Georg Cantor: His Mathematics and Philosophy of the Infinite [Paperback]

Cantor's struggle (ultimately destined to be unsuccessful) to prove the so- called "Continuum Hypothesis" (see below) is a continual theme running throughout the narrative. So too, is his 'battle' against fellow mathematicians such as the 'finitist' Kronecker (his main protagonist who described Cantor as a charlatan) to get his mathematical ideas - and indeed his philosophy (for the two were tightly bound) - acknowledged and understood by the wider mathematical community of his peers.

Prof. Dauben's book charts the 'road to Damascus' in Cantor's mind of the Transfinite Set Theory realm for which Cantor truly believed he was a Divinely inspired channel or 'medium' even. A deeply religious man, he was so concerned that this aspect of his philosophy be recognised that, despite being from the Lutheran tradition, he tried to get the Roman Catholic Church (under Pope Leo XIII) to give it the Church's 'Imprimatur', if not it's 'Nihil Obstat'! [I shall hereby resist the pun that he called his first Transfinite Number, 'Aleph #0', the first CARDINAL number. OK - I give in!]

Cantor showed that the counting numbers had an ordinal type he labelled (lower case) 'omega' but so too did all the rationals and algebraic numbers! Moreover (to paraphrase Orwell) 'some infinities were more equal than others' in that he showed the cardinality of the Real Numbers (c) was, by his famous 'diagonal argument', a greater 'power' or 'degree of infinity' than was the counting numbers (the denumerable sets). He also showed c = '2 to the power Aleph #0'; it was his lifelong struggle (that was the "Continuum Hypothesis") that this c = 'Aleph #1', the 'next' Aleph after 'Aleph #0'. His work culminated in two great opuses - the Grundlagen (1883) and the Beitrage Parts I & II (1895 & 1897).

Prof. Dauben paints a picture of Cantor as a family man and polymath. He came from a famous Russian musical family and was an accomplished violinist; he was an excellent illustrator in pencil. He was interested in Literature too, lecturing often about his belief that Bacon & not Shakespeare was the author of the latter's plays!

He was continually railing against what he saw as his lowly status as a professor at Halle as opposed to more prestigious tenures in the likes of Berlin. He was, tragically, subject to oft-repeated bouts of depression during which he had to be confined to the Halle 'Nervenklinik' (Sanatorium). Although much of Cantor's correspondence was lost, a poignant letter from his father, whilst Cantor was a young man, is reproduced in this book and shows what spurred Cantor on in his career 'against the odds'.

In conclusion, Prof. Dauben writes an interesting chapter on Cantor's legacy and how mathematical posterity (such as Zermelo and Bertrand Russell) picked up & developed his ideas. Indeed, later in the 20th Century, Hilbert himself claimed Cantor had created a 'new paradise' for mathematicians.

This is a book which I enjoyed immensely and would recommend it to anyone interested in the history of mathematics and who wishes to have some insight into the 'birth' of a new mathematics.

PS: Cantor preferred the phrase 'free' mathematics as opposed to 'pure' - from which one can, perhaps, infer that his divinely-inspired mission was drawn from the biblical phrase: 'The Truth will set you Free'! "

(Finis)