This is weird. To me, in mathematical contexts, "formally" means something like "rigorously", i.e. the opposite of informally/heuristically.
And yet, I very often read papers very the word seems to mean something else, much less defined. Example:
The treatment, though formal, is given in a continuous, functional analytical, setting.
Given the rigor associated with functional analysis, this sentence seems at odds with the perception of "formal" as denoting rigor.
Or, from Jazwinksi, who uses the word a lot [searched for "formally"]:
White Gaussian noise sample functions may be formally regarded as delta functions of vanishingly small area.
This sentence concludes a heuristic derivation of white noise as the derivative of Brownian motion.
So, are people just using "formally" very sloppily (or shall we say informally), or am I missing something?
Ok, I'm starting to understand. It seems that mostly when people write "formal" it is implied that we don't necessarily have to be very rigorous. For example, Jazwinski says, "...white Gaussian noise is the formal derivative of Brownian motion".
It must be said that this is quite confusing, as both "formal" and "informal" then suggests a lack of rigor. I.e. they are not antonyms!