Sunday, June 7, 2015

Two Books on Fermat’s Last Theorem

By the fifth grade, kids learn enough arithmetic and geometry to appreciate the challenge in Fermat’s Last Theorem.  FLT stood for about 350 years as an unsolved problem. Then, on June 23, 1993, Andrew Wiles concluded a three-day lecture by proving it – almost. Closer inspection revealed some difficulties, and Wiles sat down again with a cup of pencils and a ream of white paper.  He also brought on Richard Taylor, a former student who had been one of the judges who condemned Wiles’ initial demonstration. On October 24, 1994, they announced the publication of two papers that achieved the proof. 

Much has been published about FLT. Just search Amazon, limited to Books, for “Fermat’s.”  Easily the two most popular works are Fermat’s Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem by Amir D. Aczel (New York and London: Four Walls Eight Windows, 1996) and Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem by Simon Singh (New York: Walker, 1997; Anchor Doubleday, 1998).   I found the Aczel book to be contaminated with errors.  I found no mistakes in Singh’s work.

Aczel earned his BA in mathematics at UC Berkeley in 1975, and finished his MS the next  year.  He then was awarded a doctorate by the University of Oregon for research in statistics. In addition to this book, he authored 17 others, plus two textbooks.  (See  So, he was fully competent to have known better.
Close up of upper right of Pierre Fermat's face with many numbers running back and forth
A book with more problems than
the publisher intended.

The first mistake is on page 6. The illustration is supposedly a reproduction of Fermat’s marginal note in a bilingual (Latin-Greek) edition of Arithmetica by Diophantus. The Greek is medieval rendering in upper and lower case with one emendation in [square brackets]. But the note is not there. In fact, no such note exists. (Simon Singh's book also offers a version of the page, but, again, no note.)  The claim was made by Pierre Fermat's son and editor, Charles.

To give the reader a feel for the life and times of Pythagoras, Aczel takes you around the Mediterranean. However, he cites places such as Alexandria and the Colossus of Rhodes that came two hundred years later.  Pythagoras might also never have seen the Temple of Artemis at Ephesus. Having been destroyed in the 7th century BCE, it was rebuilt, only started in 550 BCE (perhaps by Croesus), when Pythagoras was 20.  

Aczel confuses modern Italy with its ancient character.  “In the barren stark surroundings of the tip of Italy...” (pg. 17).  Greeks colonized Italy to relieve overcrowding and political frictions in their home cities. They would not have chosen barren countryside. Croton, Taras, Sybaris, and the other towns were in green and fertile lands back then.

Leaving the ancient world for the substantive subject, Aczel claims (pg. 37): “The Fibonacci sequence appears everywhere in nature.”  Maybe it really is in the Stefan-Boltzmann Law and I just do not see it.  

Aczel falls into sophomoric humor when he writes “It is believed…” (pg. 27) and “The cossists were considered…” (pg. 42) “…our normal three-dimensional world…” (pg. 49; a nice pun, as those three of our four common dimensions are at right angles–normal–to each other).  “Mordell was unable to prove his discovery, and it became known as Mordell’s conjecture.” (pg. 87) The last is interesting because Aczel denigrates Denis Diderot for “knowing no mathematics” (pg. 48) though clearly Aczel seems deficient in philosophy if he believes that lack of proof leaves us with the Lock Ness Conjecture.  See the Addendum below for similar humor.

Pierre Fermat's face framed in a triangle, inside a circle, framed by text in rectangles
No complaints
Aczel writes past significant steps in his story.  He never stops to explain Galois theory or Galois representations (pg. 72, pg. 120). He never says what a Zeta function is (pp. 99-100).  The Horizontal Iwasawa Theory (pg. 121, pg. 132) may be very hard to explain briefly, even for an accomplished writer of popular mathematics books, but he gives no context whatsoever; he just drops the name and moves on.

Some of the problems with this book can be blamed on the editors.  I am not always sure about that versus which (pg. 121).  However, I am pretty sure that if Leonhard Euler went to St. Petersburg in 1827 (pg. 47), then he was 120 years old. 

Fermat's Last Theorem is one of very many topics in mathematics that could entertain and motivate school children. (For others, see The Man Who Loved Only Numbers reviewed here on this blog.) Appreciating the story of the Taylor-Wiles Proof may require more than an eighth grade education.  But you do not have to do anything special to "make mathematics fun."  It either is or it is not.  However, an educational system that rewards itself for the outcomes of statewide standardized tests will have much less room for the fun in mathematics

"It has long been known" ... I didn't look up the original reference.
"Three of the samples were chosen for detailed study" ... The other results didn't make any sense.
"In my experience" ... Once.


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