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Mathematics for the Adventurous Self-Learner

1232 points by nsainsbury 7y ago

hackernews7643 7y ago
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.
- yuanshan 7y ago
  Usually, some people will post some or almost all of the solutions online if the textbook is really famous, for instance, baby rudin. Although rarely, some authors do provide solutions, like Knuth's books, Stephen Abbott's Understanding Anaylsis, etc. For immediate feedback, maybe you can checkout [0] to learn some formal proof by doing interactive proving. [0] http://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game/ http://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game... BTW, you can always ask questions on https://math.stackexchange.com https://math.stackexchange.com
- clircle 7y ago
  It's not immediate feedback, but you do learn when you right and when you are wrong. You learn when you are bullshitting yourself. You learn that you need to be able to justify every step in a proof, and if you can't do, then you are wrong. Trying to bullshit the right answer is the most common way to end up with a faulty proof. Of course, you can also end up with a faulty proof because you can't differentiate your own bullshit from truth, but this is less common.
- chobytes 7y ago
  Learn proofs well and you get pretty good and knowing when you’re right. Enough for almost any problem you’ll be likely to encounter in a math textbook anyway.
  - TheTrotters 7y ago
    I strongly disagree. It's a little like saying "learn programming well enough and you'll know if some piece of code works as expected without running it."
    - inimino 7y ago
      > "learn programming well enough and you'll know if some piece of code works as expected without running it." That is 100% true. Obviously the code needs to be self-contained (not calling into other unknown code) but so do mathematical proofs.
    - abvr 7y ago
      That's what seperates them both, it's not about immediate formal checking from a compiler here, but rather if our proof is being justified adequately according with the rules of deduction and reasoning, it's a more intuitive approach in math, although you do end up knowing whether something is right or wrong, akin to programs.
ipnon 7y ago
The perenial self-learning mathematics curriculum is hbpms.blogspot.com.
gavinray 7y ago
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 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)
- photon_lines 7y ago
  I'm actually self-studying as well, and I try to compile everything I learn and the notes that I take into 'Intuitive Guides' which I'm going to make available on my github repository. I actually have a guide on Linear Algebra which you can find here: https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals https://github.com/photonlines/Intuitive-Overview-of-Linear-... I'm going to release one on Maxwell's equations next week, and I started working on a Calculus and General Relativity guides as well, so hopefully it helps!
- kyawzazaw 7y ago
  khanacademy has content for those school work level.
- quaquaqua1 7y ago
  SAT prep books such as those from Kaplan or Princeton Review tend to be extremely cheap, if not free, and cover almost all of those topics pretty succinctly :) Speaking as someone who was kicked out of precalc and then never touched math again despite working as a dev, your point resonates with me :)
melling 7y ago
A couple prior HN discussions with great comments related to learning mathematics: https://news.ycombinator.com/item?id=19811715 https://news.ycombinator.com/item?id=19811715 https://news.ycombinator.com/item?id=20446796 https://news.ycombinator.com/item?id=20446796 https://news.ycombinator.com/item?id=11469494 https://news.ycombinator.com/item?id=11469494
deleted 7y ago
[deleted]
baby 7y ago
Aren’t applications like Brilliant actually really good to do this?
daxfohl 7y ago
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.
- polyphonicist 7y ago
  Shared a similar thought here: https://news.ycombinator.com/item?id=22401420 https://news.ycombinator.com/item?id=22401420 Yes, it is really important to learn math with study-mates. Just like in code, we do reviews, in math too, we need someone else who can review our proofs. It is even easier to make an error in a proof and believe that something is proven when it isn't. A study-mate helps to prevent us from fooling ourselves.
- l_t 7y ago
  I've tried a few things recently that help with that: 1. Don't do exercises unless you want to. Completionism is a trap. 2. Take notes. Rewrite things in your own words. Imagine you're writing a guide for your past self. 3. Ask questions. Anytime you write something down, pause and ask yourself. Why is this true? How can we be sure? What does it imply? How could this idea be useful? 4. Cross-reference. Don't read linearly. Instead, have multiple textbooks, and "dig deep" into concepts. If you learn about something new (say, linear combinations) -- look them up in two textbooks. Watch a video about them. Read the Wikipedia page. _Then_ write down in your notes what a linear combination is. Anyway, everyone's different of course, but these practices have been helping me get re-invigorated with self-learning math. Hope they help someone else out there. I welcome any feedback! (edit: formatting)
  - throwlaplace 7y ago
polyphonicist 7y ago
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.
- dorchadas 7y ago
  Ha - I actually suggested the exact same thing before seeing your post. It's definitely much better to have a group. Since I've found this group I'm currently in, I've also been much more motivated, but also able to get feedback from more advanced people, and pare the problem numbers down to only the ones that are useful and will help me build concepts, limiting how many "calculation" problems I have to repeat.
  - integerclub 7y ago
    Made a group announcement on the same topic on this thread: https://news.ycombinator.com/item?id=22401750 https://news.ycombinator.com/item?id=22401750 See the user profile of my account for more details.
- dentalperson 7y ago
  It's not super clear to me how this actually works in practice. I've seen there is one public math meetup in SF, but the topic is usually different from the one I want to study. I'm glad to see there are online options for groups like Stack Exchange or tighter group's like the one integerclub mentions, but I still seem to run into the same problem. For example, I'm not sure how to get a group of people that are interested in reading book X when I want to start it. If anyone has advice on that, please share.
