Sunday, November 14, 2010

"Monoids,R-Modules And Nonassociative Rings-These Are Some of My Favorite Things: A Suggested Reading List For Graduate Algebra"


Brief political comment first.

November 2nd, 2010 may go down as one of the blackest in American history in coming decades and centuries. People who have been predicting a corporate takeover leading to a plutocracy in America -a “third world” complete with utopian gated communities defended by mercenary private security interspersed with regions of abject poverty where starving mobs of disease riddled peasants try and stay alive into their 30’s if possible-may have seen the first step towards that future on that day.

Not that our precious President and his party have been anything even remotely resembling heroic figures-between half-ass health care reform, continuing to allow the slaughter of an entire generation of young Americans, as well the slow bankruptcy of out nation to satisfy whatever mysterious powers now profit from it-they've been as cowardly and subservient to the special interests as the other side on most days. In many ways, that’s partly what led to his party’s downfall.
But with the election shenanigans with The Chamber Of Commerce-after the worst Supreme Court decision for the country at large since Plessy Vrs. Ferguson - effectively rendered the Republican party a wholly owned subsidiary of the corporations. Unless a dramatic opposition to this systematic subversion of the government occurs in the next 2 years, the very best we have to look forward to from this decision is eternal gridlock until the Commerce can buy the Presidency in 2012 and put the best candidate money can buy in there.

The poor ignorant slobs who marched for the Tea Party are in for a very rude awakening indeed-that is, if any of them have the intelligence to realize they’ve been used and discarded.

And then the future I predicted in my opening preamble-coupled with environmental collapse with only the wealthy having food and water fit for human consumption, let alone other resources-will come to pass within a generation.

That’s all I wanted to say on this for now. There’ll be much more to come when I can steal myself to a full analysis and discussion.

But now onto a promised, much more academic matter.

For those who don’t know, I’m sort of an unofficial bibliophile for mathematical education. I inherited this love of textbooks and monographs from my inspiration, friend and unofficial mentor, Nick Metas. I was 18 years old when out of simple curiosity I called him in his office to ask him for direction in independent studies of mathematics beyond calculus-and he went on for 4 hours, naming just about every textbook and describing the subject of mathematics. That long-ago conversation is what started me on the path to becoming a mathematician.

Nowadays, the influence of Nick is very clear in my life: I have an extensive library of textbooks and monographs, people ask me all the time for references on subjects and I review books for the Mathematical Association Of America’s website.(which can be found here). I have an opinion on most commonly used texts and monographs for all subjects-and I’m reading more every year. In fact, I have a private dream of beginning my own small publishing company someday.
(Of course, that’ll depend on the prediction above being dead wrong. We can all hope the country comes to it’s senses before its too late. They’re angry-that’s a good beginning Now they just have to develop enough intelligence to channel the anger constructively. Hope it happens in time. So far it doesn‘t look promising……..)

I’ve been asked many times to compose a master list of my favorite textbooks and/or monographs. The list will probably undergo many revisions and additions before it reaches final form-but more importantly, I’ve decided to compose it in modular form i.e in components. This way, it’s broken into bite-sized components of manageable length that I can post here. It seems to me if I wait and try to compose it all at once-well, I’ll end up writing a 2,500 page book from the old age home I’ll be dying of cancer in. So let’s get started and hope that what little insights I can give can help neophyte students looking to broaden their knowledge base in subfields of math or are just looking for a little help in coursework they’re struggling in. Comments, input and suggestions are, of course, very welcome.

The first module here is my favorite subject in all of mathematics: algebra. (A ludicrous but sadly mandatory clarification: When a mathematics student or mathematician says ‘algebra’; it’s supposed to be understood he or she means abstract algebra. High school algebra is, of course, the simplest special case of this wondrous arena. )

How do we define abstract algebra? Like most branches of modern mathematics, attempting a simple nonmathematical definition for non-mathematicians is a nearly impossible Catch-22 since it requires mathematical concepts to even attempt a meaningful definition. Entire philosophical treatises could probably be written attempting to answer the question and would probably fail. But I think we can try for a reasonable working definition here.

I think the best way to define algebra is that it is the general study of structures in mathematics. By a structure, we mean some kind of set -by which we mean naively a collection of objects-and a function f closed on S (the range of f is a subset of S) with a specified list of properties that characterizes that structure. For example, a group is a nonempty set S with a binary operation f such that f is associative, there is a unique element e in S such that for all elements a in S, f(e,a) = f(a,e)= a and for every a in S, there’s a unique a* such that f(a, a*)= f(a*,a)=e. Algebra deals specifically with these kinds of objects.

The pervasiveness of algebra in modern mathematics in the 21st century is astonishing. It’s more then the sheer scope of algebra itself, but the fact that most of the active areas of mathematics would not even exist without it. And I’m not talking about high-tech fields where algebra’s role is obvious-like deformation theory and higher category theory. I’m referring to the fact that most areas of mathematics are formulated in the 21st century in terms of algebraic structures. To give just one possible example of a legion, modern differential geometry would be unthinkable without the language of vector spaces and R-modules. Without tangent spaces and their associated local isomorphisms, it would be impossible to generalize calculus beyond Euclidean space. It would also be impossible to precisely define differential forms, without which most of the most interesting developments of manifold theory fall to dust. As a result, a student that’s weak in algebra needs to seriously reassess a career in mathematics.

So the least I can do is give my 2 cents on the current crop of books available.

The actual direct impetus for me writing up and posting this list was Melvyn Nathanson at the City University Of New York Graduate Center. This semester, the eminent number theorist is teaching the first semester of the year-long graduate algebra sequence there. I began this semester sitting in on his lectures in order to begin preparations for the algebra half of my oral qualifiers for the Master’s Degree in pure mathematics at Queens College. Unfortunately, a combination of personal and financial issues prevented me from attending regularly. So that was the end of that.

I found Dr. Nathanson’s (he hasn’t told me it’s ok to call him Melvyn yet, so I’m going to be extra cautious as not to offend him) comments on the subject very interesting, as he has his own unique take on just about any subject. As proof, I offer this excerpt from the course’s syllabus:

In 1931, B. L. van der Waerden published the first edition of Moderne Algebra,
two classic volumes, written in German, that were based in part on lectures by
Emil Artin and Emmy Noether and that became the canonical work in \abstract
algebra." The second edition appeared in 1937, and an English version, Modern
Algebra, translated by Fred Blum and Theodore J. Benac, was published in the
United States in 1949 and 1950. I and many other American mathematicians
learned algebra from the original English edition of van der Waerden. It is still a
great work and I strongly recommend it for intensive study. The first volume of the
seventh German edition of van der Waerden is also available in English translation,
but I prefer the original. Van der Waerden's algebra begins with introductions to different
algebraic structures. The first seven chapters are “Numbers and Sets," “Groups," “Rings and
Fields, "Polynomials"“Theory of Fields," Continuation of Group Theory," and
The Galois Theory." As proof of van der Waerden's influence, this continues to
be the starting sequence of topics in most algebra courses and most algebra books,
including the contemporary classic, Serge Lang's Algebra, which I also recommend.
This course is different, not just in the sequence of topics, but in its philosophy. It
emphasizes themes in algebra: Divisibility, dimension, decomposition, and duality,
and the course enables algebraic understanding and technique by developing these
themes. The book includes all of the theorems expected in a graduate algebra
course, but in a nontraditional order. The book also includes some important
topics that do not appear in van der Waerden or Lang.”

The affirmed originality of the course, I don’t doubt. I’m still hoping to obtain a complete set of handwritten notes from some of my friends in the course, which is the official text for the course. The clear implication from the preface and his subsequent remarks is that Professor Nathanson hopes to eventually expand these notes into a textbook for a graduate algebra course.

Privately, I’m hoping to work with Dr. Nathanson as a PHD student eventually and if he follows through on this, perhaps I can be involved in the book’s drafting process. But that’s for the future.

His comments got me thinking about the current state of algebra courses and the textbooks that form the basis of them. Nathanson’s experiences are not unlike those of most mathematicians of his generation: van der Waerden’s classic was the source from which he learned his algebra. Later mathematicians; particularly algebracists-such as my undergraduate algebra teacher, Kenneth Kramer-learned algebra from the earlier editions of Lang’s tome. (In fact, it was more personal for Kramer. As an honors undergraduate at Columbia in the late 1960’s, he was a student in the graduate algebra course taught by Lang himself-whose resulting lecture notes ultimately evolved into the classic text.) Most of the better universities’ graduate programs adopted Lang as the gold standard of first year graduate algebra, for better or worse, after the 1960’s. With a very few exceptions, this was the story until after the turn of the 21st century, when a host of graduate algebra texts came onto the market within a 5 year period. What was once a very sparse set of choices for this course is now a wide field of markedly diverse texts, many authored by very eminent mathematicians.

What follows is my attempt to form an amateur’s guide to these texts and my corresponding brief commentary to each. As a reviewer of textbooks, it seemed under the circumstances, that providing such a list to my erstwhile classmates in Nathanson’s course-as well as the mathematical world in general-would be a very positive undertaking. I don’t know if it would be WISE, merely positive. I must add the disclaimer that I am by no means an expert; I’m merely a serious graduate student. Therefore, this reading list must be taken with a salt lick of caution as coming from an amateur and as such, it is seriously subject to revision as my knowledge grows and my mathematical style tastes change.

