A couple of interesting articles: In Why the Left Needs Religion, Elizabeth Stoker Bruenig argues that religion as religion (not just people who happen to be religious) is an essential component to left organizing, citing Christian religious doctrines and practices that are frankly Marxian. In Not God’s Politics, Susan Jacoby disagrees with Bruenig, citing the... diversity... of religious ideology on the right and left, and the propensity of Christians to impose their religion on everyone, including non-Christians. Naturally I much prefer Jacoby, and while I know many Christians I'm happy to have as allies, it is because of their politics, not their religion. But I think there's a larger point that's deserves highlighting.
We on the left should not, I think, be too focused on implementing a particular political-economic regime, e.g. welfare capitalism, social democracy, democratic socialism, or communism. The regime does matter, a lot, but the regime is not the fundamental issue. The fundamental issue is to change how people see the world and each other, to change our political and social psychology. A particular political-economic regime might be the consequence of that change, or might be a means to effect that change, but a change in "human nature" must be the fundamental goal of the left. I don't mean to say we shouldn't think carefully about political and economic issue at the deepest level (I spend a lot of time thinking and writing about the economics and finance of communism), but these issues are not at the deepest level.
The fundamental struggle of the left is to inculcate the social psychology of democracy. We really like the word here in the United States; we are perhaps not quite so enthusiastic about the actual practice. Democracy is not about holding periodic elections, even with a comprehensive franchise and open candidacy. Democracy is not about letting people vote, it is about the people ruling. Elections are about people "choosing" our rulers; democracy is about people ruling themselves. Again, I want to say that although economic and political democracy is a regime, and requires specific kinds of political and economic institutions, the regime is not the fundamental, deepest, point; the deepest point is our social, cultural, political, and economic psychology, our consciousness.
Having sampled the Christian scriptures, the character of Jesus seems to me like a decent fellow.* But that's alarming right there: I have a favorable opinion of Jesus without believing for a second that the character or the narrative in which he appears has any sort of divine imprimatur. If you think Jesus is a decent character, why is that not enough to emulate him? What work does apotheosizing him do?
*I have much less familiarity with Islamic scripture. I wouldn't be surprised if apart from his egregious pedophilia, Muhammed (the man or the character of the narrator), given his time and place, was also rather decent. My point, though, is not the decency of the characters but the nature of religion.
The point of democracy, as an element of consciousness, is to act decently because we are decent; if we are not decent, we want to become decent.* If we act decently because some god demands we do so is to miss the fundamental point of democracy. More importantly, if we demand that others act decently not because they are decent, but because some god demands they do so, we don't just miss but actively undermine the whole point of democracy.
*What do I mean by "decent"? Good question. It's a vague word for a vague and complicated idea. I'll write more on this topic later.
One might argue that to persuade their readers to become decent is the Gospels authors' whole point, their real project. Perhaps so, but if that is their point, after almost two thousand years, they have decisively failed. And, I would argue, they have failed precisely because they have located the impetus to decency in the divine, rather than the human. I'm sorry, racist white European authoritarian neoliberal capitalism has captured a substantial fraction of nominal Christians, Christians who have in their homes an actual copy of the writings about the brown Middle-Eastern democratic communist, who say that what this brown Middle-Eastern democratic communist (supposedly) said is literally the most important thing in the world. If people actually believed what Jesus says, they would have greeted the writings of Marx with a collective, "Well, duh!" No, that's wrong: if people really believed Jesus, Christians never would have invented capitalism, and Marx would be known for his literary criticism. The fault is not in the content, so the fault must be in the location in the divine, not the human.
Democracy and communism are not about income equality or inequality. They are not about the Ministry of Planning or nationalizing the banks. Democracy and communism are about power, with the people taking power away from this or that self-selected elite, hereditary, economic, or theocratic, and wielding it themselves. Indeed, democracy is about abjuring power over others and privileging each person's individual power over him- or herself. By its very nature, no religion can ever give us that.