# Review: Iconoclasts and mathematics

The Crest of the Peacock: Non-European Roots of Mathematics by George

Ghevarghese Joseph, I. B. Tauris, pp 368, £14.95

The Mathematics of Plato’s Academy: A New Reconstruction by David H.

Fowler, Oxford University Press, pp 401, £20 pbk

What impression of the mathematical past is given to schoolchildren?

It is a fair bet that the hazy notions with which many children leave school

have included at least two assumptions. One is that the mainstream of modern

mathematics is something that comes down to us from the ancient Greeks,

with maybe a minor custodial and transmissional role played by some medieval

Arabs. The other is that a really important part of that classical Greek

heritage is the contribution of Pythagoras and his followers, who dis-covered

the awful secret that not all lengths are commen-surable – that some numbers

are ‘irrational’.

These two books aim to dislodge each of these assumptions from its accepted

place in the popular history of mathematics. After reading them, we cannot

see the past in the same comforting haze of age-old stories, faithfully

and uncritically retold from teacher to pupil down the years.

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George Ghevarghese Joseph is well equipped to understand the diversity

and richness of mathematical traditions. Born in southern India to a family

of Syrian Orthodox Christians, he grew up in East Africa and received his

higher education in Britain, where he now lives. His experience is a microcosm

of the story he tells, one in which mathematics is a universal cultural

pursuit, in which important and influential work has been done well away

from the narrow line from ancient Greece to modern Europe. He is especially

aware – as he shows with several choice quotations – of how the apparently

objective judgments of historians and teachers owe much more to deep prejudices

and assumptions than they would have us believe.

Joseph gives enthralling accounts of the mathematics we find on ancient

Egyptian papyri and Babylonian clay tablets, both dating from at least a

millennium and a half before the Greek miracle. He deals thoroughly with

the mathematical developments of ancient and medieval China and India, before

summarising the Arab contributions to the consolidation, synthesis and creative

passing on of earlier mathematical traditions.

Joseph is careful to distinguish his account of mathematical developments

in various cultures from his running argument with historians who have elevated

Greek mathematics by diminishing the significance and role of the mathematics

of other cultures. His book is of value and interest, therefore, even for

readers with less commitment to uncovering the Eurocentric judgments of

historians. He helps today’s reader by presenting most of the past mathematical

work in an accessible modern algebraic or geometric language rather than

in its original form. Perhaps in a subsequent book Joseph will satisfy the

curiosity of readers wanting to put in a little more effort to work through

past problems and solutions in something closer to past forms and methods.

There is an increasing awareness today that schoolchildren in a multicultural

society need exposure to a full range of cultural riches of the past. This

helps them to develop pride in their individual traditions, as well as understanding

how the world has been drawing together. Joseph’s book is an important contribution

to this movement, and will be invaluable for mathematics teachers at all

levels.

Of course, extraordinary contributions to mathematics were indeed made

in ancient Greece, as no one would deny. How to characterise them? The peculiarly

Greek approach to the proof and development of axiomatic structures of mathematical

knowledge has been especially influential. It is hard to pin down the full

story here, though, in part because of the unsatisfactory state of the surviving

textual evidence. Curiously, we have many more extant primary sources from

Babylonia and Egypt, in the form of clay tablets and papyri, than we have

from the Greek mathematical culture of 1500 years later. What we know of

ancient Greece is through copies of copies of copies of ancient texts, and

allusive remarks in texts dealing with other matters. This is true even

of the most famous of Greek mathematical works, Euclid’s Elements, whose

earliest extant complete manuscript dates from nearer our time than that

of Euclid.

David Fowler’s ‘new reconstruction’ of Greek mathematics of the 4th

century BC takes such textual concerns very much to heart, and shows an

admirable refusal to speculate beyond what is consistent with an honest

and careful – but highly readable – evaluation of the evidence we have.

With this process he helps us to re-evaluate the well-loved stories of Pythagoras:

they date from so long after Pythagoras’ time, and are so consistent with

the educational and political concerns of the period when we first learn

of them, that their historical accuracy is open to considerable doubt.

The mathematics that Fowler tells us about, which explores different

ways of characterising the concept of ratio, is of a kind that has, on the

whole, dropped out of sight subsequently. One of the things he has to explain

is why the mathematical tradition he has fascinatingly reconstructed had

apparently so little impact on later mathematical texts. His account gains

plausibility when one realises that mathematics is dominated by fashion,

characterised by stops and starts, and prone to losing past knowledge, contrary

to its ideological standing in popular history.

The great value of The Crest of the Peacock and The Mathematics of Plato’s

Academy is the bringing of fine critical intelligences to bear on popular

conceptions of the mathematical past. The particular story they each have

to tell is indeed full of interest and useful information, but above all

they set an example by addressing historical issues keenly. The reader emerges

not only with fresh information, but with a renewed sense that alert critical

judgment is the really important attribute to be developed in young people.

John Fauvel lectures at the Open University.