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So you want to study mathematics

• Py-o7 5y ago
  This felt like it was written by a physicist or engineer. Too much emphasis on differential equations and not enough on things like topology, functional analysis and/or non-introductory parts of algebra like say representation theory.
  • susanrigetti 5y ago
    guilty as charged! :)
    • bitexploder 5y ago
      As someone with a keen interest in learning Engineering part time, I found your write ups really helpful though! I enjoy learning math but like to have an angle towards a practical and useful application. It keeps me a little more motivated than pure math learning. With ADHD the concept of being able to build cooler things always keeps me going. But somewhere along the way of learning purely theoretical things for too long my brain just loses interest (not enough reward), even though I enjoy it in the moment it is hard to get to the starting line and take the first step after a while :)
  • mrjangles 5y ago
    Yep, I came here to say this. I think it is important that anyone who wants to study math understand that real math is not at all like what you learn in a physics or engineering department. In these departments you will always hear people say things like >"proofs are not useful, all you have to do is memorize the 'trick' they use. Once you know which trick to use, it is easy" or you will hear them say. >"Math isn't about understanding, it is just about learning rules and symbols on paper". This is not mathematics. These things do happen.... in a physics and engineering department. It is, in fact, a descriptions of a physics education, not a description of a mathematics education. For this reason I would be careful taking mathematics advice from physicists too seriously as they may, unintentionally, lead you very far astray.
    • susanrigetti 5y ago
      For what it’s worth, the curriculum in this guide is modeled after the math major maps of many universities, including the one I attended (Penn). I would be curious to know what part of an undergraduate math curriculum will lead people very far astray…
      • mrjangles 5y ago
        It's just that it is very much a "mathematics for engineers" style course. I think very few of the subjects outlined there give you a flavor for what "real" mathematics is really all about at all (except for algebra, which you do mention). Apart from the applied stuff you mention, the real core of a mathematics education involves, I think, 4 main areas with significant overlap Group A: number theory, graph theory, combinatorics which shares concepts with Group B: Algebra, Topology, complex analysis, differential geometry, metric spaces...etc which shares concepts with Group C: Functional analysis, measure theory which shares concepts with Group D: probability and statistics. As for the applied math that you mention, you should really need to add vector calculus and I'd highly encourage anyone to take a course on fluid mechanics (from a mathematics department instead of an engineering department) to get a real feel for vector calculus in action.
        • susanrigetti 5y ago
          I suggest taking another look at the list and comparing it to the required courses of the undergraduate math majors at the top 20 universities in the USA. Real analysis, complex analysis, topology, and number theory are there (topology and number theory are both listed as electives since most math programs categorize them as such). Graph theory, functional analysis, differential geometry, probability, and statistics are almost always either electives or graduate courses. It’s funny, because most of the things you mention as “real math” are things that many math undergraduates don’t learn (not until graduate school at least) but that physics students learn as undergraduates (differential geometry, measure theory, functional analysis, etc.).
          • 5y ago
  • BeetleB 5y ago
    > This felt like it was written by a physicist or engineer. I just compared it to my undergrad's math curriculum and it matches up pretty well. Everything you mentioned is an elective.
  • Jap2-0 5y ago
    As someone studying teaching math, I found it interesting to compare her suggestions: - Calc I-IV - Intro to proofs - Linear algebra - (Abstract) algebra I-II - Real analysis - Complex analysis - Ordinary differential equations - Partial differential equations - (Others) to my program plan: - A couple teaching courses (including one for roughly grades 5-8) - Calc I-IV - Statistics (one without calc, one with) - Linear algebra - Discrete - Geometry - Number theory - History of math (apparently not just a history class, I haven't taken it yet) - Abstract algebra and into to topology
• musgravepeter 5y ago
  I've been on a Math journey since I retired a couple of years ago and I agree with all the books mentioned that I know and look forward to picking up some of the one I do not know. I agree baby Rudin is essential, but I find it tough going. Some books I liked for self study because they have answers: Introduction to Analysis, Mattock. Elementary Differential Geometry, Pressley. There is also recently Needham's Visual Differential Geometry and Forms, which is great. Edit: I should also mention Topology without Tears (free, online, very good) https://www.topologywithouttears.net/ https://www.topologywithouttears.net/
  • auggierose 5y ago
    Very pretty book (Needham's), will check it out! I think over 20 years ago I actually attended a house party that Needham was giving in SF. It's a small world.
  • vermarish 5y ago
    I think learning Real Analysis from baby Rudin is like learning Probability Theory from Wikipedia. It's so encyclopedic that if it's your first look at real analysis, it will be too dense to understand, but if it's your second or third look, you will find beauty in its brevity.
  • susanrigetti 5y ago
    Agree that Baby Rudin is VERY difficult to study on its own. I recommend only studying it alongside the other two books I listed: Abbott's Understanding Analysis and Spivak's Calculus (which has a solutions manual). Abbott in particular is very straightforward (at least in comparison with baby Rudin haha)
    • tzs 5y ago
      Another point for Abbott is that it was one of the ~400 books Springer made available for free download near the start of the pandemic. I remember there were a few scripts here on HN back then to grab all those books, so many here probably already have a copy.
  • 5y ago
• Mimmy 5y ago
  Going from Strang to D&F seems like a steep jump. The former is an applied textbook for non-mathematicians and the latter is a proof-based text for advanced undergraduate / graduate-level math students. I would suggest working through a proof-based linear algebra book in between to ease the transition. Axler's is a good one. Alternatives include Hoffman and Kunze and the more modern Friedberg, Insel, and Spence.
  • tptacek 5y ago
    Both Strang and D&F are extra-relevant for cryptography (I was struck by how much the earliest parts of D&F --- which I haven't gotten much further beyond --- read like the mathematics background chapter of a cryptography book), and I've been in study groups for both of them with non-mathematicians that went OK. But the D&F study group fell apart for logistical reasons, so maybe it would have hit a wall after a couple more months.
    • pvg 5y ago
      read like the mathematics background chapter of a cryptography book A lot of maths-related books, especially ones intended as textbooks will read like that in part because they aren't kidding about the 'abstract' in the title - they're trying to teach/re-summarize key concepts of mathematical abstraction. It's a good and true thing to notice.
