I think the shift goes something like:
Pre-modernist thought was hierarchical, intellectual authority matched other existing hierarchies (e.g., religious authority).
Modernism depersonalized that knowledge, creating intellectual institutions instead of intellectual thought following external institutions. Modernism emphasizing big ideas, big narratives, all-encompassing theories. Marxism strikes me as very modern.
Postmodernism is mostly skeptical. Ezra Klein went as far as to say that postmodernism's purpose was to seed doubt.
The thing everyone is familiar with is the idea of multiple narratives. Some read that as relativism, I think it's a bit more modest, just making explicit what is obvious but conveniently ignored: everything is viewed by people from individual and diverse perspectives, and action (including intellectual action) follows from those perspectives.
Metamodern is maybe a combination of doubt along the lines of postmodernism, but applied to intellectual approaches themselves. Thus it fuses modernism and postmodernism and whatever else.
They started the podcast talking about the idea: "all models are incorrect, but some are useful." And it feels like that's the underlying theory of metamodernism, as presented.
This metamodernism feels a bit like a utilitarian intellectualism. It doesn't matter what is right, only what is useful. There is no claim of truth, but now that we are all steeped in postmodernism it's no longer necessary to emphasize the doubt. Instead it's a project to construct something out of this doubt.
It feels a little like the TED Talk version of intellectualism. Life hacks as a philosophy.
And metamodernism leaves the question: useful for _what_? How do we decide what to select? Metamodernism only tells us to pick and choose according to some utility that metamodernism cannot itself define.
This emptied-out philosophy is the postmodern curse that I do not believe metamodernism is able to address.
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