Aristotelian realist philosophy of mathematics

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.

[1] 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.

[2] Paul Thagard describes Aristotelian realism as "the current philosophy of mathematics that fits best with what is known about minds and science.

"[3] 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.

[11] However not all mathematical discourse needs to be interpreted realistically; for example Aristotelians may regard the empty set and zero as fictions,[5]: 234–40  and possibly higher infinities.

Aristotelians regard non-numerical structural properties like symmetry, continuity and order as equally important as numbers.

One Aristotelian philosopher of mathematics who denies the instantiation principle on the basis of Frege’s distinction between sense and reference is Donald Gillies.