Customer Reviews

Mathematical Fallacies and Paradoxes

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Being a physicist who has always had an interest in math, I've found this book a good way to learn about aspects of mathematics usually saved for metamathematical philosophy courses. The book is not a textbook but it has enough depth for you to understand why paradoxes like Godel's, Grelling's and Russell arise and what their implications are. Each chapter gets more interesting and more complicated than the previous. Even though the book does not have a completely polished feel, I recommend it to any recreational mathematician.

This is a great book for people who love mathematics, including: recreational math enthusiasts, math teachers, professors and other university level math instructors, curious and self-motivated students, etc. This book provides numerous examples of how seemingly logical steps can lead to mathematically fallacious results. The level of math ranges from advanced high school to college level math, but the level is not really important ... what is important is the insights one can get from looking at common mathematical mistakes.

This book may also be of interest to neuroscientists, cognitive scientists, and psychologists who are interested in how human beings learn and apply mathematics. On a somewhat related note, I have noticed that (for some strange reason) this book has attracted a set of rather bizarre reviewers (see below). Please ignore them and buy this inexpensive and insightful book on math.

This is a great informal treatment of some of the more notable paradoxes and fallacies of mathematics and mathematical reasoning, old and new. Bunch's prose style is clear and unencumbered and his presentation of each topic - from his easily resolved fallacies and paradoxes of basic algebra and geometry to the deeper and unresolved paradoxes of logic and analysis - is always clean, well-illustrated and engaging.

At a glance, he treats:
The Liar paradox and Godel's Incompleteness theorems
Zeno's and the Sorites paradoxes and the conceptual difficulties associated with the continuum
The existence of irrational magnitudes and some basic philosophical issues associated with existence proofs
The Petersburg paradox
The paradoxes of Infinity and the Formalist and Intuitionist responses to them
The set theoretic paradoxes of Cantor, Russell, and Burali-Forti
The paradoxes of the axiom of choice including the Cantor diagonilisation, Skolem, Hausdorff and Tarski-Banach parodoxes

and a range of thought experiments which highlight the difficulties that may be asociated with applying abstract reasoning to the real world - notably those of the Thompson lamp experiment and Tarski-Banach golden sphere manufacturing plant.

If you want a good popular treatment of the subject matter with a detailed and informal emphasis on the key themes mathematical logic, then this is the book for you. The informal description Godel's first Incompleteness theorem is excellent, as is the discussion of the paradoxes of self reference as they appear in set theory and logic. As such, I would recommend it as excellent recreational reading for anyone with a budding interest in mathematical logic, whether they be math graduates or high-school students.

Most causual users of math consider it to be the most unassailable of endeavors.

After, 2 + 2 always has to equal 4 doesn't it?

It turns out that at the periphery of math there are certain inconsistencies that can arise either owing to the use of faulty methods in arriving at a conclusion (what Bunch calls "fallacies") or inconsistencies owing to the limits of math itself (what Bunch calls "paradoxes").

Though one would need recourse to the book itself in order to completely understand what Bunch means by each category, what follows are a couple of examples to help illustrate the kinds of issues this book will treat.

In relation to fallacies, an early example used by Bunch is Aristotle's paradox wherein Aristotle tried to use a deceptively simple experiment to measure the perimeter of two circles. For ease of convenience, let's say he used two coins of different denominations...say a dime and a half dollar.

Obviously, the coins by their size have to have different measures of distance around their perimeters. And yet, according to Aristotle's experiment, they turn out equally. They turn out equally because Aristotle simply placed one on top of the other and rolled them to see which would make a complete turn the earliest. As you may have gleaned they both turned at the same time owing to the particular mathematics of circles.

Bunch's point is that by applying incorrect reasoning Aristotle's "paradoxical" result was simply a fallacy.

In terms of true paradoxes, Bunch discussed Kurt Godel's incompleteness theorem which says that any consistent system will produce so called "formally undecideable propositions." In other words, to the extent that a consistent system produces self referential statements, those statements can defy formal proof.

An oft used English language example is "This sentence is false." Obviously, the sentence is neither be bracketed with all true statements or all false statements owing to its category defying nature.

In turns out that Kurt Godel was able to stand over two millenia of math philosophy on its head by showing that math had its logically limits of proof.

As can be seen from the previous examples, this book is thought provoking even for casual readers who admittedly will have to struggle cracking the hard nutshell of some its more dense arguments. However, those who do so will be richly rewarded for the heightened understanding of the limits of math they have thereby gained in the process.

If you've never encountered the problems in MFaP before, you're sure to find MFaP a decent and generally easygoing introduction to the subject matter. If you have encountered them before, you're sure to find little new between the covers of this slim volume. MFaP is an able and by-the-numbers overview of an exceedingly complex and fascinating topic. Should you read to the end of it, I'd highly recommend having a look at the brief bibliography Bunch assembled. Not all of it represents 2008 state-of-the-art, but there are several outstanding titles on this list to whet the appetite for further study. Read MFaP to taste some choice bits. But be sure to read elsewhere if you it's a feast you're after.

I would recommend this book to anyone interested in Mathematics. The fallacies are interesting, including the author's. For example, on page 94 regarding Oscar Wilde's epigram : "The only way to get rid of temptation is to yield to it". Mr. Bunch suggests this to be a fallacy due to the key word "only", and offers an example such as suicide to show "only" to be invalid. But would not suicide be a temptation as well? Or for that manner, anything one would try?

It is the paradoxes that keep us honest in mathematics. Tarski with Banach found a basic flaw in the axiom of choice in set theory. Zeno has puzzled children for two thousand years... Time travel paradoxes are the modern "new" problem of tacyonic loops and the Hawking conjecture. Without examples of critical thinking doctrine rules and men become fools!

My 8 year old niece loves her schoolteacher but as she has a bit of devilment in her ... yesterday she ask the school m'arm, "what is 1 divided by 1 - 1"? And where did the little sweetheart get that lovely bit of nothingness? Seems her uncle was reading this book ... : Hey, this is a swell book fun to read with calculator at hand and niece [with her dog ] at knee.