A major motivation in the evaluation of each of these books has been student-friendliness. Let me clarify greatly what I mean by that. A lot of top-notch mathematicians and students have an elitist, almost snobbish reaction to a textbook when you say its’ friendly. “Oh,you mean it spoon feeds the material to the brainless monkeys that pass for mathematics majors at your pathetic university? How amusing. Here at Superior U, we use only the authentic mathematics texts. Rudin.Artin Hoffman and Kunze. Alfhors. We propagate the True Word. Math is SUPPOSED to a struggle for those truly gifted enough to be worthy of it. “

Or something equally narcassistically pretentious.

I have a LOT to say on this and related issues-but if I started going in depth about it here, I’d write an online book here. In future installments, I’ll begin to outline them in detail.
But in plain English, this is a bunch of crap.

The reason a lot of those “classic” texts are difficult to read isn’t because their authors were first-rate mathematicians and as such, their lessons are beyond the reach of mere mortals. In a lot of cases, it was simply because most of them never really thought about teaching; of being able to organize their deep understanding of their chosen fields -and as a result, they were very poor communicators. This lack of communication skill is reflected not only in their poor reputations as teachers, so often inversely proportional to their reps as researchers-but also in the resulting textbooks. Why don’t they? Well, again, it’s too complicated to fully go into here. But I WILL say that PART of the reason, as any research mathematician of any prominence will tell you-is that they don’t get paid the big bucks and get the fancy titles based on how well students learn from them.

The sad part is that this myth has been perpetuated by the canonization of certain textbooks as The Books for certain classes, despite the fact that most students almost overwhelmingly despise them. And the reason why is simple: They just aren’t clear and well-organized. That makes the very act of reading them unpleasant, let alone actually learning from them. For the serious mathematics major or graduate student, this makes studying from such books virtually an act of psychic self mutilation.

To the elitists, I only have the following to say: Charles Chapman Pugh’s Real Mathematical Analysis, Joseph Rotman’s An Introduction To Algebraic Topology, ANYTHING by John Milnor, J.P.Serre or Jurgen Jost, Loring Tu’s An Introduction to Manifolds, John McCleary’s A First Course In Topology: Continuity And Dimension, George Simmons’ An Introduction To Differential Equations With Historical Notes,2nd edition. Charles Curtis’ Linear Algebra, 4th edition. and John And Barbara Hubbard’s Vector Calculus, Linear Algebra And Differential Forms: A Unified Approach ,3rd edition.

I challenge them to consider any of these wonderful books to be spoon feeding students-and yet, they are eminently readable and wonderfully written books. In short, they are books students ENJOY reading and therefore will not only learn from them-but will WANT to learn from them.

But an interesting trend has resulted from this myth. The students who are talented enough to learn from these texts who go on their careers to become mathematicians- and who care enough about teaching- recall their experiences as students. They don’t want to subject their students-or ANYONE’S student-to the same torture. As a result, they try and write alternative books for students that do what they wish those texts had. The Computer Age has magnified this effect hundredfold as such books have become ridiculously easy to produce. As a result, we’ve gotten “backlash waves” of texts as alternatives to those classic tomes that created the large diversity of texts that currently exist in the various subfields of advanced mathematics. Where once there was a bare handful of such texts to choose from, a generation later, the “backlash” creates a myriad of them.

Some examples in the recent generations of math students will illustrate. Once, Alfhors’ ridiculously difficult Complex Analysis was the standard text in functions of a complex variable at U.S. graduate programs after the early 1960’s. There were a few alternatives available in English-such as Titchmarsh or Carathedory-but not a lot. This lead to an explosion of complex analysis texts in the 1970’s onward: Saks/Zygmund, Rudin, Bak/Newman, Conway, Heins, Greene/ Krantz, Jones/Singerman, Gamelin,- well, that list goes on and on. A similar backlash occurred in the 1960’s and 1970’s in general topology after an entire generation had suffered through John Kelley’s General Topology wrote a legion of such texts, including the classics by Willard and Munkres.
This effect has further been enhanced by progress in those fields at the research level-which results in the presentations of the standard texts of a generation becoming outmoded. The result is the “backlash” presentations can also be “upgraded” to current language. A good example is the incorporation of category theory into advanced algebra texts post-1950’s.

I strongly believe the current large crop of graduate algebra texts is the result of a similar backlash against Lang.

I’ve gone on to some length about this because I think it’s important to keep these 2 ideas in mind- the elitist conception of Great Books and the backlash against it-when considering my readability criteria for judging such texts. So without further ado, my reading list. Enjoy.

And remember-comments and suggestions are not only welcomed, but encouraged.

Part I- Graduate Warmup: These are texts that are a little too difficult for the average undergraduate in mathematics, but aren’t quite comprehensive or rigorous enough for a strong graduate course. Of course, a lot of this is totally subjective. But it’ll make good suggestions for those struggling in graduate algebra because their backgrounds weren’t quite as strong as they thought.

Topics in Algebra by I.R. Herstien, 2nd edition: This is the book I first learned algebra from under the sure hand of Kenneth Kramer at Queens College in his Math 337 course. It’s also the book that made me fall in love with the subject. Herstien’s style is concise yet awesomely clear at every step. His problem sets are legendarily difficult yet doable (mostly). If anyone asks me if they’re ready to take their algebra qualifier and how to prepare-I give them very simple advice: Get this book. If you can do 95 percent of the exercises, you’re ready for anything they throw at you. They’re THAT good. Warning: In true old European algebraicist fashion, Herstien writes his functions in the very un-Calculus like manner on the RIGHT in composition i.e. fg= gof. This confused the author of this blog initially and no one corrected him until several weeks into the course-which lead to difficulties later on. A couple of quibbles with it-the field theory chapter is really lacking. Also. Herstein tends to present even the examples-which are considerable-in their fullest generality. This makes the book harder for the beginner then it really needs to be. For example, he gives the dihedral group of rigid motions in the plane for the general n gon where n is an integer. he could start with the n=4 case and write out the full 8 member group table for the motions of the quadrilateral and THEN generalize. Still-I fell in love with this book. The presentation is considered outdated by most mathematicians now, who prefer the more geometric approach of Artin. Still, the book will always have a special place in my heart and I recommend it wholeheartedly for the talented beginner.

Algebra by Micheal Artin : The second edition of this book finally came out in September. For awhile, it looked like it might emerge posthumously-it was so long in gestation. But fortunately, this wasn’t the case. I haven’t read it carefully yet, but from what I’ve seen of it, it looks very similar to the first edition. As for the first edition- well, I got really mixed feelings about it. Artin’s book has many, many good qualities. It’s primary positive qualities are the heavily geometric bent and high level of presentation. The shift in emphasis from the permutation groups to matrix groups is an extremely smart one by Artin since it gives one a tool of much greater generality and simplicity while still preserving all the important properties of finite groups. (Indeed, permutations are usually explicitly represented as 2 x n matrices with integer valued bases-so the result is just a slight generalization. ) This also allows Artin to unify many different applications of algebraic structures to many different areas of mathematics-from classical geometry to Lie groups to basic topology and even some algebraic geometry (!) All through it, Artin brings an infectious love for algebra that comes through very sharply in his writing. So why the hesitant recommendation?
Because it’s easy to like the book when you’ve already learned the material from other sources.
Would even a talented beginner find the book so appealing? I don’t think so. Firstly, Artin assumes an awful lot of background in his prospective students-primarily linear algebra and basic Euclidean geometry. It might have been reasonable to assume this much background in the superhuman undergraduates at MIT in the early 1990’s, but I think that’s a stretch for most other students-even honors students. Especially nowadays. Secondly, the book is organized in a very idiosyncratic fashion that doesn’t always make sense even to people who know algebra. Nearly half the book is spent on linear algebra and group theory and rings, modules, fields are developed in a very rushed fashion. Some of these sections really needed more fleshing out. Also, a lot of the group theory chapters are confusing. His discussion of both cosets and tilings in the plane are particularly discombobulated. Lastly-his choice of topics for even good undergraduates is bizarre sometimes. He writes a chapter on group representations, but leaves tensor algebra and dual spaces “on the cutting room floor”? It’s a very strange choice. The book’s main flaw is there are too few exercises and most of these are ridiculously difficult. (In all fairness, I understand this was the main problem Artin is trying to rectify in the new edition. ) That being said, for all it’s flaws, a text of this level of daring by an expert of Artin’s stature is not to be ignored. I wouldn’t use it by itself, but I’d definitely keep a copy on my desk. Apparently,though, in the 2 decades since this book was written, Artin has rethought the course and it’s structure-it remains to be seen if the new version has gotten most of the bugs out. If it has, the book will be a must-have, hands down, for students of algebra.

A Course In Algebra by E.B.Vinberg This is very rapidly becoming my favorite reference for algebra. Translated from the Russian by Alexander Retakh, this book by one of the world’s preeminent algebracists is one of the best written, most comprehensive sources for undergraduate/graduate algebra that currently exists. Vinberg, like Artin, takes a very geometric approach to algebra and emphasizes the connections between it and other areas of mathematics. But Vinberg‘s book begins at a much more elementary level and gradually builds to a very high level indeed. It also eventually considers many topics not covered in Artin-including applications to physics such as the crystallographic groups and the role of Lie groups in differential geometry and mechanics! The most amazing thing about this book is how it manages to teach students such an enormous amount of algebra-from basic polynomial and linear algebra through Galois theory, multilinear algebra and concluding with the elements of representation theory and Lie groups, with an enormous number of examples and exercises that cannot be readily found in most other sources. All of it is done incredibly gently despite the steadily increasing sophistication of the material. The book has a very “Russian” style-by which I mean the author does not hesitate to both prove theorems and give applications to both geometry and physics (!) throughout. Those who know me personally know this is a position I am very sympathetic to-and for there to be a major recent abstract algebra text that takes this tack is very exciting to me.
For anyone interested in writing a textbook on advanced mathematics, this is a terrific book to study for style. It is one of the most readable texts I have ever read. An absolutely first rate work that needs to be owned by any student learning algebra and any professor considering teaching it.

