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ByMr Ron 25 February 2014

I bought this book on the basis of 2 references: one of them pointed out some of its many mathematical errors, but generally praised it; the other, by an acclaimed mathematical author, described it as being an intelligent account, and immensely enjoyable. I don't believe the latter can possibly have read the book attentively. Here's why.

First, some of Szpiro's ridiculous statements (in quotation marks):

Page 118, [concerning Hilbert's first problem], "It is not either/or. One answer and its opposite are true".

Also, "Is the number of points in the smaller section 'countable', that is, somewhat less than a continuum?"

Page 119, "Both a statement and its opposite may be true simultaneously".

Page 33, "A dual problem is, in some sense, the opposite of the original problem".

Page 44, "He [Lagrange] is considered the foremost mathematician of the eighteenth century". No mention of Euler.

Page 122, "Some readers may have expected a convex tile, while Heesch's shape is concave".

Page 166, "Eight balls placed at the corners of a regular octahedron" [on p170 six balls suffice]. This is probably a very rare instance of a typo.

Page 192, "For example, with Carl Louis Ferdinand von Lindemann's proof in 1882 that pi is a transcendental number, it was established that this number has infinitely many digits". [What about the square root of 2? Or even 1/7?]

Page 77 is so full of garbage that my remarks must suffice. Szpiro thinks that Newton single-handedly invented calculus, in particular the differential calculus (no mention of Barrow, Fermat, and others who were differentiating before Newton was born); and most tellingly there is no mention of the magnificent 'Fundamental Theorem', which Newton (and Leibniz) did discover.

According to Szpiro, Newton "ESTABLISHED" the inverse square law, rather than demonstrated its compatibility with Kepler's Laws. The historical truth behind this matter is utterly alien to Szpiro's version: Newton was almost embarrassed by his use of universal gravitation (action at a distance) as a means of explaining natural phenomena, and virtually apologises for its bizzareness in his 'Principia'.

Page 54, "Minkowski's invention of the 'space-time continuum' laid the mathematical foundations for the theory of relativity". No mention of the fact that Minkowski's space-time appeared 3 years AFTER Einstein's theory!

In places the book reads as brilliantly as E T Bell's famous book of mathematical biographies, and although even Bell made some pretty egregious biographical oversights and assumptions, at least he was a very capable mathematician, and his book as a whole is a masterpiece that has stood the test of time. Szpiro, alas, is no mathematician (never be fooled by academic degrees!), as the above extracts amply demonstrate. He gets excited about the merits of proofs by "brute force", using computers, and a disproportionate amount of space is used on this. On page 216 alone, we read, "One of his teachers was Tom Hales, who showed him the power of brute force"; and 3 lines on we have, "Hales suggested that the two of them use brute force..."; and halfway down that page, "Applying even more brute force, the thirteen exceptional cases were then also excluded one by one.....QED". Anyone interested?

I found Szpiro's treatment of some mathematicians at best ignorant, and at worst unkind. Although briefly acknowledging Landau's high praise for one of Axel Thue's theorems in number theory, he spends the next 2 or 3 pages banging on about Thue's "lack of originality". Thue deserves better.

Even worse than this is Szpiro's treatment of Henry John Stephen Smith, with, " It is ironic that nowadays the undistinguished Smith is mostly remembered for having shared the Grand Prix with the great Minkowski". Undistinguished? Has Szpiro even seen Smith's Collected Papers? Smith, who was a modest man, is one of the most neglected mathematicians of his time, and he achieved much on quadratic forms, modular equations, elliptic transformations, etc etc etc. Not to mention the fact that it was he who was commissioned to compile the invaluable 'Report on the Theory of Numbers' (in several parts, 1859 - 1865).

And then, on page 219, we read, "William Thomson was one of the most brilliant minds of the nineteenth century". This man, later to become Lord Kelvin, was loud and made himself heard; and history has been kinder to him than it has been to Smith and others. Thomson, a theoretical physicist, actually made a large number of unbelievably opinionated, unsubstantiated, and downright incorrect statements (eg opposing the Theory of Evolution, stating that aeroplane flight was impossible, denying the existence of X-rays, etc).