  • dwohnitmok 5y ago
    Second Axler. "Linear Algebra Done Right" is probably the pure mathematics textbook I've most enjoyed reading ever (but be warned you will learn very little about applied methods from it if that's what you care about). Also enjoyed Artin's Algebra.
    • musgravepeter 5y ago
      +1 for Artin's Algebra. I think is very under appreciated.
    • cgriswald 5y ago
• daxfohl 5y ago
  I loved last year being able to take university courses online. I knocked out analysis, topology, and quantum mechanics as a non matriculated student. I'd had those books for years but never could get through them alone. (The main thing being, you really don't have anything to gague whether you know it well enough or not). I really wish there was more opportunity for that. I'd love to take a few more classes, mostly in pure math, but there's simply nothing on offer for remote study past the 200ish level. (There are some remote masters programs in applied math, but nothing for pure). I don't think I'd enjoy doing a PhD full-time. One or two classes per semester while working seems just about right. But the closest university is an hour away, so in-person isn't a realistic option.
  • elteto 5y ago
    Where did you take your classes?
    • daxfohl 5y ago
      University of Washington
  • adamsmith143 5y ago
    Texas AM has a program that gets somewhat close though it definitely has a computational focus. Here's a list of their recently offered courses: https://www.math.tamu.edu/graduate/distance/openletter.html https://www.math.tamu.edu/graduate/distance/openletter.html
• irrational 5y ago
  I never got beyond algebra/geometry in High school. I think I had to take one 100 level math class in college, but it was basically a review of HS math. Oh, and I had to take a stats class for non-technical people in graduate school. That was my worst graduate class by far. But, I would like to learn some more math, like calculus. I’m hoping to get to it when I retire in a decade or so.
• mbustamanter 5y ago
  > My goal here is to provide a roadmap for anyone interested in understanding mathematics at an advanced level. Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics. NO, NO, NO. There is no real way to go up to the real deal without having understood elementary Functional Analysis, which the article doesn't even mention. FA is roughly what Linear Algebra would look like if instead of finite dimensional vector spaces we considered infinite dimensional vector spaces. It opens the rigorous path to non-linear optimization, analysis of pdes, numerical analysis, control theory, an so on. What this article mentions is a way to work around things, but nowhere near an undergraduate degree in mathematics. I'm astonished that the PDE section has such books, they look like the calculus aspect of partial differential equations. A more appropriate book would be L. C. Evans' Partial Differential Equations. Same with ODEs, no mention of Barreira's or Coddington & Levinson's books.
  • ratzkewatzke 5y ago
    I'm a fan of functional analysis, but even in my (very competitive) undergraduate curriculum, it wasn't required for a bachelor's in mathematics. I think Susan's guide covers most of what the undergraduate programs I've seen require.
    • mbustamanter 5y ago
      It was for me (french school of math), that and also measure theory.
  • davidmr 5y ago
    This is certainly not universally the case, even in very well-regarded departments. The University of Chicago, for example, does not require it: http://collegecatalog.uchicago.edu/thecollege/mathematics/ http://collegecatalog.uchicago.edu/thecollege/mathematics/.
    • secabeen 5y ago
      For those of you interested in the Chicago approach, a bibliography of textbooks used in Chicago UGrad math is maintained here: https://github.com/ystael/chicago-ug-math-bib https://github.com/ystael/chicago-ug-math-bib
• ghufran_syed 5y ago
  I went from only having done high school math 10 years ago to completing an MS in math and statistics at my local state university while working in an unrelated field. I would recommend NOT starting with calculus if you haven’t done it, instead, just learn how to do proofs - I used Chartrand “Mathematical proofs” - You don’t need to know any math beyond algebra in order to do that most of this book. If you need to revise or learn Algebra, then I would do Stroud “engineering math” first which is designed for self-learners with lots of solutions and feedback. At some point, it would be good to get a a copy of Lyx and start to learn to write math in LaTeX - Then you can get feedback on your proofs online at math.stackexchange.com if you don’t know any math people locally. Feel free to get in touch with me if you want to discuss further, happy to help!
  • criddell 5y ago
    I looked up the Chartrand Mathematical Proofs book and it's been a while since I had to buy a textbook, but $175 for hardcover and $75 for paperback or ebook? That's nuts. If I were a student today, I'd pirate that and feel absolutely no remorse for doing so.
    • dunefox 5y ago
      Thing is that I just can't read PDFs.
      • fuzzythinker 5y ago
        Try borrowing a good e-ink reader to see if it helps. If so, get a good one with larger screen if you'll be using more for textbooks.
      • rg111 5y ago
        Use an iPad, or any tablet, rather than smartphones. I also read PDFs full-screen on my high-res, high-dpi laptop. Try these two. I have read tens of thousands of pages in PDF. (Yes, I checked)
        • 5y ago
• parsd 5y ago
  If you're interested in both mathematics and physics, does it make sense to learn both concurrently? If yes, what areas complement each other? Or is there no overlap to warrant concurrent study of the essentials? By essentials I mean what a college student must know, or really anyone who pursues self-education without a background in these areas. Beautiful website, by the way!
  • susanrigetti 5y ago
    Check out my physics guide: https://www.susanrigetti.com/physics https://www.susanrigetti.com/physics. It has both the physics core curriculum AND the math essentials you need to know in order to understand the physics essentials. (And thank you!)
• fjfaase 5y ago
  I am bit surprised there is nothing about graph theory in there. Also nothing about combinatorics or knot theory to mention two other subjects. If you want to make people dive into mathematics, it might be a good idea to show a broad range of subjects instead of focusing on the traditional subjects.
  • travisjungroth 5y ago
    It’s amazing how different the subjects of mathematics are. It’s like the difference between a drum and flute. You listed some of my favorite stuff. Weirdly, when I was 11, my math tutor told me I’d probably really like finite mathematics. She turned out to be right.
