Does the fact that our perceptions can be represented geometrically and that geometry consists of eternal truths independent of the mind prove that an external reality underlies our perceptions?

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I don't think that such an argument would rationally compel external-world skeptics (who say that no one can know that there's an external world) to abandon their view. External-world skeptics think that no one can know that solipsism is false, where solipsism is the claim that nothing external to oneself and one's mind exists. The solipsist won't grant that geometry consists of truths that are independent of his own mind, because he thinks nothing is. The solipsist could admit that his perceptions have a geometric character to them without having to attribute that character to something external. So I don't think solipsism can be disproven in the way you suggest.

All of this assumes that solipsism is otherwise intelligible. But one might argue that solipsism is unintelligible because it relies on the incoherent idea of a 'private language', an idea explored in detail in this SEP article.

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