mikorym 7y ago
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.
ww520 7y ago
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?
dorchadas 7y ago
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.
- nubb 7y ago
  Could you share the discord server? Thank you.
  - dorchadas 7y ago
    I'm not the owner, and it's a small server, so I don't feel comfortable. Sorry. I think I found the owner through /r/math, so you might be able to find his old post there though.
- mathgenius 7y ago
  I would say that learning mathematics is virtually impossible without a teacher. Like, would you try learning Karate without a teacher? How about if you get with a bunch of friends and you all try to learn it together? No way. Even the professionals try to find someone to learn a new concept from. There's something about mathematical writing that is too fragile/brittle for wetware. One other strategy: I've noticed the really smart (arrogant?) people just don't bother reading new mathematics, they re-invent it themselves. It's actually worth trying if you can stomach it.
laichzeit0 7y ago
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.
- nsainsbury 7y ago
  Heh, nice to find someone who walked a similar path! :-) Ah Spivak...yes, I absolutely agree it's one of the best books to build up that mathematical maturity everyone talks about. For me Spivak took about 6 months and I managed to do almost all of the starred exercises - Gifted? No. Brutally determined: yes. And I was quite fortunate to be in a place in life where I could put serious hours in to it at the time. After that, I learned to relax a bit more as I realised I had pushed myself way too hard and was close to burning out. I still love looking back at that damn book though. There's just something that's so special about it...the way the exercises build upon each other and connect together. It's really unique.
mhh__ 7y ago
http://www.goodtheorist.science/ http://www.goodtheorist.science/ this is a step by step for theoretical physics.
integerclub 7y ago
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 https://webchat.freenode.net/#integerclub Slack: https://bit.ly/integerclubslackinvite https://bit.ly/integerclubslackinvite Mailing list: https://groups.google.com/d/forum/integerclub 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.
jdkee 7y ago
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.
peatfreak 7y ago
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.
- laichzeit0 7y ago
  > I recently purchased volume 1 of my favorite calculus and analysis book. Which book would that be if I might ask? I'm wagering... Courant? ;)
  - 7y ago
emmanueloga_ 7y ago
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.
deostroll 7y ago
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...
t_mann 7y ago
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/ https://courses.maths.ox.ac.uk/
wyqydsyq 7y ago
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)
- ducaale 7y ago
  This is why I love to do game development without using a game engine. It gives you a reason to learn math, optimize your code down to the metal, all while having fun playing your game.
  - wyqydsyq 7y ago
    Yeah I've been self-learning 3D "the hard way" and have been really enjoying it. Keeping it as low-level as possible, I'm using CycleJS for dataflow management and Regl.js for drawing via a CycleJS-Regl.js driver. All state is explicitly managed observables/streams in CycleJS, which maps out to Regl.js draw commands, which are basically raw frag/vert shaders with some bindings mapping my state from CycleJS to appropriate uniforms/attributes. I probably would be able to produce some usable output much faster if I used an engine like Unity or a framework like Three.js, but I feel like I would have missed out on gaining so much knowledge by only working with high-level abstractions and never having to even touch GLSL code.
thorn 7y ago
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?
- cammikebrown 7y ago
  Not everyone is at a startup. I’m a bartender with a physics degree who’s learning more math in my spare time for fun, and to eventually switch to some sort of data science career if I tire of bartending.
- FranzFerdiNaN 7y ago
  I have my doubts that the people who write these kinds of posts truly did everything they say. As you say, it just does not seem possible to thoroughly work through all those math books + raising multiple kids + maintain friendships + working out daily (as he claimed he did) + work full time + things like cleaning and shopping and other chores.
  - the_watcher 7y ago
    He talks about spending ~10/week on math, with more dedicated when he was between gigs. If I replaced reading for pleasure and all television (not that I watch a ton, but maybe an hour a day or so), I could carve out 10 hours a week without making other changes.
angry_octet 7y ago
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.
- tprice7 7y ago
  I emphatically agree. There is something that walking does to your brain that really helps you see the big picture.
- injb 7y ago
  I'm glad you posted this, because I use walks this way too. And because it reminds me of William Rowan Hamilton and the quaterions!
chrischattin 7y ago
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.
sampo 7y ago
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.
- nsainsbury 7y ago
  To be clear, I don't at all advocate that people work through all the Calculus books. Likewise for the Linear Algebra books. My aim was to provide alternative options (which are easier, cheaper, etc.)
dr_dshiv 7y ago
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.
- dr_dshiv 7y ago
  The best historical treatment I've yet found is the Time-Life book on mathematics. https://www.amazon.com/Mathematics-David-Bergamini/dp/B0007G5WYG https://www.amazon.com/Mathematics-David-Bergamini/dp/B0007G... I know that Freeman Dyson attributes his proficiency in math to his love of math, which he claims was kindled as a teenager by reading Bell's "Men of Mathematics"
codeisawesome 7y ago
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.
kumarvvr 7y ago