    • Someone 5y ago
      > It’s amazing how different the subjects of mathematics are. It’s like the difference between a drum and flute. I think it’s amazing how connected the fields are. It’s almost like “pick any two of analysis, algebra, geometry, number theory, topology, turn one into a adjective and you’ve got a new subject area”. Topological algebra? Check (https://en.wikipedia.org/wiki/Topological_algebra https://en.wikipedia.org/wiki/Topological_algebra) Algebraic topology? Check (https://en.wikipedia.org/wiki/Algebraic_topology https://en.wikipedia.org/wiki/Algebraic_topology). Geometric topology? Check (https://en.wikipedia.org/wiki/Geometric_topology https://en.wikipedia.org/wiki/Geometric_topology). Geometric algebra? Check (https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra) Algebraic geometry? Check (https://en.wikipedia.org/wiki/Algebraic_geometry https://en.wikipedia.org/wiki/Algebraic_geometry) Geometric number theory? Close (https://en.wikipedia.org/wiki/Geometry_of_numbers https://en.wikipedia.org/wiki/Geometry_of_numbers) Mix algebra, number theory, and topology, and you may end up with arithmetic topology (https://en.wikipedia.org/wiki/Arithmetic_topology https://en.wikipedia.org/wiki/Arithmetic_topology) And don’t confuse that with arithmetic geometry (https://en.wikipedia.org/wiki/Arithmetic_geometry https://en.wikipedia.org/wiki/Arithmetic_geometry)
      • sdenton4 5y ago
• pphysch 5y ago
  IME (as a math-degree-haver) the value of mathematics is in improving one's ability to mentally model and reason about complicated real-world phenomena. A lot of folks lose sight of the reality and get lost in the mysticism, especially within the academic regime. > [Mathematics] is the purest and most beautiful of all the intellectual disciplines. It is the universal language, both of human beings and of the universe itself. [...] That doesn’t mean it’s easy — no, mathematics is an incredibly challenging discipline, and there is nothing easy or straightforward about it I am always, always going to condemn this unnecessary mystification and idealization of mathematics. It's exclusive and misleading.
  • susanrigetti 5y ago
    You cut out the middle of that paragraph, which says: "Sadly, there is all sorts of baggage around learning it (at least in the US educational system) that is completely unnecessary and awful and prevents many people from experiencing the pure joy of mathematics. One of the lies I have heard so many people repeat is that everyone is either a “math person” or a "language person” — such a profoundly ignorant and damaging statement. Here is the truth: if you can understand the structure of literature, if you can understand the basic grammar of the English language or any other language, then you can understand the basics of the language of the universe." :)
    • pphysch 5y ago
      I'm not sure what your point is. Are you implying that you are not contributing to the mystification and idealization of mathematics? In other words, I do not see how you are dealing with the "baggage" of learning mathematics beyond name-dropping it. In my opinion, the mysticism is the baggage. And then the rest of the blogpost reads like a conventional curriculum within the conventional academic regime with which we associate that baggage.
      • dang 5y ago
        Please don't post in the cross-examining style. We want curious conversation here. This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html https://news.ycombinator.com/newsguidelines.html.
• tzs 5y ago
  Overall a pretty decent list, although I would suggest considering some tweaks. For real analysis it recommends as essential Abbott's "Understanding Analysis" and Rudin's "Principles of Mathematical Analysis". If you "haven't gotten your fill of real analysis" from those it recommends Spivak's "Calculus". I'd consider promoting Spivak to essential, but using it for calculus rather than real analysis, replacing their recommendation of Stewart's "Calculus: Early Transcendentals". By doing calculus with a more rigorous, proof-oriented introductory calculus book like Spivak, there is a good chance you won't need a separate introduction to proofs book so can drop the recommended Vellemen's "How to Prove It: A Structured Approach".
  • jeffreyrogers 5y ago
    I'll second this. "How to Prove It" gets recommended a lot, but I couldn't get through it. I found it terribly boring and unmotivated. Some people can power through dry material but I'm not one of them. I found it much easier to learn to write proofs when they were related to topics I was interested in.
    • jackthetab 5y ago
      I loved How to Prove It. Not for the proofs - which are interesting in a gazing-at-your-navel kinda way - but rather all the little _practical_ tidbits. "So THAT'S what a partially ordered set looks life in real life!" And the last(?) chapter where he uses induction to determine how to place an L-shaped figure on a grid...I never knew how to even approach that kinda' problem. So yeah, I want actual practical applications ("exercises" != "applications") for math. <climbs down from soapbox>
  • l33t2328 5y ago
    Spivak is a better analysis book than Abbot.
• da39a3ee 5y ago
  I am fairly confident that Susan Rigetti is a future president of the USA. In addition to becoming somewhat well-known as a household name early on in her adult life, she has achieved so many difficult and impressive things (publishing multiple books, studying physics and philosophy at graduate level, working for a top-tier tech company, taking down the CEO of a top-tier tech company and damaging the company's reputation, being asked to work for the New York Times, publishing curricula in graduate Physics, graduate Philosophy, and undergraduate mathematics). Furthermore, she seems to have a gift for or knack with the public eye.
  • rscho 5y ago
    POTUS is a very low bar...
  • triyambakam 5y ago
    I really wonder how some people can be so productive and prolific. In comparison I feel like an uneducated slob.
• brimble 5y ago
  Are there any "math for people who just want to use it" tracks in math pedagogy? I don't care a bit about proving any of it's true, or even reading others proofs of same. "Recognize which tool to apply, then apply tool", all focused on real-world use (so, yes, it wouldn't be "real" mathematics). That's the math education I'd like—try as I might, I just can't make myself care even a little about math for math's sake. I've got Mathematics for the Nonmathematician by Kline and that's kinda heading the right way, but what about whole courses of study? More books? It's more of an introduction than a thorough resource or course, and feels like it needs another four or five volumes and a lot more exercises to be really useful. I want a mathematics education designed for all those kids (likely a large majority?) who spent math from about junior high on wondering, aloud or to themselves, why the hell they were spending so much time learning all this. One that puts that question front and center and doesn't teach a single thing without answering it really well, first.
  • mjfl 5y ago
    The list presented here, calculus, odes + pdes, linear algebra, is essentially that, mathematics for people who just want to use it. It's all undergrad level. There's several layers on top of this - set theory, rigorous probability theory, algebraic geomeetry, topology, that are less useful but interesting to mathematicians.
