All hidden assumptions must be flagged and made explicit.
This is over optimistic even for honest applied mathematicians. The problem is one has often not even thought about (or enough knowledge about) all the assumptions that then should to stated. At least if it is going to be used for something applied outside pure mathematics.
Good applied mathematics and probability theory must ultimately be rooted in good physics. And also other way around, but if part of the depth of reality is discrete then for example probabilities based on continuous mathematics will fail in the limit?
The Scientific Method might be useful here
https://www.youtube.com/watch?v=EYPapE-3FRw
One of the main problems is experiments often are ignored for decades. Early 1900 series of empirical research showed returns in commodity prices and stocks where far from normal distributed. This was for example ignored by people like Osborne who even himself found the same in his empirical data in 1959, but claimed the data where wrong and the Gaussian model (GBM) had to be correct.
Exactly same happened in studies of galaxy velocity distributions even 1990´s, the observations was ignored as incorrect data for many years before one admitted data correct, model wrong. Actually such loosely scientific attitude in astronomy (ignoring the data) was pointed out by swedish mathematician already early 1900.
We unfortunately see the same in other scientific disciplines even to this date, observations and experiments (that even are relatively cheaply to repeat) are simply ignored if not fit main frame model framework. Instead we often see the model more and more wrapped in with complex math and multiple layers of buzz words.
this is totally against the scientific method outlined by Feyman.
series of scientist have invested too many years in building bubble models to even admit thy ignore observations.