Ask HN: Math books that made you considerably higher at math?

* Calculus by Spivak. This was used in my intro calculus course in university. It’s very much a bottom-up, first-principles construction of calculus. Very proof-based, so you have to be into that. Tons of exercises, including some that sneakily introduce pretty advanced concepts not explicitly covered in the main text. This book, along with the course, rearranged by brain. Not sure how useful it would be for self-study though.

* Measurement by Lockhart. I haven’t read the whole thing, but have enjoyed working through some of the exercises. A good book for really grokking geometric proofs and understanding “mathematical beauty”, rather than just cranking through algebraic proofs step by step.

* Naive Set Theory by Halmos. Somewhat spare, but a nice, concise introduction to axiomatic set theory. Brings you from nothing up to the Continuum Hypothesis. I read this somewhere around my first year in university and it was another brain-rearranger.

* Geometry and the imagination by Hilbert and Cohn-Vossen

* Methods of mathematical physics by Courant and Hilbert

* A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds)

* Fourier Analysis by Körner

* Arnold’s books on ODE, PDE and mathematical physics are breathtakingly beautiful.

* The shape of space by Weeks

* Solid Shape by Koenderink

* Analyse fonctionnelle by Brézis

* Tristan Needhams “visual” books about complex analysis and differential forms

* Information theory, inference, and learning algorithms by MacKay (great book about probability, plus you can download the .tex source and read the funny comments of the author)

And finally, a very old website which is full of mathematical jewels with an incredibly fresh and clear treatment: https://mathpages.com/ …I am in love with the tone of those articles, severe and playful on the identical time.

Do you know a good path or book that’s suitable for that?

Gives a bird’s eye view of math very nicely. Even from a skimming it was very useful to help me understand the gaps I have, and the shape of those gaps, and partially filling them.

I bought myself a Remarkable 2 and signed up to Khan Academy. Now I’m revising algebra basics and I plan to go as advanced as Khan Academy lets me.

I was really bad at maths in school (UK A Levels). But I’m a successful software developer today. I felt like knowing more advanced maths could make me a better developer and not feel intimidated by a lot of the things I see.

I’m actually enjoying it as well. Maths isn’t just something I have to do to get out of school, now it’s something I want to do. And it gives me the same satisfaction as solving puzzles like sudoku.

I’d recommend it to anyone. The Remarkable 2 is actually really nice to write on too, since I want to store my notes digitally. And I make so many mistakes when writing, so undo is great.

also: the remarkable 2 is great, we have one, but the screen broke and the refurbished replacement arrived with a screen that’s not functioning correctly at all, making it a unusable device. Good reminder to reach out to them again.

Thanks!

The Remarkable, at least for me, is good because I can organise my notebooks into folders by certain math lessons or concepts. And I can undo any mistakes, so my notes are clean. Even if I am quickly working something out I am clean it up and make it a good note for future me. The feel of writing on it is much nicer as well versus my laptop’s pen or my iPad’s pen.

Also, I like that it basically just does notes. There’s no Android bullshit, it’s just no nonsense note taking. Some competitor tablets have Android and all that baggage.

It can OCR you writing, but I don’t know how good that would be for math.

The Remarkable isn’t the only tablet that can do this, but it’s the one I bought because I like the style and the simplicity of the software.

There are lots of of movies, organised in playlists, from undergraduate lectures ( https://www.youtube.com/playlist?list=PL55C7C83781CF4316 ) and analysis seminars ( https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ) all the way in which to primary fundamentals like how to consider counting (e.g. https://www.youtube.com/watch?v=Puk-ipOTiD4&list=PL5A714C94D… )

The explanation I discover them fascinating is that Wildberger would not agree with a number of the typical approaches, particularly with using infinity and taking limits. This leads him down attention-grabbing paths (e.g. Rational Trigonometry and Algebraic Calculus), which (a) present the course of of arithmetic (exploring, making definitions, increase in numerous instructions, and so on.), while (b) remaining principally grounded and approachable (e.g. no appeals to inscrutable lemmas from summary analysis areas).

For instance, he is lately been making movies about “multisets” (laptop scientists would name them Luggage), their arithmetic (the place “including” is union, and “multiplying” is pairwise/cartesian product of the weather), and the way this generalises: from an algebra containing solely empty baggage (trivial, however self-consistent; behaves like zero), to baggage of zeros (behaves like pure quantity arithmetic), to baggage of pure numbers (behaves like polynomial arithmetic), to baggage of polynomials (behaves like polynomials in arbitrarily-many variables) https://www.youtube.com/watch?v=4xoF2SRp194

So, I decided I wanted to study maths for the maths.
I was in the fortunate position of being able to self fund myself through the Open University (uk based) Maths and Statistics BSc.
One module at a time I’m now on my last module.
There many things I’d studied before (calculus, sequences) and many new to me (group theory, graph theory)

I like khan academy back in 2010 because all the videos were in one place and you could see everything right there in front of you

And my impression was that Saxon Math was the worst.
What I mean by worst is that it just make you practice an algorithm by doing lot of repetition but doesn’t force you to have a deep understanding or problem solving skill.

