More basically, can any assumption at all be put into purely mathematical terms? I would contend not.
Things can be bound to mathematical formalisms by theories, but math itself is not about reality, and cannot really be rendered so. It can only codify interpretations, it cannot make them or express them.
The actual assumptions are made in the framing of the theory that makes the math seem to say something, and they cannot really be captured precisely at all, because of the theory-embeddedness of language. The math can pull out and summarise the systematic parts of a theory, but you already have to understand the linkage between reality and the measures involved. Those are the assumptions. There is no content in the mathematics not implicit in the unstated bond between the math and the science involved.
Edit: I should be explict about the framing. I adopt a kind of specific neo-Intuitionist model of mathematics as the creative process of exploring and elaborating the contents of shared intuitions about thinking that are not optional.
By reality I mean external, shared reality, not including mental contents.
If mathematics is all the built-in machinery necessary for us to apply thought to reality, then it is all about us, and not about external reality.
And by assumption, I mean something that could be decided otherwise and matter. The axioms of mathematics are not assumptions in this sense, they could be made otherwise, but to the extent they were, they would not matter anymore, at least to the problems at hand. They would apply to a different set of problems.
In that sense math is not made of assumptions we make, it is made of assertions that are made for us, that we inherit by virtue of having our minds, or that we creatively contrive by observing those basic assumptions.
The linkage that actually makes the application of mathematics to reality is not mathematics, it is made by the assumptions of some theories, and the mathematics would be there whatever assumptions you did make or theories you held instead.
To me, the language metaphor is good here -- English is not a discourse, it does not have real story or argumentative content, and it would be there no matter what discourse you chose to have in it. It is a part of our communications other than any discourse. Likewise, math is a part of all out descriptions other than any theory or decision...
So theory and mathematics are pretty much two independent aspects of a single process.