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Mathematical Fallacies and Paradoxes (Dover Books on Mathematics) Paperback – July 1, 1997
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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.
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.
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.