My experience is that I didn’t really feel like I was memorizing an algorithm. Because the problem set includes assignments from all of the old sets, it is hard to memorize all of the algorithms. So you instead memorize the different moves that are allowed and have a general idea of what types of moves might be useful.

I dunno. I went on to do engineery stuff as an undergrad rather than pure math stuff, it seems like a good match because engineering problems are also often in the “no need to be super clever, just don’t mess up” vein, so it could be just a lucky match.

I think anything that results in

1) actually reading some textbook

2) actually working through any type of problem for a couple hours a week

will compare well to the typical US math education pretty well anyway.

I’ve been using Professor Leonard’s Youtube video series[1] mostly, along with some of those “workbook” type books by Chris McMullen, and a variety of books with titles like “1001 solved problems in $SUBJECT”, “The Humongous Book of $SUBJECT problems”, and the like. The nice thing about Professor Leonard is that he has videos on everything starting from pre-algebra, middle-school math, up through Differential Equations. Note that his diff-eq class isn’t quite complete but he just announced he’s about to start recording new videos to finish that, and he’s also going to be starting a Linear Algebra sequence. And he’s a great lecturer who does a really good job of explaining things and making them understandable.

I also use Khan Academy sometimes, and stuff on Youtube from The Math Sorcerer[2]. Oh, and of course there is 3blue1brown[3], whose videos are also useful. And for Linear Algebra I’ve been using Gilbert Strang’s OCW videos[4] on Youtube.

FWIW, I’ve evolved the way I study math, and what I do now works for me, even though it’s 100% not the way you’d ordinarily see suggested. That is, I watch math videos fairly passively and don’t work problems at the same time and treat it like being in a class per-se. I used to do the thing of treating it like a class, pausing the video to work examples, and what-not, and that does work. But it’s very slow and tedious.

Now, I just watch the videos, acknowledging that I won’t absorb everything and that I also need to work problems for long-term retention. So now what I do is watch passively to a certain point (which I determine fairly subjectively) then I stop with the videos for a while, pick up a textbook or one of those “workbook” type books I mentioned earlier, and work problems for a while. Then I review the parts that I find myself struggling with. I’m also just now starting to add “creating Anki cards” as something I do during that second pass.

Once I start getting a decent Anki deck built up, I’ll be reviewing that regularly as well to help build retention. I only create cards for things that seem amenable to rote memorization, and TBH, I’m still working on figuring out what things are best to include, and how to structure those cards. What I don’t intend to do is include specific problems where all I’d be doing is memorizing the answer to a problem. So far it’s just formulas and things are are very obvious candidates to be memorized, and “algorithm” things like the “chain rule” from calculus, and similar.

Also, mathematics is a massive field. The first question would be what kinds of mathematics would you like to get better at. There are great books in analysis. If you are starting out with a solid calculus knowledge, try Abbott’s Understanding Analysis [1] or Duren’s Invitation to Classical Analysis [2]. For asymptotic methods in PDEs, try Bender and Orszag [3] which is a wonderful book. But again, this might not be your cup of tea at all and there are more abstract or formal books like Rudin’s.

If you want to approach fields without a lot of machinery, graph theory books by Bollobas are great (but difficult). See his Modern Graph Theory book [4] as an example.

For linear algebra, one of my favorites (but it was after I already learned the subject) is Trefethen’s Numerical Linear Algebra book [5]. Another beautiful topic is at the intersection of linear algebra and combinatorics. See Babai and Frankl’s lectures freely available online.

Then there are wonderful topics in geometry. A massive mountain to climb would be algebraic geometry. For one starting point, see [6]. Differential geometry (Spivak’s multi-volume work or Needham’s differential forms book) is another wonderful area. I would recommend Crane’s discrete differential geometry course at Carnegie Mellon [7] if you want a concrete introduction.

You might want to demystify a topic you have heard about. E.g. Galois theory and the unsolvability of quintic equations. You could look at [8] which guides your way through wonderful problems.

We haven’t even touched huge swathes of mathematics including anything topological or number theory. Even within the topics mentioned above, once you start, your journey will take a life of its own and you’ll encounter multiple books and papers opening up new sub-fields.

The only approach that worked well for me in the past was to get completely consumed by what one topic one was studying. This meant not getting distracted by multiple topics. Once one enters the workforce, this is very hard (or at least has been for me). Without knowing someone, it’s hard to recommend anything but the advantage with topics like graph theory and combinatorics is that one needs less machinery (as opposed to something like algebraic geometry). These fields lead you to interesting problems very rapidly and one can wrestle with them part-time.

