Hilbert's aim to reduce all mathematics to finite logical system was shown impossible by Goedel. He did mathematical analysis of logic itself (Goedel numbering). Turing defined algorithms, and mathematised (algorithms are objects of study in mathematics/cs) them as well -to solve halting problem, and formalise computation (and there goes the possibility of reducing mathematics to algorithms).
The issue which bothers me is what is "mathematical" analysis (or treatment)? What makes one treatment/analysis mathematical ? Now it is possible to do mathematics in natural language (it will be very long and tedious, but in principle, one only has to follow reasoning as outlined in the write-up). So is mathematical treatment a "style"? Is it the rigour? Is it formalisation?
Edit: I would call Turing's analysis as mathematical -bringing algorithms under the purview of mathematical techniques. This is the sense of "mathematical analysis" in this question (not to be confused with real analysis, etc.).