Aristotelian realist philosophy of mathematics

In the philosophy of mathematics, Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be immanently realized in the physical world (or in any other world there might be). It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. It contrasts with nominalism, fictionalism, and logicism in holding that mathematics is not about mere names or methods of inference or calculation but about certain real aspects of the world. Aristotelian realists emphasize applied mathematics, especially mathematical modeling, rather than pure mathematics as philosophically most important. Marc Lange argues that "Aristotelian realism allows mathematical facts to be explainers in distinctively mathematical explanations" in science as mathematical facts are themselves about the physical world. Paul Thagard describes Aristotelian realism as "the current philosophy of mathematics that fits best with what is known about minds and science." Although Aristotle did not write extensively on the philosophy of mathematics, his various remarks on the topic exhibit a coherent view of the subject as being both about abstractions and applicable to the real world of space and counting. Until the eighteenth century, the most common philosophy of mathematics was the Aristotelian view that it is the "science of quantity", with quantity divided into the continuous (studied by geometry) and the discrete (studied by arithmetic). Aristotelian approaches to the philosophy of mathematics were rare in the twentieth century but were revived by Penelope Maddy in Realism in Mathematics (1990) and by a number of authors since 2000 such as James Franklin, Anne Newstead, Donald Gillies, and others. Aristotelian views of (cardinal or counting) numbers begin with Aristotle's observation that the number of a heap or collection is relative to the unit or measure chosen: "'number' means a measured plurality and a plurality of measures .