Do you have any special math books that you hold close to your heart because of the value they delivered specifically to you and your mathematical thinking and skills?
* The Art of Probability by Hamming. An opinionated, slightly quirky text on probability. Unlike the text used in my university course its explanations were clear and rigourous without being pedantic. The exercises were both interesting and enlightening. The only book in this list that taught skills I've actually used in the real world.
* 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.
These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is:
* Basic Mathematics by Lang. Covers basic algebra and geometry at high school level.
Then one of these two, depending on your interests, or both:
* Vector Calculus, Linear Algebra and Differential Forms by Hubbard and Hubbard. Takes you through linear algebra, single-variable calculus and multiple variable calculus. Analysis is discussed in an appendix. All proofs have a constructive bias, so it's very algorithmic and natural for a CS-minded student. Solutions are in a separate volume.
Meh, Hubbard and Hubbard is good, but it certainly does not take you through single-variable calculus. For example, the following topics are assumed and not treated in any detail:
- power series and analytic functions
- the various properties of exponentials, logarithms, trigonometric functions, etc.
- L'Hopital's rule
- Integration by parts
You should definitely work through something like Spivak or Tao in addition to Hubbard.
I don't think this is a big concern. Hubbard discusses L'Hôpital and integration by parts, but it doesn't place a great deal of emphasis on them because the way topics are presented is a bit unusual.
If this is a concern, you can always use another text. I don't think this is a problem. Some freshman courses jump straight into Hubbard.
I disagree. L'Hôpital's rule is not discussed, it is simply presented without proof. And I think it's very important to carefully construct logarithms, exponentials and trigonometric functions, e.g. through power series - else the properties of these functions are just arbitrary axioms. Don't Hubbard and Hubbard claim themselves that the reader should be familiar with power series (somewhere in the chapter on Taylor series)?
Don't get me wrong, I really like the book, but I don't believe it replaces a book on single-variable calculus. I think the authors would agree.
Hubbard & Hubbard, which I referred to in my previous comment, discusses the basics of probability up to the Central Limit Theorem and a bit of Monte Carlo methods. Probability and statistics are really broad. What areas are you interested in?
If you are trying to do inference on large datasets, I think Andrew Gelman's books are superb and will teach you invaluable skills using multi-level Stan models, i.e. generative models and Bayesian inference.
If you want to study basic probability theory and stochastic processes, I really like Elementary Probability Theory by Chung. His more advanced book is really famous, but has a measure-theoretic approach and that's not very practical unless you are interested in developing theories a bit further.
Taleb is really fond of Probability, Random Variables and Stochastic Processes by Papoulis. But I find the typesetting in later editions to be really disorganized and confusing. Take a look. It's a good alternative to the first Chung book.
There is a large number of html format books on bookdown.org, mostly related to Data Science and R, lots of Bayesian too.
There are some more theoretical math books there, one of which I found to be very well written: Theory of Distributions by Peter K. Dunn
* 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'm in love with the tone of these articles, serious and playful at the same time.
I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school.
Do you know a good path or book that's suitable for that?
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.
Thinking of doing the same thing. The last math class I had was at 16, and the most advanced classes was on binary, so not very complex stuff.
I've mostly been winging it for another 16 years and seemingly picked up things here and there. But math is definitely been trial and error, and I definitely do not know the lingo in math, which I'm now starting to feel is a big disadvantage.
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.
If you don't mind the question how is the Remarkable helping you in this. Just to avoid the clutter of paper? Or does it some how OCR you're handwriting?
I hate the clutter of paper and how hard to organise it is for me. Plus, how messy I am writing on paper and crossing things out all the time.
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.
A similar solution might be https://getrocketbook.com/ RocketBook. It can be a $0 dollar solution if you download their app and then print out the free PDF pages that are pre-formatted.
I have a remarkable 2 and a Microsoft Surface Pro, intended to de-clutter math coursework. Both work, but I found that the real estate on the remarkable was too limited, even though its a great device, so I tried the Surface Pro. I can fit just about any sized work onto it, and you can endlessly scroll down which was something I couldn't figure out on the remarkable. It makes doing math easy or at least takes away some housekeeping which I find really distracting. And saving and organizing work and being able to import and export files is a bonus.
Sorry, out of topic. May I know what's your definition of success?
I've been doing software development for 10 years but success seems to be always on the horizon.
The reason I find them fascinating is that Wildberger doesn't agree with some of the conventional approaches, in particular with the use of infinity and taking limits. This leads him down interesting paths (e.g. Rational Trigonometry and Algebraic Calculus), which (a) show the process of mathematics (exploring, making definitions, building up in different directions, etc.), whilst (b) remaining mostly grounded and approachable (e.g. no appeals to inscrutable lemmas from abstract research areas).
For example, he's recently been making videos about "multisets" (computer scientists would call them Bags), their arithmetic (where "adding" is union, and "multiplying" is pairwise/cartesian product of the elements), and how this generalises: from an algebra containing only empty bags (trivial, but self-consistent; behaves like zero), to bags of zeros (behaves like natural number arithmetic), to bags of natural numbers (behaves like polynomial arithmetic), to bags of polynomials (behaves like polynomials in arbitrarily-many variables) https://www.youtube.com/watch?v=4xoF2SRp194
"The reason I find them fascinating is that Wildberger doesn't agree with some of the conventional approaches, in particular with the use of infinity and taking limits."
So no transfinite ordinal analysis or large cardinals? Hard to take him seriously.
> It's the hypotenuse of a right triangle with legs of length one.
Yes, we can construct such a line segment; but line segments are not numbers.
We don't actually need "legs of length one" (which pre-supposes some system of units); all we need is the ratio of the lengths of the sides. However, finding lengths requires the ability to take square roots, which would either make this a circular definition (e.g. that √2 = √2 / 1), or requires the limit of an infinite process (like Newton's method, or equivalent).
Instead, it's much easier to count the areas of the squares on each leg (1 and 1), and add them together to get the area of the square on the hypotenuse (1 + 1 = 2). No need for lengths, so no need for square roots, so no need for √2.