Is there a list that inclines towards abstract math that can be helpful to solve programming problems?
nilsocket 7y ago
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 https://en.wikipedia.org/wiki/Indian_mathematics
tinyhouse 7y ago
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.
leto_ii 7y ago
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.
- manu_ss 7y ago
  Seems to have some intro to logic and math language, if you have never read an math books as the ones referred in this post, that book should be a nice way to ease into it.
  - leto_ii 7y ago
    I see, thanks. I do have some experience with proofs and maths in general, but I never went through a text dedicated explicitly to developing proving techniques. I might give it a go anyway, but do you know of a more advanced version of this book as well?
    - manu_ss 7y ago
      Not really. I'm a math major, what you usually do is have an intro to logic course and then just jump into those books.
      - leto_ii 7y ago
        Thanks!
- strls 7y ago
  I worked through this book to learn how to do proofs. It turned out way more fun than I expected. The book really did demystify proofs for me. It took several months of studying - there are many exercises. But completely worth it. I'm glad I have read this book before studying Group theory and Real analysis.
  - leto_ii 7y ago
    Thanks!
Koshkin 7y ago
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.
tildedave 7y ago
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.
dwrodri 7y ago
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.
- watwatinthewat 7y ago
  In my opinion, the issue here is notation and a bit more. I did about eight years of college in math, changed paths, changed careers, changed careers again to ML/DL research, and now will finish a CS undergrad degree this month. I put it in context because it's not quite a direct comparison since I have been in greatly different situations and ages between studying math and CS, but putting that aside, I have to say I have greatly enjoyed the computer science means of teaching more than math, doubly when it comes to self-learning. Concepts in math are generally taught entwined with the means of proving those ideas. That's important if you're a grad student looking to be a math researcher, but (IMO) it is not so great if you're a newer student or learning on your own and trying to grasp the concept and big picture. A proof of a theorem can be (and too often is) a lot of detail that really doesn't help you grasp the concept the theorem provides or is used towards, often because it involves other ideas and techniques from higher levels or just different types of math, both of which are out of the scope of the student learning the topic. Worse yet, it is standard for a proof to be written almost backwards from how it would be thought out. Anyone from a math educational background has the experience in homework of solving a problem, then rewriting it almost in total reverse to be in the proper form to submit. This means not only is the proof of the theorem not useful towards conceptual understanding, reading the proof doesn't show you chronologically how you would discover it yourself. That is a lot of overhead cost to break through to get to real understanding, real learning. As you mention, notation as well is another thing you need to break through. I have found computer science and related classes to be taught more constructively. Concept is given first, and then your job as student learner is to construct it. Coming from the ML field, I love comparing math and CS proofs of topics here. Explanations from CS people of back propagation, for example, are always visual, and books/courses will have you construct a class and methods to do the calculations. Someone with a bit of programming knowledge can follow along in their language of choice. Math explanations get into a ton of notation from Calc 3+, and it's going to take a lot of playing around and frustration to get a working system out of the explanation. Even the derivation section on Wikipedia is not something most people will understand and be able to turn into useful output. The more I see other ways concepts are taught, the more I wish math had been taught a different way. There is a lot to break through in order to get to real understanding, just by the way it's formed and taught.
zerubeus 7y ago
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?
deleted 7y ago
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tmpmov 7y ago
Not sure if mentioned, another good book: Analysis by its History
jcurbo 7y ago
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.
kevstev 7y ago
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.
- strls 7y ago
  Sounds like you might simply not understand the definitions for these operators and symbols. In other words, it's not a notation problem. I find that it's helpful to mentally replace the symbols like dy/dx, sum, lim, integral, and so on with the concepts they represent. That is, go from operators to definitions.
- bo1024 7y ago
  I think usually the problem is "almost getting it" and trying to move forward, which means small uncertainties add up and all the sudden one is totally lost without being sure exactly why. So it's important to go back and make sure each piece of notation is crystal clear before moving forward. Any statement in math is meant to be directly translatable to human language. You should be able to read it out loud in English and know exactly what you mean when you say it. Unfortunately, sometimes math uses awful notation. For example, df/dx. This is a case where df doesn't mean anything (or at least it's not normally well-defined), and dx doesn't mean anything either (same comment). But the notation as a whole means something. If we write g = df/dx, then we can understand that g is a function whose input is x and output is the slope of f at x.
- proc0 7y ago
  3blue1brown channel on calculus: https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...
gshubert17 7y ago
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/mode/2up https://archive.org/details/HogbenMathematicsForTheMillion/m...
piggybox 7y ago
Thank you for such an inspiring piece
dustfinger 7y ago
Buy a chalk board! There is no more enjoyable way to work through problems. This is especially true as the problems become more complex.
wendyshu 7y ago
Not quite sure why everyone needs to learn so much math.

