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> Lisp is not an implementation of "lambda calculus".

Maybe you could explain that one for the benefit of myself and the other mortals.

I'm not who you replied to, however:

- The original Lisp was based on mutable singly linked lists. Lambda calculus has no lists, except for Church-encoded ones (just like lambda calculus only has Church encoded booleans and numbers). It also doesn't have mutability.

- Later, Common Lisp (which was a unification of the Lisp variants that had descended from Lisp 1.5) also grew an object system, and it was implemented using dynamic scoping and dynamic typing. That stuff definitely has nothing to do with lambda calculus.

If you want an implementation of the lambda calculus, you could try Haskell 98 or Standard ML. Those are based on System F [0], a kind of typed lambda calculus.

[0]: https://en.wikipedia.org/wiki/System_F

Though Lisp was originally implemented with lists that suported RPLACA and RPLACD functions for practical reasons, the interpretation of Lisp semantics in Lisp was specified (before Lisp was implemented) without using these functions.

> The original Lisp was based on mutable singly linked lists. Lambda calculus has no lists, except for Church-encoded ones (just like lambda calculus only has Church encoded booleans and numbers).

While Lambda calculus only has 1 argument functions, you can use those to encode lists [1] and numbers in many ways, including unary, binary, and ternary [2].

[1] https://en.wikipedia.org/wiki/Church_encoding#List_encodings

[2] https://bruijn.marvinborner.de/std/Number_Ternary.bruijn.htm...

Encodings built up within lambda calculus are not known to lambda calculus itself, and are ambiguous. There is no way to tell whether some encoding isn't just supposed to denote itself (the lambda calculus functional term that it is) rather than a number that it represents by convention.

> Encodings built up within lambda calculus are not known to lambda calculus itself, and are ambiguous.

Which is why it's strange for trealira to single out Church encoded booleans and numbers.

I wasn't trying to single them out; I was saying that lambda calculus has only lambda terms and applications, whereas Lisp always natively supported mutable linked lists from the beginning.

Coming back to that argument after a day, though, it admittedly seems like a weak argument; after all, Standard ML supports mutability and linked lists natively as well, and I gave that as an example of typed lambda calculus. Maybe a better argument is that it's dynamically typed, whereas I don't think there are dynamically typed formations of lambda calculus.

The original lambda calculus had no type system at all, it was just a term rewriting system of variables and lambda abstractions with two rules: (\lambda a.M)N --> M[N/a] (\lambda a.M a) --> M In this sense we can say it is "dynamically typed" in that it has a single type, the type of lambda terms, although this seems like it sells short the memory safety guarantees associated to types in modern dynamically typed languages.

This is the system that comes to mind for me when I think of "lambda calculus" because it is the one that was most important in the history of computability and logic, it can express the same computable functions as Turing machines. System F is not Turing complete.

You and the other mortals picked up the idea somewhere that Lisp is an implementation of lambda calculus. It was almost totally incorrect.

1. Lambda calculus has only function terms. Lisp has many types: symbols, strings, conses, vectors, characters, integers, floating-point numbers, ...

2. Lambda calculus has no list processing.

3. Lambda calculus has no quote operator to operate on pieces of its own syntax as data. There is no straightforward way to write a meta-circular interpreter for lambda calculus in lambda calculus. (There are papers about it if you want to see how hard this is.) Lisp evaluation defined in Lisp before it was even implemented, in a small number of definitions.

4. Lambda calculus has no functions with optional arguments, or variadic functions. Only functions of one argument, which is required (using currying to simulate more arguments).

5. Lambda calculus has no dynamic control transfers: throw/catch, restarts; no object system; no interactivity.

6. Lambda calculus has no symbols and no named entities: no global function environment. No dynamic/global variables. No mutable variables. No "goto" analogous to Common Lisp tagbody/go.

Here is a nice article about lambda calculus self-interpreters: http://anthonylorenhart.com/2021-09-04-Extremely-Simple-Self...

tromp is the local expert around here and may have something more enlightening to say.

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

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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.

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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.

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