Informed, measured, and warranted skepticism, Sep 16 2003

The book is organized like a dictionary with an alphabetical listing of various words that Carroll sets out to explore in depth. I think the book is better described as an encyclopedia, however, because of the length and style of the articles, which are not terse definitions, but mini-essays. Here is a sampling of the "A" words to give you an idea of the range of topics that Carroll addresses: acupuncture, agnosticism, alien abductions, ancient astronauts, angel therapy, anthroposophy, argument from design, argument to ignorance, aromatherapy, astral projection, astrology, atheism, automatic writing, and avatar. (This is roughly one third of the entries under "A"). Even within this short list there are some whacky ideas (angel therapy and alien abductions), some borderline ideas (acupuncture and anthroposophy), and some words that are simply in want of a careful definition (agnosticism, atheism, and avatar). Carroll deals with them all rather even-handedly, at least from the perspective of a naturalistic worldview. Other topics covered in the book include Bible codes, Bigfoot, chiropractic, confirmation bias, crystal power, ESP, holistic medicine, karma, levitation, magnet therapy, miracles, Noah's Ark, etc. I think Carroll did a rather good job in selecting his topics as they cover such a panoply of beliefs; he is just as likely to find fault with one cult as any other.

With respect to Carroll's intentions, as he states in the introduction, "this book is a Davidian counterbalance to the Goliath of occult literature. I hope that an occasional missile hits its mark." Thus, Carroll apparently intends to instill a bit of healthy skepticism into those minds willing to accept it. And who might that be? Carroll identifies his intended audience as those uncommitted to occult claims (open-minded seekers), those who believe in them but have doubts (believing doubters), those who are more prone toward doubt than belief (soft-skeptics), and those who strongly disbelieve in occult ideas (hardened-skeptics). But, "The one group this book is not aimed at is the 'true believer' in the occult. If you have no skepticism in you, this book is not for you." I suspect that Carroll is quite right in his assessment; if you are an ardent believer in any of the cultish ideas that Carroll debunks, then you are unlikely to find his arguments compelling; The reason for that, of course, remains open to debate.

Here are a few short snippets from Carroll's entries. Under "acupuncture," Carroll first describes the history of the technique, its variants, and the types of claims made for it. His brief analysis suggests that there is little reason to believe that the anecdotal successes of the technique amount to anything more than regression toward the mean. In Carroll's words, "An alternative treatment such as acupuncture is sought only when the pain is near its most severe level. Natural regression will lead to the pain becoming less once it has reached its maximum level of severity." Under "agnostic," Carroll carefully defines the often-misunderstood word, explaining that "The agnostic holds that human knowledge is limited to the natural world, that the mind is incapable of knowledge of the supernatural. Understood this way, an agnostic could be either a theist or an atheist." Under "numerology," Carroll explores the idea of ascertaining a person's characteristics from numerical data based on name and birth date, and exposes it as a total sham. He links the perceived success of numerology to the "Forer effect", which he defines in another entry as "The tendency to accept vague and general personality descriptions as uniquely applicable to oneself without realizing that the same description could be applied to just about anyone."

This is a fun book to read, both because of its wide range of topics, and because of Carroll's no-nonsense pragmatic approach. I hope that the book will cajole at least a small minority of readers into critically examining some of the outlandish beliefs that surround us, and to wisely insist on something more than anecdotal evidence, wishful thinking, or arguments from ignorance before accepting them.

Got Matrices?, Aug 1 2003

For each important algorithm discussed, the authors provide a concise and rigorous mathematical development followed by crystal clear pseudo-code. The pseudo-code has a Pascal-like syntax, but with embedded Matlab abbreviations that make common low-level matrix operations extremely easy to express. The authors also use indentation rather than tedious BEGIN-END notation, another convention that makes the pseudo-code crisp and easy to understand. I have found it quite easy to code up various algorithms from the pseudo-code descriptions given in this book. The authors cover most of the traditional topics such as Gaussian elimination, matrix factorizations (LU, QR, and SVD), eigenvalue problems (symmetric and unsymmetric), iterative methods, Lanczos method, othogonalization and least squares (both constrained and unconstrained), as well as basic linear algebra and error analysis.

I've use this book extensively during the past ten years. It's an invaluable resource for teaching numerical analysis (which invariably includes matrix computations), and for virtually any research that involves computational linear algebra. If you've got matrices, chances are you will appreciate having this book around.

