Mathematical mountaintops. The five most famous problems of all time (Q2753574)

According to the book jacket, in this book ``John Casti brilliantly recreates the solutions to the five greatest mathematical problems of all time: The Four-Color Map Problem, Fermat's Last Theorem, The Continuum Hypothesis, Kepler's Conjecture, and Hilbert's Tenth Problem.'' NEWLINENEWLINENEWLINEUnfortunately, the attempt fails badly. In the facts, the book is unreliable; mistakes ranging from typos to major abound. The book is sloppily and carelessly written. Thus it gets the mathematical facts wrong, it gets the history wrong, and then spreads major misunderstandings about the nature and the problems of the proofs and solutions that are discussed. NEWLINENEWLINENEWLINEAs for the mathematical facts, \textit{for example}, in the chapter on Hilbert's tenth problem the discussion of solvability of Diophantine equations starts with the wrong direction of the reduction to solvability in non-negative integers. (The usual proof of the other direction, as needed here, invokes the famous ``four squares theorem'' and would have been nice to explain!) On p.~117 for the definition of a lattice the reader is asked to envision ``an infinite checkerboard'' and construct a sphere packing from it; only the cubic lattice fits that description. Two pages later we learn that the author also has not understood the connection between coding theory and sphere packings. On p.~124 ``These edges divide space into simplices, called Delaunay simplices.'' Also, the author seems to believe that the nature of hyperbolic geometry is that it is four-dimensional (p.~150). NEWLINENEWLINENEWLINEAs for historical facts, \textit{for example}, in respect to Kepler's conjecture he gets Hsiang's role completely wrong, when claiming that in his proof ``after more than a hundred pages of esoteric geometry, cracks started to emerge'' (p.~124). On p.~151, Casti talks about ``the elliptic curves underlying FLT, which date to antiquity'', which eliminates the centuries of mathematics history that led to the crucial insight by Gerhard Frey. A few pages later (p.~155) he writes that ``Taylor suggested they take just one more look at the Euler systems'' and further down reports ``the quick acceptance of the revised paper prepared together with Taylor'', flatly contradicting the well-documented history of the proof of Fermat's last theorem (in this case, a report by Wiles that Casti quotes on the same page, resp.\ the main 109-page \textsl{Annals of Mathematics} paper by Wiles alone, Zbl 0823.11029). NEWLINENEWLINENEWLINEAs for the nature of the proofs/solutions discussed, \textit{for example}, Casti misunderstood that an essential point in the criticism of the Appel-Haken 1977 proof of the Four Color Theorem concerned the fact that it was hard or impossible to verify and recreate; it was never completely and independently verified (see the introduction of the new 1997 proof by Robertson, Sanders, Seymour \& Thomas, Zbl 0883.05056). Instead, Casti puts forward the false claims ``Lending still more support to the correctness of the result is a second proof found a year later by Frank Allaire'' (p.~69) and ``many different programs have now been written using different testing procedures on different hardware --- and all have come up with the same result'' (p.~73). In the Kepler chapter, he fails to mention that the Hales proof is still in the review process, and has not appeared in print yet --- so that chapter is not closed, yet. NEWLINENEWLINENEWLINEFurthermore, on other proofs Casti ``just doesn't do it'': so for Cohen's result on the Continuum Hypothesis we learn ``He managed to do this using a very powerful idea called \textit{forcing}. Unfortunately, the method of forcing is a bit to technical to go into here.'' Beyond this, no sketch, intuition or illustration is given. In the discussion of Kempe's (wrong) 1879 proof of the four color theorem, he claims that ``To describe the flaw in Kempe's argument, we need the notion of an unavoidable set of configurations.'' This red herring is then expanded on, while the key concept of ``Kempe chains'' and the mistake Kempe made with them don't appear. NEWLINENEWLINENEWLINEIn view of these shortcomings the reviewer feels that Casti is plainly arrogant in his characterization (pp.~175/176) of the books by Singh and Aczel about Fermat's last theorem and their lack of ``explaining the basic nature of the proof.'' One shouldn't complain that ``like Singh, what Aczel softpedals is any coherent account of the structure of the actual proof produced by Wiles and Taylor. Pity.'' without offering a glimpse of such an account; it seems that Casti did not grasp another essential characteristic of the proof by Wiles (and Taylor, and Ribet, and Serre, and many others): that it builds on many phantastic levels of theory, and that it is \textit{really complicated}. NEWLINENEWLINENEWLINEFinally, most of the (many) figures of the book under review seem to be copied from other publications, none of them with any acknowledgment of the sources. \textit{For example}, the color plates are reproduced from Heinrich Tietze's ``Gelöste und ungelöste mathematische Probleme'' (Zbl 0032.10101) resp.\ from the American translation (``Famous Problems of Mathematics'', Baltimore 1965, Zbl 0133.24102), which appears in Casti's references (p.~171) with ``Good graphics, too.'' One side-effect of this is that many of the figures don't illustrate what they are supposed or claimed to show; for this see, e.g., the sphere packings pp.~114/115, where the ``hexagonal packing'' does not match Casti's description on p.~113. Incidentally, these drawings are reproduced (without acknowledgment) from the British edition of Singh's famous book on Fermat's Last Theorem (Zbl 0930.00001), as one can see from the text and figures that shine through the paper of the source of the reproduction. NEWLINENEWLINENEWLINEThe reviewer feels that in view of all these shortcomings, Oxford University Press would be well-advised if they stopped selling this publication, and had it thoroughly rewritten before any republication. NEWLINENEWLINENEWLINEPS: The reviewer was informed by the author per email that a revised printing of the book is in preparation. He also says that a credits page, which ``\textit{does} list credits for all the figures you mention was inadvertently omitted (accidentally deleted, actually) from the files I sent the printer. It will certainly reappear in the next printing, too.'' NEWLINENEWLINENEWLINEPPS: After this review was finished, the reviewer's attention was drawn to the fact that not only many figures, but also various parts of the text of the book under review are copied or ``adapted'' from other writers' books or articles, some of them referenced in the bibliography, some not. To give just one example, the account of Taylor's role in Wiles' proof of Fermat's last theorem (see my comments above) is directly based on Barry Cipra's own interview with Wiles, as documented on pp.~7/8 in his ``What's happening in the Mathematical Sciences, Vol.~3: 1995-1996'' [Am.\ Math.\ Soc.\ 1996, Zbl 0858.00005]. NEWLINENEWLINENEWLINEPPPS: The reviewer has learned now (``Plagiarism doesn't add up,'' by Edward Rothstein, \textsl{New York Times}, March 9, 2002) that Laura N. Brown, the president of Oxford University Press, has announced that because of ``unattributed passages'' the book would be recalled.