watching this series by 3blue1brown: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... There is a series on calculus as well. It is honestly beautiful.
  • jgalt212 9y ago
    I concur. The calculus series is amazing.
• Naa4 9y ago
  The first time I really grasped the very basic concepts of linear algebra instead of just being able to calculate things like eigenvalues was after watching this series by 3blue1brown: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... There is a series on calculus as well. It is honestly beautiful.
• Tomminn 9y ago
  It's totally cultural. The culture comes with some costs, and some benefits. In physics, there are so many times where we present the same mathematical ideas in a manner that's way more ergonomic for the minds we're talking to-- other physicists. Inexperienced mathematicians tend to see these presentations and are kind of gobsmacked, "why did no-one tell me it was that simple and concrete?" Experienced mathematicians tend to see these presentations and cringe, "don't they understand it's not that simple and concrete? For instance, look at this pathological example here that contradicts everything they just said." The slick presentations are great if you're a physicist or anyone just wanting to use the ideas. And they're great if you're an inexperienced mathematician trying to get your bearings as fast as possible. But once you're a research mathematician, trying to build new ideas, too many slick presentations of ideas just creates groupthink. You want as many people to generate their own mental pictures as possible. That said, few people studying mathematics go on to be research mathematicians. The mathematics culture should be much, much friendlier to people whose endgame is not being a math prof. Apathy toward those whose ambitions lie outside the ivory tower is a problem endemic to most academic subcultures. Finally, when reading anything generated by academia, remember most of it is crap. Academics are fired if they don't make enough noise. This is truer than you think, even taking into account metrics like "impact". You can make a lot of "impact" just by making enough noise. And you generally can't make "impact" on a short timescale without making enough noise. Once your name shows up enough in a field, you'll get known as "a name", and start being cited in that field. Even when your work is drivel. If your name only shows up a couple of time in a field-- unless your work incredibly well written and you get a good dose of luck-- it'll take years before it gets uncovered and appreciated, if it ever does. Think of how much more attention you pay to what someone like maxxxxxxxx says here. Sure. He's good. But he's also familiar. It's primarily his familiarity, more than his quality, that makes you feel like he's one of "names" in this little subfield of the internet called hacker news.
  • jimhefferon 9y ago
    > culture Yes, I think you are right. I offer a free Linear Algebra book that appears on the first page of a Google search. The book's approach is to give a lot of explanation and motivation, so I often get emails from folks about this topic. From students often the mails say something like, "I couldn't understand my prof but by using your book as a supplement .." while from the prof I get (much less often) "way too chatty." So I agree that a lot of it is a question of culture and taste. On my first day in grad school a prof who I admire a great deal told me what he liked best about Differential Equations is that you don't have to say how you found the answer. You just state the answer and verify it satisfies the equation. I think that's the taste of people who enter the field. (I also think that is the taste of the moderators of MO, which is why questions like this have been closed over time.)
• eecc 9y ago
  Back when I was in Uni I flunked Calculus II. The following year I switched Uni and - somewhat fortunately - I re-followed the same class. I say fortunately because during my first round at Calculus II I couldn't fathom the slightest underlying motivation for all the statements and proofs that were regurgitated during class. I had no bearing - I nearly panicked with the Cauchy Problem - and was doubting whether or not I was cut for Engineering at all. The following year the other Prof took a digression and for two weeks laid out Inf-dimentional spaces, Banach spaces, functionals, contractions, fixed points and so on. Then one morning, over the course of 2 hours he plugs in differential equations, the Cauchy problem and deals with it with such elegance, clarity and insight that I can still recollect the broad strokes. I had first hand proof of the blessing and the curse of mathematics...