Wildberger abbreviates 'area of the square on a segment/vector' as the 'quadrance' of that segment/vector (defined as the dot-product with itself). Likewise we can avoid angles by taking ratios of quadrances (e.g. 'spread' is defined via a right-triangle as the quadrance of the opposite side / quadrance of the hypotenuse); together this gives rise to a whole theory of Rational Trigonometry, which gives efficiently computable, exact answers; works in arbitrary fields (except for characteristic two), and with arbitrary dot-products/bilinear-forms (e.g. euclidean, relativistic, spherical, etc.). Here's Wildberger's textbook on the subject http://www.ms.lt/derlius/WildbergerDivineProportions.pdf
no computer can calculate that exact distance, which is kind of Wildberger's point.
Infinities are very interesting but the non-infinite maths have kind of got neglected over the past 100 years. I had to memorize Laplace transforms in college but never heard of Fairey sequences until I watched his videos.
People get upset at him but he's basically just having fun seeing how far you can go in Math without infinity. It's quite interesting to a certain audience (like myself).
I’m reading and like Thomas Garrity’s “All the mathematics you missed (but need for graduate school)” which is this but for people who did a bachelors degree but missed certain areas (or forgot them).
Something else I’ve found extremely useful in getting into maths topics is the Princeton Companion to Mathematics - it doesn’t have exercises but gives excellent overview essays of a wide range of maths topics - expensive to buy (mine was a present) but should be available in academic libraries, say.
I bought and read it (more like, skimmed) and liked it a lot too.
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.
3Blue1Brown videos seem like a good resource to use along any book. My experience as a math major (in the distant past) is that the kind of visualization the author shows you is also something you want to imitate in your head when you are learning new concepts. I find things I learned in this level tended to stick in my head 10+ years later, other stuff less.
I should mention focusing on doing a few interesting problems, rather than many not so interesting ones, is also one way to help yourself understand more deeply.
Lots of easy problems is a good way to build up muscle memory, though. IMO the brute-force method of, say, Saxon Math really makes sense for things like basic elementary school algebra and probably intro calculus, where the student is sort of learning the math equivalent of how to walk. Not sure where the switch over ought to be, though.
we tried "Saxon math", Singapore math dimensions, and Beast Academy.
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.
Saxon math worked out for me, although we didn’t shop around as far as I remember, so maybe Singapore would have worked fine as well.
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. This is what I mean by muscle memory — I’ll use the famous names theorems when necessary but sometimes you just need to bash the math until the thing you want is on that side of the equal sign and the other stuff is on the other side.
I think anything that results in
1) actually reading some textbook
2) actually working through problems for a couple hours a week
will compare well to the typical US math education pretty well anyway.
I'm doing this and am starting with Linear Algebra on MIT OCW (taught by Gilbert Strang). My current plan is to relearn Linear Algebra, Calculus, Probability, and Statistics and actually focus on retaining the knowledge in my memory using something like SRS learning. I think planning past that is pointless since by the time I'm done I will have a better ability to plan my future coursework.
I did this same exact thing back in 2010. I used khan academy for it. Started with positive and negative numbers, arithmetic, through trig and algebra.
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
I had a similar thought back in 2014. I had only studied the maths required for various engineering courses I’d taken.
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’ve been doing a similar thing with Brilliant and really enjoying it. It feels like every course is orientated around teaching maths from a problem solving perspective so you actually get why you’re learning stuff rather than teachers just trying to brute force things into your head which unfortunately seems to be the default at schools nowadays.
Do you know a good path or book that's suitable for that?
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.
I will be checking out the ones I don't know, because the ones I do (Weeks, Arnold's classical mechanics and Spivak's differential geometry) are fantastic IMHO
Out of this list, the books I am familiar with, are great (Hilbert-Courant, Spivak, Korner's books). At the same time, even with extensive mathematical training, I haven't read them from start to finish. I wouldn't even like to say "read". For someone who's not used to mathematical reading, some of these books require careful study. That means generating examples to understand results (theorems), trying your own conjectures, proving things yourself etc. Over time, one becomes familiar with most/all the material in a book but the knowledge might have been acquired through various books (and courses) over time.
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.
"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.
To add one more thing: the "thing" that has helped me most lately isn't a specific book or video or anything, but rather simply committing to spending 1 hour every day on math. I even set up a Google Calendar task to remind me of this every. single. day.
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.
Yeah, I've been doing the same thing, but with the rule that I have to do "a math problem". Right now I'm going through a stochastic processes book a bit at a time that way and really enjoying it.
In my experience, nothing beats solving problems. Even if you get help via solution manuals. I went from being a B/A student to top of my classes (engineering) by solving many problems. I would do my homework and then go back before exams and redo the homework twice for a total of solving the problems 3 times. Never failed me. I actually managed to get a perfect score from a professor who wrote notoriously difficult exams where a 50% to 60% was curved to be a B
Yep, a book with proofs and worked exercises is great for this because you can try to do the proof and then look at the solution to see if you got it reasonably correct. In my case the book covers a lot of stuff that I passively "know", but working problems has helped get be back to an active understanding.
This is different from the other answers, but it does answer your question: When I was a kid I had tons of math and logic puzzle books. Two I remember specifically are "Aha! Insight" and "Aha! Gotcha" by Martin Gardner. Decades later, when a math problem comes up in my work, I have an apparently unusual ability to cut to the heart of it ("by symmetry, we must have X" or "looking at this extreme case, we must have Y" or "this looks like a special case of Z" sort of things) instead of starting by soldiering through equations, and I credit a lot of that to all the puzzle-solving I did as a kid.
“Mathematical Notation: A Guide for Engineers and Scientists”[0] really changed my abilities with being able to read papers and decipher what was going on. I had university math experience but it was a long time ago. When I started reading papers for algorithms later in my career I couldn’t get past the notation. Once the symbols are explained, as a programmer, I was able to grok so much more. This should be on everyone’s shelf.
As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.
I find one of the biggest mistakes programmers have about mathematical notation is that it's somehow just a terse, badly implemented programming language. But this is a very poor understanding of what mathematical notation is doing.
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.
> I find one of the biggest mistakes programmers have about mathematical notation is that it's somehow just a terse, badly implemented programming language. But this is a very poor understanding of what mathematical notation is doing.
No, we think that because proofs and programs are isomorphic[1]. It's not a mistake: traditional mathematical notation provably is a terse badly implemented programming language. Actually it's worse than that, because oftentimes it doesn't even parse. Now I'm not going to say I can't on some level see the appeal. After all I think Perl is a lot of fun to code in.
