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(Edit #1 after Carlo's response)

It is often claimed that the notion of natural transformations existed in mathematical vocabulary long before it had a definition. In fact, I quoted the statement in italic from [1, p. 2]. As another example, in [2, p. 70] Ralf Kromer says: The claim is that there was, at the time when [3] was written, a current informal parlance consisting in calling certain transformations natural and that Mac Lane and Eilenberg tried (and succeeded) to grasp this informal parlance mathematically.

Three years after [3], Eilenberg and Maclane discovered the fact that this notion could be mathematically defined in [4].

However, Ralf Kromer casts doubt on the above mentioned claim, due to lack of evidence (see: [2, p. 70]).

(Edit #2 after Eric's comment) My question is, can you supply an evidence of use of the phrase "natural transformations" or its variants in mathematical literature prior to [3]? I must add that in [2] Kromer gives a number of examples of use of phrases such as "natural homomorphism" or "natural projection" in the literature prior to or around the same time as [3], but in each case they turn out to have different meanings. So I am asking for example(s) of use of the phrase "natural transformations", which are really natural transformations.

References:

  1. Peter Freyd: Abelian Categories (1964).
  2. Ralf Kromer: Tool and Object: a history and philosophy of category theory (2007).
  3. Samuel Eilenberg and Saunders Maclane: Group extensions and homology, Annals Math. (2) 43, p.p. 757–831 (1942).
  4. Samuel Eilenberg and Saunders Maclane: General theory of natural transformations, Trans. AMS, 58, p.p.: 231-294 (1945).
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  • $\begingroup$ it can be argued that the determinant is a natural transformation; does that count? (natural transformation in the sense of "not depending on choice of basis" obviously goes back way before 1940...) $\endgroup$ – Carlo Beenakker Dec 6 '17 at 21:29
  • $\begingroup$ Did anyone refer to determinant as a natural transformation before 1942? $\endgroup$ – Mahdi Majidi-Zolbanin Dec 6 '17 at 21:38
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    $\begingroup$ Are you asking about use of the phrase "natural transformation" and its syntactic variants, or are you asking about instances of things which could now be called natural transformations? When you say "use of the notion", you seem to be asking for the latter, but your comments to the current answer suggest that you are pursuing the former. $\endgroup$ – Eric Towers Dec 7 '17 at 0:33
  • $\begingroup$ @Eric I am asking about the use of the phrase "natural transformation", not things that could now be called natural transformations. I will edit to make it clear. $\endgroup$ – Mahdi Majidi-Zolbanin Dec 7 '17 at 0:46
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    $\begingroup$ See Whitney's paper from 1935 where he defined tensor products of abelian groups (projecteuclid.org/download/pdf_1/euclid.dmj/1077490789). There you will find the terms natural homomorphism and (especially) natural isomorphism. Whitney makes no attempt to give absolutely rigorous definitions of those concepts, as the motivation to do so was lacking, but his sense of "naturality" is what Eilenberg and Mac Lane were making precise in their introduction of natural transformations. $\endgroup$ – KConrad Dec 7 '17 at 1:28
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See Whitney's paper from 1935 where he defined tensor products of abelian groups. There you will find the terms natural homomorphism and (especially) natural isomorphism. Whitney makes no attempt to give absolutely rigorous definitions of those concepts, as the motivation to do so was lacking, but his sense of "naturality" is what Eilenberg and Mac Lane were making precise in their introduction of natural transformations.

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Natural transformations were first introduced by Eilenberg and MacLane in the context of group theory, in Natural isomorphisms in group theory (1942) -- three years before their definition in the context of functors or categories.

Indeed, "three years before" is not "long before", but in the development of a concept it is a significant delay.

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  • $\begingroup$ Functors are defined on the second page of that paper (though only for the category of groups). $\endgroup$ – Robert Furber Dec 6 '17 at 20:15
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    $\begingroup$ Thank you for pointing that out, Carlo. I have edited my question and made it more accurate. The claim is that "natural transformations" were already in use in mathematics even before the 1942 paper, without having a precise definition. $\endgroup$ – Mahdi Majidi-Zolbanin Dec 6 '17 at 20:44
  • $\begingroup$ Actually Kromer does mention the 1942 paper. $\endgroup$ – Todd Trimble Dec 6 '17 at 20:54
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    $\begingroup$ @ Carlo: Even Eilenberg and Maclane themselves in the paper that you linked say: "Frequently in modern mathematics there occur phenomena of "naturality": a "natural "isomorphism between two groups or between two complexes, a "natural" homeomorphism of two spaces andthelike. We here propose a precise definition of the "naturality" of such correspondences, as a basis for an appropriate general theory." It seems to me that they are saying that the notion of "naturally" existed already and that they "propose a definition" of it here. $\endgroup$ – Mahdi Majidi-Zolbanin Dec 6 '17 at 21:48
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    $\begingroup$ perhaps its like the person who learns that they "have been speaking prose all their life without knowing it" ... $\endgroup$ – Carlo Beenakker Dec 6 '17 at 22:02
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The words "natural homomorphism" and "natural isomorphism" are also used (mainly in the context related to the First Isomorphism Theorem) in Pontryagin's "Topological groups" (Russian edition 1938, English translation 1939). Google books confirms my memory of this here.

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