Abstract Algebra, 3rd edition by David S. Dummit and Richard M. Foote : Ever seen a movie or read a book where based on your tastes, everything you think and what you see in it, you should love it-but just the opposite? You don’t like it one bit and you couldn’t explain on pain of death why? THAT’S how I feel about this book, one of the most popular and commonly used books for algebra courses-both undergraduate and graduate. It has a good, very comprehensive selection of material, good exercises and lots of nice examples for the serious student. So what’s my problem with it? Well, first of all, it’s WAY too expensive. You could get both Vinberg AND a used copy of Artin for the same price as this book. Second of all-it’s pretty dry and matter-of-fact. It just doesn’t excite me about algebra. Everything’s presented nicely and clearly-but it comes off almost like a dictionary. Lastly-the level the book is pitched at. It has pretty comprehensive coverage of the standard topics: groups, rings, field, and modules. It also contains some topics that are better suited for graduate courses- homological algebra and group representations, for example. The problem is the book tries to cover all these topics equally. As a result, it doesn’t succeed in developing all of them in enough depth for a graduate course and it ends up covering way too much for any one-year undergraduate course. And to be frank, a lot of the presentation of the undergraduate material is very similar to that of Herstein- except the only about half the exercises are anywhere near as interesting as the ones there. I think this is probably what annoys me the most about this book-it comes off as a bloated, watered down version of Herstien. It’s nice to have handy for looking stuff up that you’ve forgotten -but for its price, see if you can borrow a copy instead.

The Big Three: These are the 3 textbooks that up until about 10 years ago, were the standard texts at the top graduate programs in the U.S. to use for first year graduate algebra courses and for qualifying exams at PHD programs in algebra. Of course, at such programs, the line between graduate and undergraduate coursework is somewhat ambiguous. But I think most mathematicians would agree with me on this assessment.

Algebra, 3rd edition by Serge Lang:

Ok, let’s get the elephant in the room out of the way first.

Lang is a good example of the kind of strange “canonization’ of textbooks in academia which I’ve mentioned before at this blog and other places. It’s funny how some mathematicians-particularly algebracists at the more prestigious programs- that get very self-righteous and uppity when you question whether or not Lang should be used as a first-year graduate text anymore with all the new choices. I can’t help but use some of the remarks of a frequent poster at Math Overflow in this regards.
Let’s call him Mr. G.
Mr. G is a talented undergraduate at one of the more prominent universities to study mathematics in the Midwestern United States. Like the author of this blog, he also has been occasionally slammed for shooting off his big mouth on MO by the moderators.
He and I have had several heated exchanged about his Bourbaki-worship: G believes that the Bourbaki texts are sacred tomes that are the only “real” texts for mathematicians and applications are for nonmathematicans. But I’ll let his own words state his position far better then I can. Here is a recent exchange between Mr. G and 2 mathematicians who are frequent posters at MO: let’s call them Dr. H and Dr. L. This was a question regarding the presentation of graduate algebra. (I obviously can’t be more specific then that-to do so would identify the participants.)

@Dr. H: The first graduate algebra course is often going to be the student's first introduction to algebra. It's supposed to be abstract and intense! If you muddy the waters with applications, your students will never get to that level of Zen you achieve after stumbling around in an algebra course. It's like point-set topology, except the rabbit-hole called algebra goes much deeper and is much more important. –Mr.G
@Mr.G: After my first algebra course I still didn't understand why I should actually much about Galois theory from a practical point of view until I saw $GF(2^n)$ in all sorts of applications. My experience has been that most students--even graduate students studying algebra--are not going to be interested in abstraction for its own sake. Mechanics can help to motivate calculus. The same can be true of information theory and algebra. –Dr.H
@Dr.H: Graduate math students shouldn't be taught things "from a practical point of view". This isn't a gen. ed. class, and the abstract perspective one gains by really engaging algebra "as it is practiced) is completely worth the "journey in the desert", as it were. This is the "Zen" I was talking about. Also, I think that characterizing algebra as "abstraction for abstraction's sake" is really missing the point tremendously. – Mr.G.
@Mr.G. The journey through algebra does not necessarily have to go through the desert, nor is that necessarily the best or most ideal path. It might be so for you, but it is certainly not the best path for everybody. There are numerous other paths to take, most of which can lead and have lead people to mathematical understanding and success. Please take a moment to consider, for instance, Richard Borcherds' recent algebraic geometry examples post. – Dr. L.

Speaking for myself, I firmly believe in heeding Lebesgue’s warning about the state of the art in mathematics: “ Reduced to general theories, mathematics would become a beautiful form without content: It would quickly die.” Generality in mathematics is certainly important, but it can and often is, overly done. But I digress. My point is that Mr.G’s attitude is typical of the Lang-worshipper: That if you can’t deal with Lang, you’re not good enough to be a graduate student in mathematics. Or to use Mr.G’s own words on another thread on the teaching of graduate algebra: ”Lang or bust.” Many feel the “journey in the desert” of Lang is a rite of passage for graduate students, much as Walter Rudin’s Principles of Mathematical Analysis is for undergraduates.
Well, there’s no denying Lang’s book is one of a kind and it’s very good in many respects. People ask me a lot how I feel about Lang’s remarkable career as a textbook author. It’s important to note I never met the man, sadly-and everything I know about him is second hand.
Reading Lang’s books brings to mind a quote from Paul Halmos in his classic autobiography I Want To Be A Mathematician .which was said in reference to another famous Hungarian mathematician, Paul Erdos: “I don’t like the kind of arithmetic-geometric-combinatorial problems Erdos likes, but he’s so good at them, you can’t help but be impressed. “
I don’t like the kind of ultra-abstract, application-devoid, Bourbakian, minimalist presentations Lang was famous for-but he was SO good at writing them, you can’t help but be impressed.
And Algebra is his tour de force.
The sheer scope of the book is stunning. The book more or less covers everything that’s covered in the later editions of van der Waerden-all from a completely categorical, commutative diagram with functors point of view. There’s also a generous helping of algebraic number theory and algebraic geometry from this point of view as well. His proofs are incredibly concise and with zero fat, but quite clear if you take the effort to follow them and fill in the blanks. The chapters on groups and fields are particularly good. Lang also is an amazingly thorough and responsible scholar; each chapter is brimming with references to original proofs and their source papers. This is a book by one of the giants in the field and it’s clearly a field he had enough respect for to know his way around the literature remarkably well-and he believed in giving credit where credit is due. Quite a few results and proofs-such as localization of rings, applications of representation theory to functional analysis and the homology of derivations, simply don’t appear in other texts. The last point is one I think Lang doesn’t get a lot of credit for, without which the book would be all but unreadable: He gives many, many examples for each concept-many nonstandard and very difficult to ferret out of the literature. Frankly, the book would be worth having just for this reason alone.

So fine, why not go with the party line then of “Lang or bust”?

Because the book is absurdly difficult, that’s why.

First of all, it’s ridiculously terse. It takes 2-3 pages of scrap paper sometimes to fill in the details in Lang’s proofs. Imagine doing that for OVER 911 PAGES. And worse, the terseness increases as one progresses in the book. For the easier topics, like basic group theory and Galois theory, it’s not so bad. But the final sections on homological algebra and free resolutions are almost unreadable. You actually get exhausted working through them.


“Yeah, I’ve heard the horror stories about Lang’s exercises in the grad algebra book. C’mon, they’re not THAT bad are they?”

You’re right, they’re not.

They’re WORSE.

I mean, it’s just ludicrous how hard some of these exercises are.

I’ll just describe 2 of the more ridiculous exercises and it pretty much will give you an idea what I’m talking about. Exercise 30 on page 256 asks for the solution of an unsolved conjecture by Emil Artin. That’s right, you read correctly. Lang puts in parentheses before it: “The solution to the following exercise is not known.” No shit? And you expect first year graduate students-even at Yale-to have a chance? I’m sorry, that’s not a reasonable thing to do!

Then there’s the famous-or more accurately, infamous-exercise in the chapter on homological algebra: “Take any textbook on homological algebra and try and prove all the results without looking up the proofs.”

I know in principle, that’s what we’re all supposed to do with any mathematical subject we’re learning. But HOMOLOGICAL ALGEBRA?!?

(An aside: I actually had a rather spirited discussion via email with Joseph Rotman, Professor Emeritus of the University of Illinois at Urbana-Champaign, over this matter-a guy that knows a thing or 3 about algebra. Rotman felt I was too hard on Lang for assigning this problem. He thought Lang was trying to make a point with the exercise, namely that homological algebra just looks harder then any other subject, it really isn’t. Well, firstly, that’s a debatable point Lang was trying to make if so. Secondly, I seriously doubt any graduate student who’s given this as part of his or her final grade is going to be as understanding as Rotman was. Actually, it’s kind of ironic Rotman thinks that since I know many a graduate student who would have failed the homology part of their Lang-based algebra course without Rotman’s book on the subject! )

My point is I don’t care how good your students are, it’s educational malpractice to assign problems like that for mandatory credit. And even if you don’t and just leave them as challenges for the best students-isn’t that rubbing salt in the wounds inflicted by this already Draconian textbook?

