Robyn Williams:Today's talk gives an unanswerable reason why girls shouldn't do mathematics. At least not in the 5th Century AD. It's an extraordinary story and here to tell it is Maths Lecturer from Monash, Dr Michael Deakin.

Dr Deakin:Imagine a time when the world's greatest living mathematician was a woman, indeed a physically beautiful woman, and a woman who was simultaneously the world's leading astronomer.And imagine that she conducted her life and her professional work in a city as turbulent and troubled as Ayodhya or Amritsar, Belfast or Beirut is today.

And imagine such a female mathematician achieving fame not only in her specialist field, but also as a philosopher and religious thinker, who attracted a large popular following.

And imagine her as a virgin martyr killed, not for her Christianity, but by Christians because she was not one of them.

And imagine that the guilt of her death was widely whispered to lie at the door of one of Christianity's most honoured and significant saints.

Would we not expect to have heard of all this? Would it not be shouted from the rooftops? Would it not be possible to walk into any bookstore and buy a biography of this woman? Would not her life be common knowledge?

You would think so, but such is not the case. And that is the reason for this talk.

For Hypatia of Alexandria was indeed, at the time she was killed by Christian fanatics, the world's foremost mathematician and astronomer and also a leading neoplatonist philosopher. Physically beautiful, devotedly celibate, she was the revered teacher of a man (Synesius of Cyrene) who, after his conversion to Christianity, helped formulate the Christian doctrine of the Trinity, using neoplatonist principles learned at her feet.

And yes, the shadow of guilt over her lynch-murder still clouds the memory of St Cyril of Alexandria, Doctor of the Universal Church, and in particular, Doctor of the Incarnation.

Hypatia is not quite the first woman mathematician of whom we know. (There are at least two earlier claimants to that title.) She is, however, the first of whom we have reasonably detailed and reliable information. She was also the most eminent female mathematician of ancient times - until the 18th century quite unmatched, and still the only woman of whom it can be claimed that she was absolutely pre-eminent in the mathematical world of her day.

I have devoted about six years to the task of finding out about her, what it was she did, separating fact from fiction, chasing up obscure works in which her life is discussed, trying to get a feel for the mathematics of her era and seeking to understand where she fitted into the intellectual life of her time.

The task is made more difficult by the fact that many legends have come to surround her, and much of what is popularly available is sourced to fancy or to fiction rather than to properly based historical record.

Reliable accounts must all derive from the primary sources that have come down to us. I have collected all these, and in English translation, they run in total to some 14 typed pages. They include Christian writings, "pagan" histories and "heretical" documents.

But perhaps a little background first. We now believe that Hypatia was born in the middle years of the fourth century of the Christian era. Around 350 say. Her home-town was Alexandria, in Egypt, nowadays an Arab state, but in those days Greek. Her father was Theon of Alexandria, also a mathematician, best remembered today as the source of our text of Euclid's Elements, but also a major commentator of the work of Ptolemy, the astronomer whose account of the solar system reigned supreme until supplanted by Copernicus in relatively recent times.

Theon had in fact succeeded Euclid (but some 600+ years later) as professor mathematics in what, using modern terminology, we might describe as "The University of Alexandria". (It is, to be technical, commonly known as "The Museum", but this gives modern eyes and ears quite the wrong impression.) In all likelihood, Euclid was the first and Theon the last of these professors of mathematics. We know of no scholar associated with "The Museum" after Theon.

Theon himself was his daughter's teacher, and later the two collaborated on an astronomical table. We aren't entirely sure quite what this was or quite what the nature of the collaboration. However, an early list of her publications gives this along with two other mathematical works. These were commentaries (probably student texts) based on the work of earlier mathematicians. One of these was a book on algebra, the other in geometry. Sadly very little of this survives, although there have been some attempts to reconstruct it.

We may also accept the many endorsements of her effectiveness, indeed charisma, as a teacher of mathematics.

Hypatia was however not only a mathematician, but also a notable philosopher and religious thinker. Her thought was neoplatonist, and although there is some dispute about detail, we can get a feel for its general outline. Neoplatonic thought contained a strand that is strongly mathematical and which indeed makes mathematics a profoundly sacred pursuit. The underlying ideas go something like this.

Quite early on in life, we learn to abstract from individual instances and so reach a general idea of concept. Thus from our experience of observing pairs of objects (socks, gloves, parents, whatever), we come to form a notion of the number 2. Once we have done this, 2 becomes quite real for us; 2 exists, although not of course in the same straightforward way that the socks, the gloves, the parents, etc. do. This is an example of what is called a platonic ideal.

Here, from a much later work, a 20th century work, is an instance of how such ideals function. "... the exact sciences [are not] based on an accumulation of statistics. In order to teach the young that four plus three makes seven, you do not add four cakes plus three cakes nor four bishops plus three bishops nor four cooperatives plus three cooperatives... . Once the principle has been intuited, the youthful mathematician grasps that four plus three invariably make seven and he does not have to prove it over and over again with chocolates, man-eating tigers, oysters or telescopes.

The platonic ideal is thus seen as more real, more meaningful, deeper, than the mere instances by which we are led to infer it.

These are of course other, non-mathematical, examples of platonic ideals: truth, beauty, goodness and suchlike. We use them every day. But perhaps the mathematical example is the clearest. Plato himself held it in high regard, and I think we can safely say that Hypatia herself did also.

But now suppose that we perform a further act of abstraction. Just as the ideal is the profound reality and the exemplars merely its manifestation, so, beyond the ideals, is an even deeper reality: the idea of ideas, called the One. For neoplatonists, the One was not at all unlike the Judaeo-Christian God, and indeed much Christian theology is today permeated with neoplatonist language and thought.