Naturally, its adherents are practiced at making a virtue out of its defects. Who wants to admit they dedicated considerable brainpower to doing something in a fundamentally suboptimal way? That doesn't really matter though. As Mathematica and other tooling shows, the formalists have already won and now it's just a matter of mopping up the stragglers, or waiting for them to age out. This isn't terribly surprising to those who know the basics of the history of mathematics. It took something on the order of two centuries before Recorde's innovation of the equal sign was generally accepted.
I get the feeling that you're stuck in Terry Tao's 'rigourous phase' of mathematical understanding, where everything in the end is a computation and has to be carried out according to a set of rigorous steps and definitions.
I get that, but it does miss a bit the cultural context of how mathematically fluent people use mathematics to communicate with each other. When you're discussing maths with colleagues in front of a blackboard, you're often not really trying to prove anything, but discussing the relationship between mathematical objects. In this context the ambiguity and implication in the notation is almost a requirement, otherwise the communication speed tanks.
Having a mathematical discussion between a group of people all fluent in the context and terminology is a wonderfully fluid thing.
Complete proofs and programs are isomorphic. Many proofs while incomplete are perfectly legible to experienced practitioners who can fill in the details without getting bogged down in trite steps. Morevoer, isomorphisms are simply provable facts between two types of objects. Most isomorphisms are generally used to convert one object into another object which is more amenable to being used in a given proof. An isomorphism does not beg the conversion and there are many trivial isomorphisms of limited use.
As a mathematician who’s only recently started to get into computation and programming, I think the difference between my thought patterns when switching hats is so fascinating.
I was so accustomed to hearing that mathematics is nothing if not rigorous, but the more I reflect, mathematics is much more dependent on social convention and agreement amongst a community. While an outsider might think that proofs rigorously establish theorems, the purpose of a proof might be better seen as having enough detail to convince a substantial portion of the prominent mathematicians in a field that the proof is correct. In fact, there are theorems (e.g. the ABC conjecture) where a “proof” has been proposed, but not enough mathematicians have expertise with the techniques used to prove it in order to agree whether the proof is sufficient or not (though I’ve heard that the general opinion is that the proof does not suffice). William Thurston wrote one of my favorite essays related to this topic: https://www.math.toronto.edu/mccann/199/thurston.pdf
Reflecting on my own experience in mathematics, a better way to think of proofs is as being composed of “thought patterns” which many mathematicians agree are likely to be correct - when I scan a proof, I don’t look through every detail to verify that it is correct, but rather run it through a series of high level tests to see if it fails in any way, then if it passes all of those I look more closely at the argument and analyze the structure and mathematical power of each statement (e.g. one is unlikely to establish a hard analytic result through purely algebraic means, so where is the magic going on?) and so on until I’ve convinced myself that the argument is probably true. Other times, the result may be “visually apparent” (e.g. in geometry) at which point it might be sufficient for me to just to connect certain canonical arguments with the pictures as I read through the proof. For an excellent overview of this process, read Terry Tao’s blog on identifying errors in proofs : https://terrytao.wordpress.com/advice-on-writing-papers/on-l....
I don’t feel as confident commenting on the programming/computational perspective, as I’ve probably developed a very idiosyncratic way of thinking from approaching the topic so late in my education, but my feeling is that they are much different, and that the types of things a mathematician wants to convey to another mathematician rely much more on “trust” rather than the kind of rigor that might be needed by a computer.
I think this would be an interesting topic to explore in longer form.
It can be made to be. Dijkstra came up with a nice and rigorous notation he used for his own proofs[1]. That page also includes some slightly spicy takes on why things are as they are. I agree that this is an area where the broader mathematical field has much to learn from computing science. The unforgiving nature of computing automata really drove that innovation. Meanwhile one can afford to be sloppy when one is trying to convince some other mathematician with a sky high IQ.
Our project IHeartLA is a language with syntax designed to closely mimic conventionally-written linear algebra, while still ensuring an unambiguous, compilable interpretation: https://iheartla.github.io/
Mathematics only clicked and became fun for me when I started using Wolfram Mathematica, because I could fairly easily mess around with the formulas I saw in books until I understood the types and arguments and what is an index vs a reference to some unnamed convention of the field.
I so so much agree. Just applying the D (differential operator) twice reveals what are we doing much more than the conventional d2y/dx2 notation.
By and large mathematics education has missed the point of invention of computers. It is only occasionally used to make a point. We should be teaching mathematics with a programming first approach: get your function code to compile, ponder on its signature, write some test cases to really understand what is going on.
Totally agree! I remember raising my eyebrow at ambiguities in math notation as early as high school, before I could articulate them as such. One specific example is the convention of cos^-1(x) referring to the inverse cosine of x, instead of the multiplicative inverse of cos(x). Similarly, the convention of cos^2(x) referring to the square of cos(x) instead of a nested cos(cos(x)). It's madness, and totally avoidable.
Couldn't agree more. Mathematician are masters at whipping up random notations and then not adhering to it rigorously. Higher level math would be an order of magnitude easier with machine checked syntax.
> Higher level math would be an order of magnitude easier with machine checked syntax.
This just isn't true, at least in terms of developing new mathematical ideas. There are already tools (e.g Coq) for providing mathematical syntax checking etc, but nobody uses these in developing new ideas because it would cripple the process of doing mathematics.
Particularly in the early stages of work, mathematics is very intuition and heuristic driven, you tend to sketch out an idea by actively using the ambiguity in the notation. Once you've got something that looks broadly correct you progressively try to shore it up by using more detailed and rigorous argument.
Maybe as an analogy, think of how architects design houses. They don't start by placing each brick according to correct civil engineering practice. They design something that broadly makes sense and then the civil engineers 'shore it up'.
You are right that it wouldn't be true in regards to new mathematical ideas. That's not what I meant, I should have been clearer. What I meant was 10x easier to learn.
Coq isn't about mathematical syntax checking. Coq is about encoding the whole proof in such a way Coq can machine verify it. That's 100 steps further then what I'm talking about. Just a simple syntactic check. Similar to what gofmt does. Is what you are writing considered valid syntax. Not whether what you are writing is correct.
My cultural reading of this notation problem is this: mathematical notation was formalised when scientists were writing papers and letters to each scientists in the era of scarce paper and no internet. So focus was on succinctness at the expense of explainability. It is akin to why commands on Unix machine were short, you were talking to an actual serial terminal to machine somewhere away. So saving a letter or two helped. Persisting with cryptic mathematical notation today, we are stuck with an idea well passed its sell by date. Mathematical notation, mostly, is not precise.
