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Both idealism and materialism have convincing arguments for me, yet within each argument are refutations of the other. How can I reconcile which to believe when they both seem equally as likely? I've thought that perhaps idealism explains our own subjective worlds, and materialism explains the objective external world, but can both be true when they contain refutations of the other?

Yuval Avnur's response to this question is of the kind that is currently most popular among philosophers; it takes the "existence" question (or, as philosophers like to say, the "ontological" question) seriously, at face value. There is another approach, though, proposed by logical empiricists such as Rudolf Carnap, in which "existence" questions are disqualified as incoherent, since there is no logically or scientifically respectable way of answering them. In this view (best exemplified perhaps in Carnap's famous paper "Empiricism, Semantics, and Ontology"), questions about existence are best understood as practical questions about choosing a language: should we use an idealistic or materialistic language for science, or to talk about moral or political problems, or for some other purpose? (The answer might vary by purpose.) Within the Vienna Circle of logical empiricists (in the 1920s and 30s) there was some controversy about this. Otto Neurath, who was a Marxist, maintained that the...