  • eecc 9y ago
    Right, the second prof's name was Francesco Saverio De Blasi, he passed away in 2012. He was a terror to students but I appreciate the hours he spent teaching us the basics with rare depth
• gjulianm 9y ago
  Honestly, every time that a discussion on mathematics comes up in HN I feel the same way: people seem to think that mathematicians are some kind of elitist people that don't know how to explain concepts, and that mathematics would be like coding (note: projecting everything into the tech world is not the best way to understand the world) if it weren't because mathematicians don't explain things well. But the reality is way different: * We know that teaching mathematics is difficult. But, contrary to what most people think, there is no magic pill to solve it. Motivation and intuition are important, yes, but so are definitions and formality. Also, even when giving them to the students, they might not be useful: sometimes each person needs a different motivation of a concept, sometimes the intuition relies on more advanced concepts... Furthermore, things don't usually click the first time you are exposed to them, but only after you have worked with them for some time. Don't think that just because you are given a motivation and intuition behind the concepts everything is just going to become easy to understand. Even motivations and intuitions need work to be understood. * The mathematics people study in high school and first-year engineering courses is very different from the mathematics that mathematicians study. Funnily enough, most mathematicians I know don't like the approach of mathematics as a tool for other disciplines, mainly because it focuses on things that are hardly interesting (here's the 123th trick for how to solve this weird integral that we are teaching you only because you will need it to solve some engineering equation) and avoids proofs that, despite being hard, most of the time helps understanding the concepts. * Related to the last points, an important number of math books are written not for laypersons but for mathematicians. They go straight to the point, avoid long motivations/intuitions that are not specially useful and assume familiarity with a lot of concepts. Sometimes they are written as reference books for people that already know the ideas. That, of course, makes them hardly approachable by non-mathematicians, but there are already books that do that. Just because a book or a way to teach math is not of your liking does not mean it needs to be changed. * Not all mathematical concepts need to have a direct application. Sometimes, things are studied just because they seem interesting or beautiful. Searching for motivation needlessly in these cases tends to be a waste of time. It's like reading a novel or solving a puzzle: you don't need to know "why", just enjoy it. If you're lucky applications can appear later, but there's nothing wrong if they don't.
  • marcosdumay 9y ago
• the6threplicant 9y ago
  This is why lectures are so important compared to a text book. In a lecture you try and emphasis how some things are important - while other things you can hand wave away. A textbook can't do that.
• Nano2rad 9y ago
  There are two kind of mathematics books, pure mathematics and problem oriented. Pure mathematics text books needs not give real life examples or context, that is the way they are supposed to be.
• erikb 9y ago
  I think it's also strategical. Math is an area of science that most of the time is not worth as much as it costs to continue. But sometimes, one of these little things turn out so huge that they change all our lives. So we really need math, but most of the time there is no logical reason to invest in it (knowledge, time, money). What to do about it? One way that seems to have begun common in Math is to make it magical. The entry burden is hard as hell, and most reasonable things in math are expressed in a way that as best sounds mystical to common people. This gives math some kind of credibility. Nobody wants to be the one to say how little they actually understand about it, so they continue to support the crazy magicians instead of losing face. And in the end humanity is better off for it. PS: You can see that it is not necessary to be so cryptic but still math continues to do so, when you compare it to programming. For instance math continues to use a greek letter ∑ (sigma) to express it. In programming it's mostly a "for loop" or a "sum() function". Most people can understand "sum([1,2,3])" or "for <something> do <something> done", but they don't know the greek alphabet. So it's possible to encode math more readable, but math continues not to do so, and as argued before, probably also has good reason to do so.
  • mathperson 9y ago
    sigma is one character versus at least 3 for "sum() function". it literally takes 20 seconds to figure out what the symbol is...
    • abecedarius 9y ago
      But the plus sign is also one character. Why not use a big plus sign instead? That would not only be clearer, it generalizes systematically: use a big multiply sign instead of a capital pi, etc. When I was a kid I was a little intimidated by those higher math books with the "Greek E's" in them. Years passed before I learned in my early teens what they actually meant (and that it was simple, and they weren't even E's).