  • importantbrian 5y ago
    I didn't take any of these classes personally, but I do remember in college there was a whole host of Calculus for Business and Economics type courses that were much more focused on practical application than theory. Maybe picking up some of those textbooks would be the way to go.
  • susanrigetti 5y ago
    If you have a solid background in calculus, I'd recommend Zill's Advanced Engineering Mathematics, which is pretty much basic math for physicists and engineers (aka for people who need to "use it").
    • 5y ago
• dekhn 5y ago
  I'm still stuck at "wait, sets can contain other sets, and sets can contain themselves?" part of Russell's paradox, and I'm close to retirement! I don't want to study math. I want to know enough of it to solve some well-understood problems I've wanted to solve for decades. Simply learning how to diagonalize a matrix (and how to use such a thing) meant more than understanding a bunch of complicated matrix theory.
  • moonchild 5y ago
    The solution to russel's paradox is that sets can not contain themselves. There is a carefully crafted set of rules describing what sorts of sets exist, and it is designed to avoid paradoxes such as that one. These rules are the ZFC axioms (https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html)
• paulpauper 5y ago
  You can get good or better at something with effort, but few will ever make to leap to being great or world class at it, no matter how hard they try.
  • atan2 5y ago
    True! But sometikes getting better at something is all people really want, and that's ok. I see that most of my CS students just want to be able to not see math as an obstacle when learning new/interesting things.
• Buttons840 5y ago
  Where's statistics? You mean to tell me I could go through all that and come out not knowing statistics?
  • Koshkin 5y ago
    Agree, but I have a feeling that statistics is more like (theoretical) physics, in the sense that it is "not math."
    • Buttons840 5y ago
      Yeah, it's more application oriented and philosophical than the pure calculation of pure math. I think it's under-taught in schools though. I think it's more useful than calculus for most people and should be taught before it.
• hintymad 5y ago
  I find it hard to believe that the author started to appreciate physics by reading The Feynman Lectures on Physics before any exposure to physics or even algebra, and in less than three years went from barely knowing high school math to enjoying advanced mathematical physics and graduate-level quantum physics. It looks this is one-in-a-million level brilliance as learning the sheer amount of requirement knowledge in such a short time is amazingly challenging: analysis, functional analysis, complex analysis, linear algebra, abstract algebra, differential equations, mathematical statistics, and all the physics: mechanics, electromagnetism, thermodynamics, optics, statistical mechanics, relativity, and of course quantum physics, all in less than three years. Kudos if the author is this talented.
  • whatshisface 5y ago
    I agree that this does not on its surface seem possible, but I can think of a few explanations. 1. I recently spent a week on one section of one chapter of a math book. I was able to follow it within an hour on the level of "these are the rules and this is the sequence of their application," but I have stuck with it since then because I wanted to understand it well enough that the proof they chose to use would seem obvious to me. If you saw "understanding math" like the peak of a mountain, you'd get there a lot more quickly, but if you want to try out every permutation of every device and condition anything can take forever. 2. Algebra seems simple in retrospect, and my teenage self was kind of dumb. Maybe with my complete adult brain I'd be able to finish highschool starting from scratch in a few months. Evidence to that point is the pacing of college remedial math classes. Maybe, to a certain extent, people have an innate math setpoint that they will snap to very quickly when given the chance. 3. Intelligence is equally distributed between genders, but most professional physicists are men, which means that for every professor there is almost exactly one corresponding woman who has equal potential but isn't in the system. If you heard that the department chair at a university sat down and read a book about topology without a lot of trouble you wouldn't be surprised at all. In other words, it's not surprising that someone can do this, it's surprising that someone who can do this is not in the social bucket for people that do it, but if you think about the other things you've heard about that, you realize you already knew. I am inclined towards #3 out of all these explanations but all may be true at once.
• tunesmith 5y ago
  How do you like to solve math problems in this day and age? I'm partial to Jupyter notebooks lately - I run it locally from a docker container, and have a folder of notebooks. Mostly markdown cells, alternating between my narrative thinking and LaTeX math output.
  • bnbond 5y ago
    I find paper and pencil works pretty well.
• pattt 5y ago
  Spivak’s Calculus reignited my interest and appreciation in math. Sad to discover the author passed away quite recently. The way of explaining principles and making you do the hard work via problems which I believe is a must with this book, is profoundly astonishing. There’s a lot of mathematical insight packed into those problems, it almost feels you can build up the entire high school and the early uni curriculum from the ground up, for instance there are a number of popular formulas you’d arrive at and derive accidentally while working on those problems. Furthermore it really works your brains by making sure you can reason within the established framework and exercise great doubt. I’m taking this book very slowly.
• itcrowd 5y ago
  Susan, I greatly appreciate this list and will definitely come back to use it as a reference if I need a book recommendation. (I don't think I'm the target audience, although who knows what the future brings..) That being said, I think you are missing out on an opportunity to reach a wider audience. It bugs me a bit that the requirements seem very American-centric. What I mean is the following bit: > A high school education — which should include pre-algebra, algebra 1, geometry, algebra 2, and trigonometry — is sufficient. And later the paragraph on "pre-calculus". I know that many places don't have such names for courses in high school. In fact, often it's just called "Mathematics" and you either take it or you don't (obviously there is a spectrum here). How is a prospective (non-American) student to know what is covered in Algebra 2 in an American high school? I'm not asking you to change the article, I just hope I can nudge you into realizing that the text as it is now is more difficult than it needs to be for non-Americans.
  • jerry1979 5y ago
    Do we have good universal descriptors for math levels? I'm a big fan of accessibility, and I think your idea about tweaking language to reach a wider audience could be a big win for increasing the article's impact. To update the article to include your recommendations, the author would probably need some kind of "cross-walk" which would map the American perspective to a more universally understood framework. Would you happen to know what "pre-calculus's" opposite number would be in the universal framework?
• rongenre 5y ago
  I have a decades-old math degree and ended up working in tech as an engineer. Are there options, like a "Math Camp for the Middle-aged" where I could get a chance to re-learn everything I've forgotten?
