McCarthy formalism is a formalism for defining functions recursively, first introduced in classic paper Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I (1960).
In his 1967 Computation: Finite and Infinite Machines, Marvin Minsky in his § 10.6 Conditional Expressions: The McCarthy Formalism describes the "formalism" as follows:
"Practical computer languages do not lend themselves to formal mathematical treatment--they are not designed to make it easy to prove theorems about the procedures they describe. In a paper by McCarthy  we find a formalism that enhances the practical aspect of the recursive-function concept, while preserving and improving its mathematical clarity.
McCarthy introduces "conditional expressions" of the form
f = (if p1 then e1 else e2)where the
eiare expressions and
p1is a statement (or equation) that may be true or false. This expression means: See if
p1is true; if so the value of
fis given by
p1is false, the value of
fis given by
The McCarthy formalism is like the general recursive (Kleene) system, in being based on some basic functions, composition, and equality, but with the conditional expression alone replacing both the primitive-recursive scheme and the minimization operator." (Minsky 1967:192-193)
I'm interested in chronology. Was there any other formalism regarding recursive function before McCarthy Formalism (in computer science)?