We're crouched on the pavement outside Starbucks, sheltering from the drizzle, and it's a meeting for the Occupy London Stock Exchange's tent city university. There are about half a dozen people organising a programme of talks for the camp. The most pressing issue is that someone's departure has left them with no library tent. They need a place for people to read. Next up is the dilemma about how to deal with the increasing number of homeless who are pitching up to take advantage of the free hot vegetarian food and want to sleep in the "university" tent.

Newcomers join the deliberations; decisions are made that must go to the finance group for approval; the meeting can only last as long as the laptop's battery.

Over the last two weeks these few hundred campers at the Occupy London Stock Exchange have attracted an extraordinary amount of publicity – much of it hostile. Yet the derision and world-weary cynicism has not inhibited the campers' striking degree of self-possession. In part, that comes from pride in what a couple of hundred people have managed to create from scratch in a few days: a school, university, craft workshop, recycling system, a canteen serving hundreds of free meals, a newspaper and even a piano tent. There is a steady flow of donations of food and money and newcomers. The level of organisation is remarkable. There is nothing chaotic about this experiment in community in which no one is turned away, and everyone has their say.

In part, the self-possession comes from the steady flow of engagement with sympathetic, curious passersby. "I just want to say that I would be with you if I wasn't too old to camp and I didn't have a full-time job," I overheard one lady tell a protester. Everywhere there is the hum of strangers talking to each other about politics, accompanied by a sense of relief that finally people have a space in which to talk about their anger and frustration at injustice.

But most importantly, the protesters have a sense that time is on their side: they are talking already of the big demonstrations planned for 9 November and 30 November. They believe that protest will be an ever bigger part of our lives, and our political culture in the next few years. The biggest message implicit in the self-evident buzz around the camp is: we are just the start.

Perhaps that is why the camp has prompted such criticism. A defensive instinctive conservatism is closing ranks. The protesters are lambasted as benefit scroungers one moment, derided as part-time protesters returning to their jobs the next. Picked at for being losers then doubted as credible because they have iPads. Damned if they do and damned if they don't. Screeds of criticism have now been written about the protest and on almost every point, they misunderstand the purpose of this form of street protest. Is this a revolution in the making? Of course not. Will it topple the government? No.

The critics complain that there are no clearly identified objectives, no manifesto. But this is not some proto-political party. Critics insist there must be leaders or representatives. But the protesters stubbornly refuse to conform to any of the conventions of our political and media culture. That is why the invitation from the Bishop of London, Richard Chartres, for representatives to join a panel discussion with business leaders was so inept. The protesters are challenging how the illusion of public debate is created through a stage-managed process that excludes all but a self-regarding elite who are largely in agreement, quibbling only over technocratic detail.

The aim of the protesters is something far more profound than having their say in a debate in which the terms of the argument are predetermined. They want a paradigm shift, or as one protester put it during a teach-in last week: "we want to change hearts and minds". To do that they don't want to win an argument with soundbites but capture the imagination; engineer experiences that prompt the slow burn of new perceptions and new questions as simple as, why not? Why not now?

That experience may be hundreds meditating in silence on St Paul's steps or tinkling piano notes, free tea and empathy on the pavement in the midst of London's rush hour, or even strangers talking to each other about politics and faith. It is about seeding questions in thousands of minds, shaking certainties and orthodoxies so that there is space for new alternatives.

"Space" is the theme that runs through much of what the protesters say. Their first agreed principle is that the current system is unsustainable, undemocratic and unjust, and they want to create the space to think of alternatives. First that means taking key symbolic public space – this is the politics of geography – to use it for conviviality, living, learning and participation. That's no easy task in a city designed to facilitate only three activities – working, transport and shopping – with as little human interaction as possible. Metal fencing is springing up around even small public spaces in the City of London to preclude new camps. The protesters' aim is to open up space, physically and socially, for people to connect and thereby open up space in people's imaginations.

The alternatives they are looking for are not something written up in a Google doc. They live them, modelling new forms of organisation and democracy. "Watch us, learn from us, join us" is the tactic. Who knows who is learning what from this experiment, and if that knowledge may feed into radical new ideas five years down the line? If it sounds vague, just consider how concrete literalism has boxed us into a very tight corner of the theory of TINA (There Is No Alternative), for a generation.

It's that combination of idealism, naivety and dreaming that is so invigorating at Occupy London, because these are crucial elements of any healthy political culture, but they have been driven to the margins in recent decades. At an evening general assembly at the camp last week, one speaker cited the English civil war and quoted Shelley: this was the English radical tradition and the speaker was a Californian, his audience every nation under the sun. His listeners shook their hands in the air in assent, a gesture passed on from the Spanish indignados that did so much to trigger the global Occupy movement. Globally and locally connected.

The greatest irony of all is that it is the hapless Church of England that has done the most to propel this obscure but interesting protest into the headlines for much of the last two weeks. The dean and chapter of St Paul's have superbly demonstrated in recent days the point Occupy London is making: that City interests have compromised and captured some of the most powerful institutions in the country. One might have hoped that an institution such as St Paul's, conscious of its own history of civic purpose and national identity, not to mention the radical gospel of a Jewish itinerant carpenter, would have grasped the symbolism of the moment more astutely. That they failed demonstrates all the more starkly the ethical bankruptcy of our age. And that, after all, is exactly what Occupy London is about.