• nyc111 5y ago
  I don't agree with this article, it as off-putting as the usual math eduacation it criticizes. I wonder how one can propose a curriculum to study math and not mention Euclid. One learns more mathematics from this article https://mathshistory.st-andrews.ac.uk/Extras/Russell_Euclid/ https://mathshistory.st-andrews.ac.uk/Extras/Russell_Euclid/ by B. Russell where he harshly criticizes Euclid than 2 years of calculus. Newton did not know calculus but he knew Euclid's Book 5, the book about ratios and proportions. Euclid's 5th Book must be the starting point for the study of math. When we say "math is the language of nature" we really mean that nature is proportional. Ratios and proportions are fundamental.
• deleted 5y ago
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• graycat 5y ago
  Calculus: I suggest just forget about "precalculus" and, instead, just get a good calculus book and dig in. There are two main parts of calculus, and both can be well illustrated by driving a car. In the first part, we take the data on the odometer and from that construct the data on the speedometer. The speedometer values are called the (first) derivative of the odometer values. In the second part we take the speedometer values and construct the odometer values. The odometer values are the integral of the speedometer values. In notation, let t denote time measured in, say, seconds, and d(t) the distance, odometer value, at time t. Let s(t) be the speed at time t. Then in calculus s(t) = d'(t) = d/dt d(t) And d(t) is the integral of speed s(t) from time t = 0 to its present time. Those are the basics. Applications are all over physics, engineering, and the STEM fields. Linear Algebra: The subject starts with a system of simultaneous linear equation. The property linearity is fundamental, a pillar of math and its applications. The STEM fields are awash in linearity. E.g., a concert hall performs a linear operation on the sound of the orchestra. E.g., in calculus, both differentiation and integration are linear. In the STEM fields, when a system is not linear, often our first step is to make an attack via a linear approximation. E.g., perpendicular projection onto a plane is a linear operator and the main idea in regression analysis curve fitting in statistics. Most of math can be given simple intuitive explanations such as above.
• potbelly83 5y ago
  As a math PhD I have to say the only way you're going to learn mathematics is if you actually have a pressing need to do so. i.e. You have a project at work that needs some math, you have a hobby that needs some math. In this case you just learn what you need. Just learning math for its own sake outside of a University STEM track is just too hard (I wouldn't be able to do it and I've tried).
  • zinclozenge 5y ago
    This is true. When I started my first job, I tried casually learning more from where I stopped after I finished school. My motivation slowly waned as I realized I vastly preferred playing video games than watching lectures and trying to do some problem sets.
  • exdsq 5y ago
    Correct me if I’m wrong, but I assume you’re looking at this through the wrong lens. As a PhD you’ve learnt something hard at a considerable depth but this isn’t what I, or most people, hope to achieve by learning maths themselves. It’s normally to a far lower depth that’s far more achievable like Calculus I or some aspects of number theory or knowing what all the funky symbols such as Summation mean.
    • bitexploder 5y ago
      I just want to do basic engineering and physics. Understand some of the higher level models we have to describe our world.
  • JimTheMan 5y ago
    I think the question needs to be asked, "What am I hoping to get from this?" If its tools to help you solve some problem, great. If its because you find it fun, great. But if its because you feel you want to 'better yourself' or perhaps feel like it would somehow prove your intellectual worth, you probably won't get a lot out of it.
  • treeman79 5y ago
    Was programming a 3-axis machine in early college back in 90s. After a few months I was mostly re-inventing trigonometry. The I actually took trig later on. That would have been super handy at time.
• RoddaWallPro 5y ago
  I read a Murakami novel in high school, 1Q84. The protagonist is a math teacher who talked about math in a way that I had never seen before. I'd been told I was "good at math" beforehand(for whatever that means, I'm not a fields medalist or anything), but for ~6 months after reading that book, I was _really good_. Like, suddenly I did not have to do any homework in my sr. year calculus class. I loved sitting in class and watching my teacher work through problems, and it seemingly imprinted directly into my brain, because while doing no homework I could still ace the exams while writing with a pen (no erasing and re-do'ing with a pencil). All because of the way this fictional teacher from 1Q84 talked about math. Has anyone else had an experience like that? (With math or other things?)
  • jackthetab 5y ago
    C'mon! With a build-up like that and you don't mention the name of the book?! Don't leave us hanging, dude!
    • strikelaserclaw 5y ago
      its literally called 1Q84
      • jackthetab 5y ago
        Oh! I thought that was when he read it, the first quarter of 1984. Off to buy another book...
  • openfuture 5y ago
    I'm doing a MSc in mathematics. I tried to read that book once and I thought it was terrible. One of the few books I dropped before finishing it and also the reason why I have been reluctant to pick up other Murakami books. With respect to your phenomena of leveling up your mathematical maturity without doing mathematics, that has happened to me a few times and I would say yoga and psychadelics are the primary source of inspiration. Of course you need to do a lot of reading and studying (or if you have courses then you can sit back and have them feed it to you). One thing is making the information known and the other is to understand it, this is where the things that can spark random enlightenments come in.
• mathgenius 5y ago
  Modern calculus (analysis) was invented because people shot themselves in the foot working with topology and wondering exactly what is a "curve" ? I am a big fan of this approach to learning mathematics, just forge ahead and when (if) things fall apart then go back and fix up the foundations. To this end I recommend a couple of books. "The Knot Book" by Adams is a very interesting exploration in topology (without requiring all the years of study at university before you are allowed to learn exactly what a topology is). And in another direction, group theory was invented because the study of symmetry gets very tricky! But if you want to dive in anyway then have a look at Conway's "The symmetries of things". It is a lot of fun. Most modern group theory (or algebra) books don't actually have any pictures of symmetric things, just endless formulas and lemmas. If you want to be a pro, then you gotta learn that stuff, but there's definitely pathways into higher mathematics that don't require you to learn that.
  • PartiallyTyped 5y ago
    Speaking of group theory, I can recommend "A book of abstract algebra". I think that it's a very approachable introduction to the topic. As a person with a CS degree doing ML, it changed my perspective on so many different topics, I can't recommend it enough. https://www.goodreads.com/book/show/8295305-a-book-of-abstract-algebra https://www.goodreads.com/book/show/8295305-a-book-of-abstra...