These exercises are why so many mathematicians have bitter memories of Lang from their student days.

A lot of you may be whining now that I just don’t like hard books. That’s just not true. Herstien is plenty difficult for any student and it’s one of my favorites. In fact, quite the contrary. You’re really supposed to labor over good mathematics texts anyway-math isn’t supposed to be EASY. An easy math textbook is like a workout where you’re not even winded at the end-it’s doubtful you’re going to get any benefits from it.

But Lang isn’t just hard; I don’t just mean students have to labor over the sections before getting them.

The average graduate student learning algebra from Lang’s book is like a fat guy trying to get in shape by undergoing a 3 month U.S. Marine Corp boot camp and having a steady diet of nothing but vitamins, rice cakes and water. Assuming he doesn’t drop dead of a heart attack halfway through, such a regimen will certainly have the desired effect-but it will be inhumanely arduous and unpleasant.

And there are far less Draconian methods of obtaining the same results.

So unless one is a masochist, why in God’s name would you use Lang for a first year graduate course in algebra?

Is it a TERRIBLE book? No-as I said above, it has many good qualities and sections. As a reference for all the algebra one will need in graduate school unless becoming an algebraicist, the book is second to none.

Would I use it as a text for a first year graduate course or qualifying exam in algebra?


Algebra by Thomas Hungerford : This has become a favorite of a lot of graduate students for their algebra courses and it’s pretty easy to see why-at least at first glance. It’s nearly as demanding as Lang-but it’s much shorter and more selective, has a lot more examples of elementary difficulty and the exercises are tough but manageable.
The main problem with this book occurs in the chapter on rings and it boils down to a simple choice. Hungerford-for some strange reason-decides to define rings without a multiplicative identity.

I know in some ring theoretic cases, this is quite useful. But for most of the important results in basic ring and module theory, this results in proofs that are much more complicated since this condition needs to be “compensated” for by considering left and right R-modules as separate cases. Hungerford could alleviate this considerably by giving complete, if concise, proofs as Lang does in most cases.

But he doesn’t. He only sketches proofs in more then half the cases.

The result is that every section on rings and modules is very confusing. In particular, the parts on modules over commutative rings and homological algebra-which I really need for my upcoming exam-are all over the place.

Still, the book has a lot of really nicely presented material from a totally modern, categorical point of view. The first chapter on category theory is probably the best short introduction there is in the textbook literature and the section on group theory is very nice indeed.

Basic Algebra, 2nd edition by Nathan Jacobson, volumes I and II: I say we should nominate Dover Books for a Nobel Peace Prize for their recent reissue of this classic. The late Nathan Jacobson, of course, was one of the giants of non-commutative ring theory in the 20th century.

He was also a remarkable teacher with an awesome record of producing PHDs at Yale, including Charles Curtis, Kevin Mc Crimmon, Louis H.Rowen, George Seligman, David Saltman and Jerome Katz. His lectures at Yale on abstract algebra were world famous and had 2 incarnations in book form: The first, the 3 volume Lectures In Abstract Algebra, was for a generation the main competition for van der Waerden as the text for graduate algebra courses. Basic Algebra is the second major incarnation- the first edition came out in the 1970’s and was intended as an upgraded course in algebra for the extremely strong mathematics students entering Yale from high school during the Space Age. The first volume-covering classical topics like groups, rings, modules, fields and geometric constructions-was intended as a challenging undergraduate course for such students. The second volume-covering an overview of categorical and homological algebra as well as the state-of-the-art (circa 1985) of non-commutative ring theory-was intended as a graduate course for first year students. The complete collapse of the American educational system in the 1990’s has rendered both volumes useless as anything but graduate algebra texts. Still, given that the second volume was going for nearly 400 dollars at one point online in good condition, it’s reissue by Dover in wonderfully cheap editions is a serious cause for celebration.

Both books are beautifully and authoritatively written with a lot of material that isn’t easily found in other sources, such as sections on non-associative rings , Jordan and Lie algebras, metric vector spaces and an integrated introduction to both universal algebra and category theory. They are rather sparse in examples compared with other books, but the examples they DO have are very well chosen and described thoroughly. There are also many fascinating, detailed historical notes introducing each chapter, particularly in the first volume.

The main problem with both books is that Jacobson’s program here absolutely splits in half algebra into undergraduate and graduate level topics; i.e. without and with categorical and homological structures. This leads to several topics being presented in a somewhat disjointed and inefficient manner because Jacobson refuses to combine them in a modern presentation-module theory in particular suffers from this organization. Personally, I didn’t find it THAT big an issue with a little effort-but a lot of other students have complained about it. Also, some of the exercises are quite difficult, rivaling Lang’s. Even so, the sheer richness of these books make them true classics. If graduate students are willing to work a little to unify the various pieces of the vast puzzle that Jacobson presents here with astonishing clarity, he or she will be greatly rewarded by a master’s presentation and depth of understanding.

The New Kids On The Block: As I said earlier, the adoption of Lang worldwide as the canonical graduate algebra text had a backlash effect that’s been felt with a slew of new graduate texts. I haven’t seen them all, but I’ve seen quite a few. Here’s my commentary on the ones I’m most familiar with.

Basic Algebra/Advanced Algebra by Anthony W. Knapp: This is probably the single most complete reference for abstract algebra that currently exists. It is also paradoxically, the single most beautiful, comprehensive textbook on it. Knapp taught both undergraduate and graduate algebra at SUNY Stonybrook for nearly 3 decades-and these volumes are the finished product of the tons of lecture notes that resulted. The purpose of these books, according to Knapp, is to provide the basis for all the algebra a mathematician needs to know to be able to attend a conference on algebra and understand it. If so, he’s succeeded beyond all expectations. The main themes of both books are group theory and linear algebra (construed generally i.e. module theory and tensor algebra) . The first volume corresponds roughly to what could possibly be covered at the undergraduate level from reviews basic number theory and linear algebra up to an honors undergraduate course in abstract algebra (groups, rings, fields, Galois theory, multilinear algebra, module theory over commutative rings). The second corresponds to graduate syllabus focusing on topics in noncom mutative rings, algebraic number theory and algebraic geometry( adeles and ideles, homological algebra, Wedderburn-Artin ring theory, schemes and varieties, Grobner bases, etc.) This is the dream of what an advanced textbook should be-beautifully written, completely modern and loaded with both examples and challenging exercises that are both creative and not too difficult. In fact, the exercises are really extensions of the text where many topics and applications are in fact derived-such as Jordan algebras, Fourier analysis and Haar groups, Grothendieck groups and schemes, computer algebra and much, much more. The group actions on sets are stressed throughout. Also, categorical arguments are given implicitly before categories are explicitly covered by giving many commutative diagram arguments as universal properties. (This avoids the trap Jacobson fell into.) Best of all-there are hints and solutions to ALL the exercises in the back of each volume. I would LOVE to use this set to teach algebra one day-either as the main texts, as supplements or just references-but if you enjoy algebra, you HAVE to have a copy. Hopefully, there will be many editions to come.

Abstract Algebra, 2nd edition by Pierre Grillet: The first edition of this book was simply called “Algebra” and it came out in 1999. To me, this book is what Lang should be. Grillet is an algebriacist and award-winning teacher at Tulane University. Interestingly, he apparently carried out the revision in the aftermath of Katrina. The book covers all the standard and more modern topics in a concise, very modern manner-much like Lang. Unlike Lang, though, Grillet is extremely readable, selective in his content and highly structured with many digressions and historical notes. The sheer depth of the book is amazing. Unlike Lang, which focuses entirely on what-or Hungerford, which explains a great deal but also is very terse-Grillet focuses mainly on why things are defined this way in algebra and how the myriad results are interconnected. It also has the best one chapter introduction to category theory and universal algebra I’ve ever seen-and it occurs in Chapter 17 after the previous 16 chapters where commutative diagrams are constructed on virtually every other page. So by the time the student gets to category theory, he or she has already worked a great deal with the concepts implicitly in the previous chapters. This is very typical of the presentation. Also, Grillet doesn’t overload the book with certain topics and give the short shriff to others-many texts are half group theory and half everything else, for instance. Grillet gives relatively short chapters on very specific topics-which makes the book very easy to absorb. The exercises run the gamut from routine calculations to proofs of major theorems. The resulting text is a clinic in how to write “Bourbaki” style texts and it would be a great alternative to either Lang or Hungerford.

Algebra: A Graduate Course by I.Martin Issacs: This is a strange book. After being out of print for over a decade, it was recently reissued by the AMS. Isaacs claims this course was inspired by his teacher at Harvard, Lynn Loomis, whose first-year graduate course Issacs took in 1960 there. Like Loomis’ course, Issacs emphasizes noncom mutative aspects first, focusing mainly on group theory. He then goes on to commutative theory-discussing ring and ideal theory, Galois theory and cyclotomy. The book is one of the most beautifully written texts I’ve ever seen, with most theorems proved and most example constructions left as exercises. Unfortunately, Issacs’ material choice seems to follow his memories of his graduate course in 1960 far too closely-this choice of topics would be a first year graduate course at a top university ONLY before the 1960’s. Issacs omits completely multilinear algebra, category theory and homological algebra. How can you call such a book in 2010 a graduate course? That being said-it is wonderfully written and if supplemented by a text on homological algebra, it could certainly serve as half of such a course-either Osborne or Joseph Rotman’s books on the subject fill in the omissions very nicely.

