...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Monday, August 19, 2013

New Math Book, Full of Mistakes

…and delightfully so!!

-- Review of "Magnificent Mistakes In Mathematics" -- by Alfred Posamentier and Ingmar Lehmann

Any volume from Alfred Posamentier is to be looked forward to, and the latest one, "Magnificent Mistakes In Mathematics" is no exception (written again with Ingmar Lehmann); a fairly quick and entertaining read for typical math buffs, with a focus not often found in math books… on famous math errors.

The book endeavors to demonstrate that even the precise, empirical field of mathematics has its share of mistakes made by prominent, knowledgeable practitioners in the road to progress. And they note that the very need to examine and explain math 'flaws' is a good thing, often leading to whole new ideas/concepts.

The book starts right off putting the reader at ease by highlighting "noteworthy mistakes by famous mathematicians," including such accomplished figures as Pythagorus, Galileo, Fermat, Leibniz, Euler, Poincare, and several others. It's as if to say 'if the remarkable Euler blundered why should YOU dread making math mistakes.'  Many of these errors are well-known, but still interesting, or in some instances even humorous (for example a blackboard mistake by Enrico Fermi that ended up saved for posterity on an Italian postage stamp). Interestingly, at the end of the chapter it is mentioned that the also extraordinary Carl Friedrich Gauss was not known to have made mistakes in his published material.

Chapter 2 embarks on the progressive journey through mathematics with a look at mistakes in arithmetic. This may be the least interesting, or most mundane of the chapters, and is followed by chapters that delve, in order, into algebra, geometry, and probability and statistics errors; a seeming natural progression from the more abstract to the more applied or real-life-type examples. 

Geometry mistakes, of course are often a matter of misperception or interpretation (moreso perhaps than algebra, where mis-computation may be more frequent). In the geometry realm I was very surprised that the volume leaves out one of my very favorite 'mistakes,' which goes around the internet from time-to-time, demonstrating that pi=4: http://www.bestwtf.com/2010/11/explaining-why-pi-is-4.html The sort of error involved, "misleading limits," is documented in the book with other classic examples, but still, the circle-inscribed-in-a-square paradox is so good it ought not be missed.

Of course probability and statistics are perhaps the cause of more slippery mathematical mistakes than any other area, even among professional mathematicians. It is often famously told that even the great Paul Erdös initially had difficulty with the 'Monty Hall problem,' so it is a good chapter to end the book with. Many deceptive probability conundrums of recent times are now pretty much classic, and continue to elicit great debate when heard for the first time.

There are LOTS of types of examples used through the book, demonstrating how varied the sources of math mistakes can be. Having said that, there is also sometimes redundancy to the many examples employed for any one sort of error; but using multiple examples to make a point is not necessarily a bad thing.

For the professional or broadly-read mathematician this won't likely be a highly substantive or weighty math read. The bulk of examples in the volume are well-known, but what is new is bringing them altogether under one cover-to-cover format… an entire book focused contrarily not on what math does right, but on where it may go wrong. I think this somewhat unique approach makes the volume a worthwhile, entertaining addition to one's math bookshelf, and it may be particularly useful to secondary school teachers, providing a lot of grist for instructive, thoughtful examples in the classroom. As the authors repeatedly note, there is a LOT to learn from mathematical mistakes.

In short, a thumbs-up for this volume! Posamentier seems to produce a new book almost every year, and each one simply leaves me wondering what will he come up with next.

By the way, if you missed it, Posamentier did a great "Inspired By Math" podcast interview with Sol Lederman late last year here:


No comments:

Post a Comment