• cathrach 5y ago
  While I understand that the author has good intentions, I strongly disagree with the general idea of this post, which is that anyone can learn math through an almost entirely analysis-focused curriculum while other topics like topology, game theory, set theory, etc. are presented as advanced and graduate-level. This is practically equivalent to saying that anyone can learn history, and they should learn all about British history in undergrad, and then graduate-level courses might teach you more about the history of South America. Some of my thoughts (mostly drawn from personal experience, feel free to disagree): 1. IMO "learning math" is really about learning how to recognize patterns and how to generalize those patterns into useful abstractions (sometimes an infinite tower of such abstractions!). So it really doesn't matter if one does abstract algebra or linear algebra or combinatorics or number theory or 2D geometry or whatnot at the beginning. Any foundational course in any branch of mathematics, or any book on proofs, will fulfill this need. People learn in different ways and have affinities for different topics, so some subjects will be easier and/or more interesting for them, so aspiring mathematicians should start with a topic they're at least initially entertained by. If you don't know where to start, one fun (for me) topic is the game of Nim; other combinatorics topics are also elementary and entertaining to think about. I'm fairly sure that if I had to take this suggested curriculum as an undergraduate, I would have picked a different major entirely, I personally find analysis quite difficult :( 2. One's first foray into a topic should be a one-semester course, not a textbook. Lecture notes for many courses are freely available online also, so you don't have to pirate the books you want if you aren't willing to pay $100 :P The reason is this: courses are curated by a mathematician to teach students the basics of a topic in one semester, so they will better highlight what you need to know, like important theorems, and have a more careful selection of problems. If you're confused, you can read the relevant textbook chapters. On the other hand textbooks are more like comprehensive references - reading a textbook through and doing all the problems will make you an expert at the material, but it's not as time-efficient (or interesting) as a course. 3. There are benefits to diving very deeply into a topic, but IMO one's mathematical experience is much richer if there's more consideration for breadth, especially when you're starting out. A student learning basic real analysis would benefit from understanding some point-set topology (not just the metric topology that usually begins these courses) and seeing how (some of the) pathologies of topological spaces disappear when you impose a metric and then you get things like being Hausdorff or having many different definitions of compactness coincide. After learning real and complex, of course one could move onto differential equations, but there are so many other ways to branch out, like exploring differential topology or learning about measures & other forms of integration, which also meshes very nicely with statistics. Exploring different branches emphasizes that there are so many directions you can go with math, even when you're just starting out, and gives you a better feel about how "math" is done, as opposed to just the techniques for a specific topic. This is my first comment on HN, so please let me know how I can improve this comment!
• philomathdan 5y ago
  The curriculum guides Susan Rigetti provides are an amazing resource for self-study. And the fact that she worked through all of this is truly inspiring. Not to be greedy, but do any of you know of other thorough curriculum guides like this? I know about https://teachyourselfcs.com https://teachyourselfcs.com already -- another amazing guide. Are there others? I would love to find one for statistics especially, but really any subject would be interesting.
  • Jun8 5y ago
    Susskind’s Theoretical Minimum is fantastic for physics self study: https://theoreticalminimum.com/ https://theoreticalminimum.com/
    • philomathdan 5y ago
      Another good one. Thanks!
  • bgroat 5y ago
    I posted recently asking for exactly this... but for medicine. I have a body, everyone I know has a body. The operate pretty much the same everywhere in the world. I would like to know how it works.
    • susanrigetti 5y ago
      I want this too. Let me know if you find anything!!
      • rg111 5y ago
        You could try these: 1. How the Body Works from DK. 2. The Machinery of Life by Goodsell. Check them out. DK books are great to get introduced to something new- get a lay of the land, learn introductory jargons, etc.
• abhisuri97 5y ago
  As a fellow penn alum, I can totally vouch for Ghrist's approach to calculus. Check out his youtube channel: https://www.youtube.com/c/ProfGhristMath https://www.youtube.com/c/ProfGhristMath
• ouid 5y ago
  If you actually want to study math, you probably shouldn't touch calculus until you've take linear algebra and a fair amount of topology, since these are the two structures on sets that (differential) calculus is founded upon. For other subjects, you can briefly substitute an intuition for the underlying structures with sufficient finesse in the presentation of the material (see the theory of knots and links, for an example), but calculus is not, in my experience, such a subject, and the early emphasis on it is harmful for the study of mathematics, which is supposedly what your list is for. For some reason this is heresy, but I have honestly no idea how you are supposed to appreciate calculus from a mathematical perspective without being able to define the large stack of terms that constitute it. The situation is potentially different for a physicist, but if you want to study mathematics, the physical world is not the object of study, rather it is precisely the definitions that we have chosen.
  • susanrigetti 5y ago
    That’s what the real analysis course is usually for, and why it comes after linear algebra and algebra.
    • ouid 5y ago
      It also comes after many semesters of calculus, which depend upon it, and before any topology, which it depends on. Even if you are just interested in these things as tools for describing physical phenomena, there is value in placing mathematical knowledge in its natural structure.
• paulpauper 5y ago
  You don't need so many books. many of the old texts will cover many important college-level concepts in a single source. https://www.gwern.net/docs/statistics/1957-feller-anintroductiontoprobabilitytheoryanditsapplications.pdf https://www.gwern.net/docs/statistics/1957-feller-anintroduc... this is linear algebra + combinatorics + probability + stats If you understand the material in this one book it's reasonable to say that you are pretty good at math
  • ChrisLomont 5y ago
    At a first glance that book completely fails for an undergrad lin alg course, and looks weak in other areas too. Examples: the words nullity and kernel don't appear, rank of a matrix is not in it, and, well, every topic I can think of for undergrad linear algebra is simply not in the book. It's equally bad for stats: no mention of many common distributions a student would learn for example.
    • wizzwizz4 5y ago
      Is that such a bad thing? I'm half-convinced that the “rank of a matrix” is just an artefact of a particular algorithm for inverting matrices. And distributions aren't everything; I can look up any distribution I want on Wikipedia, just as soon as I need it, but a proper foundation in what statistics means is much harder to come by. (I have enough of a foundation to know when it's being taught very wrong, but not enough to actually be very useful in day-to-day life.)