I’m not sure why, but I think the fact that it’s an appendix meant the author had no motivation to inflate the content unnecessarily. So it’s more like a pamphlet; only about 30 pages IIRC, and it’s really just the bare-bones definitions and facts. The full-on textbooks dedicated to category theory have way too much superfluous content IMO, unless your aim is to be a researcher in that field specifically.

Deconstruct: Break down the math you want to know into big problems and concepts. Pick a math-related goal that is Measurable and Time-Bound

Selection: What the 20% of math concepts, that if made really strong, would solve 80% of math problems

Sequencing: What order of material should you study for maximal progress

Stakes: Find some incentive to complete the problem. Some nice view of mathematical terrain, as part of a masters program, applications to another field, a prize, a cookie. Anything that motivates you to actually make progress towards the goal

This approach helped me learn a bunch of high level math like abstract algebra, analysis, linear algebra, etc.

“Concrete Mathematics: A Foundation for Computer Science” by Knuth, Graham, and Patashnik – solid foundation in mathematical concepts and techniques, and it helped me develop a deeper understanding of mathematical notation and problem-solving.

“Introduction to the Theory of Computation” by Michael Sipser – introduced me to the theoretical foundations of computer science, and it helped me develop a strong understanding of formal languages, automata, and complexity theory.

“A Course in Combinatorics” by J.H. van Lint and Wilson – provided a comprehensive introduction to combinatorics, and it helped me develop a strong understanding of combinatorial techniques and their applications.

“The Art of Problem Solving” by Richard Rusczyk – This book is a comprehensive guide to problem-solving, with a focus on mathematical problem-solving strategies. It helped me develop my problem-solving skills and learn how to think critically about mathematical problems.

I think this error in thinking comes from the fact that Sigma notation can often be trivially implemented as a for loop.

Programming languages are designed to describe a specific computation, whereas mathematical notation is typically trying to describe an idea (one that might not even have a implementation!) Notation only sometimes and coincidentally describes computation as well.

The ambiguity, implied variables etc are an essential part of mathematical notation in the same way it is in common spoken language. Mathematical notation exists to help abstract and work out very hairy ideas, and often that ambiguity is necessary to show connections.

> code should be optimized for readability, not writtability

Mathematical notation is readable if you’re literate in it. It takes lots of practice to become fluent in it, but once you become more familiar it’s much easier to read than text (which is why it’s used in the first place). Mathematical notation is an extension of mathematical writing, not computational implementation.

Reading mathematical notation is much closer to reading poetry than reading code.

And so far this year I haven’t missed a day yet. Now what constitutes that hour can vary. It can be watching math videos, it can be solving problems on paper, and I might even let myself count futzing around with numerical computing stuff or something at some point. In practice so far it’s basically always either watching videos, reading books, or doing exercises (from books).

I won’t claim that everybody must do this, or that you need to commit 1 hour every day. Maybe 30 minutes would be fine. Or maybe some people who can spare the time would be well served to commit 2 hours a day. Who knows? But having some kind of routine strikes me as something that most people would probably find valuable.

Here’s a good starting point for philosophy of mathematics
:

https://plato.stanford.edu/entries/philosophy-mathematics/

The problem is that it’s in Latin and quite impenetrable.

We have some geniuses here and they would no doubt be able to take away a lot from these texts, but for you normals out there: don’t optimize too much, you’re quite alright in taking the normal approach of just taking a class at a community college, doing the exercises the teacher assigns, etc.

I haven’t read Mathematical Principles of Natural Philosophy (the English title) but I have read Euclid and it definitely doesn’t require a genius to understand. here is an online edition with great illustrations:

https://www.c82.net/euclid/

Anyone knows how long it would take to learn enough latin to undertake such a task?

I bet you could start this week if you used machine translations as a crutch.

I’d start working with a publication that exists in Latin and a good translation, so you can compare your work.

My fear with machine translations is that subtle errors here and there might throw me off in something like math where things are precisely stated.

A year of part time study sounds doable though

In fact, you’ll find that many philosophers will just use the Latin words directly, and not bother translating them – Latin qua jargon if you will.

Once you’ve learned the various forms of “is” (sum, very irregular) you can kinda survive reading without conjugations, just like this sentence can be worked out:

he to go to store and to buy cheese yesterday

In general I find that if someone is insisting that you study the philosophy of someone in their original language, they don’t have a good enough translation yet.

Edit: Also, to state the obvious, it’s been translated into English

This will surely make you more appreciative of subjects and concepts you are learning.

For me, a watershed book was Introduction to Analysis by Rosenlicht [1]. Proof-based, very “mathy”, small and compact (so to speak) but with a massive scope. A great introduction to a really important topic, and it’ll put your brain through its paces.

Again, I recommend working nearly every problem.

A note: this isn’t a resource for higher-level, proof based maths. It will give you a solid foundation and a pragmatic understanding to build upon. Very useful for STEM.

Link: https://tutorial.math.lamar.edu

– I was revisiting a topic in greater depth, which is a common theme in university-level math courses.

