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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?

Edit:

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!

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This is just an English language quirk. There are two basic meanings of 'formal' -- something related to 'form', which is not what you have in mind (despite being what Jazwinksi has in mind), and being rigorous. Mathematicians typically use the word rigorous when they mean rigorous. –  Glen Wheeler May 31 '12 at 14:32
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Formal has many meanings. Sometimes it refers to a symbol-manipulation calculation that might be wrong, because of issues such as unjustified switching of order of summation and integration. –  André Nicolas May 31 '12 at 14:39
    
@GlenWheeler I can't believe I didn't realize that. It certainly makes sense in most of Jazwinksi's usages, although I am still not too happy with its usage in the two specific examples cited in my post. –  Patrick May 31 '12 at 14:45
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up vote 12 down vote accepted

As André says in the comments, there is a second meaning of "formally" which means roughly "symbolically." For example when we talk about formal power series we ignore issues of convergence and work only with the symbolic form of various power series. That is, we only look at the form of the things we're manipulating rather than thinking particularly hard about what exactly they are, what spaces they live in, etc.

Regarding your edit, "formal" is more specific than a lack of rigor. It specifically means paying attention to form. Many formal manipulations can be made rigorous but one has to do extra work to do so, and after one has made them rigorous one often finds that there was essentially nothing wrong with the formal manipulations. For example, the Dirac delta function can be treated formally, but it can also be treated rigorously using the theory of distributions. But engineers and physicists didn't need distributions to use the Dirac delta to solve differential equations. In this sense "formal" is related to the notion of moral truth in mathematics.

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This makes a lot of sense. Thank you! –  Patrick May 31 '12 at 15:29
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Formal descriptions emphasize a concrete realization of something (emphasizing the form.)

For instance, an informal description of the rational numbers is "all fractions you can make with an integer in top and a nonzero integer in bottom". While this is easy to understand, it does not actually pin down what a rational number is.

Formally, you can realize the rationals as equivalence classes of a relation on $\mathbb{Z}\times (\mathbb{Z}\setminus{\{0\}})$. This is very rigorous compared to the informal version.

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