Then Mr Livingstone said: "This is not an ideology or even a perverted faith." Why did he want to say that? How - if, as the authorities tell us, the attacks were carried out by Islamist extremists - could this be true?
The main spokesman for the Metropolitan Police on Thursday was Deputy Assistant Commissioner Brian Paddick. He also complained about attacks on "purely innocent members of the public", thereby making one think that there might be other people (police? soldiers? politicians?), who are not purely innocent and should have been attacked instead. Asked about the nature of the terrorists, Mr Paddick said: "Islam and terrorism don't go together."
It is true that the vast majority of Muslims are not terrorists, or involved in terrorism, and this needs to be said strongly if people assert otherwise. But if the Metropolitan Police really believe what Brian Paddick says, if they really, truly think that the words "Islam" and "terrorism" must not be linked, then we have little hope of catching the killers, of understanding how the terrorism works, or of preventing new atrocities.
You can show this with a simple comparison. When Britain was afflicted by Irish republican terrorism, most Irish people repudiated that terrorism. It was nevertheless the case that the great majority of the terrorists - more than 95 per cent - were Irish, or of Irish origin, and they drew overwhelmingly on Irish people to help and hide them.
This was not a funny coincidence. It was because the IRA preached a doctrine about Ireland and called on the loyalty of a perverted version of Irishness. Therefore, the words "Irish" and "terrorist" went together, hard though this was on the majority of Irish people. The Brian Paddicks of the day would have been appallingly negligent if they had not concentrated their investigations among the Irish. And the vigilance of the public, which the police then and now rightly call for, inevitably directed itself towards Irish neighbours, Irish accents, Irish pubs.
So it must be with Muslims in Britain. In fact, the situation is more serious because we are dealing with a religion, not merely a national aspiration, and the demands of a religion are more absolute than anything else. If fanatics can persuade people that their religion insists that they kill others (and often themselves) in its service, then they will obey. And whereas the IRA, though utterly sadistic and fanatical, kept in mind a political aim which, once achieved, would mean that they need kill no longer, the religious fanatic lacks even this check on his behaviour.
From time to time, perhaps, he will kill for a specific reason - to take power in one country, to drive foreign troops out of another - but, in principle, there is no end to his killing until everyone who does not share his particular version of truth is exterminated.
What strikes one again and again about the reaction of the public authorities, of commentators, of the media, is the terrible lethargy about studying what it is we are up against. We are dealing with an extreme interpretation of one of the great religions of the world.
We flap around, looking for moderates and giving them knighthoods, making placatory noises, putting bits of Islam on to the multi-faith menu in schools, banishing Bibles from hospital beds, trying to criminalise the expression of "religious hatred", blaming George Bush and Tony Blair. But if we do not know the way the faith in question works, its history, its quarrels, its laws and demands, we will not have the faintest chance of distinguishing the true moderate from the fellow-traveller or of bearing down on the fanaticism.
If you look at the Koran, you will find many glorifications of violence. In Sura No 8, for example, God is quoted as saying: "I shall cast terror into the hearts of the infidels. Strike off their heads, strike off the very tips of their fingers!" This punishment comes to them for having "defied God and His apostle". It seems reasonable to ask Muslims what this sort of remark means in the modern world.
Some will counter that there are plenty of equally nasty dictums in the Old Testament. This is true - though it is surely significant that they are very much harder to find in the New Testament. History is full of violent deeds done in the name of the Christian God.
But it is an important fact about Christianity in the past two or three centuries that it has conducted a great reinterpretation of these texts and of how the faithful should follow them. The struggle against the enemy in the Book of Joshua, say, or in Judges is now seen as a strictly spiritual one. The idea that these are divine 007 licences to kill has been explicitly repudiated.
Has the equivalent happened in Islam? Certainly, most Muslim leaders advocate peace and most are surely sincere in doing so. But push a bit harder, and you encounter some interesting problems.
