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.
* 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.
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
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).
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'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
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.
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.
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.
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.
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.
Readit News
* 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.
* 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.
* Program = Proof by Mimram. Discusses logic and computation, and takes you from the basics to depedent type theory and beyond. Uses OCaml and Agda. Freely available at: https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/teaching...
- 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.
Do you happen to have any on probability or statistics and or both?
* 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.
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.
There are hundreds of videos, organised in playlists, from undergraduate lectures ( https://www.youtube.com/playlist?list=PL55C7C83781CF4316 ) and research seminars ( https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ) all the way to basic fundamentals like how to think about counting (e.g. https://www.youtube.com/watch?v=Puk-ipOTiD4&list=PL5A714C94D... )
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
https://www.amazon.co.uk/All-Math-You-Missed-Graduate/dp/100...
I don't know how good it is, but her earlier entries on Physics and Philosophy were well-received.
HN thread: https://news.ycombinator.com/item?id=30591177&p=2
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
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 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.
[1]: https://www.youtube.com/@ProfessorLeonard
[2]: https://www.youtube.com/@TheMathSorcerer
[3]: https://www.youtube.com/c/3blue1brown
[4]: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8
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.
[1] https://www.amazon.com/Understanding-Analysis-Undergraduate-...
[2] https://www.amazon.com/Invitation-Classical-Analysis-Applied...
[3] https://www.amazon.com/Advanced-Mathematical-Methods-Scienti...
[4] https://www.amazon.com/Modern-Graph-Theory-Graduate-Mathemat...
[5] https://www.amazon.com/Numerical-Linear-Algebra-Lloyd-Trefet...
[6] https://www.amazon.com/Algebraic-Geometry-Approach-Mathemati...
[7] https://www.cs.cmu.edu/~kmcrane/Projects/DDG/
[8] https://www.amazon.com/Through-Exercises-Springer-Undergradu...
"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.
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.
[0] https://a.co/d/gQmDIo7
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.
[1] https://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/E...
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.
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Another great book on this topic is "History of Mathematical Notations" https://www.amazon.com/History-Mathematical-Notations-Dover-...
It’s actually listed as a technical report from Lawrence Livermore National Lab, but the only online PDF I can find is here: https://www.academia.edu/28253460/Handbook_for_Spoken_Mathem...
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.
This will surely make you more appreciative of subjects and concepts you are learning.
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.
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 ;)