Overall quite readable, and the author stays on topic. Some concerns about content accuracy. Concern first arose with the p. 31 assertion there are 10 permutations for a certain number of letters in a name. There is no natural number for which the number of permutations is 10.

Perhaps more importantly, Bolzano’s conclusion that there are as many numbers in the continuous range 0 to 1 as in the continuous range 0 to 2 requires not only a one-to-one correspondence from 0 to 1 on the x-axis to 0 to 0 to 2 on the y-axis, given by y=2x on p. 61, but also the correspondence from 0 to 2 back “onto” the range 0 to 1. A relation that is both one-to-one and onto is called a bijection. No idea why it wasn’t included because it is easy to establish in this case since the function f(x)=2x has an inverse function of x/2. Hence, it is easy to see how every point in the range 0 to 2 maps back onto a point with half its value in the range 0 to 1.

Since the book is fundamentally about Cantor’s work, it was particularly disappointing to see the claim on p. 115 that because one could show, by Cantor’s diagonalization method, a number not in a given mapping between naturals and reals that this “proved” the reals were uncountable. No, it doesn’t. It only proves the given mapping is a bad one. It is necessary to prove that _all_ mappings are bad, which Cantor did using a simple proof by contradiction, with diagonalization as a lemma supporting the proof. Again, easy to include so no idea why it was excluded.

In both of these cases, one is simply reading and then one hits a point in the exposition wherein it doesn’t seem like the given explanation proves what is claimed to be proven. Not a good look for a math book. Still, I appreciated the author’s explanation on p. 150 of why there are 2 to the aleph naught real numbers in the [0, 1) continuum, based on the binary format for each possible (countably infinitely long) number.

This 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.

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.

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.

The odds of hitting a rational number between 0 and 1 by throwing a dart at the number line are zero.
SV

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 mainly starts with the mathmeticial George Cantor, who later went mad. It's a book about the different ways that people have tried to figure out the concept of infinity, including the Jewish group the Kabbalah and the Infinite Hotel Riddle. This is mainly about the different people who have tried to recognize the concept. Many of them wern't mad or were persecuted and died. Some of the versions were very ridiculus. For instance, there was one greek group that worshipped numbers.

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!

Tahun 2014 Amir Aczel menulis buku Why Science Does Not Disprove God, dan di buku itu jelas sekali kejengkelannya pada para "New Atheists" (dengan tokoh-tokohnya seperti Richard Dawkins dan Sam Harris), yang dengan berbekal sains yakin menyatakan Tuhan itu tidak ada.
(Sebelumnya perlu dicatat bahwa 'new atheist' itu lebih sebagai gerakan ekstrim di kalangan atheis, dan tidak mewakili semua atheis)

Membaca buku The Mystery of the Aleph, yang ditulisnya tahun 2000, jadi paham kenapa dia jengkel: karena sains itu tempatnya di level yang terendah dari samudera ilmu yang tak terbatas, bahkan matematika pun (yang lebih mendasar dari sains) tidak bisa mencapainya karena tidak lengkap! Di posisi itu mana bisa mengklaim bahwa sesuatu yang 'masih belum terlihat ujungnya' itu PASTI tidak ada?
(Kalau kata Dalai Lama di bukunya, 'tidak menemukan' itu sangat berbeda dengan 'menemukan bahwa sesuatu itu tidak ada').

Buku ini berbicara tentang sejarah manusia mencari infinity (ketakterhinggaan), terutama tentang bagaimana Georg Cantor mempelajari set theory (teori himpunan), menelusuri jejak bilangan sampai ke infinity, bertemu dengan berbagai paradoks, tenggelam dalam kontemplasi mengenai continuum hypothesis, dan menemukan ternyata infinity itu jumlahnya tak terhingga.

Sebenarnya konsep infinity sudah dipikirkan oleh filsuf jaman kuno, seperti Zeno (dengan paradoksnya). Eudoxus dan Archimedes juga meminjam konsep infinity untuk menghitung luas dan volume. Newton dan Leibniz dalam kalkulus juga memanfaatkan konsep infinity, tetapi menurut Aczel ini semua baru menyentuh 'potential infinity'. Galileo-lah ilmuwan pertama yang menyelidiki 'actual infinity' dengan memasangkan bilangan bulat dengan kuadratnya, dan menyatakan bahwa jumlah keduanya sama. Lalu Bernhard Bolzano, menemukan bahwa jumlah bilangan antara 0 dan 1, sama banyaknya dengan jumlah bilangan antara 0 dan 2. Kok bisa ya? Pokoknya kalau sudah menyentuh infinity, banyak yang aneh-aneh.