I have asked, for example, if the Muslim Council of Britain, the mainstream umbrella organisation in this country, will condemn the killing of British troops in Iraq. They will not do so in absolute terms. They prefer instead to condemn the war itself, which is by no means the same thing.
Take a case from the dramas on Thursday. One heartening tableau was of the Bishop of Stepney, Stephen Oliver, appearing with Mohammed Abdul Bari from the East London Mosque, both condemning the attacks. But if you look up Mohammed Abdul Bari, you find that he welcomed to the opening of the London Muslim Centre Sheikh Abdul Rahman al Sudais, the Saudi-government-appointed imam of the Grand Mosque in Mecca.
In Mecca two years ago, al Sudais described Jews as "scum of the earth", "rats of the world" and "monkeys and pigs who should be annihilated". Yet, criticise al Sudais, and Mohammed Abdul Bari leaps furiously to his defence.
As I write, I have beside me an article that appeared during our recent election campaign in Muslim Weekly. By Sheikh Dr Abdalqadir as-Sufi, it calls for the replacement of British parliamentary democracy with "a new civilisation based on the worship of Allah", attacks the Conservatives for being "in the hands of an illegal Jewish immigrant from Romania" and speaks of the "near-demented judaic banking elite".
These views are expressed by an educated Muslim in a Muslim publication. Are these Muslim views, non-Muslim views, anti-Muslim views?
The mayor of our bombed city has himself got involved with Muslim leaders who say some interesting things. Last year, Mr Livingstone extended a warm welcome in London to Yusuf al-Qaradawi, a mainstream, world-famous spiritual leader based in Qatar.
Qaradawi has supported suicide bombing against Israelis, the treatment of all Jews as legitimate targets, the whipping of homosexuals and the killing of all Americans - civilian and military - in Iraq. Surely, Ken recognises an ideology here, and a faith of sorts? Yet he praised, rather than condemned, and so now, when the logical extension of such ideas hits King's Cross and the Edgware Road and kills dozens of his voters, he has to say that such deeds arise from no belief at all.
There seem to be two broad reasons why many Muslim leaders appear unable or unwilling to break absolutely with the teachings that give cover to violence. The first is that their religion is much more literal and much more political than modern Christianity. Its Prophet was a political and military leader.
The faith Mohammed taught does not just hope that the world will become Muslim. It wants all human society and politics to be governed by religious law: it draws no distinction between the secular and religious sphere (except to condemn the secular). Therefore, Muslim leaders find it very difficult to resist the hotheads who say that Sharia - the divine law - should be imposed wherever possible.
In addition, the religion is absolute in its attitude to particular bits of territory. It is forbidden, for example, that any other religion be practised in the Arabian peninsula, because that land is considered sacred to Islam. Therefore, it is hard for a "moderate" to oppose the second-class citizenship of Christians or Jews in Muslim lands, or to say that "infidels" fighting in Muslim countries should not be murdered - even when they are his fellow citizens in a Western country.
When someone like bin Laden says that Islam should confront the "Cross-worshippers" and the "Zionists", he is making a claim in which politics and religion dangerously reinforce one another - a claim which most Muslims might not like, but which most of their leaders cannot find quite the right words to resist.
The second reason is that the leaders are frightened. In private conversations with the moderates, one is always told that they are under "enormous pressure", that they risk losing control of their own people, and therefore they cannot say very fierce things against the extremists. One must accept that this pressure exists, which only goes to show how serious the problem is.
The Bishop of Stepney, say, would not have to look over his shoulder before he dared to condemn Christian suicide bombers (if there were any). But if his friend Mohammed Abdul Bari wants to condemn Muslim ones in Israel, then his life - or certainly his career - might be threatened.