And now-my very favorite algebra book of all time. Drum roll,pleeeeeeeeease................

Advanced Modern Algebra by Joseph J. Rotman: I haven’t seen the second edition, but I’m very familiar with the first. Rotman may be the best writer of algebra textbooks alive. Hell, he may be the best writer of university-level mathematics textbooks PERIOD. Serious. So when his graduate textbook came out, I begged, borrowed and cajoled until I could buy it. And it was one of the best textbooks I ever bought. The contents of the book are, as the AMS’ blurb discusses:

“This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic K-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization.”

That says what’s in the book. What it doesn’t tell you is what makes this incredible book so special and why it deserves a second edition so quickly with the AMS: Rotman’s gifted style as a teacher, lecturer and writer. The book is completely modern, amazingly thorough and contains discussions of deep algebraic matters completely unmatched in clarity. As proof, read the following excerpt from the first edition, how Rotman explains the basic idea of category theory and it’s importance in algebra:

Imagine a set theory whose primitive terms, instead of set and element, are set and function.
How could we define bijection, cartesian product, union, and intersection? Category theory
will force us to think in this way. Now categories are the context for discussing general
properties of systems such as groups, rings, vector spaces, modules, sets, and topological
spaces, in tandem with their respective transformations: homomorphisms, functions, and
continuous maps. There are two basic reasons for studying categories: The first is that they
are needed to define functors and natural transformations (which we will do in the next
sections); the other is that categories will force us to regard a module, for example, not in
isolation, but in a context serving to relate it to all other modules (for example, we will
define certain modules as solutions to universal mapping problems).

I dare you to find a description of category theory that would serve a novice better. The book is filled with passages like that-as well as hundreds of commutative diagrams, examples, calculations and proofs of astounding completeness and clarity. Rotman presents algebra as a huge, beautiful puzzle of interlocking pieces-one he knows as well as anyone in the field. The one minor complaint is the book’s exercises-they’re a little soft compared to the ones in Hungerford or Lang. And the sheer size of the book-1008 HARDBACK BOUND PAGES!- is a bit daunting. ( Rotman joked with me via email that more then a few times, he mistakenly carried it to his calculus class and had to go back to his office to switch books.) But these are very minor quibbles in a book destined to become a classic. If I had to choose one textbook for graduate algebra and it’s qualifier and couldn’t pick any others-THIS is the one I’d pick, hands down.
Word from the AMS and those who have seen it that the second edition is even better-the index has been greatly improved and entire sections have been rewritten to emphasize noncom mutative algebra-which is appropriate for a graduate course.

I suggest you all place your orders now. You’ll thank me later, I promise.

I now return you to your regularly scheduled lives.

Thank you for your attention.


Student Of Fortune

Sunday, October 31, 2010

A Sidebar On My Willing Exile From Math Overflow

Crap, took WAY too long between posts again. And this is going to have to be a short one because of the lateness of the hour.

Need to do something about that. But between chronic insomnia and a sinus infection, it’s been all I can do to think.

To the main point of this post:
The guys at Math Overflow have finally had it with my shenanigans.

After my 5th suspension from the board for…well, to be honest, I’m still not completely sure. According to moderator Ben Webster ( formerly MIT C.L.E. Moore Instructor, now at the University of Oregon-you may also recognize him from his days blogging at The Secret Blogging Seminar ) , the reason was as follows:

Andrew- No one has ever been suspended on MO for the contents of mathematical statements, even if we disagree with them. The issue is your rude comments on other answers; I would call it "bad sportmanship," but MO is not a game. For example "I can't believe this guy puts down a high school slogan and gets 13 points for it and I got downvoted for "Probability is real analysis with the concept of an expectation." " on Michael Lugo's answer.

As far as I'm concerned, this is equivalent to jumping up after a seminar and shouting "You guys are clapping for that? That was a terrible talk!" which I think we can all agree would not be socially acceptable behavior.

Note to the audience: generally the moderators have adopted a policy of not arguing with Andrew on meta, since it just seems to create more drama. In this case I thought it was important to point out that the issue was not Andrew's mathematical statements (which as I said before, we would not suspend people over), but rather his behavior in comments.

Uh, ok, Ben.

My “behavior” ,as he so puts it, was simply being myself. For those who know me, that seems to be more then enough. Despite my best efforts to tone it down for the board, things just deteriorated further and further. At one point, I was emailed and messaged by several of the members telling me-politely but in no uncertain terms-that my antics were making a bad impression on the mathematical community in general and that I was thus endangering my future career and job prospects.

I really don’t like being threatened.

And make no mistake, as nicely as it was delivered, that’s what it was. A threat.

There was a time I’d have told the whole bunch of them to go fuck themselves for openers and go on for several HTML pages about them and their mothers and wives.

But if I had-well, I would have deserved what I got.

Firstly, when you’re dealing with people like Webster, Andy Putnam at Rice University and Pete L.Clark at the University of Georgia-and this drama is all going on in front of frequent posters Terence Tao, Tom Gowers and Richard Stanley (!) - well, it’s pretty obvious you’re not going to win this one.

And it’s more then that. This isn’t MY site, it’s THIERS.

This wasn’t really about right and wrong, it was about me trying to make their site something it’s not because I wanted to be able to say these things in front of professional mathematicians and get their feedback.

But that’s not what MO is for. It’s for research level and academic questions regarding the mathematical community. They set it up, they police it, they make the rules. I was just a guest.

And they decided I was messing up their furniture and would rather I left. That’s entirely within their rights to do.

And contrary to what some people may think, I don’t enjoy offending people. I wasn’t following the rules and I was dead wrong here. No matter how morally indignant I might want to look.

So after this last incident, I left my pride on the floor and reopened shop here, where I can have no rules but my own.

In closing regarding this incident, I wanted to let everyone over at Math Overflow I never intended any offense. I’m a passionate, opinionated guy. I was the proverbial bull in a china shop.
Not that that’s anything new for me. Once again, I want to apologize to everyone there and hope one day I can return.

More importantly,I hope to be posting here at a regular basis as I prepare for my oral qualifying exams in algebra and topology-hopefully,to be taken around Christmas, no later. I want to let everyone there know they are more then welcome to respond here. They are also free to say WHATEVER THEY WANT ABOUT THE POSTS OR ME. The first amendment is very much alive here.

(For as long as the government allows it after Tuesday,of course. More on that later in the week.)

Well,it’s nearly 3 am now, so the planned posted reading list for graduate algebra courses-sadly-is going to have to wait.

But hopefully-not very long at all!


EDIT 11/6/2010: As Terry Tao and another poster commented after the first draft if this post went online, I wasn't fair in my blanket description of those aforementioned comments as all threats. In fact, many were by well intentioned posters who didn't want me blow my career by shooting off my stupid mouth. For those posters, I should and will apologize. They meant well and I shouldn't be lumping them in with the idiots-"trolls" they call them on MO-who emailed me and warned me "they'd fix my career when I apply for jobs." THAT was a threat.But those brave enough to countermand me were very well intentioned.As such,I apologize once again.
See,I'm not above admitting an error. Rare as it is........

Monday, August 23, 2010

I HAVE RETURNED! Hide the women and children. Well, maybe not the women…………..

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After a disappointing year of graduate school, I’ve returned at the close of summer and the beginning of my last semester (I HOPE) of schoolwork in my life. Flecks of premature grey adorn my beard as the analogous summer of my life ends and the chill of winter slowly but inevitably approaches-and a PHD still nowhere in sight due to the collapse of the American empire.

I will develop my observations on this matter in depth in future posts.
But forget the barbarians being at the gate: The guard was their all-too-willing accomplices as they walked right in and enslaved the entire kingdom under the pretexts of “free market economy“ and “lobbying“.

I fear for the future of this country. The Radical Red Staters are poised to take advantage of 2 generations of steadily degenerating education of our citizens to polarize and divide this nation, so that it’s no longer a battle of liberal and conservative ideas. Rather, it’s now between thinking, logical, scientifically objective people and obsessively tribal, dogmatic, racist, exclusionary Christian fundamentalist extremists-with the latter group being frighteningly agitated into a frenzy by a paradoxically self righteous and simultaneously self-servingly duplicitous leadership who blasts their inflammatory beacons on the internet and talk-radio.

And just think: All it took was the election of the first African-American president-and the first real attempts at social progress in over a decade as a logical response by his administration to the worst economic disaster in our history-for the populace of the US to revert socially to the 1930’s. Socialism. Illegal aliens. Doubts over evolution being forcibly imposed in school textbooks by a few concerned educators in Texas.

Somewhere in Hell, Adolf Hitler, Joseph Stalin and Joseph McCarthy are laughing their asses off. I see the cliff coming and no one seems to want to slam on the brakes before we go over………

Again-MUCH more on this in future posts. But I can’t justify or contextualize this first post’s purpose without some preamble of the desperation felt by my aging self as the world I knew as a teenager slips away-to be replaced by a darker jungle of uncertainty.

