The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity

Have already a lot of books on popular mathematics in my book-case. Never bothered too much about infinity. However, when you study topics like series for instance, it is infinity all over the place. I always took it for granted, something ending in nothing and being a long, long distance away.Till I started to read this book from Amir Aczel. This is mathematics in another way. Not too much equations, formulas, integrals, etc. No, this is mathematics one may do by just sitting in a comfortable chair and playing with the thoughts bubbling up inside the brain.This is almost about what Georg Cantor did. Besides describing many great scientist of his time, as Weierstrass, Riemann, Dedekind and others, the book describes thoroughly the life and work of Cantor. His successes and the serious problems he encountered. From what I read in the book I started really to admire Cantor. Most people would have given up with the severe opposition he faced during his life. But not Cantor, each time he went down, he stood up to fight for his ideas again.Besides interesting mathematical topics, going back to the ancient Greeks, the book describes very well the atmosphere of the end of the nineteenth century. It also gives us an idea of life in the town of Halle in the eastern part of Germany, where Cantor lived and worked most of his life. I once had to stay a few days there. Taking the exit Halle I suddenly found myself in the middle of the nineteenth century. Rainy cobble stoned streets, apartment buildings from Cantor's time, it all was still there. That may change, lots of new roads and buildings are under construction.The book not only describes the work done by Cantor on infinity, but it also continues with the scientists building further on the foundations laid by Cantor, as for instance, Kurt Gödel. So, the book provides the reader with a general and thorough view on all what was found, stated and developed on infinity up to the second half of the twentieth century.Now I have read Aczel's book, do I know what infinity is? No, not really. But sometimes, when I sit in my comfortable chair, with Aczel's book close by, playing a little with this topic in my mind, I am sure I almost get it ...

I thoroughly enjoyed this book. It blends history, mysticism, mathematics and psychological interest into an engaging story. Though some of the mystical emphasis might be a little hokey, this was an excellent and very readable account of how Cantor came to his 'continuum hypothesis' and his deterioration into madness. Having an MS in abstract mathematics, and being an actuary by profession, I thought the author made the technical issues very conceptual and stimulating, retaining the accuracy they deserve. But, he did not just present technical material - he wove the psychological aspects related to the issue right into the mix. This made it quite fascinating. Not to be left hanging, he brings the story to a conclusion with Gödel's and Cohen's roles in solving the problem that drove Cantor (and presumably Gödel) mad. I highly recommend this title to anyone interested in the history of math, it's foundations, the continuum hypothesis or a story on how an important intellectual problem was brought to light and tackled over a period of centuries.

I read this book because I am intrigued by Cantor's thoughts, concepts and theorems on infinity. When Newton and Leibniz independently invented the modern infinitesimal calculus in the late 17th century the need for more rigor around the concept of infinity became obvious. Fortunately the 19th century mathematicians Weierstrass, Dedekind and Cantor established a whole new mathematics of infinity. This part of mathematics is the study of "singularities". Singularities are currently the "unknown unknowables" in mathematics, physics and philosophy. To date, only religion claims to have found an (the?) answer on singularities, but I think there must be more to "know". Cantor's Aleph sets of transfinite numbers may provide the key to the answer. Aleph is the reconstructed name of the first letter of the Proto-Canaanite alphabet but in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets."

By profession, I am an engineer and we tend not to take (or have time for) the pure math courses as we progress through our university eduction. Our time is spent with ODE, PDE and numerical analysis, so most engineers are a bit lacking when it comes to say, number theory, abstract algebra, group theory, etc... Having received this book as a gift, it turned out to be very interesting. It gives an educated laymen introduction to actual infinity and the continuum hypothesis, without being overly technical or obtuse. Can be read by (and understood) by anyone having done decently in high school. No special knowledge of calculus, number theory or linear algebra is needed. To a small degree, a bit murky in the last few chapters where Godel's work in logic is discussed in reference to Cantor's proofs, but then those are very high level logic concepts and not easily encapsulated in one single analogy or paragraph. Excellent read and highly recommend.

I have read a number of Aczel's works. They're decent in attempting to combine a readable by the layperson history of a mathematical concept with some very heavy mathematical implications. In this book, elements of both the history and the concept are lacking.The book teases the reader with elements around the aleph and the kabbalah, and yet really never explores them in any detail - especially the assertion of Cantor's potentially Jewish heritage. The concepts of infinity take the reader to amazing places, and Aczel does a great job with examples - but fails to follow through when it comes to set theory and tying all the concepts together. The book slowly builds and the first 150 pages are great. The last 75 seemed hurried and disjointed as he tried to bring the elements together.It was OK - and worth the read if only to ponder that which is infinite - and the mental challenges it invariably seems to cause. However, it's not on my top shelf.

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. Infinity and the Mind: The Science and Philosophy of the Infinite (Princeton Science Library)Probleme des Unendlichen. Werk und Leben Georg CantorsGeorg Cantor: His Mathematics and Philosophy of the Infinite

Este livro é uma obra-prima!! Recomendo a todos (especialistas em matemática ou não) que desejam compreender ou aprofundar suas pesquisas na natureza dos infinitos e de seus níveis.

Appearing below the title of this book is a reference to the Kabbalah, something which is referred to ad nauseum throughout this book. Glance through the index, where you will find dozens of references to God, the Kabbalah, Jewish mysticism, prayers, etc. Worse still, the page numbers given in the index represent only some of the references: in truth, the word 'God' permeates the text, and this is regrettable in a book purporting to be mainly about mathematics. Here's a sample (pp145-146): "...a verse of this prayer, recited by Jews several times daily, is that God rules the universe [there follows a sentence in Hebrew]. Hence the concept of infinity was known to anyone with a Jewish background". Or on page 156 we read, "Was Cantor's fate very different from those of the rabbis of the second century who tried to enter God's secret garden...?" These examples are not the worst excesses: they are typical.There is some neatly explained mathematics, however, and it is only fair to say how entertainingly Aczel writes: he makes the subject accessible to non-mathematicians, and in this respect provides a great service. Chapters 3 and 20 are a joy to read. There are one or two careless errors though: for example, on page 24 we read, ".. the volume of a cone inscribed in a sphere with maximal base equals a third of the volume of the sphere" [in fact it's a quarter]. Again, on page 155, Aczel uses the word 'converse' when 'negation' would be appropriate.

an excellent and entertaining book

無限は無限にあるとか、実無限(無限をあるまとまったものとみなすこと)とか、実数は連続濃度で自然数は可算濃度であり実数のほうが自然数より多いとか、壮大な話といいますか、素人はただ驚き呆れるばかりの世界が広がっているなぁ、という印象です。　　あと、個人的に面白かったのは数密主義ではテトラクテュス＝四つの数の和が10になることから、10が神聖な数とされていた(1+2+3+4=10、 p.23)とか、ヘブライ語の神の名であヤハウェＹＨＶＨも十通りに並び替えることができるが、カバリストたちはさらに「エン・ソフ」という無限概念をもってきた、というあたりも興味深かったです(p.43-)。　　ワイエルシュトラスの1, 14/10, 141/100, 1414/1000....という有理数の数列はルート2という無理数に収束する、というあたりも、すごいなぁ、と(p.86)。