What do you think?
Rate this book
The Mystery of the Aleph
In the late 19th century, a brilliant mathematician languished in an asylum. His greatest accomplishment, the result of a series of leaps of insight, was his pioneering understanding of the nature of infinity. This is the story of Georg how he came to his theories and the reverberations of his work, the consequences of which shape our world. Cantor's theory of the infinite is famous for its many seeming for example, we can prove there are as many points on a line one inch long as on a line one mile long; we can also prove that in all time there are as many years as there are days. According to Cantor, infinite sets are equal. The mind-twisting, deeply philosophical work of Cantor has its roots in ancient Greek mathematics and Jewish numerology as found in the mystical work known as the Kabbalah. Cantor used the term aleph-the first letter of the Hebrew alphabet, with all its attendant divine associations-to refer to the mysterious number which is the sum of positive integers. It is not the last positive number, because . . . there is no last. It is the ultimate number that is always being just as, for example, there is no last fraction before the number 1.
258 pages, Hardcover
First published January 1, 2000
27 people are currently reading
1150 people want to read
About the author
Amir D. Aczel
48 books158 followersAmir Aczel was an Israeli-born American author of popular science and mathematics books. He was a lecturer in mathematics and history of mathematics.
He studied at the University of California, Berkeley. Getting graduating with a BA in mathematics in 1975, received a Master of Science in 1976 and several years later accomplished his Ph.D. in Statistics from the University of Oregon. He died in Nîmes, France in 2015.
He studied at the University of California, Berkeley. Getting graduating with a BA in mathematics in 1975, received a Master of Science in 1976 and several years later accomplished his Ph.D. in Statistics from the University of Oregon. He died in Nîmes, France in 2015.
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 30 of 77 reviews
November 12, 2021
Seeing the World In Numbers
Kabbalah is an ancient form of textual deconstruction, a technique whose purpose is to undermine the accepted conventions of biblical language, thereby promoting interesting new hypotheses about their meaning. A principle function of Kabbalah is therefore literary; it is a method which can be expanded to the creative interpretation of all texts (See: https://www.goodreads.com/review/show...). But mathematics is also such a technique which, quite apart from its analytical use, allows creative synthetic interpretations of natural language and its conventions. In The Mystery of the Aleph, Aczel shows how mathematics and Kabbalah perform similar cultural duties.
Kabbalistic interpretation is, in a sense, intended to penetrate beyond the barrier of given language not by intensifying description (phenomenology) or by making more precise definitions of the components of natural language (words and grammar), but rather by allowing language entirely free rein. Language is recognized in Kabbalah as necessarily circular (only words can define other words); and, if not arbitrary, at least stiflingly conventional (bird, oiseaux, and Vogel have cultural connotations which prevent their straightforward equivalence). Some words, of course, are simply untranslatable from one natural language to another. For example, the Mesoamerican Nahuatl glyph tlacuilolli fuses the concepts of letter, art and mathematics. Or try to make an accurate translation of the Hebrew ‘waw consecutive’ tense; it’s not possible. So Kabbalah doesn’t ‘fight’ language, it takes it as all there is and doesn’t even attempt to connect language with material things in the world, or indeed even with independent meaning (See: https://www.goodreads.com/review/show...).
One of the techniques of Kabbalah is called gematria, the use of numbers to substitute for and to analyze texts. The technique effectively translates the text from one language - a natural one - to another - mathematics. The immediate effect of gematria, therefore, is to transform the text from the realm of discrete, finite letters, words and phrases to a very different domain of what is technically called the mathematical continuum. This continuum, unlike natural language, by definition, has no gaps or breaks such as those which exist between letters and words. It is both infinitely precise and infinitely extensive. In other words, the transformed language is dense in the sense that there are many more elements of expression than the mere 26 letters of the alphabet (in English) and than in the contents of the Oxford English Dictionary.
In fact the domain of gematria, because it is expressed in numbers, overwhelms its natural language ‘host’. Since It is infinitely dense (there are an infinite collection of numbers between say 0 and 1), and infinitely extended (there is no greatest nor no least number), it can potentially express far more than a somewhat ramshackle vocabulary. Therefore the use of numbers as a linguistic tool expands interpretive possibilities without limit. There is literally no end to the exploration of texts in which to discover, to innovate, interesting meanings. Intuition and systematic investigation are equally valid and both may result in any number of fruitful new interpretations (See: https://www.goodreads.com/review/show...).
It is in their consideration of the infinite that Kabbalah and modern mathematics touch each other rather intimately. This is the realm of the Aleph - the designation of infinity in both. As Aczel notes “The Kabbalists were apparently aware of the fact that infinity exists both as an endless collection of discrete items and as a continuum. God was viewed as both these infinities, as well as infinities so complex that they could not be conceived by the human mind.” Kabbalists in fact anticipated many important concepts of the infinite several hundred years before academic mathematicians. Conversely mathematical infinity has always had a religious connotation, from the ancient Greeks through the 20th century Catholic Church.
Aczel provides a great deal of detailed and interesting mathematical history to demonstrate the prescience of the great Kabbalists. But what strikes me most about his narrative is something that he never makes explicit, namely that mathematics as a professional discipline performs a very similar function to Kabbalah but on a broader scale. A direct result of mathematics, particularly the mathematics of infinity, is the profound de-centering of many ‘obvious’ truths about the world in a manner which is very Kabbalistic indeed. Or perhaps the comparison should be made the other way round: Kabbalah is, it could be argued, a specialized form of mathematical research.
Aczel implicitly makes a strong case that mathematics has been a language used to challenge natural language from the beginning of formal mathematical study. For example, the ancient Pythagoreans, by considering the entire world as constituted by numbers, created not just geometry but also initiated a cultural revolution about the nature of inquiry as something independent from commercial, practical or otherwise ‘useful’ purposes. Galileo’s and Kepler’s acute ability to see the world as numbers rather than material things provided the insight which literally displaced the minds of human beings about their significance within the universe.
And the challenges by mathematics to the concepts and relationships embedded in natural language, if anything, have increased in intensity as the discipline has progressed. It was the mathematical demonstration of relativity and quantum physics which showed, and continues to show, how wrong our conventional views about the basic structure of the universe have been. In other words, as in Kabbalah, the use of mathematics is in the first instance a way to overcome fixed intellectual prejudices embedded in language (the sun rises) and to formulate new ‘guesses’ about the character of reality (orbiting planets sweep elliptical orbits of equal area in any period of time).
Most remarkably, from the mid-nineteenth century onwards, the mathematical investigation of infinity has undermined conventional wisdom about the world and shown the constraining character of language in rather startling ways. For example, in the mathematics of infinity there are as infinitely many numbers between say 0 and 1 as there are between 0 and 2, despite the extended domain of the latter. Put another way: any sub-set of an infinite set has as many elements as the whole set. Even stranger, it has been demonstrated that the number of points on say one edge of a cube is exactly the same as the number of points in the entire cube.
Thus the distinction between parts and wholes becomes very blurry indeed. And without that distinction, all descriptions of the world, words themselves, become, at best, poetic. What they might signify is indeterminate. Descriptive theory, therefore, is certainly not something that could be reliably called scientific. The problem of infinite sets even threatens the language of mathematics itself with formidable paradoxes like those of Russell and Godel; these paradoxes make physical issues like quantum entanglement look like child’s play.