– It is a rigorous book, written in the style of definition, proposition, theorem, etc.

– It was the first math book where the exercises don’t just reinforce what you learned in the chapter, but teach you new material (another common theme in advanced math textbooks).

– Linear Algebra is arguably the most important math subject these days.

But I’d like to mention two books I read as a child which had a life-altering effect. They probably wouldn’t do any good for an adult, but might really help your kids… Unfortunately, I don’t remember the specific titles or authors (I was probably around 10 yrs old). The first was similar to this book:

“Speed Math for Kids: The Fast, Fun Way To Do Basic Calculations.” This gave all sorts of advice and tips to quickly do math in your head… simple things, mostly. For example, to multiply by 18 just double, multiply by 10, and subtract 10%; or how it’s frequently faster to multiply numbers by moving from most significant digits to least, which is opposite of how we’re taught; or how to quickly estimate square roots. This really didn’t teach new concepts, but by making routine and tiresome math operations faster and easier, it made the entire field more enjoyable to engage with.

The second book was a guide to slide rulers, and I couldn’t even find a similar book on Amazon. But learning advanced slide ruler techniques can trigger an epiphany; you learn mathematical relationships, how you can transform how numbers are represented. It was the first time I really saw an elegant structure behind the math.

Art of problem solving(https://www.amazon.in/Art-Problem-Solving-Basics/dp/09773045…) can also be a fantastic begin, particularly should you don’t have a lot expertise.

Maybe I’m unique in this regard, but I always took it as sort of implied that “reading a math book” entails “reading the book and working (at least some of) the exercises”.

The tricky part is once you get to math where you can’t trivially check your answer by “substituting back in” or “using a calculator” or whatever. Doing proofs, for example. Without a teacher, how do you know if your proof is correct? So far the only thing I’ve really found to do for that is to post on MathOverflow or one of the “learn math” related sub-reddits. I’ve often wondered if learning to use an automated theorem prover / proof assistant of some sort would be helpful, but that’s such a huge undertaking in its own right…

I guess if you want to learn thinking but not necessarily math “thinking mathematically” above mentioned is your friend.

For more applied or computational branches of math, I’d also add how to check your answers by using numerical methods or a computer algebra system if possible.

It is probably the best pair of books ever written for what I like to call “plug and chug” maths. Strouds books cover the whole of engineering mathematics.

I turned to them for calculus in first year, and for Fourier and Laplace in my final year and masters.

But it will not teach you how to solve and develop proofs.

For example, the book uses “the box model” all over the book but is not used anywhere else, and every else uses the phrase “i.i.d” which is not used in the book.

Still, it’s been really useful at my job in reasoning about timeseries data from Prometheus, especially in canary analysis. Far more useful than the whirlwind tour of distributions my 1 semester “Statistics for Engineers” course in college undertook.

Waiting to get my hands on his book ‘Measurement’ and approach it more like art.

If what he says is true, perhaps many who would have turned out great at math are locked out by how it’s taught in school.

For now, I have a test subject of one 🙂

I have had life things interfere with my learning now for the past month or so, but I hope to get back to it soon.

Chapter 4 is a great place to learn about topology for the first time.

In general, it kicks up the mathematical rigor you’re used to a notch. Seeing “>” defined as “not <” really blew my mind when I first read it! “<” is just something that satisfies some axioms, like anything else in math.

IMO, programming is “easier” to learn on your own for a few major reasons:

1) The sorts of things most people are interested in building just aren’t unforgiving intellectually in the same way that math is.

2) You have a compiler to check if you’re right, and your code will often still work even if it’s “wrong” (not as efficient as it could be, has unwanted side effects, etc.). In some sense the compiler is a bit like a teacher this way.

3) With programming, you can upload and make it available for free, and whether it’s legit or not is largely disconnected from your pedigree or how “correct” it is. This makes programming far more accessible. This makes sense considering that programming is primarily a practical tool. On the other hand, mathematics is primarily a field of scientific inquiry and is judged by different standards. If you learn a bit of math, try to write a paper, submit it to the arXiv… well, people will probably think you’re a crank.

On the other hand, if you’re just interested in math for the love of the game… you can certainly pick up a book and read it, maybe work some problems, but I think at this point it’s quite easy to fool yourself into thinking you understand more than you actually do. I guess there’s no real harm in being a charlatan, but probably the average person is interested in having some kind of real relationship with mathematics that they can be confident has a firm foundation. I’m very skeptical most people can truly pull this off by just reading books and not actually going to school.

—

As an aside, I think the fetishization of math in programming communities is very interesting…

Your point (2) about compiler/interpreter in programming giving you rapid objective feedback is spot on and a vital component for deliberate practice that most people don’t think on. You can kind of get this in math, in particular when you have some familiarity with the subject matter so the machinery isn’t “too abstract” for you to sort through. (I.e. you should be able to confirm whether your proof/answer is accurate the vast majority of the time.) This is much trickier for first exposure to a subject though and the checking effort is on you, not the compiler.

