John Derbyshire, author of the immensely popular Prime Obsession (2003), has done it again with Unknown Quantity (2006). This is a masterful history of algebra, which might be described as the mathematics of unknowns. Wandering across many curious headlands and byways, Derbyshire focuses on the growing abstraction that has brought us from geometry and counting stones to recondite theories that only highly specialized mathematicians understand, a world where propositions take years to evaluate. Literate and folksy, the book treads the dangerous path that separates facile from abstruse. There is a troll in these waters. But let’s take the upside first.

 

This book is perfect for the reader with high school math. It’s even better for those with a course or two at university level, who occasionally peruse math books for fun. Granted there aren’t many of us, but the list is growing partly because of writers such as Derbyshire who love their trade, avoid jargon and explain detail both graphically and with a poet’s gift for the striking image. One is left with a miraculous grasp of the subject and a sense of the pleasure in store if one investigates further. What more could a reader want?

 

Beginning in Mesopotamia and Egypt, Derbyshire takes us carefully through the mathematical centres of the ancient world: Babylon, Alexandria and Baghdad for starters. We see the authors of cuneiform texts struggling with the limits of their language to describe concepts they skirted and yet likely perceived. Derbyshire gives us primary material as well as summaries and links to authors who grappled with similar issues. The impact of language on thought is a fascinating side-route in Derbyshire’s journey.

 

The early Christian era was mathematically fertile and so were the middle ages. Derbyshire weighs in heavily with Khayyam, Fibonacci, Tartaglia and Cardano, Viete and Descartes, Newton and Leibnitz, while resolutely maintaining his focus on algebra. The thematic core is a great strength of this book. Derbyshire is able to visit and revisit issues from different vantage points as problems from the past are taken up by younger mathematicians, given updated slants, and solved or laid aside for a new generation to tackle. Thus with the roots of numbers (of unity especially) and the properties of polynomials. Derbyshire gives these all a romantic flare; human beings seem destined to grapple with certain mathematical issues till they’re understood. Only then can we move on.

 

So to the 19^th century’s soaring mathematical imagination in Europe (Riemann, Klein, Lie, Jordan, Listing and Poincare) and the tragic dispersion or worse that flowed from prejudice and, later, the death camps. Derbyshire somehow never quite rises from the ashes of the Nazi era. True, he makes a magnificent effort, but the chapters on recent trends are short and highly selective, as though the subject matter is too fresh for evaluation. Nevertheless, the contribution of mathematics to modern physics and our view of the universe is clearly and incisively described. Without the matrix algebra, no Heisenberg quantum leaps. Without Lie groups in three complex dimensions, no predicted particles that helped organize the hadrons.

 

Now the troll. It started when Derbyshire describes William Rowan Hamilton. In his youth during the early 19^th century, Hamilton conceived a flaming passion for someone whose family promptly married her off to another person. So far, so good. We have biographical information and it might lead anywhere, perhaps to withdrawal from the world. Are we going to see a Goethe-like Werther? No, Mr Derbyshire says that Hamilton married later, ‘more or less at random, a sickly and disorderly woman and suffered all his life from an ill-managed household.’ Poor fellow. But was the ‘sickly and disorderly’ a function of Hamilton’s personality, was he cursed with a compulsion to control, did he have a father and mother who intruded on his married life? Mr Derbyshire leaves us hanging in midair with what becomes a gratuitous criticism of this woman who otherwise plays no role in history. Rather unfair, but even Homer slips you might think. Yet earlier we encounter a potted narrative of the last 17 months of Galois’ life that ended with a duel in 1832. There seems no reason to treat us to this list of events in his life, at least none that I could discern. Another slip?

 

Later we find a host of coy remarks about Emmy Noether. Women are rare among mathematicians, it’s true. Mr Derbyshire doesn’t mention that they’re rare among all the valued occupations, because of – well, we need no reminding, do we. Yet we see a list, which appears out of nowhere, of many derogatory and sexist comments made by various people about Ms Noether. Why? No reason that I can see. Unless our author is making a political statement of which his editor has blue-penciled the conclusion. Possible. We then see numerous references to mathematicians being Jewish and the impact of the Nazis. Where are we going here? Undoubtedly true and important as these facts are, the book has wandered from mathematical to social terrain. Are we going to see an absorption of social forms into abstract mathematical functions? We’re encouraged to believe this, because Derbyshire emphasizes that algebra has moved from arithmetical to abstract structures that are independent of particular content. But no, Derbyshire touches a social issue and moves on. What are we to conclude: must we stop at the fact that Jews like women weren’t allowed to enter most professions? Okay, this is a book about mathematics, we say to ourselves. And yet, Derbyshire allows himself a discursion about the modern great, Alexander Grothendieck, whose interest in communes and other popular themes of the sixties culture Derbyshire describes in condescending terms, withering because of their eloquence. The difficulty of course isn’t the meld of mathematics and history, but the uncritical history that contrasts uneasily with the rigor of Derbyshire’s mathematical segments. Might the reader think that mathematics is related to the real world unless Derbyshire strictly curtails our vision and changes subject before we draw conclusions? There’s a right-wing flavour, a conservative scent, to this book that sometimes interferes with its thrust.

 

Let us imagine an editor. This person might be Derbyshire. It might be someone else. The editor fails to excise those parts of Unknown Quantity, mentioned above, that complicate the reader’s enjoyment. Also, in biographical and historical passages, the text often refers to mathematics described elsewhere, which the tenacious reader must then flip back and forth to find, frequently dropping his pen in the process. Finally, there occasionally seem gaps between exposition and conclusion. An example lies in Derbyshire’s description of Galois’ insight on the structure of abstract groups. It seemed to me that we were talking about elements; then came a sudden wind that blew us into a discussion of coefficients. Did I miss a key transition? Was one omitted? A humble reviewer takes the blame and calls it a failure to understand. So be it. Mea culpa. But in a book destined for the general public, perhaps the layout and conceptual underpinnings might attract greater care. At the same time, it’s fair to say that Derbyshire’s mathematics almost always reward the reader and his diagrams are wonderful. We are lured into the details of history while retaining belief in abstruse truths among which human beings somehow are privileged to walk. Mathematics may lead us further into abstraction. Where we’re ultimately heading is a mystery. But at least through writers such as John Derbyshire, a wider audience can appreciate the jungle that mathematicians have explored, the effort needed, and some of the progress in other sciences to which mathematics has contributed. Moreover, the book has great footnotes and a first-class index!