Sussman (who wrote the famous SICP book) wrote another book structure and interpretation of classical mechanics. Tough book to go through. But they start with the same premise: mathematical notation is confusing (and hand wavy at times). A better symbolic notation should reveal enough details to be able to code up the mathematics in a program. I found this approach to be bang on target, but could never get enough time to actually go through the book.
And I realised why the 'let us build it up from scratch' books work. They force you to think about the function signatures and shape of objects passed to each function. This approach reveals gaps in our understanding much better. For example, F=ma is looks like an algebraic statement, hiding the fact that `a` on the right is about time evolution of the system through the derivative.
Steven Strogatz made a funny quote in his infinite powers book. (I'm paraphrasing), if Newton was doing this in today's era he might create a flipbook animation to make this point and not symbols.
Wow, I wish I had known about this book (and had a license to Mathematica) when I was in college. I always got hung up on the notation and my inability to visualize the concept.
Plus one for this!
I bought two copies of the referenced book…and for the exact same reasons; I’m a programmer and being able to explain my algorithms using mathematical notation helps validate a program as well as troubleshoot a program…
An oldie but goodie is “Mathematics for the million”
I would not call myself great at math – I struggled with it in school, in fact – but in recent years I’ve begun “correcting” my lack of mathematical knowledge. The single best decision I’ve made is to first start with the philosophy of mathematics. Maybe it’s because my background is in philosophy, but I also think that for certain people like myself, understanding what math is makes me far more interested in understanding how it works, rather than just doing context-less calculations using formulas I don’t know the history or deeper purpose of. When I learned math in school, it was entirely cut off from any of these deeper questions.
Here’s a good starting point for philosophy of mathematics
:
Reading Euclid's Elements and Newton's Principia really helped me get an intuitive feel for geometry and calculus. They may not be entirely easy (at least the second) without some commentary, but well worth the study.
While it's laudable that you sought those texts and profited from them, I worry about what others might take away from this. When I was young I knew some geniuses who highly spoke of Principia and how it gave them great insights. And the teenager me said, okay cool, I'll have a go!
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.
It might make sense to read a translation in a language you understand. Many of the books that are considered classics are specifically because they ARE accessible. That doesn't necessarily mean that they are easy, but there is a big difference between reading Euclid and learning how to create mathematical proofs, and taking a class focused on calculating the area of various shapes or determine angles.
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:
From what I've heard, Euclid is fairly accessible and was for centuries the standard geometry textbook for children; the Principia is incredibly daunting, and Newton even admitted that he made it extra confusing on purpose to deter readers who weren't already experts.
I've been flirting with the idea of working through all of Leonard Euler's publications (as a life goal). Many of them are still not translated from Latin, so there's a possibility I may have to learn it.
Anyone knows how long it would take to learn enough latin to undertake such a task?
If you're trying to understand a domain work (such as Euler's) you could probably get a working knowledge in a month of strong study, a year of off-and-on.
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.
That's the advantage of it being a particular mathematic domain, you'll learn the terms relatively quickly and be able to catch errors in the math parts; the prose is where you will need the machine.
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:
Use the book "Lingua Latina per se illustrata" to learn Latin. It's quite magical, you just start reading Latin which is comprehensible due to similarities to English and it stacks on this without using anything but Latin. It's also much faster and more thorough than other textbooks.
I didn't read Euclid in Greek or Newton in Latin; there are quite good translations available - even free!
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.
This is sort of like recommending the art of computer programming as a way to learn how to code, isn’t it? Starting very far down the stack if you’re working through a 2000 year old book in Ancient Greek!
In that vein I highly recommend Foundations and Fundamental Concepts of Mathematics by Howard Eves. I think it might be a little dated, but it gives an amazing overview of the most important developments in mathematics that were relevant at the time. It's less focused on practice (though there are some problems) and more on the history and motivation behind the ideas. This book introduced me to axiomatics, non-Euclidean geometry, quaternions, and abstract algebra in my senior year of high-school.
I have a very similar background, did my undergrad in Philosophy and feel that I need to learn some basics. Do you have any pointers on where to move after this?
I just started with that SEP article and then googled around for some other books and videos. There are some excellent lectures on YouTube, this one for example:
Also, you might find that symbolic logic is a good introduction to thinking mathematically. I used Klenk’s Understanding Symbolic Logic for a course a decade ago and really enjoyed it.
For actual mathematics lessons, Khan Academy is pretty solid.
The best resource I've found is this random, somewhat obscure website (though I've learned that it has grown in popularity) called Paul's Online Notes. The professor has a real knack of pedagogy, and the problems are perfectly structured in terms of their difficulty. His explanations are clear and without jargon, and it goes from algebra to diff eq.
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.
In my experience, the best way to get better at math is to do a lot of it. Find some book that's "good enough" for some topic you're interested, and work many many problems from the book. You'll learn about the topic, but more importantly you'll learn problem solving skills. I recommend working the problem until you're sure the answer is right -- in grad school problem sets didn't have answers you could check, and full understanding was necessary to get the problem sets correct.
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.
This is a bit of an odd suggestion, but I learned the basics of category theory from the appendix to Weibel’s “An Introduction to Homological Algebra”.
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.
This happens a lot! For example, at the appendix of an advanced book on PDE (e.g. Evans') you find a three-page summary of main definitions and results in integration theory and L^p spaces. Or at the appendix of a book on differential geometry (e.g. do Carmo's) you find a succinct compendium of elementary differential calculus, explained in the most efficient way. These kind of condensed summaries, or fascicules de résultats, are rarely found on books that deal with the subject matter directly.
Lot's of good books already here!
In the same spirit as Polya's book "Thinking mathematically" by J.
Mason, L. Burton and K. Stacey
I learned a lot in the early days with Demidovich book on 5000 problems on mathematical analysis.
Tom Apostol books on calculus, but for me his book on analytical number theory. Alongside alan baker's thin book on number theory.
Gilbert Strang book(s) on linear algebra.
Rudin book on functional analysis.
Oh Hardy's book on divergent series!
Ian Steward books on transcendental numbers and Galois theory.
Elements of algebraic topology by Munkres is a fantastic book.
So many books are invaluable to me in teaching not only math but mathematical thinking.
I guess if you want to learn thinking but not necessarily math "thinking mathematically" above mentioned is your friend.