This is more than mildly disconcerting to anyone who considers the logic of natural language seriously. Quite simply language can’t cope with the strange character of infinity. Take the fact that there are various orders of infinity, perhaps an infinity of increasingly infinite sets of numbers. This is certainly sufficient to undermine one’s certainty about the accuracy of settled scientific not to say religious opinions. Things get really loopy when it’s shown that “given any number, there is no ‘next’ number.” Any purported next number will have one before it in sequence. Mathematics, in other words, compromises much of what we know or can express through natural language (Failure to recognise that mathematics constitutes a distinct linguistic universe has been the source of much philosophical anguish; See: https://www.goodreads.com/review/show...).
Both Kabbalah and mathematics have been criticized for being ‘disconnected’ from reality. And of course they are. The mathematician Georg Cantor developed a whole class of so called transfinite numbers which are literally beyond reality. The result is a language of unreality, in which many unreal things can be expressed. This is very Kabbalah-like indeed. In Kabbalah, for example, the apparently spurious correlations between random biblical words and phrases have no certain referents in the world outside the text at hand. In principle this is no different from the apparent ‘un-testability’ of mathematical formulations in serious physics like String Theory; these too may have no physical referent (See this review by another GR contributor: https://www.goodreads.com/review/show...).
These are consequences of treating language as if it were all that existed and quite independent of any material referents. The criticism of disconnection, therefore, is absolutely justified. Claims by a Madonna in the popular press or even by reputable professionals in scientific journals may in fact be entirely without merit and deserve to be dismissed either out of hand or after careful assessment (for an entry into the fun of celebrity pseudo-science, See: https://www.theguardian.com/lifeandst...). This is a necessary cost which must be paid for treating language - either natural or mathematical - as an isolated, enclosed entity.
However, neither the silliness of a Madonna nor the extreme abstractness of String Theory invalidates Kabbalah or mathematics as a source of interesting insights about the world. Both are in effect alternative languages of parallel fictional universes, with their own distinctive rules and embedded relationships connecting their elements. They are different from natural languages because every element and relationship is defined unambiguously - the Kabbalah through the ten Sefirot or names of God, mathematics through its fundamental axioms. And although they are both finite in terms of their underlying generative principles, they have infinite expressive power. They can combine and re-combine their elements without limit and without the need to demonstrate either the truth or usefulness of any combination.
Quite apart from the role they might have in the analysis of material objects, therefore, Kabbalah and mathematics have the capacity to provoke new thought about what might be the case about the world, including what objects it might contain, such as distinct levels of infinity. These objects may then become part of a previously unrecognized reality in natural language. And this applies as much to scientific research as it does to literary interpretation. Thus the disconnection from ‘things’ is not a flaw but the primary functional characteristic of Kabbalah and mathematics. This characteristic is what links them both historically and in terms of purpose. They catch language at its own game and wring its neck until it yields, even if only marginally and temporarily (an outstanding literary example of the power of mathematics to transcend language in order to promote human communication may be found here: https://www.goodreads.com/review/show...).
Aczel also has an interesting hint - which he also mentions in his book Finding Zero - about the relationship between Kabbalah and mathematics - essentially that they challenge each other in a way analogous to their respective challenges to natural language (See: https://www.goodreads.com/review/show...). It is clear for example that mathematics is rather adept at explaining the unstated rationale for many of the intuitions of Kabbalah. But Aczel also believes that Kabbalah is potentially useful to mathematics as a way out of the paradoxes of infinite set theory. I am far from sufficiently competent to assess the merits of such a suggestion. Nevertheless, I find it an intriguing possibility which might fit comfortably in an overall theory of semiotic development that includes mathematics and mystical meditation as well as natural language.
Call it directed meditation, out of the box thinking, structured insight, paradigm-shifting, or skunk works, the profound epistemological point of both Kabbalah and mathematical formulation is to break out of whatever constraining linguistic state we happen to be in. Much of the resulting thought may turn out to be junk. In fact, as in Borges’s Library of Babel, most of it is junk, hiding important kernels of knowledge in an infinite labyrinth. But like genetic mutations in living things, that is the nature (and the cost) of creativity; most hypotheses formulated through ‘disconnected’ language will be useless or trivial. The role of science is to sort out which is which. However without the useless and trivial, the profound would never evolve to the level of scientific scrutiny (See: https://www.goodreads.com/review/show...).
Seeing the world as constituted by numbers, in other words, is a way to overcome the power that language has over us in limiting and distorting our understanding of reality. I think that just as Kabbalah works to deconstruct texts in order to break out of conventional interpretations, so mathematics, particularly number theory and the mathematics of infinity, has an important function in undermining the fixed meaning of natural language and the pattern of thought it imposes. For me, Aczel’s narrative leads precisely to this point, one I find fascinating and, who knows, perhaps one that is worth more than Madonna’s or String Theory’s predictions.
Postscript 18Sept18: This piece gives a confirming perspective on the language-like character of mathematics: https://www.popsci.com/what-does-math...
Kabbalah is an ancient form of textual deconstruction, a technique whose purpose is to undermine the accepted conventions of biblical language, thereby promoting interesting new hypotheses about their meaning. A principle function of Kabbalah is therefore literary; it is a method which can be expanded to the creative interpretation of all texts (See: https://www.goodreads.com/review/show...). But mathematics is also such a technique which, quite apart from its analytical use, allows creative synthetic interpretations of natural language and its conventions. In The Mystery of the Aleph, Aczel shows how mathematics and Kabbalah perform similar cultural duties.
Kabbalistic interpretation is, in a sense, intended to penetrate beyond the barrier of given language not by intensifying description (phenomenology) or by making more precise definitions of the components of natural language (words and grammar), but rather by allowing language entirely free rein. Language is recognized in Kabbalah as necessarily circular (only words can define other words); and, if not arbitrary, at least stiflingly conventional (bird, oiseaux, and Vogel have cultural connotations which prevent their straightforward equivalence). Some words, of course, are simply untranslatable from one natural language to another. For example, the Mesoamerican Nahuatl glyph tlacuilolli fuses the concepts of letter, art and mathematics. Or try to make an accurate translation of the Hebrew ‘waw consecutive’ tense; it’s not possible. So Kabbalah doesn’t ‘fight’ language, it takes it as all there is and doesn’t even attempt to connect language with material things in the world, or indeed even with independent meaning (See: https://www.goodreads.com/review/show...).
One of the techniques of Kabbalah is called gematria, the use of numbers to substitute for and to analyze texts. The technique effectively translates the text from one language - a natural one - to another - mathematics. The immediate effect of gematria, therefore, is to transform the text from the realm of discrete, finite letters, words and phrases to a very different domain of what is technically called the mathematical continuum. This continuum, unlike natural language, by definition, has no gaps or breaks such as those which exist between letters and words. It is both infinitely precise and infinitely extensive. In other words, the transformed language is dense in the sense that there are many more elements of expression than the mere 26 letters of the alphabet (in English) and than in the contents of the Oxford English Dictionary.