Georg Cantor mulai menyelidiki infinity di paruh kedua abad 19. Usaha Cantor banyak menemukan halangan, termasuk dari gurunya sendiri, Leopold Kronecker, yang berkali-kali menyabotase usaha Cantor menerbitkan paper tentang temuannya di jurnal-jurnal matematika.

Apa sebenarnya yang ditemukan oleh Cantor, dan mengapa sampai begitu banyak perlawanan terhadapnya di kalangan matematikawan jaman itu?
Cantor menemukan bahwa infinity tidak sesimpel menghitung 1,2,3,sampai tak terhingga. Ia ternyata bertingkat-tingkat, satu lebih besar dari yang lain, dan tingkatannya pun tak terhingga. Kalangan matematikawan 'konservatif' tidak menyukainya. "Infinity itu urusan filsafat dan teologi, bukan matematika!"
Mungkin mirip dengan ketidaksukaan fisikawan jaman sekarang terhadap konsep multiverse dan parallel universe, karena tidak ada jalan untuk mengeceknya, meskipun hal itu adalah implikasi dari teori Cosmic Inflation dan Schrödinger equation.

Cantor memberi simbol Aleph (huruf pertama dalam abjad Ibrani) untuk menyatakan kardinalitas (kardinal: menunjukkan jumlah/banyaknya elemen dalam himpunan) suatu himpunan tak hingga .
Menurut temuan Cantor, yang selama ini kita anggap 'infinity' (infinity bilangan asli: 1, 2, 3,...) berada di infinity tingkat terendah.
Kardinalitas bilangan asli adalah Aleph-null (ditulis Aleph 0), bilangan Aleph terkecil.
Sementara untuk lambang bilangan ordinal puncak infinity, the Absolute Infinite, Cantor memberi simbol Omega.
Omega adalah titik puncaknya, the One.
Aleph adalah besarnya, the Many.

Lalu apa hubungannya dengan Kabbalah (tasawuf Yahudi)?

Aleph adalah huruf pertama dalam kata Ein Sof, yang berarti "Yang Tak Terbatas". Dalam Kabbalah, Ein Sof merepresentasikan Tuhan.
Aleph juga huruf pertama dalam kata Elohim (Tuhan, seperti Alif dalam Allah di bahasa Arab), dan Echad (Satu, seperti Ahad di bhs Arab). Aleph adalah representasi Tuhan, Yang Satu dan Yang Tak Terbatas. The One and The Many.

Pada tahun 1931, Kurt Gödel membuktikan dalam Teorema Ketidaklengkapan bahwa setiap sistem matematika yang konsisten dan mengandung aritmetika maka pasti tidak lengkap, dan konsistensinya tidak bisa dibuktikan di dalam sistemnya sendiri.

Artinya apa? Matematika dan seluruh ilmu pengetahuan manusia ada limitnya (sains, yang menggunakan bilangan Real, berada di level Aleph-2. Jadi mana bisa sombong sok tahu hanya berbekal sains?). Jika ingin tahu lebih banyak, harus naik ke level yang lebih tinggi. Melalui teoremanya Gödel menunjukkan pentingnya infinity Cantor.

Kalau boleh saya analogikan, ibaratnya dalam sebuah ruangan gelap pengetahuan manusia, Cantor menemukan pintunya, mengintip melalui lubang kunci, dan melihat cahaya terang ilmu yang tak terbatas. Kurt Gödel membuka pintunya dan menunjukkan jalan menuju cahaya itu.

Terimakasih Cantor dan Gödel.

-dydy-

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.

Nice pleasure reading that gives a soft introduction to the history of some areas in mathematical philosophy. Very pro-Cantor/infinite sets and anti-Kroncker / finitism. Does not discuss the reasons for skepticism about infinity other than explaining some of the paradoxes that arise from infinite set theory. This is overall a good thing since it creates a clear protaganist narrative. The weaving in of Kabbalah in a very soft and preliminary way is a fun touch considering how challenging Kabbalah is to actually understand.

Made me wonder if we can think of Infinite Sums purely algebraically using 1/0 as infinity. If so what is the expressive power of its grammar?

Intriguing and great for listening on a road trip.