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The Ishango Bone (2012)
Paul Hastings Wilson

Amiele becomes the first female student at Trinity College and goes on to disprove the Riemann Hypothesis at the age of 26, but is denied the Fields Medal. Written as if it were her life story recorded by a colleague shortly after her suicide, the book includes many footnotes that fill in mathematical and historical details.

As a Novel

Near the very beginning of the book there is some plot that might interest a reader who is not fascinated by mathematics and philosophy. For example, we see that she was abused as a child and that she cleverly (but cruelly) extracts some small measure of revenge. (The "narrator" supposedly gets this information from her diaries.) The beginning also relates the stereotypical account of a young genius: she reproduced Gauss' "adding" trick in class (though using a somewhat novel geometric method) angering a teacher but attracting the notice of others who recognize her potential and send her at a young age to study with a math professor. In the middle of the book, it starts seeming more like a novel with an omniscient narrator, and I had trouble believing that it was being written and annotated by a colleague. However, by this point the reader really needs to be interested in the mathematics because little else is happening. Embedded in the story we hear about such things as Grothendieck and his dessins d'enfants and of course about her work on the Riemann Hypothesis. By the end of the book, the narrator's story of Amiele's life essentially disappears to be replaced by purely philosophical excerpts directly from her "diary" in which she speculates about mathematical intuition, the anthropic principle, Roger Penrose's theories of the mind, and the ancient bone that gives the book its title.

Some would say that, in a work of fiction, all that matters is the story, the emotion, the humanity, and the beauty that the book conveys. Even though this book does contain at least some of each of those things, I think they are too sparse in this work to please such readers. Others, myself included, can be satisfied with ideas in the book, both facts and philosophical musing. However, there are too many misleading statements and too little of a coherent philosophical thesis for me to strongly recommend it to this category of reader either. Consequently, I am not sure whether there is a type of reader who would really appreciate Wilson's The Ishango Bone. (Being a mathematician, I would be happy to have this question resolved through a "constructive existence proof". That is, if you are someone who really liked this book, please write and tell me why and I'd be happy to post the information here.)

The Riemann Hypothesis and Amiele's Proof

The Riemann Hypothesis is indeed one of the most famous open problems in mathematics and was posed by one of the 19th century's most famous mathematicians. Unfortunately, The Ishango Bone never gives a good description of what the Riemann Hypothesis is. (It talks about "periodicity" in the "origination points" for "curves" associated to the zeta function, but I'm not sure that is even correct let alone informative.) And, Wilson is even farther from the truth when he claims that "a formidable number of scientific and technological developments in other fields depend on Riemann's Hypothesis being true". (I'm not sure what he means when he says that primes are "fundamental to the calculating power of the computer itself, which depends on the ability to estimate the probability of the path an electron will follow through a silicon gateway".) This all sounds to me like babble and hype, and may leave a reader with more misunderstanding than useful new knowledge.

Let me try to offer my own description of the Riemann Hypothesis and what we would gain by knowing that it is true:

Contributed by Alex Kasman

Berhard Riemann studied a formula that can be written in lots of interesting ways. The same formula can be written as an integral with exponential functions, as a product involving all of the prime numbers, or as a sum involving a function that counts all of the primes less than a given integer. The point being that if we understood this function well we could learn about prime numbers from it because despite the fact that we can write down these formulas we don't know all of the primes or the number of primes less than a given number!

The Riemann Hypothesis is just an unproven conjecture Riemann had about where the zeroes of this function (like x=+1,-1 are the zeroes of p(x)=x2-1) are located. His conjecture is that all of the non-obvious zeroes lie in a certain location ("on the critical line") and if they do not then his conjecture is wrong. What, if anything, would knowing which is true tell us about the distribution of prime numbers? Well, there is a well-known formula that returns a good approximation to that function which counts all of the primes less than a given integer. If RH is true, then a consequence would be that this approximation is even better than we previously knew. But, regardless of whether it is true or false, it does not directly give us a better way to approximate it and certainly does not straight out give us total knowledge of the distribution of primes. And, contrary to what Wilson says, I am not aware that any applications or results outside of number theory depend upon the answer, not in DNA sequencing or cryptography or anything like that.