In fact the domain of gematria, because it is expressed in numbers, overwhelms its natural language ‘host’. Since It is infinitely dense (there are an infinite collection of numbers between say 0 and 1), and infinitely extended (there is no greatest nor no least number), it can potentially express far more than a somewhat ramshackle vocabulary. Therefore the use of numbers as a linguistic tool expands interpretive possibilities without limit. There is literally no end to the exploration of texts in which to discover, to innovate, interesting meanings. Intuition and systematic investigation are equally valid and both may result in any number of fruitful new interpretations (See: https://www.goodreads.com/review/show...).
It is in their consideration of the infinite that Kabbalah and modern mathematics touch each other rather intimately. This is the realm of the Aleph - the designation of infinity in both. As Aczel notes “The Kabbalists were apparently aware of the fact that infinity exists both as an endless collection of discrete items and as a continuum. God was viewed as both these infinities, as well as infinities so complex that they could not be conceived by the human mind.” Kabbalists in fact anticipated many important concepts of the infinite several hundred years before academic mathematicians. Conversely mathematical infinity has always had a religious connotation, from the ancient Greeks through the 20th century Catholic Church.
Aczel provides a great deal of detailed and interesting mathematical history to demonstrate the prescience of the great Kabbalists. But what strikes me most about his narrative is something that he never makes explicit, namely that mathematics as a professional discipline performs a very similar function to Kabbalah but on a broader scale. A direct result of mathematics, particularly the mathematics of infinity, is the profound de-centering of many ‘obvious’ truths about the world in a manner which is very Kabbalistic indeed. Or perhaps the comparison should be made the other way round: Kabbalah is, it could be argued, a specialized form of mathematical research.
Aczel implicitly makes a strong case that mathematics has been a language used to challenge natural language from the beginning of formal mathematical study. For example, the ancient Pythagoreans, by considering the entire world as constituted by numbers, created not just geometry but also initiated a cultural revolution about the nature of inquiry as something independent from commercial, practical or otherwise ‘useful’ purposes. Galileo’s and Kepler’s acute ability to see the world as numbers rather than material things provided the insight which literally displaced the minds of human beings about their significance within the universe.
And the challenges by mathematics to the concepts and relationships embedded in natural language, if anything, have increased in intensity as the discipline has progressed. It was the mathematical demonstration of relativity and quantum physics which showed, and continues to show, how wrong our conventional views about the basic structure of the universe have been. In other words, as in Kabbalah, the use of mathematics is in the first instance a way to overcome fixed intellectual prejudices embedded in language (the sun rises) and to formulate new ‘guesses’ about the character of reality (orbiting planets sweep elliptical orbits of equal area in any period of time).
Most remarkably, from the mid-nineteenth century onwards, the mathematical investigation of infinity has undermined conventional wisdom about the world and shown the constraining character of language in rather startling ways. For example, in the mathematics of infinity there are as infinitely many numbers between say 0 and 1 as there are between 0 and 2, despite the extended domain of the latter. Put another way: any sub-set of an infinite set has as many elements as the whole set. Even stranger, it has been demonstrated that the number of points on say one edge of a cube is exactly the same as the number of points in the entire cube.
Thus the distinction between parts and wholes becomes very blurry indeed. And without that distinction, all descriptions of the world, words themselves, become, at best, poetic. What they might signify is indeterminate. Descriptive theory, therefore, is certainly not something that could be reliably called scientific. The problem of infinite sets even threatens the language of mathematics itself with formidable paradoxes like those of Russell and Godel; these paradoxes make physical issues like quantum entanglement look like child’s play.
This is more than mildly disconcerting to anyone who considers the logic of natural language seriously. Quite simply language can’t cope with the strange character of infinity. Take the fact that there are various orders of infinity, perhaps an infinity of increasingly infinite sets of numbers. This is certainly sufficient to undermine one’s certainty about the accuracy of settled scientific not to say religious opinions. Things get really loopy when it’s shown that “given any number, there is no ‘next’ number.” Any purported next number will have one before it in sequence. Mathematics, in other words, compromises much of what we know or can express through natural language (Failure to recognise that mathematics constitutes a distinct linguistic universe has been the source of much philosophical anguish; See: https://www.goodreads.com/review/show...).
Both Kabbalah and mathematics have been criticized for being ‘disconnected’ from reality. And of course they are. The mathematician Georg Cantor developed a whole class of so called transfinite numbers which are literally beyond reality. The result is a language of unreality, in which many unreal things can be expressed. This is very Kabbalah-like indeed. In Kabbalah, for example, the apparently spurious correlations between random biblical words and phrases have no certain referents in the world outside the text at hand. In principle this is no different from the apparent ‘un-testability’ of mathematical formulations in serious physics like String Theory; these too may have no physical referent (See this review by another GR contributor: https://www.goodreads.com/review/show...).
These are consequences of treating language as if it were all that existed and quite independent of any material referents. The criticism of disconnection, therefore, is absolutely justified. Claims by a Madonna in the popular press or even by reputable professionals in scientific journals may in fact be entirely without merit and deserve to be dismissed either out of hand or after careful assessment (for an entry into the fun of celebrity pseudo-science, See: https://www.theguardian.com/lifeandst...). This is a necessary cost which must be paid for treating language - either natural or mathematical - as an isolated, enclosed entity.
However, neither the silliness of a Madonna nor the extreme abstractness of String Theory invalidates Kabbalah or mathematics as a source of interesting insights about the world. Both are in effect alternative languages of parallel fictional universes, with their own distinctive rules and embedded relationships connecting their elements. They are different from natural languages because every element and relationship is defined unambiguously - the Kabbalah through the ten Sefirot or names of God, mathematics through its fundamental axioms. And although they are both finite in terms of their underlying generative principles, they have infinite expressive power. They can combine and re-combine their elements without limit and without the need to demonstrate either the truth or usefulness of any combination.
Quite apart from the role they might have in the analysis of material objects, therefore, Kabbalah and mathematics have the capacity to provoke new thought about what might be the case about the world, including what objects it might contain, such as distinct levels of infinity. These objects may then become part of a previously unrecognized reality in natural language. And this applies as much to scientific research as it does to literary interpretation. Thus the disconnection from ‘things’ is not a flaw but the primary functional characteristic of Kabbalah and mathematics. This characteristic is what links them both historically and in terms of purpose. They catch language at its own game and wring its neck until it yields, even if only marginally and temporarily (an outstanding literary example of the power of mathematics to transcend language in order to promote human communication may be found here: https://www.goodreads.com/review/show...).
Aczel also has an interesting hint - which he also mentions in his book Finding Zero - about the relationship between Kabbalah and mathematics - essentially that they challenge each other in a way analogous to their respective challenges to natural language (See: https://www.goodreads.com/review/show...). It is clear for example that mathematics is rather adept at explaining the unstated rationale for many of the intuitions of Kabbalah. But Aczel also believes that Kabbalah is potentially useful to mathematics as a way out of the paradoxes of infinite set theory. I am far from sufficiently competent to assess the merits of such a suggestion. Nevertheless, I find it an intriguing possibility which might fit comfortably in an overall theory of semiotic development that includes mathematics and mystical meditation as well as natural language.
