#### 28 Dec 2012 oubiwann»(Journeyer)

The Secret History of Lambda

Being a bit of an origins nut (I always want to know how something came to be or why it is a certain way), one of the things that always bothered me with regard to Lisp was that no one seemed to talking about the origin of lambda in the lambda calculus. I suppose if I wasn't lazy, I'd have gone to a library and spent some time looking it up. But since I was lazy, I used Wikipedia. Sadly, I never got what I wanted: no history of lambda. [1] Well, certainly some information about the history of the lambda calculus, but not the actual character or term in that context.

Why lambda? Why not gamma or delta? Or Siddham ṇḍha?

To my great relief, this question was finally answered when I was reading one of the best Lisp books I've ever read: Peter Norvig's Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp. I'll save my discussion of that book for later; right now I'm going to focus on the paragraph at location 821 of my Kindle edition of the book. [2]

The story goes something like this:
• Between 1910 and 1913, Alfred Whitehead and Bertrand Russell published three volumes of their Principia Mathematica, a work whose purpose was to derive all of mathematics from basic principles in logic. In these tomes, they cover two types of functions: the familiar descriptive functions (defined using relations), and then propositional functions. [3]
• Within the context of propositional functions, the authors make a typographical distinction between free variables and bound variables or functions that have an actual name: bound variables use circumflex notation, e.g. x̂(x+x). [4]
• Around 1928, Church (and then later, with his grad students Stephen Kleene and J. B. Rosser) started attempting to improve upon Russell and Whitehead regarding a foundation for logic. [5]
• Reportedly, Church stated that the use of  in the Principia was for class abstractions, and he needed to distinguish that from function abstractions, so he used x [6] or ^x [7] for the latter.
• However, these proved to be awkward for different reasons, and an uppercase lambda was used: Λx. [8].
• More awkwardness followed, as this was too easily confused with other symbols (perhaps uppercase delta? logical and?). Therefore, he substituted the lowercase λ. [9]
• John McCarthy was a student of Alonzo Church and, as such, had inherited Church's notation for functions. When McCarthy invented Lisp in the late 1950s, he used the lambda notation for creating functions, though unlike Church, he spelled it out. [10]
It seems that our beloved lambda [11], then, is an accident in typography more than anything else.

Somehow, this endears lambda to me even more ;-)

[1] As you can see from the rest of the footnotes, I've done some research since then and have found other references to this history of the lambda notation.

[2] Norvig, Peter (1991-10-15). Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp (Kindle Locations 821-829). Elsevier Science - A. Kindle Edition. The paragraph in question is quoted here:
The name lambda comes from the mathematician Alonzo Church’s notation for functions (Church 1941). Lisp usually prefers expressive names over terse Greek letters, but lambda is an exception. Abetter name would be make - function. Lambda derives from the notation in Russell and Whitehead’s Principia Mathematica, which used a caret over bound variables: x( x + x). Church wanted a one-dimensional string, so he moved the caret in front: ^ x( x + x). The caret looked funny with nothing below it, so Church switched to the closest thing, an uppercase lambda, Λx( x + x). The Λ was easily confused with other symbols, so eventually the lowercase lambda was substituted: λx( x + x). John McCarthy was a student of Church’s at Princeton, so when McCarthy invented Lisp in 1958, he adopted the lambda notation. There were no Greek letters on the keypunches of that era, so McCarthy used (lambda (x) (+ xx)), and it has survived to this day.

[4] Norvig, 1991, Location 821.

[5] History of Lambda-calculus and Combinatory Logic, page 7.

[6] Ibid.

[7] Norvig, 1991, Location 821.

[8] Ibid.

[9] Looking at Church's works online, he uses lambda notation in his 1932 paper A Set of Postulates for the Foundation of Logic. His preceding papers upon which the seminal 1932 is based On the Law of Excluded Middle (1928) and Alternatives to Zermelo's Assumption (1927), make no reference to lambda notation. As such, A Set of Postulates for the Foundation of Logic seems to be his first paper that references lambda.

[10] Norvig indicates that this is simply due to the limitations of the keypunches in the 1950s that did not have keys for Greek letters.

[11] Alex Martelli is not a fan of lambda in the context of Python, and though a good friend of Peter Norvig, I've heard Alex refer to lambda as an abomination :-) So, perhaps not beloved for everyone. In fact, Peter Norvig himself wrote (see above) that a better name would have been make-function.

Syndicated 2012-12-28 17:34:00 (Updated 2014-11-21 21:12:30) from Duncan McGreggor