Of course, it's a letter that we all want to write sometimes. "I didn't want to come to your party, anyway." "Well, I'm glad you rejected my novel, because I've thought about it and I wouldn't let you publish it, anyway." "Oh, good, because to be honest I don't really want to marry you, either." No doubt sometimes there's a fair assessment that it would be better to withdraw from a situation that you altogether misread: on the other hand, it seems likely that there is often an element of self-protection here in that triumphant "anyway".

A lady from Hampshire, Elly Nowell, applied to Magdalen College, Oxford, to read law. After the interview, but before she heard whether she had been accepted or not, she decided that she wouldn't take their mouldy place, anyway, and wrote them a letter informing them that she had rejected them. "I realise that you will be disappointed by this decision, but you were in competition with many fantastic universities, and following your interview I am afraid you do not quite meet the standard of the universities I will be considering."

Miss Nowell's reasons included the fact that Magdalen held its interviews in what she called "grand formal settings", which she said favoured applicants from public schools, and disadvantaged state school applicants like herself. "The gap between elitism and discrimination," she concluded in an epigrammatic way, "is a narrow one."

She now hopes to study at University College, London, which, I must point out, is itself a "grand formal setting" built by William Wilkins, who also built four Cambridge colleges and at least three public schools. I wish Miss Nowell well in her search for unremarkable architecture where she can pursue her studies, but perhaps Wilkins's massive Greek edifice in Bloomsbury is not it.

Oxford, in its characteristically pained way, rushed to point out that Magdalen offered seven places to read law, of which only one was to a student from an independent school. Whether it might, in fact, have offered Miss Nowell a place at all it would not say.

No one can doubt that the social proportions at Oxford and Cambridge are seriously skewed. The percentage of offers that went to UK state school students at Oxford was 58.5 for entry in 2011. This, the university told me, is roughly in accord with the proportions gaining the best grades in each education sector. It isn't, however, at all in accord with the relative numbers who attend state and private schools; private schools educate 7 per cent of all children, rising to 18 per cent of those over 16. Small wonder that, even among those who do not get anywhere near Oxbridge, people sometimes say, these days, "I went to a private school", as if that on its own were evidence of educational attainment.

I spent longer than most people at Oxford and Cambridge – seven years in the end – and still count some Oxbridge dons as my best friends. It is almost inconceivable to me that anyone involved in these institutions is remotely concerned with perpetuating or rewarding privilege, and, at the highest levels, they are looking out not for solid educational training, but for the original mind, wherever it may be found.

A lot of amusement is expended at the Oxford interview process, which certainly is intimidating and can seem eccentric. I read the other day of a don who asked a candidate, "Why are you wearing a watch?" It's an extremely good question about the relationship of the individual to time and society, inviting all sorts of ruminations. The candidate, who could only think of saying, "To tell the time" had not really considered the matter, and, in the Oxbridge term, "failed to shine". The merit of this sort of question is that, really, you don't have to have had an expensive education; only to have spent some years, like me, staring out of the window in chemistry lessons at a Northern comprehensive and wondering about things.

Miss Nowell is perfectly within her rights to think that Oxford was not for her, but it is a great shame if brilliant young people think that it's best to reject it before they reject them; to walk away from the prospect; to conclude that it's just a posh place with astonishingly impractical architecture and not to apply at all. The virtues of the place are overwhelming; the close daily encounters with brilliant minds, the historic resources, the sense, which everyone feels, of horizons being broadened. Of course a good university will offer many of these, and UCL is a world-beating institution, too. There is, too, the point that it is not far from a duty for those clever students from ordinary backgrounds to go to these places and make them their own.

Oxford and Cambridge, on their own, can't create an egalitarian society without lowering their standards. The problem with their acceptance rates is not a problem with their snobbery, but the very mixed standards in state education. They can only take what's on offer. As for clever, awkward, argumentative people, like Miss Nowell: it would be a mistake not to throw themselves at the very best institutions in the world, and see what happens. After all, they can only say no.