Call it directed meditation, out of the box thinking, structured insight, paradigm-shifting, or skunk works, the profound epistemological point of both Kabbalah and mathematical formulation is to break out of whatever constraining linguistic state we happen to be in. Much of the resulting thought may turn out to be junk. In fact, as in Borges’s Library of Babel, most of it is junk, hiding important kernels of knowledge in an infinite labyrinth. But like genetic mutations in living things, that is the nature (and the cost) of creativity; most hypotheses formulated through ‘disconnected’ language will be useless or trivial. The role of science is to sort out which is which. However without the useless and trivial, the profound would never evolve to the level of scientific scrutiny (See: https://www.goodreads.com/review/show...).
Seeing the world as constituted by numbers, in other words, is a way to overcome the power that language has over us in limiting and distorting our understanding of reality. I think that just as Kabbalah works to deconstruct texts in order to break out of conventional interpretations, so mathematics, particularly number theory and the mathematics of infinity, has an important function in undermining the fixed meaning of natural language and the pattern of thought it imposes. For me, Aczel’s narrative leads precisely to this point, one I find fascinating and, who knows, perhaps one that is worth more than Madonna’s or String Theory’s predictions.
Postscript 18Sept18: This piece gives a confirming perspective on the language-like character of mathematics: https://www.popsci.com/what-does-math...
May 11, 2025
A wonderful history of mankind's grappling with the concept of infinity where mathematics, philosophy and religion intersect in amazing ways. This is a relatively short book, but it's packed with fascinating stories about ancient Greek philosophers, kabalists, Galileo, Descartes, Georg Cantor, Kurt Gödel and many others. An excellent read. I'll be sure to read all the other books by Amir D. Aczel.
Read in 2013.
I'll be sure to read all the other books turned out to be an exaggeration. Since then, I read two more books by Amir D. Aczel (Entanglement and Descartes' Secret Notebook : A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe) and bought five or six more, which I still hope to read someday...
Read in 2013.
I'll be sure to read all the other books turned out to be an exaggeration. Since then, I read two more books by Amir D. Aczel (Entanglement and Descartes' Secret Notebook : A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe) and bought five or six more, which I still hope to read someday...
December 2, 2010
The book covers the idealized history of mathematical ideas related to infinity, while telling in parallel the biographies of the various people who were a part of it. More than one of the stories fall into the "tortured genius versus crusty establishment" trope.
The book breezes through fairly advanced topics without much explanation, which is fine if you already have some idea about set theory and related ideas. Yet in another place, he mentions pi and e, parenthetically explaining that 'e' is the base of natural logarithms. That is, the author seems to be a little confused about whom he's writing for: the mathematics student who already knows what is meant by "multiplying two sets" will probably not be unfamiliar with 'e'.
Despite being in the sub-title, there's not much about Kabbalah, and the connection to Transfinite numbers is strained at best. It comes across as another instance of the now cliche practice of trying to link mysticism and hard science. (If you like this kind of thing, Douglas Hofstadter does a much better job in GEB:EGB)
The book implies Some sort of connection between the contemplation of infinity and the insanity of Cantor and Godel. (There are also others who worked in similar fields that suffered similarly.) It should be mentioned that there are ten thousand mathematicians who understand the theories very well and are totally sane. My own suspicion is that the causation is perhaps the reverse: it is the disconnect with convention (or reality) allows the great mathematician to make new advances.
The edition I had was in some bad need of editing and proofreading; in my library copy, a previous borrower had thoughtfully made some corrections in pencil.
March 6, 2013
This is the third time I've read this book in the last 18 months. It has given me much to ponder on and reflect about in life and our search for the infinite and what that is.
March 31, 2020
This book was DEEP, a little above my paygrade! A fascinating premise that math and religion can explore the same topics (e.g. the infinity of God also contains nothingness vs. Set theory/math: an infinite set also contains the empty set.). I really enjoyed the brief side story, synopsis of Gödel's contributions and narratives about his relationship with Einstein. *His logic to the extreme discovered a loophole in the US constitution to allow for a dictator!? Anyways good survey of the history of math & interesting subject.
September 30, 2017
Although I am no mathematician, I enjoyed this book. Of course, there were things I didn't understand, but Amir D Aczel avoided shoving our faces into strings of equations. The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity is one of those books that show us that not everything that is true can be proved.
Particularly interesting with the detailed portraits of two great mathematicians (George Cantor and Kurt Godel) who went insane trying to deal with the so-called continuum hypothesis in which one tries to manipulate sets containing an infinite number of objects, whatever they might be.
Particularly interesting with the detailed portraits of two great mathematicians (George Cantor and Kurt Godel) who went insane trying to deal with the so-called continuum hypothesis in which one tries to manipulate sets containing an infinite number of objects, whatever they might be.
June 13, 2012
This wouldnt seem to be a book that would go under cross-cultural spirituality. It is.
It depends on how much you are trying to understand. Ultimately, deep study takes you to almost every discipline and science, hard and soft.
It depends on how much you are trying to understand. Ultimately, deep study takes you to almost every discipline and science, hard and soft.
Read
December 22, 2019This will probably be the first of several readings of this dense, fascinating book — but knowing that both Cantor and Godel went insane pondering the mysteries of infinity, perhaps not too soon.
September 18, 2025
3.5 probably for a solid, well written book on math and infinities that had chapters fit for reading at bus stops. turns out studying mathematical infinities can make you insane even if you can afford a fancy sanitarium and have a hot wife. i liked how far back in history we went and i think all the little detours actually were tied together quite well in the end! the only thing i didnt love that probably is an unsolvable problem given the audience of this book (general readers who are not at all familiar with set theory or abstract algebra) is that the language surrounding the math is not actually very exact. i don’t know how well you could do this and remain digestible for general audiences, but i wanted some more formal theorems or proof outlines, because i did know the material, and writing without it felt a little lacking. i also think when you are dealing with specific logic and math you do need to be pretty careful with what you say and how it’s worded. and i found two typos hence this is not being rounded up. infinity and completeness and choice is crazy when you think about it though. very cool.
February 23, 2009
This is an interesting book about the mathematics of infinite sets in general, and about Gregory Cantor and some other mathematicians who have studied this field in particular. Before one blanches at the phrase "mathematics of infinite sets", be advised that this book is written for a public audience and is really quite readable. This is because several of the basic facts about infinite sets, for instance that there are as many even integers as integers, and as many rational numbers as integers, but that the set of all real numbers is infinitely more numerous than the set of integers, are all quite easy to demonstrate. Amir Aczel presents these proofs in a very readable manner.
The book includes lots of background color -- Aczel shows how the "mystery" of the infinite has a very long pedigree, ranging from the ancient Pythagorean Greek school and the Jewish Kabbalah to the forefront of modern mathematics. Of particular interest in his detailed biographical sketch of various mathematicians involved, especially Gregory Cantor, the "father" of modern set theory.
My only objection is that the author, perhaps in an effort to "spice up" the book with melodrama, went too far in associating the study of the infinite with mental illness. For example, Aczel noted that the Kabbalah was studied only by a few elect because it was considered "too dangerous" for the inexperienced. He also wrote at length about Cantor's increasing mental instability and his eventual death in a mental hospital, and also about Godel, who also had some difficulties in this area. But there are lots of very "normal" people who have also studied and published in this area, including several mentioned in this book. Making too much of such illnesses leads to a reinforcement of the stereotype, brought to the public view with the movie about John Nash, that mathematicians are "crazy". I wish this aspect of the book had been more "subdued".
