What does "formal" mean?

I see formal used in at least two senses in mathematics.

• Rigorous, i.e. "here is a formal proof" as opposed to "here is an informal demonstration."
• "Formal manipulation," that is, manipulating expressions according to certain rules without caring about convergence, etc.

Confusingly they can mean opposite things in certain contexts, although "formal manipulations" can be made rigorous in many cases.

When I was learning about logic as an undergraduate, I recall being told that the word "formal", with respect to "formal languages" meant that the "form" of expressions written in that language had primacy.

In other words, rules for manipulating expressions in a formal language could be given in terms of the form of the expression only, without needing to know to what values the variables in the expression were bound.

So a formal language permits us to use relatively simple pattern-matching algorithms to decide which transformations of an expression are valid at any given time.

In this context, formality is linked to the simplicity of the rules that define the set of valid transformations of an expression.

As an example, formal power series is analyzed without regard to convergence. Really, what is of interest is the sequence of coefficients.