The biggest issue I’ve seen with people self studying or in small math groups is your final (non-aside) paragraph which is perhaps more a psychological problem than and aptitude problem. When things get tough there’s an enormous temptation to delude yourself to think you understand something that you are clueless about. The typical, schoolroom, way of mitigating this is via a final exam and you can check your grade at the end of the class; this gets typically gets short circuited in self guided study. Exams, btw, are essentially validation data sets you compare your math knowledge/model against. (We can call them ‘test’ sets if you prefer). The most important step really is repeatedly seeing how your knowledge works out of sample i.e. on ‘new’ stuff that comes out of the wood works and math.stackexchange is a perfect place for this when dealing with undergrad to mid-grad level problems. I do this all the time to get a sense of my understanding of a new subject I’ve recently acquired. But most people refuse this final step. People will tell me its ‘too hard’ and ‘takes too much time’ (meanwhile they start a new math book) but I strongly suspect it’s in large part due to cognitive dissonance. (Another kind of out of sample test comes up when working on a subject matter that uses something you just “learned” as a pre-req, though there’s a recursive element here and at some point they basically need to interact with 3rd parties.)

I suppose my relatively minor quibble is how much effectiveness depends on being “very gifted” [in some sort of math specific sense] vs understanding the basics of self-learning and being psychologically aware (astute?) enough to not go into denial. Insert quote from Feynman or whomever about how easy it is to fool yourself.

Edit: just to add, I struggle with math, lest there is any misinterpretation of my perspective. I went through phases where I thought I just needed to find the right books or the right teachers who could explain it in a way that meshed with my “learning style,” but ultimately concluded I just don’t have a very strong innate ability in the subject.

Wouldn’t that depend on what level of skill we’re talking about? I believe that most people who need to learn just, say, high-school algebra/geometry/trigonometry/etc. can probably do so with “just books” if they are motivated.

Heck, I’d even venture that most anybody who is motivated enough can learn at least through Calc III on their own using online resources. I’m going through Calc III right now actually, using a combination of books and online videos, and there hasn’t been anything that has struck me as particularly challenging so far. This after only making it to Calc I before I dropped out of school “back in the day”.

Getting a degree in math is a good start, but even then it will be limited if you don’t work with other people, go to office hours, form relationships with professors—embed yourself in the culture, so to speak.

I think these things would be critical if one wants to become a mathematician. But for somebody who just wants to learn more math for the purpose of reading (non math) research papers (maybe in machine learning to pick an obvious, if possibly trite, example), solving problems in everyday life or at work, or possibly even for doing research in another (non math) field and then writing up their research, I would think of all of those things as being “nice, sure, if you have access, time, money etc. But not even close to absolutely essential”.

Unfortunately the OP didn’t really say what their goal is, so it’s hard to say what advice makes the most sense for them.

Also unfortunate is that for probably most people who are working career professionals, there likely isn’t enough free time available to go back and do an actual math degree in school. So learning on ones own from books, videos, etc. is probably the only viable choice.

My response was based on the reader saying something about “mathematical thinking” in their post. In my experience, programmers talking about “mathematical thinking” usually have math insecurity and are referring to more advanced topics, thinking they need to consult tomes of deep mathematical wisdom to correct this deficiency (wrong). Could be totally off base, of course. But note some of the other replies here, suggesting very sophisticated books. Are they appropriate for a random HN poster who is soliciting random book suggestions to improve their “mathematical thinking”…? Seems unlikely to me…

One caveat, though:

… Also unfortunate is that for probably most people who are working career professionals, there likely isn’t enough free time available to go back and do an actual math degree in school. So learning on ones own from books, videos, etc. is probably the only viable choice.

I’m not so sure about this. If, like you say, someone is in a job where they need to read some papers with some math in them, they should be organizationally near someone who can help them out. If possible, I think a better strategy (better even than getting a degree) would be to find these people and develop a relationship with them to the point where you can ask them questions. They should then be able to explain any unclear notation, unfamiliar (simple) concepts, etc. Possibly some of that person’s advice might come in the form of “watch a Khan Academy video on Topic X”. But this will be a far more productive use of time than self-directed learning in this case.

On the other hand, if they aren’t near anyone with those skills but they’re reading these papers… something is probably amiss. For example, if they’re reading a machine learning paper which requires a significant knowledge of “engineering math” (Calc 3, linear algebra, etc.), and there is no one with that knowledge nearby… having them read that paper is probably a waste of time from an organizational perspective.

There’s also the question of why they’re reading that paper when they don’t have those basic mathematical skills. Without those skills, it is unlikely they will be able to do very much that is useful with it. If they want to implement the algorithm because they think it will be suitable for some task, I would argue that without those skills they are not in a good position to be able to accurately assess whether the algorithm will perform well. Part of what you learn in an advanced degree is how to read a paper—i.e., how to sniff out the bull shit, what to be wary of, etc.