Hang on to your right hemisphere!, Jul 24 2003

In contrast, the right hemisphere has been something of an enigma, and is consequently called the 'minor' hemisphere. But, "it is the right hemisphere which controls the crucial powers of recognizing reality which every living creature must have in order to survive." For example, the right hemisphere is responsible for "proprioception", which allows us to feel our bodies as "proper to us"; that they belong to us. This is so basic that it is difficult to even imagine what it would be like to have impaired proprioception. Sacks is keenly aware of this challenge; in a sense, the entire book is an attempt to give us a glimpse into such an incomprehensible world.

Sacks quotes Wittgenstein:, "The aspects of things that are most important for us are hidden because of their simplicity and familiarity. (One is unable to notice something because it is always before one's eyes.)" Those things that are most basic, most obvious, have a deeply mysterious foundation in the brain. One can begin to appreciate this when one considers those unfortunate individuals who have lost some of these basic perceptions due to injury or illness. As Sacks points out in the introduction, "It is not only difficult, it is impossible, for patients with certain right-hemisphere syndromes to know their own problems... And it is singularly difficult, even for the most sensitive observer, to picture the inner state, the 'situation', of such patients, for this is almost unimaginably remote from anything he himself has ever known."

Sacks presents detailed and compassionate accounts of numerous patients whose worlds are indeed unimaginably remote from our own. He tells us of patients who have difficulty distinguishing between people and inanimate objects, those who have perfect "vision" yet cannot discern the purpose of an object without tactile feedback, those who fail to recognize their own limbs as belonging to them, and those who have lost fundamental spatial concepts, such as the distinction between left and right. One of the most intriguing cases that Sacks presents is that of a woman who had "totally lost the idea of 'left', both with regard to the world and her own body," a condition known as hemi-inattention. To this woman, everything in her left visual field simply ceased to exist, in analogy to the way each of us fills the blind spots in our visual field. This unfortunate woman would eat half her lunch (that on the right side of her tray) and was incapable of turning to the left (since left did not exist) to discover what remained. In time, she learned to turn herself around, always to the right, until she found the rest of her lunch.

This book is not only engrossing, it is challenging; it forces one to acknowledge that what we take as so plainly obvious about the world is intimately tied to basic brain function. Oliver Sacks demonstrates beautifully that the brain is still deeply mysterious, particularly in how it creates our sense of reality. There are profound implications here for those interested in psychology and philosophy. It's a great read.

A classic text on the theory of computation., Jul 24 2003

The book begins with a brief introduction to the relevant discrete mathematics (such as set theory and cardinality) and proof techniques, then introduces the concepts of finite automata, regular expressions, and regular languages, describing their interrelationships. It proceeds to context-free languages, pushdown automata, parse trees, pumping lemmas, Turing machines, undecidability, computational complexity, and the theory of NP-completeness. (These are all standard topics.) Along the way, Lewis and Papadimitriou also introduce random access Turing machines and recursive functions, and do a nice job of explaining the halting problem and how this translates into undecidable problems involving grammars, various questions about Turing machines, and even two-dimensional tiling problems. All of these topics are covered with an appropriate mix of formalism and intuition.

Perhaps the feature I like best is the discussion of composing simple Turing machines to obtain more complex and interesting machines. The authors even introduce a convenient graphical notation for combining Turing machines and spell out specific rules for composition. While most authors are forced to immediately employ heuristics in reasoning about complex Turing machines (lest the notation become overwhelming), Lewis and Papadimitriou are able to keep the discussion more formal and structured by virtue of their Turing machine "schema". I believe this makes their arguments more rigorous and even easier to follow.

This is clearly one of the best books on the theory of computation. However, be aware that there have been very significant changes from the first edition, which was lengthier and more thorough. I confess that I actually prefer the first edition, as it contains nice sections on logic and predicate calculus (which have been removed from the 2nd edition), and is a bit more formal (albeit with some fairly awful notation). The 2nd edition is definitely crisper, with much cleaner notation; it is clearly more student-friendly, which was presumably the aim of the new edition.

If you wish to teach an introduction to theoretical computer science, or wish to learn it on your own, this would be a fine book to use. It's hard to go wrong with this classic.

Witty + Concise + Useful = Excellent, Jul 24 2003

This book introduces the idea of hypergeometric function, the Swiss army knife of combinatorial mathematics, and proceeds to develop algorithms for their computation as well as numerous applications. The authors also reveal what, exactly, computers can help us to decide, what is a "closed form" solution, what are "canonical" and "normal" forms, and inject relevant philosophical digressions that keep the discussions lively and entertaining. The authors also present snippets of "Mathematica" code so that you can try out many of the basic operations yourself.

