A Twitter user called Conservative MP Louise Mensch "a typical soulless rich whore"; TV critic AA Gill said Professor Mary Beard is "too ugly for TV"; a tweet allegedly described the 19-year-old victim of convicted rapist and footballer Ched Evans as "a money-grabbing little tramp". These are just a few examples of misogynistic abuse levelled at women in traditional and non-traditional media recently. Journalist and TV presenter Miriam O'Reilly, who won a landmark ageism case against the BBC, discusses media misogyny with Nina Power, a senior lecturer in philosophy at Roehampton University and the author of One Dimensional Woman, a discussion of the contemporary portrayal of women. Laura Barnett listens in.

Miriam O'Reilly: The assault on Louise Mensch on Twitter this week was shocking and insidious. I know what it's like to be attacked by an internet troll. I've only been on Twitter for two months, but already a man – and I know who he is – has been getting in touch with my followers and sending them lies and distortions about me. It was nothing like as bad as the abuse Louise was subjected to, but it's still very disturbing. You feel you should rise above it – but if someone is trying to undermine your standing and your character, then I do think you have to fight back.

Nina Power: That's horrific, Miriam. It used to be harder to say that sort of thing – or it was certainly creepier: you had to get letter-writing paper and sit down with your typewriter. It's much easier to compose something awful behind an anonymous handle.

MO'R: People seem to think the internet gives them the freedom to say things they would not normally say – certainly not to your face. Online, they can just spit their venom in any direction they like and hide behind a screen name.

NP: That's true. In the early days of the internet people were quite utopian about it; it was all about dissolving identities, getting rid of gender binaries. But the internet is obviously a central feature of contemporary life, and contemporary life is, in general, sexist, racist and so on. So why wouldn't the internet also be those things?

MO'R: But why aren't the same horrible terms used about men? Of course men are abused online, but not with the sort of four-letter words used about Louise. Is it because people are threatened by strong women who speak out?

NP: Yes, partly. Mary Beard was right, and very funny, in the way she handled [AA Gill's] stupid comment about not being pretty enough for TV [Beard responded with an article saying men fear clever women]. But we still live in a sexist world. With the recession, too, I wouldn't be surprised if we see more attacks on women as a prelude to pushing them back into the home, or making them suffer the cuts more harshly.

Laura Barnett: So you think there's a tangible link between media misogyny and the effects of the recession?

NP: Definitely. This happens when there's a perceived austerity: you start attacking various groups. It comes from the top and filters down into the media.

MO'R: What shocks me is the apparent media consensus that it's OK for Samantha Brick to write about Mary Beard in the terms she used in the Daily Mail [Brick wrote that women on TV should conform to a certain ideal of beauty]. But there were about 1,600 online responses to that article from people outraged she was attacking Mary. I was really heartened by that – and that's not something I've often said about a Daily Mail comments stream.

NP: There are also people on Twitter who always attack misogyny and racism wherever they see it. Depending on who you follow, you can still get quite a utopian vision of the world there.

MO'R: Yes. I'm not altogether depressed about the attacks that women get on social media. I think we are being pulled back from that by a clear voice from women saying: "No, we're not going to put up with it any more."

LB: Presumably, Miriam, you feel the criticism levelled at Mary chimes with what you've been saying for a long time about the emphasis placed by television on women's youth and looks?

MO'R: It's a great example of what women in TV face: we have this ideal of what's acceptable, which is to be young and pretty, with beautifully done hair and nails. I was really pleased that Mary fought back; it's tough, because you leave yourself open to even more abuse. But you just have to realise we have an opportunity to change things. I feel we're starting to turn a corner – but maybe I'm an eternal optimist.

NP: But it's still overwhelmingly white, middle-class men in the media who are calling the shots. Only 7% of people go to private school, and yet the media is still filled with them. It takes confidence to push a vision – to say: "No, we don't want to see only that kind of woman on TV." That's class confidence, instilled by private schools.

MO'R: I don't agree. I left school at 17; my confidence came through my work. TV execs are just entrenched in this view that they're giving the viewers what they think they want. It's as simple as blokes fancying young, pretty women. Even when they do use older women, they go for caricatures: they can't accept that people will want to watch an older woman who knows her subject, and it doesn't matter whether she has grey hair or she's leaping around being ridiculous on Strictly Come Dancing.

NP: Maybe this is TV's last gasp. A lot of the interesting stuff that's happening online doesn't have to rely on these gender stereotypes. It's truly audience-driven, and much more balanced.

LB: That's a much more positive depiction of the treatment of women online than the one we started with.

NP: Yes. The internet is a place for all these interesting things – investigative journalism, blogs – but it's also a place where you're attacked. But women have to keep going with what they're doing, and not be put off by the abuse.

MO'R: You're exactly right. It would be shocking if a woman were bullied into silence. Mensch was right to go public and expose these people – and with the naming of the rape victim in the Ched Evans case, it was absolutely right that the police moved in quickly. I definitely think that outing the bullies, naming and shaming them, is the way to go.

NP: My answers are more structural, I'm afraid: we're not going to solve any of these things in a society that is misogynistic and racist to its heart. It's so obvious and boring to attack Louise Mensch for being a woman: why not attack her policies? Still, the internet contains these other images of worlds that are not sexist and misogynistic. The internet may ultimately change itself.

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