So we have in our midst a religious minority in a state of ferment, and somewhere inside it a number of people (though a tiny proportion of the whole) who want to kill the rest of us. Now, it would seem, they or their foreign allies have succeeded. This country has suffered a greater land-based terrorist death toll than it has ever known before. Instead of subjecting our entire population to the loss of liberties and increase of bureaucratic power which identity cards involve, we should develop a strategy that works out much more precisely where the danger lies, and seeks it out.
Are we satisfied that our immigration and asylum system, and our ceding of much of it to European conventions, keeps a proper check on who comes in? Do our own laws give too ready an entitlement to people to join or marry family here? Do our judiciary now interpret the rights of immigrants and asylum-seekers so generously as to give the country almost no protection from those who abuse those rights?
What about the methods of the police? Sir Ian Blair, the Metropolitan Police Commissioner, has shown himself so obsessed with the implementation of the recommendations of the Macpherson report that followed the Stephen Lawrence case that he has been officially criticised for "hanging out to dry" three officers falsely accused of racism.
His approach to policing Muslims appears to be to seek the consent of those he supposes to be community leaders before "going in". It is surely not right that they should have a veto on whether or not an inquiry is pursued, and it must be asked whether all of them could be trusted not to protect some of those who merit police attention.
The methods matter, too. Although offence should always be avoided if possible, if the police will not use dogs in their investigations of Muslims (as they may do with almost anyone else), and if they undertake never to go into the religious parts of Islamic buildings, then some people with things to hide will hide them.
If the Blairs and Paddicks won't look at the link between faith and terrorism, how can they ever learn from the evidence in the websites and madrassehs and sermons which incites the trouble and brings like-minded extremists together?
And what about public vigilance? Yesterday, the Met's press conference called for public vigilance - but would you want to go and tell Sir Ian your anxieties about a Muslim neighbour? Might you worry about being turned away as a racist?
The most important question is for Muslims, and the authorities' attitude towards them. Embedded in modern government are too many advisers who believe in a quietist policy. To them, the most important thing is to avoid a "backlash" against Muslims. But the truth is that the backlash only threatens because the terror strikes. Mired in ignorance, our Government (let alone the Opposition) has little idea how to find the trends in Islam that could really improve the life of our country, and run with them.
It is only when you start thinking about what we are not getting from leaders of British Muslims, and indeed Muslim religious leadership throughout the world, that you start to see how much needs doing. The moderates are not pressed hard for anything more than a general condemnation of the extremists.
When did you last hear criticisms of named extremist groups and organisations by Muslim leaders, or support for their expulsion, imprisonment or extradition? How often do you see fatwas issued against suicide bombers and other terrorists, or statements by learned men declaring that people who commit such deeds will go to hell?
When do Muslim leaders and congregations insist that a particular imam leave his mosque because of the poison that he disseminates every Friday? When did a British Muslim last go after a Muslim who advocates or practises violence with anything like the zeal with which so many went after Salman Rushdie?
Why is not more stigma attached to the Muslims who are murdering other Muslims every day in Iraq and the Middle East?
What communal protection is offered to those Muslims who really are brave and confront Islamist violence, or the poor treatment of women, or call for democracy in the Middle East? How much do mainstream political parties with Muslim councillors and candidates really insist on their religious moderation and co-opt them to extrude the bad people lurking within their communities?
I understand and accept that there are many moderates among British Muslims, but I want to know why Britain gets so pitifully little to show for their moderation.
When a nation, a race, a political movement, a group of workers, the followers of a religion have legitimate grievances, there generally arises amongst them a champion who can command respect for his advocacy of peace, his willingness to fight without weapons and to win by moral authority. There may be many such grievances for Muslims in Britain, and in the West, but we are still waiting for the Gandhi or the Martin Luther King to give them the right voice.
We all love it when the British people shrug their shoulders and move stoically on in the face of attack. It is a powerful national myth, and a true one. But it contains within it a great danger - a self-fulfilling belief that there is nothing to be done to avert future disaster. That's not the Blitz spirit - what made London's suffering in 1941 worthwhile was that, in the end, we won.
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