1232 points by nsainsbury 7y ago

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.

yuanshan 7y ago
Usually, some people will post some or almost all of the solutions online if the textbook is really famous, for instance, baby rudin. Although rarely, some authors do provide solutions, like Knuth's books, Stephen Abbott's Understanding Anaylsis, etc. For immediate feedback, maybe you can checkout [0] to learn some formal proof by doing interactive proving. [0] http://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game/ http://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game... BTW, you can always ask questions on https://math.stackexchange.com https://math.stackexchange.com
clircle 7y ago
It's not immediate feedback, but you do learn when you right and when you are wrong. You learn when you are bullshitting yourself. You learn that you need to be able to justify every step in a proof, and if you can't do, then you are wrong. Trying to bullshit the right answer is the most common way to end up with a faulty proof. Of course, you can also end up with a faulty proof because you can't differentiate your own bullshit from truth, but this is less common.
chobytes 7y ago
Learn proofs well and you get pretty good and knowing when you’re right. Enough for almost any problem you’ll be likely to encounter in a math textbook anyway.
- TheTrotters 7y ago
  I strongly disagree. It's a little like saying "learn programming well enough and you'll know if some piece of code works as expected without running it."
  - inimino 7y ago
    > "learn programming well enough and you'll know if some piece of code works as expected without running it." That is 100% true. Obviously the code needs to be self-contained (not calling into other unknown code) but so do mathematical proofs.
  - abvr 7y ago
    That's what seperates them both, it's not about immediate formal checking from a compiler here, but rather if our proof is being justified adequately according with the rules of deduction and reasoning, it's a more intuitive approach in math, although you do end up knowing whether something is right or wrong, akin to programs.

