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?
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.
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.
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 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.
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.
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.
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 :-)
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.
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.
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.
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
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).
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.
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.
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 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.
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...
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.
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.
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).
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.
322 comments
[ 3.1 ms ] story [ 400 ms ] threadFor 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.
https://mathoverflow.net/questions/31655/statistics-for-math...
Here’s a good starting point for philosophy of mathematics :
https://plato.stanford.edu/entries/philosophy-mathematics/
The problem is that it's in Latin and quite impenetrable.
We have some geniuses here and they would no doubt be able to take away a lot from these texts, but for you normals out there: don't optimize too much, you're quite alright in taking the normal approach of just taking a class at a community college, doing the exercises the teacher assigns, etc.
I haven't read Mathematical Principles of Natural Philosophy (the English title) but I have read Euclid and it definitely doesn't require a genius to understand. here is an online edition with great illustrations:
https://www.c82.net/euclid/
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.
Anyone knows how long it would take to learn enough latin to undertake such a task?
I bet you could start this week if you used machine translations as a crutch.
I'd start working with a publication that exists in Latin and a good translation, so you can compare your work.
My fear with machine translations is that subtle errors here and there might throw me off in something like math where things are precisely stated.
A year of part time study sounds doable though
In fact, you'll find that many philosophers will just use the Latin words directly, and not bother translating them - Latin qua jargon if you will.
Once you've learned the various forms of "is" (sum, very irregular) you can kinda survive reading without conjugations, just like this sentence can be worked out:
he to go to store and to buy cheese yesterday
Edit: Also, to state the obvious, it's been translated into English
This will surely make you more appreciative of subjects and concepts you are learning.
See https://www.newyorker.com/magazine/2005/02/28/time-bandits-2
https://youtu.be/UhX1ouUjDHE
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.
In a similar spirit, but with a much more geometrical flavor, there is Evans-Gariepy.
History of mathematics will give you a very subtle entry into the minds of mathematicians and the motivation behind their theorems.
This will surely make you more appreciative of subjects and concepts you are learning.
It's touching many areas. For some it explains how they were developed or the controversy around them (e.g. the definition and use of infinity).
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.
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.
- 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.
Here [1] you can find Sheldon Axler himself explaining the topics of the book in his YouTube channel! How wonderful is that!
Here [2] you can find the solutions to the exercises in the book.
This [3] Lectures might help as well, among the books this course follow is Algebra Done Right.
Good luck learning the subject of Linear Algebra you'll have fun doing so.
[1] https://www.youtube.com/playlist?list=PLGAnmvB9m7zOBVCZBUUmS...
[2] http://linearalgebras.com/
[3] http://nptel.ac.in/courses/111106051/
Gödel, Escher, Bach: an Eternal Golden Braid - Hofstadter
Euclid's Elements
A note: this isn't a resource for higher-level, proof based maths. It will give you a solid foundation and a pragmatic understanding to build upon. Very useful for STEM.
Link: https://tutorial.math.lamar.edu
I'm eternally grateful ;)
Art of problem solving(https://www.amazon.in/Art-Problem-Solving-Basics/dp/09773045...) is also a great start, especially if you don’t have much experience.
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...
Very true. For maths, it is drill to win.
A lot of the advice seems obvious in retrospect but being systematic about a problem solving framework is enormously helpful.
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.
(Not to mention it's where I first saw this really intuitive way of doing matrix multiplication: https://blogs.ams.org/mathgradblog/2015/10/19/matrix-multipl...)
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 proofs https://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.
Beginner: NL Biggs, Discrete Mathematics, Oxford University Press
Intermediate: PJ Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press
Advanced: JH van Lint & RM Wilson, A Course in Combinatorics, Cambridge University Press
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).
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.
[0]https://jdhsmith.math.iastate.edu/class/BookOfProof.pdf