Reddit mentions of Mathematics: A Very Short Introduction

Expanding upon the important question of 'What do mathematicians even study', I strongly recommend Prof Tim Gowers' book 'Mathematics: a Very Short Introduction'. It's concisely and elegantly written by a well-regarded mathematician.

Geometry is a beautiful subject and you can study it right now. Have you already read Euclid's Elements? It may take a while to understand but it's a very nice book. I'd also suggest you study more algebra and possibly trigonometry on your own so you may tackle Calculus earlier. Almost any text book or Khan Academy may help you there. Set theory can also be very nice but Wikipedia's articles are probably not the right place to go for a beginner. Wikipedia likes to focus on rigor rather than good explanations. I wish I could recommend a set theory book or web page but I do not experience with it. I learnt most of my set theory form college-level discrete math textbooks so I'm afraid I can't help you there.

EDIT: Although I have only skimmed through it, Mathematics: A very short introduction is an interesting an quite accessible book.

Mathematics: A Very Short Introduction by Timothy Gowers. From the product description:

> The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as "Is it true that mathematicians burn out at the age of 25?") It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.

I would suggest picking up a copy of Mathematics: A Very Short Introduction by Timothy Gowers (he has one of those fancy field's medals). Gower's explains in much more detail the points that root45, yesmenapple, etc are all making. It's a pretty cheap book too.

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Once I got started ....whew. good luck!

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.

You should look into the [Very Short Introduction] (http://www.oup.com/us/catalog/general/series/VeryShortIntroductions/ series) series. About 342 books that are all in the range of 100 to 200 pages. They cover anything from [History] (http://www.amazon.com/History-Introduction-John-H-Arnold/dp/019285352X/ref=sr_1_2?s=books&ie=UTF8&qid=1346605671&sr=1-2&keywords=a+very+short+introduction) to [Mathematics] (http://www.amazon.com/Mathematics-Short-Introduction-Timothy-Gowers/dp/0192853619/ref=sr_1_4?s=books&ie=UTF8&qid=1346605671&sr=1-4&keywords=a+very+short+introduction) and even really specific topics like [Relativity] (http://www.amazon.com/Relativity-Short-Introduction-Russell-Stannard/dp/0199236224/ref=sr_1_15?s=books&ie=UTF8&qid=1346605803&sr=1-15&keywords=a+very+short+introduction). It's a gold mine of information and the best part is that all of the books are usually under $10.00's. They're perfect for what you're looking for.

I picked up [Mathematics: A Very Short Introduction by Tim Gowers] (https://www.amazon.com/Mathematics-Short-Introduction-Timothy-Gowers/dp/0192853619) a few years ago. It talked about the types of problems mathematicians worked on, had some puzzles, and I think talked about the job of a math professor (I have not read it in a while). It was an interesting read and seemed like it would be accessible to most people (also, it's super short, which is always a plus when trying to get people into things).

Absolutely the best book as an introduction to mathematics is "A Very Short Introduction to Mathematics" By Timothy Gowers.

I was a math major in college and so already had a bit of a background in math and was certainly fairly good at all high school level math stuff. However, I cannot tell you how powerfully this short little book made me look at math in a whole new way. I highly recommend everyone, whether you are into math or not, reads it.

It doesn't look at math the way you are taught it in school. It starts with the Axioms (the smallest basic assumptions you have to make in order to study something) and constructs math from there. Every body of knowledge has axioms, but we often don't think about them.

The thing that makes math so incredibly precise and logical is that the Axioms in math are completely known. We know exactly what baseless assumptions we have to make in order to know anything else. Things like "there is a multiplicative identity (namely 1)" or that there is an "additive identity" (namely 0).

You can see how important these axioms are. For example, look at the axiom of equality: x=x. We can't prove x=x but if we didn't assume it to be true then none of math would make any sense.

So once we have the axioms it turns out that every single thing that mathematicians have ever done can be constructed in logical steps from the Axioms. These are the often untaught and unspoken atoms that make up the mathematical universe.

If you truly want to understand math this is definitely the place to start though it's not necessarily the easiest and it won't immediately help you out in class. I believe however that the time you spend trying to wrap your mind around this and work through the brief introduction is worth 20x that same time spent trying to just learn PEMDAS or synthetic polynomial division.

For me, the point of studying math isn't to learn how to solve equations or actually do math. It's more about giving your mind additional tools in order to think properly. Learning the particular logical processes through which math works is like adding a tool to the toolbox that it uses to deconstruct and solve any problem. And I'm not talking about rote memorization. I swear to you if you make it through this short little book and start down the lifelong road of training your mind to work this way, then every other thing you try to think about, no matter what you end up doing with your life, will be easier.

Don't let anyone tell you that how smart you are has anything to do with the way you were born. Some people seem to learn things quicker or slower, but it's not so much a matter of their brains so much as a matter of how many logical tools they've noticed and stored in their mind growing up. The kids who happened to notice certain logical laws by which the universe abides have additional abilities in the way that think. It seems like their minds work quicker or better or something, but it's just a matter of what they know and practiced.

Doing exponents or any equation shouldn't be a matter of just sitting down and learning. A really long time ago somebody first figured out that x^2 * x^5 = x^7. He didn't learn it from a teacher since no one knew it yet to teach him. But he had the logical tools in his mind to become the first person to notice. You should give yourself the tools to be like that person. You can train your mind to do anything, it's more often than not just a matter of figuring out how to look at something.

If you revisit, might I suggest you start with this 160 page masterpiece : Mathematics: A Very Short Introduction

Thank you very much for taking the time to reply to me.

My auxiliary approach of learning the maths as I work my way through Machine learning may not be as practical as I would like, the book in question - Machine Learning: The Art and Science of Algorithms That Make Sense of Data, (Peter Flach 2012) is supposedly very light.

I am considering reviewing my approach and at least building some foundation before jumping in. Starting with this book: Mathematics: A Very Short Introduction, of which I've heard positive things about.

That aside, I found your response very reassuring. As a non-mathsy developer, I can certainly attempt to derive some logic from the formulae, but not without a lot of self-doubt!

Thank you again.

http://www.amazon.com/Mathematics-Short-Introduction-Timothy-Gowers/dp/0192853619

http://www.amazon.com/Honors-Class-Hilberts-Problems-Solvers/dp/1568812167

A good answer to the third question is to compare Hilbert's millennium address with the book Mathematics: Frontiers and Perspectives.

Hilbert's address and list of problems did a fairly good job of capturing the mathematical zeitgeist at the turn of the previous century. It took a book, and there will certainly be those who feel the book has missed something vital, to make an attempt at capturing the mathematical zeitgeist at the turn of this one. The book, and a few articles in particular, make attempts at sketching the next 100 years, and the personal accounts of the process of doing mathematics should give you an idea of an answer to your first.

Unfortunately, this book isn't very accessible to a non-math person. To really understand the answers to your question you'll first have to learn some math. A very good first step, especially if you want just enough of a taste to figure out why Frontiers and Perspectives was written, is Gowers' Mathematics: A Very Short Introduction (Tim Gowers also contributed to F&P). From the introduction,

> ... I do presuppose some interest on the part of the reader rather than trying to drum it up myself. For this reason I have done without anecdotes, cartoons, exclamation marks, jokey chapter titles, or pictures of the Mandelbrot set. I have also avoided topics such as chaos theory and Gödel's theorem, which have a hold on the public imagination out of proportion to their impact on current mathematical research, and which are in any case well treated in many other books.