ipnon 7y ago

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

gavinray 7y ago

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 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)

photon_lines 7y ago
I'm actually self-studying as well, and I try to compile everything I learn and the notes that I take into 'Intuitive Guides' which I'm going to make available on my github repository. I actually have a guide on Linear Algebra which you can find here: https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals https://github.com/photonlines/Intuitive-Overview-of-Linear-... I'm going to release one on Maxwell's equations next week, and I started working on a Calculus and General Relativity guides as well, so hopefully it helps!
kyawzazaw 7y ago
khanacademy has content for those school work level.
quaquaqua1 7y ago
SAT prep books such as those from Kaplan or Princeton Review tend to be extremely cheap, if not free, and cover almost all of those topics pretty succinctly :) Speaking as someone who was kicked out of precalc and then never touched math again despite working as a dev, your point resonates with me :)

melling 7y ago

A couple prior HN discussions with great comments related to learning mathematics: https://news.ycombinator.com/item?id=19811715 https://news.ycombinator.com/item?id=19811715 https://news.ycombinator.com/item?id=20446796 https://news.ycombinator.com/item?id=20446796 https://news.ycombinator.com/item?id=11469494 https://news.ycombinator.com/item?id=11469494

deleted 7y ago

[deleted]

baby 7y ago

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

daxfohl 7y ago

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.

polyphonicist 7y ago
Shared a similar thought here: https://news.ycombinator.com/item?id=22401420 https://news.ycombinator.com/item?id=22401420 Yes, it is really important to learn math with study-mates. Just like in code, we do reviews, in math too, we need someone else who can review our proofs. It is even easier to make an error in a proof and believe that something is proven when it isn't. A study-mate helps to prevent us from fooling ourselves.
l_t 7y ago
I've tried a few things recently that help with that: 1. Don't do exercises unless you want to. Completionism is a trap. 2. Take notes. Rewrite things in your own words. Imagine you're writing a guide for your past self. 3. Ask questions. Anytime you write something down, pause and ask yourself. Why is this true? How can we be sure? What does it imply? How could this idea be useful? 4. Cross-reference. Don't read linearly. Instead, have multiple textbooks, and "dig deep" into concepts. If you learn about something new (say, linear combinations) -- look them up in two textbooks. Watch a video about them. Read the Wikipedia page. _Then_ write down in your notes what a linear combination is. Anyway, everyone's different of course, but these practices have been helping me get re-invigorated with self-learning math. Hope they help someone else out there. I welcome any feedback! (edit: formatting)
- throwlaplace 7y ago

polyphonicist 7y ago

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.

dorchadas 7y ago
Ha - I actually suggested the exact same thing before seeing your post. It's definitely much better to have a group. Since I've found this group I'm currently in, I've also been much more motivated, but also able to get feedback from more advanced people, and pare the problem numbers down to only the ones that are useful and will help me build concepts, limiting how many "calculation" problems I have to repeat.
- integerclub 7y ago
  Made a group announcement on the same topic on this thread: https://news.ycombinator.com/item?id=22401750 https://news.ycombinator.com/item?id=22401750 See the user profile of my account for more details.
dentalperson 7y ago
It's not super clear to me how this actually works in practice. I've seen there is one public math meetup in SF, but the topic is usually different from the one I want to study. I'm glad to see there are online options for groups like Stack Exchange or tighter group's like the one integerclub mentions, but I still seem to run into the same problem. For example, I'm not sure how to get a group of people that are interested in reading book X when I want to start it. If anyone has advice on that, please share.

mikorym 7y ago

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.

ww520 7y ago

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?

dorchadas 7y ago

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.

nubb 7y ago
Could you share the discord server? Thank you.
- dorchadas 7y ago
  I'm not the owner, and it's a small server, so I don't feel comfortable. Sorry. I think I found the owner through /r/math, so you might be able to find his old post there though.
mathgenius 7y ago
I would say that learning mathematics is virtually impossible without a teacher. Like, would you try learning Karate without a teacher? How about if you get with a bunch of friends and you all try to learn it together? No way. Even the professionals try to find someone to learn a new concept from. There's something about mathematical writing that is too fragile/brittle for wetware. One other strategy: I've noticed the really smart (arrogant?) people just don't bother reading new mathematics, they re-invent it themselves. It's actually worth trying if you can stomach it.

