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Mathematics for the Adventurous Self-Learner

by nsainsbury on 2/23/20, 11:39 PM with 210 comments

  • by gavinray on 2/24/20, 2:36 AM

    I know this is going to be the case for likely nobody, but I have browsed most of the self-study math threads that pop up here as a forever-on-my-todo-list thing and I have a remark to make:

    I have yet to find a guide that does not start with the assumption that you graduated highschool.

    That is a very reasonable assumption to make. We are in a community of technology and engineering, it would be a bit ridiculous to assume the people you are surrounded by did not have a fundamental base of mathematics.

    But the times I have tried to go through these teach-yourself materials, it went from zero to draw-the-rest-of-the-fucking-owl real quick. [0]

    I have been programming for 14 years, but stopped doing schoolwork around age 12, and never did any math beyond pre-algebra.

    Does anyone know of materials for adults that cover pre-algebra -> algebra -> geometry -> trigonometry -> linear algebra -> statistics -> calculus? At a reasonably quick pace that someone with a family + overtime startup hours could still benefit from?

    [0] https://i.imgur.com/RadSf.jpg

    (Also, curse the Greeks for not using more idiomatic variables. ∑ would never pass code review, what an entirely unreadable identifier)

  • by laichzeit0 on 2/24/20, 4:22 AM

    I pretty much followed the same route as OP re-studying mathematics seriously after 10 years in industry after initially doing a CS degree and doing mostly software engineering but transitioning into Data Science the last 3 years. When I saw Book of Proof then Spivak then Apostol on his list I chuckled because that’s exactly the route I ended up following as well. Studying from 04:30 to 06:30 in the week and about 8 hours split up over the weekend, Spivak took 8 months to complete (excluding some of the appendix chapters) but if you can force yourself to truly master the exercises - and Spivak’s value is the exercises - then you’re close to having that weird state called “mathematical maturity” or at least an intuition as to what that means. You can forget about doing the starred exercises, unless you’re gifted. Spend a lot of time on the first few chapters (again, the exercises), it will pay off later in the book. It was a very frustrating experience and I had so much self doubt working through it, it’s an absolutely brutal book. Some exercises will take you literally hours to try and figure out.

    If you do Book of Proof first you will find Spivak much easier, since Spivak is very light on using set theoretic definitions of things. Even the way he defines a function pretty much avoids using set terminology. Book of Proof on the other hand slowly builds up everything through set theory. It was like learning assembly language, then going to a high level language (Spivak) and I could reason about what’s going on “under the hood”. Book of Proof is such a beautiful book, I wish I had something like it in high school, mathematics would have just made sense if I had that one book.

    I read a quote somewhere, think it was Von Neumann that said, you never really understand mathematics, you just get used to it. Keep that in mind.

  • by melling on 2/24/20, 2:40 AM

  • by deostroll on 2/24/20, 5:56 AM

    Initially I heard about Euler's famous Basel problem. Years later I got to solving it for my self (for curiosity and fun). I guess what intrigued me was to think of trigonometric sine as an infinite polynomial...After I worked it out, I had indeed seen the fire in Euler's own eyes...I could see how excited he was at having discovered something amazing...But this got me into hooked into math history. What I really wanted was how people came about discovering the Taylor's series...the intuition behind it. So that is how I came across John Stillwell's book. I have to warn people it is rather academic. But if, you, as a self-learner, is excited about mathematics, I would suggest Norman J Wildberger's youtube lectures on mathematics history. I find the buildup to calculus quite fascinating. J. Stillwell's book was the recommended reference in those lectures...

  • by hackernews7643 on 2/24/20, 1:27 AM

    One thing I don’t think is discussed enough is the process of how self-learners in math get critical feedback. Most advanced level math textbooks do not have solutions to check their work against nor do they have a way to get feedback by an expert and this is essential for learning. Least with programming, you can get immediate feedback and know whether what you did is correct or not.

  • by daxfohl on 2/24/20, 3:24 AM

    This is so difficult. I've been doing it off and on for twenty years and not made much of a dent in things.

