A groundbreaking film about the academic Hannah Arendt has just been released in the UK, enterprisingly titled “Hannah Arendt”. What makes it so important is just this: it is about a clever woman.

The biopic tracks Arendt's experiences reporting on the trial of Nazi SS-Officer Adolf Eichmann, unapologetically centralising her role as a public intellectual. Relationships with friends and her husband, including notable author Mary McCarthy, are also rather sensitively depicted. But they are subplots to the real deal, Arendt's ideas.

Unlike the majority of films or television shows when they deign to give airtime to women, “Hannah Arendt” refuses to replicate the stereotypical territory of acceptable-woman-characters. It is not about Arendt the lover (see: almost all women in films), Arendt as a writerly version of the oh-so-“normal” woman hung up on weight and boyfriends (see: Bridget Jones), or Arendt the supposed high-flyer beseiged by mental health issues (see: Scandinavian drama).

“Hannah Arendt” is, instead, about Hannah Arendt the thinker. It rightly credits Professor Arendt, responsible for some of the most publicly enduring theories in 20th century philosophy, with an intellectual interiority mostly reserved – at least in the public eye – for white men.

Public intellectuals are a dying breed in the UK. You might say they went out with the move from print to new media, the corporatization of the higher education system, and the ascendence of the ideology that value can only be calculated in economic terms. I may be inventing a romantic vision of the past in which people - supported by state-subsidised child-care and a mandatory living wage - spent their evenings in coffee shops discussing the New Left Review, but it's also true that today's tyranny of the hit-counter is a dumbing-down tool that past intellectuals did not have to worry about.

For clever women and girls across the country, this is a problem. Last week Miriam González Durántez, who appears to have better politics than her husband Nick Clegg (not much of an accolade), bashed the “absurd and demeaning stereotypes” today's women still face. Commenting on the perception that girls lack women role models, she wrote: “If we succeed in our professional lives, we’re branded 'scary'; if we follow fashion, we’re 'shallow'; if we like science, we’re 'geeks'; if we read women’s magazines, we’re 'fluffy'; and if we defend our rights, we’re 'hard'.”

She's right, and this negative labelling includes a grotesque opprobrium levelled against those who dare to demonstrate intellect. Clever women remain 'ugly' almost by definition, while attractive women are often stereotyped as 'stupid'. As women are still predominantly valued when beautiful – just compare the women on TV and in film to their male counterparts if you doubt this - those who aim to enter the public sphere solely on intellectual grounds face marginalisation.

The treatment of clever women made headlines in 2011 when classicist Mary Beard received misogynist abuse after an appearance on Question Time, with a denigrating webpage about her hosting comments like “ignorant cunt” and “a vile, spiteful excuse for a woman”. In a characteristically badass response she suggested that the page be counter-trolled with floods of Latin poetry. But it wasn't an isolated incident. Bomb threats made this year against prominent women, including the journalists Hadley Freeman and Grace Dent, showed this is a culture that fights against women's right to intellectual territory.

Why should anyone care about public intellectuals? It's not the kind of career you can get an internship for – not even an unpaid one – bound up as it is in class, race and gender privilege. And I'm not a fan of the strand of thought which identifies “feminism” with getting more white women into boardrooms. Despite an abundance of talent, clever people who are not middle-class white men are allowed limited scope to flourish, a few Audre Lordes and Stuart Halls notwithstanding. Social media's democratising tendencies aside, it's still true that if you write for a national paper, for example, many more people will read your work than if you write a personal blog. As research shows, this means mainstream society's most influential voices are still of the maler, paler ilk.

But our public intellectuals should be celebrated despite the elitism associated with them, simply because their ideas are vital. A quick walk around the libraries of British universities will reveal all-too-many academics who only write books for other academics, whose books countering the original books will only be read by the first lot of academics. I'm pretty sure nobody has ever actually understood David Lewis's On the plurality of worlds - god knows I tried - but it's clear that it wasn't written for anybody who isn't a logic lecturer (although maybe in another possible world it becomes a bestseller? No, I still don't get it).

Public intellectuals bring this stuff out of the ivory tower and into mainstream consciousness in an intelligible form. They are one of the many groups of people who think up new ways to live, the people who use theory to challenge the status quo. We need them.

And the unrepentent intellectuals who are also women are additionally important, particularly those who are non-white, non-straight, or disabled. In refusing to cater to stereotypes about what women are allowed to be, in failing to be subdued by the abuse, pressurizing and denigration thrown at women who step outside of acceptable female deference, they mark out a wider territory for woman and girls as a whole.

Young girls need to be able to turn on the telly, or open the newspaper, and see adult women speaking as experts on politics, philosophy, science, art, or any other important topic. What's more, young girls need to see they don't have to downplay their own intelligence: it's OK for them to be clever too.