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

Amir D. Aczel

New York: Washington Square Press, 2000. ISBN: 0-7434-2299-6. Pp. 258, index

Review © 2009 Branislav L. Slantchev

Overwrought and overblown, The Mystery of the Aleph is an attempt to provide a story of the idea of infinity as it appears in mathematics interwoven with mini-biographies of the often colorful personalities that dealt with it, mostly Cantor and Gödel, although there are vignettes on Galileo, Bolzano, Weierstrass, and even Descartes. That infinity is easily beyond what our physically-bound minds can understand is not much of a news. Does anyone really understand how big the universe is? Is that even a valid question? If it is expanding, how come it's not expanding into something? How can there be as many real numbers between 0 and 1 as there are between 0 and, say, 100? It is easy to see that there are infinitely many integers and infinitely many rational numbers, but not that there are as many integers as there are rational numbers. After all, the latter are more densely packed, intuition tells us there should be more of them. But there aren't, as Cantor proved. Once you've convinced yourself that this is so, you find out that although there are also infinitely many real numbers, there are more of them than the also infinitely many integers. What!?!?

The logic of the proofs for these counter-intuitive findings are not difficult to follow but even when one does accept the conclusion, can one still claim to understand it? I would say that for most of us, the concept of infinity is too elusive, like Zeno's paradox of Achilles and the tortoise. The tortoise is given a head start, and after a while the fast-footed Achilles takes off after it. By the time he reaches the point where the tortoise was when he started, the tortoise would have moved on. By the time he reaches that new point, the tortoise would have moved a bit further. Because it always takes Achilles some time to cover any positive distance, the tortoise will always be able to move some positive distance in that amount of time. Therefore, Achilles will never catch up with the tortoise.

Now, sure, calculus (even algebra) will tell you that's not the case. We can compute exactly when he will overtake the tortoise given their speeds and the initial distance. But this will not help you grapple with the fundamental issue of the paradox: if we assume infinity (so that distance and time are continuous), then Zeno seems to be right. One possible way out of this is to argue that there is no truly continuous entities in the physical world: at some point, perhaps far down at the subatomic level, we hit an end to divisibility -- the world is fundamentally discrete -- so Achilles would have to overtake the tortoise because at some point the tortoise would need more time to advance from one of these positions to the next than it would take Achilles. But this dispenses with infinity, so it really is a copt out... although Kronecker would have probably supported it.

Infinity is an instance of a concept that was easy to conceive but difficult, if not impossible, to actually understand. We often indulge in a labeling game, as if naming a concept would somehow make it real, but as anyone who has tried to understand the Christian concept of the Trinity, for instance, would know, labeling is one thing, comprehending it, quite another. It is perhaps this that makes studying the concept so fascinating to many people, and it is here that Aczel should have focused. Unfortunately, instead of trying to convey the depth of the difficult ideas that arise from the notion of infinity and the logical paradoxes inherent in mathematics, Aczel opts for cheap sensationalism that is designed to appeal to the same anti-reasoning faculties that make people indulge in séances or practice numerology.

Aczel's basic thesis appears to be that the idea of actual (as opposed to potential) infinity is so bizarre, so contrary to everything our mind tells us about the universe, that if one contemplates it for too long, one is bound to lose one's grip on reality and eventually go insane. Cantor himself apparently decided to label that incomprehensible as God substituting one inexplicable notion with another. I very much doubt that the author actually believes that tripe. But he does spend a lot of time in the book on Cantor's mental problems, on Gödel's paranoia, on Zermelo's health problems, and what not. Everybody who touches infinity seems to go bonkers. And, somewhat incredibly, this idea of infinity can be traced to the Jewish mystical (and secret) teachings known as the Kabbalah. You read that right, meaningless numerology which, at best, can be a tool for meditation, is equated with a mathematical concept. Maybe. But I find this rather detracts from the latter. And at any rate, it has absolutely no bearing on our understanding of it. After all, the Kabbalah is quite content to stop with Ein Sof, the infinite God that nobody can understand or even describe. Of course, they labeled it. Of course, that did not help things.

This is not to say the book is all bad. I really enjoyed some of the biographical sketches. I had not heard about Cantor's battle with Kronecker, and I found the academic wars quite amusing. But then again, I am an academic, so perhaps that just goes with the territory. It was also mildly entertaining to read that both Cantor and Gödel went nuts in much the same way: each spent years trying to prove that the work of a famous person was actually another's (Cantor wanted to prove Bacon wrote Shakespeare's plays, and Gödel wanted to show that Leibniz develop someone else's theories). I would have liked to see more detail in some of the proofs because some of them left me scratching my head in bewilderment. This is not a math book, I know, but then its subject is too important to leave in vague generalities.

While reaching for the mystical, Aczel is often cavalier with logical assertions. For instance, in the section "Do numbers actually exist?" (225-25), he argues that they do. The proof: if "numbers form a language or a convention invented by people," then "we should know everything about numbers and their interrelations since we have invented them"; but since "in mathematics we constantly discover the properties of numbers... through hard work---often finding truths that go against what our intuition tells us," they "cannot be our invention." Hmmm... let's see. Chess is a human invention. But the possible ways to play is so immense, no human can ever hope to know them all. We are constantly coming up (through hard work) with new combinations and counter-combinations, and we always learn new things, some of which surprise us. Therefore, chess cannot have been invented. Yes, I know Aczel said learning "truths" but "truth" is defined by the system itself and has no meaning outside out of it, it is just an indicator of whether a statement is consistent with the axioms in that system in the sense that it can be derived from them using a series of operations also defined by the system. Every strategy you invent in chess is "true" if the pieces follow the rules. And I picked chess because is one of our simpler inventions. How about Go?

Aczel tries another argument, this time about the existence of truly continuous entities. As he admits, "we have no evidence from the physical world that anything can be subdivided ad infinitum." He then says that because calculus, which "makes essential use of the assumption of infinite divisibility and the existence of the continuum, works astonishingly well in giving us precise answers to real world problems," the continuum must exist because otherwise a continuum-based approach could not have been so effective. Unfortunately, this "proof" also fails. How do we know that calculus gives us "precise answers" to real world problems? "Precise" to what degree? If we are limited in the precision we can measure, are we really dealing with a continuum? For instance, does it matter that our number theory says π is a transcendental number whose decimal expansion is infinite and non-repeating? That might be interpreted to say that our number theory is inadequate in capturing the ratio of the circumference of a circle to its diameter -- with our numbers, we just cannot quantify it very well. Or it might be that for all "real world" purposes, what happens after the 2947th digit is immaterial, so for all practical purposes, π might just as well "end" there. Either way, without knowing just how precise answers calculus gives us, we cannot claim that it is anything more than a useful enough approximation. But then again, we can probably get useful enough approximations with discrete entities: just how many terms do we need in a Riemann sum before the improvement in the approximation contributed by a successive term outpaces our ability to detect it?

I do not know the answers to any of these questions. I wish Aczel had spent more time probing the paradoxes and less time on the Kabbalah or some half-baked babble about minds going blind when trying to touch the "haunting" intensity of the continuum.

May 16, 2009