    The hardest part I think is understanding and measuring your progress. In school you've got exams and classmates to compare against, profs to talk to. Alone it's much harder. "Do I understand this well enough?" "Did I do the problems right?" (Especially with proof problems, how do you know you're right?). "I can work through some problems one by one, but it feels like something fundamental I'm missing. Am I, or is this chapter really just about some tools?"

    Then it's way too easy to say well I'm never actually going to use any of this so why am I doing it ... and take a few months off and come back forgetting what you'd learned.

  • by integerclub on 2/24/20, 4:52 AM

    For all the adventurous self-learners out here, we would like to invite you to our self-study group named Integer Club.

    IRC: https://webchat.freenode.net/#integerclub

    Slack: https://bit.ly/integerclubslackinvite

    Mailing list: https://groups.google.com/d/forum/integerclub

    We pick up old concepts from popular textbooks and literature as well as new stuff from new literature in both mathematics and computer science. We plan to have online meetings periodically to share what we learn, work through popular literature, and have a few talks on interesting topics.

    It is a tiny community right now that hangs out at Freenode IRC but the Slack channel is there too if you are more comfortable with that.

  • by dorchadas on 2/24/20, 4:05 AM

    I think it's great that people are posting book links like this, however, what I've found most helpful is actually having someone to help guide you.

    I realize how lucky I was that I found a Discord server ran by a math PhD graduate who is willing to help us guide our learning. From this, I've started learning Algebra and Analysis (just starting with the latter). It's nice to have someone to discuss problems with when you get stuck and to guide you. Likewise, he can suggest exactly which problems I should do for a give chapter, that way I don't spend my time doing ones that just repeat the same simple things over and over and can focus on nice, conceptual ones. So, if you can, please try to find someone to help guide you, or be that guide for someone else! Having it has made me seriously consider going back for a mathematics masters (and maybe PhD), switching from my physics background.

  • by wyqydsyq on 2/24/20, 6:34 AM

    As someone who dropped out of highscool after 10th grade and never went to university/college one great way I've found for learning mathematics without any foundational basis is trying to learn CG/3D programming.

    I always felt like maths was too abstract to keep me engaged, but when the output of your work is immediately observable visually it becomes a lot more engaging. There's just something so much more satisfying being able to "see" the results.

    Plus as a self-taught programmer, I find it much easier to learn front-to-back by deciding on a desired outcome and working towards it, rather than progressively building up abstract fundamental skills that can later be combined to achieve a desired outcome (which is essentially the traditional academia path for learning STEM fields)

  • by zerubeus on 2/24/20, 3:00 PM

    I came to the IT industry after a bachelor degree in math, 5 years in and all the math I know is gone I still remember some Fourier, signal processing and probability statistics that I never used in my day job, or anywhere else.

    Time is valuable, it's the most valuable thing a human being has, I understand it's the hobby of OP to learn all this math, but unless you are going to use it why wasting all the time?

  • by tildedave on 2/24/20, 2:07 PM

    I've been pursuing mathematics as a hobby for the last 2 years or so. I got a mathematics major in undergrad so my motivating factor was mainly to explore some areas that I hadn't done coursework on, primarily algebra and number theory. (I focused more on logic in undergraduate/grad.)

    I really enjoy how the subject is divorced from a lot of the modern attention demands and encourages more of a 'zen' thinking style.

    As others have highlighted, it can be difficult. I work full-time as a software engineer and at the end of the day there's usually not much left in the tank in terms of "creative work". The morning is usually more productive for me - generally I'll spend 10-15 minutes on the commute in reading over the proof of some lemma or working through some computational exercise.

    Things that have helped me:

    - Focusing on a particular problem area rather than just "mathematics". The classical problems of Gauss and Euler tend to be more my speed than the modern mathematical problems of Hilbert or beyond. What started my journey was looking into the insolubility of the general quintic polynomial equation, something you learn in high school as a random factoid but has a lot of depth.

    - Studying from small textbooks that I can fit in a backpack, so I can "make progress" during my commute. Dummit + Foote might be a great algebra reference but it's just too bulky to transport.

