Reddit mentions: The best mathematics history books

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.

My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.

Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.

General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.

Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.

Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.

Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.

Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.

Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.

Computer scientist here... I'm not a "real" mathematician but I do have a good bit of education and practical experience with some specific fields of like probability, information theory, statistics, logic, combinatorics, and set theory. The vast majority of mathematics, though, I'm only interested in as a hobby. I've never gone much beyond calculus in the standard track of math education, so I to enjoy reading "layman's terms" material about math. Here's some stuff I've enjoyed.

Fermat's Enigma This book covers the history of a famous problem that looks very simple, yet it took several hundred years to resolve. In so doing it gives layman's terms overviews of many mathematical concepts in a manner very similar to jfredett here. It's very readable, and for me at least, it also made the study of mathematics feel even more like an exciting search for beautiful, profound truth.

Logicomix: An Epic Search for Truth I've been told this book contains some inaccuracies, but I'm including it because I think it's such a cool idea. It's a graphic novelization (seriously, a graphic novel about a logician) of the life of Bertrand Russell, who was deeply involved in some of the last great ideas before Godel's Incompleteness Theorem came along and changed everything. This isn't as much about the math as it is about the people, but I still found it enjoyable when I read it a few years ago, and it helped spark my own interest in mathematics.

Lots of people also love Godel Escher Bach. I haven't read it yet so I can't really comment on it, but it seems to be a common element of everybody's favorite books about math.

This is a very loaded question, but I'll do my best to answer it.

Disclaimer: I'm a lowly undergrad.

Math is made up of many different fields, so it's hard to assess its development in a linear fashion. The main areas are: algebra, geometry, analysis, logic/set theory/foundations, number theory, and applied math.

Now, the calculus that you study was mostly invented by the end of the 17th century, but interestingly, ideas about limits didn't really get formalized until the 19th century. Cauchy was instrumental in formalizing the definition of the derivative, and Riemann in formalizing the equivalent definition of the definite integral.

If you were to continue to take math classes in an undergraduate curriculum, you might take any of the following: linear algebra, abstract algebra, more calculus (multivariable, etc.), differential equations, discrete math, real analysis, complex analysis, basic functional analysis, topology, set theory, non-euclidean geometry, graph theory, number theory, probability and statistics, etc.

But even if you took most courses available to you as a college student, you would still be about 100 years behind mathematics as it stands today. You could argue the exact number of years, as different fields have evolved at different rates, but the point is that math has exploded in variety over the last century.

The last person to truly grasp "all" areas of mathematics of his time is probably someone like Gauss or Euler. Today, you're lucky if you can master a subfield of your subfield.

I'll now try to give you a quick overview of how math has developed since the invention of calculus. Apologies if it sounds vague (that's because it is).

The invention of calculus itself was a gradual process. Leibniz and Newton are credited with the invention largely because they collected the known information at the time and managed to come up with the Fundamental Theorem of Calculus, along with a ton of notations. Newton also applied it to his physics, which began a long relationship between calculus and the natural sciences. Up until this time, most math was geometric, rather than algebraic. Complex geometric proofs were drawn up to solve problems that would later be solved simply with algebra. Analytic geometry had only been invented a few years prior by Descartes, and though it seems obvious to us today, it was revolutionary at the time.

Progress in calculus was made very quickly in the 17th century - before long, all of the calculus you learn (and then some) was invented, though it lacked the formal foundations that it has today. The Bernoullis helped to found the calculus of variations, which is a topic usually reserved for grad school. That gives you an idea of how much math there truly is. In the 18th century, most of the progress was made in further developing the calculus, while other fields of math were being born. Euler and Gauss helped to popularize the notion of complex numbers. Euler considered the Königsberg bridge problem which would later inspire graph theory and topology. Lagrange and others made discoveries in what would become the field of abstract algebra.

The 19th century brought rigor to mathematics. While mathematicians were proving things up until then for the most part, there was often an element of intuition (and sometimes hand-waving) going on, especially back in the 1600s. A discovery by Fourier that functions could be approximated by series of sines and cosines (later, Fourier analysis) made the mathematical community worried. With the foundations of calculus on such empirically-strong, but theoretically-shaky grounds, how could they be sure if these wild claims were really true? This quest to put Fourier analysis on solid ground led to a rigor revolution in mathematics. What we now call "analysis" (the more rigorous version of calculus) was developed, and many other discoveries were made as a result. The other revolution going on in the 19th century was in the field of abstract algebra. Many of the great algebraists lived during this time: Galois, Abel, Noether, and others. Noether actually lived about 80 years after Abel and Galois.

Towards the end of the 19th century, Cantor developed his theory of sets, which led to a second revolution in mathematics. Although they are ubiquitous now, sets were relatively unused at the time. The introduction of sets into mathematics, combined with the revolution in rigor discussed previously, had a huge impact on mathematics. Progress in nearly every field accelerated like crazy in the new set-based more rigorous framework. The 20th century brought a deluge of mathematical information unlike any time before it. Some of the key fields that were invented in the last century -- or really started to get going -- were topology, algebraic geometry, linear algebra (for computers), computational logic, model theory, set theory, graph theory, combinatorics, various applications of math to advanced physics like relativity theory, etc.

But these are just the tip of the iceberg. If you really want to know how far math has progressed in an intuitive sense, I invite you to open a recent math journal (or look at one online) and see how much you can understand. It's truly mind-blowing.

I hope this helped more than it hurt. I'm no historian, but I think I got at least the general shape of things right. If you want a more in-depth look, especially one that considers the more modern developments, try Kline's series of books:* link to the first one

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

Hey! This comment ended up being a lot longer than I anticipated, oops.

My all-time favs of these kinds of books definitely has to be Prime Obsession and Unknown Quantity by John Derbyshire - Prime Obsession covers the history behind one of the most famous unsolved problems in all of math - the Riemann hypothesis, and does it while actually diving into some of the actual theory behind it. Unknown Quantity is quite similar to Prime Obsession, except it's a more general overview of the history of algebra. They're also filled with lots of interesting footnotes. (Ignore his other, more questionable political books.)

In a similar vein, Fermat's Enigma by Simon Singh also does this really well with Fermat's last theorem, an infamously hard problem that remained unsolved until 1995. The rest of his books are also excellent.

All of Ian Stewart's books are great too - my favs from him are Cabinet, Hoard, and Casebook which are each filled with lots of fun mathematical vignettes, stories, and problems, which you can pick or choose at your leisure.

When it comes to fiction, Edwin Abbott's Flatland is a classic parody of Victorian England and a visualization of what a 4th dimension would look like. (This one's in the public domain, too.) Strictly speaking, this doesn't have any equations in it, but you should definitely still read it for a good mental workout!

Lastly, the Math Girls series is a Japanese YA series all about interesting topics like Taylor series, recursive relations, Fermat's last theorem, and Godel's incompleteness theorems. (Yes, really!) Although the 3rd book actually has a pretty decent plot, they're not really that story or character driven. As an interesting and unique mathematical resource though, they're unmatched!

I'm sure there are lots of other great books I've missed, but as a high school student myself, I can say that these were the books that really introduced me to how crazy and interesting upper-level math could be, without getting too over my head. They're all highly recommended.

Good luck in your mathematical adventures, and have fun!

Rudy Rucker's Infinity and the Mind changed my life when I read it in high school. It's a supremely approachable introduction to set theory and the mathematics of infinity, delightful in its eager concept-evangelism and sometimes loopy backgrounding but absolutely rigorous when it comes to the actual mathematical content. I come back to it every few years and still love it.

Edit: I also loved Marcus du Sautoy's Symmetry ("Finding Moonshine" in some parts of the world), which is a historical and popular survey of the title topic, intertwined with relevant travelogues from the life of a professional mathematician. James Gleick's The Information is an outstanding book about the development of information theory and his landmark book Chaos: The Making of a New Science is also terrific — and if you value lucid, gripping nonfiction writing, his prose is an utter pleasure to read. He has an almost supernatural facility to pick out the perfect analogy to explain an idea. None of these have chapter-ending exercises like the Rucker book, but they do develop some very sophisticated ideas that'll leave your head thrumming for months afterward.

Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.

Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.

Intro to Math:

1. Book of Proof by Richard Hammack. This free book will show/teach you how mathematicians think. There are other such books out there. For example,
   
   

• How to Prove It: A Structured Approach by Daniel Velleman
  
  
• Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand et al
  
  
• Discrete Mathematics with Applications by Susanna Epp
  
  These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
  
  Intro to Abstract Algebra:
  
  
• Linear Algebra: Step by Step by Singh
  
  
• Abstract Algebra: A Student-Friendly Approach by the Dos Reis
  
  
• Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman
  
  
• Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil
  
  
• A Friendly Introduction to Group Theory by Nash
  
  
• Linear Algebra Done Right by Sheldon Axler
  
  There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
  
  
• Algebra: Chapter 0 by Aluffi
  
  
• Algebra by Maclane/Birkhoff
  
  
• Algebra by Lang
  
  
• Advanced Linear Algebra by Steven Roman
  
  Intro to Real Analysis:
  
  
• The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg
  
  
• Understanding Analysis by Abbot
  
  
• Writing Proofs in Analysis by Kane
  
  
• The Lebesgue Integral for Undergraduates by Johnston
  
  Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
  
  Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
  
  
• Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by the Hubbards
  
  At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
  
  From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
  
  Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
  
  Good luck.

Hello, I think you're spot on about it making your life easier after struggling, and by taking this class and putting in the time, it will make other math courses much easier for you. Because of what you gain from the struggle, I would really recommend you take this over 142, if you have the time. I took 140A last fall, and although I only got a C, it took an immense amount of effort to even get that. The class is set up so that if you put in the hard work to understand the concepts, the homework, the proofs and so forth, you're gonna do well, and If you truly understand how to solve the homework problems, then the tests will be familiar (doesn't mean it will be easy).

