Intentionalism

The Meaning of Pure Mathematics
Jan Mycielski
Journal of Philosophical Logic
Vol. 18, No. 3 (Aug., 1989), pp. 315-320
jstor.org/stable/30227216

Platonism is unsatisfactory because it violates our instinctive drive to obey Ockham's principle of parsimony.

Intentionalism says that pure mathematics is a description of finite structures consisting of finitely many imagined objects.

The term intentionalism is chosen for its contrast with extensionalism which accepts actually infinite sets and leads naturally to Platonism.

This is intentionalism as developed by Jan Mycielski. The following are links to what I have written about this mathematical philosophy:

Persons without infinities
Closing Plato's Cave
Three paths to becoming a mathematical anti-platonist
Transfinity
Mathematica materialis, or How not to be lured into Plato's cave

Also by Jan Mycielski:

Locally Finite Theories
Analysis Without Actual Infinity

(Mycielski's approach is presented in two chapters of Understanding the Infinite by Shaughan Lavine: "The Finite Mathematics of Indefinitely Large Size", "The Theory of Zillions")