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 > 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 > 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 > Date : lundi, 18 décembre 2023 à 14:18 À : Michael Barr, Prof. > Cc : 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. > 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