The book includes lots of background color -- Aczel shows how the "mystery" of the infinite has a very long pedigree, ranging from the ancient Pythagorean Greek school and the Jewish Kabbalah to the forefront of modern mathematics. Of particular interest in his detailed biographical sketch of various mathematicians involved, especially Gregory Cantor, the "father" of modern set theory.
My only objection is that the author, perhaps in an effort to "spice up" the book with melodrama, went too far in associating the study of the infinite with mental illness. For example, Aczel noted that the Kabbalah was studied only by a few elect because it was considered "too dangerous" for the inexperienced. He also wrote at length about Cantor's increasing mental instability and his eventual death in a mental hospital, and also about Godel, who also had some difficulties in this area. But there are lots of very "normal" people who have also studied and published in this area, including several mentioned in this book. Making too much of such illnesses leads to a reinforcement of the stereotype, brought to the public view with the movie about John Nash, that mathematicians are "crazy". I wish this aspect of the book had been more "subdued".
September 13, 2025
Intertwining biographical pieces about mathematicians involved in the conceptualizations of infinities—of note, Cantor and Gödel—my heroes. Along with (very likely) Jewish origins behind the “Alephs,” I especially enjoy the discussions of going from “potential” to “actual” infinity.
Clear, intuitive explanations, pretty illustrations of mathematical paradoxes and theorems, yet not bogged down on the details. At least at my beginner’s level, I can see the author uses correct terminologies. Save for a strange error that YHVH should have 12 permutations of the letters, not 10? That’d cause a problem connecting with Ein Sof etc. tho…
Really appreciate the short foray into modern developments, i.e., large cardinalities.
(Thanks Bookworm Hanoi for the superb discount :p )
Clear, intuitive explanations, pretty illustrations of mathematical paradoxes and theorems, yet not bogged down on the details. At least at my beginner’s level, I can see the author uses correct terminologies. Save for a strange error that YHVH should have 12 permutations of the letters, not 10? That’d cause a problem connecting with Ein Sof etc. tho…
Really appreciate the short foray into modern developments, i.e., large cardinalities.
(Thanks Bookworm Hanoi for the superb discount :p )
January 31, 2013
This book feels cut from the same mold as Darren Aronofsky's movie "Pi" though being more of a history of the mathematical pursuit of infinity than a story of one just one person's encounter with it. The author discusses the specific mystery cults and important researchers from the Pythagoras to winner of the Fields Medal, Paul Cohen, chronicling the development of our understanding of the irrational. He spends most of the book discussing the key contributions of mathematicians Georg Cantor and Kurt Gödel. Like Aronofsky's protagonist from "Pi", they paid the price for their time spent on the frontiers of infinity with recurring neurosis, but their Promethian acts brought back knowledge of the absolute infinity that had a major impact on modern mathematics.
The book is biographical at times and in other sections it is a discussion of mathematical concepts at a textbook level. It's understandable that some mathematical theory must be taught along the way for the reader to have an appreciation for the life's work of Cantor and Gödel, but it was still difficult to understand the concepts. If like me it's been years for you since studying math, this book may be difficult to get through (or at least to fully appreciate, which I feel I don't). Mathematicians or students fresh off of college level math courses may get the most out of this book, but for the rest of us if the book simply provokes in us an interest in mathematics or slightly shames us into feeling the need to be better educated in mathematics, I'm sure the author would feel he made a positive contribution.
The only disappointment I had with the book is the unfulfilled expectations that the teachings of the Pythagorean mystery school and the Kabbalah would be expounded on throughout the book, perhaps with a similar level of detail shown to the book's exploration of post-Renaissance/modern mathematics. The first two chapters are exclusively devoted to those the mysterious mathematical cults, and while later the author had some nice tie-ins, they fizzled out as he got farther into discussions of Georg Cantor.
Those occasional hints at similarities between esoteric number cults and esoteric theoretical mathematics made it seem early on that was the aim of the book. It's an interesting type of hypothesis to see tie-ins between modern science and ancient science/mysticism and it's neither a new idea or one still on the fringes of accepted thought. One example is in some beliefs on astronomy held by a couple of ancient cultures that we long believed to be inaccurate until recent centuries when we developed telescopes that showed us otherwise. There are also books and papers that explore the tie-ins between molecular biology's understanding of the 64 codons (each made up of combinations of 3 of 4 possible nucleobases) of the genetic code and the I-Ching's 64 hexagrams (whose trigrams are made up of combinations of 3 of 4 possible moving/resting yin/yang lines). I find the intersections between ancient and modern science to be a fascinating subject and I think there's enough of a market out there for those who similarly think it's possible for mystics, sages, and ancient versions of scientists to arrive at an impressive understanding of higher science through inner explorations, particularly of the unconscious and its mythological motifs.
It's understandable that the author not return to the subject of the esoteric Greek mathematics simply because - unlike the Kabbalists - they directly affected the mathematicians that followed them from then until now. They were the early building blocks on which the later mathematicians placed their own ideas. The chapters that discuss the contributions of Galileo, Kepler, Newton, Descartes and others of that era makes sense to follow the chapter on the Greeks because those men were studied in Pythagoras, Plato, Archimedes and other early mathematicians. Same with later chapters on the Victorian era mathematicians of Germany who no doubt studied Descartes, Newton and others preceding them. One derives from the other and you end up with a historical sequence that takes you closer to modern times and deeper into higher math; makes sense for the book to flow this way. However, the second chapter on the Kabbalistic contributions is out of place because it's not part of that "family tree" of mathematicians. The Renaissance, Victorian, and early/mid 20th century mathematicans discussed in this book weren't studied in or affected by the Kabbalah, so the only purpose for devoting an opening chapter and portion of the book title to the sect excluded from that whole sequence is if you are going to bring the two unrelated groups together and develop in detail your belief that they coincide. Aronofsky alludes to the numerical structure of the Kabbalah and the ancient Hebrew language in "Pi", and it would just have been nice for this author to have discussed the subject on the tie-ins in further detail, or at the very least, not given the book a title and opening chapter that sets up expectations that he fostered such an hypothesis.
The book is biographical at times and in other sections it is a discussion of mathematical concepts at a textbook level. It's understandable that some mathematical theory must be taught along the way for the reader to have an appreciation for the life's work of Cantor and Gödel, but it was still difficult to understand the concepts. If like me it's been years for you since studying math, this book may be difficult to get through (or at least to fully appreciate, which I feel I don't). Mathematicians or students fresh off of college level math courses may get the most out of this book, but for the rest of us if the book simply provokes in us an interest in mathematics or slightly shames us into feeling the need to be better educated in mathematics, I'm sure the author would feel he made a positive contribution.
The only disappointment I had with the book is the unfulfilled expectations that the teachings of the Pythagorean mystery school and the Kabbalah would be expounded on throughout the book, perhaps with a similar level of detail shown to the book's exploration of post-Renaissance/modern mathematics. The first two chapters are exclusively devoted to those the mysterious mathematical cults, and while later the author had some nice tie-ins, they fizzled out as he got farther into discussions of Georg Cantor.