Came here to recommend Thinking Mathematically (text only) by J. Mason. If only every high school maths teacher had this book, the world would be a better place
This is a great book. Unlike any other book I've read. It helps with defining a process that I think every person working in a technical field uses one way or another. It helps you develop a way of thinking that reduces anxiety when you are stuck and helps develop paths to get unstuck. This book made me love math and I wish more people knew about it. It's also never too early or too late to pick it up.
Linear Algebra Done Right by Sheldon Axler for the following reasons:
- 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.
I have to second this. It's very well written and presents a clear view of what Linear Algebra is. Although it might be best used as a second book in Linear Algebra (depending on your preparation).
I wonder how good you can get at maths just by casually reading books. You need to work on problems for hours and hours to get a grasp on the theories. Programming is different in the sense that it's something people routinely do as a hobby because it's quite fun and addictive. But maths? maybe if you have already strong foundations you can pick up a new topic and develop your culture. But I doubt one can get these foundations without actually graduating in maths as it's an extremely strong commitment.
I second this strongly, you can’t get better at math just by reading books. You need to hone your problem solving skills, you need to fight with the problems, have the mindset of a warrior, a conqueror, only then you’ll get the juice out of it and have a clear understanding of the subject. I’ll suggest starting with Concrete Mathematics by Donald Knuth, it’s a beautiful book that catches the essence of mathematics.
I wonder how good you can get at maths just by casually reading books. You need to work on problems for hours and hours to get a grasp on the theories.
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 don't think you can get good at doing calculations without practicing the calculations, but reading books that discuss the higher-level aspects of math and the philosophical underpinnings can help you look at it in a different way that may inspire more interest as well as an easier time grasping the difficult parts.
Not a direct answer, but I once read that the best book about a technical topic is the third book you read on it. Often you'll see things in comments sections like: "I have heard this explained so many times by others, but this explanation finally clicked!". The assumption is that that's the case because the explanation is better, rather than assuming it's the case because you've struggled with the material before and you're still going at it.
Statistics by Freedman, Pisani and Purves. Don't know if I got better but loved the real world examples and cartoons. Does not have too many pre-requisites. Each section presents a tiny concept which is followed by plenty of exercises that have answers at the end. The furthest I got in a book in recent days, Math or not.
I taught from this book (it wasn't my choice, it was the standard book where I was teaching). It's really good for intuition, but because it doesn't use standard notation I think it might have done a disservice to students who were going to go on to learn more.
My pandemic project in 2020 was to finally read through the used copy I bought a decade ago. I agree it was really useful at building foundational intuitions. And that it doesn't use professional jargon which sometimes makes Stats Wikipedia's "death by integrals" approach a dense barrier to entry.
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.
Any suggestions for more conventional Statistics books? (Math-oriented, with proofs if possible). I'm reading Devore's one for Engineering and the Sciences and it's pretty good but I'm having a bit of hard time with p-values, hypothesis tests, etc. and wanted a second book as reference for those topics.
* The Language of Mathematics: Utilizing Math in Practice by Baber https://www.amazon.com/Language-Mathematics-Utilizing-Math-P... really helped me "get it", as I always found programming natural but math hard. This one is written by a CS professor and it really makes all the difference.
Measure and Category by John Oxtoby. This book studies duality results between different notions of "small" sets in measure theory and topology. It's the first (and to some extent the only) math book where things just clicked and I didn't feel like I was drowning in a sea of notation and ideas. Here are some more thoughts on it: https://bcmullins.github.io/Top-Books-2019.
Honestly, unless you're very gifted, a book on its own is not going to be enough to really develop your skills. You need to work interactively with a teacher. 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 getting engaged with an online community like MathOverflow or similar could be a substitute for this.
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...
Honestly, unless you're very gifted, a book on its own is not going to be enough to really develop your skills. You need to work interactively with a teacher.
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.
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.
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.
It's interesting. My first instinct was to disagree with this post, but on reflection I think I mostly agree with it. A couple useful mental models are (i) deliberate practice and (ii) train-validation-test(/out of sample) sets from machine learning
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.
> As an aside, I think the fetishization of math in programming communities is very interesting...
I studied a fair amount of math in school, then worked for many years as a programmer without using one bit of it. More recently though, it has become important again. I try to make sense of the posts here about AI and generally find I don't know enough math, despite knowing more than most programmers do (though far less than actual mathematicians do). So I do feel like I have to study more math if I want to understand that stuff.
Similarly, trying to learn Haskell (a worthwhile endeavour for any programmer) sent me into a deep rabbit hole in mathematical logic. Maybe it wasn't necessary, but I do think it solidified my understanding of Haskell, such as it is. I still have no idea what a left Kan extension is though.
I'm not trying to humblebrag, I think maybe I'm just stupid, but I find it hard to trust my understanding of anything unless all those details are nailed down.
Generally agree. There are books, and some are better than others, but unless you have both the passion and the aptitude, there is no book that will magically make everything understandable. If you struggle with math, it's probably you and not the book you are using.
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.
Calculus Made Easy and Probability Through Problems. I'm not sure that I'd have gotten through either my university Calculus courses or Probability and Statistics without these two books. I used them as supplementary material to the course textbooks and homework. They both have a style that is approachable and helped me build an intuition for the material unlike anything else I found.
I second this suggestion for Calculus Made Easy by Thompson. It's become a bit of a classic...was published in like 1915. Super unique approach to teaching calculus. It's an excellent supplement...lots of good insights. It may be particularly good for people who believe they're bad at math. His style may convince people otherwise.
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.
Velleman's How to Prove It greatly helped my ability to construct set theoretic proofs, which better prepared me for Spivak's calculus and Baby Rudin. Hamkins' Proof and the Art of Mathematics is designed as a a good, less set-theory heavy, introduction to proof writing that leads more naturally to analysis. OpenStax books are FREE.
Surprised to see Velleman's book so far down. It taught me that proofs are fun and do not in general require clever tricks. As a bonus, it provided plenty of practice with foundational objects such as sets, relations and functions. All this made me much better at doing mathematics and prepared to texts in real analysis, CS, algebra.
I have been going through Velleman, and it has been significantly helping me understand various CS papers and books, for example, I struggled understanding through proof outlines in PFPL, but working through just part of this book has helped.
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.
