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?

I'm not sure what arguments you have in mind for these two views, but I understand, I think, the main feature that raises your question: each implies that the other is false. And yet, as you say, you find that each is equally likely. If each implies that the other is false, then they can't both be true, to answer your last question. One of them, at least, must be false. Perhaps they are both false, if they both rest on a false presupposition, for example. This seems to be the situation with the two views, (a) that the present king of France is bald and (b) that the present kind of France is not bald. They contradict each other, so one, at least, must be false. In fact, it seems they are both false, since there is no present kind of France at all--though I should note that, at one time, more philosophers thought the two views lacked a truth value at all (that is, they were neither true nor false, so on such a view we could say that the two theories you are considering might lack a truth value at all).
So much for that last question, whether they can both be true. They can't. However, there's another interesting question in what you wrote. If you find both theories equally likely, and assuming that there isn't some third theory that is also a competitor, then why not just be equally confident in one as you are in the other? For example, if you think that Idealism and Materialism are the only options, and they are equally likely, then you should think there is a 50% chance, or probability, that each of them is true. This is what you would do when it comes to a coin flip. So why not split your confidence in this way with these theories? Not every contradiction that is unsettled by our evidence is a paradox. Sometimes it makes sense to simply be as confident as our evidence calls on us to be, and no more, in each option. This doesn't mean stop thinking about it. You may, for example, soon discover that the case for Idealism is not as good as you are thinking it is now, and that the case for Materialism is actually better. So you'll get more confident in Materialism.

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 materialistic language was practically and politically superior; Carnap agreed, but didn't think that we should therefore become materialists in any literal sense. We can agree to use the materialistic language, he said, but unlike Marx or Lenin, we don't need to make our practical convictions depend on a metaphysical materialism, which would have to be an article of faith that goes beyond any possible way of finding out whether it's true.

This question was also one of the issues at the root of the later controversy between Carnap and Quine, who revived the respectability of ontology (and the word itself, in fact) in the 1950s. Philosophers today have largely gone with Quine, and think that deciding on one's ontology is one of the most important things philosophers do. There are still a few defiant hold-outs, though, who think otherwise.

Read another response by Yuval Avnur, André Carus
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