Siobhan Roberts’ biography of John Conway, “Genius at play”, is an extraordinary feat.

Roberts manages to tell us enough about the maths to be interesting and yet not to baffle readers who know little maths, and to do it without “talking down”. (Although there’s the occasional clunky passage, I suspect she knows more maths than she lets on).

She tells us about:

• Conway’s research into finite groups (the mathematical structures of symmetries), culminating in the discovery and description of “The Monster”, which is the structure of symmetries of an object in 196,883 dimensions

You can see an introduction to the ideas behind the Atlas of Finite Groups and the Monster at:

• His invention, or discovery, of the “surreal numbers”, a number system which includes infinitesimals and infinities as well as the usual real numbers, but which (unlike Abraham Robinson’s “hyperreals”) is constructed independently, not as an extension of the real numbers. For an introduction to surreal numbers:

• His famous “Game of Life”, which has some importance for the theory of automata. This is Conway himself on the “Game of Life”:

• His “Free Will Theorem”, also fairly famous, but maybe less consequential.

Partly because a lot of Conway’s work has opened new directions in mathematics, rather than pushing back distant frontiers, as most research does, it’s easier to get at least a rough idea of what it’s about than with, say, the Langlands program.

Still, Roberts has done an amazing job. Conway has attracted a biographer because he is an unusually flamboyant mathematician, but there’s not a lot to tell about his life apart from maths. He went to Cambridge to study in 1956, stayed there until 1987, and then moved to Princeton University in the USA, where he remains.

A lot of the book is more the story of writing the biography than straight biography. She sits in a corridor in Princeton University maths department as Conway rattles on and tackles and expounds mathematical puzzles. She takes him back to England to visit his elderly sister and an old schoolmate, in an effort to pin down the truth about facts long ago. Conway draws a blank on his schoolmate’s stories, and sits oblivious, doing a mathematical puzzle, while his sister rattles on.

Conway proposes marriage to her, jokingly, but, the reader suspects, only half-jokingly.

Conway tells Roberts, sensibly I think, that “given the right education… almost everybody could do mathematics pretty damn well”.

Yet he does not seem to demur from Roberts’ calling him an egomaniac. Roberts quotes Conway’s third wife, Diana, describing him as “the most selfish, childlike person I have ever met”, though also “the most interesting person I have ever met”.

Conway may be fairly typical of mathematicians in his informality. (Roberts cites Conway’s colleague Peter Goddard speaking at a rather formal international congress: “‘Since I’m going to get down to work, you’ll excuse me if I become a mathematician” he said, taking off his jacket and tie to applause”.)

Conway is not typical in his flamboyance, nor in his egomania. But in Roberts’ hands, those traits make for an interesting story.

Conway was a junior lecturer at Cambridge when I studied there, at first not at all famous. He was known to be flamboyant and maverick, but he wasn’t even the “star” lecturer. The lecturer in group theory (which was then Conway’s own area) who could have students sitting in the aisles and avidly perched on the edge of their seats was not Conway, but the not-all-all-famous-except-among-mathematicians Jim Roseblade. Conway started to become famous around the same time that I finished at Cambridge, and seems to have developed many of his maverick and self-indulgent traits as a function of fame rather than mathematics.

Amir Alexander’s book describes the arguments about the idea of “infinitesimals” – quantities smaller than any real number, but still bigger than 0 – as that idea came into increasing use in maths in the 16th and 17th centuries, culminating in the development of calculus by Newton and Leibniz.

There were not just arguments, but furious conflicts. The Jesuits, a powerful factor in the Counter-Reformation, vehemently dismissed infinitesimals as nonsense. So did Thomas Hobbes, who was probably in private an atheist but in political philosophy an advocate of absolute monarchy.

Alexander identifies the advocacy of infinitesimals with more liberal, democratic, scientifically open-minded and inquiring trends.

I think he makes too much of his thesis. He ends by arguing (with qualifications) that the anathematising of the idea of infinitesimals in Italy, and its relative welcome in England, determined the economic and intellectual stagnation of Italy in the next centuries, and the rise of England as a scientific and industrial centre.

He doesn’t deal with the fact that, in the terms argued at the time, the Jesuits were right. Infinitesimals, as discussed by 16th and 17th mathematicians, were a contradictory and illogical concept.

He is right that mathematics often moves forward by building provisionally on dodgy foundations, and hoping for later research to make those foundations sound.

In the 19th century Cauchy, Weierstrass and others put a firm foundation under calculus, essentially by replacing the concept of infinitesimal by the concept of limit.

In the 1960s Abraham Robinson turned the tables by introducing a logically sound concept of infinitesimals with his hyperreal numbers. There is a clear and easy-to-understand introduction to his ideas at http://mathforum.org/dr.math/faq/analysis_hyperreals.html and a video on how that fits in with the history at:

Conway’s surreal numbers also include a logically sound concept of infinitesimals.