The book has concise chapters on Sister Celine's method, Gosper's algorithm, Zeilberger's algorithm, and the WZ algorithm, with sufficient detail that you will likely be able to apply the algorithms yourself (perhaps by downloading the Mathematica packages that the authors point you to). The techniques are invaluable in proving identities in combinatorial mathematics; that is, identities involving binomial coefficients, factorials, rational functions, etc. By means of such techniques, computers "not only find proofs of known identities, they also find completely new identities. Lots of them. Some very pretty. Some not so pretty but very useful. Some neither pretty nor useful, in which case we humans can ignore them."

This is a well-written and highly accessible book about an important (albeit very narrow and specialized) branch of mathematics. If you have little experience with hypergeometric functions, yet deal with combinatorial mathematics, you will likely read this book in one (long) sitting; and you'll be glad you did.

epsilon math humor, where epsilon > 0., Jul 22 2003

1) Here is a typical amusing anecdote: "G. H. Hardy was about to return from Denmark to England, by boat, in appalling weather. So he sent a postcard ahead to announce to the world that 'I have proved Riemann's Hypothesis', which was then as now the Holy Grail of professional mathematicians. Hardy reasoned that God (in whom Hardy did not profess to believe) would not allow the boat to sink, thereby leaving open the suspicion that Hardy had achieved this remarkable feat."

2) Here is a limerick by Paul Halmos, a famed contemporary mathematician:
"If you think that your paper is vacuous,
Use the first-order functional calculus,
It then becomes logic,
And, as if by magic,
The obvious is hailed as miraculous."

3) Here is an example of a puzzle with some actual mathematical content: "A medical researcher does a carefully controlled experiment whose result is that new medicine X is more effective on male patients than a placebo. The experiment is then repeated on female subjects, with the same result... The data from the two experiments are then added together, and they prove that overall the medicine is LESS effective then the placebo. Is this possible? Yes." [explanation provided in the back of the book]

While I like the idea of this book, and found a few entries that were really funny or interesting, over all I thought the collection was mediocre. It can be fun, if you are in to math, but don't expect too much. It's very light stuff.

Great for the coffee table, Jul 22 2003

It turns out that questions of the form "Does A always imply B?" entail proofs with two very different flavors, depending on whether the answer is affirmative or negative. The affirmative variety can be very difficult, as it usually deals with an infinity of things. But a negative answer requires only one solitary example of an A that is not a B; this is affectionately known as a "counter-example". These are the slickest little proofs around--often a one liner--and they can provide a lot of insight. Here's a trickier one: Are all linear functions continuous? Surprisingly, the answer is "no", which means there is a counter-example. Gelbaum and Olmsted show how to construct a discontinuous linear function. Case closed. They also provide examples of

A perfect nowhere dense set

A linear function space that is a lattice but not an algebra

A connected compact set that is not an arc

A divergent series whose general term approaches zero

A nonuniform limit of bounded functions that is not bounded

I won't give away any more (although there are hundreds). The book has chapters on real numbers, functions and limits, differentiation, sequences, infinite series, set and measure on the real axis, functions of two variables, metric and topological spaces, and more. Each section begins with a brief summary of the basic concepts and definitions, then launches into a list of terse counter-examples. This is simply indispensable for students of mathematical analysis, as it can help to explain why you cannot weaken those seemingly stringent hypotheses to various theorems; if you do, one of these quirky counter-examples will rush in and ruin your day. This is a great book to have on hand. I highly recommend it. (I won't tell you how it ends.)

A must for engineering and science students., Jul 15 2003

This is about as tame a book on vector calculus as you could ever hope to meet, which is part of the reason it's been so popular for so long. It's very easy to read (as far as math texts go), it has many simple but effective illustrations, it has ample exercises (most of which have solutions in the back), and it avoids excessive formalism, instead focusing on the nuts-and-bolts of vector calculus as it most commonly arises in electrostatics, for example.

Math majors will not be so enamored of this book, simply because of its heuristic approach (hence the word "informal" in the subtitle) and its close ties with applications, which it uses as motivation. Moreover, Schey does not develop differential forms or exterior calculus, which logically subsume and extend the material in this book (at the expense of far greater abstraction, which the majority of engineering and science students will prefer to avoid or at least delay). Instructors, if you teach electrostatics or fluid dynamics, you may wish to consider having this as a supplementary text for your students. It's such a clear and helpful little book your students will really appreciate it. (But, you already knew that.)

