The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity: Mathematics, the Kabbalah, and the Human Mind [Gebundene Ausgabe]

 

 

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Infinity is a baffling concept which has bewildered countless thinkers ever since the dawn of human civilisation. It may appear in many guises: A straight line seems to be divisible into infinitely many intervals, space may extend beyond any imagination, or God may be what is beyond everything finite. The first consideration lead to one of Zeno's paradoxes, the second is inherent in speculations about the nature of the universe, the last is contemplated in the Kabbalah. Aczel starts from these roots of thought on infinity to develop a story which takes the reader all the way from ancient Greece to nineteenth century Germany, where Georg Cantor first developed a rigorous mathematical approach to infinity. Aczel from there leads on to seminal discoveries by Gödel and later Cohen who completed our current picture of the foundation on which builds the theory of the infinite: set theory.

Aczel tells hist story swiftly, he spares the reader detailed mathematical reasoning (but some hints are provided which give a taste of the riches of set theory for the mathematically inclined), and he fleshes out the story with the personal stories of the protagonists involved in uncovering the mysteries of the infinite.

The book starts at the roots in ancient Greece: The discoveries of the Pythagoreans are the starting point. Next comes the Kabbalah and its notion of infinity, Ein Sof. Galileo and Bolzano feature as two precursors to the actual hero of the story: Georg Cantor. His biography between his promising beginnings, his major discoveries, and his eventual descent into depression and hospitalization is vividly summarized. Cantors life story is embedded in the history of set theory, which is so tantalizingly marked by the discovery of the paradoxes which at the time dealt a severe blow to the very foundations of mathematics. Gödel's famous incompleteness theorems are placed in the context of this development, and the account stops with Cohen's independence proofs. The theories are not presented in any detail, rather Aczel uses easily comprehensible analogies to give the reader an inkling of what these theories are about.

Azcel tries to argue two points in his book which, I think, he fails to prove convincingly. The first is the analogy between the mystical insights of the Kabbalists and the later speculations of Cantor and his visions after his descent into depression. Analogy may be very well when it comes to explaining issues beyond the grasp of the listener, but it is a far fetched claim that a chance similarity between statements and experiences by mystics and mathematicians in mental distress is of any significance. Azcel also makes much of the fact that after Cantor other set theorists, and most prominently among them Gödel, suffered (and died) from mental breakdown. This fact Aczel wishes to attribute to the blinding power of infinity which is too much for the human mind to contemplate. Needles to say, he does not substantiate such a claim with proof (such as, for instance, a rigorous statistical argument of a suitable sort). But such claims go down well with the credulous reader and add mysterious drama to the book.

Aczel has no need to force his hand, for there is enough drama to the story of infinity as it is. The book tells that story and the story of the people grappling with it in a splendid and entertaining fashion. It leaves out the mathematical detail and instead presents the human factor behind the equations. If you wish to read an account of infinity which presents more technical details (but which stops far short from being a forbidding mathematical treatise), you might want to turn to Rudy Rucker's Infinity and the Mind: The Science and Philosophy of the Infinite (Princeton Science Library). A rather more serious introduction with similar content to the present book and more mathematical detail is Probleme des Unendlichen. Werk und Leben Georg Cantors. The unbeaten biography of Cantor is Georg Cantor: His Mathematics and Philosophy of the Infinite. But if you are happy with the juicy side of things and enjoy Aczel's journalistic style, then this book is a splendid read.

Excellentes Buch für alle Mathe-Begeisterten, die eigentlich keine Mathematiker sind (i. e. yours truly). Für das Verständnis sollte ein wenige Grundlagenwissen und die Fähigkeit (und das Interesse) zu logischem Denken vorhanden sein. Auch dann kann es erforderlich sein, a) sich ausreichend Zeit beim lesen zu nehmen oder b) das Buch mehrmals zu lesen (war bei mir der Fall).

Ansonsten ist das Buch - obwohl von einem Mathematiker geschrieben - erstaunlicherweise gut verständlich.

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