    - Limiting the scope of how I think about the activity - my goal isn't to master these concepts on the level of a mathematics graduate student, it's more on the order of Sudoku. If I don't get something, that's okay. People spend their whole lifetimes learning this material and I'm just trying to fit this into whatever creative time I have left after the full-time job is done.

  • by kevstev on 2/24/20, 5:30 PM

    Do any of you all have some tips for understanding mathematical notation? I feel this is often poorly explained, and it feels like a language all its own that just does not speak to me. I did pretty well in calculus, but I still don't really understand what the dx was supposed to represent and in reality I was just really good at pattern matching when it wasn't supposed to be there anymore.

    I try to read papers now and again with a math orientation, and I quickly get lost when trying to translate the concepts into cryptic formulas, and often when they make the "obvious" transition from step 3 to step 4 I just have no idea how they got there.

    I feel this is by far my biggest barrier to understanding most mathematics, and I have thus far found no way to overcome it.

  • by angry_octet on 2/24/20, 8:14 AM

    The most key piece of advice is to take walks. Walking is essentials for mathematics. Many times when walking with my father he would turn for home and start walking faster, and by that sign I knew that he wanted to get home and write down a lemma.

  • by jcurbo on 2/24/20, 4:40 PM

    This is a solid read, with good book recommendations. After several years of tinkering with self-learning I bit the bullet and applied to a MSc in Applied Math program. (via ep.jhu.edu) I've had to take some pre-reqs to get started since it's been almost 20 years since I have been in a college math class, but it's been an enlightening journey re-learning calculus and now dipping my toes into differential equations. I don't think I could have gotten this far with self-learning, but I realize YMMV.

    I will say I don't feel like single-variable real number calculus tells the whole story. I had taken that and linear algebra in undergrad but never any further, and now that I've taken single and multiple variable calculus, with real and complex numbers, plus integration of linear algebra ideas, the mathematical model feels a lot more like a cohesive whole to me, highlighting fundamental ideas that only barely peek through in a typical Calculus I class. I would encourage anyone talking to calculus to at least do the typical Calc II class, if not Calc III/multivariate. There is a beauty and structure to building up from calc I through III that I was missing before.

  • by peatfreak on 2/24/20, 5:43 AM

    I'm pretty skeptical about these "best of" lists of books for self-directed mathematics education.

    I have my own "best of" list that is very different to this list, although there are a couple of crossovers.

    If you are fortunate enough to have access to a university library (or libraries) I would _highly_ recommend inquiring about access to their general collection. I was also fortunate enough to study mathematics to a university-level three-year degree at a research university. So I had an excellent head start.

    A HUGE part of my journey of collecting my "perfect library" of mathematics self-tuition and reference books (and course books) was to do my own research on collecting the perfect titles. I started when I was in the early days of my mathematics degree and I used resources like Amazon, Usenet, libraries (already mentioned), and ... that was about it.

    Another important question to ask yourself is the following:

    "Why am I doing this?"

    Life is short and by the time you hit middle age, if you have a family or bills to looks after, are you REALLY going to want to lock yourself away in your study room to learn Lebesgue integration instead of focusing on the rest of your life?

    Consider that people fail to emphasise is that mathematics is a social activity much more than many people realize.

    Exercise: Find the topics of mathematics that are important to your goals and are missing from the list and find your favorite books or two that cover/s these topics.

    Exercise: Consider whether your interest in (self-directed) mathematics is so sincere such that you have a serious application in mind, that you might be better off enroling in a course? Even if it's a night course that last a couple of years, you will meet a LOT of people who can help in ways that are immensely more productive than trying to do this all by yourself.

    I recently purchased volume 1 of my favorite calculus and analysis book. It's an incredible masterpiece. The coverage of topics is much broader and more interesting than Aposotol or Spivak. The latter books are both very good but they also have myopic, one-track pedagogical approaches and limited themes in their coverage.

    Exercise: Find your own favorite introductory calculus book that is suitable for the motivated student.

