Two More Books

Just mentioning a couple of volumes today for your consideration (both currently available in paperback, BTW)....

Won't have time to write a full review, but will recommend Alfred Posamentier's (with Bernd Thaller) latest book, Numbers: Their Tales, Types, and Treasures."  It's a typical Posamentier offering... clearly written and organized, without oversimplifying or dumbing-down the wide-ranging material; some interesting and fun things mixed in with historical and classic material.
I enjoyed the second half of the book, and especially the last few chapters, more than the first half with its focus on history (for whatever reason, math history, pre-1800 has never much held my interest). Again it is a great refresher for some adult math enthusiasts and an especially good read for the young person already inclined toward mathematics.
The final chapter of the book focuses on foundations and philosophy of mathematics, although near the end the authors seem to favorably quote Steven Weinberg's portrayal of philosophy as something that does not "provide... any useful guidance" to scientists. They even cite another philosopher as saying that philosophy is a "waste of time... from the point of view of the working mathematician." (Most of the book is definitely more mathematical than philosophical.)

Anyway, if you want to get a greater sense of the volume's overall content, here's a longer review from the Web:
http://tinyurl.com/o97tzvr

    A book I'm currently perusing (haven't finished, but willing to recommend) is "Professor Povey's

Perplexing Problems" by Thomas Povey, who will be familiar to many from his "Perplexing Problems" website:
http://perplexingproblems.com/

The book is a nice compendium of amusing problems with varying difficulty;  a little more challenging mix than often found in standard puzzlebooks. However, only about a quarter of the problems are strictly mathematical. The remainder are more physics-related (though often, of course, still requiring math), so for someone with little interest in physics this may not be a good book choice. Luckily, most math fans probably enjoy physics as well, and young, budding physicists should definitely enjoy.

In the last chapter Povey tells the story of Larry Walters who took flight from his backyard in 1995, in a lawnchair powered by helium balloons, just one of the typically entertaining segments of the book. You can read more about Larry here if you like:
http://www.markbarry.com/lawnchairman.html

You can even view news of Larry's oddball flight on YouTube here:
https://youtu.be/CFFVVo9usFY?t=33s 

And here's a couple of lines that cracked me up for some reason, from chapter 2 of the volume:

"Over dinner once I was told what I believe is a true story about the principal of a Cambridge college taking out a calculator to multiply a number by 100.  In a rare moment of lucidity I quipped, 'was it a difficult number that was being multiplied?' Only the scientists got the joke."

Anyway, you get the idea, in-between the puzzles are some entertaining bits.
To finish out, I'll adapt one of the simpler math problems from the volume for inclusion here (I'm telling it less entertainingly than Povey's rendition):

Captain Fishmonger goes on a treasure-hunting voyage.  He arrives at the deserted island for which he has a treasure map showing just two trees and the instructions, "Walk 50 paces from one tree AND also 50 from the other. There lies the treasure."
But as Fishmonger peruses the island he finds that in the time since the treasure was buried and the map drawn, now 14 more trees have grown up. There are now 16 trees all less than 100 paces from one another.
In the WORST CASE scenario, what is the MAXIMUM number of spots the Captain will need to dig to find his treasure?
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. answer:  240  ...drawing 50-pace circles around any two tree-pairs gives you TWO possible digging (intersecting) points, and combinatorics can be used to calculate the total number of possible tree pairings (120).  2 x 120 = 240 as maximum number of digs required.

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:

http://wildaboutmath.com/2012/12/16/alfred-posamentier-inspired-by-math-13/