Paul Mason wants to replace capitalism with a "gift economy". As if to explain, I am greeted at the door of his Victorian cottage in Kennington, south London, by Lottie, his 11-month-old puppy. She brings me a pink toy rabbit. I take it politely and give it back to her. "She understands the gift economy," says Mason.

I'm not so sure I do. Mason, a hero of the anti-capitalist left and author of Why It's Kicking Off Everywhere, has written a new book called Postcapitalism, which explains what he thinks will happen to the economy next and how the change will come. I've read it, and have some sceptical questions for him. But first, he welcomes me into his lovely home and offers me a cup of tea. I say hello to his wife Jane, a nurse, who is getting ready to go out. It's a domesticated setting for the reinvention of Marxism in which Mason is engaged.

I begin by asking about these neoliberals who control the world. They sound like terrible people. They believe in "uncontrolled markets", says the book, and that "inequality is good". Who are these people? Who says inequality is good? Nobody says it. "They do," replies Mason. Not in this country they don't. "Ah. Well. I could find you several quotes. They don't say it in that way, they say it is natural, it is a necessary product of society."

In pictures: Greek referendum Show all 28 1 /28 In pictures: Greek referendum In pictures: Greek referendum Greece EU Referendum People celebrate in Athens after the first exit-polls of the Greek referendum Getty Images In pictures: Greek referendum Greece EU Referendum A "No" supporter flashes a victory sign before a Greek flag atop the parliament in Athens, Greece July 5, 2015.Greeks voted overwhelmingly "No" on Sunday in a historic bailout referendum, partial results showed, defying warnings from across Europe that rejecting new austerity terms for fresh financial aid would set their country on a path out of the euro. REUTERS/Yannis Behrakis In pictures: Greek referendum Greece EU Referendum Supporters of the No vote react after the first results of the referendum at Syntagma square in Athens AP In pictures: Greek referendum Greece EU Referendum Supporters of the No vote wave Greek flags after the referendum's exit polls at Syntagma square in Athens AP In pictures: Greek referendum Greece EU Referendum Greece's finance minister Yanis Varoufakis casts his vote in the country's referendum EPA In pictures: Greek referendum Greece EU Referendum A ballot box is emptied by a voting official at the closing of polling stations in Athens, Greece July 5, 2015. Greece voted on Sunday on whether to accept more austerity in exchange for international aid, in a high-stakes referendum likely to determine whether it leaves the euro-currency area after seven years of economic pain. 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He has called on voters to reject creditors’ proposals for more austerity in return for rescue loans AP In pictures: Greek referendum Greece EU Referendum Protesters march holding a torn European Union flag during a demonstration for the 'NO' campaign in Thessaloniki Getty Images In pictures: Greek referendum Greece EU Referendum Greek Finance minister Yanis Varoufakis has pledged to resign if his country votes “yes” to the bailout plan Reuters In pictures: Greek referendum Greece EU Referendum Pensioner wait to get their pensions outside a National Bank of Greece branch in central Athens. Banks only opened for pensioners to allow them to get their pensions, with a limit of 120 euros. AFP/Getty Images In pictures: Greek referendum Greece EU Referendum Protesters attend an anti-austerity pro-government rally in front of the parliament building Getty In pictures: Greek referendum Greece EU Referendum The possibility of Greece leaving the Eurozone is increasing by the day. Merchandise already exists to accompany the event EPA In pictures: Greek referendum Greece EU Referendum An elderly man waits to receive his pension outside the closed National Bank of Greece headquarters in Athens AP In pictures: Greek referendum Greece EU Referendum People stand in a queue to use an ATM outside a closed bank in Athens AP In pictures: Greek referendum Greece EU Referendum A banner supporting the NO vote in the upcoming referendum hangs from the offices of the Greek Finance Ministry Getty Images In pictures: Greek referendum Greece EU Referendum Greeks will be asked whether they accept the austerity terms demanded by the country's creditors Getty In pictures: Greek referendum Greece EU Referendum Eurozone finance ministers expressed disappointment at the Greek decision to hold a referendum on the bailout terms Getty In pictures: Greek referendum Greece EU Referendum Alexis Tsipras, the Greek Prime Minister, said bailout conditions had ‘asphyxiated’ his country EPA

I remember Margaret Thatcher and Ronald Reagan saying things like that. Which is odd, as Mason says that neoliberalism began in 1989, with the fall of the Berlin Wall. But my impression is that since Thatcher, the Conservatives have given up on that kind of thing. Mason talks about how people say different things inside the Bilderberg Group and outside of it. I've always regarded any mention of the Bilderberg, an elite US-European discussion forum, as the mark of a conspiracy theorist – but I decide that to challenge him would waste our time. "What is said on the decks of mega-yachts that will sail past Kefalonia this summer," he goes on, "is not the same as what those people will say when they are speaking through their public-affairs agency on behalf of their company."

When he mentions "the salons of the elite", I balk. How does he know what goes on in such places? "I actually have been in the salons of the elite, where they defend inequality," he says. "I've met enough of them to know that that is how they think."

Surprised by my mental picture of Mason reclining on a chaise longue in the Prime Minister's salon, I ask: "You think that's how David Cameron thinks?" His reply is unexpected: "I certainly don't think that's how David Cameron thinks. At all. Because I think that Cameron is an unusual elite politician, who actually gets it, from a paternalistic point of view." Cunning people, these neoliberals. Before you can get a grip on them, they rebadge themselves as sincere paternalists.

We are not going to agree on whether neoliberalism exists or who espouses it, but what is interesting about Mason's book is his analysis of how information technology is going to lead to the abolition of the market – and what he calls the "supersession" of capitalism by a new form of economic organisation.

