'Curious minds'
But how should public thinkers make big ideas open and accessible? How do they avoid being sidelined in an era of fake news and attention-seeking punditry?
This is the challenge for the British Academy, the national body for the humanities and social sciences, which is throwing open the doors of its London headquarters for a summer showcase, promising "ideas for curious minds".
Image caption Dame Mary brought the history of ancient Rome to a mainstream television audience
Getting people across the threshold is the first challenge, she says. The British Academy is based in an elegant building near St James's park in central London. It's a place dedicated to public culture, but wants to stop looking like a private club.
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"You can see why people might be a bit intimidated," says Dame Mary, a fellow of the academy.
"But what the television shows you is that ideas are not the preserve of posh white men."
'No sloganising'
She wants to open up academic debates to a wider public, but she says a really important part of the message must be to make people realise that things can be "complicated".
"You don't need to be an expert to have a view of Roman politics," she says, such as whether it was right to assassinate Julius Caesar.
Image copyright British Academy Image caption The academy's summer showcase will present research on music, including South African jazz
But whatever your view, she says there has to be a recognition of the moral and political complexity of such decisions.
"Everyone can join in, but it doesn't mean you can shoot your mouth off." Opinions shouldn't be untroubled by thought, she says.
A by-product of the advance of social media, she says, is the tendency to "sloganise" and to turn every argument into a one-dimensional sound-bite.
Image caption Dame Mary wants people to look further than their noses
Ideas can be complicated and divisive, she says, and she takes umbrage at the type of people who think everyone should agree about everything.
Dame Mary says she gets annoyed when people on Twitter say they are "disappointed" in her views. She says they're "patronising" and can't stand people having their own opinions.
'Death threats'
The British Academy's august building was once the home of 19th Century prime minister William Gladstone. It's hard to think what such a high-minded statesman would have made of being shouted at on Twitter.
Dame Mary has been subject to dreadful online abuse, but she still argues for the benefits of engaging with people and their thoughts on social media.
Image caption The British Academy wants to open its doors to show big ideas are not a private club
"You don't think - great, death threats," she says.
But she's still ready to get into arguments, even though there is a likely risk of it all descending into "lots of sexist rants".
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"It's a combination of anger, outrage and appalling self-righteousness," she says of too many social media exchanges.
It's not the rudeness that really rankles, she says, because "being rude is part of the human condition".
What really annoys her is the over-simplification and the refusal to recognise that some problems can be complicated and nuanced and not sorted out by a shouting match.
Such public thinkers might feel that they're not running with the grain of the times.
Fake news
The academy's president, historian professor Sir David Cannadine, warns against a populist mood in which decisions are made on "gut instincts" rather than evidence.
"In the world of fake news there is a distrust of experts which is wholly regrettable," says Sir David.
Institutions such as the British Academy might be engaged in advancing knowledge, but he says they might not be "very good at explaining themselves".
Image copyright Benedict John Image caption "Ideas change society," says Sir David Cannadine, president of the British Academy
Sir David wants the open days to persuade people of the relevance of the humanities - to show that they are at the heart of "private recreations and public culture".
The summer showcase will have talks and presentations and displays on history, music, art, literature and the future of work.
It's also making a case for internationalism - as he says ideas and knowledge and expertise "know no international boundaries".
"Ideas change society," he says.
British Academy Summer Showcase is on 22 and 23 June, 10-11, Carlton House Terrace, St. James's, London SW1Y 5AH
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