Reddit mentions of An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

It depends on where they are and what the purpose is. If you are trying to discourage them (and there might be valid reasons to do that), I'd say try measure theory.

Maybe use the Bartle book.

That would give them a taste for how abstract things can get and also drive home the point tiny books can require a lot o work.

On the other hand, if you want to do something that will help them, they An Introduction to Mathematical Reasoning.

It won't break the bank and, despite a few small typos, covers a lot material fairly gently.

It sounds like you might perhaps want a background in Number Theory and/or Basic Logic and/or Set Theory. The thing about math is that there is a lot...

My advice for a text that might serve you well is N.L. Biggs' Discrete Mathematics (http://www.amazon.com/Discrete-Mathematics-Norman-L-Biggs/dp/0198507178). If you are at all interested in computer science, this is also a great book for that because it introduces some of the mathematical rigor behind it. Some people have a smidgen of difficulty with this text because it doesn't give some names to proofs/algorithms that maybe you've heard whispered (e.g. Dijkstra's shortest path and Prim's minimal spanning tree). A text that I tend to think is on par with Biggs', but many think is vastly superior (I love both, but for different reasons) that covers some (most) of the same topics is Eccles' An Introduction to Mathematical Reasoning (http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=pd_sim_b_4?ie=UTF8&refRID=1BB6VKRP59S2420M132F). This book has a wonderful focus on building from the ground up and emphasizes clearly worded and mathematically rigorous proofs.

You seem genuinely interested in mathematics, but I do want to warn you about some more ahem esoteric (read: improperly worded, perhaps?) problems that ask such things as why 1 is greater than 0. The mathematics here is largely armchair - lacking any fundamental logic. There would be no issue with redefining a set of bases such that "" is greater than "1". However, if you want to have rationale of the concept of things being greater than another, that's more like number theory. You can learn the 10 axioms of natural numbers and then build from there.

Both of the books I mentioned will cover stuff like this. For example, they both (unless I'm not remembering correctly) delve into Euclid's proof of infinite primes, something which may interest you.

Briefly (and not so rigorously), assume that the number of primes, p1, p2, p3, ..., pN, is finite. Then there exists a number P which is the product of these primes. Based on the axioms of natural numbers, since all primes p1,p2,...,pN are natural numbers P is a natural number and so is P+1. Consider S = P+1. If S is prime than our list is incomplete, assume S isn't prime. Then some number in our list, say pI, divides S because any natural number can be written as the product of primes. pI must also divide P because P equals the sum of all primes. Therefore if pI divides S and pI divides P, then pI divides S-P = 1. That's a contradiction because no prime evenly divides 1.

Stuff like this is super cool, super simple, and super beautiful and you absolutely can learn it. These two books would be a great place to start.

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

If you are still an undergrad and your school offers a "how to prove stuff and how to think about abstract maths" course take it anyway. No matter how far along you have come.

An example text for such a course is this one:

https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188

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As for Linear Algebra (the most useful part of all higher mathematics for sure (R/math: if you disagree, fight me on this one...i'll win) ) I will tell you i learned a LOT from these two texts:

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https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995

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https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=APH3PQE76V9YXKWWGCR9

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This was the class as it was last quarter (Spring 2012), they used this textbook. I live off-campus and only go to the UW to drop off homework and almost never talk to anyone, so you almost definitely don't know me, but perhaps we walked right by each other one time without ever knowing it. EXCITING!

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).

It depends a bit on what your college offers and what you think you would benefit from.

On the math side an intro to proofs course would be helpful because it teaches you a lot of math and reasoning that undergirds a lot of cs. A book like this is good for self study: https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188

If your school has a good philosophy dept, then check out the classes on critical reasoning and logic too. Herrick's logic book is not bad for a first course.

take the most advance math courses you can. do undergraduate research...summer programs, independent studies. make sure to write a math research paper. it doesn't have to be published, but a published paper would look great. give a talk about your research at an undergraduate math conference. go to many math conferences. many schools require the math subject test gre, which is difficult and requires a fair amount of study outside of coursework.

that being said, since you are still a beginner, be warned that upper level math is very different than high school math. after a certain point, computations are no longer of use and all math is theoretical and abstract. you will be focusing on "proofs" and generally these are much more logic based and theoretical than any math you do before university. any proofs you did proofs in a highschool geometry class are also not relevant. to get a better idea, look at an elementary proof-writing book. for example http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=sr_1_2?s=books&ie=UTF8&qid=1320289226&sr=1-2#reader_0521597188

more specifically, once you are enrolled in a phd program, you will have to take at least 2 years of coursework. you will also need to pass one or two sets of "qualifying exams", the number and style of testing is based on the university. these test you on your basic knowledge of math, and also on the subject of your research. to obtain a phd you have to do NEW mathematical research and then write a dissertation about it. the research part of the phd can take 2-4 years on average.

An Introduction to Mathematical Reason - Peter Eccles. Very good book.

http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188

I'm currently in my first year of undergraduate Maths and our course uses the book 'An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions' by Peter J Eccles. It's such a helpful book aimed at introducing first year university students to pure mathematics, the book has definitely helped me feel confident in my pure module.

It states propositions and theorems and proves them and gives problems for you to solve or prove with the solutions at the back.

The class I took "Math Reasoning", which I believe is equivalent to the one you're taking used this book. www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/.

I cannot, however, give you any advice on how it compares to your book, as I'm not familiar with yours.

I used this book in my first mathematical reasoning class at my university.

Start with a book like this:

http://www.amazon.com/books/dp/0521597188

or this:

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995

or the one teuthid recommended. When you're doing self-study, it's doubly important to be able to read and follow most of the material.

There is actually a book called An Introduction to Mathematical Reasoning.