Although this book is readable, and in a way it could be enjoyable for a non-mathematician, I would not recommend it at all. It's author writes with a mixture of superficial authority, arrogance, and plain ignorance.

First, some of Szpiro's ridiculous statements (in quotation marks):

Page 118, [concerning Hilbert's first problem], "It is not either/or. One answer and its opposite are true".

Also, "Is the number of points in the smaller section 'countable', that is, somewhat less than a continuum?"

Page 119, "Both a statement and its opposite may be true simultaneously".

Page 33, "A dual problem is, in some sense, the opposite of the original problem".

Page 44, "He [Lagrange] is considered the foremost mathematician of the eighteenth century". No mention of Euler.

Page 122, "Some readers may have expected a convex tile, while Heesch's shape is concave".

Page 166, "Eight balls placed at the corners of a regular octahedron" [on p170 six balls suffice]. This is probably a very rare instance of a typo.

Page 192, "For example, with Carl Louis Ferdinand von Lindemann's proof in 1882 that pi is a transcendental number, it was established that this number has infinitely many digits". [What about the square root of 2? Or even 1/7?]

Page 77 is so full of garbage that my remarks must suffice. Szpiro thinks that Newton single-handedly invented calculus, in particular the differential calculus (no mention of Barrow, Fermat, and others who were differentiating before Newton was born); and most tellingly there is no mention of the magnificent 'Fundamental Theorem', which Newton (and Leibniz) did discover.

According to Szpiro, Newton "ESTABLISHED" the inverse square law, rather than demonstrated its compatibility with Kepler's Laws. The historical truth behind this matter is utterly alien to Szpiro's version: Newton was almost embarrassed by his use of universal gravitation (action at a distance) as a means of explaining natural phenomena, and virtually apologises for its bizzareness in his 'Principia'.

Page 54, "Minkowski's invention of the 'space-time continuum' laid the mathematical foundations for the theory of relativity". No mention of the fact that Minkowski's space-time appeared 3 years AFTER Einstein's theory!

In places the book reads as brilliantly as E T Bell's famous book of mathematical biographies, and although even Bell made some pretty egregious biographical oversights and assumptions, at least he was a very capable mathematician, and his book as a whole is a masterpiece that has stood the test of time. Szpiro, alas, is no mathematician (never be fooled by academic degrees!), as the above extracts amply demonstrate. He gets excited about the merits of proofs by "brute force", using computers, and a disproportionate amount of space is used on this. On page 216 alone, we read, "One of his teachers was Tom Hales, who showed him the power of brute force"; and 3 lines on we have, "Hales suggested that the two of them use brute force..."; and halfway down that page, "Applying even more brute force, the thirteen exceptional cases were then also excluded one by one.....QED". Anyone interested?

I found Szpiro's treatment of some mathematicians at best ignorant, and at worst unkind. Although briefly acknowledging Landau's high praise for one of Axel Thue's theorems in number theory, he spends the next 2 or 3 pages banging on about Thue's "lack of originality". Thue deserves better.

Even worse than this is Szpiro's treatment of Henry John Stephen Smith, with, " It is ironic that nowadays the undistinguished Smith is mostly remembered for having shared the Grand Prix with the great Minkowski". Undistinguished? Has Szpiro even seen Smith's Collected Papers? Smith, who was a modest man, is one of the most neglected mathematicians of his time, and he achieved much on quadratic forms, modular equations, elliptic transformations, etc etc etc. Not to mention the fact that it was he who was commissioned to compile the invaluable 'Report on the Theory of Numbers' (in several parts, 1859 - 1865).

And then, on page 219, we read, "William Thomson was one of the most brilliant minds of the nineteenth century". This man, later to become Lord Kelvin, was loud and made himself heard; and history has been kinder to him than it has been to Smith and others. Thomson, a theoretical physicist, actually made a large number of unbelievably opinionated, unsubstantiated, and downright incorrect statements (eg opposing the Theory of Evolution, stating that aeroplane flight was impossible, denying the existence of X-rays, etc).

Although this book is readable, and in a way it could be enjoyable for a non-mathematician, I would not recommend it at all. It's author writes with a mixture of superficial authority, arrogance, and plain ignorance.

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