Expect to put a lot of work in. This statement needs to be taken seriously for this class, I've talk to some people in the class who say they put in 40 hours a week. This is usually because the concepts do not come immediately and you have to constantly repeat and approach at different angles to find a good understanding.

I recommend having a supplementary text while you are studying from the dreaded Rudin. For 140A, you should be looking at compactness and chapter 2 very early on as this is a big hurdle in that class. Other concepts will be more familiar but still challenging.

​

Some recommended texts (definitely find your own that works for you)

https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935 (If you prefer "casual" explanations of the concepts, this help me survive chapter 2 of Rudin. There are useful book recommendations in the very back)

https://www.amazon.com/Elementary-Analysis-Calculus-Undergraduate-Mathematics/dp/1461462703 (Ross is used for the 142 series, and I find it is very helpful if you are struggling. If you are having trouble, start with the easier version of a problem and build up from there. The book mainly stays within the R\^2 metric, which is what makes it simpler)

https://minds.wisconsin.edu/handle/1793/67009 (at some point, you're gonna get stuck and you will have to look at the solutions. This is ok, but don't become reliant on it, that really hurt me in the end when I did that. Some of the questions are fuccckkkiiinngg hard, so when you hit that wall, take a look here. They give solutions that skips over a ton of steps, or might not be that good of a way to solve the problem, but this is a great resource)

https://www.math.ucla.edu/~tao/preprints/compactness.pdf (Who doesn't know who Terence Tao is? This is very helpful for giving an answer to "what is compactness used for?". It gives some intuition about what it is, and you should read it a couple times during 140A.)

​

So this is advice that I would give myself when entering the course, and maybe it won't apply to you. Since you got an A in 109 without too much trouble, you are definitely very ready for 140, and you have a very chance of succeeding. Stay curious, and don't stop at just the solution. Really question why it is true. You probably won't have this problem, but when it hits you (probably when you get to chapter 2) you have to keep at it and don't give up. Abuse office hours, ask lots of questions, study everyday etc. and you'll do well. If you want to get better at math then the pain is worth it.

Mathematical Literature is a genre I don't think many people are aware of, I'm glad you're interested.

The Mathematical Experience is a great survey of mathematical ideas. This book toes the line perfectly - someone not knowledgable of advanced mathematics can follow easily yet the book does not dumb down complicated ideas. This is my top recommendation for anyone thinking about studying mathematics.

If you love geometry, then check out Geometry Revisited by H.S.M Coxeter. Coxeter is one of the greatest mathematicians of his time - he single handedly brought geometry back into vogue as a serious study.

Maybe for lighter reading, Ian Stewart has a bunch of good Mathematical survey books for the "layman" - I'd recommend if you have minimal mathematic knowledge.

There's a yearly collection of mathematical writings that you might like too. I've only bought and read the 2010 edition, but I assume the followups have been great. The essays collected vary from finance, game theory, geometry, social sciences, literature, etc. with connections to mathematics.

Hope you have a fun time with math, good luck!

There are a fair number of popular level books about mathematics that are definitely interesting and generally not too challenging mathematically. William Dunham is fantastic. His Journey through Genius goes over some of the most important and interesting theorems in the history of mathematics and does a great job of providing context, so you get a feel for the mathematicians involved as well as how the field advanced. His book on Euler is also interesting - though largely because the man is astounding.

The Man who Loved only Numbers is about Erdos, another character from recent history.

Recently I was looking for something that would give me a better perspective on what mathematics was all about and its various parts, and I stumbled on Mathematics by Jan Gullberg. Just got it in the mail today. Looks to be good so far.

When I was his age, I read a lot of books on the history of mathematics and biographies of great mathematicians. I remember reading Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.

Any book by Martin Gardner would be great. No man has done as much to popularize mathematics as Martin Gardner.

The games 24 and Set are pretty mathematical but not cheesy. He might also like a book on game theory.

It's great that you're encouraging his love of math from an early age. Thanks to people like you, I now have my math degree.

Oh, I'm terrible at calculus, haha. I teach discrete maths and logic, and never have to touch calculus at all, thank goodness :)

But a younger friend of mine is doing calculus just now, so I'll find out what he found useful and PM you. He did say that some of the books I'd recommended him were immensely useful for maths generally (not necessarily calculus in particular). In roughly ascending order of difficulty:

• Gowers, Mathematics: A Very Short Introduction (<http://amzn.to/1TQNEWY>)
  
• Devlin, The Language of Mathematics (<https://www.amazon.com/Language-Mathematics-Making-Invisible-Visible/dp/0805072543>)
  
• Stewart, Concepts of Modern Mathematics (<http://bit.ly/1L7rY4i>)
  
• Liebeck, A Concise Introduction to Pure Mathematics (<http://amzn.com/1439835985>)
  
  So you could pick one of those, and see if it meets the level you're after. There are also some standard books on "how to write proofs" which often get recommended: Velleman, How to Prove It: A Structured Approach <https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-ebook/dp/B009XBOBL6>, and
  Solow, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes <https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024>, and they might be useful. I've never read either, shamefully, tho I probably should.
  
  Also: something that can be handy once you're past the basics is the amazing (but large and expensive) Princeton Companion to Mathematics. Which has something useful and interesting to say about pretty much every major area of modern mathematics.
  
  I hope that helps!

Maybe a bit off topic, but I think that if you have a "math phobia" as you say, then maybe you need to find a way to become interested in the math for math's sake. I don't think you'll be motivated to study unless you can find it exciting.

For me, The Universal History of Numbers was a great book to get me interested in math. It's a vast history book that recounts the development of numbers and number systems all over the world. Maybe by studying numbers in their cultural context you'll find more motivation to study, say, the real number system (leading to analysis and so on). That's just an example and there are other popular math books you could try for motivation (Fermat's Enigma is good).

Edit: Also, there are numerous basic math books that are aimed at educated adults. Understanding Mathematics is one which I have read at one point and wasn't bad as far as I can remember. I am sure there are more modern, and actually for sale on Amazon, books on this topic though.

I'm an autodidact and currently studying computer science. Usually I'm not good at learning with videos or websites, I prefer to study books, so I will give you the first great books in the order that I studied. I'm assuming that you're fairly good at math, if not read this book before since it's a comprehensive survey about the subject.

• Computer Science Illuminated by Nell Dale
  
• Structure and Interpretation of Computer Programs - 2nd Edition
  
• Land Of Lisp (optional but very fun book)
  
• C Programming Language by Brian W. Kernighan
  
• Introduction to Algorithms by Thomas H. Cormen
  
• Concepts, Techniques, and Models of Computer Programming
  
  Now, these 5 books are going to teach the you the basics and you'll learn to program with Lisp and C, which are great languages that will improve your way of thinking about computing; and since all modern languages come from these two, after you learn them it will be easy to you pick up new languages in a matter of days, just buy a good reference book about the particular language and you're on business.
  
  After that the next step is to learn about networks, and learn HTML, CSS and Javascript:
  
  
• Computer Networking: A Top-Down Approach by Keith W. Ross or Computer Networks by Andrew S. Tanenbaum
  
• HTML and CSS: Design and Build Websites by Jon Duckett
  
• JavaScript and JQuery: Interactive Front-End Web Development by Jon Duckett
  
  It's also a good idea to learn Python since is heavily used for all sorts of development and a lot bitcoin apps use it:
  
  
• Learning Python, 5th Edition
  
• Programming Python by Mark Lutz
  
• Flask Web Development: Developing Web Applications with Python
  
• Creating Apps in Kivy
  
• Python Cookbook by David Beazley
  
  Now for cryptography:
  
  
• Understanding Cryptography by Christof Paar
  
• Applied Cryptography by Bruce Schneier
  
• Mastering Bitcoin By Andreas M. Antonopoulos

Initially I'd avoid books on areas of science that might challenge her (religious) beliefs. You friend is open to considering a new view point. Which is awesome but can be very difficult. So don't push it. Start slowly with less controversial topics. To be clear, I'm saying avoid books that touch on evolution! Other controversial topics might include vaccinations, dinosaurs, the big bang, climate change, etc. Picking a neutral topic will help her acclimate to science. Pick a book related to something that she is interested in.

I'd also start with a book that the tells a story centred around a science, instead of simply trying to explain that science. In telling the story their authors usually explain the science. (Biographies about interesting scientist are a good choice too). The idea is that if she enjoys reading the book, then chances are she will be more likely to accept the science behind it.

Here are some recommendations:
The Wave by Susan Casey: http://www.amazon.com/The-Wave-Pursuit-Rogues-Freaks/dp/0767928857

Fermat's Enigma by Simon Singh: http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622

The Man who Loved Only Numbers by Paul Hoffman: http://www.amazon.com/Man-Who-Loved-Only-Numbers/dp/0786884061/ref=sr_1_1?s=books&ie=UTF8&qid=1405720480&sr=1-1&keywords=paul+erdos

I also recommend going to a book store with her, and peruse the science section. Pick out a book together. Get a copy for yourself and make it a small book club. Give her someone to discusses the book with.

After a few books, if she's still interested then you can try pushing her boundaries with something more controversial or something more technical.

Awesome stuff! Let me volunteer my time; if you ever want to ask a question of someone (who you will soon outstrip in mathematical ability if not interest), PM me.

One thing that might help you is to go through a history of math. This way you're learning math the way the world has been learning math. It may give you a better understanding of why things are done the way they are. Mathematics for the Non-mathematician is something I'd like to work my way through (anyone interested in doing an online reading group this summer?) that may help you. Johnson/Mowry: Mathematics: A Practical Odyssey is a text I worked through that discussed some of the history and went in something resembling historical order (I had the third edition). As far as getting texts for self study goes, keep an eye out for older editions; you'll save a lot on a subject that doesn't change over time.

