Editing Philosophy of mathematics (section)

====Aristotelian realism<!--linked from 'Structuralism (philosophy of mathematics)'-->====
{{Main|Aristotelian realist philosophy of mathematics}}
{{See also|In re structuralism|Immanent realism}}
[[Aristotelian realist philosophy of mathematics|Aristotelian realism]] holds that mathematics studies properties such as symmetry, continuity and order that can be literally 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. 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.<ref>{{cite book |last=Franklin |first=James |date=2014 |title=An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure |url=https://web.maths.unsw.edu.au/~jim/franklinaristotelianrealistphilosophyofmathematics.pdf |publisher=Palgrave Macmillan |isbn=9781137400727}}</ref><ref>{{cite journal |last1=Franklin |first1=James |date=2022 |title=Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics |url=https://rdcu.be/chatd |journal=Foundations of Science |volume=27 |issue=2 |pages=327–344|doi=10.1007/s10699-021-09786-1 |s2cid=233658181 |access-date=30 June 2021}}</ref> Aristotelian realism is defended by [[James Franklin (philosopher)|James Franklin]] and the [http://web.maths.unsw.edu.au/~jim/structmath.html Sydney School] in the philosophy of mathematics and is close to the view of [[Penelope Maddy]] that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world).<ref>[[Penelope Maddy|Maddy, Penelope]] (1990), ''Realism in Mathematics'', Oxford University Press, Oxford, UK.</ref> A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world.

The Euclidean arithmetic developed by [[John Penn Mayberry]] in his book ''The Foundations of Mathematics in the Theory of Sets''<ref name="Mayberry-2001"/> also falls into the Aristotelian realist tradition. Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". Whether or not there are definite multitudes of units for which Euclid's Common Notion 5 (the whole is greater than the part) fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.