The illegitimate open-mindedness of arithmetic

We are often told that we should be open-minded. In other words, we should be open to the idea that even our most cherished, most certain, most secure, most well-justified beliefs might be wrong.
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We are often told that we should be open-minded. In other words, we should be open to the idea that even our most cherished, most certain, most secure, most well-justified beliefs might be wrong. But this is, in one sense, puzzling. After all, aren’t those beliefs that we hold most dearly–those that we feel are best supported–exactly the one’s we should not feel are open to doubt? If we found ourselves able to doubt those beliefs – that is, if we are able to be open-minded about them–then they aren’t all that cherished, certain, secure, or well-justified after all! This has led some philosophers to treat open-mindedness, not as an attitude that applies to particular beliefs, but rather as a second-order attitude that applies to our body of beliefs as a whole. I can’t do full justice to this sort of approach here, but the following should give one an idea of what is going on. To make things concrete, let’s let Φ(x) be a predicate that applies to numbers, and let’s say. . .

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News source: OUPblog » Philosophy

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