On the etymology of the word "functor"

On the etymology of the word "functor"

[Fosco Loregian]

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It seems surprisingly difficult to trace back the precise origin of the word "functor" imported by Mac Lane in category theory from philosophy. I wonder if someone more experienced than me can find a better answer to this mystery.

- As it is well-known, Mac Lane says in the historical notes of Chapter 1 in CWM, that the name "functor" is borrowed from Carnap's "Logische Syntax der Sprache"; Carnap writes the book in 1934.

In his book "Tool and object", Ralf Kr¨omer partially rectifies this claim in that he says: " The somewhat arrogant account [of Mac Lane's review of Carnap's LSS] obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language".

So, whence did Carnap borrow the term? Was it also a "current informal parlance" [CWM, p. 30] in the logical analysis of language, as much as "natural transformation" was in Mathematics? (cf. again Kr¨omer, where he makes a good point of how the term was employed by Lefschetz and Hurewicz).

It seems that Haskell Curry, in his "Some logical aspects of grammatical structures", attributes the term to Tadeusz Kotarbiński, where it was introduced in his 1929's Elementy teorji poznania -in a somewhat clumsy translation, edited in English as "Gnosiology" by O. Wojtasiewicz, and only in 1966. It seems then reasonable that Mac Lane, not knowing Polish (or did he?) just wasn't aware of this much older occurrence. Curry's paper is behind a paywall for me, but in his 1979's book "Foundations of Mathematical Logic" Curry summarized some ideas from his "Logical aspects" (it's the paper where Curry introduces the toy language of szám, tetél and tantét), and the word "functor" itself occurs multiple times in the latter book, with no particular introduction -reasonable, in 1979.

Too bad that, if one is stubborn enough to find a copy of "Gnosiology" (the original in Polish seems quite difficult to recover, but I'd happy to see it), they will notice that yes, the term "functor" is explained to some extent in the text, but it is not introduced in proper detail, as if the concept was already there and Kotarbiński just borrowed it from someone else. (Kotarbiński speaks of a functor as an abstract "sentential connective" at page 259, at the very start of his second chapter "The deductive method".
"Gnosiology" comes with an appendix containing the review that Adjukiewicz wrote on Kotarbiński's book; Adjukiewicz uses the word functor quite liberally (see for example: "if that of which mathematics speaks is the objective correlates of some functors occurring in mathematical theorems, correlates which in turn have no arguments, then mathematics speaks of numbers, as, for example, in the arithmetical statement <<3 + 2 = 5>> such ultimate arguments are numbers, and nothing else"). There is no mention about Kotarbiński being the first to use the word, be it as formalization of a concept from common parlance, or as a word coined for the first time.

This is where I start getting lost, and my skills are not enough any more.

Kotarbiński seems in fact to attribute the coinage of the word "functor" to Łukasiewicz or Leśniewski, but never explicitly links any of the two to the term (I quote: p 244, "Lukasiewicz, in his system of the sentential calculus, places the functors directly before the functions to which they pertain" and p 403 "other logical types can be formed by sentence-forming or term-forming functors of the various kinds (Lesniewski)"). I find very little evidence that this attribution can be confirmed; this paper http://www.numdam.org/article/CM_1968__20__153_0.pdf<https://protect-au.mimecast.com/s/-z-bC6XQ68fKKO5MCpjk5b?domain=numdam.org> talks from the very beginning of "the implication and negation functors of Łukasiewicz" referring to Rosser and Turquette's book "Many-valued logics". Unfortunately, the only edition of the book I could find has zero occurrences of the word "functor". From what I can find on the internet, Leśniewski seems to widely employ the _concept_ of a functor, and he is taken as the most ancient philosopher doing so (Leśniewski dies a few years after Carnap's book is published!) but there seems to be no proof that he was the first to employ the _name_.

I'd like to get to the end of this. Any help?



Fosco


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Re: On the etymology of the word "functor"

[Johannes Huebschmann]

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Frege used the term "functor". Perhaps it then was
lingua franca in philosophy circles.
Also we should be aware of the notion of function
being a recent idea (Euler ...).

Best

Johannes



________________________________
De: "Fosco Loregian" <fosco.loregian@gmail.com>
À: "categories" <categories@mq.edu.au>
Envoyé: Samedi 17 Février 2024 13:09:21
Objet: On the etymology of the word "functor"

It seems surprisingly difficult to trace back the precise origin of the word "functor" imported by Mac Lane in category theory from philosophy. I wonder if someone more experienced than me can find a better answer to this mystery.

- As it is well-known, Mac Lane says in the historical notes of Chapter 1 in CWM, that the name "functor" is borrowed from Carnap's "Logische Syntax der Sprache"; Carnap writes the book in 1934.

In his book "Tool and object", Ralf Kr¨omer partially rectifies this claim in that he says: " The somewhat arrogant account [of Mac Lane's review of Carnap's LSS] obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language".

So, whence did Carnap borrow the term? Was it also a "current informal parlance" [CWM, p. 30] in the logical analysis of language, as much as "natural transformation" was in Mathematics? (cf. again Kr¨omer, where he makes a good point of how the term was employed by Lefschetz and Hurewicz).

It seems that Haskell Curry, in his "Some logical aspects of grammatical structures", attributes the term to Tadeusz Kotarbiński, where it was introduced in his 1929's Elementy teorji poznania -in a somewhat clumsy translation, edited in English as "Gnosiology" by O. Wojtasiewicz, and only in 1966. It seems then reasonable that Mac Lane, not knowing Polish (or did he?) just wasn't aware of this much older occurrence. Curry's paper is behind a paywall for me, but in his 1979's book "Foundations of Mathematical Logic" Curry summarized some ideas from his "Logical aspects" (it's the paper where Curry introduces the toy language of szám, tetél and tantét), and the word "functor" itself occurs multiple times in the latter book, with no particular introduction -reasonable, in 1979.

