The Foundation of Matematics consist of mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory.Thre are related to philosofi of matematics.
Philosofi of matematics consist of :
“Platonists, such as Kurt Gödel, hold that numbers are abstract, necessarily existing objects, independent of the human mind”
“Formalists, such as David Hilbert (1862–1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...” 
“Intuitionists, such as L. E. J. Brouwer (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them.” 
The Foundation Of Matematics :
1) Strong : example geometri in Ingland to be foundation of matematics
2) Viewless :
Epistimologi : learn the science of mathematics resources.(Emanuel Khan)
Based on Critical Thinking
1. Syntetic apriory : not see the person can understand. The science obtained from Kontradiktion.