Product Description

One of the greatest revolutions in mathematics occurred when Georg Cantor (1845-1918) promulgated his theory of transfinite sets. This revolution is the subject of Joseph Dauben's important studythe most thorough yet writtenof the philosopher and mathematician who was once called a "corrupter of youth" for an innovation that is now a vital component of elementary school curricula.

Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest. Cantor's own faith in his theory was partly theological. His religious beliefs led him to expect paradoxes in any concept of the infinite, and he always retained his belief in the utter veracity of transfinite set theory. Later in his life, he was troubled by recurring attacks of severe depression. Dauben shows that these played an integral part in his understanding and defense of set theory.

Product Details

• Amazon Sales Rank: #124520 in Books
• Published on: 1990-09-20
• Original language: English
• Number of items: 1
• Binding: Paperback
• 424 pages

Review
Historians of mathematics can only be grateful for the effort Professor Dauben has expended to create the synthesis of Cantor scholarship found in this book. But the book can, and I hope will, be read with profit by a far more extensive audience. Any student, mathematician, philosopher, theologian, or general historian with an interest in Georg Cantor and the wondrous revolution in mathematical and philosophical thought that his work did so much to precipitate will find this book of considerable interest.
(Thomas Hawkins Historia Mathematica )

Customer Reviews

A superb exposition of Georg Cantor and his Mathematics
"In a truly massive book of some 400 pages (of which 100 are Appendices, Notes, Bibliography and an Index) Professor Dauben has written a fascinating, empathetic &, at times, poignant survey of Georg Cantor's (1845-1918) Mathematics and Philosophy of the Infinite. An interesting and detailed biographical chapter about Cantor is postponed to the 'Epilogue' (Chapter 12) leaving the bulk of the book to the mathematical origins and development of Cantor's Transfinite mathematics - his Ordinals and Cardinals.

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)