Origins of lambda notation (was: Allen's lambda syntax proposal)

David-Sarah Hopwood david.hopwood at
Sun Dec 21 12:46:32 PST 2008

Dave Herman wrote:
> In Principia Mathematica the notation for the function f with f(x) = 2x + 1 is
>      ^
>     2x + 1.
> The typesetter could not position the hat on top of the x and placed it
> in front of it, resulting in ^x.2x + 1.

That's not quite right. In Principia Mathematica, x̂ was used for
class abstraction. For example x̂.x = y or x̂(x = y) would mean the
singleton class, in this case also a set, {y}. Gottlob Frege had
earlier used a half-ring or reversed-c above, e.g. x͗(x = y), for
essentially the same thing [Frege, 1902].

Since this is not the same as functional abstraction (but was similar
in the sense of being a variable-binding operator), Church deliberately
made the change to using a prefix operator; that step was not due to
typesetting considerations. As I posted before,

Felice Cardone, J. Roger Hindley,
History of Lambda-calculus and Combinatory Logic
2006, Swansea University Mathematics Department
Research Report No. MRRS-05-06.

# (By the way, why did Church choose the notation "λ"? In [Church, 1964, §2]
# he stated clearly that it came from the notation "x̂" used for
# class-abstraction by Whitehead and Russell, by first modifying "x̂" to
# "∧x" to distinguish function-abstraction from class-abstraction, and
# then changing "∧" to "λ" for ease of printing. This origin was also
# reported in [Rosser, 1984, p.338]. On the other hand, in his later years
# Church told two enquirers that the choice was more accidental: a symbol
# was needed and "λ" just happened to be chosen.)

[Church, 1964] A. Church, 7 July 1964. Unpublished letter to Harald Dickson.

[Rosser, 1984] J. B. Rosser. Highlights of the history of the lambda
calculus. Annals of the History of Computing, 6:337–349, 1984.

[Frege, 1902] G. Frege, 1902. Letter to Russell. Reproduced in
Van Heijenoort, J. From Frege to Gödel: A Source Book in Mathematical
Logic, 1879-1931. AuthorHouse, December 6, 1999. ISBN 158348597X.

[If your email client doesn't display the above correctly:
  "x̂" is an x with a hat operator (similar to a circumflex) above it.
  "x͗" is an x with a reversed 'c', or half-ring open on the left, above it.
  "∧" is an upward-pointing wedge.
  "λ" is a lowercase Greek lambda.]

David-Sarah Hopwood

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