Aristotelian realist philosophy of mathematics

In the

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.^[2] Paul Thagard describes Aristotelian realism as "the current philosophy of mathematics that fits best with what is known about minds and science."^[3]

History

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.^[4] 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).^[5]

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, ^[6] Anne Newstead,^[7] Donald Gillies, and others.

Numbers and sets

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 ... the measure must always be some identical thing predicable of all the things it measures, e.g. if the things are horses, the measure is 'horse'."^[8] Glenn Kessler develops this into the view that a number is a relation between a heap and a universal that divides it into units; for example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots.^{[9][10][5]: 36–8}

On an Aristotelian view,

Aristotelians regard non-numerical structural properties like symmetry, continuity and order as equally important as numbers. Such properties are realized in physical reality, and are the subject matter of parts of mathematics. For example

Since mathematical properties are realized in the physical world, they can be directly perceived. For example, humans easily perceive facial symmetry.

Aristotelians also accord a role to abstraction and idealisation in mathematical thinking. This view goes back to Aristotle's statement in his Physics that the mind 'separates out' in thought the properties that it studies in mathematics, considering the timeless properties of bodies apart from the world of change (Physics II.2.193b31-35).

At the higher levels of mathematics, Aristotelians follow the theory of Aristotle's Posterior Analytics, according to which the proof of a mathematical proposition ideally allows the reader to understand why the proposition must be true.^{[5]: 192–6}

Objections to Aristotelian realism

A problem for Aristotelian realism is what account to give of higher

"Set theory is committed to the existence of infinite sets that are so huge that they simply dwarf garden variety infinite sets, like the set of all the natural numbers. There is just no plausible way to interpret this talk of gigantic infinite sets as being about physical objects."[12]

Aristotelians reply that sciences can deal with uninstantiated universals; for example the science of color can deal with a shade of blue that happens not to occur on any real object.