This is a story about intersectionality. It's going to displease a few people who don't know what intersectionality is, annoy a few people who do, and enrage a load of people who don't use Twitter. But I checked with my privilege, and my privilege said it was OK. (Don't know what "check your privilege" means? This might turn out to be a problem for you, too).

In January, an argument on Twitter started in the manner characteristic of, possibly unique to, that medium. Someone called historian Mary Beard a racist. Helen Lewis, the deputy editor of the New Statesman, asked what made Beard a racist. A small but persistent Twitter intersectionality-core rounded on Lewis, accusing her of mindlessly defending the establishment against outsiders, effectively using her platform in the mainstream to defend racists within feminism from the critical voices whom feminism ought properly to champion and defend.

That precis doesn't quite evoke the tone of the attack: another Twitter feminist defended Lewis later with: "It is never OK to call another woman a vicious rancid bitch." The fact that this needs to be said, in an argument between one feminist and another, makes me chuckle, though of course I won't be chuckling if (when) it is said to me.

A racist feminist just wouldn't make sense. You can't fight for equality on the basis of one innate characteristic without signing up to the precept that we're all born equal. The problem was – and this happens quite a bit on Twitter – a mistake at the outset. Beard is not a racist. Lewis got annoyed and left Twitter this week, though only temporarily.

It could be taken as an unfortunate misunderstanding, except for the obvious pattern; Suzanne Moore left Twitter after essentially the same argument, though it started not with perceived racism but with a remark that was taken to be transphobic.

Times columnist Caitlin Moran got on the wrong side of intersectionality when she said she "didn't do race". This made her a racist; also the mindless beneficiary of middle-class privilege, said critics. I weighed in, and said that not all feminists had to represent every perspective of feminism all the time. And middle class? She was raised on benefits. She's rich now, came the reply, plus she has a platform; ergo, she's part of the white, middle-class, straight, able-bodied, cis(gender) hegemony. To remain a true and respectful feminist with those privileges (never mind check them, it will take you long enough just to count them), your work must essentially be an act of atonement to all the people who are more marginalised than you are. As a feminist, you are occupying the space of the marginalised; to do so thoughtlessly is an act of trespass.

What makes me doubt this idea is its striking similarity to a technique of the right, the hyper-individualisation of every argument. Unless you are penniless right now, this second, you can't complain about inequality. Even more exclusively, unless you were born poor you can't take the side of the poor. I dislike the argument because it's anti-intellectual, dismissing reason and systems – all the tools of discursive progress – and attempting to replace them with the power of personal testimony.

But on a purely pragmatic level we can all see, presumably, what the real goal is in this ad hominem play: if only the authentically poor are welcome on the left, that considerably depletes our numbers. If only the truly marginalised can speak as feminists, that depletes our numbers too. And if people "with a platform" are disqualified for being part of the power structure, that leaves us without a platform. This criticism started on the right for a reason – because it withers the left. We should think a bit more strategically before we internalise it.

But then I heard Helen Belcher of Trans Media Watch speak at a public meeting this week. She said the media had three ways of portraying trans people: "The first is that they're fraudulent. They're not really who they say they are. We'd better humour them in their delusion. The second is trans as undeserving deviant. The number of times you get costs – usually inflated – set against the money you could have spent on kiddies. The third is trans as comedy."

In other words, all the prejudice that has been disallowed by modern standards is now concentrated on this one, pretty small group. It is very extreme, these days, to refer to gay people as deviant, but still allowable to make this insinuation about transsexuals. It is apparently permissible, in our mean-spirited age, to talk about how much disabled people cost the state, but I can't imagine it would be OK to laugh at them. Transsexuals are dealing with a prejudice way out of proportion to their number, facing not only the people who hate the idea of transsexuality but all the people who wish they were still allowed openly to hate gays, openly to laugh at the disabled – hell, probably a few who wish they could still openly despise women.

Women of colour, likewise, when they call white feminists "colour-blind", are not saying every conversation about misogyny must start and end at the point where it bisects racism, rather that battles white feminists assume to be over have merely been shifted elsewhere (when the Equal Opportunities Commission existed, they did some research and found that 80% of black and ethnic minority women had been asked at their last job interview – illegally, needless to say – whether they intended to get pregnant). And that's the better reason to "check your privilege" – not from some restrictive idea about how authentic you are, or whether you've endured the hardship to qualify as a progressive voice, but because not all prejudice is extinguished – some of it is just displaced. If someone else is taking the flak you would have got, in eras past, that flak is still your problem.

Twitter: @zoesqwilliams

• This article was amended on 19 April 2013, to clarify that the argument referred to in the second paragraph did not start last week, as the original article said, but in January, and that Helen Lewis's temporary departure from Twitter happened this week. Lewis says that when she left Twitter for about 48 hours earlier this week, it was not just because of the ongoing argument relating to the false claim against Mary Beard, but a series of incidents over several months.