I'm a mathematician (PhD in numnber theory...) and I took a look. It's a basic textbook about some concepts like functions, relations, and basic proof techniques. It seems okay. I was expecting something different from the title, though. It's not too different from many basic books introducing such concepts but obviously a lot of effort was put into it.

Personally, I think if you want an introduction to the "art" of mathematics, it would be a lot better to pick up a more idiosyncratic book that doesn't aim to cover the basics of the standard curriculum in a textbook-style way, which in my opinion is rather tedious. That could either be a more high-level book like Ian Stewart's "From Here to Infinity" or one of Raymond Smullyan's fun texts on logic.

Or for a more basic book, something like "The Mathematical Universe" by William W. Dunham is a much more interesting introduction to the "art" of mathematics than a textbook-style intro.

I’m ABD in math for discolosure purposes. I strongly disagree with your recommendations if the purpose is to get an introduction to higher math. The book in question is much, much better for introducing one to higher math than any of the books you recommended.

Smullyan’s books are great but one isn’t going to go from Smullyan to abstract algebra, point set topology, or real analysis.

Meh, well different strokes for different folks I guess. I got into higher math while reading Ian Stewart's book in grade school but I guess some people are going to want to go the standard way.

My problem with the book we are discussing is that it seems rather prosaic -- it doesn't really give a sense of the true reason to practice math: the asking of interesting questions and creating new universes. It's just the same old stuff that we're taught because it's a convention.

The language comes first then the applications. There’s a reason the order of topics evolved the way it has.

Well, I'm just going by empirical evidence: what has worked for me and what has worked for many of my fellow colleauges that have done actual research in mathematics.

Before one can do advanced mathematics (I have too have done research in math) one needs to learn the basics. Reading Smullyan did not in any way help you learn advanced mathematics. It may have helped you to get motivated to learn math and want to learn advanced math but it didn’t help you accomplish this. There’s a reason just about every mathematics department teaches classes on how to do proofs and on basic set theory but almost none of them teach from Smullyan’s book.

The overwhelming empirical evidence is that having a course on proofs and basic set theory is much better preparation for advanced mathematics than reading Smulyan’s recreational math books. I guess you’d rather your students read Martin Gardner and then do Fraliegh’s Abstract Algebra book. No one does it that way but go with your so called empirical evidence.

Having a Ph.D. in math ought to have taught you to reason better than to use “actual research” as part of your reasoning when discussing learning topics that are not cutting edge. One doesn’t need to have done research in mathematics to know about Gorenstein rings or projective dimension or other such stuff. It also has nothing to do with teaching basic math.

I could be wrong in my opinion but attack it on its merits without using superfluous things like “actual research” when research has nothing to do with the topic.

Well said! You are absolutely right and "vouaobrasil" is wrong.

As a person interested in self-learning Mathematics, i have read and amassed a lot of "popular mathematics" books by authors like Ian Stewart, W.W.Sawyer, E.T.Bell, George Gamow etc. all of which were great motivators but none of which taught me the basics of "Modern Mathematics" which i could only get from Textbooks. The quality of Textbooks are of course all over the map and so i am always on the lookout for the simplest, clearest and yet rigourous explanations available. The book under discussion seems to check all such boxes for a beginning student.

Do you have any evidence that Euclid disagreed with me? Elements starts off with a list of postulates, definitons, and common notions. Then he proceeds to proposition 1. He does not motivate why one would want to come up with proposition 1 or why would should care about it. Have you read Elements?

How do you propose one do applications of point set topology without knowing about sets and mappings? Before the applications one must know the language. We don’t teach the quadratic formula and solving simple velocity problems before teaching students how do the basics of manipulating algebraic expressions.

One must first be a baby before being an adult.

The purpose of the book as stated in the preface is to be used for a course in proof writing ie it is a sort of bridge to higher maths which the author defines as "defining axiomatic systems and proving statements within them" vs "elementary maths" which he defines as "solving problems".

So I think the idea behind the title is get students to see this as the gateway to the good stuff as opposed to a lot of proof texts which might be seen as irrelevant.

One the same subject and as accessible, I love the two books from Jay Cummings: Proofs and Real Analysis. Each just $16 on Amazon. It is a joy to read these books and try some of the exercises. I wish PDF versions were also available...

