A deep dive into the search for infinity—both mathematical and philosophical. Still not sure if chasing infinity drives people mad or if you have to be a little mad to truly grasp it. Either way, it’s a fascinating read.

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.

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.

Not a bad book at all. The book itself is just a little too "intro" to mathematics and not very deep the cultural idea of infinity, as implied in the title.

One of the most spiritual books out there for me. Lots of mathematical and logical content to enjoy, but also enough history and mysticism (described from our modern perspective) to keep your mind stimulated.

This was a very interesting read - intriguing glimpses into the lives of mathematicians, some rather inaccessible (but expected) maths, and some hauntingly mindblowing aspects that hint at the nature of "infinity" to exist at the curled edges of our own consciousness, for something to exist outside of what we can humanly conceive of. That's slightly nuts, TBH.

Would suit anyone with an interest in the weirder side of numbers and philosophy - maths background helps, but this isn't about formulae as much as how human concepts get turned into explanations. The 'battle' between real numbers and continuous number lines, for example, is one which is something that most kids will happily puzzle away at.

I will probably list this for sending/swaps, so let me know if you're interested in it.

There were some 4/5-star moments in this book but I kinda think this coulda been better off as an essay. I like some of the ideas about the connection between studying the continuum and mental illness, and love diving into some math especially now that I'm just doing data science with my brain. But I could feel the author stretching to fill in the story with biographical notes, and getting into word holes just writing about whether findings on infinity make humans closer to God or prove the eternal seperation of our knowledge from that of a God. Like that is all the stuff I cut from my thesis. Ok I never included it in the first place, and I also didn't really cut anything. Basically just turned in a first draft but it got the job done. This book is not overly mathy so it could be a fun pickup if you don't have a big depth of math knowledge but wanna play with some ideas, but there are better stories in that realm. Benji's summer crusade against nonfiction continues.

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.

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.

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...

Fascinating story of a man who "went mad" while studying the properties of infinity that is also an accessible introduction to advanced mathematical concepts. It's not without flaws - the Kabbalistic correspondences are widely known to be incorrect (and frankly the connection to the Kabbalah felt superficial). There are some funny (funny-strange and unintentionally-funny-haha) transitions. But the biographical and math content was enough to keep me engaged.

Impressive, well-researched, easy to follow. Good explanations of the math.