I've never been quite sure of the exact meanings of the terms formal, informal, rigorous and non-rigorous in mathematics.
For example, I've read a set of notes in which the author speaks of a particular derivation as being "formal, but non-rigorous", does this simply mean that the derivation was written down using purely mathematical expressions and is thus formal, but that the author has neglected as to whether a particular limit exists in the derivation, or whether two mathematical objects can be commuted, for example, and hence is non-rigorous (as issues such as convergence have not be checked explicitly).
If someone could clarify these points for me I'd much appreciate it.
To me formal means working with algebraic or analytic tools without bothering with technical hypotheses. A representative example : assume you want to show that $exp'=exp$. You write :
$$exp(x)'=(\sum_n\frac{x^n}{n!})'=\sum_n(\frac{x^n}{n!})'=0+\sum_{n\geq 1}\frac{x^{n-1}}{(n-1)!}=\sum_n\frac{x^n}{n!}=exp(x) $$
Of course, here I exchanged derivation and summation without invoking any theorem whence "formal".
This meaning is well represented by the use of "formal power series" (power series where we do not care about convergence). It takes actually some rigorous mathematics to make sure that reasoning with formal power series is valid...
Informal thinking basically means that you need some insight before going into a problem. The solution does not come out of nowhere but is the translation of an intuition (an overview of "the way things work out"). For instance, if one thinks about a very simple result, if $A$ and $B$ two subsets of a finite subset $C$ and one tries to show that $|A\cup B|=|A|+|B|-|A\cap B|$.
The informal thinking would be here to draw $A$ and $B$ count all elements of $A$, all elements of $B$ and one sees that actually the elements of $A\cap B$ (and only those ones) will be counted twice so that we need to discard them once.
The rigorous counter-part here would be to use indicator functions.
To sum up formal and informal reasoning are both non-rigorous.
The difference between both is that for the formal reasoning you just have faith in the mathematical formalism and assume that everything works as good as it can, go as far as you can and try to see if you reach an interesting conclusion (to be proven later).
Whereas in an informal reasoning you try to go as far as you can from the mathematical formalism and try to express the clearest way possible (with English sentences or pictures or whatever...) the idea behind the mathematical formalism.