Good luck! I'm rooting for you!

Like justrasputin says, there usually is quite a lot of work to be done before you start to really see the beauty everyone refers to. I'd like to suggest a few book about mathematics, written by mathematicians that explicitly try to capture the beauty -

By Marcus Du Sautoy (A group theorist at oxford)

1. Symmetry
   
2. The Music of the Primes
   
   By G.H. Hardy,
   
3. A Mathematician's Apology
   
   Also, a good collection of seminal works -
   God Created the Integers
   
   And a nice starter -
   What is Mathematics
   
   Good luck and don't give up!

I am a Strange Loop is about the theorem

Another book I recommend is David Foster-Wallace's Everything and More. It's a creative book all about infinity, which is a very important philosophical concept and relates to mind and machines, and even God. Infinity exists within all integers and within all points in space. Another thing the human mind can't empirically experience but yet bears axiomatic, essential reality. How does the big bang give rise to such ordered structure? Is math invented or discovered? Well, if math doesn't change across time and culture, then it has essential existence in reality itself, and thus is discovered, and is not a construct of the human mind. Again, how does logic come out of the big bang? How does such order and beauty emerge in a system of pure flux and chaos? In my view, logic itself presupposes the existence of God. A metaphysical analysis of reality seems to require that base reality is mind, and our ability to perceive and understand the world requires that base reality be the omniscient, omnipresent mind of God.

Anyway these books are both accessible. Maybe at some point you'd want to dive into Godel himself. It's best to listen to talks or read books about deep philosophical concepts first. Jay Dyer does a great job on that

https://www.youtube.com/watch?v=c-L9EOTsb1c&t=11s

No it was a great tool utilized most effectively first by Gauss and Riemann but had been around for quite a while. Thats the whole point of this comic, today these things are accepted without even thinking about it, but when new ideas like removing the parallel postulate are introduced, people get emotional about it and do push back (even if those conversations get lost in the history of time as people accept the proofs, as it should be)

However, as the history of Mathematics is not my specialty I can't remember the specific sources but I'm def sure its not as clear cut as you suggest where people just immediately accept a proof without emotion.

For Godel: most of what I know is from biographies of Godel and survey books on Incompleteness and various things I've read relating to the connection with the halting problem and Turing Machines that I'm sure have mentioned it. I could list about 20 books here, but I can't remember the specific sources.

As to the rest of my knowledge of mathematical history, most of it comes from the classic four volume set http://www.amazon.com/The-World-Mathematics-Four-Volume-Set/dp/0486432688 that I found for a great deal in a used book store a long long time ago. There are copious examples of new ideas and the push back generated due to emotional fixations beyond those offered in proofs.

Ideally I'd like to live in your world too in all kinds of scientific and rational endeavors, but that doesn't seem to be the way most of the world works and to pretend that mathematics is immune is ignoring a lot of history and human psychology imo.

I have a few books I read at that age that were great. Most of them are quite difficult, and I certainly couldn't read them all to the end but they are mostly written for a non-professional. I'll talk a little more on this for each in turn. I also read these before my university interview, and they were a great help to be able to talk about the subject outside the scope of my education thus far and show my enthusiasm for Maths.

Fearless Symmetry - Ash and Gross. This is generally about Galois theory and Algebraic Number Theory, but it works up from the ground expecting near nothing from the reader. It explains groups, fields, equations and varieties, quadratic reciprocity, Galois theory and more.

Euler's Gem - Richeson This covers some basic topology and geometry. The titular "Gem" is V-E+F = 2 for the platonic solids, but goes on to explain the Euler characteristic and some other interesting topological ideas.

Elliptic Tales - Ash and Gross. This is about eliptic curves, and Algebraic number theory. It also expects a similar level of knowlege, so builds up everything it needs to explain the content, which does get to a very high level. It covers topics like projective geometry, algebraic curves, and gets on to explaining the Birch and Swinnerton-Dyer conjecture.

Abel's proof - Presic. Another about Galois theory, but more focusing on the life and work of Abel, a contemporary of Galois.

Gamma - Havil. About a lesser known constant, the limit of n to infinity of the harmonic series up to n minus the logarithm of n. Crops up in a lot of places.

The Irrationals - Havil. This takes a conversational style in an overview of the irrational numbers both abstractly and in a historical context.

An Imaginary Tale: The Story of i - Nahin. Another conversational styled book but this time about the square root of -1. It explains quite well their construction, and how they are as "real" as the real numbers.

Some of these are difficult, and when I was reading them at 17 I don't think I finished any of them. But I did learn a lot, and it definitely influenced my choice of courses during my degree. (Just today, I was in a two lectures on Algebraic Number Theory and one on Algebraic Curves, and last term I did a lecture course on Galois Theory, and another on Topology and Groups!)

Start with Khan Academy. You start with a test that determines where you're at, and I think it's pretty damn good, if only slightly conservative, and that's the way it should be.

The video lessons are particularly good at teaching you how, and basically why, but most math materials, across the board, don't really do much, if anything, to help you with insight.

"What does it mean?"

There is an answer. You just need to know where to find it.

I recommend books on math history. I find I get a greater intuitive sense of math learning it like this than just learning the pure process and concepts in a math education book.

Journey Through Genius is one of my favorites. It's 26 proofs, the history behind how they were developed or discovered, why, and what they mean, in a sense. And you don't need to know shit about math for this book to make sense. The author really breaks it down for you. Sometimes the raw proof is right there, and he goes over it, sometimes he skips it and explains it by example.

A History of Pi is also fantastic, and follows it's earliest known history to relatively modern day. An anthropologist friend taught me "follow the money" when studying history, to understand the rise and fall of nations and empires. This book taught me "follow the knowledge," and it's equally telling, but in ways following the money won't.

This may not exactly be an answer to your question but I would recommend buying this book: https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192

It's not quite a textbook nor it is a pop-sci book for the layperson. The blurb on the front says " "A lucid representation of the fundamental concepts and methods of the whole field of mathematics." - Albert Einstein"

In and of itself it is not a complete curriculum. It doesn't have anything about linear algebra for example but you could learn a lot of mathematics from it. It would be accessible to a reasonably intelligent and interested high-schooler, it touches on a variety of topics you may see in an undergraduate mathematics degree and it is a great introduction to thinking about mathematics in a slightly more creative and rigorous way. In fact I would say this book changed my life and I don't think I'm the only one. I'm not sure if i would be pursuing a degree in math if I had never encountered it. Also it's pretty cheap.

If you're still getting a handle on how to manipulate fractions and stuff like that you might not be ready for it but you will be soon enough.

I have Mathematics:From the Birth of Numbers and it’s excellent.

Highly recommend

> This extraordinary work takes the reader on a long and fascinating journey--from the dual invention of numbers and language, through the major realms of arithmetic, algebra, geometry, trigonometry, and calculus, to the final destination of differential equations, with excursions into mathematical logic, set theory, topology, fractals, probability, and assorted other mathematical byways. The book is unique among popular books on mathematics in combining an engaging, easy-to-read history of the subject with a comprehensive mathematical survey text. Intended, in the author's words, "for the benefit of those who never studied the subject, those who think they have forgotten what they once learned, or those with a sincere desire for more knowledge," it links mathematics to the humanities, linguistics, the natural sciences, and technology.

Archimedes created a method in which you start with 2 shapes (like a square/triangle/hexagon), one drawn on the inside of the circle so its points touch the edge, the other drawn outside of the circle so its edges touch the sides of the circle.

Lets use squares as an example (He actually used hexagons, because they fit a circle better, but anyway). So you have a circle with a square on the inside and another on the outside. You can find the lengths of the edges of the squares, and the average of these two numbers will be close to the circumference of the square. Knowing that Pi is simply the ratio of circumference to diameter, you can solve for Pi.

This will give you a very rough estimate. You can improve the accuracy of this by increasing the number of sides of the shapes you draw inside and outside of the circle. For example, we could double the number of sides, and then we'd have an octagon instead of a square, which fits a circle a lot better. As you add more and more sides the numbers begin to converge to a single value. It takes a polygon with about 96 sides to get a value of Pi accurate to 5 decimal places.

There is a very interesting book on this subject,
A History of Pi

What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".

There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&qid=1486754571&sr=8-1&keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!

I only skimmed your post but I'd advise against beginning at arithmetics. It'll be boring as fuck, you'll mostly find material intended for children and you're probably gonna lose interest. Also there really isn't much to it.

One book that paints a bigger picture is this:

https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248232/

Though old, it's a pretty interesting and well written book and it covers the basics of many topics. It has countless real applications of mathematics and even a lot of history. You can find it on Library Genesis but the physical copy is 8 bones right now so I'd just go for that tbh.

From there you might want to dive deeper into whatever topic interested you most, if that's Calculus you might want to get some kind of precalculus book and then "Calculus - A physical approach" which was written by Morris Kline as well. I personally really enjoy this guy's style, can recommend his stuff, but there are a lot of other good textbooks out there. Spivak and Rudin might be suitable alternatives.

For those that just think it's funny because it might be something you see in a textbook, it's not just that. This joke is a direct reference to Fermat's Last Theorem, which was proposed in 1637 and the text above was scribbled in the sidebar of the paper. It wasn't actually proven until 1994, 350+ years later.

Interestingly enough, it's unlikely that the current proof, which I think was around 300 pages, was anything like Fermat's proof that he alluded to (and possibly never had, which makes him an amazing troll). The current proof used methods that were not developed until recently, and I believe the author of the proof even developed some new mathematics in order to solve it.

https://en.wikipedia.org/wiki/Fermat's_Last_Theorem

Here's a great book on it, and the guy that finally provided the proof. Definitely worth reading, it's not boring at all: http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622

This isn't a website but I'm really enjoying Mathematics for the Non-Mathematician by Morris Kline at the moment. It goes into the history of math which gives you a much better understanding of why math is the way it is rather than just how to do it. The history of mathematics is surprisingly fascinating. I just want to go back in time and hug the Greeks now.