      • mathperson 9y ago
        THE BIG PLUS SIGN IS USED IN MATHEMATICS! It has an entirely DISJOINT meaning from simple summation... https://en.wikipedia.org/wiki/Direct_sum https://en.wikipedia.org/wiki/Direct_sum
• Ericson2314 9y ago
  Formal math in theorem proper with copious marginalia +1
• teddyh 9y ago
  See also “A Mathematician’s Lament”, written by Paul Lockhart in 2002: http://www.maa.org/external_archive/devlin/devlin_03_08.html http://www.maa.org/external_archive/devlin/devlin_03_08.html “Sadly, our present system of mathematics education is precisely this kind of nightmare. In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”
  • brutus1213 9y ago
    I took a minor in Math because I enjoyed some aspect of it but completely agree with the quote. All my math courses were taught in a vacuum or worse, taking problems that bore no connection to situations I cared about. Over the years, as I explored various hobbies, I found mathematical ideas having application - that's when I truly made an effort to understand concepts fully. For my kid, I plan to introduce math concepts in the context of these hobbies .. if any one cares .. these are astronomy and radio.
• kwhitefoot 9y ago
  See A Mathematician's Lament: https://news.ycombinator.com/item?id=14331752 https://news.ycombinator.com/item?id=14331752
• doener 9y ago
  In may cases better than reading the book: https://www.blinkist.com/ https://www.blinkist.com/ It's only the gist of it.
• moron4hire 9y ago
  My pet theory is that there are financial incentives to make bad text books. Text books are usually written by university professors, often the professors from whom you are taking the class. There are absurdly many entry-level textbooks for a subject whose beginning parts probably haven't changed in over 100 years. They do this at the behest of the university, which will be capturing their royalties under an IP clause in their employment contract. The university intends to sell this book for upwards of $200 to a captured audience of teenagers who don't yet have the life experience to know they can opt out of putting up with bullshit. If the university doesn't have a constant churn of new editions, then the books will very quickly filter into the secondary market where they will be drastically discounted. So universities pressure professors to write textbooks quickly. And university professors aren't exactly the best-paid vocation. Invoke "good, fast, cheap: pick two" and you see it's "good" that is getting dropped on the floor.
• lois 9y ago
  This is something I think about often. Maths is one of those subjects which is taught repetitively rather than philosophically, and yet there couldn't be a worse subject to teach in such a manner. It forms a barrier-to-entry which mathematicians probably care for, glancing at some of the quotes already posted here, but doesn't help people who would make excellent mathematicians if only they had the encouragement. I always found mathematics self-explanatory. From all the repetitions the understanding came naturally, and I think there is definitely a subset of the population that has what you might call 'aptitude' for maths. It makes sense, because philosophically it is self-explanatory by definition. LHS = RHS - the trick is to prove it or fill in the blanks. But I dare say the majority of people aren't 'apt' for maths in the way in which it is taught, but at the same time it's totally unfair to rule them out as maths-stupid. Many people who have no mathematical knowledge whatsoever still prove rather deft at deducing when their partners are lying or when an argument in a debate is self-contradictory. People who failed their mathematical education go on thinking they suck at math, but when they play video games and figure out optimal strategies and formulae for success in the game, it's not all that different. Sometimes the mathematical understanding comes later to people as a direct result of being forced to reason about mathematical problems in an applied setting and discover that they could have been good at maths all along if they had only understood the 'why' of the exercises and formulae they were doing. I think there are two big stumbling blocks that stop people from taking a bigger interest and investing more concentration and care into maths: firstly people mix up arithmetic with maths, and assume from the fact that they take a long time to divide a couple of numbers (or get the wrong answer when they do so) that they are doomed to be crap at maths. But that's utter BS. I'm hopeless at mental arithmetic, but maths isn't about adding numeric values, its about deduction and reasoning, and I'm certainly not alone in being a terrible arithmetician and a good mathematician. The other issue that people struggle with is that the notation itself isn't clearly explained or appears daunting, especially if they didn't go beyond compulsory education. It can prove quite distracting because a person might have questions as to why limit notation looks the way it does or what the hell the integral sign means, when really it's no different to choosing commas and brackets to denote semantic delineation in written language. Take limit notation or summation. The placement of the various numbers is arbitrary on a fundamental level, it's just the standard everybody agreed to use. If