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<mailto: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<mailto:doctorwes@gmail.com>> Date : lundi, 18 décembre 2023 à 14:18 À : Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> Cc : categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto: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... 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<mailto: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<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>

Just do it, Jean-Pierre. It will be very interesting to see it. Generally speaking, we do such things all too little. When we wrote our book on quantales we came across a few such historical pathways, one of which was the monad and e.g. also the star composition of natural transformation. We spent a "reasonable portion of time" unravelling that history, and it comes down to less than one page, so we could have done more. We also had a chapter on finite machines and modules, and we look back on finite machines, where CS fixes the "origin" to some paper or papers, but when we looked at it, these "original papers" were quite sparse in their bibliography, which kind of "boosted their originality". For a while I've been looking into the categorization of foundations, i.e., not the foundations of category theory. For decades we have had a formal term monad which has shown to be useful for categorization of formulas, entailment, proof rules and proofs, etc. Categorical constructivity is very diametral to the traditional "putting symbols one by one in sequence" approach in logic and foundations, which leads to antinomies and many other strange situations. This work, which is all too much for me personally, and I can do only fractions, involves many historical pathways, e.g. Schröder tp Peano, Peano to Principia; Hilbert's lectures 1917-1922; deduction by Hertz and Post; Bernays; Hilbert's and Ackermann's logic foundations; Gödel 1931 and just before; Hilbert-Bernays Grundlagen 1934/39; Church, Turing, lambda calculus and computing; etc. just to name a few. Category theory existed at that time, because of set theory, even if it wasn't "discovered" just by the end of this "mathematical foundations pathway". So I'm fascinated by the thought that that categories would have been "discovered" already before Principia. Principia said that "logic must be freed from the burden of algebra", and foundations never recovered from that attitude. Bernays and Gödel wrote letters in the early 1960's about "something new called category theory" and "there was a guy speaking about something in Poland" (it was Mac Lane), but they simply thought a bit on foundations of such categories rather than such categories being used in constructions in their foundations, that had puzzled them for almost half a century. My point here is that "tracing far back" is very important. Categorization of foundations (Hilbert wasn't even close, but he could have been) connects subpathways in this history and e.g. makes criticism against antinomies (like e.g. Gödel 1931) quite precise. So, Jean-Pierre, just do it. Or anyone else thinking similarly about "deep tracing", just do it, because nobody else will, most probably. We need to strengthen the culture of "deep tracing", rather that desperately seeking for "this was the moment, this was the breakfast, these were the persons around that table at that moment in time, and that person must be named as the originator, for that person we must create a bust". "Deep tracing" opens up more research, "creating busts" does the opposite. Best, Patrik On 2023-12-18 22:08, Jean-Pierre Marquis wrote: I guess it is a paper like this one that justified Eilenberg & Mac Lane's choice of terminology. It would be interesting to trace how far the terminology goes back, who used it and where. Jean-Pierre De : Evgeny Kuznetsov <jenkakuznecov@gmail.com> Date : lundi, 18 décembre 2023 à 14:43 À : Jean-Pierre Marquis <jean-pierre.marquis@umontreal.ca> Cc : Wesley Phoa <doctorwes@gmail.com>, Michael Barr, Prof. <barr.michael@mcgill.ca>, categories@mq.edu.au <categories@mq.edu.au> Objet : Re: Modification of what I said 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<mailto: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<mailto:doctorwes@gmail.com>> Date : lundi, 18 décembre 2023 à 14:18 À : Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> Cc : categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto: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... 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<mailto: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<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>

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<mailto: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<mailto: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<mailto: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<mailto:doctorwes@gmail.com>> Date : lundi, 18 décembre 2023 à 14:18 À : Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> Cc : categories@mq.edu.au<mailto:categories@mq.edu.au> <categories@mq.edu.au<mailto: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... 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<mailto: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 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<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=6bf90c14-94d1-45b7-a0b5-9dd447734d27> | Learn more about Microsoft 365 Groups<https://aka.ms/o365g>

Permit me to perturb this modification further in the Whitney direction. As a grad student I knew about Hassler Whitney in connection with homological algebra and cohomology. These are very categorical subjects! In 1980 André Joyal was visiting Sydney. He was lecturing us in our Seminar on using toposes for differential geometry. [André's night job then was writing ``Une théorie combinatoire des séries formelles, Adv. Math. 42 (1981)''.] In connection with smooth functions \phi on the reals satisfying \phi(-x) = \phi(x), André quoted a result of Whitney. I was not conscious of Whitney's work on manifolds before that. In any case, around that time, Hassler Whitney appeared in our staff common room. [In those days, people met in the ``tea room'' for lunch as a matter of course.] Whitney was not there because of math research, he was visiting our Numeracy Centre. When I was at Wesleyan University in 1976-77, I learnt about the ``Math Anxiety Clinic'', a concept which, I believe, started there. We had a need for such a centre at Macquarie because a math course was mandatory for students planning to teach primary school and many of them, through no fault of their own, were terrified of that. I took an immediate liking to Whitney. He had grandchildren and I had two children. Whitney had taken a keen interest in Math Anxiety and in teaching math to the young. He pointed out that, when his grandchild was setting the table for supper, they would not say there are 6 people, so I need 6 forks. They would say: one fork for Mommy, one for Daddy, one for sister, one for brother, one for me, and so on. A bijection! My point is that we have heard in this forum about Whitney's categorical thinking in his early career. I am claiming that he was still at it throughout his interesting life. [By the way, our Numeracy Centre is still in demand and thriving under the leadership of Carolyn Kennett.] Ross

On 19 Dec 2023, at 4:44 pm, Dusko Pavlovic <duskgoo@gmail.com> wrote:

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...

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