Many people would argue that his job is an impossible one at the best of times, and these are very bad times indeed. A crisis in the Church of England over female priests has coincided with a national upsurge of interest in spiritual matters. In the background loom financial difficulties as the investments on which the Church have long lived seem less certain than for a long time. But Dr Carey has risen to neither opportunities nor problems.
The immensely difficult negotiations to establish a settlement that will allow supporters and opponents of female priests to coexist within the Church of England are largely being handled by the Archbishop of York, Dr John Habgood.
Even members of Dr Carey's own staff agree that Dr Habgood is the man for this job, on which much of the future of the Church of England depends. He has all the qualities that the Archbishop of Canterbury lacks: personal authority, political experience, and intellectual stature.
One member of the Ecclesiastical Committee of Parliament, which is examining the legislation to enable the ordination of women, rang Lambeth Palace recently with a difficult question for the Archbishop of Canterbury. In due course, his question was answered - by the Archbishop of York, to whom Dr Carey's staff had automatically passed it.
The Bishop of London, Dr David Hope, is the man who will have the greatest responsibility for making a settlement over female priests work on the ground. He is reported to be infuriated by Dr Carey's inadequacies and clumsiness.
Friction between Lambeth Palace and London House, where the Bishop of London is based, is nothing new. In the late Eighties a state of open warfare raged between the then Bishop of London and Dr Robert Runcie.
However, London is the diocese with both the largest number of female deacons, and the largest number of irreconcilable opponents of women's ordination. Any settlement that can be made to stick there will work throughout the country; failure there could deal the Church of England a great blow.
Dr Hope, unlike his predecessor, Dr Graham Leonard, actually wants the Church of England to be able to accommodate female priests. It would matter if he had lost confidence in Dr Carey's ability to lead the Church of England out of the toils of the women question.
The difficulties here have been compounded because the Vatican seems now to have decided to encourage dissatisfied Anglican priests to leave the Church of England. It is probable that no more than a couple of hundred priests will do so in the first instance, but as many as a thousand could go over 10 years. Dr Carey, in an unguarded moment, has suggested he feels that these people would be no loss.
Almost the first public utterance of his primacy was to accuse opponents of women's ordination of 'a very grave heresy', a remark for which he later apologised. However, he has never appeared to understand how ridiculous and hurtful the phrase seemed, coming from the leader of a church whose bishops are not called 'heretical' even when they pronounce themselves to be agnostic on the virgin birth or on bodily resurrection.
Dr Carey's extraordinary inability to grasp imaginatively the reasons which motivate many opponents of female priests is displayed again in a television programme that will be shown on Channel 4 at the weekend. In the programme he says that people who leave the Church of England 'just because their own particular point of view is not upheld by the Archbishop . . . are doing very serious damage to the Christian faith'.
This remark is glossed by Lambeth Palace as meaning that people who are bitter or divisive about female priests are weakening the Christian Church. It is possible to read the remarks that way. But that is not their plain sense.
This problem raises the other great difficulty that Dr Carey faces. What he would probably like to do is to present himself as a public figure: an ordinary Christian who can leap through the offputting formality and pomp of the Established Church and relate to other ordinary blokes.
'The People's Archbishop' was an epithet applied to his early period, to distinguish him from the supposedly elitist Dr Runcie. But the contrast is not entirely accurate. Dr Runcie was not born elitist. He rose into the elite. His mother was a ship's hairdresser, and he was himself a scholarship boy at Oxford.
In fact Dr Runcie had had a much wider experience of the world outside the Church of England than George Carey, born poor in Barking, whose social rise has all taken place inside the Church of England. And Dr Runcie's style of studied ambiguity was very different from the unschooled clumsiness of his successor. He would have not have offended opponents of women's ordination by accident.
Dr Carey's instinct is to turn away from this row, which is incomprehensible to most of those outside the Church of England and even many inside it, and to speak directly to the people on the social and moral issues that really concern them.
His great problem here is addressing his audience in a language that they can understand: working out what he wants to say and then saying it clearly. After his first, disastrous interviews, he was sent on a BBC course to learn public speaking and television manners, and for some time he kept away from the spotlight of the media.
Now he has learnt caution without concision. When he says something memorable, he must often apologise for it later. To read an interview in which he perpetrates no gaffes, however, is like making a substantial meal of foam rubber.
In the last year, he has delivered 72 speeches or sermons; and by far the greater part of these were his own work. This would be a considerable burden by itself. But he also undertook 10 overseas visits. He is off to Korea after Easter: in May he will visit Moscow, Georgia, and Armenia.
In Korea, the Anglican Church thrives, as does every sort of Christianity, but there are few, if any, Anglicans in the Caucasus, and Dr Carey is not going there to succour them. He sees himself as a world spiritual leader in a way that few of his predecessors did. In fact his habit of lecturing the Roman Catholic Church at a time when many Roman Catholics believe he is presiding over the disintegration of the Church of England has caused more embarrassment than almost anything else he has done.
His role as leader of the Anglican Communion is one he takes increasingly seriously. The Anglican Communion, however, is an extraordinarily disorganised thing, with hardly more substance than the Commonwealth, and to critics, his travels signify a desire to escape the pressures of the job in Britain to foreign countries where huge crowds will turn out to cheer him.
No such heady delights are available at home. But he does receive quiet praise from the House of Bishops, where he is spoken of as a chairman who gets the business done.
With one or two exceptions he is liked by his colleagues. They have come to terms with his limitations and they appreciate his energy. So long as the really difficult decisions are in the hands of Dr Habgood and Dr Hope, the Church of England will do about as well as it can in the management of female priests.
In the long term, Dr Carey's encouragement of a more outward-looking church concerned to gain members can hardly do damage. No one doubts that he was an admirable diocesan bishop; it is just that no one outside Lambeth Palace sees him as a world spiritual leader either.
The irony in all this is that Dr Carey, like John Major, has delivered to his supporters all the qualities for which he was chosen. In both cases, the essential qualification was to be as unlike his predecessor as possible.
Where Dr Runcie, with his delicate political antennae, could seem paralysed by the ambiguities and limitations of an archbishop's job, Dr Carey has no such doubts. He knows where he wants the Church to go: he wants a streamlined, efficient, Protestant organisation.
'Jesus was a management expert,' he once famously said. Such a church might be a good thing, but it would differ in almost every respect from the Church of England that he inherited.
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