someone good enough at math sat down and tried to build up math from first principles, with no knowledge of modern mathematical standards and notation, he/she could come up with the same formulae and concepts but an entirely different way of representing them. That's exactly what happened really. you had Leibniz and Newton just organizing information in a way that made sense to them, and it stuck. An explanation of how arbitrary the notation is on a fundamental level would probably make a board full of symbols and algebraic letters considerably less daunting for those who could-but-daren't understand it. I think there's a big problem in education as well. I'm from Great Britain and growing up I think the biggest mistake in our education system is the dire failure to explain mathematics in a way that is friendly for people who aren't tip-top at abstract thinking or imagining. Take the humble function, for example. teacher says "a function takes an input and produces an output. so f of x equals y minus 5x..." - and half the class stares blankly, daydreams for 15 minutes, chatters through the "work your way through the textbook" phase and does one of two things before the teacher goes through the solutions: A. copy answers from the back of the textbook B. copies an adjacent boffin. I think the much maligned set theory actually provides a very good way to teach about functions, and yet it doesn't appear until you hit college, and then suddenly a function is referred to as a mapping, without any preparation or explanation of whether there's a difference between 'function' and 'mapping'. I was endlessly helping classmates in school and I often think about how the education system could have been better and made math more accessible. The conspiracy theorist might say that society cannot afford for everybody to grasp mathematics and get a /good/ job, but hopefully that's not possible. Geometry in education is a big fail as well. when I was growing up, trigonometry wasn't explained at all, it was just repetition of applying trig tables. Nothing visually relating right-angled triangles and circles like Thales was even touched upon until college. Secondary education in UK tries to cover to many things. Not that there's any way of chopping math down to a single textbook or subject area, but there's often little in the way of structure and it's about cramming as many formulae down the student's throats and then examining them on their ability to apply what they have repeated. but you can't apply what you repeat, you apply what you learn and grok. I'm not sure why there is this enormous failure in education. It could be because the people who write the books and lesson plans already grok mathematics and forget to give people aids to intuition or intrinsically understanding what they're doing. Or maybe it's because a mathematical education excludes the philosophy of math and set theory which is a mistake. Set theory is perfect to start off with, it gives insight into functions, probability, everything really. Can always discredit the /project/ of founding mathematics on set theory later. Of course, maybe I went to a bad school, and it was a few years ago now so perhaps things have changed - but I don't think so. most of my younger sister's friends have a worse understanding than the disinterested kids when I was growing up, but that could be down to the whole fast-food, instasnaptwit culture and a hundred other things that seem to be distracting people from anything academic.
• baby 9y ago
  I think it depends what book you're looking at. Pedagogy in education is a pretty recent thing IMO. As of the teaching side, I went through my entire math bachelor without understanding what were the applications of what I was learning. Now that I do cryptography I finally understand some of these. There was also this course by Gilbert Strang on matrices that taught me that they were used in compression. None of my teachers could have told me that...
• badpenny 9y ago
  I've often thought that one of the main problems with pedagogy, at least the kind that tends towards the abstract, is that it tries to give you answers without prompting the questions. I suppose you could argue that it's the student's responsibility but maybe that would require insight and inspiration on a par with that of the person who came up with the answers in the first place and I think it's fair to say that that's an unrealistic and unreasonable expectation.
  • marcosdumay 9y ago
    > I suppose you could argue that it's the student's responsibility You'd then argue that it's the job of the less experienced people around to deeply understand the world's problems. I don't think it's a good argument at all.
• FabHK 9y ago
  A wonderful book that doesn't suffer the discussed flaw is Visual Group Theory by Nathan Carter, discussed on HN earlier. It provides ample motivation and examples before diving into the theorems. https://news.ycombinator.com/item?id=11745486 https://news.ycombinator.com/item?id=11745486
• neom 9y ago
  I really really wish I could do math. I have very high dyslexia and dyscalculia and I really really struggle to understand numbers, it's just so difficult for me to think about them, think about putting them together, even basic addition can be a complete struggle. I'm not sure why, but I can somewhat understand geometry.
