TITLE: What is the meaning of "formal" in math-speak?
QUESTION [8 upvotes]: I have been encountering the word "formal" in several math books, and I don't understand what it means in context.  For example, Hatcher describes n-chains as "finite formal sums" on p. 105 and p. 108. What is a "formal sum?"  Are there "informal sums?"
Similarly, Steve Awodey on p.15 of "Category Theory" says opposite (or dual) categories are derived from other categories "with all the arrows formally turned around."  What does "formally" mean here?
This may be more of a language question than a math question, but I'd appreciate any clarification.

REPLY [2 votes]: "Formal" here is meant in the sense of "relating to form".
In regards to the first example, given a set $X$, a "Formal sum of things in $X$" means an element of the free abelian group generated by $X$.
There's a general construction in universal algebra of free algebraic structures where the element of a free algebra on $n$ elements are precisely the algebraic expressions one can write down in $n$ variables. (modulo the relations such structures satisfy)
For example, elements of the free commutative ring generated by one element are polynomials — the kinds of expressions you can construct from a single variable using just addition and multiplication (and $0$, $1$, and negation), simplified by imposing the algebraic identities defining rings.