Standard mathematical developments, be they set theoretic, type theoretic, synthetic, etc. all follow the same basic pattern:

>Lay out a language, assume some stuff in this language, then prove that other things are true relative to what we assumed and some agreed-upon logical system.

Modern developments in set theory (and other fields) have begun exploring what happens as we vary the stuff we assume, induced to do so by statements $\varphi$ that are independent of whatever our current assumptions are but still of interest to us.

More generally, it seems that each mathematician $M$ has some collection of statements $\Phi_M$ which they hold to be 'true' independent of which axiomatic system they're working in, and the validity of a given foundation $\mathfrak{F}$ to mathematician $M$ is measured to some extent by the degree to which $$\mathfrak{F}\vdash\Phi_M,$$ that is the extent to which the given foundation can prove that the things we hold to be true are actually true, and how easily the given foundation can establish these 'facts'. Things begin to get hairy when we have statements $\varphi,\varphi'\in\Phi_M$ which are independent of some standard agreed upon foundation $\mathfrak{F}_{reasonable}$, and are mutually exclusive in the sense that $$\mathfrak{F}_{reasonable}+\varphi\vdash\neg\varphi' \hspace{5mm} \text{and} \hspace{5mm} \mathfrak{F}_{reasonable}+\varphi'\vdash\neg\varphi,$$ for example taking $\mathfrak{F}_{reasonable}=ZF$, with $\varphi={\sf axiom\ of\ choice}$ and $\varphi'={\sf axiom\ of\ determinacy}.$

I am curious if there has ever been an attempt to codify this pattern of interaction between mathematician and foundation, taking $\Phi_M$ to be the 
'primitive thing' and then exploring which foundations satisfy the above relationship. This obviously runs into some issues at primitive levels, i.e. where are we carrying out this investigation if what we are investigating are the places where we usually investigate stuff, and I have some ideas for how to address this issue but am curious how others might have remedied it. Further, we run into the question of why $\Phi_M$ itself should not always just form the foundation that mathematician $M$ uses.  Any pointers or relevant discussion are appreciated.