The wake of my father’s death-now 4 years distant-has financially crippled what remains of my family. My father’s 18 year battle with cancer-which he ultimately lost-has now left us with a debt that barely allows my mother to buy food. I know-we’re hardly alone in that. But this has taken it’s toll on my career. I’ve had to take 3 graduate courses in the last year just to maintain my eligibility for health insurance as well as student loans to PAY for that insurance. (Of course, this being America, you’d think this would be the primary place for free health care for all it’s citizens. Of course, we don’t live in a logical or moral reality, do we?)

That alone would make struggling through these courses hard enough. But then in a moment of desperation-I made a very foolish choice regarding my health. I really can’t get into the specifics here-I hope to do so in a future post in connection to a larger issue.
I will say this: I’ve officially learned the hard way that if you have no other available methods of controlling psychological illnesses other then drugs and someone tells you to go off them cold turkey and “you’ll feel so much better, all you need is to man up!”-then if you actually listen to them, the REAL imbecile in the conversation is YOU.

As you can imagine, the long term result was somewhat less then expected. I rectified the result, but the damage was done. And now I’m stuck with 3 subpar grades in courses I didn’t even fucking WANT. It’s entirely possible now my dreams of an Ivy League PHD programs-unless I solve the Riemann Hypothesis-are out the door and my career ends here with a Master’s degree and a future as a high school teacher.

The wonderful destiny awaits of dying of cardiac complications from prostate cancer when I’m 75-face down in puddle of cold decaf tea grading high school algebra papers from student who can barely read.

So it goes.

I haven’t given up yet. I can’t. Why not? That’s what a rational person of my intelligence would do.

Well, we’re all going to die of something. I lost my youth long ago-sacrificed it for my family. I think I’ve finally made my peace with that. It hurts like hell what I lost. But we’re all losing so much as the second decade of the 21st century opens. Parents losing their children in 2 pointless wars that have bankrupted the country the survivors are coming back to. PHDs mopping floors because those educated in other nations or of higher pedigrees are preferred for the vanishingly small number of jobs available.

Again, the cliff approaches for all but the top 1% of us.

I dunno about all of you, but I’d rather go off the cliff screaming my rage at this pointless, Godless reality and fighting with my last breath against the dying of the light.

Which is why I’m taking the initiative and beginning my own online business. (Nice segway, huh?)

I began this Ebay store on whim. I’ve been selling textbooks online for nearly 8 years on and off for pocket cash-and all told, it’s worked out pretty well. But “working out” has meant what’s basically chump change by selling second hand books and cannibalizing my own library.
So I started thinking: What kind of cash can I make with a REAL inventory of books to sell?

So one thing lead to another and viola! Parthenon Academic Books.

Actually, it wasn’t that simple. History never is.

So what’s the scoop behind it?

I’d love to tell you guys, but I’m passing out at almost 3 am. So it’ll have to wait-along with several other things-for the next post.

After all, how can you build a following to a blog without good cliffhangers?


In the meantime, please check it out! Yes, I’m begging. Begging is something we all should get used to-most of us will be making a living doing it for the next few decades. So it’s good practice for all of us.

Wait, wait, don’t go! At LEAST give me a chance to let you see the place before you blow it off. And yes, you can SEE what you’re buying. That’s the first main advantage of buying from me. I’ll show you EVERYTHING before you buy-you don’t have to take my word for it like Amazon and the rest of the corporate text machines.

What are the others? Well, since I have a webpage, why should I waste my time just repeating what’s there?

My About Me Page:

And honestly-how mamy stores HAVE THIER OWN BLOG? No,not THIS one. I can't have anyone upstaging me,let alone filling this air with complaints about the dustjackets of purchased books. Nope-I got my own blog just for that. And even better,it's got more photos of the inventory and is updated by me semiweekly. Does Amazon do that or let you speak directly with thier CEO? I think not.

The blog:

Come on in and browse-any questions,feel free to ask.

Be back soon,same blogsphere plane and frequency.

Beware, Citizens...........

Saturday, July 11, 2009

Re: arXiv:The Modern Cure For B.A.D Mathematicans?

This'll be a short one. But hopefully something I'll comment on more in the future.

I've found a terrific article by Professor Melvin Henriksen published in The Mathematical Intelligencer in 1993 and republished online at the Topology Atlas by him.Henriksen-if you're unfamiliar with him-is currently Professor Emertis at Harvey Mudd College and he's one of the few remaining active-sort of-mathematicans in one of my favorite areas of mathematics:point set topology. He's also published quite a bit in algebra,has been very active in historical aspects of mathematics and is currently one of the major overseers of the mammoth virtual site Math Forum supported by Drexel University.

The elitism in mathematics is nothing new-nor,sadly,is it unique to mathematics among academic endevors.Henriksen-never one to keep his opinions to himself-summed up the situation-and the reasons behind it-beautifully and informatively,in this article:

If anything,the situation has WORSENED since he wrote this article.When you mention certain branches of mathematics at some unmentionable universities that believe only the half a dozen places on Earth called "Ivies" are where civilized humans exist and everywhere else is untamed jungle with blood drinking,grunting barbarians with pieces of paper masquerading as educated humans-you actually get audible laughter. They don't even try to be kind about it. Why should they? Anyone who can't see that they're right is a fool and won't get promoted anywhere. At least,not if they have anything to say about it-and sadly,they do.

I'm currently investigating the possible role of general topology in additive number theory. To be honest,if arXiv-simply called "Archive" by most of us-didn't exist,I doubt I could publish it ANYWHERE. This remarkable tool has changed publishing forever through open access and free publishing in mathematics and physics with no formal refereeing-and therefore no monkey buisness. Any attempts to shut it down or seriously regulate it should be met with savage resistance. Such attempts at regulation has already begun in the form of the vote in 2004 by the Archive board to allow only preprints-presumably because copyright issues could arise that could jeapordize corporate profits. The bottom line is that it's begun-the attempts to control it.
(The copyrighting of concepts as property is a terrifying phenomenon that I hope to tackle in depth another time.Suffice to say this is a Pandora's Box that threatens all original thinking if it's not strictly controlled.)
It's a wonderful reality we live in currently in this regard-free publishing. Let's not let the B.A.D. crew wreck it like they've wreaked whatever doesn't serve thier purposes.
In other words,buisness as usual on Planet Earth.

Monday, July 6, 2009

A Brief (Partial) Apology For Speaking out of Turn: Calculus, Cirricula And Constudents...............

I'm not usually one to apologize when I feel someone is being a dick. Anyone that knows me knows that. But my guilt has gotten the better of me and I think I need to amend my swipe at McMasters' University professor James Stewart.I think I was angry at the decaying civilization around me and I took it out on him.
I'm really apologizing just for one small part of the rant that I felt was beneath me. It was simply untrue and would be very unfair for me to say about someone I've never even heard lecture or speak once.I referred to Stewart as a "grotesquely overpaid hack without an ounce of mathematical talent".
Well,that was completely untrue and unfair:Professor Stewart is actually a very fine teacher and mathematician from what I know of him.(It turns out he's the mathematical grandson of the famous Oxford mathematician E.C.Titchmarsh.I didn't know that and found that kind of interesting in and of itself.)
I dug out my copy of the third edition of his textbook to act physical evidence in this trial of my conscience.I also borrowed a copy of the 6th edition.He's made a lot of improvements in the text since I used it-a lot more pictures, the exponential and logarithmic functions are introduced and discussed MUCH earlier (in the first chapter,in fact),and in general a lot more explicit focus on the overall process of problem solving,which was only indirectly stated in the edition I'm familiar with. Stewart was a graduate student of George Polya before moving on to get his PHD-the influence of the Stanford problem solving master is all over this textbook in both editions. Stewart approaches calculus as a problem solving enterprise first and foremost-such an approach is bound to be pragmatic and will intentionally sacrifice rigor where it obscures understanding.
In short, Stewart is trying to teach his students how to become intelligent problem solvers above all else. As teachers (and speaking for myself as an aspiring teacher at the college level), discouraging the good intentions behind such an approach is the LAST THING we should want to do.
It's easy to forget how confusing calculus and physics is when one first seriously tackles it as a college undergraduate-or for the more fortunate and/or talented, high school. As a result, it's easy to get on your high horse and badmouth a text like this from the viewpoint of someone who's mastered a good portion of rigorous mathematics. Stewart offers to take the student by the hand and walk him or her step by step through the fog-showing them tricks of the trade along the way and tried-and-true methods of attacking problems in ways that not only obtain solutions,but a complete understanding of the MEANING of what's being asked of them. "What do they want from you?What will satisfy the question?" This is what Stewart is trying to teach with his book.
It's really informative in this regard to read Stewart's own comments on the text from an interview done by the MAA on July 6th,comparing it to the texts he used as a student at Stanford University and The University Of Toronto in the 1960's:

IP: How have mathematics textbooks changed over the years?
JS: Compared with the textbooks that I had as a student, textbooks are so much better now. I don’t know how kids learned from these old books. There was no motivation. It was very austere. You can go too far in the other direction, but the state of the exposition of mathematics is just so much better than it was three decades ago.
As an author of the high school textbooks in the 70s, I kept my eye on trends in education. The new math had been well ensconced by then. But what I observed and decried was the waves, the extremes, the pendulum going back and forth from the new math back to basics. You still see this, especially in the U.S., especially at the high school level, where it is much more virulent. At that time, I longed to get hold of that pendulum and stop it somewhere in a sensible middle. People get too dogmatic.

