The preface to this book says, "I believe that the reading of part or all of this book would be a good project for the summer vacation before one begiThe preface to this book says, "I believe that the reading of part or all of this book would be a good project for the summer vacation before one begins graduate school in mathematics."

Well, I tried to do this. Unfortunately, for someone with my mindset about mathematics, this is essentially impossible. It's just so much boring stuff to wade through.

However... this book is still really good. An example of what's so good about it: the discussion on nets and filters. This book gives the "right" definition of subnet, compares it to the other possible definitions, explicitly gives the connection between nets and filters... For some reason this information isn't really available in other books. But I have no desire to just sit down and read this book (or a significant portion of it) straight through. ...more

You might agree with me that this statement is true:

"It is impossible for a human to run a mile in thirty seconds."

It's so far beyond the ability ofYou might agree with me that this statement is true:

"It is impossible for a human to run a mile in thirty seconds."

It's so far beyond the ability of anyone alive that it seems clear it's impossible. But in some philosophical sense, this is nonsense. There is no physical barrier making it absolutely inconceivable that some mutant will some day run a mile in thirty seconds. In that sense, it's not impossible, it's just so hard that we seriously doubt anyone will ever do it.

The word "impossible" means something different in mathematics. In mathematics, we work with very simple systems that have clearly specified rules. These systems are so simple that we can prove properties about them in a rigorous fashion, such that there is no doubt these properties hold. In particular, we can prove that certain things are impossible. Just as in chess, one could easily demonstrate that a bishop that begins on a black square can never end up on a white square during a chess match played according to the rules.

A problem the ancient Greek mathematicians set out to solve was to trisect an angle, using only the tools of compass and straightedge. This problem was unsolved until the 19th century, when it was demonstrated that it is impossible. The proof that this construction is impossible is not particularly difficult as these things go: it requires just some plane geometry, some trigonometry, and a smidge of so-called modern algebra.

Nevertheless, a baffling number of people throughout the 19th and 20th centuries set out to trisect the angle, and in fact believed they had done so. Dudley documents many attempted trisections here in this very interesting book. Amazingly, many of these trisectors are fully aware that there exists a proof that their goal is impossible, but don't seem to understand the concept of mathematical proof, and continue anyway. Many of them seem to hold mathematicians in contempt for "giving up" on the problem. Meanwhile mathematicians see the trisectors about the same way they'd see people who spend their days deep in thought over a chess board, trying to find a sequence of chess moves to get a bishop from a black square to a white square.

The arrogance of many trisectors is staggering. Then again, the whole enterprise can only be born out of arrogance. How else could someone with barely a high school education who can't follow simple proofs believe that every mathematician in the world is wrong about a simple, elementary fact known for over a century? ...more

"It is primarily through experience that the combinatorial significance of the algebraic operations of [formal power series] is understood, as well as"It is primarily through experience that the combinatorial significance of the algebraic operations of [formal power series] is understood, as well as the problem of whether to use ordinary or exponential generating functions." - Stanley's Enumerative Combinatorics

While I'm sure experience helps, all this can be made precise in the theory of species, which really illuminates things.

Unfortunately I've had to set this book aside for now because of, you know, my actual classes. But hopefully I will come back before long and finish this book, which I've really enjoyed so far. ...more

This book is very clearly written and I like Kaplansky's style.

On the other hand, it provides no motivation at all and no connection to geometric ideaThis book is very clearly written and I like Kaplansky's style.

On the other hand, it provides no motivation at all and no connection to geometric ideas. It also gives hardly a single example. This is a perspective that is still very strange to me: why would someone write a book about rings without actually mentioning any rings? Surely one will fairly often need to give a counterexample to an obvious conjecture, or want to illustrate the application of a theorem in a concrete case, or what have you... but not here. ...more

I don't recommend this. The problems just aren't closely aligned with the actual test. Some of them are way too difficult or time consuming. Some of tI don't recommend this. The problems just aren't closely aligned with the actual test. Some of them are way too difficult or time consuming. Some of them are just not relevant, on topics that would never appear on the actual test, like game theory. ...more

Eh. It's OK. Several ridiculous errors in equations and whatnot that have apparently been known about but not corrected since the first edition. The pEh. It's OK. Several ridiculous errors in equations and whatnot that have apparently been known about but not corrected since the first edition. The problems aren't completely like those of the actual tests, but they are OK. ...more