[vouaobrasil]

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.

[skhunted]

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.

[vouaobrasil]

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.

[skhunted]

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

[vouaobrasil]

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.

[skhunted]

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.

[rramadass]

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.

[lupire]

It's weird that everyone from Euclid to Gauss disagrees with you.

The language is a baby compared to the applications.

[skhunted]

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.

[seanhunter]

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.

[beryilma]

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