Even more insightful into Stewart's thinking is his comments on teaching and what he's doing lately:

IP: Are you still teaching?
JS: Although I am Professor Emeritus at McMaster, a year ago I was appointed professor of mathematics at the University of Toronto, and I have twice taught first-year calculus. Although I don’t teach fulltime anymore, I love teaching. Being an author is a pretty solitary, sedentary occupation, so I miss the social aspect—which is teaching. I do it partly to keep in touch with kids, because it brings out the best in me, and to give me new ideas for new editions of my books.
This fall I am introducing a new course at the University of Toronto on problem solving. I introduced such a course at McMaster quite some time ago.
When I was a graduate student at Stanford I fell under the spell of George Polya, who was retired but used to come in and give these problem-solving talks. He had all of us—teachers and students alike—literally sitting on the edges of our seats with mathematical excitement, presenting data, asking us to make conjectures.
The idea is: Suppose you’re faced with a problem that you have never seen before. How do you get started? The first few lectures introduce some basic principles of problem solving. The remaining lectures start with a “problem of the day.” How would you solve it? What strategy would you use? What about trying a special case or solving a simpler problem first? It’s my favorite course to teach.
I’m doing that this fall, working with some of the faculty at the University of Toronto so that they can carry on after me. It will be a kind of capstone course. You’re drawing on everything that you’ve learned up to that point, putting it together. There’s no new content whatsoever. But once you take a problem out of the context of a specific course, it becomes harder.

Now THAT'S a book I'd love to read-a problem solving textbook by Stewart that emerges from that course!
But sadly,Stewart seems to miss that the problem with this approach to calculus which has made his book so successful is also why it's damaging to students used by itself. The result of the "practical" nature of the text is that THE FACT THAT IT'S A BOOK ON CALCULUS BECOMES COMPLETELY INCIDENTAL.He never asks the all important "why" questions that brought the real number system and the structure of real analysis into focus for mathematicans in the 19th century.Everything's given a name-Sum Rule,Product Rule,Method of Secants,etc.-which makes them tailor-made for memorization rather then learning.He gives quite good "geometric" explainations-such as a good discussion of motivating the definition of the derivative as the limit of a sequence of secant lines to a point on a curve.But such a discussion is completely independent of the definition of a derivative as a limit.It might as well appear in a book on physics or geometry. As a result, it's all completely mechanical-the fact that it's a book on calculus almost become irrelevant!
And this is the problem he fails to see:TO MOST OF TODAY'S STUDENTS, IT IS IRRELEVANT. YOU MAY AS WELL BE TEACHING THEM HOW TO PLAY CHECKERS AND THEY MEMORIZE THE RULES. Sure, a few students will really look at the very nice geometrical arguements and walk away really learning something. But most students-who make up most of today's colleges and whom the university administrators are aiming to sell calculus to-couldn't care less. I call such students constudents-a hybrid of conmen and students. They aren't interested in learning-in fact, like thieves excited about stealing and not getting caught or cheating husbands who call thier wives to tell them they're going to be late while getting oral sex from thier mistress-getting an A while never learning a damn thing is exciting to them.
I know what some of you are thinking: "Andrew,come on-that's human nature,there's always going to be students like that!" Sure,of course.But the big advantage of the rigorous calculus texts of the past was that it was almost impossible for such students to con thier way to a good grade-the fact that rigorous mathematics was an ESSENTIAL part of the structure of the course ensured they actually had to learn something to do reasonably well. And the course acted to ensure that students with impure motives who didn't even try DIDN'T get good grades.
Books like Stewart's have eliminated this failsafe altogether.
I remember as a premed sitting around with a number of students taking calculus using Stewart and the discussion of the exam was like they were talking about a football game and how they were going to "beat" the exam. They came up with codes,mnemonics,word games-not a single theorum or concept or proof. I made the idiotic mistake of asking if anyone actually learned the material and the whole table erupted with laughter. The President of the Student Medical Association smiled at me like The Grinch.
"Winning is about APPEARING to know what you're doing,not actually doing it.Don't worry,Andrew-you can always work taking out the trash in my office on 5th avenue."
Our society rewards this kind of behavior.Why?Becuase letting these monsters use Stewart and get thier A's without learning anything is good for buisness,that's why. The university gets to pack the classes with 200 PAYING students by making this a required course,the students get thier A's which the college can use to improve it's ranking standing so that administrators get promoted for making so much money and helping public relations and off they go to Ivy League medical schools thinking urea is made in the kidney-and worse,not giving a shit.
And 5 years later they're killing and crippling patients left and right and being aquitted at malpractice trials because the only one in the room who's a better liar then they are is the son of a bitch defending them. A book like Stewart's ENABLES this kind of system.
I have no problem with Stewart wanting to make the book of a problem solving nature-as I've said,this leads to the book having many positive qualities.My problem is that including mathematical rigor need not be contrarian to this intention and for someone claiming to be so devout to teaching, Stewart refuses to acknowledge this.
Sadly, I think he's too smart not to see this. I think his position is one of willful ignorance in a corrupt academic culture that's made him not only very wealthy for his occupation, but very famous. I doubt anyone outside of McMaster would have ever heard of him without this text. And I stand by my earlier criticism of Stewart of his ridiculous excess with his own concert hall. He loves music, fine. Bless him. But spending more on his hobby then 5 families spend on thier homes is nauseating and he should be ashamed of himself. Of course,he's hardly alone in that in this day and age.

But he's an academic. He should know better.

Frankly,I think he DOES and his own words betray this:

When I started writing my first book, I had no idea you could make any money writing books. That was not a motivation at all. It was a surprise, but it enabled me to build this house. And I’ve got to continue to work to pay for the house. The house’s cost [$24 million] is double the original estimates.

It sounds like he has a very strong motivation for continuing to enable the sharks. Amazing what people are able to justify to themselves.

I hope Dr.Stewart keeps making money and succeeds in paying for his house so his heirs have the proceeds from using it as a tourist trap when he passes away. I hope all his kids and grandkids go to Harvard from it and maybe follow in his footsteps as a teacher instead of becoming criminal defense attorneys and bankers as the later generations usually do when the first generation creates a fortune for them. And I hope a lot of teachers of calculus use it as a supplementary reference or secondary source for thier calculus courses and as the main text in high school courses.

I just hope one day someone has the balls to challenge the American way someday and writes the text that replaces Stewart by combining mathematical rigor with his teaching skills to give us a calculus text for students and not constudents.

And I hope I and my loved ones are never at the mercy of enabled in a hospital with thier lawyers' number constantly in thier back pocket.

Welcome To The Twilight Jungle. Abandon All Honesty And Integrity Ye That Enter Here.................

Sunday, July 5, 2009

A Brief Ode To Stewart's Calculus-NOT.........................

I just read a really funny post at Ars Mathematica and had to share it with all of you with commentary. Apparently, the question's come up with what McMaster's University's self-made gazillionare James Stewart did with all his royalites from the infamous calculus book every other university's department uses. Apparently, he built a gigantic house with his own personal concert hall in the middle of it. You don't believe me? See for yourself :

This was so he-a trained violinist-could perform with his friends in the privacy and comfort of his own mansion. Talk about hubris worthy of being struck down by the Gods with the Ceres asteriod.Apparently the only way Stewart's fragile self esteem could make it as a violinist in a concert hall was to have one built for himself where he'd be the star of the show every single night...............LOL
Sigh. Only in America would that seem like a logical action and not a gigantic excess of self-centered indulgence. I pass-on my way to the bus-recently homeless families of 4 living in thier cars with thier 4 year old daughter crying to the mother, "Mommy,what happened to my bed?" Meanwhile, this grotesquely overpaid hack without an ounce of mathematical talent is spending 3 times what these poor people's former house was worth because he doesn't want to embarrass himself in public with his violin playing.................