Fair point.

One could also form a math “study circle” of some sort if you are in an area with enough mathematically oriented folks to find people to participate. I did this briefly and it was a valuable thing. It kind of fell apart for different reasons, but I could see doing it again at some point.

Also, Vibrations and Waves, by AP French. Granted, this is a physics book, but I appreciate his style so much. He makes use of a lot of geometric methods to solving problems. It definitly expanded my math horizons! His other books are good too.

First time I’m hearing about this one, thanks for the recommendation. Unlike Calculus or even a typical one semester Statistics course, probability is one of those topics where you need to see a lot of problems to really grok anything. The only way is to see a lot of solved problems and think about why that’s the right answer.

Even highly recommended books (e.g. by Blitzstein) don’t have enough solved problems, so it’s nice to there’s a problem focused book out there.

History of mathematics will give you a very subtle entry into the minds of mathematicians and the motivation behind their theorems.

This will surely make you more appreciative of subjects and concepts you are learning.

In a similar spirit, but with a much more geometrical flavor, there is Evans-Gariepy.

I am not sure if this book is particularly good or better than other books. (Well, it still looks like a very gentle introduction to the topic.) But as per your question, this was the book at the right time for me.

“Fourier Series and Orthogonal Functions” by Harry S. Davis. https://www.amazon.com/dp/0486659739/

To become a better problem solver with high-school level maths:

  - Polya's How to Solve It.
  - Books of your choice about math contests.
  - Concrete Maths. I understand that this book is taught in college, but it requires very little advanced maths, and its techniques are hugely useful for high school students too.

To hone my intuitions. I learned it the hard way that college maths were different from high school math: in high school, my teachers painstakingly drilled intuitions into us with very targeted explanations and tons of well designed exercises. In college, we won’t get such luxury. So, it’s really up to us to understand mathematical concepts intuitively before diving into technical details. For that matter, the following books helped me a lot:

  - The visual series. Visual Complex Analysis and Visual Group Theory, for instance
  - Pinter's A book of Abstract Algebra
  - Strichartz's The Way of Analysis
  - Linear Algebra Through Geometry by Wermer. The book offers a comprehensive geometric interpretation to linear algebra concepts. It's especially helpful for me to understand quadratic forms.

To understand Analysis better. This area is vast, so I’ll skip recommendations of excellent text books:

   - _Counterexamples in Analysis_. Those counterexamples in Analysis play a huge role in helping me truly appreciate the intricacies of Analysis. Similarly, books like _Counterexamples in Probability and Real Analysis_ are of great help too.
   - The Way of Analysis by Robert S. Strichartz. This books is AMAZING for laymen like me. You'd want someone to *explain* how concepts emerge, and how intuitions evolve.

To become good at maths by doing maths, so the following books used to help me a lot:

  - Problems and Proofs in Real Analysis
  - Putnam and Beyond. I still suck at maths, but those well designed problems in Putnam really taught me how to seek insights in higher maths.
  - Piotr's Problems in Mathematical Analysis. But really, any problem books that challenge you will do. I'd recommend you find problem sets from the website of university courses. They cover essential techniques, and will not be as overwhelming as the books.

For those looking to delve into discrete mathematics, I highly recommend the lecture notes from L. Lovasz and K. Vesztergombi (Yale University, Spring 1999) and from Eric Lehman, Tom Leighton, and Albert Meyer (MIT, 2010).

Sarason’s book is only 160 pages long, with legible text and clear examples. It covers the length of an undergraduate university class, and explains Holomorphic functions perfectly. The proofs are crystal clear, and so are the motivations. I haven’t seen a better introduction.

This is an introduction written by the original author of the list:

“Somehow I became the canonical undergraduate source for bibliographical references, so I thought I would leave a list behind before I graduated. I list the books I have found useful in my wanderings through mathematics (in a few cases, those I found especially unuseful), and give short descriptions and comparisons within each category. I hope that this list may serve as a useful “road map” to other undergraduates picking their way through Eckhart Library. In the end, of course, you must explore on your own; but the list may save you a few days wasted reading books at the wrong level or with the wrong emphasis.

The list is biased in two senses. One, it is light on foundations and applied areas, and heavy (especially in the advanced section) on geometry and topology; this is a consequence of my interests. I welcome additions from people interested in other fields. Two, and more seriously, I am an honors-track student and the list reflects that. I don’t list any “regular” analysis or algebra texts, for instance, because I really dislike the ones I’ve seen. If you are a 203 student looking for an alternative to the awful pink book (Marsden/Hoffman), you will find a few here; they are all much clearer, better books, but none are nearly as gentle. I know that banging one’s head against a more difficult text is not a realistic option for most students in this position. On the other hand, reading mathematics can’t be taught, and it has to be learned sometime. Maybe it’s better to get used to frustration as a way of life sooner, rather than later. I don’t know.” – by original author.