Too bad that, if one is stubborn enough to find a copy of "Gnosiology" (the original in Polish seems quite difficult to recover, but I'd happy to see it), they will notice that yes, the term "functor" is explained to some extent in the text, but it is not introduced in proper detail, as if the concept was already there and Kotarbiński just borrowed it from someone else. (Kotarbiński speaks of a functor as an abstract "sentential connective" at page 259, at the very start of his second chapter "The deductive method".
"Gnosiology" comes with an appendix containing the review that Adjukiewicz wrote on Kotarbiński's book; Adjukiewicz uses the word functor quite liberally (see for example: "if that of which mathematics speaks is the objective correlates of some functors occurring in mathematical theorems, correlates which in turn have no arguments, then mathematics speaks of numbers, as, for example, in the arithmetical statement <<3 + 2 = 5>> such ultimate arguments are numbers, and nothing else"). There is no mention about Kotarbiński being the first to use the word, be it as formalization of a concept from common parlance, or as a word coined for the first time.

This is where I start getting lost, and my skills are not enough any more.

Kotarbiński seems in fact to attribute the coinage of the word "functor" to Łukasiewicz or Leśniewski, but never explicitly links any of the two to the term (I quote: p 244, "Lukasiewicz, in his system of the sentential calculus, places the functors directly before the functions to which they pertain" and p 403 "other logical types can be formed by sentence-forming or term-forming functors of the various kinds (Lesniewski)"). I find very little evidence that this attribution can be confirmed; this paper http://www.numdam.org/article/CM_1968__20__153_0.pdf<https://protect-au.mimecast.com/s/kanWCxngGkfA6NrkH8OfKY?domain=numdam.org> talks from the very beginning of "the implication and negation functors of Łukasiewicz" referring to Rosser and Turquette's book "Many-valued logics". Unfortunately, the only edition of the book I could find has zero occurrences of the word "functor". From what I can find on the internet, Leśniewski seems to widely employ the _concept_ of a functor, and he is taken as the most ancient philosopher doing so (Leśniewski dies a few years after Carnap's book is published!) but there seems to be no proof that he was the first to employ the _name_.

I'd like to get to the end of this. Any help?



Fosco


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Re: On the etymology of the word "functor"

[Fosco Loregian]

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Interesting, do you have a reference for Frege's usage of the word (and his definition, if any)?

"Perhaps it then was
lingua franca in philosophy circles."

I am starting to believe that, yes.



Il giorno lun 19 feb 2024 alle ore 10:32 Johannes Huebschmann <johannes.huebschmann@univ-lille.fr<mailto:johannes.huebschmann@univ-lille.fr>> ha scritto:
Frege used the term "functor". Perhaps it then was
lingua franca in philosophy circles.
Also we should be aware of the notion of function
being a recent idea (Euler ...).

Best

Johannes



________________________________
De: "Fosco Loregian" <fosco.loregian@gmail.com<mailto:fosco.loregian@gmail.com>>
À: "categories" <categories@mq.edu.au<mailto:categories@mq.edu.au>>
Envoyé: Samedi 17 Février 2024 13:09:21
Objet: On the etymology of the word "functor"

It seems surprisingly difficult to trace back the precise origin of the word "functor" imported by Mac Lane in category theory from philosophy. I wonder if someone more experienced than me can find a better answer to this mystery.

- As it is well-known, Mac Lane says in the historical notes of Chapter 1 in CWM, that the name "functor" is borrowed from Carnap's "Logische Syntax der Sprache"; Carnap writes the book in 1934.

In his book "Tool and object", Ralf Kr¨omer partially rectifies this claim in that he says: " The somewhat arrogant account [of Mac Lane's review of Carnap's LSS] obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language".

So, whence did Carnap borrow the term? Was it also a "current informal parlance" [CWM, p. 30] in the logical analysis of language, as much as "natural transformation" was in Mathematics? (cf. again Kr¨omer, where he makes a good point of how the term was employed by Lefschetz and Hurewicz).

It seems that Haskell Curry, in his "Some logical aspects of grammatical structures", attributes the term to Tadeusz Kotarbiński, where it was introduced in his 1929's Elementy teorji poznania -in a somewhat clumsy translation, edited in English as "Gnosiology" by O. Wojtasiewicz, and only in 1966. It seems then reasonable that Mac Lane, not knowing Polish (or did he?) just wasn't aware of this much older occurrence. Curry's paper is behind a paywall for me, but in his 1979's book "Foundations of Mathematical Logic" Curry summarized some ideas from his "Logical aspects" (it's the paper where Curry introduces the toy language of szám, tetél and tantét), and the word "functor" itself occurs multiple times in the latter book, with no particular introduction -reasonable, in 1979.

Too bad that, if one is stubborn enough to find a copy of "Gnosiology" (the original in Polish seems quite difficult to recover, but I'd happy to see it), they will notice that yes, the term "functor" is explained to some extent in the text, but it is not introduced in proper detail, as if the concept was already there and Kotarbiński just borrowed it from someone else. (Kotarbiński speaks of a functor as an abstract "sentential connective" at page 259, at the very start of his second chapter "The deductive method".
"Gnosiology" comes with an appendix containing the review that Adjukiewicz wrote on Kotarbiński's book; Adjukiewicz uses the word functor quite liberally (see for example: "if that of which mathematics speaks is the objective correlates of some functors occurring in mathematical theorems, correlates which in turn have no arguments, then mathematics speaks of numbers, as, for example, in the arithmetical statement <<3 + 2 = 5>> such ultimate arguments are numbers, and nothing else"). There is no mention about Kotarbiński being the first to use the word, be it as formalization of a concept from common parlance, or as a word coined for the first time.