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!

Simon Singh explains.

edit: Hey, I didn't expect this to become the top comment. Neat. Might as well abuse it, by providing bonus material:

This is the same Simon Singh discussed in this recent and popular Reddit post; he is a superhero of science popularization. He has written some excellent and highly rated books:

• Big Bang: The Origin of the Universe
  
• The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography
  
• Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem
  
• Trick or Treatment: The Undeniable Facts about Alternative Medicine
  
  (full disclosure: I have nothing to disclose, other than that I'm a fan of his work.)

I can post a few links from some books about numbers. I haven't read a few of them, but the history of some numbers like phi, pi, zero... all of them are fascinating.

• http://www.amazon.com/Joy-Pi-David-Blatner/dp/0802775624
  
  
• http://www.amazon.com/Golden-Ratio-Worlds-Astonishing-Number/dp/0767908163/ref=pd_sim_b_4
  
  
• http://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347/ref=pd_sim_b_2
  
  
• http://www.amazon.com/Imaginary-Tale-Story-square-minus/dp/0691127980/ref=pd_sim_b_1
  
  
• http://www.amazon.com/Zero-Biography-Dangerous-Charles-Seife/dp/0140296476/ref=pd_sim_b_3
  
  
• http://www.amazon.com/History-Pi-Petr-Beckmann/dp/0312381859/ref=pd_sim_b_5
  
  Those six are all the history of the five most important constants in mathematics. If you're looking for the history of some of the most brilliant minds in mathematics, I'm afraid I haven't the resources or expertise to help you out.

I'll throw out some of my favorite books from my book shelf when it comes to Computer Science, User Experience, and Mathematics - all will be essential as you begin your journey into app development:

Universal Principles of Design

Dieter Rams: As Little Design as Possible

Rework by 37signals

Clean Code

The Art of Programming

The Mythical Man-Month

The Pragmatic Programmer

Design Patterns - "Gang of Four"

Programming Language Pragmatics

Compilers - "The Dragon Book"

The Language of Mathematics

A Mathematician's Lament

The Joy of x

Mathematics: Its Content, Methods, and Meaning

Introduction to Algorithms (MIT)

If time isn't a factor, and you're not needing to steamroll into this to make money, then I'd highly encourage you to start by using a lower-level programming language like C first - or, start from the database side of things and begin learning SQL and playing around with database development.

I feel like truly understanding data structures from the lowest level is one of the most important things you can do as a budding developer.

I'm not sure I understand your concern, but if you struggle with math, it may help to start with coding. It can make things a little more concrete. You might try code academy, a coding bootcamp, or MIT open courseware.

An Emory prof has a great intro stats course online: https://www.youtube.com/user/RenegadeThinking

Linear algebra is the foundation of the most widely used branch of stats. This book teaches it by coding example. It's full of interesting practical applications (there's a coursera course to go with it): https://www.amazon.com/Coding-Matrix-Algebra-Applications-Computer/dp/0615880991/ref=sr_1_1?ie=UTF8&qid=1469533241&sr=8-1&keywords=coding+the+matrix

Once you start to feel comfortable, this book offers a great (albeit dense) introduction to mathematics. It used to be used in freshman gen ed math courses, but sadly, American unis decided that actually doing math/logic isn't a priority anymore: https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192/ref=sr_1_1?ie=UTF8&qid=1469533516&sr=8-1&keywords=what+is+mathematics

Gödel proved several theorems; I'm guessing you're referring to the incompleteness theorems, which are the most well-known. The key point is that Gödel's incompleteness theorems are precise mathematical statements about certain formal systems — not vague philosophical generalities about the nature of truth or anything like that.

In particular, the content of the first incompleteness theorem is essentially:

>In any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true (in the standard model of arithmetic), but not provable in the theory.

This statement, as with any other statement mathematicians call a "theorem", has been formally proven. Philosophical questions like whether mathematical objects are "real" in whatever sense are irrelevant to the question of whether something is a theorem or not.

By the way, if you want a good introduction to the details of what Gödel's incompleteness theorems say and how they can be proved, I highly recommend Gödel's Proof by Nagel and Newman.

Study next in what sense? Pursue another degree? You'll have some trouble pursuing a mathematics degree with just the mathematics you've studied. Other readers of /r/math might have better answers. Perhaps try a master's in engineering focused on mathematical methods?

Study as in read for your own pleasure? There is /r/learnmath as a starting point. I can also recommend the four volume set "The World of Mathematics" compiled by James R. Newman (this is the Dover paperback reprint, hardcover first printings can sometimes be found in used & rare bookstores for $50 for the set, a steal). The books offer just what is promised and should give you a good sense of what topics you (personally) would like to direct further self study to. The bibliography stops in 1956 and many great textbooks have been written since then, but once you have an idea of where you'd like to go the book lists in the FAQ will probably be very helpful.

I know I must be missing some, but these are all that I can think of at the moment.

Fiction:

Collected Fictions by Jorge Luis Borges

The Stranger by Albert Camus

The Amazing Adventures of Kavalier & Clay by Michael Chabon

White Noise by Don Delilo

A Visit from the Goon Squad by Jennifer Egan

The Waste Land by T. S. Eliot

Everything that Rises Must Converge by Flannery O'Connor

His Dark Materials by Philip Pullman

The Crying of Lot 49 by Thomas Pynchon

Cryptonomicon by Neal Stephenson

Brief Interviews with Hideous Men by DFW

Infinite Jest by DFW

Of these, you can't go wrong with Infinite Jest and the Collected Fictions of Borges. His Dark Materials is an easy and classic read, probably the lightest fare on this list.

Non-Fiction:

The Music of the Primes by Marcus du Sautoy

Chaos by James Gleick

How to be Gay by David Halperin

Barrel Fever by David Sedaris

Let's Explore Diabetes with Owls by David Sedaris

Secret Historian by Justin Spring

Of these, Secret Historian was definitely the most interesting, though How to be Gay was a good intro to queer theory.

I would recommend geometry or number theory.

Also, not a text book, but definitely made me excited about maths and at the time I read it, Music of the Primes - Marcus du Sautoy. It's a very accessible and well written book about the Riemann Hypothesis.

It would be helpful to know what your background is to recommend a suitable text.

• The Elements of Statistical Learning
  
  It's available free online, but I've def got a hard cover copy on my bookshelf. I can't really deal with digital versions of things, I need physical books.
  
  
• If you're looking for something less technical, try The Lady Tasting Tea
  
  
• You haven't mentioned how old your sister is. If she's on the younger side of the spectrum, she might enjoy Flatland.
  
  
• Also, you mention how much your sister loves proofs. Godel's Proof is a really incredible result (sort of brain melting) and the book I linked does a great job of making it accessible. I think I read this book in high school (probably would have understood more if I read it in college, but I got the gist of it).

You might want to try "What is Mathematics?" by R.Courant and H.Robbins. The book is written for people new to the field of theoretical mathematics and is intended for those who wish to develop a solid foundation on the topic.

I had started college as an engineer, switched to English, and now work as an ESL instructor. However, my love of math never died (despite my university professors' best attempts). So, I picked up that book a little while ago. It's a good read (albeit a dense one), and it covers a little bit of what you have listed.

[Amazon link here] (http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192)

Edit: some words

It doesn't go into much depth in explaining the concepts but The Math Book is an entertaining little book. Each page briefly describes a significant mathematical achievement and it's ordered chronologically. You can just open the book anywhere though and there should be something interesting. Great coffee table book.

this book is quite short but perfect for an aspiring mathematician that is going to start hearing about Gödel's proof in casual conversation. This provides a concise easy treatment of it's importance and how the proof works. Also, see it's reviews on goodreads

I'm a lot like you, very self directed learning (I spent as little time in HS as I possibly could, and nearly flunked out because of it). This book really sparked my interest in math. Everything from why zero is so exceptional and how hard it was for our species to realize it, to how to figure out a square root by hand (maybe boring, but I was interested in the method, since just pushing the square root button on a calc was dissatisfying).

Calculus is something that is damn near impossible to get without help (you can do it, but you probably won't understand it). Finally, it's pretty important to talk to people to see what's worth learning and what you haven't considered yet. Speaking of programming, if you fail to get yourself out in there and talk to other people (people that are better than you at something) you are liable to feel proud of inventing something like bubble-sort.

First, consistently solving A1 and B1 is a great start! Puts you well above the typical. Be sure to pay attention to how you write it up: Putnam graders are very strict and solutions most often get 0, 1, 9, or 10 points. Be also aware of what your goals are and don’t get anxious, you’re not looking to solve everything, so it's good to fully solve one problem before moving on. Putnam problems in particular often have short clean solutions that are really satisfying to find.

You also can't beat just working through problems. Putnam 1985-2000 by Vakil, Kedlaya, Poonen is fantastic as it gives many ways of solving or approaching each of the problems. It also gives just the right level of hints. This way you can learn both by working through the problem and by seeing the different perspectives. For example, with a single problem there may be a long brute-force solution, a quick but hard to discover solution, and a quick solution based on advanced math (you can use most things that come up in an undergrad math curriculum, even elliptic curves).

The Art and Craft of Problem Solving is a great read for general strategies and practice, and will remain relevant throughout any later work.

Mathematical Olympiad Challenges by Andreescu and Gelca shows off a few major problem solving styles and has a great selection of problems. I studied it in high school and it ended up being very important for me getting Putnam Fellow.

