The reasons it can’t happen, apparently, are too many to number. Teachers are too busy, facilities are already fully-used, campuses are booked-up throughout the holidays, and, anyway, who’s going to fund it?

Or, as one father finally put it, with liberating honesty: “What I just can’t see, is why anyone should be able to get for free what I have to work my backside off to pay for.”

This is the bottom line of private schooling, and no parent who pays out Pounds 6,000 or more a year for their child’s education, would be telling the truth if they said they didn’t recognise the sentiment. Which is why any plans to bridge the divide between the two sectors are unlikely ever to amount to much more than a rebraiding of the flimsy walkway that the Assisted Places Scheme tried to throw across the chasm.

However willing some of those working in independent schools might be, however much their hearts be in the right place, the bottom line is that they are in business, and must please their clients, who are not their students but the kind of parents sitting around our dinner table - parents who have not only paid good money to send their children to their schools, but often done so precisely to get them away from what they perceive as problems in the state sector.

My children have, at various times, attend three very different independent schools and at each of these schools there have been a few parents who have been uncomfortable about this rift in society, but many more who have been either accepting or strongly supportive of the public-school tradition. However there has also been one overwhelming majority - that of parents who can’t write fee cheques without wincing at the number of noughts at the end, many of whom have had to make real sacrifices to be able to do so.

The kind of uncontentious co-operations these parents might readily countenance - a cautious loaning-out of the school swimming pool, a joint musical venture with local schools, swapping theatre props - are often already in place; but suggestions of anything further raises hellish visions of Bash Street kids chewing up the cricket pitch or (far, far worse) their own precious cash being diverted away into something that is not of direct and immediate benefit to their own children.

Yet there are things that could be profitably traded in both directions. Independent schools might have the edge when it comes to great facilities, and to sporting and academic excellence, but they don’t always achieve this through the liveliest teaching, and could have things to learn from talented colleagues in the state sector, whose classroom skills are - often of necessity - more honed by challenge and informed by the mainstream of educational debate and training.

And, on all sides, at the moment, the desire to close gulfs and smooth edges seems to be paramount.

Whether it is a secondary school trying to offer a seamless transition from primary school for struggling readers, or a television programme bringing together a public school head and an east London comprehensive; whether it’s teenage girls catching up with the boys at school, or pupils embarking on their degree courses while still in the sixth form; the idea of old barriers being broken down and new flexibilities emerging seems much the tenor of the times.

In fact, the day that Jackie Lang, head of Walthamstow Hall, in Sevenoaks, and current president of the Girls’ Schools Association was in Bristol being told by Stephen Byers, the school standards minister, that independent schools needed to cosy up to comprehensives, I, in turn, was in Sevenoaks visiting a ground-breaking scheme to bring together the caring of the very young and the very old under one roof.

This is the kind of cross-generational world that Jemima Khan is waxing lyrical about having found in Pakistan, but which we in the West have so lost sight of that we can only get it back to with a huge imaginative leap, and an enormous amount of effort. It took the approval of more than 20 interested parties before this project could go ahead (a bureaucracy worthy of Pakistan itself), so high are the barriers we have built for ourselves, so calcified the social educational and welfare structure we have created.

Yet, such is human nature that for every gap we try to close, another seems to open. We bring down the number of boy smokers, and girls start to puff away as if there’s no tomorrow. We bring in a unified examination system, and up pops a school like Ampleforth College declaring its intention to break loose and go for something different.

Nowhere more so than in the fast-moving world of technology where, by the time they hit their teens, some children are living by e-mail and knock up computer pie charts and bar graphs as if it’s a skill they were born with, while others can still only peck away at a keyboard like timid hens in a foreign farmyard.

Last month I spent an evening helping a primary teacher friend devise an IT project for her class of eight-year-olds. The idea was that she would bring her class round to my house so they could first send an e-mail to Australia, and then get some hands-on experience sending a fax from our machine, in the study, to her fax, set up in the kitchen, and back again. Not exactly sophisticated, she said, but most of them hadn’t ever seen either e-mail or a fax in operation, and there wasn’t much she could do in school when it came to technology because they didn’t have the space or the equipment for anything ambitious.

In the event, her fax refused to cooperate with ours, and the project had to be abandoned, but as I watched my children watching the two of us struggle, I thought of the nice, new, carpeted computer room at their school, their specialist IT teacher, their computers in every class.

Wouldn’t it be nice, I thought, if these eight-year-olds could get their hands on equipment like that. But then, almost immediately, I remembered why they couldn’t. After all, the teachers are all so busy, the facilities always in use, the campus is booked-up in the holidays, and then, of course, who’s going to pay for it?