If you go back to Whitney's announcement in PNAS, and the subsequent paper, titled "On products in a complex", you can see where he started using the phrase, even an even more non-technical manner:

"In recent years the existence of certain products in a complex K has been much studied, combining a p-chain and a q-chain to form a (p+q)-chain.

For such products to have topological significance, certain conditions should be satisfied; it seems that (PI) and (P2) below (§5) are the most natural."

"If we wish to study products of L^r and L^s into L^{r+s}, using coefficient groups G in L^r and H in L^s, then we most naturally use the coefficient
group GH in L^{r+s}; the theory with any other coefficient group in L^{r+s} follows from this"

(from the 1937 PNAS paper https://doi.org/10.1073/pnas.23.5.285)

The second quote is the first hint of the tensor product of groups, denoted here by GH, which are defined (apparently with their universal property!) thus:

"11. Tensor products of groups. - The definition of "group pairs" may also be split into two parts, as follows. Given the (abelian) groups G and H, the *tensor product* GH is the group with generators gh (defined abstractly), and, for relations, all those obtainable with the two distributive laws. Then the possible definitions of group pairs G and H with respect to the group Z are obtained merely by choosing the possible homomorphisms of GH into Z."

In the subsequent fuller article (submitted mid-1937) we find:

"p-cell times a q-cell far away from the p-cell should certainly give nothing; hence (P1) of §5 is a natural assumption."

"Hence \delta(A\cup B) must be expressible in terms such as \delta A \cup B and A \cup \delta B. (P2) is the natural form."

"The maps Sd and Sd* of L^p(K) and L^p(K*) into L^p(K') are defined in the natural manner;"

(from the 1938 Annals paper https://doi.org/10.2307/1968795)

As a bonus, we have the sentence "It is natural to call two parametrized curves equivalent if one can be obtained from the other by a change of parameter (preserving orientation)." in Whitney's 1937 Compositio paper "On regular closed curves in the plane" http://www.numdam.org/item?id=CM_1937__4__276_0

And, even "If we are satisfied with a function F of class C^r, we would naturally use (e') in place of (d), (e) and (f)." from his 1936 Annals paper "Differentiable Manifolds" https://doi.org/10.2307/1968482. In this paper Whitney also uses "required nature", "general nature" when talking about certain maps he wishes to construct.

On this issue of "canonical" maps, earlier this year I managed to trace the origins back to the late 1930s, and have a hypothesis as to how it was chosen, based on the term from matrix theory. I hope to write up a short article on this before I am too much older. The chain is roughly: Ehresmann -> Bourbaki -> Eilenberg/Cartan/Weil -> Serre -> Grothendieck, though the middle steps are somewhat nonlinear as all of the named mathematicians were involved in Bourbaki at various stages. Certainly Eilenberg and Cartan's book helped disseminate the notion, as did Grothendieck's Tôhoku and then EGA1, as well as Bourbaki's texts. I didn't track the usage *forward* through 1960s CT literature, though, to see who was using it (eg Reports of the Midwest seminar, La Jolla/Batelle proceedings etc)

All the best,

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Tue, 19 Dec 2023 at 22:56, Dusko Pavlovic <duskgoo@gmail.com> wrote:

CAUTION: External email. Only click on links or open attachments from trusted senders.

The nice thing about this reference to Whitney is that it explains why Eilenberg and MacLane's Kantian naming taste was applied to *categories* and *functors* but abandoned when it came to 'natural transformations'. I was always wondering why we have been deprived of the pleasure of talking about, say, *transcendental* transformations all these years...

Whitney's intent with "natural transformations" seems to have been similar to Godement's intent with "standard constructions". In a similar vein, people used to talk about "canonical isomorphisms"...

It seems that Whitney's natural homomorphisms and natural topologies were natural in the sense that their definitions were the only thing that you could write down in the given context. Programmers call such definitions polymorphic. Now we know that you can do that precisely when what you are writing down is preserved under the homomorphisms induced by the type constructors. Eilenberg and MacLane noticed this phenomenon in some form, perhaps by aligning their homological and homotopical constructions mentioned by Mike. To capture the homomorphisms induced by the constructions, they had to define the homomorphism part of the constructions: the functors. And to capture the homomorphisms... etcetc. They said themselves that they introduced categories to define functors and functors to define natural transformations.

Whitney did use natural transformations, and he even defined the tensors between which his natural transformations occur, but he doesn't seem to have considered what his tensor constructions do with the homomorphisms. So he didn't have the tensor *functors*. So the naturality *idea* was in the air but it hadn't quite landed yet. He talks about a group R "operating" on a group G (as per Murray-von Neumann). Could his idea of naturality account for the difference between the families of isomorphisms V~V* and V~V**, viewed as a field R "operating" on a suitable group G (as von Neumann did)?

All students of physics learn how Galileo introduced the idea of relativity with respect to frames of reference. But he didn't have Lorenz transformations and Riemanian geometry, so the mathematical realizations of the idea of relativity had to wait... Whitney didn't have functors.

I think the message is that concepts are in the air people breathe before they figure out a way to write them down for other people. Other people figure out some other ways. We shouldn't take concepts personally.

2c,

-- dusko

On Mon, Dec 18, 2023 at 9:49 AM Evgeny Kuznetsov <jenkakuznecov@gmail.com> wrote:

Here is a copy of the paper by Hassler Whitney of 1938 titled "Tensor products of abelian groups"

On Mon, Dec 18, 2023, 23:32 Jean-Pierre Marquis <jean-pierre.marquis@umontreal.ca> wrote:

Most likely.

Whitney uses the terms ‘natural isomorphism’ and ‘natural homomorphism’ as well as the terms ‘natural topology’ and ‘natural neighborhood’ at many different places in the paper. But these terms are never explicitly defined.

Cheers,

Jean-Pierre

De : Wesley Phoa <doctorwes@gmail.com>
Date : lundi, 18 décembre 2023 à 14:18
À : Michael Barr, Prof. <barr.michael@mcgill.ca>
Cc : categories@mq.edu.au <categories@mq.edu.au>
Objet : Re: Modification of what I said

Was he referring to the paper “Tensor products of abelian groups”, cited in this discussion? https://mathoverflow.net/questions/287869/history-of-natural-transformations

I don’t have access to it either, but it’s on Scribd: https://www.scribd.com/document/172981416/Hassler-Whitney-Tensor-Products-of-Abelian-Groups

The terms “natural isomorphism” and “natural homomorphism” are used on pages 500-501, and these do turn out to be natural transformations, but it’s not obvious that he intended to explicitly define a new formal concept.

Wesley

Sent from my iPad

On Dec 18, 2023, at 10:00 AM, Michael Barr, Prof. <barr.michael@mcgill.ca> wrote:

Peter Freyd claims that Hassler Whitney defined natural transformation in a 1938 paper. I no longer have access to Math. Reviews (except by going to McGill, which I have done only once in the last four years) so I cannot supply a reference.

Michael

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

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

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