Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire
My rating: 4 of 5 stars

Gift or Neurosis?

Obsession is something with irresistible psychic power. I take Derbyshire’s use of the term literally. Explaining this power is like explaining any other mental abnormality. From the outside, obsession appears irrational. From the inside its logic is compelling and justifies itself entirely. But only because some unstated fundamental premise is where the logic starts. This is the only really interesting thing about obsession; everything else provokes only a sort of mild voyeurism. Finding that premise is what much of classic biography, as well as psychoanalysis, is about. Without the premise, the obsession is merely strange.

I can understand fascination with the topic of prime numbers and the mathematicians who devoted their lives to their unusual properties. I am fascinated by their fascination, as I am by the experiences of the great mystics. With fascination, one can take it or leave it, however; obsession drives one’s being. And as far as I can tell the great mathematicians are remarkably similar in their obsessions to the great mystics. But we know considerably more about what goes on in the minds of the latter, I suppose because their fixation is more common in the general population.

With mathematicians as with mystics, what interests them most, and what they report about to the rest of us, is technique, that is how they get from one state of knowledge or awareness to another. How could it be otherwise? Without technique they are left with nothing really to communicate except subjective reports of their own psychic states. But as we know all too well, the subjective reports of any of us are probably the least reliable source of information about personal development, spiritual or scientific (See postscript below). So they, and we, construct ‘methods’ to hide our lack of understanding of our real motivations. The main function of these narratives is not their general applicability, but their role as evidence that we are in fact sane.

The foundational premise behind the logic of the mystic and the mathematician is obscured by the description of their technique, their methods, as it is by their bare biographical details. We learn nothing about the reason for their obsession, nor what justifies it in their own minds. We learn nothing about the event or sequence of events that led to their discoveries. They therefore appear as prodigies, freaks, automatons, professional celebrities or idols to be worshipped rather than human beings intent on achieving some intellectual or spiritual objective. Their talent is apparent; what they’re trying to do with it is not.

Derbyshire is quite candid about the almost complete lack of documentary information regarding the intimate life of Riemann. Aside from the remarkable fact that he began life as essentially a rural hick and ended as one of the most productive mathematician of the 19th century, there is little that can be said about him which is personal. His travels, his appointments, his professional habits explain nothing about the man’s mind and his intentions. To say that his purpose in life was to solve mathematical problems is simply uninteresting, and probably false.

So Derbyshire is forced to create a sort of Whig history of the man by, on the one hand giving the history of relevant mathematical thought and Riemann’s place in it; and on the other hand telling us about other mathematicians and their equally odd obsessional idiosyncrasies. Everything inevitably leads precisely where it had to. The stars converge, and out pops Riemann and his Hypothesis, which has puzzled every generation of mathematicians since, just as Ignatius Loyola’s Spiritual Exercises have puzzled generations of aspiring religious novices.

The combination of mathematical derivations and historical vignettes in Prime Obsession is frustrating because neither ‘track’ explains Riemann’s obsession. So it is also with others who constitute Riemann’s oeuvre. A man like Gauss, for example, had no ambition for influence or power. Born an uneducated peasant, he was likeable but taciturn to the point of rudeness. Are we to think that he was simply taken over by his talent as a mystic is taken over by faith? Was there some fundamental youthful insight, perhaps, that guided his life and found its place in academic mathematics? Could these people have decided to be great novelists instead?

I can understand Derbyshire’s excitement when he writes, “The Riemann Hypothesis was born out of an encounter, ...a great fusion, between counting logic and measuring logic. To put it in precise mathematical terms; it arose when some ideas from arithmetic were combined with some from analysis to form a new thing, a new branch of the mathematical tree, analytic number theory.” I get it; it is an important historical moment. But this is how it is interpreted after the fact, not as Riemann viewed the situation. What was Reimann proving about himself, not just with his Hypothesis but with all his work?

It is of course unfair to burden Derbyshire with these sorts of issues. He has his own obsessions. But I am left wanting something which Derbyshire doesn’t provide, perhaps no one can, namely, a reason for why Riemann did what he did, not merely a description of the deeds. Riemann himself noted that there are an infinite number of mathematical hypotheses which are difficult to prove or disprove. Why make this one? And why has it become central? Is there an unstated aesthetic among mathematicians which is provoked by it? Or does it now simply represent a possibility for professional recognition?

Riemann’s obsession, and Derbyshire’s, sit there, therefore, as the behemoth in the room. As Derbyshire says clearly, “mathematical thinking is a deeply unnatural way of thinking, and this is probably why it repels so many people.” This makes someone like Riemann even more interesting. If he is so unnatural, perhaps that’s part of the explanation for the obsession. My suggestion is that this is what needs to be explored, perhaps even more urgently than the Riemann Hypothesis itself.

Postscript: I was reminded while reading this of something written by an academic acquaintance almost 50 years ago. Ian Mitroff in his book The Subjective Side of Science studied the moon rock scientists while they studied the moon rocks. What he found, as I recall, was an ability to predict their conclusions based on their personalities. At this point I have no idea about the validity or follow-on from his work. Perhaps, however, it is he who is responsible for creating my expectations of Derbyshire! See: https://www.goodreads.com/book/show/1...

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