He says that the internet is already allowing a "human revolution" in lifestyles – "We are preparing ourselves to be able to live this life that is now possible." I think it sounds like early Marx writings about what the communist life would be like – "Hunt in the morning, fish in the afternoon, rear cattle in the evening, criticise after dinner" – and Mason is enthusiastic: "It is exactly early Marx. Exactly that."

But isn't it also pie in the sky?

"Why is it pie in the sky?"

You can't abolish market forces; you might as well try to abolish maths.

Mason at home in Kennington, south London (Immo Klink)

"Market forces operate in conditions of scarcity, but these days I can copy and paste the entire back catalogue of the Beatles and it has a zero reproduction cost. Therefore the market forces will drive the cost of the entire back catalogue down to zero."

So market forces will continue to exist, I say. It is just that the cost of some forms of information will be zero. "If what I am predicting comes about and large parts of society move to a form of non-money exchange, the interesting thing will be what happens. There are two possibilities. One is that we move to a substantially gift economy. OK, you say it is 'pie in the sky' for the market to disappear. I think it has begun to disappear. The market in the ideas space is very fragmented and difficult to enforce and non-spontaneous."

He is full of his recent experience in Greece, where he sees the postcapitalist economy emerging from inside the mess of existing structures: "As you move through it, from one beleaguered souvlaki place to the next, what you are mapping is the informal economy – where the default way of running things is through barter, sharing, non-market, some market, the market injecting the liquidity."

But if that new economy doesn't emerge from the back streets of Greece or the Raval district of Barcelona, what is the other possibility?

"One of the exhilarating things about trying to write about technology is that it keeps on changing while you're writing. It literally makes me gasp that Facebook is 10 times bigger than it was when Lehman Brothers collapsed," he says. The book makes much of Wikipedia as a non-market model of the sharing economy, but, since he finished the draft 18 months ago, other supposedly "democratic" ways of using the internet have become successful, replicating the rent-seeking, tax-avoiding features of existing capitalism. "You've got what are, unfortunately for me, bad examples of it becoming really prominent – specifically Airbnb and Uber."

According to his book, the transition will be achieved by building an open-source computer simulation of economic reality: 'We take away Apple, Microsoft, Google's ability to monopolise information. Then we build a strong, resilient counter-power.' (Immo Klink)

It would seem that postcapitalism is as elusive as the neoliberal ideology it will replace. And the means by which it will come about is – to me – equally unclear. But it won't be by violence, he says. "The revolution I'm talking about is the creation of an economic counter-power. And that is all I'm talking about, let me be clear about that. Because in my youth I was a revolutionary, I was a hardline Marxist. But this book is definitely an intellectual journey away from that."

Indeed, its long central section goes into some detail about how the Bolshevik revolution in Russia went wrong. "Twentieth-century Marxism was built on a misunderstanding of what the working-class wants," he says.

According to his book, the transition will be achieved by building an open-source computer simulation of economic reality. "Once we have reliable predictions, we can act." But what would "we" actually do? "We take away Apple, Microsoft, Google's ability to monopolise information. Then we build a strong, resilient counter-power."

Mason is a powerful and fluent polemicist, and I admire the ambition of his book. Rewriting Marx is quite ambitious, isn't it? "I'm not setting myself up as the new Marx. I would say that somebody has to write this, because there is this big subcultural world of economists and social theorists who are already engaged in complex discussions of these very same ideas."

He has certainly caught the mood of a new generation of the left, the same tide carrying Jeremy Corbyn so high in the Labour leadership election. Not that he can talk about that, because of the impartiality requirements of public- service broadcasting and his employer, Channel 4: "It's bizarre. We can discuss whether or not Camus is a better novelist than Sartre, but I can't give you my opinion on whether Jeremy Corbyn would be a better leader than Andy Burnham."

Postcapitalist economy: Mason reporting from Greece last month

I'm interested, though, in how he has a platform in the mainstream media, which is supposedly so much a tool of neoliberalism. "I am quite happy to call myself a Marxist at the level of method," he says, "because historical materialism as a method is a great tool for understanding history" – but he doesn't like being labelled simply "a Marxist".

His assumptions and prejudices come from his experience, he says, not just his political experience as a revolutionary socialist in his twenties. "I was a Catholic schoolboy, schooled in Catholic social teaching. I was an atheist rebel. A priest tried to teach us biology, and mis-described the female reproductive system so badly that we all looked at each other and said, 'How can any of the rest be true?' We were all 14 years old and we had already explored the female reproductive system with the Catholic girls' school next door."

As a young man in Leigh, Greater Manchester, he was into northern soul – in 2013, he made a documentary for the BBC, "a journey back into my past". He studied music at college, specialising in Schönberg and atonalism – avant-garde discordance, in other words.

"I was a raging leftie. I was a supporter of a group called Workers Power. But I wasn't getting anywhere with the music creatively, so I went into journalism at the age of 30, confident that because I could write polemical propaganda leaflets, news wouldn't be a problem."

Now, he says, "My interests are classic petit bourgeois. I'm a fascinated and dedicated classical-music person. I'm a big Wagner fan. I'm a big fan of Late Romanticism. I go to the Proms." He has a full-size piano-style keyboard in his front room. "Wish I had more time to play."

We are back full circle. The man who wants to start an economic revolution goes to the petit-bourgeois Proms and, although he "loves running around in slightly revolutionary places", seems quite at home in this gentrified corner of south London. During our interview, the doorbell rings, and Lottie the puppy goes to the pile of toys and takes a different one to the postman. I'm not convinced the gift economy will replace capitalism, but Paul Mason has a lovely dog.