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 A mathematical novel! Is this a new genre? Is there any justification for such a subtitle? Well, this is a novel and a novel with a captivating plot at that. Then it is also a book about mathematics, about its philosophy, its beauty and about its relevance to the human understanding of the surrounding world. There is not a page where mathematics or mathematicians are not mentioned. Mathematics is woven inextricably into the story line itself and I would say that the plot evolves with the mathematical precision. The story begins with a nostalgic flashback experienced by the main character, Ravi Kapoor, to the time his mathematician grandfather gave him a math problem to try on a calculator. The manner in which the problem was given suggests gentleness in the grandfather's relationship with his grandson and appreciation of the magical effect a solution might have on the boy. The grandfather died the next day, but the reader is left with the realization of the importance the memory of the grandfather played in Ravi's life. It is noteworthy that in the absence of the grandfather's wise guidance the boy grows indifferent to mathematics. Next, Ravi is accepted to Stanford and is inclined towards a career in economics. In due time, Ravi takes a course "Thinking about Infinity" and befriends the course professor Nico, who incidentally specializes in the field of Ravi's grandfather. Nico finds a paper with a footnote to the effect that the grandfather conceived the main ideas developed in the paper while in a New Jersey jail. At this point the narrative proceeds in two intertwining threads. On one hand, Nico's lectures present math topics where infinity plays a prominent role. On the other, Ravi's research into his grandfather's imprisonment produces transcripts of philosophical discussions on the nature of truth, certainty and mathematics that the grandfather conducted in jail with a judge assigned to investigate his case. The grandfather had landed in jail under a blasphemy law. (Unbelievably, New Jersey was not the only state that at the time - early 1900's - had a blasphemy law on its books.) The math topics discussed in the book range from Zeno's paradoxes and infinitude of primes through Godel's Incompleteness and Paul Cohen's Consistency theorems. Both authors have master's degrees in mathematics and write about the subject confidently and competently. (I have found just one misprint which I am sure is due to an editorial oversight in the proof of Cantor's theorem that ℵo < 2ℵo.) The backbone of the philosophical discussions is the essence and importance of the axiomatic method. Do Euclid's axioms truthfully express the geometrical structure of the world or are they just articles of faith laid down more two thousand years ago. If the former is the case, how can one explain existence of non-Euclidean geometries? And, if the validity of the axioms stems from our beliefs, why not to adjoin an axiom natural to many, viz., Everything has been created by somebody? One is the viewpoint of Ravi's grandfather, the other of the judge. Both have been deeply affected by the 1919 confirmation of Einstein's General Relativity by Sir Arthur Eddington's experiments. This is a rough outline of the scope of the discussions threaded through the book, but the skillfully laid out plot allows the authors present additional view points, in particular that adopted by the present day mathematical community. There is so much in this book, of philosophy, of mathematics, even of pedagogy, there is no way a short review may give a due credit to this masterpiece. But one additional observation is well warranted. The authors have managed to present mathematics as a human endeavor by many timely excerpts from the diaries and correspondence of great mathematicians and scientists. These, as the rest of the book, are mostly a work of fiction, but succeed infallibly in endowing mathematics with a human face. The book is a delightful and informative read. In the Epilogue, we learn that Ravi eventually preferred a career in mathematics to a probably more prosperous one in economics. We are given to understand that he had married Claire, a math student he met at the infinity course. Truth be told, I was a little disappointed. For, while reading the book and sensing the evolving romance between Ravi and Claire, I'd been hoping that the authors would expand the side story into a sequel in the same genre. It's not yet late for that; and if they do, they may count me among the readership crowd eagerly awaiting a continuation of the story. A Certain Ambiguity - A Mathematical Novel, G. Suri, H. Singh Bal, Princeton University Press, 2007
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