Call the explanation of one mathematical fact by another an intra-mathematical explanation. To date, there has been a tendency to approach the topic of intramathematical explanation by investigating the distinction between explanatory and non-explanatory proofs (see, for instance, [14; 22; 31]). This is very natural since it is widely acknowledged that some proofs are explanatory while others are not [16, p. 879]. Still, focussing exclusively on proofs as the only locus of explanation in mathematics is a mistake [15; 24]. That would be to prejudice the question of where explanations in mathematics are to be found. As with other cases of explanation, we should be asking “what makes a particular explanation explanatory?” Jumping to the question “which proofs are explanatory?” introduces a restriction on the theoretical options available for understanding intra-mathematical explanation. For the time being, at least, we’d like to remain open minded about where explanations in mathematics reside. Ultimately, then, what we seek is a theory of intra-mathematical explanation that is capable of telling us how such explanations work, one that avoids restricting itself from the outset to asking only after proofs. In this paper, we explore a counterfactual approach. However, we will not be offering a full-blown counterfactual theory of intra-mathematical explanation just yet. Instead, we will offer a preliminary theory and show that the explanatory structure of intra-mathematical explanations can be modelled using counterfactuals. This clears the way for the

中文翻译：

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数学解释的反事实方法

将一个数学事实对另一个数学事实的解释称为数学内解释。迄今为止，有一种趋势是通过研究解释性和非解释性证明之间的区别来接近数学内解释的主题（例如，参见 [14; 22; 31]）。这是很自然的，因为人们普遍承认，一些证明是解释性的，而另一些则不是 [16, p. 16]。879]。尽管如此，仅仅将证明作为数学中唯一的解释中心是错误的[15; 24]。那会影响到数学中在哪里可以找到解释的问题。与其他解释案例一样，我们应该问“什么使特定的解释具有解释性？” 跳到“哪些证明是解释性的？”这个问题。引入了对可用于理解数学内解释的理论选项的限制。至少就目前而言，我们希望对数学中的解释存在于何处保持开放的态度。最终，我们寻求的是一种数学内解释理论，它能够告诉我们这种解释是如何工作的，避免从一开始就限制自己只在证明之后才提问。在本文中，我们探索了一种反事实方法。然而，我们还不会提供一个完整的数学内解释的反事实理论。反而，我们将提供一个初步理论，并表明可以使用反事实对数学内解释的解释结构进行建模。这为我们扫清了道路