A nicely written story about Abdul Karim, a mathematics teacher at the local municipal school, set against the backdrop of the religious turmoil between Hindus and Muslims in India. I couldn’t quite understand the final message the author was trying to convey through her use of a mathematical theme but a pleasant read.
Abdul, as a precocious child, wants to scale the mountains of mathematics like Cantor and Ramanujan - He “wants to see infinity” - but gets pulled back by the untimely death of his father, relegating him to the life of a school master. His childhood friend, Gangadhar, and he, take up mathematics once again in their late maturity. Abdul worries about Godel’s Incompleteness Theorem and wonders if problems like Riemann’s Hypothesis might be undecidable. He “pores over the proof of Godel’s theorem, seeking to understand it, get around it.”.
While they are engaged in their regular discussions over games of chess, religious tensions break out, leading to riots. When the dust settles, a very emotional Abdul appears to renounce his religious beliefs: “It is in Mathematics, only in mathematics that I see Allah”.
At this point, the story veers off into fantasy of which I couldn’t make much sense. Abdul either has a vision or gets swept off into another dimension, the land of mathematics, “the space where primes live, the topology of the infinite universes”. He sees that “some transcendental numbers are marked as doorways to other universes, and each is labeled by a prime number” (not sure what that means. The transcendentals form a dense set and are not indexable by natural numbers). Enthralled, he feels Allah has given him a deep insight into the workings of the cosmos. But as he goes to tell his friend about his wonderful vision, his world is torn asunder by more religious riots. He tries to save a young woman who has been mauled by the rioting savages by taking her to the mathematical realm but the beings there are of no help and he realizes that mathematics (and the denizens of the realm ) are quite impersonal when it comes to human affairs. However, he finds his friendship with his Hindu friend, Gangadhar, remaining strong as always, giving him eternal hope.
There are a couple of errors in the story which need to be pointed out since they are mathematical in nature. There is quite an incorrect description of the Continuum Hypothesis in the following:
“The Continuum Hypothesis, which states that there is no infinite set of numbers with order between Aleph-zero and Aleph-one. In other words, Aleph-one succeeds Aleph-zero; there is no intermediate rank. But Cantor could not prove this.”
CH actually conjectures something different (in the context of the Zermelo-Fraenkel set theory, augmented by the Axiom of Choice - "ZFC"): The power-set of a set of cardinality / size aleph-zero has cardinality aleph-1. In terms of real numbers, this amounts to saying that the cardinality of the set of real numbers is equal to aleph-1.
Indeed, the fact that the alephs can form a well-ordered set so that they can be indexed as aleph-0, aleph-1, aleph-2, etc is due to Cantor’s theorem (simplified by Bernstein and Schroeder without using Choice, and now called Cantor-Bernstein-Schroeder theorem.). A priori, there is no reason why the alephs may not form a collection such that it does not have a second smallest element (like the set of rational numbers between zero and one inclusive). So Cantor really did prove that there is no infinity between aleph-zero and aleph-1 by virtue of the fact that the C(BR) theorem makes it a tautology.
At another place, the author mentions that Godel proved CH to be an undecidable statement. More accurately, Godel showed that CH was consistent with ZFC and Paul Cohen later proved that not-CH was also consistent with ZFC, thus proving CH to be undecidable in ZFC.