Bottom line for engineering and science students: You need to know this material, and it simply won't get any easier than this. Don't wait for the audio edition!

The consequences of numerical ineptitude, Jul 14 2003

But why is it a problem? What does it matter if people get a little confused over numbers and probabilities? Well, just consider how many important decisions we are faced with, almost on a daily basis, that involve numbers and probabilities. Nearly all political discussions these days involve mention of millions, billions, or even trillions of dollars, so it would be prudent to have some familiarity with those concepts. Nearly all discussions of product safety and drug efficacy involve the results or studies, which require some understanding of probability and statistics. Yet another reason to get a grip on numerical reasoning, Paulos claims, is that innumeracy is linked with belief in pseudoscience and other outright nonsense. As Paulos points out, "...a significant portion of our adult population still believes in Tarot cards, channeling mediums, and crystal power." With a bit more quantitative savvy, these ruses would not hold sway.

Paulos writes in a very engaging style, although he admits at the outset that "Because the book is largely concerned with various inadequacies--a lack of numerical perspective, an exaggerated appreciation for meaningless coincidence, a credulous acceptance of pseudosciences, an inability to recognize social trade-offs, and so on--much of the writing has a debunking flavor to it." Yet, Paulos's method of debunking is friendly and even comical. He typically uses social scenarios to make his points. For example, he relates the following story: "...we were watching the news, and the TV weathercaster announced that there was a 50 percent chance of rain for Saturday and a 50 percent chance of rain for Sunday, and concluded that there was therefore a 100 percent chance of rain that weekend." Paulos notes that he had to explain the error to others who were listening to the weathercaster; and even then, one of the listeners "wasn't nearly as indignant as he would have been had the weathercaster left a dangling participle." Given the generally sad state of literacy in this country, anecdotes like this should make one wonder just how deplorable the state of numeracy is.

If you are looking for a friendly and readable book that will help you to make sense of the ever-increasing use of numerical data and probabilistic thinking that is appearing in public discussions of all forms, I recommend this book very highly. If you are yourself a "numerate" person (perhaps even an instructor of mathematics), it behooves you to understand just how ill-prepared the majority of our population is to deal with such reasoning; you need to know what you are up against. It will shock you.

A concise catalog of theorems and conjectures., Jul 14 2003

To give you an idea of the level of the discussion in the text, here is an excerpt from page 1: After a terse definition of vertex coloring and "chromatic number", the authors state that "The existence of the chromatic number follows from the Well-Ordering Theorem of set theory... However, even if it is not assumed that every set has a well-ordering, but maintaining the property that every set has a cardinality, then the statement 'Any finite or infinite graph has a chromatic number' is equivalent to the Axiom of Choice...". If you are unfamiliar with concepts such as well-ordering or the axiom of choice, such a discussion will be of little value to you. However, if you are familiar with these ideas, you will appreciate how quickly the authors jump into meaty discussions. As another example, the chapter on planar graphs begins with a number of excellent questions: "Does there exist a short proof of the four-color theorem...?", "Is there a short argument to demonstrate that the four-color problem is a finite problem?", and "Is there a short argument that proves the existence of a polynomial algorithm to decide if a given planar graph if 4-colorable?", to mention three. The chapter then proceeds to discuss what is currently know about these problems, and many others.

I suspect that the book would be of great interest not only to mathematicians but also computer scientists, as there are numerous discussions/problems on computational complexity. For example, "Does there exist a function g and a polynomial algorithm that for any given input graph G will find a number s, such that the chromatic number of G satisfies s <= X(G) <= g(s)?" (Here X(G) is the chromatic number of graph G.) The authors state that the question was answered affirmatively by Alon in 1993 if X(G) is replaced by "list-chromatic number". This is typical of the problems cataloged in this book: a terse but formally correct statement of a problem followed by what is currently know, with full citations.

The bibliography at the end of each section is extensive, if not daunting, so there should be little problem looking up all relevant literature concerning a given problem. The authors have also set up an on-line archive for up-to-the-minute research results on these problems. This is an excellent reference for those who are interested in serious research in graph coloring. I would not recommend it to undergraduates in computer science or mathematics, nor to those seeking accessible discussions of classic graph algorithms; this is not an introductory text.