Tools of the Trade (Monograph Book) 0th Edition

If you are a beginner or a person who wants to self study some basic advanced math, the book might not be for you. However, if there is someone who knows math really well and who can help you when you get confused, you will find that this is a wonderful book. It has got a nice structure and the proofs, though not easily understood sometimes, are very succinct.
In short, a great book for instructors.

in very good condition

It's hard to believe Paul Sally is still teaching classes. I don't mean that he's old (he is), or that we never hear from him (we do), but that he seems as much a legend as a working math professor. For those who haven't heard of him, Paul Sally is the University of Chicago's famous "Math Pirate." He gets the nickname from his eyepatch and two prosthetic legs, apparently due to years of uncontrolled diabetes. The name is used by students and faculty alike, and it commands as much awe (and sometimes fear) as if it were that of Blackbeard himself. He seems out of place ambling down the halls of modern-day Eckhart Hall, and passers-by grant him a wide berth.

His book is just as eccentric. Instead of the usual "QED box" at the end of a proof, Paul Sally uses a sketch of his own head, complete with eye patch and pipe. The presentation is likewise distinctive: most of the mathematical structures (groups, metric spaces, relations) are built up from sets. Most authors, for instance, would say that a relation on a set E is a statement which is either true or false for every pair of elements in E (for instance, if X is the real numbers, a relation could be the statement a < b for real numbers a and b). Sally says that a relation is a subset of the Cartesian product of the set with itself (E x E) (in this case the relation "less than" becomes the set of all pairs of real numbers (a,b) such that a is less than b. The statement "a<b" means that the ordered pair (a,b) is an element of that subset). This is an extreme of formalism: not even Rudin uses such strict definitions, and they can be confusing the first time around.

Other peculiarities: he makes frequent use of decimal expansions in proofs, something I've not seen done in most books. It's clumsy, but it gets the job done, and you see how they fit in to formal math. Sometimes he will state a theorem, and for the proof write the word "Obvious." Most of these proofs really are obvious if you give them some thought, but the practice drives beginning math students crazy.

But despite these sometimes-frustrating conventions, this book can be extremely useful. It covers a lot of topics that are out of place in books on calculus or abstract algebra (the books most undergrads will read). Countable versus uncountable sets get clear and thorough coverage, as do two different constructions of the real numbers. If you've never seen vector spaces, Sally provides a good introduction. It's not a book about any particular area of mathematics (the title hints at this), and the topics sometimes seem thrown together haphazardly, like the tools on the cover, but they are almost all important. The ideas here (with the possible exception of the last chapter, which covers Paul Sally's area of research and which I have not read) show up over and over in later math classes.

Still, there are some topics which don't get proper treatment: the Axiom of Choice seems to come out of nowhere (and without any other axioms), and Zorn's lemma fares even worse. He insists on the fully general definition of the determinant, without any logical motivation whatsoever: to quote the mathematician Vladmir Arnol'd, "The determinant of a matrix is an (oriented) volume of the parallelepiped whose edges are its columns. If the students are told this secret...then the whole theory of determinants becomes a clear chapter of the theory of poly-linear forms. If determinants are defined otherwise, then any sensible person will forever hate all the determinants, Jacobians and the implicit function theorem." Tools of the Trade suffers from just this flaw.

But if you are buying this book, chances are good that you are a student in Mr. Sally's math department, and therefore subject to all his whims and idiosyncracies. My advice, then, is this: read the book and do the problems- it will be time well-spent. But if something seems frustrating or unclear or even downright unintelligible, don't despair. Ask a professor, or a TA, or try another book. The ideas in Tools of the Trade are important, but their presentation isn't always the best.

For learning introductory analysis, this is the gold standard--there is no better text. Sally's book covers both the basics (from elementary set theory through metric space topology, with linear algebra and the construction of the reals along the way) and atypical/enriching projects on related content, all in an extremely clear and invitingly casual presentation. It is true that Sally leaves some proofs as exercises, but nothing mission-critical. And after all, mathematics is learnt by doing, not by head-nodding: this is a book for students of mathematics, not an encyclopedia for those who've forgotten a step in a proof. As such, it contains many excellent exercises and investigations, in which Sally weaves in algebra, topology, and logic with the main theme of analysis in a natural way that reinforces the student's learning, stresses important connections across the main branches of mathematics, and ideally prepares the student for future study.

The book is also remarkably well-polished. It contains very few typos (and none that cause any lack of clarity)--I think I've spotted only two or three after multiple times teaching from the text. It also features the most intuitive (and honestly the only logical) chapter-section-statement numbering scheme of any mathematics text, wherein theorems, definitions, etc. all increment sequentially with the chapter and section, so there's none of the nonsense you get in texts written by less thoughtful authors, i.e. where the reader ends up in a futile search for Theorem 6.3, which comes after Lemma 6.4, and before Definition 6.2....all in Section 6.5.

In short, this is the right book to transition a beginning student of mathematics into one capable of picking up any other undergraduate text and making headway. The reader who finishes it should just prepare to be disappointed when picking up the next book in her mathematical odyssey, as odds are it will be written by a different author neither as clear nor as enriching as Sally.

Mr. Sally's book starts with basic, almost obvious elements of the set theory that provide the logical foundation of essentially all modern mathematics. In this way, he mines the best of the 20th century's progress in solidifying those foundations. But fear not, he does not become mired in Russelian logic.

Instead, he goes on to derive and define the properties of myriad basic mathematics, up through some of the basics of topology, complex analysis and so on. A variety of statements are left for the reader to prove, though all important theorems at this level are at least stated explicitly.

The reader will go through essentially the same mental development undergone by students fortunate enough to take the famous U of Chicago Math 207 course. Budding mathematicians have been inspired by that course for generations. The proofs, where provided, are pithy and crystal clear while the exercises have a fine distribution of difficulty from trivial to tricky.

This is not a reference text by any means, but rather, as the title implies, a means to give you the skill set necessary to understand and invent real mathematics.

Very unclear when it comes to proofs.