With its historical range and intellectual clarity, Gordon Brown’s speech on 16 August showed why he is one of Labour’s greatest sons. At its core was a lesson that his party has learned before and may have to learn again: “Making what we want – the desirable – possible means making the desirable popular and electable.” Late in the day, perhaps too late, he warned Labour not to let its epitaph be “pure but impotent”.

The Labour Party is in crisis. It has been destroyed in Scotland and routed in England and is threatened by insurgents of the left and the right. In common with many other social-democratic parties in Europe, it is struggling to adapt to the challenges, as well as the opportunities, of globalisation, mass migration, slimmer public sectors and strained government finances. After two successive general election defeats, Labour is in danger once again of becoming the natural party of opposition. Worse still, it could become irrelevant.

The drama and division that have characterised its leadership contest are the result of this malaise. Tony Blair has advised supporters of Jeremy Corbyn to have a heart “transplant”. Dave Ward, the general secretary of the Communication Workers Union, has described Blairism as a “virus”. Many seem happier fighting the enemy within than the one without. When the leadership contest began, it seemed destined to follow an unremarkable path. The candidates – Andy Burnham, Yvette Cooper, Liz Kendall – all occupied roughly the same political ground. It was Mr Corbyn’s arrival on the ballot, with two minutes to spare, that changed everything. With impressive speed, as private polling obtained by the New Statesman website first reported, he became the front-runner. In the absence of a late upset, he will win. It would be one of the most startling results in British political history.

Those who lament Mr Corbyn’s success should try to understand it. An unapologetic socialist, Mr Corbyn has run an excellent and disciplined campaign, rich in policy proposals. He believes what he says and says what he means and people like this. His anti-austerity rhetoric, comparable to that of Nicola Sturgeon, has attracted many. It is heartening to see such a large number newly engaged or re-engaged in politics.

None of this is happening in isolation. Throughout Europe, anti-establishment candidates and parties of the left and right are thriving: Syriza in Greece, Podemos in Spain, Beppe Grillo’s Five Star Movement in Italy, the Scottish National Party, Marine Le Pen’s Front National in France. Even the fiercely two-party United States has broadened the mainstream to encompass the socialist senator Bernie Sanders on the left and the tycoon (and buffoon) Donald Trump on the right. “The age of party democracy has passed,” wrote Peter Mair in Ruling the Void, an analysis of democracy’s crisis. “Although the parties themselves remain, they have become so disconnected from the wider society, and pursue a form of competition that is so lacking in meaning, that they no longer seem capable of sustaining democracy in its present form.”

The presence of Mr Corbyn in the contest has inspired many thousands to join a selectorate that now numbers 610,000. The revised rules, which allow anyone prepared to pay £3 to vote for the leader, have significantly weakened the influence of the party’s MPs but have empowered activists and supporters. In this way, Labour has made itself vulnerable to possible outside manipulation by those who wish it ill, as well as “entryism” from the extra-parliamentary left. In the circumstances, this is an extraordinary act of self-harm.

Yet the surge of support for Mr Corbyn cannot be attributed to entryism alone (the combined membership of Britain’s far-left factions and groups is too low). Many who were alienated from the party in the Blair years and demoralised by the Iraq war have returned. The Labour right, having endorsed the new rules, should also ask itself why it has proved so unsuccessful at persuading sympathetic voters to register.

Mr Corbyn, who will be 70 at the next general election in 2020, has surpassed all expectations, including his own. The role of leader is one he neither wanted nor expected. More troubling for him is that, if elected, he would struggle to command the authority and respect of the parliamentary party after breaking the whip 534 times since 1997. He has never been a minister. Many senior shadow cabinet members have said they would not serve under him; many other MPs have indicated that they would seek to undermine his leadership from the beginning. Labour is in open revolt.

Meanwhile, the Tories, who have a slender majority of 12 in the Commons, watch what is happening with quiet contentment, scarcely able to believe their luck. Chancellor George Osborne has stated that he aspires to occupy the centre ground of British politics. He must be delighted by the disarray within Labour and that the party is being pulled to the far left at a time when the Liberal Democrats are so weak.

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One of our principal objections to Mr Corbyn is on foreign policy, so little discussed in the contest. We live in an era of crumbling world order. The Middle East is riven by civil war and the Syrian tragedy is one of the great humanitarian catastrophes of recent decades. The emergence of the barbarous Islamic State and a revanchist Russia pose powerful threats. Yet Mr Corbyn is a resolute unilateralist. He advocates withdrawal from Nato – which would leave Britain vulnerable, our peace and defence alliances in tatters. We believe Britain’s interests are best served by seeking to reform the European Union from within the bloc. By contrast, many on the radical left consider the EU to be a “neoliberal” institution and support “Brexit”.

As for Mr Corbyn’s economic programme, the policy of a “people’s quantitative easing” would risk rampant inflation and is not a sustainable means of financing infrastructure programmes. We believe that a 7 per cent rise in National Insurance for those earning £50,000 or above to fund a return of the student grant is the wrong policy, and would not win support in those marginal seats Labour must win in the south of England if it is ever to form another government. The proposed renationalisation of the energy sector rests on tax ­projections that have been shown to be heroically optimistic. For these reasons and others, though we recognise his qualities as a principled campaigner, we cannot support Mr Corbyn’s candidacy. Labour’s next leader must be drawn from the party’s mainstream and must command the loyalty and respect of his or her MPs.

Liz Kendall has shown courage and tenacity in emphasising the need for the party to appeal to Conservative voters. We admire her commitment to pluralism and decentralisation but her pitch has relied too heavily on the language and concepts of the mid-1990s and she lacks the stature and experience necessary to command authority. However, she has enhanced her reputation through these summer months.

Andy Burnham has advocated important policies such as the establishment of a national health and care service and the transformation of vocational education. But he has frequently resorted to platitudes about “the Westminster ­bubble” (of which he is an archetypal member, though he protests otherwise). His positioning has been inconsistent. Too often, he gives the impression of merely arguing for whatever seems politically profitable on any given day.

Yvette Cooper, the shadow home secretary, has run a cautious campaign that was designed to maximise second-preference votes from her rivals’ supporters, but in recent days, responding to the Corbyn surge, she has been bolder. Her experience of government, intellect and credibility mark her out. She has made a passionate and persuasive case for policies such as universal childcare and has forcefully resisted the Conservatives’ fiscal dogmatism. Labour, which purports to be the party of progress and opportunity for all, has never had a better chance to elect its first woman leader.

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Labour remains traumatised by an election defeat that it never saw coming and by Ed Miliband’s hasty resignation, which plunged the party into a leadership contest for which it was not prepared. The temptation for some is to write the next election off in advance and elect the candidate who provides most consolation. This must be resisted. History teaches that even the strongest governments can unravel with remarkable speed, which is why Labour must be in a position to offer a credible alternative. The best hope of it being able to do so, in the present circumstances, is the election of Yvette Cooper.

We say this knowing that Mr Corbyn is the clear front-runner and the likely next leader. Yet there are important elections next year in London, Wales and Scotland. If Mr Corbyn’s Labour Party performs poorly in each of these and its present dire position in the polls remains unchanged – and if civil war has broken out in the parliamentary party – his leadership will be in grave danger. Ms Cooper’s moment may yet come.