There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized.

有多种现实主义但非柏拉图主义的数学哲学-自然主义或亚里士多德主义现实主义。它们由亚里斯多德和密尔（Aristotle and Mill）持有，在20世纪数学哲学中几乎没有发挥任何作用，但最近又得到了复兴。他们将数学与其他科学同化。他们认为数学是X的科学，其中X是（物理或其他非抽象的）世界的某些可观察特征。X的选择包括数量，结构，样式，复杂性，关系。本文列出并比较了这些选项，包括它们对X的解释，支持每种理论的示例以及用（大部分或全部）数学识别X科学的原因。对这些选项进行了一些比较，但主要目的是展示柏拉图主义和名义主义的可行替代方案的范围。