This is where I start getting lost, and my skills are not enough any more.

Kotarbiński seems in fact to attribute the coinage of the word "functor" to Łukasiewicz or Leśniewski, but never explicitly links any of the two to the term (I quote: p 244, "Lukasiewicz, in his system of the sentential calculus, places the functors directly before the functions to which they pertain" and p 403 "other logical types can be formed by sentence-forming or term-forming functors of the various kinds (Lesniewski)"). I find very little evidence that this attribution can be confirmed; this paper http://www.numdam.org/article/CM_1968__20__153_0.pdf<https://protect-au.mimecast.com/s/2CKiCnx1Z5Uo0LrLH9JZls?domain=numdam.org> talks from the very beginning of "the implication and negation functors of Łukasiewicz" referring to Rosser and Turquette's book "Many-valued logics". Unfortunately, the only edition of the book I could find has zero occurrences of the word "functor". From what I can find on the internet, Leśniewski seems to widely employ the _concept_ of a functor, and he is taken as the most ancient philosopher doing so (Leśniewski dies a few years after Carnap's book is published!) but there seems to be no proof that he was the first to employ the _name_.

I'd like to get to the end of this. Any help?



Fosco


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Re: On the etymology of the word "functor"

[Johannes Huebschmann]

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Perhaps I misremember something.
I will try to check this.
Frege extensively uses the term "function".

Originally
"functio" is a classical Latin word, derived from
the verbe " fungor, fungi" (be engaged in, perform).


Best

Johannes



________________________________
De: "Steve Awodey" <awodey@cmu.edu>
À: "Johannes Huebschmann" <johannes.huebschmann@univ-lille.fr>
Envoyé: Lundi 19 Février 2024 15:06:47
Objet: Re: On the etymology of the word "functor"



On Feb 19, 2024, at 05:22, Johannes Huebschmann <johannes.huebschmann@univ-lille.fr> wrote:

Frege used the term "functor".

Do you have a reference for this?

Perhaps it then was
lingua franca in philosophy circles.
Also we should be aware of the notion of function
being a recent idea (Euler ...).

Best

Johannes



________________________________
De: "Fosco Loregian" <fosco.loregian@gmail.com>
À: "categories" <categories@mq.edu.au>
Envoyé: Samedi 17 Février 2024 13:09:21
Objet: On the etymology of the word "functor"

It seems surprisingly difficult to trace back the precise origin of the word "functor" imported by Mac Lane in category theory from philosophy. I wonder if someone more experienced than me can find a better answer to this mystery.

- As it is well-known, Mac Lane says in the historical notes of Chapter 1 in CWM, that the name "functor" is borrowed from Carnap's "Logische Syntax der Sprache"; Carnap writes the book in 1934.

In his book "Tool and object", Ralf Kr¨omer partially rectifies this claim in that he says: " The somewhat arrogant account [of Mac Lane's review of Carnap's LSS] obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language".

So, whence did Carnap borrow the term? Was it also a "current informal parlance" [CWM, p. 30] in the logical analysis of language, as much as "natural transformation" was in Mathematics? (cf. again Kr¨omer, where he makes a good point of how the term was employed by Lefschetz and Hurewicz).

It seems that Haskell Curry, in his "Some logical aspects of grammatical structures", attributes the term to Tadeusz Kotarbiński, where it was introduced in his 1929's Elementy teorji poznania -in a somewhat clumsy translation, edited in English as "Gnosiology" by O. Wojtasiewicz, and only in 1966. It seems then reasonable that Mac Lane, not knowing Polish (or did he?) just wasn't aware of this much older occurrence. Curry's paper is behind a paywall for me, but in his 1979's book "Foundations of Mathematical Logic" Curry summarized some ideas from his "Logical aspects" (it's the paper where Curry introduces the toy language of szám, tetél and tantét), and the word "functor" itself occurs multiple times in the latter book, with no particular introduction -reasonable, in 1979.

Too bad that, if one is stubborn enough to find a copy of "Gnosiology" (the original in Polish seems quite difficult to recover, but I'd happy to see it), they will notice that yes, the term "functor" is explained to some extent in the text, but it is not introduced in proper detail, as if the concept was already there and Kotarbiński just borrowed it from someone else. (Kotarbiński speaks of a functor as an abstract "sentential connective" at page 259, at the very start of his second chapter "The deductive method".
"Gnosiology" comes with an appendix containing the review that Adjukiewicz wrote on Kotarbiński's book; Adjukiewicz uses the word functor quite liberally (see for example: "if that of which mathematics speaks is the objective correlates of some functors occurring in mathematical theorems, correlates which in turn have no arguments, then mathematics speaks of numbers, as, for example, in the arithmetical statement <<3 + 2 = 5>> such ultimate arguments are numbers, and nothing else"). There is no mention about Kotarbiński being the first to use the word, be it as formalization of a concept from common parlance, or as a word coined for the first time.

This is where I start getting lost, and my skills are not enough any more.

Kotarbiński seems in fact to attribute the coinage of the word "functor" to Łukasiewicz or Leśniewski, but never explicitly links any of the two to the term (I quote: p 244, "Lukasiewicz, in his system of the sentential calculus, places the functors directly before the functions to which they pertain" and p 403 "other logical types can be formed by sentence-forming or term-forming functors of the various kinds (Lesniewski)"). I find very little evidence that this attribution can be confirmed; this paper http://www.numdam.org/article/CM_1968__20__153_0.pdf<https://protect-au.mimecast.com/s/g-aaC3QNl1SQ1EPPhgjVPX?domain=numdam.org> talks from the very beginning of "the implication and negation functors of Łukasiewicz" referring to Rosser and Turquette's book "Many-valued logics". Unfortunately, the only edition of the book I could find has zero occurrences of the word "functor". From what I can find on the internet, Leśniewski seems to widely employ the _concept_ of a functor, and he is taken as the most ancient philosopher doing so (Leśniewski dies a few years after Carnap's book is published!) but there seems to be no proof that he was the first to employ the _name_.