One reason we know that Hypatia's philosophy gave a central place to mathematics is that her reliance on mathematics is the source of one of the few critical comments on her that have come down from antiquity. the philosopher Damascius compared her unfavourably with his own idol and mentor, Isidorus: "Isidorus," he wrote, "greatly outshone Hypatia, not just because he was a man and she a woman, but in the way a genuine philosopher will over a mere geometer."

Women and mathematicians alike will warm to the gloss put on this by one modern scholar: "It means in plain language that Isidorus knew nothing of mathematics."

However, it was not a good time to be a mathematician. Nor a neoplatonist. By the late fourth century the Roman empire was divided and beset, officially Christian, but holding within its sway various others: Jews, heretical sects, diverse schools of neoplatonists and other assorted "pagans" - and all of them at one another's throats. Alexandria in particular was seething with intercommunal rivalry and sectarian bitterness. In either 391 or 392, the Christian archbishop, Theophilus, obtained imperial permission to raze to the ground the temple of Serapis, the particular deity of many Alexandrian pagans. After a series of bloody battles, he succeeded in this aim and set out to establish on the site a church dedicated to St John the Baptist, of whose body he had custody of some alleged relics. One theory has it that this action also and finally put paid to the Museum and its once glorious library.

In such a climate, mathematicians came to have a bad name, and it's worth taking a little time to see why this should be so. The ptolemaic system of the universe (despite its later overthrow by the copernican) was one of the world's earliest and still greatest scientific achievements. By its use, the positions of the planets could be foretold, and even more tellingly, eclipses could be predicted with great accuracy.

Then as now, the business of science was, in large measure, the quest to foretell the future. But for ordinary people in the street, the future that most interested them was not so much the state of the heavens, as their own immediate future. Into this void came astrology. The sun and the moon ruled the calendar and the seasons, or even more spectacularly, gave rise to eclipses. The other variable heavenly bodies, the planets, it was thought, likewise had effects. They presided over the detail of our daily lives. (Some people still believe this.)

And so there arose numerous charlatans: astrologers and numerologists in today's terminology. The bad currency drove out the good. Reputable astronomers and geometers like Theon and Hypatia got confused in the popular and in the ecclesiastical mind with these fly-by-nights. All were lumped together as "mathematicians".

The council of Laodicea in the mid-4th century outlawed "divination" and forbade priests to practise mathematics; at about the same time. the emperor Constantius promulgated a law that "No one may consult a soothsayer or a mathematician". Apparently not everyone complied. Hypatia's contemporary, St Augustine of Hippo tried a neat bit of casuistry before disillusionment set in: "Those imposters," he wrote, "whom they call 'Mathematicians' I consulted without scruple; because they seemed to use no sacrifice nor to pray to any spirit for their divinations" which art, however, Christian piety consistently rejects and condemns."

This is the background to Hypatia's murder. In the year 412 Archbishop Theophilus died and was succeeded by his nephew Cyril. Although Theophilus had razed the temple of Serapis, he had never, in over 30 years, moved against Hypatia. In part this may very well have been a result of his friendship with Hypatia's influential and adoring pupil, Synesius of Cyrene. Synesius himself died in 413 or thereabouts, and so Hypatia was suddenly left without her powerful protectors.

Cyril, making use of a 500-strong private militia, began to exert his authority in the temporal as well as in the spiritual sphere, and thus he came into conflict with the civil governor, Orestes, in the course of a series of increasingly violent confrontations between the various factions in the city. In 415 and as the violence escalated, his powerful militia made a direct assassination attempt on Orestes himself. However this failed and the governor was able to apprehend one of the ringleaders, whom he tortured so severely that the man died. Cyril was furious and, managing to get hold of the body, pronounced over it a ceremony of canonisation, enrolling the would-be assassin in the calendar of the saints (where incidentally he remained until this century).

However the militia were not about to leave matters there. Hypatia was close to Orestes and there was a rumour that it was Hypatia's influence that prevented the Christian Orestes from accepting Cyril's spiritual direction and so becoming reconciled with his rival. Moreover she was seen as one "devoted at all times to magic, astrolabes and instruments of music, [who] beguiled many people through her satanic wiles, and the governor ... through her magic." That this was not in fact true is beside the point; it was believed and was bruited abroad.

So somewhat later, as Hypatia was returning home, she was set upon, torn from her carriage and dragged into a church, where she was stripped naked and battered to death with roofing tiles, "and while she was still feebly twitching they beat her eyes out". They then orgiastically tore her body limb from limb, took her mangled remains out from the church, and burned them.

The question of Cyril's complicity in this affair has long been hotly debated. Assessments are normally made along "party lines". Roman Catholic commentators normally stoutly defend him; anti-clericals gleefully denounce the man. Perhaps the closest we find to a balanced view is that of the Anglican historian Canon Bright, who wrote: "Cyril was no party to this hideous deed, but it was the work of men whose passions he had originally called out. Had there been no [earlier such episodes], there would doubtless have been no murder of Hypatia."

But even this moderate line may perhaps be queried. Cyril is especially venerated by the Coptic church, and last century, among their archives, an Ethiopic translation of an old Greek text was discovered. So if more needs to be said, let us take it from the mouth of Cyril's most vigorous defender, the Coptic bishop John of Nikiu: "[After Hypatia's death] all the people surrounded the patriach Cyril and named him 'the new Theophilus'; for he had destroyed the last remains of idolatry in the city."

With this as the defence, what further need have we of witnesses?

Robyn Williams:What need indeed. That was Michael Deakin who lectures in mathematics at Monash University in Melbourne. And Hypatia also featured in Margaret Wetheim's excellent book "Pythagoras's Trousers". Next week Okham's Razor comes from Canberra where Colin Groves has been thinking all about Okham's Razor, and Charles Darwin.

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