Reddit mentions of Concepts of Modern Mathematics (Dover Books on Mathematics)

It really depends which direction in mathematics you want to go. Even as a math major, I didn't really understand how vast it was until I got into abstract math.

My favorite way to learn is browse Amazon for "Dover Books on Mathematics." They are generally had for a penny + shipping if you don't mind buying used.

A good intro into modern mathematics: https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247

Here's an easy read that I liked: Concepts of Modern Mathematics by Ian Stewart. It gives a pretty broad overview. And you can't beat the price of those Dover paperbacks.

You may also be interested in a more thorough exploration of the history of the subject. Try History of Mathematics by Carl Boyer.

This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.

General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.

Many thanks for the suggestions!

For the interested, I bought this book for GT:

http://www.amazon.com/Introductory-Graph-Theory-Gary-Chartrand/dp/0486247759

I also was tempted by the following book:

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247



I think buying a book feels better than sex. (I can compare.)

Interviews with mathematicians from MIT (haven't read it, but it is leisurely):
http://www.amazon.com/Recountings-Conversations-Mathematicians-Joel-Segel/dp/1568817134

Some good magazines from AMS:
http://www.amazon.com/Whats-Happening-Mathematical-Sciences-Mathermatical/dp/0821849999

If you want to learn some math in a leisurely way (although it does get pretty deep at times):
http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247

A good book on the history of mathematics:
http://www.amazon.com/Mathematics-Nonmathematician-Dover-explaining-science/dp/0486248232

I'll definitely check out that Poincare book, it looks good!

Hi,

I'm a TA in my school's CS theory course (a mixture of discrete math, and the automata, languages and complexity topics most CS theory courses cover).

As others have said, "theory" is pretty broad, so there are an awful lot of resources you could look at. As far as textbooks go, we use two - Sipser's Introduction to the Theory of Computation (which others have recommended), and the freely available textbook Mathematics for Computer Science, by Lehman, Leighton and Meyer - which concentrates more on the "discrete math" side of things. Both seem fine to me. Another discrete-math–focused set of notes is by James Aspnes (PDF here) and seems to have some good introductions to these topics.

If you feel that you're "terrible at studying for these types of courses", it might be worth stepping back a bit and trying to find some sort of an intro to university-level math that resonates for you. A few books I've recommended to people who said they were "terrible at uni-level math", but now find it quite interesting, are:

• Keith Devlin, The Language of Mathematics: Making the Invisible Visible
  
• Ian Stewart, Concepts of Modern Mathematics
  
• Martin Liebeck, A Concise Introduction to Pure Mathematics (rather terser than the previous two, but still a lot gentler than some uni-level texts)
  
  These aren't specific to computer science, but might be of help. Good luck!
  
  
  

Mathematics for the Nonmathematician, disgustingly eurocentric but still good, Concepts of Modern Mathematics gives an overview of some higher maths, and I have the set The World of Mathematics which I occasionally read a random chapter, it covers lots of ground.

This one is pretty good

Check out:

• Concepts of Modern Mathematics by Stewart
  
• Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov & Lavrent’ev
  
  

Before the Princeton Companion to Mathematics, there were:

What Is Mathematics? by Courant and Robbins

Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev

Concepts of Modern Mathematics by Ian Stewart

These are different fields (programming vs math etc) however I will ask you, do you like math or programming ? If not maybe you need to get to know these quite interesting fields better. For math I would recommend one of the Dover introduction books, such as Ian Stewarts' concepts of modern math.

This is just my perspective, but . . .

I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.

​

Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.

​

I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:

​

How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics

Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.

Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.

​

As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.

​

Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:

​

A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )

​

Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).

​

If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.

Concepts of Modern Mathematics by Ian Stewart is an excellent book about modern math. As is Foundations and Fundamental Concepts of Mathematics by Howard Eves I would recommend these two along with the far more expensive Naive Set Theory by Halmos

It's a pretty difficult question to answer because only you know what you want out of this (or maybe, you don't know yourself!)