I'd like to get to the end of this. Any help?



Fosco


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Re: On the etymology of the word "functor"

[Michael Barr, Prof.]

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I don't know what is mysterious about the origin of functor.  It is a 2-function and they surely wanted to suggest a variant of function.

But this illustrates a point I have been trying to make for decades to so-called mathematical historians.  While they have been grinding the origins of calculus finer and finer, they are allowing contemporary history to disappear.  If someone had interviewed Eilenberg or Mac Lane at length 30 years ago we would know why they chose functor.  And much much more.  Now they are gone.  Bill Lawvere is gone.  There are still a few of the older category theorists left, but probably not for long.  But this is why I have been posting these historical notes.

Michael
________________________________
From: Johannes Huebschmann <johannes.huebschmann@univ-lille.fr>
Sent: Monday, February 19, 2024 3:32 AM
To: Fosco Loregian <fosco.loregian@gmail.com>
Cc: categories <categories@mq.edu.au>; Johannes Huebschmann <johannes.huebschmann@univ-lille.fr>
Subject: Re: On the etymology of the word "functor"

Frege used the term "functor". Perhaps it then was
lingua franca in philosophy circles.
Also we should be aware of the notion of function
being a recent idea (Euler ...).

Best

Johannes



________________________________
De: "Fosco Loregian" <fosco.loregian@gmail.com>
À: "categories" <categories@mq.edu.au>
Envoyé: Samedi 17 Février 2024 13:09:21
Objet: On the etymology of the word "functor"

It seems surprisingly difficult to trace back the precise origin of the word "functor" imported by Mac Lane in category theory from philosophy. I wonder if someone more experienced than me can find a better answer to this mystery.

- As it is well-known, Mac Lane says in the historical notes of Chapter 1 in CWM, that the name "functor" is borrowed from Carnap's "Logische Syntax der Sprache"; Carnap writes the book in 1934.

In his book "Tool and object", Ralf Kr¨omer partially rectifies this claim in that he says: " The somewhat arrogant account [of Mac Lane's review of Carnap's LSS] obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language".

So, whence did Carnap borrow the term? Was it also a "current informal parlance" [CWM, p. 30] in the logical analysis of language, as much as "natural transformation" was in Mathematics? (cf. again Kr¨omer, where he makes a good point of how the term was employed by Lefschetz and Hurewicz).

It seems that Haskell Curry, in his "Some logical aspects of grammatical structures", attributes the term to Tadeusz Kotarbiński, where it was introduced in his 1929's Elementy teorji poznania -in a somewhat clumsy translation, edited in English as "Gnosiology" by O. Wojtasiewicz, and only in 1966. It seems then reasonable that Mac Lane, not knowing Polish (or did he?) just wasn't aware of this much older occurrence. Curry's paper is behind a paywall for me, but in his 1979's book "Foundations of Mathematical Logic" Curry summarized some ideas from his "Logical aspects" (it's the paper where Curry introduces the toy language of szám, tetél and tantét), and the word "functor" itself occurs multiple times in the latter book, with no particular introduction -reasonable, in 1979.

Too bad that, if one is stubborn enough to find a copy of "Gnosiology" (the original in Polish seems quite difficult to recover, but I'd happy to see it), they will notice that yes, the term "functor" is explained to some extent in the text, but it is not introduced in proper detail, as if the concept was already there and Kotarbiński just borrowed it from someone else. (Kotarbiński speaks of a functor as an abstract "sentential connective" at page 259, at the very start of his second chapter "The deductive method".
"Gnosiology" comes with an appendix containing the review that Adjukiewicz wrote on Kotarbiński's book; Adjukiewicz uses the word functor quite liberally (see for example: "if that of which mathematics speaks is the objective correlates of some functors occurring in mathematical theorems, correlates which in turn have no arguments, then mathematics speaks of numbers, as, for example, in the arithmetical statement <<3 + 2 = 5>> such ultimate arguments are numbers, and nothing else"). There is no mention about Kotarbiński being the first to use the word, be it as formalization of a concept from common parlance, or as a word coined for the first time.

This is where I start getting lost, and my skills are not enough any more.

Kotarbiński seems in fact to attribute the coinage of the word "functor" to Łukasiewicz or Leśniewski, but never explicitly links any of the two to the term (I quote: p 244, "Lukasiewicz, in his system of the sentential calculus, places the functors directly before the functions to which they pertain" and p 403 "other logical types can be formed by sentence-forming or term-forming functors of the various kinds (Lesniewski)"). I find very little evidence that this attribution can be confirmed; this paper http://www.numdam.org/article/CM_1968__20__153_0.pdf<https://protect-au.mimecast.com/s/0x1pCnx1Z5Uo0ommT9TXvI?domain=numdam.org> talks from the very beginning of "the implication and negation functors of Łukasiewicz" referring to Rosser and Turquette's book "Many-valued logics". Unfortunately, the only edition of the book I could find has zero occurrences of the word "functor". From what I can find on the internet, Leśniewski seems to widely employ the _concept_ of a functor, and he is taken as the most ancient philosopher doing so (Leśniewski dies a few years after Carnap's book is published!) but there seems to be no proof that he was the first to employ the _name_.

I'd like to get to the end of this. Any help?



Fosco


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Re: On the etymology of the word "functor"

[Julian Rohrhuber]

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Another snippet.

Reading up in German literature on Carnap e.g. [1,2] suggests that
Carnap may have chosen the word "Funktor" to rhyme with "Junktor".
Both rhyme with "Operator": a sign which allows us to derive an expression from another expression:
a Junktor (logical connective in propositional calculus) derives a proposition from propositions, and
a Funktor (placeholder for a function) derives a terms from terms.