Those occasional hints at similarities between esoteric number cults and esoteric theoretical mathematics made it seem early on that was the aim of the book. It's an interesting type of hypothesis to see tie-ins between modern science and ancient science/mysticism and it's neither a new idea or one still on the fringes of accepted thought. One example is in some beliefs on astronomy held by a couple of ancient cultures that we long believed to be inaccurate until recent centuries when we developed telescopes that showed us otherwise. There are also books and papers that explore the tie-ins between molecular biology's understanding of the 64 codons (each made up of combinations of 3 of 4 possible nucleobases) of the genetic code and the I-Ching's 64 hexagrams (whose trigrams are made up of combinations of 3 of 4 possible moving/resting yin/yang lines). I find the intersections between ancient and modern science to be a fascinating subject and I think there's enough of a market out there for those who similarly think it's possible for mystics, sages, and ancient versions of scientists to arrive at an impressive understanding of higher science through inner explorations, particularly of the unconscious and its mythological motifs.
It's understandable that the author not return to the subject of the esoteric Greek mathematics simply because - unlike the Kabbalists - they directly affected the mathematicians that followed them from then until now. They were the early building blocks on which the later mathematicians placed their own ideas. The chapters that discuss the contributions of Galileo, Kepler, Newton, Descartes and others of that era makes sense to follow the chapter on the Greeks because those men were studied in Pythagoras, Plato, Archimedes and other early mathematicians. Same with later chapters on the Victorian era mathematicians of Germany who no doubt studied Descartes, Newton and others preceding them. One derives from the other and you end up with a historical sequence that takes you closer to modern times and deeper into higher math; makes sense for the book to flow this way. However, the second chapter on the Kabbalistic contributions is out of place because it's not part of that "family tree" of mathematicians. The Renaissance, Victorian, and early/mid 20th century mathematicans discussed in this book weren't studied in or affected by the Kabbalah, so the only purpose for devoting an opening chapter and portion of the book title to the sect excluded from that whole sequence is if you are going to bring the two unrelated groups together and develop in detail your belief that they coincide. Aronofsky alludes to the numerical structure of the Kabbalah and the ancient Hebrew language in "Pi", and it would just have been nice for this author to have discussed the subject on the tie-ins in further detail, or at the very least, not given the book a title and opening chapter that sets up expectations that he fostered such an hypothesis.
November 15, 2018
An interesting history of people and ideas that I really wish were better written. I found the writing stripped down and formulaic, which is too bad since the topic itself seems filled with interesting possibilities.
March 22, 2020
About the mathematical development of infinity. A few irrelevant bits about the Kabbalah jammed in, sillyness. Otherwise mildly interesting- Then stupidly stopped mid-stream after the end of the work of Kanter.
Overall a stupid book written on behalf of a publisher
Overall a stupid book written on behalf of a publisher
September 29, 2015
A somewhat flawed, magical, fascinating read
Aczel's fascinating book is a short narrative history of the concept of infinity (the aleph) with a concentration on its mathematical development, especially through Galileo, Cantor, Gödel, Paul Cohen and others, meshed with some very interesting material from the ancient Greeks and the Kabbalists who associated infinity with their ideas of God. He includes some material on how strikingly difficult it was for Cantor and others to go against established ideas. I think it was also Aczel's intent to force the reader to think about infinity as something spiritual. At least his book had that effect on me.
God is infinity, the ancient Kabbalists proclaimed, and indeed an all-powerful, all-knowing, immovable yet irresistible God may be something akin to infinity. God is perhaps a higher order of infinity, above the infinity of the transcendental numbers: infinity to the infinite power, one might say, and having said that, one might dismiss it all from the mind as being hopelessly beyond all comprehension. Yet, here, Amir Aczel brings us back. Cantor showed that we can think about infinity, at least to the extent that we can prove differences among infinities. We can, as it were, and from a distance, make distinctions about something we cannot really grasp. In a sense it is similar to contemplating what is beyond the big bang, or imagining the world below the Planck limit. Our minds were not constructed to come to grips with such things, yet maybe we can know something indirectly.
Maybe. In science what we know is forever subject to revision; but in mathematics we are said to have eternal knowledge. When it is proven (barring error) it is settled. Still, might mathematics exist beyond even the furthest reach of the human mind with a higher order affecting our proofs? Beyond the infinities might there exist something more "irrational" more completely "transcendent" than we can imagine even in our wildest fantasies?
At any rate, reading Aczel's magical book, I am persuaded to think so. And I can understand how New Agers and Kabbalists can become so enamored of numbers that they slip quite imperceptibly into numerology. (Numerology being to mathematics what astrology is to astronomy.)
Where I think Aczel is off the mark is in suggesting that it was concentration on the continuum that led to the ill mental health of Georg Cantor and Kurt Gödel. The old saw about thinking so long and hard on a subject leading to madness is something however that won't go away. In chess we have the preeminent examples of Paul Morphy and Bobby Fischer, both towering genius like Cantor and Gödel, who slipped into delusion and paranoia some believe after plummeting the depths of Caissa. With the great strides being made in neuroscience today, we might one day understand what these men had in common besides great intelligence and the ability to concentrate to an extraordinary degree.
There is a lot of interesting material throughout the book. I was especially intrigued with an implication of the fact that an infinite number of steps (e.g., 1/2 + 1/4 + 1/8...etc.--convergence) could lead to a finite sum. (p. 12) This really implies to my mind that we can relate in some sense to the idea of infinity. I contrasted this with Aczel's assertion on page 90 that if one could choose at random a number on the real line, that number would be "transcendental with a probability of one" (missing by force any of an infinity of rational numbers). However, as Aczel points out elsewhere, one cannot actually choose a number randomly out of an infinite collection!
I also liked the report about the exasperated Paris Academy in the nineteenth century passing "a law stating that purported solutions to the ancient problem" of squaring the circle "would no longer be read by members of the academy." (p. 89) This reminded me of the action by the U.S. Patent Office some many years ago of refusing to accept patent applications for perpetual motion machines.
Aczel gives Cantor's proof of a higher order of infinity for transcendental numbers on page 115. It is a very beautiful proof that can be understood with very little knowledge of math. On page 112 he gives Cantor's equally beautiful proof that rational numbers are as infinite as whole numbers. However his gloss at the top of the next page I think contains some typographical error that makes it not helpful. But perhaps I am wrong. (Maybe somebody knows and would tell me.) There is also some confusion about when Gödel married Adele on pages 198 and 200, and there are perhaps too many typos in the book, e.g., on the first sentence of page 162 the word "of" is missing, and on page 164 the word "way" (or something similar) should follow the word "humiliating." Also note Michael R. Chernick's correction in his review below showing the two missing permutations for the Hebrew word for God that Aczel left out on page 32.
Despite these flaws, this is overall an extremely engaging book and a delight to read.
--Dennis Littrell, author of “The World Is Not as We Think It Is”
Aczel's fascinating book is a short narrative history of the concept of infinity (the aleph) with a concentration on its mathematical development, especially through Galileo, Cantor, Gödel, Paul Cohen and others, meshed with some very interesting material from the ancient Greeks and the Kabbalists who associated infinity with their ideas of God. He includes some material on how strikingly difficult it was for Cantor and others to go against established ideas. I think it was also Aczel's intent to force the reader to think about infinity as something spiritual. At least his book had that effect on me.