laichzeit0 7y ago

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.

nsainsbury 7y ago
Heh, nice to find someone who walked a similar path! :-) Ah Spivak...yes, I absolutely agree it's one of the best books to build up that mathematical maturity everyone talks about. For me Spivak took about 6 months and I managed to do almost all of the starred exercises - Gifted? No. Brutally determined: yes. And I was quite fortunate to be in a place in life where I could put serious hours in to it at the time. After that, I learned to relax a bit more as I realised I had pushed myself way too hard and was close to burning out. I still love looking back at that damn book though. There's just something that's so special about it...the way the exercises build upon each other and connect together. It's really unique.

mhh__ 7y ago

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

integerclub 7y ago

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 https://webchat.freenode.net/#integerclub Slack: https://bit.ly/integerclubslackinvite https://bit.ly/integerclubslackinvite Mailing list: https://groups.google.com/d/forum/integerclub 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.

jdkee 7y ago

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.

peatfreak 7y ago

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.

laichzeit0 7y ago
> I recently purchased volume 1 of my favorite calculus and analysis book. Which book would that be if I might ask? I'm wagering... Courant? ;)
- 7y ago

emmanueloga_ 7y ago

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.

deostroll 7y ago

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...

t_mann 7y ago

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

wyqydsyq 7y ago

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)

ducaale 7y ago
This is why I love to do game development without using a game engine. It gives you a reason to learn math, optimize your code down to the metal, all while having fun playing your game.
- wyqydsyq 7y ago
  Yeah I've been self-learning 3D "the hard way" and have been really enjoying it. Keeping it as low-level as possible, I'm using CycleJS for dataflow management and Regl.js for drawing via a CycleJS-Regl.js driver. All state is explicitly managed observables/streams in CycleJS, which maps out to Regl.js draw commands, which are basically raw frag/vert shaders with some bindings mapping my state from CycleJS to appropriate uniforms/attributes. I probably would be able to produce some usable output much faster if I used an engine like Unity or a framework like Three.js, but I feel like I would have missed out on gaining so much knowledge by only working with high-level abstractions and never having to even touch GLSL code.

thorn 7y ago

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?

cammikebrown 7y ago
Not everyone is at a startup. I’m a bartender with a physics degree who’s learning more math in my spare time for fun, and to eventually switch to some sort of data science career if I tire of bartending.
FranzFerdiNaN 7y ago
I have my doubts that the people who write these kinds of posts truly did everything they say. As you say, it just does not seem possible to thoroughly work through all those math books + raising multiple kids + maintain friendships + working out daily (as he claimed he did) + work full time + things like cleaning and shopping and other chores.
- the_watcher 7y ago
  He talks about spending ~10/week on math, with more dedicated when he was between gigs. If I replaced reading for pleasure and all television (not that I watch a ton, but maybe an hour a day or so), I could carve out 10 hours a week without making other changes.

angry_octet 7y ago

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.

tprice7 7y ago
I emphatically agree. There is something that walking does to your brain that really helps you see the big picture.
injb 7y ago
I'm glad you posted this, because I use walks this way too. And because it reminds me of William Rowan Hamilton and the quaterions!

chrischattin 7y ago

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.

sampo 7y ago

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.

nsainsbury 7y ago
To be clear, I don't at all advocate that people work through all the Calculus books. Likewise for the Linear Algebra books. My aim was to provide alternative options (which are easier, cheaper, etc.)

dr_dshiv 7y ago

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.

dr_dshiv 7y ago
The best historical treatment I've yet found is the Time-Life book on mathematics. https://www.amazon.com/Mathematics-David-Bergamini/dp/B0007G5WYG https://www.amazon.com/Mathematics-David-Bergamini/dp/B0007G... I know that Freeman Dyson attributes his proficiency in math to his love of math, which he claims was kindled as a teenager by reading Bell's "Men of Mathematics"

codeisawesome 7y ago

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.

kumarvvr 7y ago

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

nilsocket 7y ago

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

tinyhouse 7y ago

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.

leto_ii 7y ago

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.

