Canons of Reasoning
Mathematics is a deductive rather than an empirical science.
Mathematicians will generally accept only those results for which they have (or could obtain) demonstrative proof.
Codifying the rules of proof is therefore an important part of laying the foundations for mathematics.
If you change the rules of reasoning you change the mathematics you can derive.
Intuitionists do this deliberately, and end up with constructive rather than "classical" mathematics.
Matters of Ontology
It is essential to have some abstract objects about if you are to do mathematics.
These can be implicit in the logical system, or provided through non-logical axioms.
As well as, and prior to, providing basic concepts such as that of a function, set theory provides a complex universe of sets (many of which are then used as representatives of functions).
Changes to the ontology are liable to affect the mathematics which is subsequently derived.
In set theory, logic and ontology appear to be separated, but in other systems this isn't so clear.
Intuitionists connect ontological excess with underlying logical principles.
Absoluteness and Universality
Logic is often said to be general and topic neutral.
The first attempts, by Frege and Russell, to provide logical foundations to mathematics were based on the idea that the whole of mathematics could be developed from a neutral logic by defining mathematical concepts and then deriving mathematical theories.
Since then many doubts have been cast upon this vision.
There remain philosophical, technical and pragmatic reasons for the belief that underlying the syntactic babel there can be located an absolute notion of logical truth and absolute foundation systems which can provide a basis for the development of any mathematical subject domain.