Fantastic book for Discrete Mathematics with lucid explanation and good exercises. The other one would be concrete mathematics.

Reading about running does not make you a better runner. You can watch 1000 marathons, sprinters, Olympians. You may get _ideas_ for running, but it will never make you a better runner. To be a better runner you have to do it. To be a better programmer/mathematician/physicist/whatever, you need to go work at it.

I suppose I am just taking action against how the question is written, but I see a lot of people seemingly hoping that “if they just found the correct book, tutorial, or video, they would be better”. A lof of those people are my students. When I ask how many problems they have worked, I typically always get the same response. Zero, or the bare minimum.

Your answer is somewhat helpful in the latter sort of world where OP didn’t know how to read a textbook (which is unfortunately common) but not in the former sort. You seem to be venting though, which is understandable. But venting with a side of helpful content would be even better.

For instance, advising OP on how to to read a maths book (generate content yourself, check dependancies, connect things to what you know etc.) or suggest books which contain advise like this alongside their main content (I think Tao’s analysis texts might do this?)

But math books are often full of exercices that you are supposed to do! Actually reading a math book means that you try to anticipate the proofs before reading them, and you work out all the details and do all the exercices. Reading a math book and working math problems are essentially the same thing.

It introduces math from a mathematician’s standpoint (full with proofs, and so on.) moderately than rote memorization and workouts, however it does so from the attitude of a programmer.

That’s because math is fundamentally arbitrary. This realization came to me late in life. Math was always presented as some aspect objective reality. However over time I’ve come to understand it as software for human brain. Using this math or that to describe something is often simply a matter of taste! Very similar to programming, in fact. Your study of history gives context to the question of utility different maths “packages” for certain problems, but does not invalidate your original impression.

It’s easy to create something new, because you get to play with the rules – but unless it is in some sense deep, effective, and usually elegant [1] it won’t stick around.

Beginner: NL Biggs, Discrete Mathematics, Oxford University Press

Intermediate: PJ Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press

Advanced: JH van Lint & RM Wilson, A Course in Combinatorics, Cambridge University Press

I can’t remember if the book addresses this, but for myself my inability to tolerate frustration really impeded my ability to work on any mathematical challenge for decades.

  How to Solve it -- Polya
  The Art of Problem Posing -- Brown and Walter

I’m not sure it made me any _better_ at math, but I did always enjoy

  How to Lie With Statistics -- Huff

My intro to abstract math… Wide range of topics, very clearly written and very well structured. Sets, groups, vector spaces, tensors, topology, differential geometry, lie groups and more.

An Introduction to Category Theory by Harold Simmons

Very enjoyable read. You cannot go wrong with this as your first book on the subject.

Chaos theory and deterministic systems are a fascinating vantage point for thinking about the dynamics of large computer systems. Thinking of them as stochastic systems is sometimes useful, but most of the systems are actually just operating in unstable periodic processes which are much closer to being a chaotic system rather than a stochastic system. This influences how I think about testing and debugging large distributed systems.

I will say, I’m not sure I could have learned it well without a class and a good professor. The author has a number of books though and is a professor at Cornell.

Sure, it’s focused in the direction of center and highschool college students. Sure, I learn it and (extra importantly) labored by many of the issues in my mid-30’s. It’s nice if, like me, you coasted/crammed by your early arithmetic training and by no means felt such as you dialed within the fundamentals. Additionally it is nice if, like me, you wanted some pen-on-paper observe and didn’t know the place to begin.

2. Differential Equations and Dynamical Systems by Lawrence Perko – Solidified for me how dynamic systems behaved and were solved. Very much helped my understanding of control theory as well.

3. A Concise Introduction to the Theory of Integration by Daniel Stroock – Helped solidify concepts related to Lebesgue integration and a rigorous formulation of the divergence theorem in high dimensions.

4. Convex Functional Analysis by Kurdilla and Zabarankin – Filled in a lot of random holes missing in my functional analysis knowledge. Provides a rigorous formulation of when an optimization formulation contains an infimum and whether it can be attained. Prior to this point, I often conflated the two.

I did horribly in math because I figured it was hard and just accepted I’d never be good at it. That quote somehow managed to dissolve my mental block.

Matrix Analysis by Horn and Johnson (perhaps the best end-of-chapter problem sets of any math book I’ve encountered!)

Matrix Computations by Golub and Van Loan

Elements of Statistical Learning by Hastie, Friedman, Tibshirani

Functional Analysis by Reed and Simon

But I’ve enjoyed the following texts to a larger extent than others:

– Algebra: Chapter 0 (Aluffi)

– Real Mathematical Analysis (Pugh)

– Mathematics and its History (Stillwell)

– An Introduction to Manifolds (Tu)

– Gauge Fields, Knots and Gravity (Baez)

– A First Look at Rigorous Probability Theory (Rosenthal)

– All of Statistics (Wasserman)

There are some authors I trust and am happy to buy so long as the topic vaguely interests me: VI Arnold, Tristan Needham and John Stillwell.

