Aczel: The Mystery of the Aleph
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Amir D Aczel, The Mystery of the Aleph : Mathematics, the Kabbalah, and the Search for Infinity. New York : Four Walls Eight Windows, 2000. 258 pages, with references, notes, and index.
Typically as I read a book I note down an occasional thought or some passages that I might like to quote here in a book note. For this interesting but odd book I wrote one comment: "About G Cantor / not about G Cantor". It seems not out of place among the idiosyncrasies of this book.
The title describes the book more accurately than seems possible. I worried a bit before I started reading that reference to "mystery" and "Kabbalah" might signal more mysticism than would suit my taste, but I was wrong. The idea of infinity is, indeed, the idea that ties everything together.
Georg Cantor (1845—1918) figures prominently because he was the singular mathematician in the nineteenth century who first investigated the arithmetic of transfinite numbers, and who introduced the convention of referring to different orders of infinity with the Hebrew letter "aleph". He proved that the infinity of counting numbers, the integers, is the same size as the number of rational numbers, all those that can be written as ratios of integers. He also proved that the infinity of irrational numbers is larger than the infinity of integers and rationals.
Irrational numbers that are not algebraic are transcendental. Most numbers on the number line are transcendental. While algebraic numbers and rational numbers are infinite, the transcendental numbers are of a higher order of infinity. If you could randomly "choose" a number on the real line, the number will be transcendental with probability one. Choosing a rational number, or an algebraic one—even though there are infinitely many of them—is just too unlikely because of the preponderance of the transcendental numbers. Thus finding a rational or algebraic number when choosing a number at random from the real line has zero probability. [p. 90]
Cantor also conjectured what is now known as the "Continuum Hypothesis", which says that there is no infinity with a cardinality between the cardinality of the integers, the countable infinity denoted by "aleph" with the subscript "0", and the cardinality of the continuum of irrationals, an uncountable infinity, denoted by "aleph" with the subscript "1" (provided that the hypothesis is true and that the two cardinalities can be ordered with nothing in between).
At the end of the nineteenth century the Continuum Hypothesis stood without proof and mathematician David Hilbert put it on his celebrated list of great outstanding problems in mathematics. In 1939 Kurt Gödel, famous for his theorems on undecidable propositions in mathematical systems ("incompleteness theorems"), showed that the Continuum Hypothesis could not be proved true or false based on the axioms of set theory as they were commonly used; that the Continuum Hypothesis was formally undecidable.
Cantor wrestled with the Continuum Hypothesis later in his life as he wrestled, too, with mental disease, spending frequent episodes in an institution. In his later years Cantor became obsessed with proving the notion that Francis Bacon was the real author of the plays of Shakespeare. The author, rather romantically, seems disposed to believe that Cantor's mental instability resulted from his struggle with trying to understand infinity. I’m not at all convinced, but I didn't find that the question intruded enough to irritate me.
So, the topic of the book is "infinity" and its history as an idea, which is why there is a chapter on the Kabbalah and it made sense. It's also at the root of why the book is not by any means a biography of Cantor and yet Cantor's life weaves in and out of the story.
Thinking of this book as an extended essay on "the infinity" and the fascinating topic of transfinite arithmetic that gives Cantor his well-deserved place of prominence is reasonable. The author's writing is good but the language is sophisticated and not really meant for the mathematically unsophisticated, although it's not written just for those with mathematical training. The author doesn't explain every technical term he uses; those left unexplained aren't really necessary and can be glossed over but it will irritate some readers.
I found it interesting to read, with interesting history and connections between ideas. I know of no other book that treats the subject, certainly not in this manner, so I can recommend it if the subject sounds interesting.
-- Notes by JNS