No specific book, but generic advise about how to use a math book. Homework, homework, homework. Read the whole thing, but focus on the exercises. Do every exercise as soon as you can manage: don't wait until you've read the whole chapter -- once you get confused and stumped, the lesson of the chapter becomes urgent and I find that sharpens my attention.
I usually enjoy Newport but... this was rather underwhelming? It's basically "do a lot of proofs and study consistently not just 48hs before the exam".
Most if not all of my math and some comp-sci courses would require this, they were very proof-heavy, specially Algebra, Logic and Graph Theory, and there was no way you could even pass just studying for a week after let alone 48hs before.
I mean it's underwhelming if you've done it before, but for people who are unsure about how to succeed in a proof-heavy course, seeing it laid out like this in concrete steps is helpful.
You've outgrown being in the target audience for it, but that doesn't mean the audience wouldn't find it helpful.
I would add understanding the reasons for definitions, how they fit together with theorems, lemmas, corollaries, proofs, and some basics of the format of proofs. You can find good explanations through Google.
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.
Rudin's Principles of Mathematical Analysis has a really special place in my heart. Chapter 3 is great- it's a great reference for derivations of a lot of fundamental identities about limits used in undergrad calculus.
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.
I don't think there is any book I've read as an adult that was particularly special. If I wasn't already good at math, I wouldn't be reading these books in the first place. Not that there weren't good and helpful books, but I wouldn't say any of them were revolutionary to me.
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.
I'm self-taught so for me it was learning how to write proofs that gave me a big boost in being able to branch out into different area of interest and not give up. :)
I have not yet become significantly better, and not a math book, but I recently read A mathematicians Lament by Paul Lockhart and it resonated so much with me that I plan to take another stab at math different from how it is taught in school.
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.
A lot of people have mentioned Keith Devlin's book "Introduction to Mathematical Thinking" by Keith Devlin but for me his other book "Mathematics: The Science of Patterns" was something that really had a huge impact on me just to put mathematics in perspective. Probably has something to do with my own personal character and education but I needed that perspective before I could take the next step. Then the "Introduction to Mathematical Thinking" is a great following read. So it depends where you are.
Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach
It's a rigorous but chatty textbook in the style of Spivak but written by someone who is sensitive to applied maths. I would not have survived my astrophysics classes without it.
For sure I got significantly(?) better with classics like Spivak, Apostol, Rudin.
"Real and Complex Analysis" by Rudin, and the two books both named "Calculus" from Spivak and Apostol. But also from Apostol his more concise and far-reaching "Mathematical Analysis". And from Spivak his small gem "Calculus On Manifolds" made quite a dent on me.
Other than more "classic math" books, I also wanted to mention two outliers that I found eye-opening and generally awesome:
* Geometric Algebra for Physicists, by Doran and Lasenby. I found the power and elegance of geometric algebra mesmerizing, and even if this book is also about physics and there may be more appropriate math-only books about geometric algebra, this is the one that made it for me.
> * Geometric Algebra for Physicists, by Doran and Lasenby. I found the power and elegance of geometric algebra mesmerizing, and even if this book is also about physics and there may be more appropriate math-only books about geometric algebra, this is the one that made it for me.
I've tried to read several of them, and, sadly, I feel most geometric algebra books fail at explaining it. It's a shame as it's part of what kindled my interest in pure mathematics and I still feel I'm nowhere nearer understanding it despite working through several other mathematics textbooks, including just plain algebra. But, it did spark my interest and now I've moved on to other interesting topics, though Geometric Algebra is still my white whale.
In addition to "Geometric Algebra for Physicists" (whose first two chapters I'd recommend to get a nice overview), I found Hestenes' "New Foundations for Classical Mechanics" to be very good and readable.
Also, there are many good resources in https://bivector.net/ , including videos, papers, presentations and programs.
Finally, an interesting paper (that got me kickstarted) is "Imaginary Numbers Are Not Real—The Geometric Algebra of Spacetime" by Gull, Lasenby and Doran.
I think you guys might find this list I found long ago very useful when deciding on a mathematics book you want to read.
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.
During my undergraduate studies, I loved "Discrete Mathematics and Applications" by Kenneth Rosen. I really enjoyed reading through the various examples and biographies of famous mathematicians included in each chapter.
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).
Not nearly as rarefied as many of the books cited here, for me it was John Saxon's excellent Algebra 1/2, Algebra 1, and Algebra 2. I didn't get a good enough grasp on basic Algebra in high-school. When I was in my early–mid 20s, a friend gave me these three algebra textbooks. In a marathon session lasting about two weeks, I went through all three books from end-to-end and really learned algebra. For whatever reasons, the Saxon books worked really well for me — better than any other learning I have ever gotten from a textbook. Although I was very motivated, I attribute a lot of my success learning algebra well to those three books. I still own them.
A lot of comments about textbooks that helped in specific topics but I don't think that answers the spirit of OP's question. Sure working through ANY Linear Algebra textbook is going to improve your Linear Algebra skills.
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.
Also, Geometry for Programmers but only because I wrote it. I had to update my skills significantly while gathering material and doing all the experiments. Not sure if reading the book would have the same effect :-)
It introduces math from a mathematician's point of view (complete with proofs, etc.) rather than rote memorization and exercises, but it does so from the perspective of a programmer.
K A Stroud Engineering Mathematics is probably the book that helped me most.
31 chapters each composed of about 60 problems. The problems are progressive and contain explanations of the new concepts that they contain.
All the answers are at the back.
I used this book and the sister book “advanced engineering mathematics” for my bachelors and masters degree in mechanical engineering.
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.
Several people who have mentioned "How to Solve It" by George Polya. It's been decades since I've looked at it, but a favorite Polya quote of mine is " "The open secret of real success is to throw your whole personality into your problem."
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.
Math always seemed a bit arbitrary to me. Why and how did all those fields and branches develop, why are some so much more intuitive than others etc. What helped me cope with that challenge (and ultimately be a better learner / user of mathematics) was digging into the history of mathematics. Many great books in that genre but a very influential one for me was the Concise History of Mathematics by Dirk Jan Struik
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.
I think "arbitrary" gives the wrong flavor here. Math is fundamentally curated.
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.
A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres
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.
Probabilistic Graphical Models by Koller & Friedman. In anything statistics and ML related, being able to deal with complicated probabilitistic things that are all related is really useful. This book gives you that toolkit. It's a "strong foundations" kind of book, rather than a bunch of methods you'll use directly.