God is infinity, the ancient Kabbalists proclaimed, and indeed an all-powerful, all-knowing, immovable yet irresistible God may be something akin to infinity. God is perhaps a higher order of infinity, above the infinity of the transcendental numbers: infinity to the infinite power, one might say, and having said that, one might dismiss it all from the mind as being hopelessly beyond all comprehension. Yet, here, Amir Aczel brings us back. Cantor showed that we can think about infinity, at least to the extent that we can prove differences among infinities. We can, as it were, and from a distance, make distinctions about something we cannot really grasp. In a sense it is similar to contemplating what is beyond the big bang, or imagining the world below the Planck limit. Our minds were not constructed to come to grips with such things, yet maybe we can know something indirectly.
Maybe. In science what we know is forever subject to revision; but in mathematics we are said to have eternal knowledge. When it is proven (barring error) it is settled. Still, might mathematics exist beyond even the furthest reach of the human mind with a higher order affecting our proofs? Beyond the infinities might there exist something more "irrational" more completely "transcendent" than we can imagine even in our wildest fantasies?
At any rate, reading Aczel's magical book, I am persuaded to think so. And I can understand how New Agers and Kabbalists can become so enamored of numbers that they slip quite imperceptibly into numerology. (Numerology being to mathematics what astrology is to astronomy.)
Where I think Aczel is off the mark is in suggesting that it was concentration on the continuum that led to the ill mental health of Georg Cantor and Kurt Gödel. The old saw about thinking so long and hard on a subject leading to madness is something however that won't go away. In chess we have the preeminent examples of Paul Morphy and Bobby Fischer, both towering genius like Cantor and Gödel, who slipped into delusion and paranoia some believe after plummeting the depths of Caissa. With the great strides being made in neuroscience today, we might one day understand what these men had in common besides great intelligence and the ability to concentrate to an extraordinary degree.
There is a lot of interesting material throughout the book. I was especially intrigued with an implication of the fact that an infinite number of steps (e.g., 1/2 + 1/4 + 1/8...etc.--convergence) could lead to a finite sum. (p. 12) This really implies to my mind that we can relate in some sense to the idea of infinity. I contrasted this with Aczel's assertion on page 90 that if one could choose at random a number on the real line, that number would be "transcendental with a probability of one" (missing by force any of an infinity of rational numbers). However, as Aczel points out elsewhere, one cannot actually choose a number randomly out of an infinite collection!
I also liked the report about the exasperated Paris Academy in the nineteenth century passing "a law stating that purported solutions to the ancient problem" of squaring the circle "would no longer be read by members of the academy." (p. 89) This reminded me of the action by the U.S. Patent Office some many years ago of refusing to accept patent applications for perpetual motion machines.
Aczel gives Cantor's proof of a higher order of infinity for transcendental numbers on page 115. It is a very beautiful proof that can be understood with very little knowledge of math. On page 112 he gives Cantor's equally beautiful proof that rational numbers are as infinite as whole numbers. However his gloss at the top of the next page I think contains some typographical error that makes it not helpful. But perhaps I am wrong. (Maybe somebody knows and would tell me.) There is also some confusion about when Gödel married Adele on pages 198 and 200, and there are perhaps too many typos in the book, e.g., on the first sentence of page 162 the word "of" is missing, and on page 164 the word "way" (or something similar) should follow the word "humiliating." Also note Michael R. Chernick's correction in his review below showing the two missing permutations for the Hebrew word for God that Aczel left out on page 32.
Despite these flaws, this is overall an extremely engaging book and a delight to read.
--Dennis Littrell, author of “The World Is Not as We Think It Is”
January 17, 2010
Nothing is worse, in my view, than coming across an error or inconsistency near the beginning of a book, especially one that I am reading in order to add to my understanding of a subject. In his discussion of the Kabbalah in chapter three Dr. Aczel states that there are ten permutations of the letters YHVH, which represent the name of God. Now I might have breezed right past this had a previous reader not drawn my attention to the error. The word 'ten' was crossed out and replaced with 'twelve', with the added marginalia "but only 10 are used because.....?" Not one to take the word of an unknown commentator I did the math. To find the number of permutations of a word take the factorial of n, denoted by n!, where n is the number of letters in the word; in this case 4! = 24. In the case of duplicate letters (there are two of the letter H), which would result in duplicate permutations, divide the permutation by the factorial of the number of letters that are identical. 4! / 2! = 12 is the correct answer! The previous reader's question is a good one. Dr. Aczel makes much of the importance of the number ten and indeed the Sefirot, the subject at hand, is composed of ten Sefira or divine qualities. Either he misunderstood the relationship between the permutations of YHVH and the Sefirot or there is none. I was left puzzled. Mysticism is a difficult enough subject for one who prefers reason over the supernatural.
July 18, 2009
This books gives a history of thinking about Infinity, both in a mathematical as in a philosophical and religious meaning. It is also a biography of George Cantor, a mathematician who developed many of the ideas of infinity in mathematics. It is an interesting story about the work of mathematicians.
No Math knowledge is needed to understand this book, but if you know a bit about classical analysis it gives a view of the people behind the theorems.
The philosophical angle is the question: what is the basis of numbers, nature or man?
The book also gives some information on the Jewish Kaballa as related to the infinite. Both Cantor and Gödel, another mathematician who worked on the infinite, have Jewish roots.
The book made me think about infinity, in a good way. It helped to put my knowledge in historical perspective.
I listened to the audio-version, which partly made it harder, because it is more difficult to go back and reread some parts. On the other hand, the voice on this audio-book is a pleasant voice to listen to.
No Math knowledge is needed to understand this book, but if you know a bit about classical analysis it gives a view of the people behind the theorems.
The philosophical angle is the question: what is the basis of numbers, nature or man?
The book also gives some information on the Jewish Kaballa as related to the infinite. Both Cantor and Gödel, another mathematician who worked on the infinite, have Jewish roots.
The book made me think about infinity, in a good way. It helped to put my knowledge in historical perspective.
I listened to the audio-version, which partly made it harder, because it is more difficult to go back and reread some parts. On the other hand, the voice on this audio-book is a pleasant voice to listen to.
February 7, 2011
This book was a captivating read... but not exactly what I was looking for when I read it. Though flavorful -- and I can appreciate that this is book is written for a specific audience that I might not be a part of -- I felt that Aczel could have dared to present a little more mathematics in a few places. There were about two or three pages devoted to silhouetting Cantor's diagonal proofs for the countability of the integers and reals, but besides occasionally inserting a statement of the continuum hypothesis he shied away from presenting anything much deeper than a layman's explanation of some very important mathematics.
I'm glad I read this book, and I still would have if I'd known more about the content ahead of time. It was well-composed and gave me lots of interesting trivia and historical context. Just know that if you're looking for something that tells you much more about infinity in the mathematical sense than the first paragraph of the Wikipedia page, you'll want to find a different book.
I'm glad I read this book, and I still would have if I'd known more about the content ahead of time. It was well-composed and gave me lots of interesting trivia and historical context. Just know that if you're looking for something that tells you much more about infinity in the mathematical sense than the first paragraph of the Wikipedia page, you'll want to find a different book.