Earlier I had also studied Problem-Solving Strategies but that may be too big and not as focused on Putnam type of problems

Here are some great books that I believe you may find helpful :)

• Elements of Algebra by Leonhard Euler
  
• Foundations of Mathematics by Ian Stewart
  
• Geometry Revisited by H. S. M. Coxeter
  
• Excursions in Calculus by Robert M. Young
  
• New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian
  
• The Calculus Gallery: Masterpieces From Newton to Lebesgue by William Dunham
  
• A Primer of Real Functions by Ralph P. Boas
  
• Understanding Analysis by Stephen Abbot
  
• Chaos: Making a New Science by James Gleick
  
• Logic: An Introduction to Elementary Logic by Wilfred Hodges
  
  and last but definitely not least:
  
  
• A Mathematical Gift I, II, III by Kenji Ueno et al.
  
  Later on:
  
  
• Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere
  
• Twelve Landmarks of Twentieth-Century Analysis by D. Choimet & H. Queffélec
  
• Theory of Complex Functions by Reinhold Remmert
  
• Algebraic Combinatorics. Walks, Trees, Tableaux & More by Richard P. Stanley

What Is Mathematics?: An Elementary Approach to Ideas and Methods

"Succeeds brilliantly in conveying the intellectual excitement of mathematical inquiry and in communicating the essential ideas and methods." Journal of Philosophy

https://www.amazon.ca/What-Mathematics-Elementary-Approach-Methods/dp/0195105192

Well, if you want something light and accessible and suitable for the layperson, I'm quite fond of Jan Gullberg's Mathematics from the Birth of Numbers. It goes over basically everything you would typically learn in primary and secondary school, and it presents everything with historical background. But it doesn't go into tremendous detail on each topic, and it doesn't provide the most rigorous development. It's more of a high-level overview.

But if you really want to learn some mathematics, on a deep and serious level, be prepared to read and study a lot. It's a rewarding journey, and we can give you book recommendations for specific topics, but it does take a lot of discipline and a lot of time. If you want to go that route, I would recommend starting with an intro to proofs book. I like Peter J. Eccles's An Introduction to Mathematical Reasoning, but there are many other popular books along the same line. And you can supplement it with a book on the history of mathematics (or just read Gullberg alongside the more serious texts).

I enjoyed this one by the same author: Fermat's Enigma. Maybe 1/3 to 1/2 of the book tells the story of Andrew Wiles trying to prove Fermat's Last Theorem (and the significance of it), and mixed in throughout is information about all sorts of mathematical history.

This is not a highly advanced or hard-to-read book. Anybody with an interest in mathematics could enjoy it. If you're looking for some higher-level mathematical knowledge, this is not the book to read. I haven't read The Code Book, so I don't know how similar it is.

EDIT: The first review starts with "After enjoying Singh's "The Code Book"..." The reviewer gave it 5 stars.

Mathematics for the Nonmathematician by Morris Kline

I swear by this. As someone who has always been a reading person, math textbooks drive me crazy. Stupid bold text, boxed problems, and cluttered graphs distracted me from the poorly written explanatory paragraphs. If I was lucky, my math teacher would be good at explaining a concept verbally as well as visually. Many have already recommended Khan, who is much better with the visual than the verbal.

Morris Kline was a professor at NYU in the 50s, 60s and 70s, a time when textbooks were still more like books than illustrated guides. His writing is clear and concise, which is a must for math, but it is also filled with examples from the real world (including history, art, and engineering).

This book was specifically written for Liberal Arts majors at NYU, not math majors. I'm a Biology major, so somewhere in between as far as technical math goes. I bought this before taking Pre-Calculus in the summer from 8am to 11am. I had never been great at math, just as good as the time I put in doing problems (which was not much). After reading through it, I was excited about trigonometric functions, imaginary numbers, exponentials, and the like. He puts things into a conceptual framework that is very attractive to a "big picture" person like myself.

Buy this book. Buy it now. 28 reviews on amazon and 4.5 stars. 10 bucks. Do it.

The Calculus Gallery - Masterpieces from Newton to Lebesgue by William Dunham it's like having the world's best math teacher lead you thru all the jaw-dropping amazing parts of the history of calculus; how at each stage someone figured out some bizarre counter-example which caused the next person to push calculus along further. With understandable equations!

...I thought you were referring to my write up on the complex numbers. Yeah GEB is definitely a project to get through. If you ever try it again, I recommend as a companion Godel's Proof by Nagel, edited by Hofstatder, the guy who wrote GEB. It gives you a really nice and short explanation, building from essentially the ground up, of how the incompleteness theorems are proved, and if you read it along with GEB then in the more technical symbolic logic parts, you get a little bit better of an idea of "okay where's he going with this".

I doubt this can be answered for a five year old, I read an excellent book on the subject and still don't really get it. I will try to recount the jist of what I remember.

Fermat left a small note scribbled in the margins of a book: a^n + b^n = c^n has no solution for positive integers greater then 2.

What fascinated everyone is that if n=2 you have the Pythagorean theorem which every knows, loves, and uses all the time. But to say that there is no solution for a^3 + b^3 = c^3 well that seems a bit crazy. You can sit down and try to plug in the first few values yourself, and low and behold you cant find any solution. Fermat had claimed that he had a proof that showed that this was true from 3 > infinity. (personally I don't think he had an actual proof, more of a very strong gut instinct and if anyone in his lifetime proved him wrong he would have laughed at them and said that he trolled them hard.)

That's the background, now to your questions, what are mathematical proofs? They show that a given formula is true in all cases, any two positive integers plugged into the Pythagorean theorem will result in a real solution for C.

Why is it hard to make them? because you have to show that the theorem works to infinity, you can plug in billions of numbers into a theorem, and prove nothing because the billionth + 1 may not be true

What was so special about Fermat's? Not much, except that it drove people insane with its simplicity, but it took hundreds of years to prove that a^3 + b^3 = c^3 had no real solutions and hundreds of years more for Andrew Wiles and Richard Taylor to discover the general proof.

From wikipedia as to whether Fermat actually had a general formula:

>Taylor and Wiles's proof relies on mathematical techniques developed in the twentieth century, which would be alien to mathematicians who had worked on Fermat's Last Theorem even a century earlier. Fermat's alleged "marvellous proof", by comparison, would have had to be elementary, given mathematical knowledge of the time, and so could not have been the same as Wiles' proof. Most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents n.

and finally my attempt at EILI5:

You know how you ask me a million questions every day, and some times I don't have the answer. Now imagine going to your teacher and asking them, and they don't know, and ten years from now you ask another teacher and they still don't know, you grow up and go to college and ask your professors and they don't know either. Your question sounds like it should be easy to answer, why doesn't anyone know the answer, then you try to answer it for yourself, and you can't figure it out. You try for thirty years to answer the question, and talk to other people who have tried to answer the question for the last 400 years and still no answer. Some people might give up, but the fact that you could be the first person in the world to know something makes you work even harder to find the answer to this simple question.

A good place to start might be Gödel's Proof. This book really guides the reader through all the different ideas that were used for Gödel's theorems, the authors give a lot of examples and really succeed at simplifying the concepts and outlining the steps in Gödel's reasonning. Very good read. (And of course they talk about Gödel's numbering in it :) )

The Mathematical Experience is really a collection of essays, so it doesn't bog you down, and you can easily skip the ones that don't interest you. It covers some actual mathematics, but is mostly focused on what it means to do math, what it is like to be a mathematician, and philosophy of math topics. If you liked A Mathematican's Apology, this is a good bet.

You need it because it makes the math work, but the term imaginary is foolish. "Imainary" numbers are numbers just like real numbers. The only difference being that you cannot have an Imaginary quantity of something (just like you cannot have a negative quantity of something, but we still use negstives). Imaginary numbers are associated with rotations and periodicity (sine waves), and they even have the geometric interpretation of a problem being "unsolvable" with real lengths, but even if you construct these unsolvable problems with the complex numbers, lo-and-behold the complex solution gives you the geometric property you wanted to construct!

If you are at all interested, this is an excellent book written by an electrical engineering professor about the history and applications of imaginary numbers.

The book that really hooked me on math (I was an undergraduate math major) was G. H. Hardy's, "A Mathematician's Apology". You can find free versions online, because over 50 years have passed since publication. But the free versions I saw don't contain the introduction by C.P. Snow that the book has. So you might consider getting the book, either out of the library or from Amazon.

Two other recommendations would be:

• James R. Newman's 4-volume, "The World of Mathematics"
  
  
• E. T. Bell's, "Men of Mathematics"
  
  All three of those kept me duly inspired before and during my undergraduate years.

I fall firmly on the discovered side of the debate although I have had this discussion many times before. I believe math exists independently of humanity and that math is simply a tool used to describe patterns in a logical and rigorous way, but that those patterns have always and will always be in the universe and did not need us in order to exist.The driving force in the history of mathematics has not been practical invention by someone determined to solve an existing problem but by investigation of someone into a pattern that they saw or suspected that they saw. Math ends up being useful to people who want to solve a real world problem, but very little math gets created to solve a real world problem. Mostly because the real world problem didn't exist before the math, or that some very clever person saw the math and realized it could be adapted to solve a longstanding problem.

A great book for non mathematicians covering the history of how math came to be what it is today, and all of the impressive fundamental changes that had to occur in the way we though about math would be The Language of Mathematics by Kevin Devlin, a professor of mathematics at Stanford University. My favorite bit from the book is that things that we think are very simple in fact took thousands of years for humans to abstract and notice the patterns. Egyptians for example basically knew the formula for the volume of a pyramid. However we have not a shred of a formula from that period. Instead they treated each set of dimensions a pyramid could have to be a separate problem and therefore failed to notice the formula that was right in front of their faces. It took the greeks (and a good deal of time) for someone to notice the pattern between all pyramid formulas and abstract it into a general solution to the pyramid volume problem.

Applications? If you're a set theorist or a logician there are applications. Otherwise transfinite numbers are of intrinsic interest and have no known applications.