On the other hand, the method of proof (whether it is proved or disproved) may end up revealing some useful information beyond simply the answer to the question of whether all of the zeta zeroes are on the critical line. Some attempts at a proof stray into areas well outside of number theory, and it is one of these that Amiele seems to implement in the book. In fact, mathematicians and physicists such as Hugh Montgomery, Andrew Odlyzko and Michael Berry have worked on the idea that the zeta function zeroes seem to be distributed something like energy states in quantum physics (or, equivalently, like the eigenvalues of a linear operator). In the book, none of their names is mentioned and instead this idea is attributed to Amiele, who apparently is later able to turn it into a proof that RH is false. I say "apparently", because I'm not sure I understand the description of the proof which appears in the book, and cannot even tell if it is indeed a proof. (It sounds as if it is either probabilistic argument, or as if it merely turns the RH into an equivalent but similarly unresolved question about particle dynamics in quantum theory. Either of those would be of interest, of course, but would not be a proof that the RH is false, which is clearly what the book implies she has.)

Like another work of fiction, Life After Genius, this one had me LOL by suggesting that the Mathematical Intelligencer would be an appropriate venue for publishing a proof of an important theorem. No offense intended to that fine publication, but it is a magazine about math, not a math journal! (Annals of Mathematics would seem more appropriate.) Of course, this is a harmless mistake...I just thought it was funny.

A more worrying misrepresentation is the implication that Amiele is denied the Fields Medal for illegitimate reasons. Almost immediately after she announces the proof, colleagues suggest her for the prize, and the selection committee apparently denies it to her either because (a) the proof involves probability (b) they don't want RH to be false (c) she is a woman or (d) they do not like her personally. Of course, prize selection committees are subject to personal bias, but this suggestion seems almost slanderous to me. (Moreover, it is silly to think that because she did not get it immediately that she never would. The Fields Medal is not for a paper that came out in the months preceding the award but rather for a body of work of an individual mathematician before the age of 40. Since she was only supposed to be 26 when she published her disproof of RH, there would be plenty of time for her to still win the prize!)

Gödel and Ramanujan and other doppelgangers

The book mostly takes place in the late 1970s and early 1980s. I was actually quite surprised when I realized this late in the book because it really does not give any clues early on. A scene in which she is kept out of the library because she is female had me imagining that it was taking place during the Edwardian era. I was later able to verify that Trinity College did indeed admit women only beginning in the late 1970s, so that was a clue on the first page that I had missed.

However, Wilson apparently wanted to include interactions between Amelie and the mathematician Srinivasa Ramanujan. Consequently, he includes a character very similar to Ramanujan, named Rabindranath Shree, a former student of G.H. Hardy who serves as an early mentor for Amelie.

I am more perplexed by his similar inclusion of a character based on Kurt Gödel named Magnus Behman. It is not just certain characteristics, but Gödel's actual work that Wilson attributes to the fictional Behman. This seems likely to confuse a reader who is not already well familiar with Gödel, and seems somewhat unnecessary since his death in 1978 would seem to allow for him to possibly have met Amelie even without creating a dopplelganger as in Ramanujan's case.

So, there are at least some seemingly unnecessary name changes. Am I to suppose then that other factual (or spelling) errors, such as the reference to American President "Reagen" or the attribution of a proof of Fermat's Last Theorem to André Weil rather than Andrew Wiles, are also intentional? And why is Shree said to have had a Buddhist wedding rather than a Hindu one?

Turing and his work are also mentioned, but for some reason he gets to keep his name.

