Path: utzoo!attcan!uunet!ogicse!emory!utkcs2!mclennan
From: mclennan@cs.utk.edu (Bruce MacLennan)
Newsgroups: comp.lang.functional
Subject: origin of lambda symbol
Summary: lambda a modified caret
Keywords: lambda, Church, lambda calculus, notation
Message-ID: <1991Feb27.165322.25377@cs.utk.edu>
Date: 27 Feb 91 16:53:22 GMT
Sender: maclennan@cs.utk.edu (Bruce MacLennan)
Followup-To: why lambda ? (Daniel de Rauglaudre)
Distribution: comp.lang.functional
Organization: University of Tennessee, Knoxville - CS Department
Lines: 53

> From: jhf@chaco.c3.lanl.gov (Joe Fasel)
 . . . . .
> Hi, Bruce.
> 
> The following query was posted to comp.lang.functional recently:
> 
> | From: ddr@margaux.inria.fr (Daniel de Rauglaudre)
 . . . . .
> | There is a question of great importance in our institute: why "lambda"
> | in "lambda calculus" ? Why not "alpha" or "dzeta" or any other symbol ?
> | Does anybody know the origin of this choice ? Thank you for you answers.
> |
> |         Daniel de Rauglaudre
> |         INRIA - France
> |         ddr@inria.inria.fr
> 
> I recall that you got the answer straight from the horse's mouth at the
> 82 LFP conference.  I think I remember the gist of it, but I might get
> the details wrong, so I wonder if you could relate the story to us.
> 
> Thanks.  I hope all is going well for you and your family.
> 
> --Joe

At the 1982 LISP and Functional Programming Conference I asked
Alonzo Church about the origin of the lambda symbol.  What I found out
is briefly summarized in a footnote on p. 357 of my book
"Functional Programming:  Practice and Theory" (Addison Wesley,
1990).  Since Church never confirmed the story in writing, I
thought it was inappropriate to attribute it to him in my book.
Nevertheless, here is the history of the lambda symbol, based 
on the notes I wrote down after my conversation.

Church said that the starting point was Russell and Whitehead's
abstraction operator (in Principia Mathematica), which they wrote
with a caret over the bound variable:  $\hat{x}(x^2+1)$.  To facilitate
mechanical manipulation of the bound variables, he began to write
the caret in front of the bound variable (because, I presume, this
made the abstraction a string rather than a two-dimensional structure):
$\hat{}x(x^2+1)$.  From there the caret symbol evolved into an
uppercase lambda, $\Lambda x(x^2+1)$, and finally a lower case lambda,
$\lambda x(x^2+1)$.  I presume the latter stages of this evolution
were under the pressure of writing convenience and to avoid confusion
with other symbols (such as the and-sign).

Bruce MacLennan
Department of Computer Science
107 Ayres Hall
The University of Tennessee
Knoxville, TN 37996-1301

(615)974-0994/5067
maclennan@cs.utk.edu