The new Labour leader Jeremy Corbyn will be in Scotland later this week and says it will be the first of many visits north of the border. "I'm going to be in Scotland a great deal as leader of the party," he said at the weekend, but how will the Scottish electorate greet him? Mr Corbyn says Labour membership in Scotland has risen dramatically in recent days, but electoral success is another matter. Is the new leader's message the right one?

Whatever happens, his strategy on the SNP will be critical to success or failure in Scotland, and, like so much else about Mr Corbyn, it is old-school socialist. He believes that nationalism cuts across a more important issue which is class solidarity. "If you're poor in Glasgow and you're poor in Birmingham, you're poor," he says. "If you need a house in Glasgow and you need a house in London, you need a house." And to remove any doubt, Mr Corbyn threw in an anti-nationalism slogan: "Flags don't build houses."

None of this internationalism should come as any surprise. The argument that national boundaries are less important than fighting for the working class in all nations was one that convinced many old Labour voters in the referendum last year and led them to vote No, but Mr Corbyn's focus on anti-austerity and fairness could also appeal to many of those who voted Yes.

The SNP did ultimately lose the referendum, but part of the reason it did so much better than expected was that it tapped into the same anti-austerity sentiments felt by Mr Corbyn and his supporters. There was a feeling of frustration that the traditional ways of doing politics were bad for working people and that the gap between rich and poor was growing unchecked. The SNP seemed to be speaking the same language and still does. A similar message from Mr Corbyn could be good for him electorally in Scotland.

The Labour leader's response to the anti-austerity message of the SNP is that it is superficial. "Yes, the SNP have a headline in that they're opposed to austerity, fine," he says. "The SNP are also privatising Calmac, also were behind the privatisation of Scot Rail, are also cutting college places, also privatising services, cutting local government funding." It is not a new argument and the obvious response is the SNP has had to rein in spending like every other government in Western Europe, but there will come a time when it will have to do more to prove its radical credentials. In just over two years time, extensive new powers will be transferred to Holyrood, including almost full control over income tax. The big test of how truly radical the SNP is will be what they do with those new powers, which will present an opportunity for Mr Corbyn and Labour.

One of the reasons the SNP has done so well electorally to date, and is likely to so again at the Scottish elections next year, is that it has espoused anti-austerity while pursuing many centrist policies in government – it is also worth remembering that, however successful the Yes campaign is judged to have been, it was Better Together that won the referendum with middle-ground arguments on the economy, security and the pound.

The electoral problem for Mr Corbyn, and the reason so many New Labour politicians have their heads in their hands, is that his policies do not attempt the same balancing act of the SNP and are unapologetically non-centrist. The re-nationalisation of the railways, for instance, will be popular with many voters, but Mr Corbyn's criticism of the SNP over ScotRail ignores the fact that services in Scotland have improved. It is not the only policy that will make him unelectable in the minds of many voters in the centre.

If Mr Corbyn is to do well, he must make another critical adjustment. For Labour voters, the issues their new leader is discussing are important – the Iraq war, public or private ownership, and so on – but Mr Corbyn must avoid the impression that he is talking only about the issues of the past. For political geeks, these are exciting times; ordinary voters, on the other hand, would like to know that Mr Corbyn is focusing on the problems of the present and providing some answers for the future.