"I want to see what kind of mathematics is out there"

Try The Joy Of X. This is a super fun "guided tour" of mathematics. Each chapter surveys a different mathematical topic with examples, intutions, and fun thought experiments. You won't learn to "do the math", but you should have more of an idea of the kinds of things mathematicians think about, and some of the history of mathematics. This is easy and enjoyable, even though no mathematical background.

There's a "Hard mode" version of this called Concepts Of Modern Mathematics. The language is still light and informal, but the concepts are dealt with in more depth and abstraction -- there are fewer "real life" examples, and you will have to follow some real mathematical arguments in your head or on paper. This is more difficult, but still requires no formal mathematical background.

The other place to check of course is YouTube. 3Blue1Brown, Mathologer, Numberphile and many other channels have great exposés of mathematical concepts for the general audience, with 3Blue1Brown being my favourite for his wonderful animations.

"I want to actually improve my mathematical knowledge and skill"

This is difficult, but doable. I'm a mature mathematics student, and I was only really in with a shot of university owing to the kindness of my then fiancée supporting me while I knuckled down and learned the basics. The first step will be to brush up on what you should know from school. I'm not really sure what to recommend here; most texts targeted at this level of mathematics are targeted at... well, bored teenagers who don't want to learn mathematics, rather than keen adults possessing of some degree of patience and perseverance. I suppose Serge Lang, probably the most prolific mathematics textbook author of all time, can offer "Basic Mathematics", but this means paying Springer textbook prices, unless you enjoy marauding on the high seas. Khan Academy is a website with dozens (hundreds?) of free videos, articles, and exercises on basic mathematics

After you're up to speed on your basic algebra and geometry, the two most widely applied and important topics in mathematics beyond school-level are calculus and linear algebra (other than maybe statistics and probability). Calculus is typically learned first, but actually, it doesn't really matter which order you do these in. Exactly how to learn these topics is also a pretty difficult question, and depends what you want to get out of it. I guess post back here if "step 1" (recovering all your school-level maths) goes well?

Maths is hard, but fun. You have to do exercises and practice. You have to think deeply about difficult and abstract concepts. If you do choose the "improve actual skill" route, I'd still recommend supplementing your learning with the books and YouTube videos from the first half of this reply. Being exposed to fun new ideas regularly helps motivate you to push through the technical difficulties of learning it "properly".

How about selected chapters from Stewart's Concepts of Modern Mathematics? It has a pretty wide range of jumping off points and is a relatively affordable Dover book. You could go into more or lesser detail on these topics based on the students' backgrounds.

Another idea would be to focus on foundations like set theory, logic, construction/progression of number systems from ℕ -> ℤ -> ℚ -> ℝ -> ℂ , and then maybe move into some philosophy of math. There could be some fun and accessible class discussion, such as having them argue for or against Platonism. [Edit: You could throw in some Smullyan puzzle book stuff for the logic portion of this for further entertainment value.]

To answer your second question, KhanAcademy is always good for algebra/trig/basic calc stuff. Another good resource is Paul's online Math Notes, especially if you prefer reading to watching videos.

To answer your second question, here are some classic texts you could try (keep in mind that parts of them may not make all that much sense without knowing any calculus or abstract algebra):

Men of Mathematics by E.T. Bell

The History of Calculus by Carl Boyer

Some other well-received math history books:

An Intro to the History of Math by Howard Eves, Journey Through Genius by William Dunham, Morris Kline's monumental 3-part series (1, 2, 3) (best left until later), and another brilliant book by Dunham.

And the MacTutor History of Math site is a great resource.

Finally, some really great historical thrillers that deal with some really exciting stuff in number theory:

Fermat's Enigma by Simon Sigh

The Music of the Primes by Marcus DuSautoy

Also (I know this is a lot), this is a widely-renowned and cheap book for learning about modern/university-level math: Concepts of Modern Math by Ian Stewart.

I think there could do some really cool analysis tracing lines of thought and how they developed or comparing what was in vogue in math to world developments at the time. This book might be a good overview for modern developments and this one has a overview of the development of math through history

I enjoyed Concepts of Modern Mathematics when I was in high school. It might be a little basic and it's a bit uneven in places. But it's a really good lay account of the basic notions in "modern" mathematics. It doesn't really mention so much the various fields. For that, surfing Wikipedia is hard to beat.