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.
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 Unpublished letter to Harald Dickson.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.]
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
Dave Herman wrote:
[...]
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. 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.]
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