*Enlightening Symbols* is mostly a historical study of the mathematical symbols we encounter in school arithmetic and elementary algebra, informed by a larger interest in the formidable utility mathematical symbols possess. What it offers is not a general or theoretical study of mathematical symbols per se (which the title might suggest), but an informative, highly readable and scholarly history of a small but fundamental subset of symbols familiar from the schoolroom in two parts, followed by a third which attempts to bring to light their hidden power.

The first part tracks the writing of numbers, chiefly by Greek, Indian, Arab and Hebrew mathematicians, homing in on the symbolic marvel of zero as a number and placeholder in the familiar Hindu-Arabic system of numerals. Joseph Mazur takes us from the first recorded known use of zero as a number in the *Brāhmasphutasiddhānta* (AD 628) by the Indian astronomer and mathematician Brahmagupta, through the work of the mathematician al-Khwārizmī, who translated Brahmagupta’s text into Arabic in AD 820, to Fibonacci of Pisa’s famous *Liber abbaci* (1202), written as an instruction manual for tradesman. Though the ground here has been covered many times, Mazur scores by offering a concise, accessible path through the material while weaving in contemporary scholarship challenging several widely held positions. As well as the cleverness of early Chinese number characters, Mazur cites work that suggests that it’s ‘not too wild to consider the possibility that the Hindu-Arabic number system might have come from the Chinese counting rods’ in use circa 1000 BC (he also records that, contrary to general opinion, the Chinese manipulated negative numbers four centuries before Brahmagupta). At another point, he questions the view, widely held by medievalists, that Fibonacci’s *Liber abbaci* was the inspiration for introducing modern – zero-based – arithmetic to the West.

The second part leaves zero-based numerals aside to narrate the transition from rhetorical mathematics – problems posed and solved in words or their abbreviations – to algebra, which replaces words by symbols. For example, al-Khwārizmī in his *Algebra* solves the following problem: ‘I have divided ten into two parts, and multiplying one of these by the other the result was twenty-one … Then you know that one of the parts is thing’. Nowadays we learn in school to write this as *x* + *y* = 10 and *xy* = 21 to get the quadratic equation *x*^{2} – 10*x* + 21 = 0 with solutions 7 and 3. Mazur charts the gulf separating al-Khwārizmī from us: from the verbal lexicon of ‘plus’, ‘times’, ‘divides’, ‘minus’, ‘square root’, ‘equals’, and ‘the thing to be found’ to the symbols +, x, ÷, -, √, = and *x* respectively. He traces the typographical and syntactic struggles and conceptual obstacles behind each of these transformations. Far from being the simple signifying moves they appear to us, each was fraught with complications and interpretive ambiguities. By the second half of the 16th century, through the work of François Viète, Rafael Bombelli, Simon Stevin and others, the symbolic replacement of rhetorical mathematics was almost complete. It needed only Descartes’s use of different letters for constants and for variables for the 17th-century ‘explosion’ of symbolic mathematics to occur.

In the final part Mazur goes after those hidden powers. He sees symbols as highly complex objects, likening them to ‘suitcases’ packed with ideas, rich in metaphorical associations, their power arising from an abstractive linking of conscious thought to ‘the collective subconscious’. Unlike the tightly focused scholarship of the previous parts, the account here travels lightly, principally through brief forays into psychology. We are presented with a cognitive-science study of eye movements of subjects negotiating complex symbolic expressions; the Invisible Gorilla effect is invoked to suggest a possible mathematical parallel of ‘inattentional blindness’ when thinking symbolically; and he gives an introspective glance of his own reflections during the course of a simple proof. He also repeatedly compares mathematical symbols with poetic ones: ‘Like any great poem, Maxwell’s equations tell us far more than what appears in the language.’ Unfortunately Mazur doesn’t supply anything of substance linking the two. He does, however, provide concrete examples where a new symbol had radical consequences. So, for example, we get the mathematical effect of the symbolising of √-1 by *i*. The move facilitates the emergence of a new type of entity – complex numbers of the form *x* + *iy* – from the token √-1, which constitutes a major extension of what had previously been understood by ‘number’.

Mazur concludes that the beauty found in mathematics – elegance of proofs, simplicity and economy of expression, ingenuity of concepts – is due ‘in a large part […] to the illuminating efficiency of smart and tidy symbols’.