Usage of the word "formal(ly)"

Question

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!

Answer 1

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.

Answer 2

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.

Answer 3

There is another definition, used in works like Thompson's (1917) On Growth and Form and Alexander's (1964) Notes on the Synthesis of Form.

For Alexander, "physical order," "organization," and "pattern" are all approximate synonyms to "form." As in rschwieb's answer, the form is a concrete realisation of something -- a simple name for this something is "function."

However, in general "function" is too specific and telelogical; another broader term in Alexander's lexicon that also describes what gives rise to form is "forces."

Some added complexity along these lines comes from, e.g., Turing, whose paper on "The Chemical Basis of Morphogenesis" is basically a continuation of Thompson's line of thinking. Morphogenesis is basically a shorthand for "the growth of form." The philosopher Simondon also thought about information this way, and Sloman describes how this philosophical usage relates to colloquial understandings of the term.

What all of this has to do with the formalism of, e.g., Hilbert is quite interesting. In this sort of mathematical formalism, what is considered is manipulation of statements according to certain given axioms and inference rules. This is similar to the usage of Jazwinksi in the question. Another example would be "formal manipulation of Taylor series," famously used by Euler. Here, certain rules are followed algorithmically, without concern for whether they are entirely legal, or for what the results "mean." See André Nicolas's comment about "symbol-manipulation" -- but keep in mind that to be "formal" there do need to be some underlying rules governing the symbol manipulations, they are not arbitrary.

Contrary to the popular mis-attributed quote, Hilbert did not think of mathematics as a "a game played according to certain simple rules with meaningless marks on paper." Russell did seem to think about it this way, since he said:

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Even if this is true (!) it is a rather extreme view. For Hilbert, rather than truth, the relevant consideration would be plausibility:

In discussing the method of mathematics, I have already stressed that when building a particular theory, it is a fully justified procedure to assume still unproved, but plausible, theorems (as axioms), provided one is clear about the incomplete character of this way of laying the foundations of the theory.

To simplify somewhat, a certain kind of elementary "formal" argument would be one along the lines of "If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck." This may be surprising, because it makes reference to observations, not just axioms or definitions (although it does refer to those). What's interesting though is that the observations explicitly refer to observed form (in the Alexander/Thompson sense described above). In this restricted setting we see, again, that the symbol manipulation is not arbitrary, and that it is guided not only by rules, but also by observations. In the same way, formal manipulation of Taylor series is only used when it is useful for the purpose at hand!