One of the most asked questions in r/math must be posted by someone, usually way past their college days, who wants to learn maths from scratch. This was also me, and surprisingly the same situation posed by a colleague. There’s a whole lot of people out there who want to learn maths, but don’t know how.
The maths you were taught in school doesn’t help, if you remember it at all, since maths shouldn’t be about rote calculation and word problems.
In this short post, that I’ll keep updated as needed, I’ll try to answer this question. But first, a little bit about my journey.
I was a 26 years old with a MA in Philosophy who landed a job in software development. One day I stumbled upon Haskell and the world became a wonder. I soon realized that I had to learn at least some maths (my least, very least, favorite subject), and so I went, posting in r/math as everyone else. I didn’t want to get stumbled by the calculations, I wanted to get the concepts down, solving equations? Can’t a machine do that? I’m a programmer afterall!
That approach made me lose so much time I cannot but facepalm recalling it. I had many, many false starts and still do. Let me help you in your journey.
Motivation, Productivity, and all that crap One of the first books I tried read: “Most mathematical problems are psychological problems”. Oh boy is it true. As I started there were many negative ideas populating my mind. On my morning commute I would see students going to school, 18 or 19 at most, and I would think they knew way more maths than I did at 28.
Good for them! There’s many studies showing tradeoffs between adult and kids in learning environment, but the short story is: you’re not doing maths to become a PhD or the next Erdos. That ship sailed. I’ve seen late starters become great. One of them is Sabino Brunello whom I met when he was 12. In 6 years he became one of Italy’s first grandmasters. Absolutely astonishing. But please note he was a late starter at 12.
Even if you know you’re not going to be a great mathematician, you should know you will suffer. Learning something new is fun for the first 10 minutes, then it becomes a chore. Cherish the little goals and move on. One of the most important aspects of this journey will be grit. A lot of it. Luckly I found that grit is something that transfers to other areas of life.
On motivation I’ll just say: I love doing maths, but I don’t do it every day. I can skip months between going through maths textbooks. In the days of the internet there’s a subtle way to avoid pain: active procrastination.
What is active procrastination? In short, all the productivity crap we’re bombarded with everywhere we go. Life hacks and other senseless noise. You don’t need it. The old adage still stands: know yourself. Know when you’re cheating yourself and avoid it. There’s a whole lot of self-manipulation that goes on in our everyday life we should be aware of. A classic one is “I’ll do that later” when you know well enough you won’t. An extremely common one that happens in maths is to avoid doing the exercises. As I flip through a textbook I would catch myself saying “oh ok, that’s trivial, I can skip it”. Nine times out of ten, if I actually try it, I fail. This is a small trap I call “the illusion of knowing”. How do you reverse such habits? Just bite the bullet. Will power is a muscle.
Another matter is time: if you’ve been to university (it’s OK if you didn’t) you have some recollection of studying for several hours a day. I assume you have a job and / or family so you won’t be able to set aside 8+ hours a day for your next hobby. That’s OK, don’t be surprised if it takes you 6 years to finish the equivalent of a BS in a European university (more on that later). It’s easy to say that you should have realistic expectations, but the situation you’re in changes as you go, so do your expectations.
Lastly, one thing that I really enjoy is trying to employ what I learn, usually in programming. If you can do this, I cannot recommend it enough. One of the highlights of 2018 to me was MuniHac where I was able to put mathematical formulas into algorithms sitting next to a Phd. One in 2017 was talking about the algebraic structure of a Rubik cube with another ex scholar. Little things that show me if I’m on the right path.
The illusion of the royal path Some people ask what are the exact steps to go from zero to hero in maths, some get quite frustrated when you cannot provide an answer. I don’t believe there’s an answer either. Some topics are foundational to others, but there’s a whole lot of layers of understanding for each topic. Even when you hear people say “there’s no prerequisite to Knot theory”, they mean no prerequisite except mathematical maturity, i.e. you need to know how to prove stuff and have the grit and intuition to attack a problem.
You can do Linear Algebra without knowing what a field is. Hell, you can do LA without knowing how to prove a mathematical statement! Just see what MIT offers in their openCourseware: no proofs, just topics of faith and a whole lot of exercises. This is not necessarily bad, but if you’re here for the maths we need to go a little deeper. You can splash around or build layer upon layer. There’s no right way to move around topics. What I will suggest later on then is what I consider good stepping stones for you to move.
One approach that doesn’t work but might appeal to people who come from humanities is to go in chronological order. Something silly like reading Euclid’s Elements before anything else, moving up to Diophantus Arithmetica etc. You don’t need to go through this. There are some great old books, but an historical prespective will help little with the actual problems. As we develop new maths we also develop better ways to teach the old one. Embrace it.
Another pitfall for fellow humanities survivors that took me a while to accept: we tend to read books fairly linearly. Textbooks are very different. More often than not for every paragraph you’re going back re-reading definitions. That’s perfectly normal, it’s supposed to be this way. Sometimes you hit the exercises and you realize you didn’t understand a topic or you don’t remember a specific technique. Just go back and forth as you learn the material. This impacts speed too. I could read a Derrida / Husserl / Heidegger classic in a week tops regardless of size, that’s completely unreasonable for a maths textbook that’s worth the paper its printed on.