Kuno Lorenz, Funktor, in: Jürgen Mittelstraß (ed.) Enzyklopädie Philosophie und Wissenschaftstheorie Band 2 C–F-J.B. Metzler (2005)
ibid., Operator, in: Jürgen Mittelstraß (ed.) Enzyklopädie Philosophie und Wissenschaftstheorie Band 6 O–Ra-J.B. Metzler (2016)

This is consistent with Haskell Curry's "Foundations of Mathematical Logic":

"There are three main classes of phrases, viz., nouns, sentences, and functors.
A noun names some object (real or imaginary); a sentence expresses
a statement; and a functor is a means of combining phrases to form other
phrases." (p. 32)

Generally, the movement from (Opera-)tion to (Opera-)tor is one towards higher-order-(opera-)tions, in linguistics, logic and mathematics.
A hidden issue is whether higher order means logical/analytic or not – this might relate to the argument MacLane makes about Carnap.

It is worth quoting the full footnote by Ralf Krömer in Tool and Object (p.59) that you refer to writes that Mac Lane "purloined" the word ("It seemed in order to take over that word for a better and less philosophical purpose.", Mac Lane, The Development and Prospects for Category Theory, p. 131)

"It was Mac Lane who reviewed the English translation of Carnap’s Logische Syntax der Sprache in the Bulletin of the AMS (1938); he mentions there that (and how) Carnap employs the term. In [1996, 131], Mac Lane writes: “Carnap […] had talked of functors in a different sense and made some corresponding mistakes. It seemed in order to take over that word for a better and less philosophical purpose”. This somewhat arrogant account obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language. Steve Awodey at a 2005 Paris meeting on history of category theory (entitled “Impact des categories. 60 ans de théorie des catégories: aspects historiques et philosophiques”, October 10–14, ENS, Paris) delivered an interesting talk about the relationship between Carnap and Mac Lane, especially on the role of Mac Lane in Quine’s reception of Carnap."


Here is a link to the talk by Steve Awodey: https://youtu.be/alLgEf0uVkg?feature=shared&t=1194<https://protect-au.mimecast.com/s/FQQPC5QP8ySGBMJDh21MBS?domain=youtu.be>
As much as I can tell, the (great) talk is about Mac Lane's critique of Carnap's analyticity (preceding Quine).
For reference, the Mac Lane review of Carnap is available here: https://www.semanticscholar.org/paper/Carnap-on-Logical-Syntax-Maclane/a5b00cabfce7c998ba830a68019014bcab252baa<https://protect-au.mimecast.com/s/-vPwC6XQ68fK5PNBTxh-Jq?domain=semanticscholar.org>


> On 19. Feb 2024, at 16:49, Johannes Huebschmann <johannes.huebschmann@univ-lille.fr> wrote:
>
> Perhaps I misremember something.
> I will try to check this.
> Frege extensively uses the term "function".
>
> Originally
> "functio" is a classical Latin word, derived from
> the verbe " fungor, fungi" (be engaged in, perform).
>
>
> Best
>
> Johannes
>
>
>
> De: "Steve Awodey" <awodey@cmu.edu>
> À: "Johannes Huebschmann" <johannes.huebschmann@univ-lille.fr>
> Envoyé: Lundi 19 Février 2024 15:06:47
> Objet: Re: On the etymology of the word "functor"
>
>
>
> On Feb 19, 2024, at 05:22, Johannes Huebschmann <johannes.huebschmann@univ-lille.fr> wrote:
>
> Frege used the term "functor".
>
> Do you have a reference for this?
>
> Perhaps it then was
> lingua franca in philosophy circles.
> Also we should be aware of the notion of function
> being a recent idea (Euler ...).
>
> Best
>
> Johannes
>
>
>
> De: "Fosco Loregian" <fosco.loregian@gmail.com>
> À: "categories" <categories@mq.edu.au>
> Envoyé: Samedi 17 Février 2024 13:09:21
> Objet: On the etymology of the word "functor"
>
> It seems surprisingly difficult to trace back the precise origin of the word "functor" imported by Mac Lane in category theory from philosophy. I wonder if someone more experienced than me can find a better answer to this mystery.
>
> - As it is well-known, Mac Lane says in the historical notes of Chapter 1 in CWM, that the name "functor" is borrowed from Carnap's "Logische Syntax der Sprache"; Carnap writes the book in 1934.
>
> In his book "Tool and object", Ralf Kr¨omer partially rectifies this claim in that he says: " The somewhat arrogant account [of Mac Lane's review of Carnap's LSS] obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language".
>
> So, whence did Carnap borrow the term? Was it also a "current informal parlance" [CWM, p. 30] in the logical analysis of language, as much as "natural transformation" was in Mathematics? (cf. again Kr¨omer, where he makes a good point of how the term was employed by Lefschetz and Hurewicz).
>
> It seems that Haskell Curry, in his "Some logical aspects of grammatical structures", attributes the term to Tadeusz Kotarbiński, where it was introduced in his 1929's Elementy teorji poznania -in a somewhat clumsy translation, edited in English as "Gnosiology" by O. Wojtasiewicz, and only in 1966. It seems then reasonable that Mac Lane, not knowing Polish (or did he?) just wasn't aware of this much older occurrence. Curry's paper is behind a paywall for me, but in his 1979's book "Foundations of Mathematical Logic" Curry summarized some ideas from his "Logical aspects" (it's the paper where Curry introduces the toy language of szám, tetél and tantét), and the word "functor" itself occurs multiple times in the latter book, with no particular introduction -reasonable, in 1979.
>
> Too bad that, if one is stubborn enough to find a copy of "Gnosiology" (the original in Polish seems quite difficult to recover, but I'd happy to see it), they will notice that yes, the term "functor" is explained to some extent in the text, but it is not introduced in proper detail, as if the concept was already there and Kotarbiński just borrowed it from someone else. (Kotarbiński speaks of a functor as an abstract "sentential connective" at page 259, at the very start of his second chapter "The deductive method".
> "Gnosiology" comes with an appendix containing the review that Adjukiewicz wrote on Kotarbiński's book; Adjukiewicz uses the word functor quite liberally (see for example: "if that of which mathematics speaks is the objective correlates of some functors occurring in mathematical theorems, correlates which in turn have no arguments, then mathematics speaks of numbers, as, for example, in the arithmetical statement <<3 + 2 = 5>> such ultimate arguments are numbers, and nothing else"). There is no mention about Kotarbiński being the first to use the word, be it as formalization of a concept from common parlance, or as a word coined for the first time.
>
> This is where I start getting lost, and my skills are not enough any more.
>
> Kotarbiński seems in fact to attribute the coinage of the word "functor" to Łukasiewicz or Leśniewski, but never explicitly links any of the two to the term (I quote: p 244, "Lukasiewicz, in his system of the sentential calculus, places the functors directly before the functions to which they pertain" and p 403 "other logical types can be formed by sentence-forming or term-forming functors of the various kinds (Lesniewski)"). I find very little evidence that this attribution can be confirmed; this paper http://www.numdam.org/article/CM_1968__20__153_0.pdf<https://protect-au.mimecast.com/s/taPuC71R63CkDzgJhRitrc?domain=numdam.org> talks from the very beginning of "the implication and negation functors of Łukasiewicz" referring to Rosser and Turquette's book "Many-valued logics". Unfortunately, the only edition of the book I could find has zero occurrences of the word "functor". From what I can find on the internet, Leśniewski seems to widely employ the _concept_ of a functor, and he is taken as the most ancient philosopher doing so (Leśniewski dies a few years after Carnap's book is published!) but there seems to be no proof that he was the first to employ the _name_.
>
> I'd like to get to the end of this. Any help?
>
>
>
> Fosco
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups



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Re: On the etymology of the word "functor"

[Steve Awodey]

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thanks for mentioning that, Julian.
I’ll just point out the nice comments from Benabou in the discussion following the talk.

best wishes,
Steve

On Feb 19, 2024, at 4:46 PM, Julian Rohrhuber <rohrhuber@protonmail.com> wrote:

Another snippet.

Reading up in German literature on Carnap e.g. [1,2] suggests that
Carnap may have chosen the word "Funktor" to rhyme with "Junktor".
Both rhyme with "Operator": a sign which allows us to derive an expression from another expression:
a Junktor (logical connective in propositional calculus) derives a proposition from propositions, and
a Funktor (placeholder for a function) derives a terms from terms.

Kuno Lorenz, Funktor, in: Jürgen Mittelstraß (ed.) Enzyklopädie Philosophie und Wissenschaftstheorie Band 2 C–F-J.B. Metzler (2005)
ibid., Operator, in: Jürgen Mittelstraß (ed.) Enzyklopädie Philosophie und Wissenschaftstheorie Band 6 O–Ra-J.B. Metzler (2016)

This is consistent with Haskell Curry's "Foundations of Mathematical Logic":

"There are three main classes of phrases, viz., nouns, sentences, and functors.
A noun names some object (real or imaginary); a sentence expresses
a statement; and a functor is a means of combining phrases to form other
phrases." (p. 32)

Generally, the movement from (Opera-)tion to (Opera-)tor is one towards higher-order-(opera-)tions, in linguistics, logic and mathematics.
A hidden issue is whether higher order means logical/analytic or not – this might relate to the argument MacLane makes about Carnap.

It is worth quoting the full footnote by Ralf Krömer in Tool and Object (p.59) that you refer to writes that Mac Lane "purloined" the word ("It seemed in order to take over that word for a better and less philosophical purpose.", Mac Lane, The Development and Prospects for Category Theory, p. 131)

"It was Mac Lane who reviewed the English translation of Carnap’s Logische Syntax der Sprache in the Bulletin of the AMS (1938); he mentions there that (and how) Carnap employs the term. In [1996, 131], Mac Lane writes: “Carnap […] had talked of functors in a different sense and made some corresponding mistakes. It seemed in order to take over that word for a better and less philosophical purpose”. This somewhat arrogant account obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language. Steve Awodey at a 2005 Paris meeting on history of category theory (entitled “Impact des categories. 60 ans de théorie des catégories: aspects historiques et philosophiques”, October 10–14, ENS, Paris) delivered an interesting talk about the relationship between Carnap and Mac Lane, especially on the role of Mac Lane in Quine’s reception of Carnap."


Here is a link to the talk by Steve Awodey: https://youtu.be/alLgEf0uVkg?feature=shared&t=1194<https://protect-au.mimecast.com/s/UR8ICoV1Y2SzO76zT1M5de?domain=youtu.be>
As much as I can tell, the (great) talk is about Mac Lane's critique of Carnap's analyticity (preceding Quine).
For reference, the Mac Lane review of Carnap is available here: https://www.semanticscholar.org/paper/Carnap-on-Logical-Syntax-Maclane/a5b00cabfce7c998ba830a68019014bcab252baa<https://protect-au.mimecast.com/s/lUgXCp81gYC8Yjp8uD0YUS?domain=semanticscholar.org>