But be that as it may-I was honestly asked:How bad IS Stewart's book and what are some of YOUR favorite texts?What would YOU use to teach calculus given the chance?
Well,sadly,since I was a complete imbecile in high school and didn't know grades mattered in life-and my parents being laborers,well,they didn't know either-I ended up at The City University Of New York instead of a REAL college.(I made many friends there and learned a lot-but let no one be decieved my lack of pedigree will give me a huge battle ahead for any degree of success.) So my first exposure to calculus WAS Stewart.
In all fairness,it's not as bad as some people make it out to be. The real positive about the book is the IMMENSE number of exercises with complete solutions. Unfortuately,that's a double edged sword and it's also the main reason it's completely unpalatable for mathematicans:It reduces calculus to a step-by-step, plug-and-chug bag of techniques without even any mathematical insight or thinking. Anything that requires more thought then a baboon is either completely omitted or shunted to a mythical "advanced calculus" course-which no longer exists,of course. The students don't have to do any real thinking at all-which is why most students love it,of course. Let's face it-THAT'S why the bottom feeding universities buy it every year-so the premeds,accounting students,actuaries,pre-law and all the rest of the master cheaters that form the vast majority of bodies filling the enormous lecture halls of the average 200 student calculus course can program the solutions of all thier exams into thier programmable calculators.
"This is AMERICA. Let the Japanese waste thier time thinking and just give me my f***ing A so I can go out and screw people over for 6 figures a year,geek."
It's also why Stewart would never have become so absurdly wealthy writing a book that is the very pinnacle of mediocrity in any other academic system BUT America's. It's why a piece of crap like CHARMED was on for 7 years while great shows like FARSCAPE vanish, why TRANSFORMERS:THE REVENGE OF THE FALLEN-with a mindless plot and racist "black" Autobots-is the #1 film in America-it's why we sold our blood won freedoms to a stupid evil Texan from a rich family we elected king for the illusion of safety while Americans lost the entire Bill of Rights for 8 years.
"Americans aren't stupid!" Really?You must be living in a different USA then I am.
So it goes.
My favorites? Well,when anyone tells you Spivak's CALCULUS is the best calculus book ever-EVER-it's really hard to argue. It's incredibly beautiful and a model of clarity. But much more then that,with every word,picture and exercise,Spivak asks the reader to THINK about the concepts before him or her before setting the task of doing it. Really THINK about it.
Is it too hard for the average student? Well,depends on what you mean by the average student. The average student cheating thier way through every homework and test and sleeping with TAs to get a 4.0 to get into Harvard medicial school,sure. But if you're talking about the average student-not necessarily a mathematics or natural science student-who reads everything with an effort and wonders and asks real questions even if they don't understand or particuarly like it because they're there to LEARN something-it would be a struggle. But with a good teacher by thier side, they could definitely get through it.
And they'd be all the better for it. For the mathematically talented, the book will become a treasured keepsake for a lifetime.The chapter on infinite series alone is worth photocopying and keeping.
I refuse to recommend soft,"applied" books.To me,the pure/applied mathematics distinction is a symptom of the problem above. There is no pure math or applied math-there is only MATHEMATICS. If you don't realize that,you're not part of the solution,you're part of the problem. That being said-the main problem with using Spivak is that he has virtually no applications-just one lame application of vector algebra to celestial mechanics late in the book. The main point of calculus is to calcul-ATE. Theory is important and all well and good, but teaching calculus as real analysis completely devoid of application is a little like teaching music students the complete mechanics of writing scores and symphonies,but never teaching them how to play!!!!
A book that fascinates me and I'd love to try to use for a basic calculus course one day is Donald Estep's PRACTICAL ANALYSIS IN ONE VARIABLE. Estep,a numerical analyst, teaches a basic real analysis course combined with a basic calculus course, using numerical methods to motivate the rigorous development of the real numbers and epsilon-delta arguements-with DOZENS of actual real-world examples from chemistry and physics!!! I'd be a little scared to use the book,though-Estep makes a couple of really strange choices. The biggest one is deciding NOT TO DISCUSS INFINITE SERIES-TO ESTEP, INFINITE SERIES IS BEST DONE WITH COMPLEX VARIABLES,SO HE DECIDES TO FORGET IT. HUH?!?
My favorite all around calculus book is a nearly forgotten one by a legendary teacher-CALCULUS,2nd edition by Edwin E.Moise-based on the course in calculus that Moise taught for many years at Harvard and won several awards for. It's completely rigorous, yet beautifully intuitive with many,many pictures and geometric insight motivated using Euclidean geometry such as lines,planes and conic sections, as well as many,many physical applications. THIS is the book I would use to teach my children calculus.Go to the library and check it out for yourself if you're disappointed with the ton of fluff the departments are trying to push on you to teach calculus with. You'll thank me later,I promise.
Stewart and his private concert hall.Yet another example we are living in the era of the barbarians at the gate. It's so frustrating-with no address,you can't even drive by and throw a firebomb through his window to burn it down...........LOL

Monday, January 26, 2009

Re:The Vampires Of American Medicine And WTF Does "Well Defined" Mean?!?

So much for posting regularly at this blog.I may as well just shut it down and start again.

But I won't. I WILL keep trying to post on a regular basis for the rest of the summer until the blog catches on. Or it doesn't. A blog is for the author,no one else.Anyone else reads it,that's a plus.

I AM hoping it does catch on,though. I have a lot of thoughts on many things ongoing-but now's not the time. If anything, small posts will begin appearing regularly.

This summer-my last one before applying to PHD programs has not gone well. Sleep has eluded me for the better part of a month-stolen by gut pain combined with frequent urination. And the wonderful health care system of America has assured my internest can't see me.

I don't have a right to live according to the AMA, you see-not enough money to buy good health.

That's why they let my father die of agonizing prostate cancer at the end-they crunched the numbers and thier profits simply outwieghed my dad's treatment. So they gave us the bullshit story that "There's nothing more we can do." The cancer metastisized througout his bones over his last few weeks, giving him a death you wouldn't wish on Bernie Madoff.

Meanwhile, if he was a drug kingpin who dropped off 5 million in CASH,I wonder if a miraculous treatment they suddenly remembered about would have appeared and extended his life by 5-10 years. Since corporations now control the publication of most medical research as well as the mass media, we'd never know if one existed no matter how much you researched.

I can get fully into this here, but I WILL say this: The fact that Yale Medical School considered seriously adding ACTING CLASSES to it's required cirriculia for the M.D. for all students entering after 2011 to "improve maximally productive patient-practitioner interaction"(translation:to make the doctors the best con-artists possible) speaks volumes of the age of medicine we live in-and why I turned my back on that world years ago. I consider myself VERY lucky to have good and trustworthy doctors-but I can't tell you how hard my family searched to find them.
100 monsters for every one like them.

"We're coming for your money and we'll GET it all. We're the only real winners.The players don't stand a chance." -from the screenplay of Martin Scorsese's CASINO

Changing the subject to something mathematical, something on the web caught my eye yesterday and I just need to share it with the house. Ever wonder what well-defined means? It's amazing how many graduate students-particularly those working in category theory and the higher altitudes of algebra,where the phrase probably comes up most-never ask what that means. It's kind of accepted everyone "sorta" knows what it means. And for most people,that's good enough.
I remember the first time I ever wondered about it-it was in Kenneth Kramer's honors abstract algebra course a few years ago at Queens College. He was sketching the proof of Cayley's theorum on the fact that every group is the same as some group of permutations on a set (i.e. they're isomorphic). ( Actually, he wasn't proving it,he just wanted to sketch the proof because he'd rather spend the classes' time developing the theory of group actions on a set, of which Cayley's theorum is a special case-i.e. a group acting on itself. But I digress.............)
He was constructing the composition map which is the isomorphism of a group G onto it's corresponding permutation group acting on it's underlying set S -I forget what he denoted it as,call it P(S). He commented the map was clearly well-defined. I raised my hand in frustration since I'd asked the question before and never gotten a straight answer from any professor (some of them actually got annoyed with it and made unkind remarks about my age as a student)
What followed was one of the most impressionable moments of my student career as Dr.Kramer and I exchanged comments on what exactly it meant to be well-defined. "It means it's not ambiguous what the value assigned is, Andrew-that we don't get 2 values for the same arguement." "Oh, you mean the relation actually specifies a function?" "Well, not exactly-if the formula IS a function, you're absolutely right. But this may not be a function and still be a well defined mathematical object." I didn't get it. After a few minutes of him giving a few examples, no progress was made. He ultimately asked me to table the question so we don't waste any more of the classes' time.
I did so,but ultimately,it disturbed me. Dr.Kramer is a gifted teacher on all matters mathematical-an early student of John Tate's at Harvard-and usually the most pleasant and patient of people with even the stupidest of students' questions. In fact,I'll be taking a course on elliptic functions with him at the City University Of New York Graduate Center this fall. The sheer wieght of the subsequent coursework-the first 6 chapters of Herstein's classic Topics In Algebra in a VERY intensive 2 semesters,plus his own notes-prevented us from broaching the subject further. All that really got settled was that it was pretty clear what "well-defined" meant if the object under consideration was a function-in fact, it's almost redundant. But how would you describle a general mathematical object as being "well defined"?

Leave it to Tom Gowers to make everybody happy.

There are several blogs online I try so hard not to miss. Peter Woit's Not Even Wrong, Terrance Tao's, John Baez's The N-Category Cafe' , The Secret Blogging Seminar and a few others. But nothing matches Gower's blog for sheer beauty of writing and thinking about mathematics. A lot of people can do mathematics, a lot more people can teach mathematics, and even more people can talk about mathematics .(Sadly, this is whether or not they know what the fuck they're talking about or not...........)

There are so few who can do all of the above.

Elias Stien can do it (sometimes).

Melvyn Nathanson can do it.

James Stasheff can do it. Better then anyone I've ever heard.

William Thurston can do it.

But for my money, no one does it better currently and consistently then Tom Gowers. His blog should be required reading for all mathematicans and serious math students. (By the way-his old teacher at Cambridge, Bela Bollabos-is also great at all of the above. I doubt that's an accident. )
Anywho, I was reading Gowers' blog and low and behold, Gowers also wanted to know, after grading the exams for the year at Cambridge and discovering NONE of his students understood it,either-what's it mean for something to be well defined?

People who know me know I'm Socratic to a fault, to the point of making people violent. I almost NEVER agree with EVERYTHING someone says.

But this is rare occasion when I'm speechless with complete conviction and agreement with someone else's analysis. As I said, leave it to Gowers to give the perfect answer to a great question.

I'll simply let the beauty,depth and simplicity of Gowers' blogpost speak for itself-I simply have nothing to add to it. Nothing at all. Anyone asks me this question in the future, I'll simply give them a copy of Gowers' post. For all basic mathematical discussions that may come up in the future, I seriously doubt anyone can debunk this discussion.

It's THAT good.

Oh,screw the self-engratiating pontification,here's Gowers. And if you don't bookmark his blog, shame on you.

Good night to all,fellow travelers. Until next time.