Clearing out a pile of old magazines while tidying up my desk, I came across an old review of "Unknown Quantity: A Real and Imagined History of Algebra" by John Derbyshire (2006).   Since I basically earn my living messing about with simultaneous equations and I like a clever title, I thought I would get hold of a copy and have a look.   It turned out to be a quite a good read with a very conversational style – as if you were listening to an enthusiastic presenter.  It's compact, too, at 320 pages - I managed it in a week.

At the start he tells you that he is writing a history of algebra for the non-mathematician – noting that there are a number of much more technical books aimed at the professionals.  In general I found that he does a very good job of outlining concepts and connecting them (bearing in mind, please, that I am NOT properly a mathematician myself).  Someone who didn'’t at least have a vague memory of secondary school algebra lessons might find it hard to connect.  He does have an annoying tendency to present an equation and some equivalent to it without walking through the steps that transform one to the other (like the joke about the professor who, on being asked to explain such a problem disappears for 30 minutes and comes back to say – its obvious), so I did have to take an “if you say so” attitude to some things.   All the sections and diagrams are well labelled so that you can easily find things when he refers back (and forward) and there are stand-alone primer sections to explain the maths in modern day terms so you know the territory before you get into the who and when.  I felt that I could follow the logic as he presented each topic, although I am not sure how much of it will have stuck.

I had been hoping to see something of the underpinnings of the linear and non-linear optimization techniques which are used to solve the planning problems I work on.  In this, I was somewhat disappointed.  When simultaneous equations were discussed in the chapter "An Oblong Arrangement of Terms", I could see the how these 19th century developments were precursors to current day planning and distribution problems.  He covers Gaussian substitution as a method (apparently also practiced by the Chinese since the Han dynasty)  for finding a solution to definite sets -  where there are an equal number of unknowns and equations.   However nothing was said about the problem of finding solutions where there are more unknowns than constraints – and the particular challenge of finding from among the multiple solutions the ones which minimize or maximize the value of one of those equations.  So I was left wondering when people started trying to solve these and how those early attempts evolved into the algorithms we used today, such as simplex and interior point methods.

There is I am sure, some dependence on the development of matrix algebra which was the next topic, although it was presented primarily as a step on the way to theories of higher order objects, rather than an aid to computation.  As Derbyshire tells it, though,  the great revolutions in algebra are not about finding answers to equations but all involve moving on from thinking of it as a topic covering numbers and equations to become the study of all sorts of weird and wonderful objects – fields, rings, manifolds, lines at infinity and more, that connect and intersect with other disciplines such as analysis and geometry.

This is very clearly primarily a history of mathematics as a subject that is studied for its own sake.  There is not a lot on applications.   Early on he proposes that the origin of humanity’s interest in squared and cubed terms comes from trying to manage land and beer production – but it is clear that he believes that the pleasure of the challenge is enough to explain the years of work on increasingly complex equations  - with a bit of academic prestige and the occasional cash prize thrown in.  As for the discovery and classifications of patterns in algebras, groups, fields and rings, - all noted with exclamation marks – it is obviously enough for the him that they are things of beauty.  In the last chapter, however, he stops to address the issue of utility, arguing that “The growth of pure algebra …was so abundant that the subject raced ahead of any practical applications to dwell almost alone in a realm of perfect uselessness.”   However, in recent decades “all the new mathematical objects discovered… have found some scientific application, if only in speculative theories.”   I'm not sure that I would count a speculative theory as an actual application but he chooses physics as an example arena, listing a number of major pieces of work that were very dependent on those advances in non-Euclidian geometry and group theory, and involved some quite concrete things like predicting the existence of particles that turned out to actually exist.  A passing mention is given to some other areas, including the comment that “matrices are now fundamental to economic analysis”  - which I think is, indeed, where we come in.

I really did enjoy this book and am glad I didn’t give up at the sight of the first set of nested cubic roots.  His enthusiasm is infectious.  There is a nice balance in the presentation between what was done and who did it (and who got credit!).  Not quite everyone mentioned qualifies as an eccentric genius, although there are a lot of interesting lives.  He gives you plenty of opportunities to stop and think and notice something for yourself. I appreciated the unembarrassed way he embraces his love of computation and pattern and shapes (some of the best bits of these are in the end notes).  I have a love of quirky terminology and obscure words and there was plenty of that, along with numerous tid-bits for collectors of trivia.    I may not retain much about the higher -maths in the last chapters, but I definitely have some new vocabulary to try out.   Starting with – I hope you will forgive me – a really awful joke -:   Why did “x3 + px + q=0”  struggle to get out of bed every morning? Because it’s a depressed cubic!.

From Kathy's Desk
6th September 2019.

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