• StillBored 9y ago
  For lower level courses (say calc and below) the books written in the early 1900's for students seem better. Of course there are plenty of stinkers there too, but it seems being able to explain a topic in a clear concise manner with applications was prized more. The text books my kids have in comparison are utter garbage in their effort to present the material in dozens of nonsensical manners to help those that didn't understand the primary methods. Plus "challenge" problems that rely on missing knowledge abound. The perfect example I had of this is when my child was in 3rd grade as part of the homework there was a set of algebraic word problems with multiple variables. When I confronted the teacher, she didn't see anything wrong with teaching a "trick" to solve problems where the underlying math was two or three grade levels ahead (said trick, was so narrow an unwieldy to basically be useless for any kind of problem more complex than ones made up for 3rd graders)... The bottom line seems to be that the "noise" level keeps increasing to the point where the core fundamental concepts are lost. Which is to bad, because teaching "tricks" which don't clarify the problems, or make them simpler to solve is IMHO just a waste of time. BTW; For abstract algebra there are a number of "applications" books which teach the core concepts necessarily to understand error correction or whatever. Use those books first, and then read the more traditional text books. The textbooks should be considered more of a reference book than something that should be read from cover to cover. (although that itself is a problem because jumping into the middle isn't easy).
• calebm 9y ago
  "Who is Fourier? A Mathematical Adventure" is a great (though rare) counterexample - it is wonderfully intuitive. It's the best math book I've ever seen.
• jgord 9y ago
  Partly this is a meme diffusion problem - eg. we have really good diagrams that explain visually why Pythagoras' theorem holds ... yet students often don't see these more intuitive forms. Another example is the Feynman Video Lectures are effectively inaccessible, due to being in an ugly format - they should be on youtube as a planetary resource. I was recently looking at reviewing University chem so I can have informed discussions with my teenage son.. Atkins has a great book, but at an exorbitant price. College texts seem to be a kind of scam. Some would argue its morally justified to pirate such content. We seem to be stuck in all these local minima for minor psychological or economic reasons. There should be open texts that cover most of the basics of highschool and freshman science, with upkeep costs shared by universities/schools or covered by a library tax. We can think of education methods as a technology itself, and the best technology, namely the best explanations, are much much better than the average - how do we propagate the best ideas ? Having ratings on quality, like those on goodreads and amazon, seems to be a practical benefit, but maybe we need smaller granularity - eg. rate a particular diagram on an open biochem textbook wiki.
• gcr 9y ago
  What are some recommendations for math books that really are deeply enlightening?
• kabanossen 9y ago
  There is an exception, "Calculus Made Easy" by Silvanus P. Thompson from 1910. http://www.djm.cc/library/Calculus_Made_Easy_Thompson.pdf http://www.djm.cc/library/Calculus_Made_Easy_Thompson.pdf https://en.wikipedia.org/wiki/Calculus_Made_Easy https://en.wikipedia.org/wiki/Calculus_Made_Easy
• m23khan 9y ago
  Frankly, I think the old World habits of people distancing themselves from distractions and to involve their self in deep thoughts relevant to their fields, experimenting tirelessly and then collaborating with like minded personalities are either dead or severely limited nowadays. With such pervasive usage of TV, internet, cellphones, telephones, material items, shopping, entertainment over the last few generations, we have gotten to the point where we are no longer producing Einsteins, or Thomas Edisons or Babbages. These days, your typical 'scientist' is a guy more concerned with dressing and appearing hip and is more concerned with being a CEO/marketing than with actual trench work in the scientific plains. These are the same types who encourage in turn for younger generations to follow their 'hearts', to be more 'social', to refrain from 'dry and boring studies' that consume one's entire lifespan.
• random_comment 9y ago
  When I taught maths at university, I would always try to include in every lecture a couple of slides showing how the lecture could be used to get a job, or to solve a real world problem in computer programming. It's weird that other disciplines seem to ground knowledge much better than maths, when it isn't hard to ground maths.