Skimming over the replies, they range from arithmetic to algebraic geometry and measure theory!
Along the lines of fascicules de résultats, I find talks are a good way to get a coup d'oeil for a field: people giving a talk tend to take a direct approach to what they want to introduce, hitting only the salient points. But that yields enough keywords to then consult any relevant texts.
1. Principles of Mathematical Analysis by Walter Rudin (baby Rudin) - I'd studied real analysis in the past, but this book is direct and rigorous and provided a good framework to move forward into things like functional analysis in a way that I was not prepared for with other books.
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 read the first chapter of “Mathematics; Its Content, Form, and Meaning” (or something close to that) and it explained the entirety of my high school mathematics. If I had been given that to read back then I might have gone into mathematics in college. Instead I got burned out and quit doing any math for a decade or more. Sigh.
A book of abstract algebra - Charles C. Pinter. Each chapter is a few pages of explanation, and the rest you solve yourself by doing exercises that introduce aspects of the theory step by step.
Differential and Integral Calculus (Volumes 1 & 2) – Nikolai Piskunov
Not as colorful and attractive but the adage "do not judge a book by its cover" applies so well to this masterpiece. With brief and precise explanations and high quality exercises with solutions, I went from struggling to getting A+
Epp's book is my favorite Discrete Math book by far. Her writing style is very clear and easy to follow. And as you say, there are good exercises. And if you buy an older edition, used copies can be had for a (relatively) reasonable price.
The Road to Reality: A Complete Guide to the Laws of the Universe
In this book, Roger Penrose a Nobel Prize winner in Physics for his contributions in mathematical physics of general relativity and cosmology, provides background math to understand the book's contents in the first half of the book.
As a child, I re-read The Number Devil constantly. It introduced me to some really cool mathematical ideas, couched in a cute story. The very last chapter includes a picture of the Principia Mathematica's proof of 1 + 1 = 2 -- mostly for shock value, I think, but also "even this is within your reach". I recently got to see a friend's copy of the Principia, and I realized just how much of it I actually did understand, which was a really nice closing of the loop.
As an adult, Imre Lakatos' Proofs and Refutations gave me a much richer understanding of definitions in mathematics -- what job they're meant to do, and when it makes sense to change your definitions instead of adding premises to your theorems.
I'd really like to find a similar book on conic sections --- my next major project seems to need them, and when I tried to solve it using trigonometry alone, I wound up 7 or 8 levels deep in triangles and wasn't much more than half-way to where I needed to be.
Going to echo the suggestions for the Art of Problem Solving books, particularly I recommend the contest books (vol 1 or 2). Several very talented people have said to me that these books taught them how to think. Maybe a bit exaggerated, but they’re very good.
In early high school in the 90s, I got my parents to buy me an (expensive) copy of Chaos and Fractals: New Frontiers of Science by Peitgen, Jurgens, Saupe. The end of each chapter was a Basic program for calculating and displaying various fractal/chaos theory images, and that's what got me started programming.
It also included a bunch of mathematics involving "neighborhoods", meaning the set of all points within a distance of an arbitrarily small epsilon from some point X. Although I never did any of the math problems from the book, that early exposure to epsilon made calculus vastly easier to understand, and for that, it's close to my heart.
Counter-intuitively, reading Tim Ferris and his DSSS approach made me much better at math
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.
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.
Yes, it is targeted towards middle and high school students. Yes, I read it and (more importantly) worked through most of the problems in my mid-30's. It is great if, like me, you coasted/crammed through your early mathematics education and never felt like you dialed in the fundamentals. It is also great if, like me, you needed some pen-on-paper practice and did not know where to start.
I was reading this book, when the ideas of function spaces, functions as vectors, functions as elements of vector spaces, functional analysis clicked on me.
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.
Donald Sarason's "Complex Function Theory".
There are bigger more complete books on complex analysis. There are even ones that are more appealing, like "Visual Complex Functions" by Elias Wegert.
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.
Mathematik für Ingenieure und Wissenschaftler I, II and III from Lothar Papula (in German). The solutions are detailed, making it perfect for self-studying.
Book of Proof by Richard Hammack. A great introduction to proofs in mathematics. The book is available free online [0], but also I bought the physical version because I really enjoyed it.
For what level and in which area? Books like _Methods of mathematical physics_ can be both too hard and irrelevant to your needs. For starters,
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.
I highly recommend working through Claude Shannon's Mathematical Theory of Communications [0]. It's originally a paper but was later restructured as a book, in either form it works quite well.
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.
Steele's The Cauchy-Schwarz Masterclass is actually quite good and seemingly designed for self study. (A lingering result of this book is I heavily use inequalities even outside of analysis.)
Artin's Algebra probably has had the most impact on my math thinking. The development of groups and rings while tightly linking them to linear algebra was rather brilliant.
Khan Academy. Not a book per-se, but have worked through the courses over the past few years, I'm confident that my college and university results would be about 2 or 3 grades higher we I to retake them.
None. As far as I'm concerned, they all suck because academia goes about teaching math in all the wrong ways.
I finally learned the point behind math thanks to dabbling in programming. All the math classes and teachers and textbooks in the world will never teach me what the importance of 1+1 is.
Books do not make you better at math. Working math problems makes you better at math. Go ahead, down vote all you want.
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.
Problem sets are useful for improving your mathematical ability, as is looking at an expert's attempt at a problem after your attempt. Both are a core part of reading books well. So when someone asks for books to improve their mathematical ability, a likely interpretation is just "they want books with good problem sets, clear presentation and elegance". Another interpretation is "what's a book which, when after I intrepret all the individual sentences into a vague impression, will make me good at maths?"
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?)
> Books do not make you better at math. Working math problems makes you better at math.
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.
So many great responses recommending all kinds of books and then there's... this. Lol. Such a jaded/cynical teacher thing to say, but sure, nothing is a substitute for what you get out of putting the effort into doing. For me, the biggest hurdle to succeeding at college-level math was a lack of motivation due to the meaninglessness of most of the content.
If I were a student, eager to solve math problems and become better at math, no book would be better than any other book at presenting and guiding me to problems that would improve my understanding?
The classic "Advanced Engineering Mathematics" by Erwin Kreyszig was absolutely 'good enough'. Maybe even more important - for me: it was one of the easiest books to follow and digest during my undergrad. See it as a solid base for other heavier/purer titles.