      • ChrisLomont 5y ago
        >I'm half-convinced that the “rank of a matrix” is just an artefact ... The rank of a matrix is fundamental to understanding linear transformations, since it given you knowledge about the dimensions of the "output space". It becomes more fundamental if the person goes on to study deeper math. It tells you how to compute the size of a basis for the target space. The uses go on and on. >And distributions aren't everything; I can look up any distribution I want on Wikipedia Yes, you can look up anything on wikipedia, but not in this book, which is why this book will not teach you the things the OP claimed it would. >a proper foundation in what statistics means is much harder to come by It's very hard to get that proper foundation from a book that does not cover those foundations. One is better served by using a proper book with a consistent and well laid out presentation of the needed concepts. Saying one can look fundamental stuff up elsewhere means the book is lacking.
• LAC-Tech 5y ago
  I really don't. For many, many years I thought I did. I'd have a brief surge of interest for a few weeks, and then get completely bored of it. I'm not someone who finds it inherently easy, so boredom + difficulty = failure. When I was foolish enough to do this in university, it meant doing great in the first few assignments, and then abysmally in the exam. So my policy now is to never study maths for its own sake. Only when there's equations in a computer science paper I don't understand.
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• dr_dshiv 5y ago
  If doing math is essential to conceptual understanding and application, could the interface of math and physics be made more human-centered? For instance, the shift from Roman numerals to Arabic numerals made doing math easier. Based on your experience, might it be possible to increase accessibility by revising some of the arcane conventions of math and physics? See Brett Victor’s 2011 proposal: http://worrydream.com/KillMath/ http://worrydream.com/KillMath/
  • andrepd 5y ago
    Why is it that every time any subject about mathematics comes up there is always a complaint about notation? Your link doesn't even exactly talk about notation, but about pedagogy. Can you be more specific about which notation your consider "arcane"?
    • BeetleB 5y ago
      > Why is it that every time any subject about mathematics comes up there is always a complaint about notation? The assumption that there is a much better notation is one I tend to see only with the HN crowd. Outside of this group, even people who dislike the notation and/or struggle with math do not claim that a better/simpler one obviously exists.
      • openfuture 5y ago
        Yep, blaming your tools is what the incompetent do.
        • dr_dshiv 5y ago
          Blaming the user is also what the incompetent do.
      • boppo1 5y ago
        I really only think calculus notation is an issue. Calc 1 books absolutely demand that you view d/dx as an operator and not a ratio, likewise with the dummy variable and integration symbol. Then later the book teaches about unit normals and it’s implicitly acceptable and treat dx/dz and dy/dz as ratios. i.e. |u X v| = (1 + (dx/dz)^2 + (dy/dz)^2)^(1/2) = (dz^2 + dx^2 + dy^2)^(1/2). This was extremely frustrating to me for a while until I accepted that this was how Leibniz did it, so if it’s good enough for him it’s good enough for me.
• foobarbecue 5y ago
  "... but make sure you get the paperback or hardcover version for readability purposes." As opposed to... the ebook?
  • susanrigetti 5y ago
    Yeah, iirc the ebook formatting made the book difficult to read.
• C-x_C-f 5y ago
  The website mentions some good courses, personally I love Richard Borcherds' YouTube channel[1] for both undergraduate and graduate courses. No frills, exceptionally clear, (mostly) bite-sized lectures that cover a good range of material (especially in geometry). Something that might interest HN's demographic is Kevin Buzzard's Xena Project[2], centered around proof systems (in Lean). The natural numbers game [3] is particularly fun IMHO. I don't know if it counts as learning materials per se but it's certainly instructive. [1] https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/playlists https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ/pla... [2] https://xenaproject.wordpress.com/ https://xenaproject.wordpress.com/ [3] https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/ https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...
• wanderingmind 5y ago
  No recommendation on probability. Thats strange given that the author is a physicist and fundamentals of modern physics rests on probability. My recommendation is the classic "Probability Theory, The logic of Science by E.T.Jaynes" which is a Bayesian formulation.
  • BeetleB 5y ago
    You may be surprised to discover that many/most physics undergrad curricula do not mandate a course on probability.
• jdkee 5y ago
  I would personally recommend The Princeton Companion to Mathematics as an excellent introduction to mathematics from someone no pursuing a degree. See https://press.princeton.edu/books/hardcover/9780691118802/the-princeton-companion-to-mathematics https://press.princeton.edu/books/hardcover/9780691118802/th...
• EoinB 5y ago
  As a mathematics major, I find it encouraging to see non-mathematicians sing the praises of the subject. So I applaud that. But I think the author is overstating/slightly wrong about a few things, perhaps because her exposure to mathematics has been through the lens of physics. Maybe a more humble rewording of some of her statements e.g., "Anyone that follows and completes this curriculum will walk away with the knowledge equivalent to an undergraduate degree in mathematics." would be helpful. Her suggested curriculum doesn't include anything from Number Theory, which is a foundational part of an advanced mathematics education. It is also one of, if not the most, beautiful topics one can study in mathematics. I find it odd to call out "Introduction to Proofs" as a topic in and of itself. Proofs aren't really a topic in the way analysis or number theory is. At advanced levels, devising theorems and theirs proofs is what mathematics is.
• polypodiopsi 5y ago
  Wow the philosophy guide is super narrowminded.
• untake 5y ago
  Wrong: math is neither hard nor difficult. It is this single belief that deters many people from learning it. Math is all about logic. It is nothing but how to go from A to B. Any person who can reason should be able to be good at math.
  • eternityforest 5y ago
    All over the world many people try to learn French. Almost any would tell you that learning a second language is hard. Nobody will fight you on this. Math is a language to express entirely new concepts thar have nothing to do with everyday thinking and often include things like recursion that makes them impossible to reason about without more than just a few bytes of mental RAM. There is no step by step algorithmic process to do it. If there were, a computer would be able to do it and far fewer people would want to learn. Even at the level where there is an algorithm, it's far too big, nobody hand executes the source code of XCas by hand. Math seems to require not just a skill that can be learned, but an inate ability to deal with multiple connected pieces of information at once, and to see abstract patterns in things. In programming, if you have to understand more than one tiny bit at a time, you might consider tossing it and starting over. In math it's just normal for lots of ideas to connect.
