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.

<

http://www-maths.swan.ac.uk/staff/jrh/papers/JRHHislamWeb.pdf>

# (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 Unpublished letter to Harald Dickson.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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