> On 19. Feb 2024, at 16:49, Johannes Huebschmann <johannes.huebschmann@univ-lille.fr> wrote:
>
> Perhaps I misremember something.
> I will try to check this.
> Frege extensively uses the term "function".
>
> Originally
> "functio" is a classical Latin word, derived from
> the verbe " fungor, fungi" (be engaged in, perform).
>
>
> Best
>
> Johannes
>
>
>
> De: "Steve Awodey" <awodey@cmu.edu>
> À: "Johannes Huebschmann" <johannes.huebschmann@univ-lille.fr>
> Envoyé: Lundi 19 Février 2024 15:06:47
> Objet: Re: On the etymology of the word "functor"
>
>
>
> On Feb 19, 2024, at 05:22, Johannes Huebschmann <johannes.huebschmann@univ-lille.fr> wrote:
>
> Frege used the term "functor".
>
> Do you have a reference for this?
>
> Perhaps it then was
> lingua franca in philosophy circles.
> Also we should be aware of the notion of function
> being a recent idea (Euler ...).
>
> Best
>
> Johannes
>
>
>
> De: "Fosco Loregian" <fosco.loregian@gmail.com>
> À: "categories" <categories@mq.edu.au>
> Envoyé: Samedi 17 Février 2024 13:09:21
> Objet: On the etymology of the word "functor"
>
> It seems surprisingly difficult to trace back the precise origin of the word "functor" imported by Mac Lane in category theory from philosophy. I wonder if someone more experienced than me can find a better answer to this mystery.
>
> - As it is well-known, Mac Lane says in the historical notes of Chapter 1 in CWM, that the name "functor" is borrowed from Carnap's "Logische Syntax der Sprache"; Carnap writes the book in 1934.
>
> In his book "Tool and object", Ralf Kr¨omer partially rectifies this claim in that he says: " The somewhat arrogant account [of Mac Lane's review of Carnap's LSS] obscures the fact that Carnap’s terminology has always since been widely employed in logical analysis of language".
>
> So, whence did Carnap borrow the term? Was it also a "current informal parlance" [CWM, p. 30] in the logical analysis of language, as much as "natural transformation" was in Mathematics? (cf. again Kr¨omer, where he makes a good point of how the term was employed by Lefschetz and Hurewicz).
>
> It seems that Haskell Curry, in his "Some logical aspects of grammatical structures", attributes the term to Tadeusz Kotarbiński, where it was introduced in his 1929's Elementy teorji poznania -in a somewhat clumsy translation, edited in English as "Gnosiology" by O. Wojtasiewicz, and only in 1966. It seems then reasonable that Mac Lane, not knowing Polish (or did he?) just wasn't aware of this much older occurrence. Curry's paper is behind a paywall for me, but in his 1979's book "Foundations of Mathematical Logic" Curry summarized some ideas from his "Logical aspects" (it's the paper where Curry introduces the toy language of szám, tetél and tantét), and the word "functor" itself occurs multiple times in the latter book, with no particular introduction -reasonable, in 1979.
>
> Too bad that, if one is stubborn enough to find a copy of "Gnosiology" (the original in Polish seems quite difficult to recover, but I'd happy to see it), they will notice that yes, the term "functor" is explained to some extent in the text, but it is not introduced in proper detail, as if the concept was already there and Kotarbiński just borrowed it from someone else. (Kotarbiński speaks of a functor as an abstract "sentential connective" at page 259, at the very start of his second chapter "The deductive method".
> "Gnosiology" comes with an appendix containing the review that Adjukiewicz wrote on Kotarbiński's book; Adjukiewicz uses the word functor quite liberally (see for example: "if that of which mathematics speaks is the objective correlates of some functors occurring in mathematical theorems, correlates which in turn have no arguments, then mathematics speaks of numbers, as, for example, in the arithmetical statement <<3 + 2 = 5>> such ultimate arguments are numbers, and nothing else"). There is no mention about Kotarbiński being the first to use the word, be it as formalization of a concept from common parlance, or as a word coined for the first time.
>
> This is where I start getting lost, and my skills are not enough any more.
>
> Kotarbiński seems in fact to attribute the coinage of the word "functor" to Łukasiewicz or Leśniewski, but never explicitly links any of the two to the term (I quote: p 244, "Lukasiewicz, in his system of the sentential calculus, places the functors directly before the functions to which they pertain" and p 403 "other logical types can be formed by sentence-forming or term-forming functors of the various kinds (Lesniewski)"). I find very little evidence that this attribution can be confirmed; this paper http://www.numdam.org/article/CM_1968__20__153_0.pdf<https://protect-au.mimecast.com/s/D8zlCq71jxf5qpJ5SQDR_Z?domain=numdam.org> talks from the very beginning of "the implication and negation functors of Łukasiewicz" referring to Rosser and Turquette's book "Many-valued logics". Unfortunately, the only edition of the book I could find has zero occurrences of the word "functor". From what I can find on the internet, Leśniewski seems to widely employ the _concept_ of a functor, and he is taken as the most ancient philosopher doing so (Leśniewski dies a few years after Carnap's book is published!) but there seems to be no proof that he was the first to employ the _name_.
>
> I'd like to get to the end of this. Any help?
>
>
>
> Fosco
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message. View group files | Leave group | Learn more about Microsoft 365 Groups



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Re: On the etymology of the word "functor"

[George Janelidze]

Dear Colleagues,

Referring to Julian Rohrhuber's message of February 20: I don't think it is
a good idea to cut a sentence in the middle and then call it "somewhat
arrogant", especially when it is from a paper of Saunders Mac Lane. The full
sentence is:

"There was also some fun with the choice of terminology. Since the
philosopher Kant had made ample use of general categories, the term was
borrowed from him for its present mathematical use, while Camap, in his book
on Die Logische Syntax der Sprachen had talked of functors in a different
sense and made some corresponding mistakes. It seemed in order to take over
that word for a better and less philosophical purpose."

