Concepts of Modern Mathematics

The art of popularising mathematics has a long history — possibly beginning with the publication of Jacques Ozanam’s Récréations Mathématiques et physiques (1694). Much later contributions include Lancelot Hogben’s Mathematics for the Million (1936), which has an historical and practical emphasis. There is another Dover publication — W.W. Sawyer’s Prelude to Mathematics (1955 and 1982) — which also explores the nature of mathematical thinking in the context of ‘modern’ mathematical ideas. Other notable writers in this field include Martin Gardner, Eugene Northrop and, of course, the prolific Ian Stewart.

The first edition of this particular book appeared in 1975, and the idea for it came from an extramural course given in 1971 by Ian Stewart for citizens who wanted to understand what was meant by ‘modern mathematics’ (in the UK) or ‘new math’ (in the USA). Some of the contents are in keeping with this aim. Because the controversy concerning the school mathematics curriculum had abated by the year of its first edition, the book’s main purpose became ‘to explain why the underlying abstract point of view had gained currency among research mathematicians’. Hence, there is continual reference to research activity throughout the book.

Early discussion dwells upon a range of general mathematical themes that form a platform of understanding to sustain a mathematical lifetime. Themes include psychological issues, such as intuition versus formalism, the role of mathematical imagery (pictures) and the importance of questioning. Later chapters address matters of ‘indirect thinking’, the role of axiomatics, the Hilbert programme and Gödel’s ideas on decidability.

The mathematical contents of this book could be grouped under the headings of algebra, number, geometry, analysis and probability. Algebra includes an introduction to groups, rings, fields and linear algebra. The coverage of geometry is very unlike a synthetic approach based on Euclid. Yet, although symmetry and transformations are explained in a lively manner, readers may not be convinced that such methods could yield the myriad of theorems provided by the synthetic arguments of Euclid.

Since the introduction to topological thinking is highly visual, it could be classified as being geometric (despite there being no metrics). Within 56 very well illustrated pages, it proceeds from notions of rubber sheet geometry, the 4-colour problem, topological invariants and the fundamental group. The narrative then extends to the nature of hyperspace leading to the Euler-Poincaré formula and the nth homotopy group. Who wouldn’t be interested in all this?

Other mathematical topics include functions, counting and cardinality, a very short chapter on aspects of real analysis, probability and the role of computers in mathematics. There is shown to be much interrelationship between most of these themes. Consequently, a holistic view of mathematics is on offer here (but an introduction to equivalence relations would have added greater cohesion)

Whatever the mathematical background of the reader, he/she would come away from this book having learned much mathematics and, more importantly, wanting to know even more about it. Some aspects of the book require a higher level of prior knowledge — and some of the material could (I don’t say ‘should’) have been developed less formally (for example, showing that \(\mathbb{Q}\) has measure zero).

The chapter ‘Language of Sets’ omits discussion of the value of sets with respect to classifying types of natural number or 2-dimensional shapes. One final quibble concerns the short chapter ‘Real Analysis’ which, apart from mention of the completeness axiom, doesn’t come across here as a ‘modern’ topic. Moreover, the assertion in chapter 1 about the power of visual imagery could have been further exemplified by concrete illustration of the infinite series \(1+\frac12 + \frac14 + \frac18 + \dots\), thereby dispelling the expressed concern that it may mean nothing at all.

Peter Ruane taught mathematics to people from the age of 5 to 55 — that is, from early school arithmetic to transfinite arithmetic.