• fn-mote 5y ago
  I question just how realistic is is to have "Proofs from the Book" at the start. While the art is wonderful, the background required to read it is not realistically at a lower level than the textbooks listed. The discussion here has been much more interesting than the actual list, to me. If you wanted to master all of that material, I think a master's program is the way to go, not self-study. I would be interested in hearing from people who _do_ successfully self-study. What makes it work? The least interesting (from my point of view) is "already successful in a related field, applied my skills". That would include CS professionals studying maths, I think. More interesting would be "unable to attend university because of X, did Y and really enjoyed it." Are you a person who completes MOOCs and get something out of them?
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• tmabraham 5y ago
  I love that the author highlighted Prof. Robert Ghrist's great material. I took his calculus classes through Coursera maybe about 8-9 years ago. He just makes everything exciting and his visuals are just beautiful. For example, within the first few lectures, he made it feel like Taylor series was like the coolest thing ever. Highly recommend checking his lectures out. Check out his website: https://www2.math.upenn.edu/~ghrist/ https://www2.math.upenn.edu/~ghrist/
• rotifer 5y ago
  To the section "Popular Math Books" I would add almost anything by Julian Havil. John D. Cook referred to him as a writer of "serious recreational mathematics" [1]. I would probably put his "Nonplussed!" and "Impossible?" books in the "Level: Easy" group, with the others at least in "Level: Medium". "Gamma" is one of my favorite serious recreational math books. If you like getting into the nitty-gritty of problem solving then check out the books of Paul Nahin. They vary between "Level: Medium" and "Level: Difficult", with many of them reveling in the solution of equations, and integrals in particular. Although he recognizes the need for proofs, he makes a point of avoiding them in his books. [1] https://www.johndcook.com/blog/2019/09/29/a-sort-of-mathematical-quine/ https://www.johndcook.com/blog/2019/09/29/a-sort-of-mathemat...
• lmilcin 5y ago
  I studied theoretical math. Step #0: make sure you know what math is I can't stress it enough. I know this sounds funny, but when I went to study math a lot of people would drop out very quickly because they did not realise that what most people call math and what is taught at high school is completely different from actual theoretical math. What most people think math is is a collection of formulas that you need to learn to "know math". Math is actually a dynamic activity and is 100% about solving problems. Just like programming is not about knowing programming languages. Programming is about solving problems (and knowing programming languages is necessary but not sufficient to be programming).
• frostblade 5y ago
  Setting aside the question of motivation, how would a working adult find the time to work through all these books? The author of this clearly leads a busy life outside of reading math textbooks. Reading a math textbook is time consuming endeavor, regardless of underlying ability. The author herself mentions this in the introduction. I can think of a few factors that might make it possible for a busy person to go through all these books in a few years: - They include books that were read partially while taking course. - Consistency: allocating 1 - 2 hours per day for a few years. - Doing exercises selectively: If you only do a handful of exercises per chapter this would dramatically increase the rate at which you go through a book. This would come at the cost of deeper understanding. I have a large backlog of math books I'd like to read, but time is a constraining factor. If people have found strategies for reading these types of books, I'd like to hear about them.
  • susanrigetti 5y ago
    For me, it’s all about consistency. It’s spending a few minutes to a few hours every day, forever, til the day I die. I usually can squeeze in 30 minutes to an hour every day to study something (whether it’s math or something else — right now I’m studying cinematography). Sometimes that’s in 15-minute chunks if it’s a busy day. Usually it’s before bed or while I’m eating lunch or if I have extra time on the weekends while my kids are napping. It’s all about just doing a little bit every day. That’s been successful for me.
• arisbe__ 5y ago
  Here is the best mathematics course on YouTube: Introduction to Higher Mathematics (Bill Shillito) - https://youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQBQ8TSWVX https://youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6KzmkUQ...
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• dmitrybrant 5y ago
  The author doesn't seem to take her own brilliance into account when composing these self-study guides. For the other 99.99% of us, it would take many lifetimes worth of free time to make a substantial dent in these materials. To me, guides like these are too intimidating to even begin. Maybe what's needed is a meta-guide on structuring one's time and developing the necessary focus to be able to do this within one human lifetime.
  • susanrigetti 5y ago
    I will tell you the secret: just jump in. Grab a pen and paper, open up the first book in any of the guides, and start reading. Read for 15 minutes, or 20 minutes, or whatever time you have on your lunch break or before you go to bed or while you’re using the bathroom. Do it again the next day. And the next. And the next. That’s how I did it. There’s no brilliance involved. It’s just jumping in. The more you do, the easier it gets.
    • __rito__ 5y ago
      Thank you for writing this. I knew consistency was important, and did not place much importance on "brilliance", although that attribute has been showered upon me since I was little. What I did not know, and still don't know is that studying only for a few minutes or half an hour regularly will make me good at something as advanced math. You seem to know your stuff and I like your approach and attitude, and I will do now what I do very rarely and upon serious consideration- take a leap of faith. I will start doing math everyday for at least half an hour, and I will see how that goes. I will let you know after a few months.
• baggy_trough 5y ago
  It's a crime not to mention Gödel, Escher, Bach.
• senderista 5y ago
  I don't think most math undergrads (as opposed to engineering or physics students) take a PDE course.
• gprs 5y ago
  whats the point of learning them, i studied them in undergrad and i dont even use them anymore
• sydthrowaway 5y ago
  Unfortunately math wont make you smarter in other fields Only reading does
• trosi 5y ago
  It's true that a standard undergraduate curriculum in mathematics will contain a lot of analysis (including calculus), a lot of ODEs/PDEs and some algebra. But if I wanted to get someone interested in math I would also point them towards some of the more fun (and less known stuff), such as combinatorics, probability, topology, differential geometry and number theory. Some of these are also much more applicable today than, say, complex analysis.
• igravious 5y ago
  Not to be too negative but I wouldn't have super high hopes for this if it's anything like, “So you want to study philosophy”[1][2] [1] website :- https://www.susanrigetti.com/philosophy https://www.susanrigetti.com/philosophy [2] discussion :- https://news.ycombinator.com/item?id=28367416 https://news.ycombinator.com/item?id=28367416 A shame I missed out on that discussion.