Let me also add a sentence from Mac Lane's paper "Samuel Eilenberg and
Categories" (JPAA 168, 2002, 127-131):

"Also the terminology was largely purloined: “category” from Kant, “natural”
from vector spaces and “functor” from Carnap. (It was used in a different
sense in Carnap’s influential book “Logical Syntax of Language”; I had
reviewed the English translation of the book (in the Bulletin, AMS) and had
spotted some errors; since Carnap never acknowledged my finding, I did not
mind using his terminology.)"

Referring to Michael Barr's message of February 19, which is:

"I don't know what is mysterious about the origin of functor.  It is a
2-function and they surely wanted to suggest a variant of function. But this
illustrates a point I have been trying to make for decades to so-called
mathematical historians.  While they have been grinding the origins of
calculus finer and finer, they are allowing contemporary history to
disappear.  If someone had interviewed Eilenberg or Mac Lane at length 30
years ago we would know why they chose functor.  And much much more.  Now
they are gone.  Bill Lawvere is gone.  There are still a few of the older
category theorists left, but probably not for long.  But this is why I have
been posting these historical notes."

So very true! However, this is not just about "so-called mathematical
historians", but also about certain mathematicians who tell historians what
is important in mathematics and what is not. And it is also about many of
us, who, for example, did nothing with the unthinkable article "Timeline of
category theory and related mathematics" in Wikipedia and a similar article
in nLab (well, both of them have a lot of good mathematics mentioned, but
putting 'selected good' and 'selected bad' together, might be the worst kind
of disinformation...).

George



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Re: On the etymology of the word "functor"

[Julian Rohrhuber]

Dear George, thank you for posting the full text, I felt this impulse too.
(just to be clear: I didn't cut the passage and called it "somewhat
arrogant"!)

On 20. Feb 2024, at 14:20, George Janelidze <janelg@telkomsa.net> wrote:
>
> Referring to Julian Rohrhuber's message of February 20: I don't think it is
> a good idea to cut a sentence in the middle and then call it "somewhat
> arrogant", especially when it is from a paper of Saunders Mac Lane. The full
> sentence is:
>
> "There was also some fun with the choice of terminology. Since the
> philosopher Kant had made ample use of general categories, the term was
> borrowed from him for its present mathematical use, while Camap, in his book
> on Die Logische Syntax der Sprachen had talked of functors in a different
> sense and made some corresponding mistakes. It seemed in order to take over
> that word for a better and less philosophical purpose."

What I would be interested in would be in what sense Carnap's mistakes, according to Mac Lane,
corresponded to the concept of functor. Or, perhaps, how Carnap's failed attempt to define analyticity
corresponds to a wrong concept of functor.




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Re: On the etymology of the word "functor"

[Steve Awodey]

> On Feb 20, 2024, at 9:57 AM, Julian Rohrhuber <rohrhuber@protonmail.com> wrote:
>
...
> What I would be interested in would be in what sense Carnap's mistakes, according to Mac Lane,
> corresponded to the concept of functor. Or, perhaps, how Carnap's failed attempt to define analyticity
> corresponds to a wrong concept of functor.
>

Dear Julian,

the “mistakes” that Mac Lane refers to are not related to Carnap's use of the word “functor”, which was just what we would now call a "function symbol” in a formal language.  Rather, they had to do with Carnap’s attempt to define logical validity syntactically, which Saunders showed in his review was mathematically defective.  The connection between that and the “purloining” of the word “functor” was, I think, just a playful reminiscence.

Steve



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Re: On the etymology of the word "functor"

[Ross Street]

Dear George and all

I fear that all writing is fiction with the possible exception of some mathematics papers.

Ross


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Re: On the etymology of the word "functor"

[Posina Venkata Rayudu]

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Dear Professor Street,

If I may, [good] fiction, as Tolkein brought into figural salience for
all to see, has "the inner consistency of reality."

Thanking you,
Yours truly,
posina
P.S. Regarding "religious wars" alluded to in earlier emails, all
wars, going by one of Kahneman's heuristics, are The USA wars, which
is to say "religious wars" is no different from Chomsky's "colorful
ideas sleep furiously" ;)
P.P.S. In the context of declining trust in science
(https://protect-au.mimecast.com/s/8aCtCxngGkfA7pODU8TfhU?domain=science.org), here's one way
to look at it: https://protect-au.mimecast.com/s/nA5rCyoj8PuM8GnKSMhOe-?domain=philarchive.org.


On Wed, Feb 21, 2024 at 5:09 AM Ross Street <ross.street@mq.edu.au> wrote:
>
> Dear George and all
>
> I fear that all writing is fiction with the possible exception of some mathematics papers.
>
> Ross
>
>
> ----------
>
> You're receiving this message because you're a member of the Categories mailing list group from Macquarie University.
>
> Leave group:
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Re: On the etymology of the word "functor"

[Julian Rohrhuber]

> On 20. Feb 2024, at 20:06, Steve Awodey <awodey@cmu.edu> wrote:
>
>
>
>> On Feb 20, 2024, at 9:57 AM, Julian Rohrhuber <rohrhuber@protonmail.com> wrote:
>>
> ...
>> What I would be interested in would be in what sense Carnap's mistakes, according to Mac Lane,
>> corresponded to the concept of functor. Or, perhaps, how Carnap's failed attempt to define analyticity
>> corresponds to a wrong concept of functor.
>>
>
> Dear Julian,
>
> the “mistakes” that Mac Lane refers to are not related to Carnap's use of the word “functor”, which was just what we would now call a "function symbol” in a formal language.  Rather, they had to do with Carnap’s attempt to define logical validity syntactically, which Saunders showed in his review was mathematically defective.  The connection between that and the “purloining” of the word “functor” was, I think, just a playful reminiscence.
>
> Steve

Dear Steve,

thank you for the clarification. My suspicion was that Carnap's use of the word functor was too syntactical, and Mac Lane found that for achieving logical validity, functors need to be mathematical objects proper. Would you say that this is a reasonable interpretation?

Julian



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