On the other hand if you google around you can find papers trying to relate set theory to physics. Not much out there but there's a little.

For a really cool introduction to transfinite numbers and their philosophy, I strongly recommend Infinity and the Mind by Rudy Rucker.

https://www.amazon.com/Infinity-Mind-Philosophy-Infinite-Princeton/dp/0691121273

Not really a "textbook", but I can highly recommend The Music Of The Primes by Marcus du Sautoy. It's a great read. http://www.amazon.com/The-Music-Primes-Searching-Mathematics/dp/0062064010

A History of Pi by Petr Beckman is a fun read. There are also several casual math books by Simon Singh that I've heard are pretty good. I also kind of like reviewing old subjects I've already learned about in bed, since I don't need to sit down with a pen and paper to get them.

Hi, here I will post some great books, some free (by Santos), some not (others).

Junior problem seminar: Santos

Number Theory for Mathematical contests: Santos

The Art and Craft of Problem solving: Zeitz

Problem-Solving Strategies: Engel

Mathematical Olympiad Treasures: Andreescu, Enescu

Mathematical Olympiad challenges: Andreescu, Gelca

Problems from the book: Andreescu

Those are more or less the "general" books, they always contain the main topics of mathematical olympiads, they usually aren't focused on just one topic, for one-topic books see here: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=319&t=405377

Jan Gullberg's Mathematics: From the birth of numbers is a great book I'd recommend: https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X

It introduces a lot of mathematical topics starting from the "simplest" (numbers you asked about) and advances to common stuff found in university studies (although not going extremely far), but what might be the biggest feat and useful to your case is that tells as a non-fictional story while at it, explaining mathematical tools, their history and how they relate to each other extremely well in a way a normal college textbook doesn't, and it doesn't assume you already know everything from school.

I really enjoyed Godel's Proof by Nagel + Newman. It's a layman's guide to Godel incompleteness theorem. It avoids some of the more finnicky details, while still giving the overall impression.

https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371/

If you like that, it's edited by Hofstadter, who wrote Godel-Escher-Bach, a famous book about recurrence.

Finally, I would recommend Nonzero: The Logic of Human Destiny by Robert Wright. It's a life-changing book that dives into the relevance of game theory, evolutionary biology and information technology. (Warning that the first 80 pages are very dry.)

https://www.amazon.com/Nonzero-Logic-Destiny-Robert-Wright/dp/0679758941/

A great book on Andrew Wiles and Fermat's Last Theorem is Simon Singh's Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.

I know this is not exactly what you had in mind, but one of the most significant proofs of the 20th century has an entire book written about it:

http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371

The proof they cover is long and complicated, but the book is nonetheless intended for the educated layperson. It is very, very well written and goes to great lengths to avoid unnecessary mathematical abstraction. Maybe check it out.

Here are some suggestions :

https://www.coursera.org/course/maththink

https://www.coursera.org/course/intrologic

Also, this is a great book :

http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_5?ie=UTF8&qid=1346855198&sr=8-5&keywords=history+of+mathematics

It covers everything from number theory to calculus in sort of brief sections, and not just the history. Its pretty accessible from what I've read of it so far.


EDIT : I read what you are taking and my recommendations are a bit lower level for you probably. The history of math book is still pretty good, as it gives you an idea what people were thinking when they discovered/invented certain things.

For you, I would suggest :

http://www.amazon.com/Principles-Mathematical-Analysis-Third-Edition/dp/007054235X/ref=sr_1_1?ie=UTF8&qid=1346860077&sr=8-1&keywords=rudin

http://www.amazon.com/Invitation-Linear-Operators-Matrices-Bounded/dp/0415267994/ref=sr_1_4?ie=UTF8&qid=1346860052&sr=8-4&keywords=from+matrix+to+bounded+linear+operators

http://www.amazon.com/Counterexamples-Analysis-Dover-Books-Mathematics/dp/0486428753/ref=sr_1_5?ie=UTF8&qid=1346860077&sr=8-5&keywords=rudin

http://www.amazon.com/DIV-Grad-Curl-All-That/dp/0393969975

http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0738204536/ref=sr_1_2?s=books&ie=UTF8&qid=1346860356&sr=1-2&keywords=chaos+and+dynamics

http://www.amazon.com/Numerical-Analysis-Richard-L-Burden/dp/0534392008/ref=sr_1_5?s=books&ie=UTF8&qid=1346860179&sr=1-5&keywords=numerical+analysis

This is from my background. I don't have a strong grasp of topology and haven't done much with abstract algebra (or algebraic _____) so I would probably recommend listening to someone else there. My background is mostly in graduate numerical analysis / functional analysis. The Furata book is expensive, but a worthy read to bridge the link between linear algebra and functional analysis. You may want to read a real analysis book first however.

One thing to note is that topology is used in some real analysis proofs. After going through a real analysis book you may also want to read some measure theory, but I don't have an excellent recommendation there as the books I've used were all hard to understand for me.

Try to find entry points that interest you personally, and from there the next steps will be natural. Most books that get into the nitty-gritty assume you're in school for it and not directly motivated, at least up to early university level, so this is harder than it should be. But a few suggestions aimed at the self-motivated: Lockhart Measurement, Gelfand Algebra, 3blue1brown's videos, Calculus Made Easy, Courant & Robbins What Is Mathematics?. (I guess the last one's a bit tougher to get into.)

For physics, Thinking Physics seems great, based on the first quarter or so (as far as I've read).

I recommend going through some of the lessons on Brilliant, and here is Brilliant's quick exposition on the set of complex numbers.

I don't know what a softer explanation would entail exactly, but I would offer you the alternative perspective that the representation of complex numbers as two real numbers a+ib for the real numbers a and b is extremely useful because of the interpretation of the extension of the one dimensional real number line into the two dimensional complex plane.

Also, I recommend reading on a simple exposition of complex numbers from Richard Courant's "What is Mathematics".

You and the other A/AS-level kids shouldn't worry about complex numbers. At university, you'll come to appreciate the usefulness and beauty of them. For example, see this post I made on /r/math earlier today. People have written entire books eulogising about them. For example, Paul Nahin has written two such books.

Fermat's Last Theorem is a pretty good story. It's an easy to understand problem that was unsolved for 300 years until ~20 years ago.

There's a book about it and a PBS documentary you can watch for free.

Try this book for help with understanding Algebra. My uncle had left a copy at my grandparents house, and I picked it up when I was there when I was in the third grade (we were working on multiplication and division). I made a perfect score in the state tests for Algebra 1, Algebra 2, Geometry, and Trigonometry.

I read this book in high school, and it really helped me figure out how to think about breaking down more complex problems.

This book made math very clear for me as well.

I think these books may help you because you could do the math he read to you. These books helps give you an understanding of what is actually happening. Foe example, most people do not understand that multiplication is nothing more than extended addition, until you explain it to them. If you can think about the problems and understand what the problem is saying, it will be easier to figure out. I did a lot of math in my head that would have taken several pages to write it out the way I did it, but if you wrote it the way they expect would only take a few lines.

I am very happy for you for finally finding someone who knew what was going on with you. I had a similar problem in elementary school, but my parents did not trust the school and had me tested on their own. They decided that I had a "social communication disorder, kind of like a really weak autism" (This is what my parents ended up telling me anyway). The school thought I was "developmentally challenged" ("borderline retarded" was the phrase that was bandied about) but when my parents had my IQ tested, it was a 141, which is not quite what was expected, they decided that the problems were elsewhere.

One thing that is very important in math is that if you do not understand, you can go back and work on fundamentals and build up your foundation, and the more advanced stuff will be easier.

Good luck, and I believe you really are an adept writer. What you wrote grabbed my interest and was compelling.

Thanks for the suggestions! Just so you are aware the Fermat's Enigma link is a duplicate of Journey through Genius.

Journey through Genius sounds really interesting. I'm curious if you've ever read Gödel, Escher, Bach? If so how would you compare the two?

Paul Nahin has published many good historical math books that don't skimp on the mathematical underpinnings. I particularly enjoyed An Imaginary Tale: http://www.amazon.com/An-Imaginary-Tale-Princeton-Science/dp/0691146004

Regarding Spivaks: I'm also working on it, and found that my proof technique was lacking. An Introduction to Mathematical Reasoning (Eccles) was helpful for me: http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188

When I first went through it, I found it very verbose and too abstract for me. I was clearly not prepared for it.

Then I happened to read Gödel's proof, by Nagel and Newman, with an updated commentary by Hofstader. What a terrific book! Having gone through it, I began enjoying GEB.

There's tremendous depth in both books, and I look forward to iterating through these two alternately and getting more and more insights.

Please, simply disregard everything below if the info is old news to you.

------------

Algebraic geometry requires the knowledge of commutative algebra which requires the knowledge of some basic abstract algebra (consists of vector spaces, groups, rings, modules and the whole nine yards). There are many books written on abstract algebra like those of Dummit&Foote, Artin, Herstein, Aluffi, Lang, Jacobson, Hungerford, MacLane/Birkhoff etc. There are a million much more elementary intros out there, though. Some of them are:

Discovering Group Theory: A Transition to Advanced Mathematics by Barnard/Neil

A Friendly Introduction to Group Theory by Nash

Abstract Algebra: A Student-Friendly Approach by the Dos Reis

Numbers and Symmetry: An Introduction to Algebra by Johnston/Richman

Rings and Factorization by Sharpe

Linear Algebra: Step by Step by Singh

As far as DE go, you probably want to see them done rigorously first. I think the books you are looking for are titled something along the lines of "Analysis on Manifolds". There are famous books on the subject by Sternberg, Spivak, Munkres etc. If you don't know basic real analysis, these books will be brutal. Some elementary analysis and topology books are:

Understanding Analysis by Abbot

The Real Analysis Lifesaver by Grinberg

A Course in Real Analysis by Mcdonald/Weiss

Analysis by Its History by Hirer/Wanner

Introductory Topology: Exercises and Solutions by Mortad

I'd only recommend Everything and More if you are a Wallace completist (and I am), but the math needs some work.