I love math books with great exercises, I just wish I could code them up in a theorem prover and solve and store my proofs that way. I've tried a bunch of tools but haven't found a language or workflow that really meets the needs for computer assisted study of mathematics.
Calculus Made Easy (1910) simply for the quote at the beginning: "What one fool can do, another can."
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.
Had to do a Calculus course in uni despite not having taken any calc or pre-calc in high school. "Precalculus Mathematics in a Nutshell" and "Calculus Made Easy" were complete lifesavers.
* Algorithms to Live By, by Brian Christian & Tom Griffiths (not really a Math book, mostly computer science, but still has some math algorithms and their implementations to real life)
Probably the most elegant math book I have ever seen is Probabilty theory a graduate course by Achim Klenke. A very nice exposition into the abstract, measure theoretic prob. thoery (but it assumes some prior knowledge).
Something that really helped me with my mathematic modules at university: Lothar Papula's "Mathematik für Ingenieure und Naturwissenschaftler" [1]. If you get the stuff in this book right, you're set for life.
More advanced: Many random one offs which I haven't yet to establish a good pattern for, Richard E Brocherds seems to have a few series though I have to take them slowly as they are more akin to a college lecture.
Number crunching examples: Michael Penn. Also should have some more theory based videos but I have only watched his problem solving ones.
Avoid: Numberphile just ends up feeling empty and devoid of any depth. Maybe it works for someone so entry level that they don't follow along with Mathologer but I don't see any value and would like a way for YouTube to stop recommending their videos.
Significantly better I don't know but when I was a child I was given Der Zahlenteufel. Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben (The Number Devil) and I liked it very much
There exist only two kinds of modern mathematics books: ones which you cannot read beyond the first page and ones which you cannot read beyond the first sentence. -- Chen Ning Yang
Spivak, Abbott, Hubbard & Hubbard, Linear Algebra Done Right, Gallian’s Abstract Algebra, and believe it or not MTW taught me more good math than many math books.
I'm going to cheat and combine a couple of books into "one book". The Manhattan Prep GMAT test prep math books were really good for me for everyday life. I learned a lot of shortcuts and quick heuristics to use and got better at estimating after going through those books. It doesn't help me "get better" in an academic sense, but those books pay dividends every day for me.
“If you want to build a ship, don’t drum up the people to gather wood, divide the work, and give orders. Instead, teach them to yearn for the vast and endless sea.” --Antoine de Saint-Exupéry
Some favorites below. Books 0-3 are accessible. The remaining books are more difficult but I'd highly recommend them to math students.
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 teaching mathematics.
5. E.C. Titchmarsh The Theory of Functions: Recommended by my undergraduate advisor because he noticed that I liked reading older books. It contains sections on complex analysis and real analysis with measure theory, but I've only read the complex analysis sections. It's not for everyone, if I recall correctly, there is not a single picture, but it is very lively and has a lot of material you won't find in a standard complex analysis book, including Dirichlet series. Excellent as a supplement to a standard complex analysis book.
6. George Polya. Mathematics and Plausible Reasoning. An excellent expansion on Polya's ideas on How to Solve it. While the goal is to seek rigorous proofs, to get there it's powerful to be able to think based on intuition, heuristics, and plausible reasoning. A lot of math exposition is theorem/proof based and doesn't help develop these skills. In a similar vein, see also Terence Tao's classic post There's more to mathematics than rigour and proofshttps://terrytao.wordpress.com/career-advice/theres-more-to-....
7. H.S.M Coxeter, An Introduction to Geometry. A book of very beautiful classical geometry. Something typically not touched on at all in a typical mathematics curriculum.
I got a lot out of "Unknown Quantity" by John Derbyshire. Subtitle a real and imaginary history of algebra. I particularly enjoyed the lead up to the Chapter "Assault on the Quintic".
Also, I hold "The Dictionary of Curious and Interesting Numbers" close to my heart for the endless fun it brought me.
The Time/Life book "Mathematics" published 1969. I was in 2nd grade, liked mathematics and saw a book with that name and pictures. It was a high level survey of a lot of mathematical concepts, explaining things in a way I could understand at that age, but also in a way that wouldn't be talking down to me today.
What if by Randall Munroe
Not rigorous but it changed my perspective that was instilled in me in middle school.
Otherwise Feynman Notes changed me academically. Easier to pickup math through physics if you arent looking for deeply pure avenues.
All the Math You Missed by Thomas A. Garrity. I have not read it, but it looks interesting and is on my list. It is aimed at new graduate students who need a quick refresher that is still detailed enough to be useful for postgraduate math.
David Bressoud’s 4 books on calculus/analysis are the most engaging math books I’ve ever read. He uses the history of mathematics to drive the narrative and it’s an enlightening approach. However, this means certain theorems often taught in undergrad analysis are delayed until his “graduate” book and his graduate book would not be sufficient for passing many universities analysis quals. But the context and history he gives is fantastic.
I always recommend Bressoud's measure theory/integration text as a supplement the first graduate course in analysis. His discussion of weird and pathological sets of reals such as fat Cantor sets are really helpful for building intuition.
Nonlinear Dynamics and Chaos by Strogatz - does a great job at explaining very complex mathematical topics with great examples. Not for beginners. Every applied math student should read this cover to cover.
It may be sacrilege but I learned a lot from "Numerical Recipes". Not in much depth, but enough to wet my feet in a number of areas that were new to me.
Maybe not "made me better at math" per-se, but definitely "made me more enthusiastic about math":
The Universe Speaks in Numbers[1] by Graham Farmelo
I found this very motivating and insightful, in terms of developing even more of an appreciation for how much math underpins other branches of science. Not that that is a novel insight by any means... but the details of the incidents where breakthroughs in mathematics allowed further advances in physics, etc. and looking at the "back and forth" between the domains, that was wildly interesting to me. Reading this book definitely helped motivate me to get serious about committing more time / focus to studying mathematics.
I also enjoyed the "counterpoint" book by Sabine Hosenfelder, Lost in Math[2]. I think these two books complement each other nicely.
Then the handful of additional (no pun intended) books that jump to mind would be:
- How Mathematicians Think by William Byers[3]
- How to Think Like a Mathematician by Kevin Houston[4]
- Discrete Mathematics with Applications[5] by Susanna Epp
- How Not To Be Wrong[6] by Jordan Ellenberg
- Introduction to Mathematical Thinking[7] by Keith Devlin
* 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.