For those interested: https://webpages.csus.edu/jay.cummings/Books.html

I'm a mathematician, currently working as a private tutor for adults who want to learn proof-based math. I had a quick glance through this book and it seems to me like a pretty nice version of this "intro to proofs" sort of book. This is a topic that's done well in a lot of different books, though, so if you really want to dig into this topic I'd maybe recommend looking at a couple different ones and finding the one that agrees with you the most.

Right now I have a student working on this material and we're using "How to Prove It: A Structured Approach" by Daniel Velleman, which so far I'm finding decent. Some others I've seen (but that I haven't looked at in as much detail) are "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings and "Book of Proof" by Richard Hammack.

As someone who tutors adults, can you suggest a more digestible book for abstract algebra?

While I was motivated, I used one of the typical college books. For me Abstract Algebra is what opened a lot of doors for me... but I am simply using applied math.

That moving away from proofs being magical across sub-topics is what I would like to share with some co-workers who are unwilling to buy a textbook and answer key.

As I didn't even mind Spivik for calc, my radar is way off for making suggestions to most people.

Pinter, "A Book of Abstract Algebra", is very nice. It's rigorous but not too terse. It divides the material into many small chapters with many exercises. Chapters are mostly around 10+/-3 pages with about 40-60% of that being text and the rest exercises.

The exercises for each chapter are split into several sections each section covering a different aspect of the chapter's material. Sometimes there is a section of exercises applying the material to some interesting area.

For example, the chapter on groups of permutations has 6 pages of text, then 5 pages of exercises divided into 9 sections. Those sections are: computing elements in S6 (5 problems), examples of groups of permutations (4 problems), groups of permutations in R (4 problems), a cyclic group of permutations (4 problems), a subgroup of SR (4 problems), symmetries of geometric figures (4 problems), symmetries of polynomials (4 problems), properties of permutations of a set A (4 problems), and algebra of kinship structures which consists of 9 problems covering how anthropologists have applied groups of permutations to describe kinship systems in primitive societies.

There are answers in the back for a decent number of the exercises.

It's a Dover republication so is not too hard on the wallet. List price is $30 at Dover but its around $20 on Amazon.

The combination of short chapters and lots of exercises make it easier than most textbooks to fit into a busy adult schedule.

Thank you, ordered it.

Abstract Algebra: A Student Friendly Approach by Dos Reis and Dos Reis [0] is like The Little Schemer but if it was a first course in abstract algebra.

[0] https://www.amazon.com/gp/product/1539436071?psc=1

I assume you're talking about an algebra book for self-study? Gallian's "Contemporary Abstract Algebra" is a common suggestion for a more accessible algebra book, and people also sometimes suggest Fraleigh's "A First Course in Abstract Algebra", but I can really only speak to what it's like to work on this stuff with a teacher --- since my students are by definition not self-studying the things I'm working on with them, my suggestions might be of limited use!

In general, I think self-studying proof-based math can certainly be done if someone's motivated enough, but it's pretty hard and takes a lot of work, especially if you're still getting used to the skill of reading and writing proofs. It's very valuable to be able to have a person available to evaluate the proofs you're writing, and I've definitely seen a few people who came to me thinking they'd mastered proof-writing on their own and were kind of mistaken about that. (I've definitely also seen people who really did learn this skill pretty well without help! It varies a lot.)

Thanks for the suggestion. I personally used Dummit and Foote's book and found it useful, but like early Calculus with Spivak, it seems most people prefer clear and concise over slightly more comprehensive and rigorous while still being introductions.

With self study I prefer a bit more breadth to make up for the realities of needing to self study which often ends up with deep but not wide understanding of topics.

Having a brother who had a PHD in complex analysis to bother probably helped with self-learning. That is the only option when you are on-call for decades at a time as higher math courses are/were always in person.

But hopefully someone will figure out a business model to help people who need to grow and adapt.

Thanks again for the suggestions, I have ordered both books to add to my lending library.

The 3rd edition of Velleman's book has an online supplement that uses Lean to work through the book so you can get feedback about your proofs.

I have read many parts of Cummings'es book, and I can vouch for its quality.

It's the ideal book for learning proofs if you are self-learning.

Putting on my Computer Scientist hat (and getting all of the tentacles out of my face so I can see), GIAM seems to be a good, if a little verbose[1], introduction to formal logic and basic set theory. Those are fundamental to CS and this is a heck of a lot better than many of the intros to those topics that I have seen in curricula without a dedicated class.