The paper http://www-formal.stanford.edu/jmc/recursive.html was published, when Lisp was already implemented, to present some ideas to a different audience. But Lisp already existed by then as a running programming language.
First and foremost Lisp was a real programming language implemented on a real computer.
http://www.softwarepreservation.org/projects/LISP/lisp15_fam...
See also: History of Lisp, John McCarthy, http://jmc.stanford.edu/articles/lisp/lisp.pdf
"My desire for an algebraic list processing language for artificial intelligence work on the IBM 704 computer arose in the summer of 1956..."
"The implementation of LISP began in Fall 1958 ... These included programs to read and print list structure. I can’t now remember whether the decision to use parenthesized list notation as the external form of LISP data was made then or whether it had already been used in discussing the paper differentiation program."
"The M-notation also used brackets instead of parentheses to enclose the arguments of functions in order to reserve parentheses for list-structure constants. It was intended to compile from some approximation to the M-notation, but the M-notation was never fully defined, because representing LISP functions by LISP lists became the dominant programming language when the interpreter later became available."
"Anyway, I decided to write a paper describing LISP both as a programming language and as a formalism for doing recursive function theory. The paper was Recursive functions of symbolic ex- pressions and their computation by machine, part I (McCarthy 1960)."
McCarthy early on proposed to write a compiler - actually this was one of the very very early papers:
J. McCarthy. Memo to P. M. Morse: A Proposal for a compiler. Memo CC-56, Computation Center, Massachusetts Institute of Technology, December 13, 1957, 19 pages.
http://www.softwarepreservation.org/projects/LISP/MIT/CC-56....
what?