Is it a common belief among philosophers that the external world does not exist independently of consciousness? That consciousness creates the material world rather than the other way around? How can anyone believe this?

I'd say it's an extraordinarily uncommon view among philosophers. Very few philosophers have believed it throughout the history of the discipline (Bishop Berkeley is the most obvious exception) and I can't think of any contemporary philosophers who do, though I'm sure there are some somewhere. Berkeley was an idealist (that's the usual name for such views) because he thought the conception of matter found in Locke, Newton and other thinkers of the time was incoherent. If you read his Three Dialogues Between Hylas and Philnous, you may find some of his arguments more interesting that you would have thought.

We could add: orthodox theological views hold that not that our minds make matter, but that God creates the world. And accordng to that same orthodoxy, God isn't a material being. So the view that some mind may be the source of matter is actually not at all uncommon. And for various reasons, contemporary New Age and Magical thinkers often favor a view that puts mind first. If you want a sophisticated treatment of magical ways of viewing the world, I recommend Tanya Luhrmann's Persuasions of the Witch's Craft, still excellent 22 years after it was published.

There's a different sort of view that isn't quite as wild. It doesn't hold that minds make matter, but it does hold that mind brings something to the table. More particularly, the idea is that the world doesn't come pre-sliced into kinds of things; what gets grouped together with what depends on how we classify things. Traditionally, this sort of view was called "nominalism"; to varying degrees and in various ways it's still with us. You can find strains of it in Kantian philosophy, in post-modern thought, under the heading of "social construction," and in various so-called "conventionalist" views in philosophy. Within Anglo-American philosophy, Nelson Goodman was perhaps the staunchest defender of such a view; see his Ways of Worldmaking if you're interested.

So no: the view that mind makes matter is not common among philosophers. But the idea that knowledge has a "constructed" element and that in any case, the world doesn't simply sort itself by itself has a wide variety of defenders.

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