Luke Heaton 2015. A Brief History of Mathematics. Key concepts and where they come from. London: Robinson, 321 pp.

This is the “foundations of mathematics” book that I’ve been seeking for a long time. Luke Heaton has an enviable grasp of, surely, the greatest topic of all and has distilled the essence of it into 13 chapters. A book to return to often.

Beginnings has some slightly unconvincing discussion of cognition, symbols, rituals in prehistoric humans. The question of whether the concept of counting and recording numbers arose independently mutliple times or was spread with human dispersal is only touched on. Egypt, Babylon and Mesopotamia too are here but not in depth. The research into the Babylonian cuneiform tablets doesn’t make it and nor does Hamurabi and his stele.

From Greece to Rome is probably not when the grand human project to discover mathematics began, but it is where the book takes off. That the Greeks had the wit and the curiosity to begin to think about philosophy and mathematics (and that society fostered it) is a thing of wonder. No doubt the telescoping of 2,000+ years and the scarce archaelogical record mean that antecedents are mostly invisible, although surely they were there. The Minoans, for example.

Ratio and Proportion: not only that and sequences but also the nugget that Leonardo Fibonacci was the first to use decimal notation in Liber Abaci (1202).

The Rise of Algebra and Mechanics and the Calculus here are given their historical and mathematical contexts with concise clarity, especially on the two protagonists of calculus, Gottfried Wilhelm Leibniz (who published the fundamental theorem of calculus in 1684) and Isaac Newton (who had known of it for nearly 20 years but couldn’t be bothered telling anyone). (Very much as would happen 200 years later with Charles Darwin and Alfred Russel Wallace and natural selection.) Leibniz in his proof was explicit about infinitesimals (and had an eye to fame) while Newton was more in the geometry line of argumenation (and didn’t care who thought what about him).

Leonhard Euler and the Bridges of Königsburg is a short account of travelling salesman-type NP complete problems as conceived by Leonhard Euler, and about Euler numbers for geometrical objects, and about dimensions.

Euclid’s Fifth and the Reinvention of Geometry almost makes sense of non-Euclidean geometry.

Working with the Infinite is a fascinating chapter on Blaise Pascal, George Cantor and investigations into infinity, the inspired diagonal argument (later used by Alan Turing and others) and the conclusion that infinity comes in different sizes.

The Structure of Logical Form is not just about George Boole but Leibniz is involved again (despite these multiple mentions and his seminal work he somehow got left our of the index).

Alan Turing and the Concept of Computation Turing Machine explained as Turing Cards (which makes the logic of the concept much more clear) and Turing’s mentor Alonzo Church given credit for a change.

Kurt Gödel and the Power of Polynomials is surely the most important and sobering chapter of all. Kurt Gödel’s theorem is given an inspired explanation in parable form

Modelling the World is strangely lightweight compared with other chapters, many of which require a lot of the reader (well, this reader). Although it did open up yet more future reading by Denis Noble and others on systems theory in biology (a reaction against gene-based reductionism?).

In the final chapter Lived Experience and the Nature of Facts I was hoping for some discussion of “Why mathematics works” but is more of a discussion of the difference between understanding something and understanding what we can say about something. As with physics (Niels Bohr: “It is wrong to think that the task of physics is to find out how Nature is. Physics concerns what we can say about Nature”). Luke Heaton: “In my view, mathematics comes into being with the rule-overned use of symbols: without the explicit rules or principles set down by ourselves or our predecessors, there can be no mathematical facts” (pp. 294-295).

There are some interesting omissions, The Reverend Bayes is not mentioned, even though there is a chapter on modelling. Chaos is mentioned briefly but not Edward Lorenz or Benoît Mandelbrot. Presumably Luke Heaton classes these as not “mathematics” or not “foundational”. Arabian contributions to understanding of mathematics are mentioned now and then (especially regarding algebra) but seem somehow undersold.

The Further Reading section is short but a high proportion of the titles listed were already, or are now, on my in-tray. They are arranged under six headings that are conceptual and don’t relate to the largely chronological arrangement of the chapters. More useful that way.