The Ishango Bone and other musings

The title of the book refers to a real archological artifact that is often touted as one of the earliest examples of mathematics. This 25,000 year old bone with 168 parallel "tally marks" on it, in three separate columns and in mysterious groupings, is certainly intriguing. Many hypotheses have been posited about its meaning and its significance. In Wilson's book, Amiele is invited to the museum where it is housed and offers her own thoughts on it as part of a diary excerpt. Several of the real theories, that it represents a menstrual calendar, that it suggests "base 10" (which makes no sense to me, since there are no digits here, only tally marks) are again attributed to her here. In addition, she goes farther finding numerological tricks that "almost" suggest a knowledge of the prime counting function.

Since this will probably be my only opportunity to voice my opinion about this intriguing discovery, let me add that while these things are interesting to speculate about, I am among those who remain unconvinced that it has any mathematical significance at all and is not merely a bone that has been notched to make it easier to grip. Moreover, although there seems to be much discussion about what the numbers on it mean, there seems to be too little discussion about what numbers they actually are. For instance, although the first column is often said to "list the prime numbers between 10 and 20", when I look at it (as in this picture stolen from Wikipedia):

I'm as likely to read it as "4+15+17+5+8+11" as "19+17+13+11". In other words, I'm asking how we're so certain where to put the blue dotted lines and plus signs?

Anyway, I have to admit that there is something emotionally potent about having this character examine this ancient object and speculate about its mathematical meaning. This is a case in which emotion, rather than logic or reason, makes this an interesting part of the novel, probably enough so that it deserves to be the title of the book.

As for the other speculations in the same "Coda", their significance to either the plot or message of the book eluded me. In my opinion, like the Ishango Bone speculation, they do not present strong arguments. The analysis of the anthropic principle seems entirely out of place, and misses the point besides. And the discussion of the role of intuition in math, while certainly an interesting topic and one that Gödel certainly thought his discoveries would address, is muddled here by the author's implication that any conjecture or hypothesis as well as any proof involving geometry or probability necessarily is "intuitive" rather than rigorous. Moreover, I did not see how they were tied to the overall story...unless we are to think that Amiele did not have a rigorous proof.

In the end, I'm afraid I have trouble strongly recommending this book. It has too much technical detail to interest someone seeking literature on a human scale, and too many of those technical details are misleadingly presented for someone interested in those details for their own sake. Moreover, I never grasped an overall philosophy or message that the book was attempting to convey. Still, I wonder if maybe I missed it. If you think you can explain it to me, I would be very grateful to hear your thoughts on it.

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Works Similar to The Ishango Bone
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. Life After Genius by M. Ann Jacoby
  2. The Mathematician's Shiva by Stuart Rojstaczer
  3. The Fairytale of the Completely Symmetrical Butterfly by Dietmar Dath
  4. Continuums by Robert Carr
  5. The Five Hysterical Girls Theorem by Rinne Groff
  6. Arcadia by Tom Stoppard
  7. Sophie's Diary by Dora Musielak
  8. Beyond the Limit: The Dream of Sofya Kovalevskaya by Joan Spicci
  9. Too Much Happiness by Alice Munro
  10. The Goddess of Small Victories [La déesse des petites victoire] by Yannick Grannec
Ratings for The Ishango Bone:
RatingsHave you seen/read this work of mathematical fiction? Then click here to enter your own votes on its mathematical content and literary quality or send me comments to post on this Webpage.
Mathematical Content:
5/5 (1 votes)
Literary Quality:
3/5 (1 votes)

GenreHistorical Fiction,
MotifProdigies, Academia, Real Mathematicians, Female Mathematicians,
TopicGeometry/Topology/Trigonometry, Algebra/Arithmetic/Number Theory, Real Mathematics, Logic/Set Theory,

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May 2016: I am experimenting with a new feature which will print a picture of the cover and a link to the page for a work of mathematical fiction when it is available. I hope you find this useful and convenient. In any case, please write to let me know if it is because I would be happy to either get rid of it or improve it if that would be better for you. Thanks! -Alex

(Maintained by Alex Kasman, College of Charleston)