The university trap There’s two aspects of the university trap. The first one is going to university. Let me be clear on this one. I’m not one of those guys who say that higher education is overvalued and kids these days should learn on their own, or on the job, or follow their passions into whatever. Nope.
One of the first things I did starting out was to look at several universities curricula and course programs to see what they were doing. Certainly maths programs have solved the problem of math learning… right? Not exactly.
What I learned is that European programs are far more rigorous than US or online programs (these last two are often equivalent). I found US programs that would get me from zero (as in multiplication tables) up to what a normal Italian high schooler gets tested on and call it a BS. I found MIT or other consolidated institutions to have a similar curricula to European universities, except they cost 100 times more. A single course in a Ivy League university would cost me as much as getting both my MA and BA did in Italy (including food, books, and commute). Italy is not a 3rd world country, I promise.
So here’s the first trap: if you can go to a good university and it won’t affect your bank balance too much, go for it! otherwise, stay home. This is my advice. So why won’t I enroll in one of these cheap Italian universities? Well, they don’t have night classes and I cannot afford to stop working. One does not NEED to attend, but at that point I would effectively do the same thing I’m doing now but with more stress and less money.
Another trap is thinking that you can do what a university student does except on your own. Sometimes it works, sometimes it fails catastrophically. You not only lack time and focus (you don’t spend your days thinking about maths, you have to often go back and re-learn things etc.) but you don’t have office hours and a support network (fellow students) either. Most importantly, you don’t have a teacher guiding you. If you look at a syllabus you’ll see no one actually covers a textbook cover to cover. How do you know which parts are important? which you can skip safely (and not just because they overlap with other courses you would be taking that semester) and which exercises are foundational?
The textbook trap As you move along your maths journey you’ll soon figure it out: not all textbooks were made for self-learners. The great majority were probably not. I’m looking at you Steward’s Calculus. If a book is a monster 1000+ pages supposed “bible”, it’s probably thought for classes where they don’t study half of it.
Another problem with books is that you might have that nagging feeling that you have to finish them. Nope. If a book doesn’t do it for you, drop it. I usually encounter three situations:
- I find that the book’s topic does not interest me that much and I don’t need to learn anything to continue with other books => drop it
- I find that the book is far too advanced, I cannot follow through even the first chapter => drop it
- I find that the book is about my level, the topic interests me but I found a stumbling block => this is a good sign! persevere, if I cannot go past after serious eyebrow sweat, I skim the rest just to get an idea of the topics as I ponder on my next move
One important thing then is to try out the book before diving in (and/or buying it). If you don’t want to commit copyright infringement, know there’s many great textbooks that are either made freely available by their authors or very cheap, as in 15$ cheap. If anything, there’s the library. I find nothing more depressing than seeing a math textbook bought but never studied. Today there’s also MOOCs one might consider. The ones I found were generally bad. Atrociously bad. One notable exception was Coursera’s “Calculus: Single variable” from University of Pennsylvania, which took me a few months and a lot of pain to complete (after about 3 false starts, so a bit more than 2 years in total). I was waking up at 4am to see if I could finally crack that integral and more often than not the answer was no.
Why are MOOCs bad? Because it’s hard to nail your audience. If you don’t want to record tens and tens of videos (as in the course mentioned above) or write your own free textbook to follow along, or write your own Khan Academy-like website for exercises (these last two were done by University of Ohio for their calculus MOOC) you’re left with pocket sized lectures where you either say too much or too little. MIT OpenCourseware is nice but it doesn’t have that lot of videos on maths topics and for lectures the silence-to-information ratio is a bit skewed making it not a good use of my time.
Opinions and experiences may vary of course, feel free to try!
Cut to the chase, give me the reading list! Not before I bore you down some more with my biased opinions! I want to distill a good method I found to select maths books I know I’ll be able to digest:
- the fact that it is a “classic” doesn’t mean it will be a good textbook to learn something from scratch or at all (looking at you Spivak’s Calculus!)
- see if it’s a recommended textbook for self-learners, see if it has any prerequisites (you’ll get better at this as you develop your math-fu)
- is it rigorous (aka, does it show proofs) or is it made for US engineers (no offense, really)? Does it have exercises?
- although USA has probably the worst maths programs I’ve seen, they do make great textbooks
- is it under ~400 pages? I cannot see myself working through a textbook for longer than 3 months and my pace is usually one chapter a week
- is it cheap? if it’s above 40€ drop it, it’s not worth it. I know knowledge has no price, but food does
- download either a sample or the whole thing and see if it sounds like something I would enjoy going through
- I simply cannot do serious maths on my commute nor at the computer. I need to be in a position where I can’t even look at my computer
So here we go! This is by no means intended to be a “path” into maths, I’m just giving my recommendations for resources that worked for me.
