See the following excerpt of a longer paper. Generally the lambda calculus is a specific mathematical formalism, LISP is quite a bit more and an actual programm

See the following excerpt of a longer paper. Generally the lambda calculus is a specific mathematical formalism, LISP is quite a bit more and an actual programming language (it has more than just functions) and implementations don't follow the rules of lambda calculus for evaluation. We also need to see differences between formal models for some of Lisp and actual implementations of a "real" Lisp with numbers, etc. We don't need to model numbers with functions, like with "church numerals". Here follows an excerpt from a paper "Some History of Functional Programming Languages" by D. A. Turner. It talks about LISP, as invent/discovered by John McCarthy: https://www.cs.kent.ac.uk/people/staff/dat/tfp12/tfp12.pdf https://www.cs.kent.ac.uk/people/staff/dat/tfp12/tfp12.pdf ----- Some Myths about LISP Something called “Pure LISP” never existed — McCarthy (1978) records that LISP had assignment and goto before it had conditional expressions and recursion — it started as a version of FORTRAN I to which these latter were added. LISP 1.5 programmers made frequent use of setq which updates a variable and rplaca, rplacd which update the fields of a CONS cell. LISP was not based on the lambda calculus, despite using the word “LAMBDA” to denote functions. At the time he invented LISP, McCarthy was aware of (Church 1941) but had not studied it. The theoretical model behind LISP was Kleene’s theory of first order recursive functions. The M-language was first order, as already noted, but you could pass a function as a parameter by quotation, i.e. as the S-expression which encodes it. Unfortunately, this gives the wrong binding rules for free variables (dynamic instead of lexicographic). If a function has a free variable, e.g y in f = λx.x + y y should be bound to the value in scope for y where f is defined, not where f is called. McCarthy (1978) reports that this problem (wrong binding for free variables) showed up very early in a program of James Slagle. At first McCarthy assumed it was a bug and expected it to be fixed, but it actually springs from something fundamental — that meta-programming is not the same as higher order programming. Various devices were invented to get round this FUNARG problem, as it became known. Not until SCHEME (Sussman 1975) did versions of LISP with default static binding appear. Today all versions of LISP are lambda calculus based. ----- A remark from me: "Today all versions of LISP are lambda calculus based.", except where they are not, like evaluation rules, dynamic binding, data types, etc. What we have now in most Lisps since the mid 80s is lexical binding and closures, but not exclusively. Scheme earlier called dynamic bound variables "fluids". CL has it, for example by default for global variables.