Rudy Rucker's Infinity and the Mind covers the same material and is more mind-blowing. I used to be comfortable with the natural numbers and he ruined that (in a good way).

When I was in seventh grade my math teacher lent me some books to read:

• Fantasia Mathematica, by Clifton Fadiman (and there's a sequel, The Mathematical Magpie);
  
• A History of Pi, by Petr Beckmann;
  
• and some introductory book about topology, explaining the Möbius strip, the four-color theorem, the Klein bottle, the projective plane, and maybe some other stuff, whose title I have since forgotten.
  
  I credit these books for sparking my interest in math.
  
  Other books I'd recommend include Imaginary Numbers by William Frucht, Flatland/Sphereland by Edwin A. Abbott/Dionys Burger (and Flatterland by Ian Stewart), The Penguin Dictionary of Curious and Interesting Numbers by David Wells, A Passion for Mathematics by Clifford A. Pickover, The Mathematical Tourist by Ivars Peterson, and any book by Martin Gardner or Raymond Smullyan. Also most books by Ian Stewart would be good, but he also writes higher-level math textbooks, so watch out for that.

A Panoramic View of Riemannian Geometry is HUGE and contains so much quirky stuff. That was the book that taught me how to really think about Riemannian geometry. I would just lie in bed and read it. No need for notes or calculating anything.

Also, Engel's Problem Solving Strategies is a classic if you are into that sort of thing.

I also like Problems and Theorems in Classical Set Theory where you get to work out pretty much all of set theory by yourself through solving problems. Very little is actually given.

I've currently been absolutely loving The Mathematical Experience. It's an amazing book thus far, and is blowing my mind more and more every day.

Try What is Mathematics?, by Courant & Robbins. It's a good overview of mathematics beyond the elementary level you've completed. Another good book like that is Geometry and the Imagination, by Hilbert & Cohn-Vesson.

Well, regarding Fermat's Last Theorem, it indeed was written by Aczel, as could easily be determined by following the link in the article. However, it looks like there are 100s of books with a similar name. The one your read by Simon Singh was called: Fermat's Enigma.

You weren't misled, you just "misremembered".

Anything specific you're searching for? Tales from the field? Methodology? Lesson planning? This isn't precisely what I think you're looking for, but if you're looking for some thought-provoking reading to remind you why you love math... here

Agree. I picked up on that from the intro to GEB, stopped reading GEB, and decided to get a better understanding of Gödel's proof by reading the book Hofstadter says introduced him to Gödel - Gödel's Proof, by Ernest Nagel and James R. Newman. I recommend it as a very approachable introduction to Gödel's incompleteness theorems. Even now I can recall moments reading that little book where I'd get a big smile on my face as the force of his argument and conclusion would bear down on me. What Gödel did is nothing short of mind blowing.

After that, if you want more, then go to Gödel's Incompleteness Theorems by Raymond M. Smullyan (You'll want to buy this one used). This one is a much more technical, though still approachable if you're prepared at an undergrad level, guide through to Gödel's conclusions. You should go into it with an undergrad level of fluency in propositional and predicate logic.

You can read GEB without all that, certainly without the second book, but I've found it a better experience having more familiarity with Gödel as I work through it.

I'm reading Mathematical Thought from Ancient to Modern Times by Morris Kline, and I really like it. It's three volumes, so it's pretty thorough. I've only finished the first volume, but he does a really good job of explaining the way the Greeks thought about mathematics, compared to the modern Western way. It's not too technical to follow, but he tries to explain the important theorems and their proofs in some detail. Volume 3 goes up through Poincare and Goedel, and then stops, so you won't learn about later twentieth century developments from him. But he says it's too soon to know what's historically important there, anyway. I'm looking forward to the next two volumes.

You can start with "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem" >> http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622 . Is a great book, I read it several times.

This whole thing reminds me of a book I read a few years back about a guy who proved Fermat's Last Theorem. Fascinating stuff. Really gives one an insight into how beautiful the human mind is.

Please take a look at What is mathematics by Courant, Robbins, Stewart. It is much leaner, yet it is accessible and was endorsed by Einstein:

"A lucid representation of the fundamental concepts and methods of the whole field of mathematics. It is an easily understandable introduction for the layman and helps to give the mathematical student a general view of the basic principles and methods."

Another vote for The Code Book, as a book targeted more towards the general public, I thought it was excellent. I read it in high school and it's one of the reasons I decided to go into math/CS in university!

Fermat's Enigma (also by Singh) is another one I enjoyed.

Not that I'm aware of. If you're interested in reading more, I recommend checking out Paul Nahin's An Imaginary Tale. It's midway between a textbook and a popular math book that explores the history and utility of imaginary numbers (believe me, there are a lot of applications of imaginaries)

same Rudy Rucker?

I really enjoyed this book when i found it in the free bin a couple years ago.

I'd recommend Mathematics: A Very Short Introduction by Tim Gowers if you'd like a fairly serious but informative book describing what mathematics is really about.

For a more fun, coffee-table kind of book, have a look at The Math Book by Clifford Pickover.

William Dunham, an author I appreciate because he succeeds in popularizing mathematics without sacrificing rigorous exposition, wrote on the Baire Category theorem in The Calculus Gallery. I too was a bit confused as to the motivation of Baire's Category theorem until I went back and read his chapter. I don't have my copy handy at the moment but I recommend checking out that book for an expository and historical explanation.

There were two books that got me completely involved in the world of mathematics.

History of Pi

Golden Ration, Phi

These two books were great when I read them when I was 16 and they got me completely wrapped up in mathematics (currently I am a Physics Grad student working on my Ph.d). Well worth reading.

James Newman collected a bunch of nice source material in a several-volume set he titled The World of Mathematics (1956). You can find them on the Internet Archive, or buy a relatively recent reprint printed copy from Amazon.

To put the history of mathematics in context, try Stillwell’s book.

Do you have any particular interests? People here can give better advice if you can narrow down the subjects you’re looking for.

It's worth grokking though. I read this the Summer b/t HS and College: http://www.amazon.com/Godels-Proof-Ernest-Nagel/dp/0814758169

The preface was actually by Hofstadter. Apparently, as a child, his parents were good friends with the author, Ernest Nagel.

https://www.amazon.com/Math-Book-Pythagoras-Milestones-Mathematics/dp/1402788290/ref=mp_s_a_1_5?keywords=history+of+mathematics&qid=1557935307&s=gateway&sprefix=bhisyory+of+ma&sr=8-5

Great book, everyone should read it, who likes math.

This was my textbook for an undergrad class I took in math history. I believe it covers everything in your list, and it's all super interesting!

https://www.amazon.com/gp/product/0030295580/ref=oh_aui_search_detailpage?ie=UTF8&psc=1

Even though most every proof technique (contradiction, induction, smallest counterexample etc) is shared by almost all branches of math, every branch of math has very specific goals or ways of going about proving statements. In your case you probably don't have time to learn how to prove statements in number theory, combinatorics or category theory or whatever so you must concentrate on analysis proofs. To that end, check out:

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by R Grinberg

Writing Proofs in Analysis by J. M. Kane

Those are/were my interests and I enjoyed [Logical Dilemmas] (http://www.amazon.com/Logical-Dilemmas-Life-Work-Godel/dp/1568812566/ref=sr_1_1?ie=UTF8&qid=1324312359&sr=8-1), a thorough biography of Kurt Godel. [Godel's Proof] (http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?s=books&ie=UTF8&qid=1324312476&sr=1-1) might be too basic but is a good read.

There's an excellent book, A History of Pi by Petr Beckmann, ^^Check ^^it ^^out ^^from ^^your ^^local ^^library that explains the cross-cultural interest in this specific ratio from the ancient pre-base 10 people to modern times.

You would probably like these two books:

• Geometry and the Imagination by David Hilbert and Stefan Cohn-Vossen.
  
  
• What is Mathematics? by Richard Courant.
  
  Neither of those are "popular math" books; the authors are famous mathematicians, and they explore various fields of mathematics without requiring too much advanced knowledge.

Book recommendation for an intro to Godel's Theorem: Gödel's Proof - Ernest Nagel and James Newman. Well written, concise and requires no prior mathematical knowledge.

Edit: Never mind. misread "I do have an introductory understanding..." as "I don't have an introductory understanding...". Still a good recommendation for anyone else who is interested!

A couple of people have mentioned it (I upvoted), but I thought maybe a top-level comment would be helpful. "Godel's Proof" by Ernest Nagel and James R. Newman is a very clear exposition of the theorem, requiring little to no mathematical background.

Fermat's Enigma by Simon Singh is an approachable history of Fermat's last theorem, various brilliant but failed proofs, and Wiles' ultimate conquest. While it's not technical, the book profiles the mathematicians tormented by Fermat's theorem and details the approaches they used. You may find it helpful as a map or a timeline. Certainly worth reading.

http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622

BTW, if you want a relatively easy description of Godel's work, this book may be useful.

I really like What is Mathematics by Richard Courant. It's aimed at the lay person and I think a 13 year old would enjoy it. It's a book you can jump around in too.

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i'd highly recommend 'an introduction to the history of mathematics' - gives a really good background on math from an cultural and evolutionary standpoint.

link

I am currently working my way through Godel's Proof by Nagel and Newman. So far, it
has been rather interesting. I would highly recommend it if you are
interested. Some of the formulas on the Kindle edition don't look the greatest, but it is still legible. Also, it is surprisingly affordable.