  • by dwrodri on 2/24/20, 2:30 PM

    I recently had the experience of taking my first graduate-level probability course. It assumed quite a strong familiarity with real/complex analysis, and I suffered quite heavily. Something of note was that once I finally managed to "peel back" the analysis, the underlying intuition made a lot of sense for the simplest cases in probability (e.g. hypothesis testing between two normal distributions is a matter of figuring out whose mean you are "closer" to).

    I am of the opinion that notation is a very powerful tool for thought, but the terseness of mathematical notation often hides the intuition which is more effectively captured through good visualizations. I would really like to take self-driven "swing" at signal processing, this time approaching it through the lens of solving problem on time-series data, since as a programmer I believe that would be quite useful and relevant.

  • by thorn on 2/24/20, 8:14 AM

    I am always astonished to learn that there are such self-learners in the world. I wonder how it is even possible to have a family and spend whole day building a startup - I cannot imagine that startup work is less than 8 hours a day - and then at evening they learn math or other complicated branch of science. What time and more especially how much energy they have for the family? Are these guys superhumans? I never was able to achieve such level of daily energy spent without trapping in burn out. I am not critiquing or being jealous here, just having genuine interest. How is it possible to be sustainable across so many years?

  • by emmanueloga_ on 2/24/20, 5:54 AM

    Wow that's a brutal list of books... I'm impressed the author could work through all of that in just six years! I feel like math is a subject you need to get back again and again to refresh in order to retain. I got some pretty good grades in linear algebra back in the day... but I don't really remember much about it right now, sigh.

    My strategy to get back to study math these days is getting to learn Wolfram Mathematica and Sage. Once I can move around those two, I feel like I will be able to create a tighter feedback loop on whatever Math subject I'm happen to be studying at the time.

  • by leto_ii on 2/24/20, 12:54 PM

    Does anybody have any experience with How to Prove It? by Velleman? Recently I was thinking of starting on it, but I'm not sure about the level of commitment necessary.

  • by mikorym on 2/24/20, 11:34 AM

    I would recommend Conceptual Mathematics: A First Introduction to Categories by Lawvere.

    It is written by a true pioneer. And also, you will impress your friends by your hipster foray into category theory.

    However, this book is far from being hipster. Also, I would not be surprised if a high school student would be able to follow this book over the course of a year or two.

    If you titled the book: Sarcastic introduction to how simple set theory is then I would actually be fooled that it were the correct title.

  • by sampo on 2/24/20, 11:38 AM

    There is 3 books listed that essentially cover the freshman (first year) courses in Calculus. And 4 for Linear Algebra. If you work your way through even 2 different books for one topic, you are going to have a broader foundation in the topic than a normal math student in a normal university after completing the corresponding course. And you will have spent much more time, too. University courses don't usually cover everything that is in a textbook. And students don't usually read books through. In fact, students usually try to skim the course notes just enough so that they can solve the weekly problem sets.

    There is maybe nothing wrong with being thorough with the elementary topics if you're studying for fun. But if you're studying for applications, I think you should cover the basics only adequately, and then quickly move on to more advanced topics. Basic Calculus is only the foundation, stuff that is actually useful in applications comes later. Basic Linear Algebra can be useful in its own right, but the advanced stuff is even more useful.

    I suggest building an adequate foundation, not a comprehensively thorough foundation, and then moving on to the more powerful stuff. Which varies depending on what you actually want to use math for.

  • by gshubert17 on 2/24/20, 5:50 PM

    Another book similar to Morris Kline's _Mathematics for the Nonmathematician_, which the OP mentioned, is Lancelot Hogben's _Mathematics for the Million_. Originally published in 1937, it has been in print ever since, through several revisions. This also takes a historical approach, beginning with numbers and counting, measure and Greek geometry; and eventually covering calculus, matrices, probability, and statistics.

    You can take a look at it, at the Internet Archive,

    https://archive.org/details/HogbenMathematicsForTheMillion/m...

  • by ww520 on 2/24/20, 3:54 AM

    I found the Real Analysis course to the really really hard back in university. I thought it was like the CS information analysis when looking at the course title. It was nothing like that at all. It didn't help that the professor teaching it was pretty bad. I remember he used to jog into the class carrying a tennis racket and in tennis sporty dress with headband, seemingly just coming back from a tennis practice and acting to be cool. People just rolled their eyes. The teaching was just reading off the book. Darn, it was one of the top schools. How did this clown get in?