Basics
Khan Academy. I think their mission is a bit naive, the typical SF view of the world where everyone has access to the internet and cultural background/money/time to prepare for school. However it’s the best tool I found to re-learn high school maths. I still find myself going back to it when I forget something
Brilliant. Great website with challenging problems ranging a variety of topics. Hoarding knowledge is useless if you can’t use it to attack problems. Going through some problem sets here will give you great ideas to apply in your journey
Calculus
Khan Academy has a section on calculus which is good. Another good resource is “Calculus: Single variable” by University of Pennsylvania that you’ll find on Coursera. It is no walk in the park, but it will introduce you to a wide array of topics and I especially like the approach of starting from series. Please note that you still won’t see any proof, until this point it’s just calculations.
Proofs
Once you get your chops back, you might want to learn how to prove simple statements or at least read proofs without pulling your hair out. I went though the usuals: how to prove it, the art of problem solving, the book of proofs… they bore me to death, until I found this little gem: Chartrand, Ping et al. - Mathematical Proofs, a transition to advanced mathematics
I recommend it to everyone who asks me on how to get started in maths. This was the single most important book to get me started. You don’t need to go through it all, you can safely skip the latter parts on applications to specific branches.
Graph theory
I really enjoyed Chartrand, Ping - A first course in Graph Theory. Please note that other books they wrote on this topic are a bit of a rehash of the same, they don’t go any deeper than the abovementioned. This book is around 300 pages and it will challenge you, as usual choose to work through it at the level of depth that you find comfortable. I couldn’t understand some advanced proofs and that’s OK for me right now.
Combinatorics
I tried several books until I found Chuan-Chong et al - Principles and Techniques in Combinatorics. This book is hands down the best introduction to Enumerative Combinatorics I found. It has very challenging exercises which I completed with the help of a maths-able colleague (thanks Age!).
Once you have some basic abstract algebra, I also suggest Bogart et al. - Combinatorics Through Guided Discovery to reinforce some topics and give you more confidence. It’s a book in the style of Moore’s Inquiry-Based-Learning, so you’ll effectively recreate the whole theory by yourself, with a few hints.
Abstract Algebra
Some people love it, others hate it. I live by it. Pinter - A Book of Abstract Algebra. Every chapter is around 3 pages of explanation and 6 pages of exercises. Some people say too much of the content is in the exercises, and I find that’s perfectly OK. Other people say it’s too narrow. Well, dear, it goes up to Galois theory in less than 300 pages, I’m not even sure what else is there for an introductory book.
Linear Algebra
I’m very hard pressed to find a good introductory book. My first introduction to the topic came with MIT OpenCourseware which was OK but pretty long and without proofs. I don’t think that’s bad, it’s hard to prove stuff you’re not even comfortable manipulating. In 2019 I plan to work more on LA so I should have at least some recommendation for a second third book on the topic. If you’re looking for a good introduction to matrices you might consider the first chapter of Artin’s Algebra (the Indian edition is half the price and the only thing missing is a stupid appendix). If you’re going for Artin: it moves fast, very fast, if I recall correctly the second chapter covers basically half of Pinter’s book.
Recreational
Do you need time to recover after finishing that monster textbook but still want to be in the maths-world? Here’s a list of books I enjoyed that are more on the pop side:
Simon Singh - The code book. A fascinating journey into cryptography
Simon Singh - Fermat's Enigma. The story of how the most famous conjecture in number theory was finally proved by Andrew Wiles. Note that it gets nowhere near the maths involved
Petzold - The annotated Turing. While the famous Turing’s paper is fairly accessible, the book does a great job at showing the zeitgeist around it and commenting it. I kid you not there’s at least a whole paragraph (more often whole pages) every two lines of the paper
Bottazzini - Il flauto di Hilbert. Sorry non-Italians, this book is just amazing. It’s a mammoth history of modern mathematics from calculus to the 30s. Sometimes it gets pretty technical, especially in the last chapters, but overall provides an astonishing view of the world of maths as it came to be. I don’t think it’s ever been translated into English, putting it here just in case
Ash, Avner - Fearless Symmetry. I must admit I couldn’t finish this one, I got stuck at around 60% where the maths became a bit too advanced. This is not exactly light-reading, but not a textbook either. It does go from trivial to “OMG we’re going to die” pretty abruptely. The highlight is supposed to be a hand-wavy understanding of Fermat’s last theorem proof. Definitely a book I want to get back to
Robin Wilson - Four colors suffice. Get the colored edition. A great read on the history of the 4-colors theorem (the chromatic number of a planar graph is 4 or less). I read it twice, first time for the story, second time to get more of the maths
Derbyshire - Prime obsession. I’m a sucker for biographies and this one was pretty good. It details the life of Riemann focusing, obviously, on the famous zeta function
Hoffman - The man who only loved numbers. A biography of Paul Erdos. Very fun read, not much maths. Actually, very little to no maths
Abbott - Flatland. I thought it was a children book until I read it and it blew my mind. Do yourself a favor: study some linear algebra and then read this book
The end?
I’ve obviously read (not always completed) other books that I can’t recommend. I’ve only put books that I actually finished (or worked through to my satisfaction). I read many many many many many other chapters of other books here and there that certainly helped me in my journey. As you can tell I’m not that far off from high school maths, 5 years later I barely completed the equivalent of a BS program with gaps and a brittle foundation.