manu_ss 7y ago
Seems to have some intro to logic and math language, if you have never read an math books as the ones referred in this post, that book should be a nice way to ease into it.
- leto_ii 7y ago
  I see, thanks. I do have some experience with proofs and maths in general, but I never went through a text dedicated explicitly to developing proving techniques. I might give it a go anyway, but do you know of a more advanced version of this book as well?
  - manu_ss 7y ago
    Not really. I'm a math major, what you usually do is have an intro to logic course and then just jump into those books.
    - leto_ii 7y ago
      Thanks!
strls 7y ago
I worked through this book to learn how to do proofs. It turned out way more fun than I expected. The book really did demystify proofs for me. It took several months of studying - there are many exercises. But completely worth it. I'm glad I have read this book before studying Group theory and Real analysis.
- leto_ii 7y ago
  Thanks!

Koshkin 7y ago

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.

tildedave 7y ago

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.

dwrodri 7y ago

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.

watwatinthewat 7y ago
In my opinion, the issue here is notation and a bit more. I did about eight years of college in math, changed paths, changed careers, changed careers again to ML/DL research, and now will finish a CS undergrad degree this month. I put it in context because it's not quite a direct comparison since I have been in greatly different situations and ages between studying math and CS, but putting that aside, I have to say I have greatly enjoyed the computer science means of teaching more than math, doubly when it comes to self-learning. Concepts in math are generally taught entwined with the means of proving those ideas. That's important if you're a grad student looking to be a math researcher, but (IMO) it is not so great if you're a newer student or learning on your own and trying to grasp the concept and big picture. A proof of a theorem can be (and too often is) a lot of detail that really doesn't help you grasp the concept the theorem provides or is used towards, often because it involves other ideas and techniques from higher levels or just different types of math, both of which are out of the scope of the student learning the topic. Worse yet, it is standard for a proof to be written almost backwards from how it would be thought out. Anyone from a math educational background has the experience in homework of solving a problem, then rewriting it almost in total reverse to be in the proper form to submit. This means not only is the proof of the theorem not useful towards conceptual understanding, reading the proof doesn't show you chronologically how you would discover it yourself. That is a lot of overhead cost to break through to get to real understanding, real learning. As you mention, notation as well is another thing you need to break through. I have found computer science and related classes to be taught more constructively. Concept is given first, and then your job as student learner is to construct it. Coming from the ML field, I love comparing math and CS proofs of topics here. Explanations from CS people of back propagation, for example, are always visual, and books/courses will have you construct a class and methods to do the calculations. Someone with a bit of programming knowledge can follow along in their language of choice. Math explanations get into a ton of notation from Calc 3+, and it's going to take a lot of playing around and frustration to get a working system out of the explanation. Even the derivation section on Wikipedia is not something most people will understand and be able to turn into useful output. The more I see other ways concepts are taught, the more I wish math had been taught a different way. There is a lot to break through in order to get to real understanding, just by the way it's formed and taught.

zerubeus 7y ago

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?

deleted 7y ago

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tmpmov 7y ago

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

jcurbo 7y ago

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.

kevstev 7y ago

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.

strls 7y ago
Sounds like you might simply not understand the definitions for these operators and symbols. In other words, it's not a notation problem. I find that it's helpful to mentally replace the symbols like dy/dx, sum, lim, integral, and so on with the concepts they represent. That is, go from operators to definitions.
bo1024 7y ago
I think usually the problem is "almost getting it" and trying to move forward, which means small uncertainties add up and all the sudden one is totally lost without being sure exactly why. So it's important to go back and make sure each piece of notation is crystal clear before moving forward. Any statement in math is meant to be directly translatable to human language. You should be able to read it out loud in English and know exactly what you mean when you say it. Unfortunately, sometimes math uses awful notation. For example, df/dx. This is a case where df doesn't mean anything (or at least it's not normally well-defined), and dx doesn't mean anything either (same comment). But the notation as a whole means something. If we write g = df/dx, then we can understand that g is a function whose input is x and output is the slope of f at x.
proc0 7y ago
3blue1brown channel on calculus: https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...

gshubert17 7y ago

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/mode/2up https://archive.org/details/HogbenMathematicsForTheMillion/m...

piggybox 7y ago

Thank you for such an inspiring piece

dustfinger 7y ago

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

wendyshu 7y ago

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