I really like the list put out by @enriquto in a separate comment, but I’ve avoided duplicating those recommendations in the list above.

* Differential Manifolds by Antoni A. Kosinski

* Introduction to Smooth Manifolds by John M. Lee

Mathematical Tapas: Volume 1 and Vol. 2.

* An Introduction to Manifolds by Loring Tu

* The Elements of Integration and Lebesgue Measure by Robert G. Bartle.

These two books were instrumental to my studying for my qualifying exams.

I find that my biggest barrier to learning math has always been how unengaging, excessively contrived, and unfun the learning material has been.

My mom was really into mathematical proofs and I being a huge loser kid with no friends naturally took to this book as well.

0. Jan Gullberg, Mathematics, From the Birth of Numbers. A highly accessible popular survey on different branches of higher mathematics. I read this over the Summer between high school and starting my undergraduate degree. It’s what made me want to study math. Previously I’d wanted to be a guitar player, but had to find a new ambition after an injury left me unable to play.

1. The high school mathematics series by Israel Gelfand. Algebra, Trigonometry, The Method of Coordinates, and Functions and Graphs. I
didn’t have much mathematics background in high school, but working through these really solidified my grasp on the basics.

2. George Polya. How to Solve it. A short book giving excellent high level advice on mathematical problem solving.

3. George E. Andrews, Number Theory. I worked through this freshman year contemporaneously with my first proof based class on simple logic and set theory. A very beautiful and accessible introduction to basic number theory. The combinatorial/geometric proofs of Fermat’s Little Theorem and Wilson’s Theorem are lovely. It also includes a very nice proof of Chebyshev’s theorem on the asymptotic density of primes and even the Rogers-Ramanujan identities for integer partitions.

4. Vladimir Arnold, Ordinary Differential Equations: Undergrad ODE classes are often taught in a cookbook fashion and if so, don’t offer much enlightenment. This book explains what’s going on at geometrical level. I didn’t appreciate ODEs until I read this. See https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html for Arnold’s views on instructing arithmetic.

5. E.C. Titchmarsh The Idea of Features: Really helpful by my undergraduate advisor as a result of he seen that I favored studying older books. It accommodates sections on advanced evaluation and actual evaluation with measure principle, however I’ve solely learn the advanced evaluation sections. It is not for everybody, if I recall appropriately, there may be not a single image, however it is rather energetic and has a variety of materials you will not discover in a normal advanced evaluation ebook, together with Dirichlet sequence. Wonderful as a complement to a normal advanced evaluation ebook.

6. George Polya. Arithmetic and Believable Reasoning. A wonderful growth on Polya’s concepts on The way to Resolve it. Whereas the aim is to hunt rigorous proofs, to get there it is highly effective to have the ability to suppose based mostly on instinct, heuristics, and believable reasoning. Loads of math exposition is theorem/proof based mostly and would not assist develop these abilities. In the same vein, see additionally Terence Tao’s traditional submit There’s extra to arithmetic than rigour and proofs https://terrytao.wordpress.com/career-advice/theres-more-to-….

7. H.S.M Coxeter, An Introduction to Geometry. A ebook of very lovely classical geometry. One thing sometimes not touched on in any respect in a typical arithmetic curriculum.

A lot of the advice seems obvious in retrospect but being systematic about a problem solving framework is enormously helpful.

In the spirit of OP’s question:

How to Solve it by G. Polya

Solving Mathematical Problems by Terrence Tao

Introduction to Mathematical Thinking by Keith Devlin

Are all amazing, How to Solve it in particular is an all time classic.

College Algebra Heineman

Discrete Math Rosen !

Linear algebra D lay

Calculus Stewart

Nonlinear Dynamics Strogatz +

Combinatorics Mazur +

Statistics *

ESLR +

Gödel, Escher, Bach: an Eternal Golden Braid – Hofstadter

Euclid’s Elements

The reason I recommend it is because it shows mathematical reasoning that is easy to follow and relevant to your daily life. It’s real math, but very easy to read through and understand. If your unfamiliar this paper is where the very idea of “bits” comes from.

One of the most important things in the paper for non-mathematicians to see is that the definition Information Entropy is derived simply from the mathematical properties Shannon desires it to have.

This is important because I find that one of the biggest questions people ask about mathematical formula and idea is “What does this mean? Why is it this way?” without realizing that math is really not engineering nor physics. When deriving his definition of Information, Shannon simply states that information should have the following x,y… properties and then goes on to show that the now standard definition of information meets all these criteria.

In mathematics it is very often the case that only after an idea is created to we start realizing the applications. This is quite different than science where a model is only adopted if it correctly describes a physical process.

Work through the paper and you will have worked through the mathematical underpinnings of the information age and will likely have understood most of it pretty well.

0. https://people.math.harvard.edu/~ctm/home/text/others/shanno…