Following on from a review of a book on Nabokov on butterflies, my second piece in a proper non-student publication was this review of Amir D Aczel’s book on George Cantor and infinity. I still find the topic of this book quietly mind-blowing. The “diagonal argument” is a wonderfully accessible “ah-a” moment. Around this time I read a lot of popularisations on maths – which may have given me an entirely false confidence in my own mathematical ability.
The French mathematician Henri Poincaré wrote that the work of Georg Cantor was “a malady, a perverse illness from which someday mathematics will be cured”; the equally legendary German mathematician David Hilbert held that “no one will expel us from the paradise that Georg Cantor has opened for us”. Cantor, working in isolation in a provincial university, was at the cutting edge of late 19th-century mathematics, discovering set theory, establishing notation for infinite numbers, and stating the continuum hypothesis, for decades regarded as the most difficult problem in pure mathematics.
Galileo demonstrated in 1638 that one can prove that the set of all whole numbers is equal in number to the set of all squares of whole numbers, which is a subset of the set of all whole numbers. How can this be so? If we list all the natural numbers 1, 2, 3… and so on, we can place each of these umbers in direct one-to-one correspondence with its square. We can also put each one in correspondence with a prime. Cantor would later use such thinking to define an infinite set as a collection of objects that can be put into a one-to-one correspondence with a part of itself. Cantor realised that the paradoxes of infinity produced weren’t just slightly bothersome games but required a new type of arithmetic. Sets that can be matched to each other like the example above are then said to have the same cardinality; Cantor dubbed such sets “countably infinite” and denoted their cardinality by “aleph-null”—the Hebrew letter aleph with the subscript zero.
Cantor proved that there are infinities larger than countable infinities by a remarkably ingenious argument—if we try to count all possible real numbers (numbers that can represented as decimals) between 0 and 1, we find we cannot put them in a one-to-one correspondence with the natural numbers of countable infinity. Suppose we list the natural numbers and correspond them with all possible decimals between 0 and 1, in no particular order, like so: and so on forever. Cantor constructed a “diagonal number” by taking the first digit from the first place after the decimal point of the first number, the second digit from the second place after the decimal point of the second, and so on. In this example we get the number 0·27267…which is made of a digit from every single number on the list. If we alter each digit in this number by adding one to it, we get a new number (in this case 0·38378…) which cannot appear anywhere on the original list, since by its very construction it differs by at least one digit from every single entry in the list. In other words, constructing the diagonal number creates a number that has at least one digit in common with every single decimal on the list—and by changing that digit we create a number that loses this common characteristic with each of the numbers on the list. So the decimals cannot possibly be put into one-to-one correspondence with the natural numbers—they are uncountably infinite and are denoted by the symbol C for continuum. The author also demonstrates how Cantor used the concept of the continuum to prove, amongst other things, that there are as many points on any given line as in any shape or volume, no matter of what size. “I see it, but I don’t believe it!” Cantor wrote (in French) of this result.
1 ………… 0.2345678 to infinity
2 ………… 0.5756037s to infinity
3………… 0.6729283 to infinity
4 ………… 0.2386412 to infinity
5 ………… 0.9877754 to infinity
The continuum hypothesis was Cantor’s next step. He wondered whether infinite sets exist that are intermediate in size between aleph-null and C. He thought they didn’t—in his own notation, he hoped to prove that aleph-one (which he defined as the next order of infinity following aleph-null) equalled C—but was unable to prove so. The problem increasingly began to haunt him. His work was under attack from the Berlin-based mathematical establishment, embodied in Leopold Kronecker, who sternly declared “God made the integers; all else is the work of man”. He longed for an appointment to the mathematical faculty in Berlin, and began to believe that his enemies were conspiring against him. Spending increasing amounts of time in the Halle Nervenklinik, he also became an enthusiastic advocate of the Baconian theory of Shakespearean authorship; Aczel represents this as Cantor’s tortured intellect taking refuge from the blinding light of infinity, which he compares to the infinite brightness of the chaluk, God’s robe in Kabbalah tradition. Increasingly Cantor gave the continuum hypothesis the status of dogma, declaring that “from me, Christian philosophy will be offered for the first time the true theory of the infinite”.
The mathematicians Kurt Gödel (who himself suffered from paranoia and hypochondria) and Paul Cohen would later show that, firstly, if we treat the continuum hypothesis as an additional axiom of set theory, it doesn’t contradict any of the other axioms of set theory, and secondly if we treat the opposite of the continuum hypothesis as an additional axiom of set theory, it doesn’t contradict any of the other axioms of set theory. Thus the continuum hypothesis is independent of the other axioms of set theory, and therefore can neither be proved or refuted from those axioms.
As he discusses Cantor’s existence in the provincial university of Halle, Aczel announces “mathematical research is best done within a community of good mathematicians. Research results can be shared and ideas exchanged, so that new theories can develop and thrive”. This is almost certainly true, yet within a few pages Aczel has discussed not only Cantor but two of his contemporaries who made spectacular advances working in isolation; the immensely likeable Karl Weierstrass (who developed the modern theory of mathematical analysis by night while working as a schoolteacher), and Richard Dedekind (who made equally important contributions to the definition of irrational numbers in the provincial University of Brunswick)—yet Aczel never even discusses the implications of this.
It is significant that a recent survey of American scientists’ attitude to the divine found mathematicians the most likely (with biologists the least likely) to believe in a God. Reading of the dizzying orders of infinity that Cantor explored, one feels perhaps that maths and music are the closest humanity can get to any sense of the divine. Aczel treats this potentially fascinating theme in a curiously perfunctory way; the Kabbalah is discussed in one chapter, belying the subtitle. There are some rather superficial references to the ability of the human mind to comprehend the infinite, with occasional references to the connection between Cantor’s fragile mental state and his work on the continuum hypothesis. Periodically Aczel announces that Galileo or Cantor or Güdel had the ability to face in full the concept of infinity, which most mathematicians and indeed human beings never do, but never explores precisely what this means.
All told The mystery of the Aleph deals with one of the most fascinating themes that mathematics holds for the general reader, and deals sympathetically with its central character. Indeed the rarefied world of infinity and its relationship with the divine is perhaps the most beguiling seductress mathematics can rely on to persuade the reflex numerophobes conditioned to see mathematics as dry, soulless, and worst of all, boring. Like Paul Hoffman’s The man who loved only numbers and John D Barrow’s Pi in the sky, this is another accessible introduction to the world of pure mathematics, although perhaps Hoffman’s work is more engaging. Aczel’s work belongs in the set of books dealing with fascinating tales and concepts that fall just barely short of greatness.