March 26, 2013
The main thing you need to know is that despite the subtitle, this is pretty much a book about Georg Cantor. There's enough historical treatment of mathematics for Cantor's story to make sense, but there's no actual mathematics. And there's much less kabbalah than the subtitle would encourage you to believe. But it's really a fine book, quite readable and informative. It's main crimes are that a) it's not as good as David Foster Wallace's "Everything and More" with respect to infinity, and b) it's not as good as Joseph Dan's "A Very Short Introduction to Kabbalah" with respect to Kabbalah. But if DFW is too mathy and too context-heavy, and/or if "A Very Short Introduction to Kabbalah" is too long, you can scratch both itches with this book. But I have to say that while the kabbalah/set-theory connection is interesting and romantically appealing, it is insufficiently argued. I haven't done the math, but I suspect that this is because it doesn't actually withstand the additional scrutiny.
March 8, 2011
I was infinitely disappointed with this book. I expected so much more. The biggest problem is that I don't believe Aczel knew what kind of book he wanted to write. The subtitle is “Mathematics, the Kabbalah, and the Search for Infinity.” What the subtitle should have been was, “Study Infinity and Lose your Mind.” Because that's really all he harped on. Cantor studied infinity, and what happened to him? He went crazy. Godel picks up the torch. Result? Crazy. I think a few more mathematicians might have gone off the deep end, but I don't remember. I found myself just wanting to get through it. What math there was contributed little to the narrative. The Kabbalah gets one chapter, and is barely mentioned again. But I did learn something about infinity. It can't be comprehended. Gee, I think I knew that before I picked up this book. Oh well, I did get one good tidbit out of it. Lord Bertrand Russell could be a real scoundrel when he wanted to. God love him.
January 17, 2010
There is a video in You Tube about “dangerous knowledge”. It is a product of the BBC which examines the lives of three mathematicians Georg Cantor, Ludwig Boltzman and Kurt Godel. The video tries to make the argument because each of these men explored the idea of infinity they went insane. I thought the video to be sensationalistic drivel and didn’t watch the whole thing.
In The Mystery of the Aleph, Aczel explores the concept of infinity from Pythagoras to Cantor. I wish I could understand all the math in the book, but I did grasp enough to get the gist of set theory and infinities. I like the idea of some infinities being larger than others. Poor Cantor did spend a lot of time in mental hospitals, but I agree with Jenna Levin, there probably are many factors related to his illness, besides his work. Overall, the whole idea of infinity is a slippery concept.
In The Mystery of the Aleph, Aczel explores the concept of infinity from Pythagoras to Cantor. I wish I could understand all the math in the book, but I did grasp enough to get the gist of set theory and infinities. I like the idea of some infinities being larger than others. Poor Cantor did spend a lot of time in mental hospitals, but I agree with Jenna Levin, there probably are many factors related to his illness, besides his work. Overall, the whole idea of infinity is a slippery concept.
December 4, 2011
A book about infinity and the man who in modern times did most to advance its study, Georg Cantor. Cantor died in a mental asylum, having been driven there, in a sense, by the maddening complexities of his work. Anyone with a mathematical bent or a certain kind of philosophical inclination who enjoyed mind-bending late-night dorm-room discussions will find much to marvel at in this book.
Side note: When I read this, I had already been intrigued by the complexities of Kabbalah as it figured into Umberto Eco's conspiracy novel, Foucault's Pendulum, so I was pretty well prepared to appreciate its role in Cantor's thinking. Such correspondences and linkages are among the pleasures of wide reading.
Side note: When I read this, I had already been intrigued by the complexities of Kabbalah as it figured into Umberto Eco's conspiracy novel, Foucault's Pendulum, so I was pretty well prepared to appreciate its role in Cantor's thinking. Such correspondences and linkages are among the pleasures of wide reading.
August 28, 2009
I feel the same way about advanced mathematics and advanced physics.
The concepts and ramifications fascinate me, but I don't want to spend the
time and energy it would take to fully comprehend the details.
The author did an excellent job focusing on the background of the
discussions about infinity, Cantor's life, and a few mathmaticians that followed
in his footsteps. He explained just enough of infinity math so that the layman
could understand the concepts without getting bogged down in the details.
If you like this one than I would recommend trying your hand at
'The Dancing Wu Li Masters' which is a layman's discussion of
quantum physics. And if you really want a brain bender try
'Godel Escher Bach' which talks about the connections between
mathematics, art and music.
The concepts and ramifications fascinate me, but I don't want to spend the
time and energy it would take to fully comprehend the details.
The author did an excellent job focusing on the background of the
discussions about infinity, Cantor's life, and a few mathmaticians that followed
in his footsteps. He explained just enough of infinity math so that the layman
could understand the concepts without getting bogged down in the details.
If you like this one than I would recommend trying your hand at
'The Dancing Wu Li Masters' which is a layman's discussion of
quantum physics. And if you really want a brain bender try
'Godel Escher Bach' which talks about the connections between
mathematics, art and music.
February 14, 2016
an interesting book about the history of math, in particular about cardinal numbers (sizes of infinity) as well as logic. i gave it a five not because it is perfect but because it is a good book for someone looking for an introduction to mathematical analysis. the density of Q and R-Q, sets of measure zero, comparing "infinities", completeness and other topics are not necessarily discussed in depth here, but notions are introduced to allow the reader to move into move advanced topics. not a substitute for a textbook and i didnt find the spiritual aspects of the book very interesting but overall it is a very informative casual math book. no math knowledge needed.
July 30, 2011
I confess I didn't hold out much hope for this book when I began to read it,several years after it came into my possession,but I ended up enjoying it despite my reservations. Much of the math,even "dumbed down" to layman levels by Mr. Aczel was still over my head. But the history of the search for an understanding of infinity,and the connections to such mystic disciplines as Kabbalah really held my interest. I'm a sucker for those seeking ultimate truth. Even if in the end they fall a bit short. The search goes on. To infinity(or in the spirit of the book infinities) and beyond.
January 29, 2014
Definitely the most philosophical or mystical of his books I've read so far. Loads about the Kabbalah and how contemplating infinity might drive you mad, like poor Georg Cantor who is featured in the book. I've always found the levels of infinity hard to grasp, but Aczel makes them as transparent as possible. Definitely worth trying to wrap your head around from either a mathematical or a religious perspective, but be warned: you might start to question everything!
December 17, 2017
This is deep. You don't need to understand the math to find this unforgettable. It made me comprehend better what mathematics actually is. Is mathematics true for example or is it made up? Since it in fact appears to describe real things in the universe, what does it say or suggest in large terms about that universe? This is quite something to get a look into. The biographies of Kantor and Godel are also fascinating. Definitely in the running for one of my favorite books.
July 9, 2009
3.5. There's really not very much about the Kabbalah in here. There's a decent amount of math, but honestly, I feel like a lot of space was wasted giving the backgrounds of all the mathematicians mentioned. I didn't really care about that -- I wanted more religion and math stuff! B/c that stuff was really interesting and cool.
July 31, 2011
A few small errors, but in all a fairly accessible text on the beginnings and subsequent development of set theory and Cantor's continuum hypothesis. Does point to some key resources for more sustained study. As an introductory text for non-mathematicians, it frames some of the major issues acceptably.
December 22, 2007
was a good read although it did not go indepth..."casual" math reading, if you will. i always gravitate towards the mathematics section in the bookstore (math nerd) and this was a good "light" read while i was doped up on vicodin after i got my wisdom teeth out.
Displaying 1 - 30 of 77 reviews