  • by ipnon on 2/24/20, 2:13 AM

    The perenial self-learning mathematics curriculum is hbpms.blogspot.com.

  • by jdkee on 2/24/20, 5:12 AM

    This is fantastic. I recently been studying set theory and discrete mathematics as a self-directed learner and it is incredibly helpful to see others hewing the same path.

  • by chrischattin on 2/24/20, 9:07 AM

    I love stuff like this. I graduated with an MS and never understood Calc II+. It was always memorization and repeating various theorems etc on the test. But, I didn't truly grok the fundamentals. I was just a good test taker and it bothers me to this day. So, learning math has been a continuing project of mine and things like this are beautiful.

    So grateful. The world is wide open to the self learner in this day and age.

    We are very lucky.

  • by codeisawesome on 2/24/20, 10:00 AM

    Completely unrelated question but: how do you go about finding out that there is an opportunity to build a small business selling Microsoft teams and slack integration apps? I’m stuck in the mindset that software companies make billions of dollars or no money at all. I’ve not seen the right kind of indie hacker post that talks about how exactly people size their ideas and how much money is possible to make.

  • by tinyhouse on 2/24/20, 12:39 PM

    It was a great read. I bookmark this post hoping one day to buy a couple of the recommended books. It's just too hard for me right now to find the time. Life is too busy with a family, a full time job, and all the distractions around. Self learning is one of the most enjoyable things.

  • by Koshkin on 2/24/20, 2:02 PM

    Practice makes a master. More importantly, the true understanding only comes through practice. Also, more often than not to "understand" something comes down to simply getting used to it (which is how we learn things in the elementary school). Practice is the key.

  • by t_mann on 2/24/20, 6:12 AM

    Fyi, for those who like working with lecture notes, much of the material for the math course at Oxford is online as well (no solutions, though):

    https://courses.maths.ox.ac.uk/

  • by tmpmov on 2/24/20, 4:26 PM

    Not sure if mentioned, another good book: Analysis by its History

  • by mhh__ on 2/24/20, 4:28 AM

    http://www.goodtheorist.science/ this is a step by step for theoretical physics.

  • by dr_dshiv on 2/24/20, 9:55 AM

    Does anyone know a good historical approach to maths? Like, start with Pythagoras?

    Even something purely in the modern era, learning about Fourier and Weiner, harmonic analysis, etc.

  • by dustfinger on 2/25/20, 11:14 AM

    Buy a chalk board! There is no more enjoyable way to work through problems. This is especially true as the problems become more complex.

  • by kumarvvr on 2/24/20, 10:07 AM

    Is there a list that inclines towards abstract math that can be helpful to solve programming problems?

  • by piggybox on 2/24/20, 6:50 PM

    Thank you for such an inspiring piece

  • by baby on 2/24/20, 3:23 AM

    Aren’t applications like Brilliant actually really good to do this?

  • by wendyshu on 2/25/20, 5:22 PM

    Not quite sure why everyone needs to learn so much math.

  • by polyphonicist on 2/24/20, 3:31 AM

    I am going to suggest something that might go against this idea of self-studying math.

    Do not do it alone. I mean, it is okay to self-learn mathematics as much as possible but don't let that be the only way to learn. Find a self-study group where you can discuss what you are learning with others.

    I think the social-effect can be profound in learning. I realized this when I used to learn calculus on my own. My progress was slow. But when I found a few other people who were also studying calculus, my knowledge and retention grew remarkably. I think the constant discussion and feedback-loop helps.

    With round the clock internet connectivity, it is easier to find a self-study group now than ever.

  • by nilsocket on 2/24/20, 10:08 AM

    In mathematical history, there is a lot of discredit to Indian mathematics and their contributions.

    Number System, Algebra, Geometry, Trigonometry, Calculus, ...

    https://en.wikipedia.org/wiki/Indian_mathematics