Re: On the etymology of the word "functor"

[Johannes Huebschmann <johannes.huebschmann@univ-lille.fr>]

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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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