I introduce an argument for Platonism based on intra-mathematical explanation: the explanation of one mathematical fact by another. The argument is important for two reasons. First, if the argument succeeds then it provides a basis for Platonism that does not proceed via standard indispensability considerations. Second, if the argument fails, it can only do so for one of the three reasons: either because there are no intra-mathematical explanations, or because not all explanations are backed by dependence relations, or because some form of noneism—the view according to which non-existent entities possess properties and stand in relations—is true. The argument thus forces a choice between nominalism without noneism, intra-mathematical explanation, and a backing conception of explanation. You can have any two, but not all three.

中文翻译：

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柏拉图主义和内部数学解释

我介绍了一种基于数学内部解释的柏拉图主义论证：一个数学事实由另一个数学事实解释。这个论点很重要，原因有二。首先，如果论证成功，那么它就为柏拉图主义提供了基础，而柏拉图主义并不是通过标准的不可或缺性考虑来进行的。其次，如果论证失败，它只能因为以下三个原因之一而这样做：要么因为没有数学内部的解释，要么因为并非所有解释都得到依赖关系的支持，要么因为某种形式的非论——根据不存在的实体拥有属性并处于关系中——这是正确的。因此，这个论证迫使我们在没有无主论的唯名论、数学内部解释和解释的支持概念之间做出选择。您可以拥有任意两个，但不能拥有全部三个。