I read Mathematics: From the Birth of Numbers in high school / early college. It's a long book, but it's definitely worth checking out.

Gödel's Proof is a good starting point for the incompleteness theorem. Covers the basics of the theorem and its impacts. Unless you are prepping for coursework in logic than this book likely has the right amount of depth for you.

I don't have a recommendation for Tarski. Hopefully someone else has something for you.

For a general overview of everything to do with the history of math, which might be what you're looking for, I recommend Mathematics: From the Birth of Numbers. Very inspiring with a little bit of "how to do everything."

One of my favorites is An Imaginary Tale: The Story of √-1. I read it when I was a freshman and couldn't wait to take Complex Analysis. The author has a sort-of-sequel about Euler's equation, which I didn't like as much but was still enjoyable.

You might want to consider reading a book about the history of maths like: https://www.amazon.com/Math-Book-Pythagoras-Milestones-Mathematics/dp/1402788290/ref=sr_1_2?ie=UTF8&qid=1502399281&sr=8-2&keywords=history+of+maths

This will help you understand how math evolved and put the various major math discoveries of the past in context.

The short path is through Kline's Mathematics for NonMathematicians. I think calling it a book for non-mathematically inclined readers is a stretch (even the inclined are going to have 600+ pages to wade through) but it's definitely a solid redux of history. Just detail-lite.

These might interest you:

Poincare's Prize


Prime Obsession


Fermat's Enigma

The Code Book

There's a great introduction to Gödel's Incompleteness Theorems, it's called and Gödel's Proof by Nagel & Newman. Hofstadter has wrote it's foreword. It's a very short book, 160 pages in total.

Amazon Link!

Mathematics: From the Birth of Numbers

It's gigantic, but really entertaining to flip around in.

Rucker's non-fiction book on infinity, Infinity and the Mind: The Science and Philosophy of the Infinite, is great, too. I read that when I was about 20 and it blew my mind.

Came to say this.
http://www.amazon.co.uk/Math-Sterling-Milestones-Clifford-Pickover/dp/1402788290

Hefty, beautiful, informative.

Highly recommend Godel's Proof for anybody looking to jump into the question of how well founded modern mathematics is.

If you're looking for a concise introductory level reference, I don't know of any at only the high-school level; additionally most undergrad level textbooks are gonna assume a certain level of sophistication w.r.t. the student.


However, if you are interested, the book "Godel's Proof" by Nagel, offers many accessible insights into the workings of mathemical logic

https://www.amazon.com/Gödels-Proof-Ernest-Nagel/dp/0814758371

I picked up the book by Simon Singh at a garage sale 10 or so years ago. Fascinating read. Looking forward to watching the doc now.

EDIT: evidently the book is now called Fermat's Enigma in the US...

Seriously this may be a great coming-of-age title for you: Infinite Jest.
Also since you got your first job check out The Wall Street Journal's Guide to Starting Your Financial Life. If you haven't yet appreciated math, I would suggest you do so as you're going to need it for any decent job these days. Detach yourself from Fallacious Thought.

These are a couple of nice old books about mathematical thinking:

• What is Mathematics? by Courant, Robbins, and Stewart
  
• How to Solve It by Polya

Morris Kline isn't a professional historian but he has a book on this that you might be interested in. IIRC, volume 2 has the most stuff on the history of calculus.

Be brave, start with http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192

Try khan academy - free site with videos about basic math. This book too: https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248232

The Math Book is full of visuals! Every two pages it has a neat bite-sized idea with a full page picture.

Godel's Proof is the original inspiration for Hofstadter. I find it a shorter but no less interesting read.

http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371

Well I don't know how interested you are in this, but if you want to understand the incompleteness theorem and its implications without learning all of number theory, I ran across this book which provides the history leading up to Gödel, the mathematical context he was working in (e.g. Hilbert's project), and a full explanation of the proof itself in just over 100 pages. I read it in a day, and while I have a background in the area, even if you didn't know anything going into it, you could probably understand the whole thing with two days' careful reading.

If you're interested check out Mathematical Thought from Ancient to Modern Times by Kline or A History of Mathematics by Suzuki.

Aside from The Princeton Companion to Mathematics, you might like to check out What Is Mathematics? An Elementary Approach to Ideas and Methods by Courant and Robbins, and Mathematics: Its Content, Methods and Meaning by three Russian authors including Kolmogorov.

I haven't read it yet, but Richard Courant's What is Mathematics? has been highly recommended to me.

If anyone is interested in serious math, but in a (somewhat) light fiction form, take a look at the books:

Rozsa Peter, Playing with Infinity [Amazon] [Goodreads]

Kline Morris, Mathematics for the Nonmathematician [Amazon] [Goodreads]

A good book on Gödel's proof is Gödel's Proof.

P.S. May I also recommend:
e: The Story of a Number,
A History of Pi, and
Zero: The Biography of a Dengerous Idea

A History of Pi. Recommended for those who like math and those who don't. Very interesting read.

Talk about a lack of substance!

A book I read way back when that was excellent was Gödel’s Proof by Nagel and Newman. http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371

They've been very bad since at least the 1970s, running one ludicrous political article in every issue, the first of the "real" articles. I watched over several years as the Kosta Tsipis group, I think it was, steadily decreased their claims of what was beyond the foreseeable future state of the art. They were so bad, they effectively said the Mount Palomar Hale Telescope's 1948 mirror was ... beyond the foreseeable future state of the art. Someone pointed out that you and I could in an afternoon, using car batteries in a small alley or the like, construct a power system they said that was ... beyond the foreseeable future state of the art (of course, that wouldn't be space rated). Really, utterly sloppy, no wonder he doesn't have a Wikipedia entry.

Someone who I consider to be reliable, Petr Beckmann, perhaps best widely known as the author of A History of pi, was a refugee from Soviet occupied Czechoslovakia (and Nazi occupied, he related getting out from a first run showing of Fantasia to learn that the British and French had sold out his country in Munich). He said in his Access to Energy newsletter that you could predict the topic of this political Scientific American article roughly 6 months in advance based on what another anti-West and science and technology Communist Czech journal published. Can't read the language, but it's a very falsifiable claim, and he didn't make up bullshit.

I'm reading Mathematics: From the Birth of Numbers right now, I recommend it for your needs.

You might like this book too. It's a condensed version of Gödel's proof for the layperson, with an introduction by Hofstadter.

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.

Nagel's book 'Gödel's Proof' is a good, intelligible summary of Gödel. I suggest reading that, even if you suck at math.

http://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687

This is one of the best philosophy of math books I've ever encountered. It's not exactly short like you wanted, but it covers a lot and it's very readable.

Fermat's Last Theorem by Simon Singh.

(I'm reasonably sure the linked book is Fermat's Last Theorm, just with a different title. It was the closest I could find on US Amazon)

I am not sure huge roadmaps are any useful. Especially if you're just starting out. That said, check out Writing Proofs in Analysis by Jonathan Kane.

For the price and material, you really can't go wrong with Mathematics for the Non-mathematician

Preview here

There is a book entitled Godel's proof that was written that was written to explain the ideas of Godel's proof without requiring too much background.

http://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/0814758371/ref=sr_1_1?ie=UTF8&qid=1342841590&sr=8-1&keywords=godel%27s+proof

It is hard for me to offer to much advice beyond that because I am in a different field of mathematics (number theory).

Meh, was a bit anti-climactic.

I preferred Fermat's Last Theorem. That took 350 years to solve, not just a quick google. Kids today etc...

Completely backwards; I learned earlier disc math in bits and pieces from various college algebra books (e.g., this, there were others but I can't recall their titles) and the bulk of my disc math/comp sci theory/graph theory/big-O from this one which I have no reason whatsoever to believe is the best way to do it and which dramatically over-emphasizes grammars and Turing machines compared to WGU curriculum and, I think, college curricula in general. It was also kind of a heavy lift since my disc math was weak coming into it, I think I spent more time on the one-chapter 'review' of discrete math than I did on any other three chapters in the book.

It's difficult to make recommendations without being certain of what you actually know and what you imagine mathematics to be like. A lot of university-level mathematics is technical and requires familiarity of high-level concepts. This is in contrast to softer popular mathematics, which is more related to solving problems and contest questions. One of the things I've noted about pre-university students passionate about mathematics is that they assume that the subject is only about problem-solving and fail to take into mind the level of technical knowledge that must be learnt and memorized to be a mathematician.

If you're simply looking for problems to solve, try The Art and Craft of Problem Solving by Zeitz or Problem Solving Strategies by Engel. Generally any book geared to the Olympiad or regional competitions will be alright. Here, you're not looking for a specific body of knowledge, but rather an approach to thinking and persevering when handling tough problems.

But if you're looking to learn more about 'technical' mathematics, you'll need to know the basics of numbers and sets. Numbers & Proofs by Allenby is a good introduction, using an approach that gets you to actively solve problems. Once you get past that, then you can try your hand on analysis or group theory or linear algebra or even basic graph theory. But keep in mind that with 'technical' mathematics, all knowledge is built on understanding of previous fields, so don't rush through it or you'll get discouraged by any difficulty or unfamiliarity you'll encounter.

Also:

• An Introduction to the History of Mathematics by Howard Eves

https://www.amazon.com/Music-Primes-Searching-Greatest-Mathematics/dp/0062064010

I just bought this, and I'm waiting for it to be shipped. I heard it is life-changing.

From Richard Courant's "What is Mathematics", page 35, a constructive method is suggested.

"What is Mathematics ? An elementary approach to Ideas and methods ( 2nd Edition) by Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor) "

maybe you mean What is Mathematics? by Courant and Robbins.

Aleksandrov, Kolmogorov, Lavrent'ev. http://amzn.com/0486409163. Foundations to applicationsl.

Courant, Robbins, Stewart. http://amzn.com/0195105192. Tour of mathematics.